Heterogeneous Reactor Design 4n5w4h

41

volume external surface area

q g ( t )-r=

volume (V,) to external pellet surface area (Sz)varies inversely with the same factor. Therefore, the product of the Thiele modulus and the effectiveness factor will approach unity for large values of the Thiele modulus if the characteristic dimension L is used in the definition of the Thiele modulus. This is shown in the last column of Table 4.1. It has been shown* that the use of the characteristic length in the definition of the Thiele modulus brings together all curves of q 4 corresponding to different pellet shapes. This is illustrated in Figure 4.2 (Rester and Aris 1969) for a first-order reaction. The agreement is very good at small and large values of the normalized Thiele modulus. The deviation of the exact value of q from the asymptotes q = 1 and q4 = 1 depends on the specific kinetic expression and the value of However, the normalized curve of r ) for the first-order reaction gives a reasonable approximation in many casa for all values of 4, and especially for small and large values of In view of these facts, the slab-like geometry will be used in what follows since the half-thicknss L is the characteristic length for this shape. This considerably simplifies the mathematics. Shape normalization based on the characteristic length L(= V,/&) enables one to relate the effectiveness factor for one shape to that for any other shape. Given the internal effectiveness factor, the global rate can be expressed as:

-+

+.

+.

2

For the general treatment of arbitrary shapes involving a first-order reaction by Mumy, scc the book by Aris, The Mathematical Theoty of Dimion and Reaction in Permeable Catalysts, Vol. I, Oxford University Press, London (1975).

f For a rate constant obeying the Arrhenius relationship, differentiation gives: d l n R ~ dlnk d(l/T,)

-=d(l/Ts)

(Err )obs = Ea

dlnq +-d(l/T*s)

dlnq dln+ - Rs d In 4 d ( l / T , )

where E, is the intrinsic activation energy, (Ea)obsis the observed activation energy and A, is the gas constant. For small values of 4, (d In q/d In 4) approaches zero, and the observed and intrinsic activation energies are the same. On the other hand, this derivative approaches ( - 1 ) for large values of and Eq. 4.24 gives:

1

REACTION/ HEAT AND

effects is one half the intrinsic value. Experimental evidence of the above result is provided by the study of Weisz and Prater (1954) shown in Figure 4.3. They obtained these results for the cracking of cumene on a silica..alurnina catalyst in which the change in Thiele modulus was effected by varying the pellet size. It is seen that the slope of the ln RG versus 1/T plot for severely diffusion-limited pellets of 0.175 cm radius is approximately one-half that of the 0.0056 crn pellets. Following similar procedures as in Eq. 4.24, it can be shown for an nth-order reaction that: (n).bs

+

105 MASS TRANSPORT

=n

+

(y) -1

dlnq

+

It follows from Eq. 4.26 that the observed order of reaction (n)ohris (n 1)/2 in the asymptotic region of strong diffusion effects. A comparison between theoretical and experimental effcctivenss factors has been made by Dente and Pasquou (1965) using the methanol oxidation reaction over an Fe-molybdenum oxide catalyst in a differential reactor. Results are shown in Figure 4.4. With excess O2 the intrinsic rate is one-half order in methanol. The experimental effectiveness factors in Figure 4.4 were calculated from:

where the temperature dependence of the effective diffusivity has been neglected. Thus the observed activation energy in the asymptotic region of strong diffusion

using the experimental differential rates while the Thiele moduli were calculated from the intrinsic kinetics. The solid line is the theoretical curve for one-half

Figure 4.3 Experimental demonstration of the effect of diffusion on apparent activation energy. (Weisz and Prater 1954. Reprinted with permission from Advances in Catalysis. Copyright by Academic Press.)

I \

I,

Figure 4.4 Comparison between theoretical and experimental effectiveness factors. (Dente and Pasquou 1965)

REACTION/ HEAT AND MASS TRANSPORT 1 07

order kinetics. Good agreement is seen in the figure. Under these experim&tal conditions, the external mass transport resistance was negligible. The pellet was assumed to be isothermal at the temperature dictated by the external heat transport proceaa. The validity of these assumpti~nswill now be examined.

the usually unknown surface quantities C . and T,. In dimensionless form, Eqs. 4.27 through 4.3 1 become:

-d2t - -fi+*F(c)

4 4 INTERNAL A N D EXTERNAL TRANSPORT PROCESSES

dv2

exp[-~(l/t

- l)]

(4.33)

Only isothermal pellets and internal effectiveness factors have been treated so far. Limitations on internal heat transfer, external mass transfer, and external heat transfer can all affect the reaction rate. Consider a pellet placed in a fluid medium. Assuming constant physical properties, one-dimensional mass and heat balances yield fot a slab-like pellet:

where c = C/Cb, r = T/Tb, y = z/L, and F(c) = f(C)/ f(Cb). The Biot numbers for mass and heat, which are a measure of the ratio of internal to external resistance, are given by: where he = effective thermal conductivity (-AH) = heat of reaction s .=:subscript denoting pellet surface The concentration and temperature are also affected by the resistances at the fluidpellet interface:

i The Damkoehler number Da, which is a measure of the ratio of chemical reaction velocity to external mass transport velocity, is an observable quantity when rnultiplied by the effectiveness factor q:

-

The dimensionless quantities fl (sometimes called Prater number) and /3, relate the concentration differences to the temperature differences within the pellet and external to the pellet, respectively, as will be seen later.

where k, film mass transfer coefficient h = film heat transfer coefficient b = subscript denoting bulk-fluid condition

These relationships state that the mass and energy transferred across the interface are equal to the fluxes at the pellet surface, which in turn are respectively equal to the global rate of reaction per unit area and the rate at which the heat is generated by the reaction. These relationships can be used if desired, to eliminate

i

REACTION / HEAT AND MASSTRANSPORT1 09

Figure 4.5 Nonisothermal effectiveness factors for irreversible first-order reactions in spherical pellets. (Weisz and Hicks 1962. Reprinted with permission

It follows from Eqs. 4.37, 4.38, 4.40, and 4.41 that:

from Chemical Engineering Science. Copyright b y Pergamon Press, Inc.)

The Arrhenius number

E

and the Thiele modulus are given by:

where k b is the rate constant evaluated at Tb. The overall effectiveness factor expressed in of bulk-fluid conditions is obtained by:

The above can be rewritten to give:

where by definition the first quantity is the internal effectiveness factor qi and the second is the external effectiveness factor q,. Although the symbol 77 was used previously for the internal effectiveness factor, the subscripts are now used to differentiate internal and external effectiveness factor. Nonisothemal effectiveness factors are considered next before proceeding to the determination of the relative importance of the various transport processes. In order to obtain the nonisothermal internal effectiveness factor, Eqs. 4.27 through 4.29, or equivalently Eqs.4.32 through 4.34, have to be salved numerically for use in the definition of the internal effectiveness factor given in Eq. 4.46. No analytical solutions exist even for a first-order reaction. Shown in Figure 4.5 are the numerical results obtained by Weisz and Hicks (1962) for an irreversible firstorder reaction in a spherical pellet. It is seen that the effectiveness factor for endothermic reactions (P C 0) is always smaller than that for the isothermal case (p = 0) at the same value, whereas the effectiveness factor for exothermic reactions (P > 0) is always larger. Furthermore, the effectiveness factor far exothermic reactions exceeds unity in certain ranges of values. This occurs when the increase in reaction rate due to temperature rise within the pellet exceeds the decrease in the rate due to concentration gradients. Figure 4.5 also reveals that the effectiveness factor for exothermic reactions is not unique in certain ranges

+

of 4 values, i.e., steady state multiplicity exists. It should be pointed out (Luss 19771, however, that the multiplicity due to the internal temperature gradient usually occurs at unrealistic values of the parameters and is mainly of academic interest. When a single reaction is involved, the external and not the internal gradient is usually responsible for any steady state multiplicity. When the catalyst is not ed, as in the case of metal wire or foil, only the external transport processes affect the reaction rate. In such a case, Eqs. 4.35 and 4.36 with 7 replaced .by q, can be used in the definition of 7, given in Eq. 4.46:

Using Eqs. 4.35 and 4.36 in 4.47 yields the following implicit relationship:

where

+

For a first-order reaction, this reduces to:

REACTIONIHEAT AND MASS TRANSPORT1 1 1

Integrating the same equation first from zero to y and then from zero to unity results in: tc

Shown in Figure 4.6 is this relationship obtained by Carberry and Kulkarini (1973). Similar behavior as in Figure 4.5 can be observed. However, multiple steady states do not exist. It is obvious that the external effectiveness factor by itself is mainly of academic interest except for those unusual cases where the catalyst has only an external surface. It is clear from the foregoing discussion that the general problem of diffusionreaction for the overall effectiveness factor is quite involved. Fortunately, however, the physical and chemical processes at work under realistic conditions favor isothermal pellets and negligible external mass transfer resistances. A more detailed examination of this is in order. Combining Eqs. 4,32 and 4.33 results in:

1 =PBr(l - c,)

I

(4.52)

(4.53)

which is the Prater relationship. Here, the subscript c denotes the pellet center. Dividing Eq. 4.53 by 4.52 gives:

where hTi = Tc - Ts,AC; = C, - Cc, ATx = T' - To, and AC, = Ca - Cs, The maximum value of ATi is that corresponding to Cc = 0. It follows from Eqs.4.35 and 4.53 that:

Also, from Eqs. 4.36 and 4.42:

Integration of this equation with respect to y once using Eqs. 4.34 through 4.36 gives: ts

- t, =P(cs - cc)

ATx = TbPt?@a = TbbBrqDa

i

Combining Eqs. 4.55 and 4.56 gives, after some manipulations:

The temperature ratios are shown in Figure 4.7 as a function of qDa(51) with Br as the parameter (Carberry 1975). It is seen that the overall temperature differIr

Figure 4.7 Ratio of external to total temperature difference for catalytic reactions. (Carberry 1975. Reprinted with permission from Industrial and Engineering Chemistry Fundamentals. Copyright by American Chemical Soci,

,

ety *)

Figure 4.6 Nonisothermal external effectiveness factors for first-order reactions. (Carberry and Kulkarini 1973. Reprinted with permission from Journal of Catalysis. Copyright by Academic Press.)

REACTION/HEAT AND MASS TRANSPORT 1 1 3

ence ATo is almost entirely due to the external temperature difference when the Biot number ratio B, is large. The various dimensionless groups and typical ranges of parameters are summarized in Table 4.2. Here, k, and h for the Biot numbers are obtained from j factor correlations (see Chapter 14 for details):

where v is superficial velocity, and K and a are correlation constants. It follows from the above relationship and the Biot number definitions that:

which can be used to arrive at the typical range of B,. in Table 4.2. The ranges of parameters given in Table 4.2 show that the Biot number ratio is large (>lo) for gas-solid systems. This suggests that the pellet can be treated as isothermal (Butt et a1 1977). In fact, the ratio is in the range of hundreds under typical reaction conditions. Kehoe and Butt (1972) measured temperature profiles of a ed Ni pellet during the hydrogenation of benzene in a singlepellet reactor. Their measurements, shown in Figure 4.8, confirm the theoretical

results. It is seen that the pellet is indeed isothermal when the Biot number ratio is large, whereas there can exist an internal temperature gradient when the ratio is relatively small. The fact that the Biot number of mass (a measure of the ratio of internal diffusion to external mass transfer resistance) is much larger than unity, implies that the major resistance lies in the internal diffusion process. A simple analysis can be made to assess the relative importance of internal and external mass transport processes now that the pellet can be considered isothermal. For an isothermal pellet, Eq. 4.32 can be written as:

Br=20 5 liters 1 min

100

10 liters / rnin 15 liters 1 min

70

60

Table 4.2 Ranges of Parameters and Dimensionless Quantities

a. Ranges of parameters

Gas

Liquid

Porous Solid

14.1 10-4-10-5 10-2-10-4 -

10'5-10-6 lov2-1Oe4 10-'-2

10-'-10-3 1014.4

Quantity

D,(cm2/sec) he(cal/cm sec O C ) pc(cal/cm3 O C ) €

P Pr

& = (BiIm/(Bi)h

-

-

-

-

-

GasSolid

Liquid-Solid

-

-

5-30 0.001-0.3 0.01-2.0 l.0-X04

5-30 0.0014.1 0.0014.05

IO-'-~O-~

b. Dimensionless groups

50 1

I

Bulk

1

I

I 0.8

I 0.6

~

I 0.4

I 0.2

I

I

I

~

0

Radial position r/e After J.J. Carberry, ~hemicaland Catalytic Reaction Engineering, @ 1976; with permission of McGraw-Hill Book Company, New York.

Figure 4.8s Measured temperature profiles in pellets for (Bi),/(Bi)h = 20 with a feed temperature of 52OC: (a) as a function of flaw rate at about 10% CsH6 (b) as a function of feed composition at 15 l/rnin (c) at 10 I/min. (Kehoe and Butt 1972)

REACTION/ HEAT AND MASS TRANSPORT

1 15

Since the internal effectiveness factor is given by:

150rK1 ,

5 liters I min

1

10 liters / min

15 liters / min

130

50

,

1

Bulk

1

1

1

1

1

1

1

1

0.8 0.6 0.4 Radial position r/e

1

0.2

1

1

0

Figure 4.8b Measured profiles in pellets far (Bi),/(Bi)h = 300: (a) as a function of feed composition (b) as a function of flow rate (c) as a function of feed temperature. (Kehoe and Butt 1972)

z.

Therefore:

,

U

1

Equation 4.60 can be rewritten as:

Integrating this equation with the boundary where k, is the rate constant at conditions of Eqs. 4.34 and 4.35 gives:

I

This result reveals that the external concentration difference is strongly dependent on the internal diffusion process: when the internal diffusion resistance is negligible, +,2yi approaches zero and so does the concentration difference, whereas this difference increases with increasing 4, and approaches +,/(Bi), in the asymptotic region of strong diffusion effects (qr+s-+ 1). As indicated earlier, the value of (Bi), is typically in the range of hundreds. Therefore, the external concentration difference is negligible unless the Thiele modulus assumes a value close to that of the Biot number for mass. It has been shown that under realistic reaction conditions, the major transport resistances are external heat transport and internal diffusion. The pellet can be treated as isothermal and the external mass transfer resistance can be neglected. While these conclusions are valid for gas-solid systems, they are not valid for liquid-solid systems as evident from the range of the Biot number ratio (lo-'10-l) given in Table 4.2 for the liquid-solid systems.

4-5

WHY IS DIFFUSION IMPORTANT?

One may wonder at this point why so much attention is being paid to the internal diffusion. If indeed the diffusional process reduces the rate of reaction, wouldn't it be better to eliminate the diffusion resistance altogether? This question was addressed at the beginning of this chapter. A systematic approach is taken here to probe the matter further. Consider the Thiele modulus for a first-order reaction defined in Eq. 4.4b:

REACTIONIHEAT AND MASS TRANSPORT 1 '1 7

where Eq. 4.1 has been used for k The Thiele modulus is a composite of three factors: the characteristic pellet size L, a measure of the catalyst dispersion g($) for a given catalyst loading, and the type of with a certain sulface area and pore characteristics that determine the value of D,. For a given , the pellet size and the method of impregnation that yields a certain dispersion ['(Sr)] determine the modulus. It is always possible to reduce the pellet size and thus to reduce the diffusional resistance. The extent to which this is practical is limited by the pressure drop in the reactor that a tight packing of small pellets can cause. This brings up an important question: Is it desirable to have a higher dispersion that will result in a higher diffusional resistance? The answer is rather obvious if it is recognized that a higher dispersion, despite the higher diffusional resistance it causes, will result in a higher apparent rate of reaction or a higher rate constant. This can be seen by examining the apparent rate constant ( k ) . for a firstorder reaction:

,

( k ) . = k q = k tanh 4/+

L

= (k)112 (De)"2tanh [ L ( ~ / D . ) " ~ ]

(4.65)

It is clear from the examination of this equation that the apparent rate constant (k). increases with increasing values of k or with increasing dispersion. Therefore, it is desirable for a given and pellet size to have as high a dispersion as practical despite the diffusion resistance that highly dispersed catalyst brings about. It shall soon be shown that this desire has to be moderated for various reasons. If a reaction is highly exothermic, too high a dispersion or too high a rate of reaction may cause a runaway situation. It can also cause faster sintering of catalyst crystallites, resulting in a higher rate of deactivation. Consider the choice of a . For a given pellet size, a with a higher surface area gives a higher rate constant, since the rate constant is proportional to the surface area of the for the same catalyst loading per unit pellet volume. On the other hand, the with a higher surface area has smaller pores, which tend to cause a reduction in effective diffusivity. Therefore, detailed knowledge of the is necessary to make definitive conclusions regarding the choice of and its effect on the apparent rate constant. If the average pore radius of a is used as a rough measure of surface area, the rate constant can be considered to be inversely proportional to the square of the average pore radius r since the surface area is inversely proportional to r. Therefore, the apparent rate constant will increase proportionally to ( l / r ) according to Eq. 4.65 if g(&) is simply &, which is the active surface area per unit volume. Now consider the effect of r on the effective diffusivity. For the simple case of equimolar counterdiffusion, Eq. 4.16 reduces to:

Here, the subscript 1 in Eq.4.16 was dropped for the counterdiffusion. The correction factor for pore characteristics, & ( E / K ) is, for instance, simply e2 if the random pore model (see Chapter 14) is used. In general, the voidage E decreases with decreasing r. Therefore, the effective diffusivity will increase with decreasing r, the exact manner in. which r affects the diffusivity being dependent on the factor & ( E / K ) . It is seen then that depending on the relative effects of r on k and D,, a with a higher surface area may or may not result in a higher apparent rate constant. This analysis shows that the choice of the with a higher surface area, which causes a higher diffusional mistance, is not necessarily undesirable for a higher apparent rate constant. ittedly, many constraints have been placed on these analyses to better focus on the point being made: diffusional resistance is not necessarily undesirable as long as the net effect of the factor causing the resistance is in the direction of a higher apparent rate of reaction. Otherwise, the diffusional resistance should be minimized. The importance of internal diffusion can also be appreciated from a different point of view: the fact that the internal diffusion plays a pivotal role in internal and external transport processes. For negligible concentration gradient in the pellet, Eq. 4.57 still holds. However, the value of qDa will be larger than that for diffusionlimited case for the same intrinsic rate since q is larger and therefore the pellet will be more isothermal as Figure 4.7 reveals. Further, a relatively large Biot number for mass under realistic conditions still ensure negligible external mass transfer resistance. It is seen then that in the absence of diffusional resistance, the pellet tends to be more isothermal and the only major resistance is likely to be external heat transfer.

4-6 GENERALIZED INTERNAL EFFECTIVENESS FACTOR

6

Z

I I

This section develops generalized internal effectiveness factors (Aris 1965; Bischoff 1965; Petersen 1965), generalized in the sense that they are applicable to arbitrary kinetics. For a slab-like, isothermal pellet, Eq. 4.21 reduces to:

where the effective diffusivity can depend on concentration. If P is defined as dC/dz, a change of variables gives:

REACTION/ HEAT AND M A S S TRANSPORT 1 1 9

Then, Eq. 4.67 can be written as:

This result is identical to the Thiele modulus derived earlier. For the unknown Cc, Eq. 4.74 can be used to arrive at:

Integrating this equation with respect to C from the pellet center to any point in the pellet yields:

which can be solved for C,. Following similar procedures for an n th-order reaction gives:

where the boundary condition at the center (P = 0) has been used and Cc is the concentration at the center. Solving this equation for P, one gets:

This generalized modulus is different from the traditional modulus 4 by a factor 1)/2]1'2. A plot of generalized effectiveness factor versus generalized of [(n modulus is shown in Figure 4.9 (Bischoff 1965). It is seen that the product qt+c approaches unity when +G 2 3, i.e.:

+

qfcQG = 1

The internal effectiveness factor is obtained from Eqs. 4.46 and 4.71:

when

+c

3

(4.75)

This fact suggests that the reaction may be considered to be in the asymptotic region of strong diffusion effects whenever +G Z 3, which in turn means that Cc in Eq. 4.72 can be set to zero for all practical purposes when 4 ~3. 3, A comparison between theoretical and experimental effectiveness factors has been made by Brown and Bennett (1970) using the approach of generalized effectiveness factors. Their results for the synthesis of methanol from carbon monoxide and hydrogen are shown in Figure 4.10. Good agreement is shown, especially in the region of strong diffusion effects. The usefulness of the generalized modulus lies in the fact that it is applicable to arbitrary kinetics. Further, the product qs+c approaches unity for all forms

A property of the dFectiveness factor is that the product of the Thiele modulus and effectiveness factor approaches unity as the value of the Thiele modulus approaches infinity. In this asymptotic region of strong diffusion effects, the pellet center concentration approaches zero. Therefore, a generalized Thiele modulus can be defined as:

The pellet center concentration C, is still an unknown quantity. It can be obtained by integrating Eq. 4.71:

which is an implicit expression for Cc. Consider a first-order reaction for which r, = kC Using Eq. 4.73 for the Thiele modulus with constant diffusivity gives:

!.

Figure 4.9 Generalized effectiveness factor versus generalized modulus for nth-orderreactions, (Bischoff 1965)

12 1 REACTIONIHEAT AND MASS TRANSPORT

Figure 4.10 Comparison between theoretical and experimental generalized effectiveness factors. (Brown and Bennet

It has been shown that, under realistic reaction conditions, the pellet can

be considered isothermal and the external mass transfer resistance can be neglected. Under these conditions, the reactor point effectiveness can be written for the arbitrary kinetics of Eq. 4.2 as:

where qi = Lrc(k,cb)[2~.1~~r~(a)da]"~

(4.72)

Here, Cshas been set to Cb since the external mass transfer resistance is negligible. The ratio of rate constants can be written as:

of rate expression in the asymptotic region of strong diffusion effect (4G)-- 3). However, it is not without its limitations, It is obvious from Eq. 4.73 that +c is indeterminate when the order of reaction is negative. If there are multiple solutions (Satterfield et al. 1967) to steady state mass balances, Eq. 4.73 also fails in the sense that it gives one single +o for the reaction. It should be recognized on the other hand that multiplicity of solutions usually occurs for unrealistic values of the physical parameters, as pointed out earlier, even when an internal thermal gradient exists. In general, the approach of the generalized modulus is not applicable to those reactions that exhibit empirical or approximate negative-order behavior.

The heat transferred across the external film is equal to the heat generated by the reaction:

The ratio of temperatures Ts/Tb is quite close to unity and therefore a Taylor series expansion of the exponential in Eq. 4.79 can be truncated and still gives an adequate representation. Taking only the first-order term in the expansion, the rate constant ratio can be written as:

4-7 REACTOR POINT EFFECTIVENESS

If all the transport resistances affecting the reaction rate can be combined into a single factor A, the global rate at any point in the reactor can be written as:

where x is the reactor coordinate. This factor A is termed the reactor point effectiveness (Lee, 1981b) since it represents the effectiveness of a catalytic reaction at any point in the reactor. If this reactor point effectiveness is expressed in of bulk-fluid conditions, Eq. 4.76 can be used directly in the overall conservation equations for the reactor without having to solve the pellet conservation equations. The reactor point effectivenessis defined by:

This ratio can be better approximated by introducing a factor of 1.2 for small values of the argument of the exponential:

Use of Eq. 4.80 in 4.8 1 gives: ,

This equation together with Eq. 4.72 can be used in Eq. 4.78 to arrive at the reactor point effectiveness expressed in of bulk-fluid quantities:


The left hand side designated as represents the rate of cooling by the bulk fluid; the right hand side designated as Q, represents the rate of heat generation due to reaction. For steady state operation, should be equal to (Zh. However, there can be no solution as shown in Figure 4.1 1 for if the operating conditions and particularly the fluid velocity are chosen in such a way that the heat transfer coefficient h is too small. When there is no solution, the heating rate is larger than the cooling rate and there will be accumulation of heat, leading to a runaway. In such cases, the reactor point effectiveness given by Eq. 4.83 would become negative, indicating that the selected operating conditions would lead to a runaway. The heat transfer coefficient h is strongly dependent on the fluid velocity, its value increasing with increasing velocity. Therefore, heat transfer between the bulk fluid and the pellet places a constraint on the fluid velocity that can be used. The velocity must be high enough to ensure there is at least one solution to Eq. 4.80a for steady state operation. With further increases in velocity, two steady states, and then three steady states will arise. The latter case is shown in Figure 4.1 1.

a

where

The reactor point effectiveness of Eq.4.83 is in a form suitable for direct inclusion in reactor conservation equations. However, the pellet center concentration Cc has to be specified. As discussed earlier, C, can be set to zero when cSc 2 3. For 4G less than 3, C, can be calculated in principle using Eq. 4.74, but this involves trial and error procedures, which are cumbersome. Instead, an approximate solution for Cc can be based on the fact that the value of the integral I in Eq. 4.84 is rather insensitive to errors in the estimated value of Cc. If we let ACc be the error in C,, the corresponding maximum relative error in the integral I for a second-order reaction, for instance, is 13 ACcC,2/CgI. A 30% error in C, will result in a maximum error of less than 10% for 4 = 1. Consider the estimation of the center concentration, Cc.As indicated earlier, the center concentration can be set to zero for cSc 2 3. On the other hand, the reaction can be considered to be free from diffusional effects for all practical purposes when cSc is less than 0.3. Therefore, estimation is necessary only for the intermediate range of &, i.e., 0.3 < # J ~< 3. As the results in Figure 4.9 show, the maximum error in using the effectiveness factor of a first-order reaction for a third-order reaction is less than 10% when the generalized modulus is used. This suggests that the center concentration corresponding to a first-order reaction may be used for arbitrary kinetics:

where 4Gis given by Eq. 4.73. Note that the generalized modulus can be determined without the knowledge of the center concentration. Although a more accurate method is available (Lee and Ruckenstein 1983) for the estimation of the center concentration, the center concentration given by Eq. 4.85 is sufficiently accurate for the purpose of obtaining the global rate as the results in Section 4.9 for multiple reactions will show. The linearization given by Eq. 4.81 needs further explanation in light of the fact that there can be as many as three solutions (steady states) to Eq, 4.80. Equation 4.80 can be rewritten as follows: Figure 4.11 Possible multiple steady states for Eq. 4.80a.

i

The steady state az is bbunstable"in the sense that a slight change in operation either way will move the steady state condition to either al or as. The steady state a3 is possible only when T,/K 1. Since Z / T b is close to unity under typical operating conditions, the steady state corresponds to the state a l . The steady state corresponding to Eq. 4.81 is that of a l . i.e., the steady state at the lowest Ts,when there are multiple solutions to Eq. 4.80. The utility of the reactor point effectiveness can be best illustrated through an example. Consider an adiabatic plug-flow reactor in which a catalytic reaction takes place. Suppose that +c is greater than 3 and that the intrinsic rate expression is:

*

rC=kf(C)= k

This global rate can be used directly in the reactor conservation equations, leading to the specification of reactor size in a straightforward manner since now the global rate is expressed in of the bulk fluid quantities Cb and Tb. Otherwise, the pellet conservation equations would have to be solved first. Details on this approach of reactor point effectiveness and comparison with the traditional approach are given in Chapter 10. It is appropriate at this time to point out that any rate expression for a single reaction can always be reduced to the rate expression of Eq. 4.2, which contains only the concentration of the main reactant, C. Consider the following reaction:

(-)l +CK C

Assuming constant velocity and physical properties, the steady state mass and heat balances are:

where A is the main reactant, and the lower case letters are the stoichiometric coefficients. Let the rate expression for the main reactant be:

where 2 = normalized reactor coordinate, x/Z T = Z(l - E ~ ) / v Z = reactor length Ee =void fraction of packed bed v = supeficial fluid velocity For reactor design purposes, these equations have to be solved along with the pellet conservation equattons 4.27 through 4.29 and the relationships at the pelletbulk fluid intedace (Eqs. 4.30 and 4.3 1). With the use of reactor point effectiveness, however, solution of these equations can be avoided. The global rate is:

where CB,CE,and Cw are the concentrations of species B, G,and W, respectively. The steady state conservation equations for species A and G, for instance, are:

1'

i

i

where D, and ( D e ) are ~ the effectivediffusivities for the main reactant and species G, respectively. Combining these two equations yields:

The integral I for A is: Integrating with respect to z from the center to z, one has:

Using this in Eq. 4.83 and then in Eq. 4.77 for the global rate Rc, gives:

REACTION/HEAT AND MASS TRANSPORT 127

where the boundary condition at the center has been used. Integrating once again, but this time from z to the pellet surface gives:

I

where (Cc), and C, are the concentrations at the surface for species G and the main reactant, respectively. For any number of species involved in the reaction, in general, the concentration of species i in the pellet can be related to that of the main reactant as follows:

where the following identity has been used:

Repeated integration by parts of the last term in Eq. 4.97 yields the following integro-differentialseries (Lee 1981a), evaluated at z = L and valid in the asymptotic region of strong diffusion effects;

where vi is the ratio of the stoichiometric coefficient of species i to that of the main reactant, with the convention that it is negative for products and positive for reactants. Since the concentrations of all species can be expressed in of the main reactant concentration C as given by Eq. 4.93, the substitution of Eq. 4.93 into Eq. 4.87, for instance, reduces the rate expression to that given by Eq. 4.2. 4-8

The right hand side can be integrated by parts to give:

NONUNIFORM ACTIVITY DISTRIBUTIONS 1 dz-l 3 ==-

So far in this chapter, the catalyst activity has been assumed uniform throughout a pellet. Nonuniform activity can result from poor impregnation, deliberate partial impregnation, or from deactivation by poisoning species. In such cases, the intrinsic kinetics can be represented by:

P m

dz

with Jo = G'/F=

(4.100)

+

where the number of times r(C) is integrated is equal to (i 2). Here, the overbar denotes evaluation at z = L, and the subscript denotes the region of strong diffusion effects. In accordance with the previous conclusions on pellet isothemality and negligible external mass transfer resistance, which still hold, the pellet surface concentration has been set to Cb. The integro-differential series in Eq. 4.99 has been shown (Lee 198la) to be a converging series. The flux at the surface can be adequately represented by the first term:

where is a normalizing factor such that G ( z ) S 1. Here, G(z) is an arbitrary but presumably monotonic function for the activity distribution. The steady-state pellet mass balance written for a slab-like pellet in of P(= dC/dz) for constant effective diffusivity is:

provided the ratio R given below is much less than unity: The boundary conditions are the usual ones. Both sides of this equation are multiplied by P and integrated from O+ to z to give: > j

I

i

k

This ratio is essentially the maximum fractional error involved in using the approximate relationship of Eq. 4.101. For an nth-order reaction, this ratio reduces to:

REACTION/HEAT AND MASS TRANSPORT 129

In the asymptotic region of strong diffusion effects, therefore, this ratio is much less than unity unless G is small. When the activity monotonically decreases toward the center of the pellet, this ratio R is usually much less than unity. When the activity increases toward the centen, however, the value of R should be checked 2 0.5 it has been shown (Lee 1981a) for various before using Eq. 4.101: for forms of the activity distribution that R is much less than unity. The generalized internal effectiveness factor in the asymptotic region of strong diffusion effects is obtained from:

t

i

1 i

j

the use of the characteristic length L for shape normalization is also valid for pellets with nonuniform activity when the surface activity is not zero. It should be noted however that for pellets of geometric shape other than slab, the like integral appearing in the denominator of Eqs. 4.104 and 4.106 must be changed to a volume integral. Suppose that a slab-like pellet of initially uniform activity is deactivited such that the activity profile is given by k[0.2(1 - z / L ) ~ 0.81 when the surface activity is reduced to 80% of its original activity. Suppose further that the intrinsic rate r, for the fresh catalyst is kc2.Then, in the asymptotic region of strong diffusion effects, one has:

+

Therefore, the internal effectiveness for this case is:

Since q*+o should approach unity in this asymptotic region, the generalized modulus can be defined as: which is the generalized effectivenessfactor for the second-order reaction multiplied by (LG~I~/JI,L Gdz). Note that the activity of the deactivated pellet was used here as the intrinsic activity for the purpose of illustration. The activity of fresh catalyst should be used as the intrinsic activity for the combined effects of deactivation and diffusion as will be seen in Chapter 5. For a pellet of nonuniform activity, the reactor point effectivenessis:

In an analogy with the case of a uniform activity distribution, a reaction may be considered to be in the region of strong diffusion effects when (+c)n is greater than 3. The analytical results obtained by Wang and Varma (1980) for a firstorder reaction confirms this limit of 3 for the asymptotic region. For all practical purposes, then, the generalized internal effectiveness factor is:

Following the same procedures as for a pellet of uniform activity, one can arrive at the following expression for A:

when (+o)n is greater than 3. This relationship follows from Eqs. 4.101 and 4.104. When the ratio R given by Eq. 4.102 is not much less than unity, more of the integro-differential series in Eq. 4.99 can be retained for the calculation of F-, which then can be used for rlt (Eq.4.104). If one more term is used, for instance, Eq. 4.99 becomes a cubic equation in Pa. and a numerical search for Fm can start with the value of P, obtained from Eq. 4.101. In general, this procedure will be required for a pellet with activity increasing toward the center for < 0.5. It has been shown for a first-order reaction (Wang and Varma 1980) that

with I(Cb) = 12f ( ~ ) d ~

i

!

b

where Eq. 4.106 has been used in 4.107. The restrictions placed here are that the maximum fractional error given by Eq. 4.102 is small and that cbG is much larger than unity.

REACTION/HEAT AND MASS TRANSPORT1 3 1 4-9

MULTIPLE REACTIONS

Consider the relationship between the concentrations of A and B. If the relationship is known, Eqs. 4.109 and 4.110 can be integrated to yield:

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). Consider the following reaction network:

where the lower case letters denote the ratios of stoichiometric coefficients, and the capital letters denote the species. This is the basic unit of any complex reaction network. Since the reaction stoichiometry allows one to express the concentrations of all species in of the concentrations of species A and B and the diffusivity ratios, as shown in Section 4-7 for a simple case, it is sufficient to consider only the concentrations of A and B for rate expressions. As was done in Section 4-6 (Eq. 4.69), the steady state mass balance equations are written on concentration coordinates:

>,

' I

Since pA = dA/dz, and Ps = dB/dz, Eqs. 4.1 12 and 4.1 13 can be integrated and then combined to give:

where the integration is from any arbitrary position zi to a position Zi.1 such that Az = - a+l)is small. If the small changes in A and B corresponding to the small change in a (Ar) are denoted as AA and AB, then the trapezoidal rule of integration can be used to expand Eq. 4.1 14 as:

AA AB ~2~ [ p A ( ~ + l ) ] ~ 1 + [ P ~ ( ~ ) ] - ' } ~ ~ ( [ ~ B ((4 ~. 1~1 5~) ) ] - 1 ~ [ p where

[ lA(o) i A( z i + ~ )

DAd -

PA(z*+I) =

~ i

2 dA

- krlf~(A,B ) + k ~ f(A, b B);

Ded ~ i

-bk&~ (A,B ) + kj& (A,B);

I

2 dB

PA(a)

Ps=dB/dz

(4,110)

112

( k h+k h ) d ~ ]

[$

(4.1 16)

( k ~ f .+ 4 k ~ f ~ ) d ~ ] ~ ' ~ (4.1 17)

s given by Eq. 4.112. Similar expressions can be written for P B (+I) ~ and Pe(zi). Now, Eq. 4.1 16 can be rewritten as: where the Di ( i = A, B, and D) are the effective diffusivities of species i and are assumed constant, and ki are the rate constants. The usual boundary conditions apply.

.

REACTIONIHEAT AND MASS TRANSPORT1 33

where

L Similarly for Pi (zi + I ) :

-

-

f ABi)

+ k&o(Aa

AAi,Bb

+ ABi) J

+

where

Substituting Eqs. 4.1 18 and 4.120 into Eq. 4.1 15 yields:

since, for instance, ( k A f ~ ) i +=~ k,&(A1,B1) = k.&(Ab - AA1,Bb AB1) for i = 0. This result uses the fact that the reactant concentration decreases toward the pellet center, whereas the product concentration increases. Equation 4.126 contains only one unknown, AB1, which is thus determined. Now that all information at i = 0 is available, one can increment i successively and repeat the whole procedure until the pellet center is reached, thereby obtaining the profile of B as a function of A. The step size AAi may be made different for each step or constant for all steps. The numerical method based on Eq. 4.125 does not involve any two point boundary value calculations: only straightforward marching is involved. Furthermore, the integrated values obtained in the process of solving Eq. 4.125 can be used directly in Eqs. 4.112 and 4.1 13 for the global rates:

Here, the following relationship has been used:

It follows from the definitions of

CA

and

6s

and Eqs. 4.122 and 4.123 that:

where Ac and Bc are the pellet center concentrations. A similar expression for ( R c ) Dcan also be written. In order to evaluate the integrals, however, the center concentrations have to be known. For an estimate of the center concentration of A, one can define the effectiveness factor for the reactions involving the consumption of A in accordance with the usual practice:

If the trapezoid rule is used again for the integration, one obtains:

This is the relationship that can be used to generate the concentration profile of B as a function of A, thus providing the relationship between A and B needed for the integration of Eqs. 4.112 and 4.113, Since the position i is arbitrary, one can start at the pellet surface where Ar = Ab and Bi = Bb, Ab and Bb being the bulk concentrations, In order to calculate the concentrations A and B at a distance Az away from the pellet surface, one can choose a small value for A A l . This immediately gives A 1 (=Ab - AA I) for i = 0.For the calculation of AB1 corresponding to the chosen value of AA1, Eq. 4.125 can be written as:

It is sufficient to estimate only Ae since all the other center concentrations can be found from it. In anticipation of the development to follow, the subscript G was used to denote the generalized effectiveness factor. As was done in Section 4-6, the generalized modulus +c is defined by:

This follows from the fact that A, approaches zero for large not. The definition of q ~ when , used in Eq. 4.130 yields:

+G

but Bc does

REACTIONIHEAT AND MASS TRANSPORT1 35

+

where r~ = kqfA kDfo. Since the relationship between A and B is now known through Eq 4.125, the generalized modulus +c can be evaluated readily, using the value of B corresponding to A, = 0 as Bc. As was done in Section 4 7 , the center concentration is approximated by:

where +G is given by Eq. 4.13 1. It is seen that the calculation of the global rates involves only the solution of Eq. 4.125, from which A,, Bc# PA(L),and PB(L)can be directly calculated with the aid of Eqs. 4.131 and 4.132. The original split boundary value problem has been transformed into an initial value problem. Consider the following reaction network for an illustration of the numerical method:

4---

The rate expressions can be substituted into Eq. 4.125 for the calculation of B as a function of A. For instance, the numerator in Eq. 4.125 become:

Center

Surface

-

0 = 7.86

Normalized Concentration,ofA

Figure 4.12 Comparison between correct and approximate solutions for concentration profiles.

Here again, one must that the concentration of species A decreases toward the pellet center (At - AAi+l), but that of species B increases (Bi ABi+l). Starting at the pellet surface (i = 0) at which A and B are at bulk-fluid concentrations, i.e., A0 = Ab and Bo = Bbr Eq. 4.125 can be solved for at each step i for a chosen value of AAr. This forward marching yields not only the corresponding values of A and B, but also those of k d A , k&, and k&. The forward marching can continue by incrementing the index i until A = 0 is reached. The values of k h , kxfo, and A thus generated are then used for the evaluation of $G using Eq. 4.131. Since the pellet center concentration of A, Ak, is given by Ak/Ab = l/cosh (pG, the calculated value of +G can be used to locate the index k, yielding the center concentrations of A and B. Now that all quantities

+

necessary to evaluate the integrals in Eqs. 4.127 and 4.128 are known, the global rates can be calculated from these equations. Results obtained from the example reaction network are summarized in Table 4.3 for various sets of rate constants and surface concentrations (see Problem 4.24), which were selected to give +o ranging from 0.1 to 8. The concentration profiles for some selected cases are shown in Figure 4.12. The example system turned out to be very sensitive at low concentrations of A and the usual trial and error method failed to give any reliable results. For the purpose of obtaining the real concentration profiles, therefore, the system equations were solved by assuming pellet center concentrations and treating the resulting surface concentrations as boundary conditions. The solutions thus obtained are listed as "real" solutions in Table 4.3. It is seen that the maximum error in global rates is about 5% at $G = 4 and the error is substantially less for other +c values.

1 36

REACTIONIHEAT AND MASS TRANSPORT137

HETEROGENEOUS REACTOR DESIGN

Table 4.3 Comparison of Global R a t e for Various +c Values

( R C ) (rnol/cm3 ~ s) $G

Real

Calculated

(RG)B(mo1/cm3 Real

The steady state mass balance for species Aj can be written as:

S)

Calculated

DA PA Z L= Aj R( j = 1 , 2 , .

% Error In (&)A

2

(Rc)B

n ) :

dAj PAj=x

(4,135)

dAj

where DAj is the effective diffusivity of the species A& If one follows the same procedures as for the basic unit considered earlier, the results corresponding to Eqs. 4.125, 4.127, and 4.13 1 are: (AA l)i

* Calculated based on Eq. 4.125.

+I

The numerical method presented for the basic reaction unit can be extended in a straightforward manner to any complex reaction network (Akella 1983). Consider the general reaction network shown in Figure 4.13. If there are n species (Al, A2, AS, . . . , An), the rate of consumption of a species Aj (j = 1, 2, . . . , n) is given by:

where vtj is the appropriate stoichiometric coefficient, and where a general rate expression for the reaction path between Ai and Ar is given by:

A complex reaction net-

where (j = 1, 2, .

. . , n)

The same procedures as for the basic reaction unit can be followed to calculate the global rates. However, the solution of Eq.4.136 involves simultaneous solution of n nonlinear algebraic equations, which is by no means trivial. One way of circumventing this problem is to use the rectangular rule of integration instead of the trapezoidal rule in Eq. 4.136, i.e.:

This approximation allows one to calculate the only unknown (AAj)i t xdirectly, but at the expense of accuracy. The rectangular rule leads to an integration error that is first-order in magnitude with respect to the step size. In order to avoid such errors, one can use the (6Aj)i+l value calculated from Eq. 4.139 only as a predictor and then use these predictor values to obtain corrected values. For this purpose, let (AAj)r+l be the predictor value calculated from Eq. 4.139. When this is used in Eq. 4.136 for the corrected value,

REACTION/ HEAT AND MASS TRANSPORT1 39

Calculations involving this equation and Eq. 4.139 are straightforward since no trial and error is involved. Further, the accuracy should be much better than that possible with only Eq.4.139. In fact, the results obtained by the above predictorcorrector method for the example problem in Table 4.3 have been found to be identical with those obtained by solving Eq.4.136 by an iterative bisection method. A major subset of the general reaction network shown in Figure 4.13 is the one in which all the reactions involved are irreversible and the rate of reaction for a given species depends only on the concentration of the species in question and those from which the species in question is formed. The steady state mass balances can then be written as:

It is obvious from the balance equations that they can be solved two at a time. The results obtained for the basic unit can be used to solve Eqs. 4.140 and 4.141, which in turn can be used to solve the entire set. This section has shown that the global rates for multiple reactions can be obtained in a straightforward manner. In order to use the global rates in reactor conservation equations, however, the pellet surface temperature corresponding to the bulk-fluid temperature has to be known. This involves satisfying the heat balance around the film between pellet and bulk-fluid (Eq.4.31) with (-AH)RGL replaced by T?f (-AHj)(R9)L for the calculation of the pellet surface temperature T,. This aspect will be explored in detail in Chapter 10.

4-10

TRANSPORT CRITERIA AND PELLET DESIGN

The developments so far are based on the premise that intrinsic rate expressions are available. As discussed in Chapter 3, these rate expressions are determined from the kinetic data obtained under experimental conditions that ensure negligible transport effects. Various criteria of negligible transport effects have been developed to provide guidance on the experimental conditions and these were summarized in Chapter 3 without derivation. Consider these criteria in light of the understanding gained in this chapter. In view of the fact that external transport effects usually

can be neglected in the absence of internal diffusion resistance, it is necessary that the internal diffusion resistance be negligible. For a first-order reaction, the internal effectiveness factor vi approaches unity when ~ 1 (= 4 ~ tanh 4) is much less than unity. On the other hand, the definition of the internal effectiveness factor gives:

+

RG - Rc rli

"

rc

--

kc.

*

Since 4* = L2k/De for the first-order reaction, it follows from Eq. 4.144 that the product qi+2 is RcL2/(DeCs). Therefore, a criterion of negligible diffusion effects for a first-order reaction can be stated as follows:

This is known as the Weisz-Prater (1954) criterion. The usefulness of the criterion lies in the fact that all quantities appearing in the right hand side of Eq. 4.145 are measurable. While this relationship has been derived for a first-order reaction, it is nevertheless applicable to many reactions for which the kinetics are not known. Mears (1971), for instance, suggested the use of l/n in place of 1 in Eq. 4.145 for an nth-order reaction. A conservative number, say 0.1, may be used in the criterion for reactions with unknown kinetics. It should be recognized, however, that the criterion does not necessarily work in all cases. This is particularly so when the reaction is strongly inhibited by one of the products (Froment and Bischoff

1979). In experiments to obtain intrinsic kinetic data, the pellet size is often chosen small enough to ensure that diffusion is negligible. This involves carrying out a series of isothermal experiments in which the pellet size starts at a very small value and is successively increased. If the initial rate decreases in successive experiments above a certain pellet size, say LJ, then the experiments to obtain intrinsic kinetics are carried out with pellets of size smaller than LF A more efficient method is an experiment in which the reaction temperature is increased progressively in a planned manner. As discussed in Section 4-3, the slope of a plot of the logarithm of measured rates (In R c ) .versus 1/T changes from (-E,/R,) to (-Ea/2R,) as the reaction regime changes from a diffusion-free to a diffusion-limited regime with the increase in temperature. In the transition regime, the plot is curved, as shown in Figure 4.14 (see also Problem 5.16). Therefore, a reaction can be considered free from diffusional effects, in a certain temperature range and for the pellet size used in the experiment, if a plot of In RG versus 11lT give a straight line for that temperature range and then a curved line at higher temperatures, the straight line representing the intrinsic kinetics. A criterion on the importance of external heat transport can be arrived at by combining Eqs. 4.80 and 4.8 1:

REACTION/ HEAT AND MASS TRANSPORT1 4 1 Figure 4.14 A test for the effect of diffusion.

where the deviation in the value of the rate constant due to the external temperature difference is restricted to less than 5%. This criterion is almost identical to that developed by Mears (1971). Much more rigorous tests of the importance of diffusional effects can be made if the intrinsic kinetics are known on the basis of the generalized internal effectiveness factor and reactor point effectiveness. Using the rate expression of Eq. 4.2 in Eq. 4.73 for one gets:

+:,

Applying the Weisz-Prater condition that qi+$ be much less than unity, the same procedures used in obtaining Eq. 4.145 give the following criterion of negligible diffusion effect (Petersen 1965; Bischoff 1967):

Here, the inverse of the denominator of Eq. 4.83 has been approximated by the first-order term in a Taylor series. Pellet design is concerned with the selection of pellet size, average pore size, pore size distribution, and the specification of the activity distribution function g(S,) in Eq. 4.1. Pellet design cannot be considered separately from reactor design. Nevertheless, certain pertinent conclusions can be made. It is almost always true, for instance, that the characteristic pellet size be chosen as small as the pressure drop and temperature rise in the reactor will allow. A has to be chosen for the desired pore size distribution. There is rarely any control over the pore size distribution. If selectivity is severely affected by diffusion, a of low surface area should be used. A of higher surface area can give a higher rate of reaction per unit pellet volume but also gives a higher diffusion resistance, which reduces selectivity. The catalyst surface area per unit volume, g(S,), can be made to vary with pellet coordinate by choosing an appropriate impregnation method. Hence, this function represents not only the level of dispersion but also activity distribution. A partially impregnated (or equivalently hollow) pellet is a typical example of a pellet with a certain activity distribution. The motivation for making such a pellet becomes obvious if it is recognized that the reactant concentration becomes almost zero at some point in the pellet when the reaction is diffusion-limited. The fraction of the volume of the pellet for which the concentration is zero is not utilized at all. If this fraction is made hollow or inert, then the observed rate on a per pellet basis should be the same as the fully impregnated pellet. Let us examine this further. Suppose that a pellet is hollow or partially impregnated for a distance Li from the center. Consider a diffusion-limited, first-order reaction. The internal effectiveness factor for this hollow pellet is:

For the fully impregnated pellet, it is:

Therefore, the internal effectiveness factor increases as a result of making the pellet hollow by the following factor: where R G / [ k J ( C . ) ]has been used for v. The reactor point effectiveness can be used as a criterion of negligible transport effects by insisting that it be greater than some value, say 0.95:

REACTION/HEAT AND

While the internal effectiveness factor increases, the reaction rate per pellet remains the same since:

where Vr and Vh are the pellet volume for the fully impregnated pellet and that for the hollow pellet, respectively. It is seen that the use of hollow pellets does not increase the amount of reactant converted but does result in a saving of catalyst in the ratio of (L - &)/L. For the same dispersion and catalyst loading, then, it is best, in the absence of attrition and poisoning, to have all catalyst deposited in the thinnest possible exterior layer of the pellet at least for the simple reactions now under consideration. The rationale behind this is that the highest possible rate constant can be attained with this arrangement. The observed rate constant is proportional to the square root of the intrinsic rate constant and this intrinsic rate constant increases with increasing surface area per unit volume. However, this approach to pellet design is not practical even in the absence of dhemical and physical deactivation since most of the catalyst will be lost due to attrition in the reactor. In practice, therefore, a compromise has to be made that is suitable for the application of interest. The optimal catalyst distribution for relatively complex reactions can depend on several factors (Morbidelli et al. 1982). While the activity profile can be tailored to suit a certain application, uniformity of activity is often an overriding concern in practice because the repeatability of catalyst performance is one of the overriding factors in plant operation. As is apparent from the j-factor correlations of Eq. 4.58, external transport effects can be minimized by a proper choice of fluid velocity. While the transfer coefficients are also affected by the pellet size, the fluid velocity is often the choice for the manipulation. As discussed earlier, the external mass transfer resistance should not pose a problem at throughputs commonly encountered in typical operation.

NOTATION

pellet surface concentration of species Aj pellet center concentration of species Aj Darcy's permeability Biot number for heat, defined in Table 4.2 Biot number for mass, defined in Table 4.2 (Bi)h /(Bi)m c/cb

CC

c

solution of Eq. 4.85 when F(c) = c fluid concentration bulk-fluid concentration concentration for species i pellet center concentration

c/cs

specific heat capacity concentration vector for all species

c.- C'

Cb - c, effective diffusivity D, for species 1 effective molecular diffusivity of species i in species j effective Knudsen diffusivity for species i effective Knudsen diffusivity for species of interest in binary system effective surface diffusivity for species i molecular diffusivity effective diffusivity of species Aj activation energy concentration dependence of intrinsic kinetics [r, = kf(C)] correction factor given in Eq. 4.11 f(C)/f(Cb) catalyst surface area per unit pellet volume catalyst activity distribution function (G 5 1)

Summary

This chapter has shown that internal and external transport processes can significantly alter rates of reaction. Since the actual conversion of reactants occurs at the global rate as affected by the transport processes, pellet conservation equations have to be solved first to relate the intrinsic rate to the global rate for reactor design. The fact that the major resistances are usually internal diffusion and external heat transfer allows one to express the global rate in of intrinsic kinetics and pertinent transport properties. This result, in turn allows one to deal directly with the reactor conservation equations for reactor design without having to solve the pellet conservation equations. For multiple reactions, however, the pellet conservation equations have to be solved as part of the reactor design. The numerical method given in this chapter allows this calculation in a straightforward manner.

MASS TRANSPORT1 43

dG/dz

kSP

G and G' evaluated at z = L film heat transfer coefficient heat of reaction integral defined by Eq. 4.84 j factor for mass given in Eq. 4.58 j factor for heat given in Eq. 4.58 quantity defined in Eq. 4.100 rate constant rate constant used for normalization of k [ k = kG(z)] film mass transfer coefficient rate constant per unit catalyst surface area


equilibrium constant; a constant in j-factor correlation characteristic length of pellet (= V,/S,), position for pellet surface as measured from pellet center for slab Lewis number defined in Table 4.2 distance from pellet center up to which the pellet is inert or hollow L for various pellet shapes MlM2 molecular weight for species i order of reaction flux for species i directional flux vector for species i dC/dz; total pressure dAj/d2 P evaluated at z = L P i n the region of strong diffusional effect Prandtl number defined in Table 4.2 rate of reaction defined in Eq. 4.94; average pore radius intrinsic rate of reaction given in Eq. 4.1 intrinsic rate of reaction for the reaction path from species Ar to species Af intrinsic rate of consumption of a species Af radius; ratio defined by Eq. 4.102 global rate gas constant Reynolds number defined in Table 4.2 . Schmidt number defined in Table 4.2 active catalyst surface area external pellet surface area temperature bulk-fluid temperature Tc - Ts Ts - Tb T/Tb supeficial fluid velocity pellet volume mole fraction of species i z/L pellet coordinate with pellet center as the orgin reactor length

Greek Letters

a

fl

dummy variable for integration; a constant in j factor correlation quantity defined in Table 4.2; dummy variable for integration

quantity defined in Table 4.2 pellet porosity; Arrhenius number defined in Table 4.2 quantity defined by Eq. 4.119 bed porosity; quantity defined by Eq. 4.121 tortuosity dummy variable for integration effectiveness factor internal effectiveness factor defined in Eq. 4.46 7 for various pellet shapes overall effectiveness factor defined in Eq.4.46 external effectiveness factor defined in Eq. 4a.46 7 in the asymptotic region of strong diffusion effect quantity defined in Table 4.2 effective thermal conductivity reactor point effectiveness defined by Eq. 4.77 fluid viscosity fluid density stoichiometric coefficient for species i Thiele modulus defined by Eq. 4.44 Thiele modulus for slab-like, cylindrical, and spherical pellets generalized modulus defined by Eq. 4.73 generalized modulus for nonuniform pellet, Eq. 4.105 Thiele modulus defined in Eq. 4.59 Subscripts

at bulk-fluid conditions at pellet center, at z = 0 fully impregnated pellet hollow pellet index for concentration grid species at pellet surface, at z = L pellet shape (m = 0, slab; m = 1, cylinder; m = 2, sphere) at pellet surface PROBLEMS

4.1. Obtain the internal effectiveness factor for a first-order reaction taking place in a spherical pellet. Use the following transformation of variables: v = Cr where r is the radial coordinate. 4.2. Suppose that a pellet is isobaric. Does this mean that the convective flux can be neglected? What is the rationale behind using Eq. 4.4a in most treatments of reaction-diffusion problems?

REACTIQNIHEAT AND MASS TRANSPORT 147 4.3. Show that Eq. 4.15 follows from Eq.4.13 for a binary system. 4.4. Extract as much information as you can on the intrinsic kinetics and diffusional effect from the results shown in Figure 4.3. rnol/(cm3 pellet sec). If 4.5. The observed rate using a 2 cm cubic pellet is what is the half-thickness of a slab that would the intrinsic rate is yield the same observed rate? Suppose further that conversion is independent of initial concentration. What is the intrinsic rate constant? The effective diffusivity is 0.01 crn2/sec. 4.6. Calculate the surface concentration and the concentration and temperature at the pellet center for the following conditions:

RG =

rnol/(cm3 pellet sec)

Cb= mo1/cm3 k, = 1 crn/sec Tb = 1l o O c B, = 60

4.12. Draw conclusions on the effect of internal diffusion on the internal and external temperature and concentration gradients from the following information for a first-order reaction: P=O.Ol,

I I

4 =5 E = 0.5 L=2cm T, = 12O0C

Suppose that Ea/&Tb = 10. What is the effectiveness factor if the external temperature difference is taken into ? 4.7. Derive the effectiveness factor for a first-order reaction when the catalyst is deposited only on the external surface of the , 4.8, When the CO content is quite low, the oxidation of CO over Pt can be represented by r, = k/(CO). Discuss this behavior and obtain the internal effectiveness factor. 4.9. Show that for a slab-like pellet, the effectiveness factor for a first-order reaction is given by:

'

=1

tanh +/4

+ 4 tanh +/(Bi),

Assume that the pellet is isothermal and that the external heat transfer resistance is negligible. What can you conclude about the relative importance of external mass transfer? What if the internal diffusion resistance is very high? 4.10. Liu (1969) gave the following approximate relationship for the nonisothermal, internal effectiveness factor: tanh 4

rli =-

4)

exp(eP/S)

Find an expression for the locus of the maximum values of qi corresponding to different values of EP. Note that both 4 and rP change with temperature. 4.11. For uniform catalyst pellets with a dispersion of 0.2, the Thiele modulus is 5. If the same pellets with a dispersion of 0.8 are used under identical reaction conditions, what is the change in the observed rate?

B,=100, Tb =4W°K

@=5, D, =0.01 cm2/sec E = 20, Cb = 0.6 X 10'%ol/cm3 (Bi), = 200

The values of 4 and (Bi), are for L of 2 cm. Calculate the gradients ( A T s and AC's) corresponding to L = 0.1 ern, 1 cm, and 10 cm. Is it true that all gradients increase as 4 increases? 4.13. Extract as much numerical information as you can from the results shown in Figure 4.8 on the values of dimensionless groups given in Table 4.2 and on the values of concentration differences (AC's). 4.14. Show that the reactor point effectiveness for a first-order reaction reduces to:

i

j 1

A=

tanh

Q,

441 - (7Da)xI

I

I

4.15. Consider a reaction, A

+B

--+

2 0 . The intrinsic rate expression is:

Rewrite the rate expression in the form of

Obtain the generalized modulus for this reaction. Assume equal Deb. 4.16. ~ktersen(1965) used the data of Austin and Walker (1963) for the reaction, C COz 2C0. Use the result of Problem 4.15 and the following data to check the importance of diffusion limitation, first using the Weisz and Prater criterion and then the criterion based on the generalized modulus.

+

+

mol/cm3, K I= 4.15 X lo9 crn3/rnol ( C C O= ~ )1.22 ~ X Kz = 3.38 x 105 cm3/mol, D, = 0.1 cm2/sec L = 0.7 cm, Rc = 4.67 x mol/cm3 sec

REACTION/HEAT AND

What conclusions can you make regarding the suitability of the criteria for l e t1.22 this reaction that is inhibited by product? Assume that ( C ~ ~ ~ ) i n = X mol/cm3. 4.17. The following has been proven by Corbett and Luss (1974): For two identical pellets in which a first-order irreversible reaction occurs, which have the same volume averaged rate constant but different rate constant density functions, the effectiveness factor corresponding to the pellet with the larger surface activity exceeds that of the second pellet if the difference between the rate constant density functions of two spherical pellets changes monotonically with the radial distance. Is the statement correct for slab-like pellets in the asymptotic region of strong diffusion effects? Use the generalized effectiveness factor for nonuniform activity. 4.18. In the region of strong diffusion effects, approximate as well as exact solutions for the effectiveness factor can be obtained by various methods for a firstorder reaction taking place in a pellet of nonuniform activity. Obtain the effectiveness factor when the activity distribution function G(z) is given by (Z/L)~,using the generalized modulus approach, the method of exact solution in the form of Bessel functions and the method of Liouville-Green approximation (WKB approximation) (Yortsos and Tsotsis 1982). Discuss and compare the results. 4.19. Calculate the effectiveness factors as a function of k for a pellet of nonuniform activity for which r. = EGCI Use the conditions given below.

MASS TRANSPORT1 49

Show that the concentration gradients at the pellet surface are given by:

4.23. The rate expression for the following reversible reaction is:

where

klfi = k l A B 2 k2fi = kzD

Obtain the internal effectiveness factor for species A. Use the generalized modulus approach. 4.24. For the example network in Section 4-9, the following parameters and concentrations were used: Use Eq. 4.101 first. Add one and then two more correction for F' (Eq. 4.99) to compare these results with those obtained from Eq. 4.101. Redo the problem for G = 0.271 (0.9 - z/L)3. 4.20. Does an increase in the Thiele modulus result in an increase, decrease, or no change in the internal and external concentration and temperature differences? Answer the question when the increase is due to an increase in pellet length, a decrease in effective diffusivity, or an increase in temperature. 4.2 1. Some s have a bimodal pore-size distribution consisting of rnacropores and micropores. Show for a first-order reaction that the isothermal, internal effectiveness factor (Mingle and Smith, 1961) is given by:

+

Calculate the global rates, (Rc)* and (Rc)., given in Section 49. 4.25. Derive Fq. 4.122 and Eq. 4.139.

using the numerical method

REFERENCES

where the subscript p denotes rnicropores. Explain why there usually is no micropore diffusion limitation unless there is also a macropore limitation. 4.22. Consider the following parallel reaction network:

Akella, L.M., Ph.D. Thesis, Dept. of Chem. Eng., University of Florida (1983). Aris, R., Ind. Eng. Chem. Fund., 4, 227 (1965).

Aris, R., Elementary Chemical Reactor Analysis. Prentice-Hall, Englewood Cliffs, N.J. (1969). Aris, R., Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Vol. I, Oxford University Press, London (1975), Austin, L.G. and P.L.Walker, AIChE J., 9, 303 (1963). Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena. Wiley, New York (1960). Bischoff, K.B., AIChE J., 11, 351 (1965). Bischoff, K.B., Chem. Eng. Sci., 22, 525(1967). Brown, C.E. and C.O. Bennett, AIChE J., 16, 817(1970). Butt, J.B., D.M. Downing and J.W. Lee, Ind. Eng. Chem. Fund., 16, 270(1977). Carberry, J.J. and A.A. Kulkarini, J. Catalysis, 31, 41 (1973). Carberry, J. J., Ind. Eng. Chem. Fund., 14, 129(1975). Carberry, J.J., Chemical and Catalytic Reaction Engineering, p 230, McGraw-Hill, New York (1976). Corbett, W.E. and D. Luss, Chem. Eng. Sci., 29, 1473(1974). Dente, M.and I. Pasquou, Chim. Ind. (Milan), 47, 359(1965). DiNapoli, N.M.,R.J.J. Williams and R.E.Cunningham, Lat. Am. J. Chem. Eng. Appl. Chem., 5, 101(1975). Feng, C.F. and W.E. Stewart, Ind. Eng. Chem. Fund., 12, 143(1973). Froment, G.F.and K.B. Bischoff, Chemical Reactor Analysis and Design, p 194, Wiley, New York(1979). Gunn, R.D. and C.J. King, AIChE J., 15, 507(1969). Hite, R.H.and R. Jackson, Chem. Eng. Sci., 32, 703(1977). Kehoe, J.P.G. and J.B. Butt, AIChE J., 18, 347(1972). Lee, H.H., Chem. Eng. Sci., 36, 1921 (1981a). Lee, H.H., AIChE J., 27, 558 (1981b). Lee, H.H.and E. Ruckenstein, Cat. Rev., 25, 475(1983). Liu, S.L.,AIChE J., IS, 337(1969). Luss, D.,in Chemical Reactor Theory, eds. Amundson and Lapidus, Chap. 4, PrenticeHall, Englewood Cliffs, N.J.(1977). Mason, E.A. and R.B. Evans, J. Chem. Ed., 46, 358(1969). Mears, D.E., Ind. Eng. Chem. Proc. Des. Dev., 10, 541 (1971). Mingle, J.O. and J.M. Smith, AIChE J., 7, 243(1961). Morbidelli, M., A. Servida and A. Varma, Ind. Eng. Chern. Fund., 21, 278(1982). Petersen, E.E., Chem. Eng. Sci., 20, 587(1965). Rester, S. and R. Aris, Chem. Eng. Sci., 24, 793(1969). Satterfield, C.N., G.W. Roberts and J. Hartman, Ind. Eng. Chem. Fund., 6, 80(1967). Thiele, E.W., Ind. Eng. Chem. 31, 916(1939). Wang, J.B. and A. Varma, Chem. Eng. Sci., 35, 613(1980). Weisz, P.B. and C.D.Prater, Adv. Cat., 6, 144(1954). Weisz, P.B.and J.S. Hicks, Chem. Eng. Sci., 17. 265(1962). Yortsos, Y.C.and T.T. Tsotsis, Chem. Eng. Sci., 37, 237(1982).

CHAPTER

5 Chemical Deactivation

5-1

INTRODUCTION

A catalyst is not consumed during a reaction and thus can be used indefinitely unless it loses its activity through deactivation. Unfortunately, however, all catalysts are to a certain extent susceptible to deactivation, the cause of which can be chemical or physical in nature. Physical deactivation is treated in the next chapter. Chemical deactivation in which the loss of catalytic activity is caused by strong chemisorption of some impurities, normally contained in the reacting mixture, is termed poisoning. Sulfur, phosphorus, and nitrogen compounds are typical poisons. Chemical deactivation is termed fouling when the loss of activity is caused by reactant or product degradation on the catalyst surface. Coke formation is the most important example. Chemical deactivation is also termed independent or dependent depending on whether the species causing deactivation is involved in the main reaction (dependent) or not (independent). While reactions on the catalyst surface are the main source of deactivation, noncatalytic reactions such as thermal cracking, where coke simply deposits on the surface, can also cause deactivation. The present level of understanding of chemical deactivation, although not complete, does allow a quantitative description of deactivation for purposes of reactor design and optimization. The unique feature of deactivation as related to reactor design is the time dependence of catalytic activity: the activity decreases with time on stream. In fact, studies on deactivation have been centered around the time dependence of activity so that it can be incorporated into reactor studies. The earliest work by Voorhies (1945), for instance, treated the correlation of coke concentration with time on stream. Various models result depending on how the time dependence is represented. Another important factor for reactor design is the interaction between diffusional and deactivational effects. Limiting cases of this interaction as well as more general cases can be treated with an understanding of noncatalytic gas-solid reactions. The first half of this chapter, therefore, deals with mechanisms of deactivation, noncatalytic gas-solid reactions, and deactivationdiffusion models. As the catalyst deactivates, the active surface area decreases. If deactivation is assumed nonselective, the effective rate of reaction is simply the intrinsic rate multiplied by the fraction of surface area still remaining active. This effective rate can then be used in pellet conservation equations to arrive at the global rate,

CHEMICAL DEACTIVATION 153

provided that the deactivation kinetics are known. It is then possible to combine the effects of diffusion and deactivation into a reactor point effectiveness, which gives the ratio of global rate to intrinsic rate in terns of bulk fluid quantities. These subjects are treated in the second half of this chapter. 5-2

Table 5.1 Influence of Electronic Configuration on Toxicity Nontoxic Types (shielded structure)

Toxic Types

NATURE OF DEACTIVATION

In Chapter 1, catalysts were grouped into three types: metals, acid oxides, and semiconducting oxides. The nature of chemical deactivation will be examined in accordance with this division. The common poisons for metal catalysts can be classified into two groups (Maxted 1951): (a) molecules containing elements of the periodic groups Vb and VIb, namely: N, P,As, and Sb from group Vb, and 0, S, Se, and Te from group VIb including the free elements with the exception of nitrogen, and (b) compounds of catalytically toxic metals and metallic ions: Cu+, Cu2', A&+,Au+, Zn2+,Cd2+, Hg+,Hg2+, In3+, TI+,Sn2+, Pb2+,BiJ+, Mn2+, Fc2+, Co2+, Ni2+. A poison that deactivates metal catalysts does so by being more strongly adsorbed than the reactant. It is not surprising then that the poison should have unshared electron pairs or empty valency orbitals available for forming stable surface compounds with the catalyst. This fact is exemplified in Table 5.1 (Maxted 1951) in which toxic and nontoxic compounds of group Vb and VIb elements are shown with their electronic configurations. The argument of unshared electron pairs can be used to explain why a substance poisons a catalyst for one reaction but not another. For example, arsenic compounds are general poisons, via arsine formation, for platinum in hydrogenation reactions:

Hydrogen sulfide

Phosphine

Phos~hateion

Sulfite ion (also selenite and tellurite)

Sulfate ion (also selenate and tellurate) 0 (R)C : s : OH 0

Organic thiol

Sulfonic acid

0

(R)C

:

s

: C(Rt)

0 Organic sulfide

Sulfone

Pyridinium ion

However, they have no effect on catalytic activity for the decomposition of hydrogen peroxide, presumably by remaining in a '+shielded" or saturated form such as the arsenate under strong oxidizing conditions

The catalytically toxic metallic ions listed above show the existence of some connection between the toxicity of a metallic ion and the structure of its d band: all have occupied d orbitals. It. is presumed that these metallic compounds or ions are involved via the d-shell in the formation of chemisorbed complexes resembling intermetallic species. While most of these rsults pertain to observations made in hydrogenations with Pd, Pt, or Ni, the arguments presented can serve as a basis for examining the nature of deactivation for other types of reactions. Acidic oxides, notably silica-alumina, suffer from both poisoning and fouling.

-

-

(Maxted 195 1. Reprinted with permission from Advances in Cat~lysis.Copyright

by Academic Press.)

While fouling is not confined to acid oxides, these catalysts suffer the most from fouling. Organic base, particularly nitrogen compounds, are effective poisons for these catalysts. In view of the fact that these catalysts owe their activity to their surface acid sites, it is expected that basic compounds would poison the catalysts and that their effectiveness as poisons would be related to basicity. In fact, the basic nitrogen compounds have been found in poisoning tests (Mills et al. 1950) to be more poisonous in the order of quinaldine > quinoline > pyrrole > pipeddine > decylamine > aniline, which is in the order of increasing basicity with the exception of piperidine, the most basic compound. It has been found, however, that a significant amount of piperidine cracked in the tests while others did not. It can be postulated, therefore, that the mechanisms of poisoning far these catalysts is chemisorption of the poison on incompletely coordinated aluminum or silicon ions, which results in a reduction of acid sites, of acid strength, or both.

The redox mechanism for catalysis by semiconducting oxides suggests that the poison would be a species that chemisorbs strongly on lattice oxygen that is presumably in the form of the 0 2 - ion. Sulfur and sulfur compounds are typical poisons. While there are other modes of fouling, such as metals deposition on catalysts in desulfurization of residual oils, emphasis here will be mainly with coking because of its importance and the relatively abundant literature available on the subject. The coke typically encountered in fouling is a carbonaceous material ranging from pure carbon to a hydrogen-deficientcarbon matrix of high molecular weight. Carbon deposition in steam reforming, for instance, occurs through decomposition of carbon monoxide, or its reduction by hydrogen, and by dehydrogenation of hydrocarbons. On most catalysts used in hydrocarbons (Butt 1972) where coking is observed, highly unsaturated species of high molecular weight are adsorbed preferentially, and in particular, polyring aromatics have been associated with coke formation. After adsorption, these aromatics undergo further condensation and hydrogen elimination on the surface to form hydrogen-deficient coke. The amount of coke thus formed can be well in excess of 50% in of weight percent on the catalyst and this can lead to pore blocking. When pore blocking occurs, the reactants cannot get into the interior of pores. Although the kinetics of coke formation have been relatively thoroughly investigated, the chemical and physical nature of coke formation has not. In particular, it is difficult to comprehend why pore blocking should occur if coking takes place on the catalyst surface. The point here is that once the whole surface is eventually covered by coke, no further coking should occur, and that one layer of coke cannot be thick enough to cause pore blocking. Some light on this aspect has been shed by the work of Baker et al. (1972; 1980). They studied the metal catalyzed decomposition of acetylene using an in situ electron microscope. They observed that flocculent amorphous carbon formed initially around metal particles. On raising the temperature, the metal particles became mobile and carbon filaments were observed to grow beneath them, eventually lifting the particles from the . The filaments grew with one metal particle at the end of each filament. The growth of the filaments stopped when the bare front of the metal particle was encapsulated by deposited carbon. The mechanism of this growth was attributed to the diffusion through the metal particle of carbon deposited on the bare front, which provided carbon for the filament growth. The s used for the metal catalysts were silica and graphite, on which the mobility of metal particles is relatively high (see Chapter 6). On alumina, however, the mobility is lower and thus the same mechanism may not be operative. While there are other questions to be resolved, the observations of Baker and coworkers do provide a basis for further studies on coking and pore blocking. Models for catalyst deactivation by site coverage and pore blockage have been proposed by Beekman and Froment (1 980; 1982). A series of coking experiments on silica-alumina catalysts by Levinter et al. (1967) demonstrated that pore blocking can occur on coke deposition to various extents depending strongly on catalyst properties and reaction conditions. Under

extreme conditions, for instance, the coking can be localized at the pore mouth, thereby completely eliminating accessible surface at very low total levels of coke. Under more moderate conditions, near-theoretical limiting amounts would be required before surface access is denied. Coking, therefore, can substantially alter physical and transport properties of catalysts as well as their activity. Deactivation by coking and its effect on the transport properties is treated in Section 5-8.

5-3

DEACTIVATION KINETICS

The earliest effort to correlate the time dependence of activity was that of Voorhies (1945). He found:

where C, is the concentration of carbon on the catalyst (which may be related to the activity), A and n are the correlation constants, and t is time on stream. This correlation fits a very large amount of catalyst deactivation data, not only on coking, and is encountered throughout the literature. Based on various correlations in the literature, the following power law form has been proposed by Szepe and Levenspiel (1970):

where a is activity. This has been extended (Khang and Levenspiel 1973) to:

where the rate of deactivation is expressed in of powers of reactant concentration C, concentration of poisoning species N, and activity. Since deactivation affects the rate of the main reaction through the activity, the deactivation kinetics are invariably tied to main reaction kinetics. When the effect of deactivation can be factored out in such a way that kinetic dependencies, which are time independent, and activity dependencies, which are not, are separable, the rate expressions are termed separable (Szepe and Levenspiel 1970). Separable kinetics can be expressed as:

where rl and r3 are the usual rate expressions without deactivation, and rz and r , are functions of activity. These kinetics in many, but not all, cases represent real catalytic deactivation mechanisms. One apparent advantage of this approach,

even when ernpiricisms are involved, over the pure activity correlations such as Eq. 5.1 is that the time dependence of activity is not tacked on only after the fact. The latter approach gives the following deactivation kinetics, when written in the separable form:

When the mechanism of deactivation is the chernisorption of poisoning species, the rate at which active sites disappear due to poisoning is equal to the rate at which the poisoning species chemisorbs on these sites. Therefore, the rate of deactivation is given by:

The term XrKiCr can always be rewritten in of the concentration of key species C and equilibrium constants. Take, as an example, a simple reaction with independent poisoning:

A+S+A*S A .S s B .S BmS+B+S

(KA)

(k,controlling step) (Ke)

which are elementary steps for the reaction A a B, and: assuming that the desorption rate is negligible. Here, s is the fraction of vacant active sites per unit volume of catalyst pellet, and kp is the adsorption constant. If it is assumed that the surface is energetically uniform, the rate of the main reaction, when subjected to deactivation, is simply:

where S represents the active site(s). These steps lead to the following rate expressions:

where k is the rate constant for fresh catalyst. A balance on the total number of sites gives: Therefore, the quantity G is: where Ct = total number of sites/pellet volume C, = number of sites poisoned/pellet volume Ct = reactant and product concentrations excluding poisoning species involved in elementary steps Ki = equilibrium constants n = constant ranging from 1 to 3 depending on the mechanism and controlling step in the main reaction The value of mi is !4 or 1 depending on the mode of adsorption (Chapter 2): M for dissociative adsorption and 1 for molecular adsorption. Rearrangement of Eq. 5.9 gives:

G(CrKi)= KACA

+ KBCH= CA(KA- K e ) -t- KNCA~

which is expressed solely in of the key species A and equilibrium constants. Here, CAsis the concentration of species A at the pellet surface. The deactivation kinetics of Eq. 5.11 apply to both dependent and independent poisoning. The chemical nature and mechanisms of fouling are not as well understood as that of poisoning. Nevertheless, deactivation kinetics can still be treated in the general framework of chemisorption. Fouling is usually a dependent deactivation process, and this dependent fouling can be represented by the following models (Butt 1972): +

Series: since by definition, Ct - C;, = Ct(l - y). Here, y is the fraction of catalyst deactivated. Combining Eqs. 5.8 and 5.10 yields:

_

A+S-+B+S Parallel: A A

A+S--+B+S B+S-+B.S

where A is the main reactant and B is an intermediate product. Following similar procedures as in poisoning, one can arrive at the deactivation kinetics:

Parallel: rp =

bCA(1 - 7 )

I! I

These simple models can be expanded in their forms to represent various situations such as fouling of polyfunctional catalysts, reversible chemisorptions, etc. An approach by Froment and Bischoff (1979) for coking is to write the deactivation kinetics as:

where (rp)Ois the initial coking rate and the deactivation function to coke concentration by:

+c

1

It is assumed that ~ j and l D are constant and that the particle is a slab. Other geometries can be handled in a similar manner. The boundary conditions are the usual ones:

is correlated

1i

where a is a constant. The various forms of deactivation kinetics show dependence not only on the mechanism of deactivation, but on the composition of reactants and products. Consequently, deactivation would not take place uniformly throughout a pellet whenever there exist concentration gradients within the pellet. In order to handle deactivation in the presence of diffusion limitations, one needs an understanding of noncatalytic gas-solid reactions.

5 4

where e p = particle porosity D = effective diffusivity r~ = rate of reaction of A on a volume basis C = concentration of species A

Two rate processes are at work here: the chemical reaction and the transportation of gaseous reactant to the solid reactant surface at which the reaction takes place. The relative magnitudes of these two rates determine the manner in which the solid reactant is consumed. Two limiting cases are of interest: one in which the potential rate of transport is much higher than the rate of reaction, which leads to uniform consumption of the solid reactant throughout the particle, and the other in which the potential rate of transport is much lower than the rate of reaction, which leads to formation of a distinguishable reaction front separating the outer product zone from the inner reactant zone (Figure 5.1). In the former, the reaction is completely free from diffusion limitations of the gaseous reactant; in the latter it is diffusion-limited. Concentration profiles within the particle are shown in Figure 5.1. It is seen from Figure 5.1(a) that the concentrations are uniform within the particle when the gas-solid reaction is diffusion-free. The mass balance of Eq. 5.17 reduces to:

NONCATALYTIC GAS-SOLID REACTIONS

Regeneration of coke-laden catalyst pellets by combustion in an oxygen-bearing gas stream, calcination, coal gasification, and oxidation of solids in blast furnaces, involve gas-solid reactions in which a component of the solid reactant phase is consumed. A transient exists because of this consumption of solid reactant. A major concern here, as related to catalyst deactivation, is the manner in which the solid reactant is consumed since there is a close parallel between this and the manner in which the catalyst is deactivated. In general, gas-solid reactions can be represented by:

where lower case letters are stoichiometric coefficients normalized with respect to the coefficient of the species of interest, A. A one dimensional mass balance for the species A can be written as:

The last boundary condition can be replaced by CIsurlaCe = G. The particle may be assumed isothermal without loss of generality. The rate at which the solid reactant disappears is obtained through stoichiometry:

i

In the other limiting case shown in Figure 5.l(b), the gaseous concentration at the reaction boundary approaches zero because of a very high consumption rate

Figure 5.1 Concentration profiles for various cases: (a) no diffusion limitation (b) diffusion limitation (shell-progressive reaction) (c) intermediate between the two extremes.

-

the completely reacted outer shell progressively moves inward with time. For gassolid reactions, this automatically results in a pseudo-steady-state situation since the speed at which the reaction front moves is much slower than the speed of transport of gaseous reactants. For this steady-state, Eq. 5.21 further reduces to:

solid I

...I

Y

"

with surface

center

Referring to the coordinate system shown in Figure 5.2, Eq. 5.19 gives:

(a)

dNA =

dr

reaction front

1 dN8 bdt

1 -d

Asps -

b d r ( M . )

where p~ and Mg are the density and molecular weight of the solid reactant B, and A is the cross-sectional area. On the other hand, one has at x = q:

Figure 5.2 Shell-progressive model for noncatalytic gas-solid reactions.

I

surface

center

I

surface

center

of the reactant at ,the reaction front. This moves inward with time, leaving behind a completely reacted zone (product zone). No reaction takes place in the product zone and the balance equation reduces to:

Equation 5.20 applies at the reaction front where a very thin band of reaction zone exists. When the potential rate of transport of gaseous reactants is much lower than the rate of chemical reaction, a shell-progressive model (SPM)results since

Product shell

Combining these equations gives:

I 1

Substituting into the right hand side of the above equation the solution of Eq. 5.22 yields:

I

The solution of this equation gives the following relationship:

I

where

The parameter z determines whether the particle grows or shrinks as the reaction proceeds: expansion occurs for z > 1, constant size for z = 1, and shrinkage for z < 1. With the SPM assumptions used in the shrinking core model, the major resistances are diffusion through the product zone and external mass transfer, the case of chemical reaction controlling being inconsistent with the SPM.The time required for a given level of conversion when one of the resistances is controlling can be obtained in the usual manner. Since these resistances are in series, the time when both resistances are important is obtained by simple addition, i.e., t = to t,, where to and t, are the times obtained when the diffusion and the external mass transfer, respectively, are controlling. Take, as an example, the case of diffusion controlling. Equation 5.22 for spherical geometry becomes:

+

since for the slab geometry the conversion y is equal to (1 - xd/L). The fact that the SPM for gas-solid reactions automatically results in a pseudosteady-state SPM can be verified by comparing the solution of Eq. 5.21 with that of Eq. 5.22, as shown by Bischoff (1963). The error in using the solution of Eq. 5.22 in place of that of Eq. 5.21 is of the order of Cb/66p~ for slab geometry. For gas-solid reactions, Cb is of the order of 10"3ps. Therefore, the error is quite small. On the other hand, the pseudo-steady-state assumption is not good for liquid-solid reactions since Cb then is of the order of ps. The change of particle size due to reaction is often of interest for noncatalytic gas-solid reactions. The SPM for the spherical particle shown in Figure 5.2 is called shrinking core model (Wen 1968). Referring to the figure, the moles of B reacted for the particle of initial radius Ro are:

assuming the reaction of Eq. 5.16. Similarly for the solid product M:

where x is radial coordinate. The solution with the boundary conditions of Eq. 5.22a is:

The equation equivalent to Eq. 5.26 for the spherical geometry is:

1I

Using Eq. 5.30 for R in the above equation and integrating, one gets:

Combining Eqs. 5.28 and 5.29 through the stoichiometry, ANs/b = ANM/m, yields: which is the time required for the conversion (1 the product zone is controlling.

- 63)when

diffusion through

5-5

DIFFUSION-DEACTIVATION

When reactions are affected by both deactivation and diffusion, one-dimensional, steady state pellet conservation equations are:

to intraparticle transport. This shell-progressive poisoning is termed pore-mouth poisoning. Uniform poisoning, in which the active surface is poisoned uniformly throughout the pellet, should occur if the rate of poisoning is low relative to the potential transport rate. The effect of diffusion on deactivation was treated concisely by Wheeler for the two limiting cases in of an activity factor defined as:

A=

rate of reaction for deactivated pellet rate of reaction for fresh pellet

(5.36)

If the pellet is uniformly deactivated, y is the same everywhere and A, for diffusionfree main reactions, is simply that given by:

The usual boundary conditions apply. Here, the kinetics of the main reaction are expressed as r = k ( l - y)f [CJ where k is the rate constant for fresh catalyst; k(l - y) is the effective rate constant for the catalyst being deactivated. It has been assumed far the energy balance that the heat of reaction for the main reaction is much greater than that for the deactivation reaction. Following the same procedures as in the case of diffusion-limited reactions (Chapter 4), it can be readily shown that the pellet is practically isothermal under typical reaction conditions. Therefore, only mass balance equations will be dealt with here. It shall soon be seen, however, that this assumption of isothermality is not good in the limiting case of shell-progressive deactivation. The fraction of catalyst deactivated, y, depends not only on time but also on the spatial position within the pellet. In order to quantify this dependence, Q is defined as the maximum uptake of poisoning species by the catalyst in moles of poisoning species per unit apparent volume of catalyst pellet. Then:

For diffusion-limited reactions, A = (1 - y)lI2 (see Problem 5.3). The devastating effect of diffusion on the residual activity left after poisoning for the pore-mouth poisoned pellet, as compared to uniform poisoning, is amply revealed in Figure 5.4 (Wheeler 1955), where the activity factor is given in of y with the Thiele modulus as parameter. This is due to the presence of the completely deactivated outer shell, as shown in Figure 5.3 for a slab-like pellet. Referring to the figure, we set the rate of the main reaction within the fresh inner core equal to the transport rate through the deactivated outer shell at steady state:

Figure 5,3 Shell-progressive poisoning, slab-like pellet.

This relationship states that the uptake rate of poisoning species by the catalyst is equal to the rate at which the poisoning takes place. The pseudo steady state assumption has been made with respect to concentrations. Validity of this assumption comes from the fact that the rate of poisoning is much slower than the rate at which the concentrations reach steady state. If this were not true, the catalyst would lose its usefulness because it would have to be regenerated too frequently. For instance, the useful life of catalyst pellets is of the order of months, whereas the time requikd to reach steady state is of the order of L/D112. For reactor design, it is important to know how the solutions of Eqs. 5.32, 5.34, and 5.35 affect the intrinsic rate of reaction. Wheeler (1955) treated this deactivation-diffusion problem for two limiting cases: uniform and pore-mouth (shell-progressive) poisoning. As described in the previous section for noncatalytic gas-solid reactions, the poison will deposit preferentially on the pore-mouth initially and grow progressively inward with time, if the rate of poisoning is rapid relative

This reduces to:

Here, the effectiveness factor for the fresh inner core based on the conditions at x = xd is given by:

and y is simply l/L. While a pore-mouth poisoned pellet is not, in general, isothermal, it has been assumed isothermal in accordance with the original work of Wheeler. The definition of the activity factor gives: 0 0

-

since the global rate for the deactivated pellet is k(1 y)qinCc on the basis of whole pellet volume. The effectiveness factor for the fresh pellet is: tanh

we--+

+ '

0.4

0.6

0.8

1.0

Figure 5.4 Activity factor for uniform and shell-progressive poisoning. (Wheeler 1955)

4 = L(ks/De)lt2

Substitution of Eq. 5.38 into 5.39 gives:

which reduces to:

+

0.2

when is large since then qJ+ 4 1 and +in/+ = (1 - 7). For slab geometry, this result is the same as that obtained by Wheeler. The activity factor derived originally by Wheeler for spherical geometry (see Problem 5.4) is shown in Figure 5.4. The approximation of deactivation-diffusion behavior by the porcmouth poisoning model has been found in many cases to be quite adequate when the initial rate of deactivation is relatively fast, although one should be judicious in applying the approximation (Petersen 1982). One would expect the poremouth poisoning model to be valid for independent deactivation when the main reaction is quite diffusion-limited since the rate constant for the independent deactivation reaction is inherently higher than that of the main reaction, and: therefore, the deactivation

reaction is likely to be more diffusion-limited than the main reaction. This, however, is not necessarily true for dependent deactivation as shall be seen shortly. Masamune and Smith (1966) solved the conservation equations of Eq. 5.32, 5.34, and 5.35 for various deactivation cases in which both the main and deactivation reactions are of first-order. They found that a catalyst with the lowest intraparticle diffusion resistance gave the maximum activity for any time on stream when a series dependent deactivation is involved; for a parallel deactivation, a catalyst with an intermediate diffusion resistance is less easily deactivated and gives a higher conversion to desirable product, particularly at long on-stream times. Their simulation results for the overall effectiveness factor representkng the combined effects of deactivation and diffusion are shown in Figure 5.5 for independent poisoning. With the assumption of negligible external transport resistances, the overall effectiveness factor can be plotted, as shown, in of time, with the Thiele moduli for the main and deactivation reactions as parameters. As elxpected, the effectiveness factor decreases with time. Carberry and Gorring (1966) treated the deactivation-diffusion problem as a noncatalytic gas-solid reaction to arrive at the time dependence of the fraction of catalyst deactivated. Following the same procedures as in arriving at Eq. 5.27, one can get:

reactor point effectiveness. In this section, the reactor point effectivenesses is obtained for three cases: general, uniform, and pori-mouth deactivation. As in Chapter 4, realistic assumptions are made of negligible external mass transfer resistance and isothermal pellets. In the case of pore-mouth deactivation, however, the nonisothermal nature of the pellet is taken into consideration.

General

For a pellet of nonuniform activity, it has been shown in Chapter 4 that the internal effectiveness factor in the asymptotic region of' strong diffusion effects can be accurately approximated by:

Dimensionlesstime,

$ = 10

Overall effectiveness factor for independent poisoning and diffusion-affected first-order reaction. (Masamune and Smith 1966)

Figure 5.5

where a =

1 ; (J31)rn

where the overbar denotes the quantity evaluated at surface conditions, and G is the activity distribution function normalized with respect to the surface activity of a fresh pellet. Note that the effectiveness factor here is based on the activity of fresh catalyst and therefore the denominator of Eq. 4.104a is simply LYc as opposed to that given in Chapter 4 (Eq. 4.104). A restriction placed on the relationship was that the following conditon be met:

klL

(BiIm = De

They found that this result for slab geometry would apply equally well to spherical geometry up to y of 0.3 if L is defined as the ratio of volume to surface area (L = R/3 for sphere). This result gives a square root dependence of y on time. In order to obtain the global rates for reactor design, Eqs. 5.32, 5.34, and 5.35 have to be solved numerically, given the kinetics of the main and deactivation reactions. In the asymptotic region of strong diffusion effects, this deactivationdiffusion problem for the global rates can be simplified conside'rably as will be shown in the next section.

REACTOR POINT EFFECTIVENESS AND GLOBAL RATES 5-6

Chapter 4 has demonstrated that the global rate, which appears in reactor conservation equations, can be obtained by simply multiplying the intrinsic rate by the

where G' = dG/dx. Since deactivation results in a nonuniform activity and this activity distribution is (1 - y ) for the deactivated catalyst, these results can be used as such with G equated to (1 - y). The left hand side of Eq.4.102 is essentially the maximum fractional error involved in using Eq.4.102. This condition is usually C< since the second term in the left hand side is smaller satisfied unless than unity and De is of the order of crn2/sec. For the deactivation being considered, which is intermediate between uniform and pore-mouth deactivation, the slope of the activity function (1 - y) at the surface is not very steep. Further, the surface activity is not likely to be less than 0.4 if the catalyst is regenerated before its average activity becomes less than 70% of its original activity. Therefore, the condition of Eq. 4.102 is likely to be satisfied in many cases. For instance, this condition for an nth order reaction becomes:

+

where is the usual Thiele modulus. In what follows, it is assumed that this condition is satisfied. The reactor point effectivness has been defined as:

The rate constant ratio for the deactivation reaction equivalent to Eq. 4.86 is:

observed rate A= intrinsic rate for fresh catalyst at bulk fluid conditions which can be expressed as: where E , is the Arrhenius number for the deactivation reaction. An assumption used here is that the heat liberated by reaction is mainly due to the main reaction. Use of Eqs. 5.48 and 5.49 in the definition of the reactor point effectiveness for the deactivation reaction gives: under the assumption of negligible external mass transfer resistance. Since the rate constant ratio in the above relationship has been shown (Chapter 4) to be well approximated by: It is seen that the global rates for the main and deactivation reactions contain only the surface activity (1 - y), and not the activity profile. Therefore, Eq. 5.35 is evaluated at the pellet surface for independent deactivation: this can be used along with Eq.4.104a in Eq. 4.8 1 above to give:

Global rates are now expressed in of bulk fluid quantities and the surface activity required for the rates is given by Eq. 5.51. These are summarized in Table 5.3. Unlike the cases of uniform and shell-progressive deactivation, the results obtained for the general case are restricted to diffusion-limited reactions. They are also subject to the limitation of Eq. 4.102.

where

Here, G has been replaced by (1 - 3). For the global rate of the deactivation reaction, Eq. 5.34 is rewritten for independent deactivation:

where Eq. 5.1 1 has been used for r,. Since the inhibition t e r n G(C) is quite a bit smaller than unity, an arithmetic average can be used to approximate the denominator: H(Cb) =

[I

+ G(Cb)] + [1 + G(C = O)]

Uniform

One would expect the rate constant of an independent deactivation reaction to be inherently larger than that of the main reaction. One would therefore expect that the independent deactivation reaction is also diffusion-limited,meaning nonuniform deactivation, if the main reaction is. Nonetheless, uniform deactivation can take place even when the main reaction is diffusion-limited. In such a case, G(C) appearing in the deactivation kinetics would be much smaller than unity so that the kinetics of independent deactivation can be reduced to:

2

Then procedures identical to those used in obtaining Eq. 4.5 lead to the following internal effectiveness factor for the deactivation reaction:

since, otherwise, uniform deactivation cannot take place. If G(C) were not much smaller than unity, y would depend on position within the pellet because of the

dependence of G(C) on the position, which contradicts the assumption of uniform deactivation. Since the fraction of catalyst deactivated is uniform throughout the pellet, the reactor point effectiveness is simply that for a diffusion-limited reaction (Eq. 4.87)with k replaced by the effective rate constant k ( l - y):

where

Combining Eqs. 5.56 and 5.57 gives:

It has been shown both experimentally and theoretically (Butt et al. 1977; Lee et al. 1978)for a pore-mouth poisoned pellet that the main intraparticle ternperature gradient lies in the deactivated outer shell and that the fresh inner core is isothermal. This is the reason why the fresh inner core was treated as isothermal in writing Eq. 5.57. The intra- and interparticle temperature gradients (Lee 1980a) are related by:

The time dependence of y is given by: where A Td [= (Td - Ts)Jis the temperature difference in the deactivated outer shell, and ATe[= (z - Tb)]is the external temperature difference. These two relationships can be combined and rearranged to give: Pore-Mouth (Shell-Progressive)

The global rate of the main reaction is obtained from: Setting the heat transferred across the external film equal to the heat liberated by the reaction gives:

The concentration at the boundary between fresh inner core and the completely deactivated outer shell (Cd)is obtained from Eq. 5.38 written for arbitrary kinetics, rc = kf(C):

In order to express one writes:

kd

appearing in Eq. 5.58 in of bulk quantities and Cd,

Using Eqs. 5.61 and 5.62 in 5.63 and then in 5.58 gives: For the nonisothermal pellet being considered, k, in Eq. 5.38 has been replaced by kdr i.e., the rate constant evaluated at the conditions at the boundary. With the assumption of negligible external mass transfer resistance, C, has also been replaced by the bulk fluid concentration Cb. The internal effectiveness factor for the fresh inner core is rewritten as:

This is an implicit expression for Cd in of bulk-fluid quantities. The global rate can readily be calculated from Eq, 5.55 and Cd obtained from 5.64, given the kinetics and bulk-fluid conditions. As was seen in the section on noncatalytic gas-solid reactions, there must be a very thin deactivation reaction zone at the forefront of the moving boundary separating the deactivated outer zone from the still active inner zone in order for pore-mouth deactivation to take place. In this small band, the rate of deactivation or the rate of chemisorption of poisoning species must be much greater than the rate of chemisorption of the other species. Therefore, the rate of deactivation is practically independent of the fraction of vacant sites in this small band and the deactivation kinetics of Eq. 5.7 can be approximated by:

Figure 5.6 Comparison of theoretical pellet effectiveness with experimental values: (a) M°C, 0.375% thiophene, (b) 1 10°C, 1% thiophene. (Lee and Butt 1982)

Run 2u

Time, min (a)

It follows then that:

t

Run 1u

Setting the rate of diffusion of poisoning species through the deactivated zone equal to the rate of deactivation yields:

Combining Eqs. 5.66 and 5.67 gives:

0

Therefore, the global rate of reaction as affected by diffusion and pore-mouth deactivation is described by Eqs. 5.55, 5.64, and 5.68. These results are also summarized in Table 5.3. When the Thiele modulus for the deactivation reaction ( $ p ) d Table 5.2

Parameters: Shell-Progressive Poisoning in the Hydrogenation of Benzene on Ni/Kieselguhr by Thiophene (Lee et al. 1978)

20

40

60

80

100

120

Time, min

(b)

is very large, which is often the case for pore-mouth deactivation, the solution of Eq. 5.68 reduces to:

D, = Dp = 0.052 cm2/s (counter diffusion) or 0.03 cm2/s (random pore) L = 0.65 cm (kp)d

= 28

s-I

Q = 4 X lo-' rnol/cm3 C. = 5.7 X 10-6 mol/crns

kd =0.15

s-I

Ns= 3.6 X lo-' rnol/cm3

}

kd = 0.10 s-1 N, = 1.35 X lo-' mol/cm3

forrun24

for run 14

A comparison (Lee and Butt 1982) between theoretical and experimental results is shown in Figure 5.6 for a single pellet undergoing pore-mouth poisoning. Parameter values for the experiments, which were determined by independent measurements, are given in Table 5.2. For the first-order main reaction being considered, the hydrogenation of benzene, Eqs. 5.55, 5.58, and 5.68 reduce to:

where C is the concentration of species A. In both cases, then, the reactor point effectiveness and global rate are given by Eqs. 5.44 and 5.45. The surface activity (1 --7) follows directly from Eqs. (5.12) and (5.13):

Parallel:

d

dr )'('

= l L

The concentration of poisoning species was kept constant during the experiment, which permits direct integration of

I:

Nbdt to Nbt. It is seen that the agreement

between theory and experiment is satisfactory. It is also seen that the effective diffusivity is a sensitive parameter in calculating the reactor point effectiveness. The upper line in Figure 5.6(a) is for the diffusivity determined from the counterdiffusion method* (D, = 0.052 cm2/sec) and the lower line is for the diffusivity obtained from the random-pore theory' (De = 0.03 cm2/sec). The result obtained so far for independent deactivation can be extended to dependent deactivation. In contrast to independent deactivation, the rate constant of a dependent deactivation must be much smaller than that for the main reaction since otherwise the rate of the deactivation reaction could become greater than that of the main reaction. Such a catalyst could not serve any useful purpose in the form of pellets. In fact, this is precisely the reason why catalytic cracking of hydrocarbons, in which the coking rate is high, is carried out in a fluidized bed with continuous regeneration and recycling of deactivated silica-alumina catalyst in the form of very small particles. Note that the same species involved in the main reaction is also involved in the deactivation reaction as can be seen from the dependent deactivation models given previously:

Series:

d

(kp)s(C*)b(f - 7) 2 (Kici) mi]

+

t

J bulk

(kp)s(C~)b(l 7)

dt

Reaction stoichiometry can be used in these equations to express (CLl)bin of ( C A ) ~ . As discussed earlier, catalyst particles instead of pellets are used when the rate of deactivation is high. The typical size of the catalyst particles used in fluidizedbeds ranges from 20 to 300 pm, and therefore, diffusional effects can be neglected. In the absence of diffusional effects, the particles are isothermal and the external mass transfer resistance is negligible (Chapter 4). The only transport resistance is external heat transfer. For these particles then, the global rate is simply:

where y depends only on time. Expressing k, in of bulk quantities with the aid of Eq. 4.82, one gets:

k

Parallel:

A+S-B+S A + S - - -kP+ A . S

The space-independent y is obtained from:

If the rate constant af the deactivation reactions k, is much smaller than that of the main reaction k, as would be the case for dependent deactivation when catalyst pellets are used, the rate of reaction for the species of interest (A) can be approximated in both cases by:

* See Chapter 14, Section

1.

It is of interest to note that the catalytic activity (1 - y) decays exponentially with time according to Eq. 5.77 if the bulk-fluid concentrations are constant. In well-mixed fluidized-beds, concentrations can be considered to be uniform everywhere, approaching CSTR behavior. This exponential decay of catalytic activity has been found (Weekman 1970) satisfactory for the simulation of a fluidizedbed catalytic cracker.

178


Table 5.3 Glabal Rates and Reactor Point Effectiveness rc = k f ( C ) A = RG/k$(Cb):

kb

b. Dependent deactivation: general, diffusion-limited: k, < k

= k(Tb)for fresh catalyst

a. Independent deactivation: diffusion-affected main reactions General (restricted to diffusion-limited reactions such that C, = 0) Parallel:

d

dl (Q')=

Series:

d

(")=

[I

[I

( k p ) s ( C ~ ) b ( l- 9) (KrCi)mi];u,k

+

(kp)s(Ce)bIl

+

- 7)

(KiG)mi]Lu,k

1

c. Small catalyst particles: diffusion-free

Uniform

where

6 =(kp)sNb(l 7 ) [l + G(Cb)ln

: independent

: parallel

: series

E=Eu/R,Tb.

r

cc/cb=

E,--(E,),/R,Tb 0 for QC8

33

forO.3<$~<3 cosh $c

Replace Cb by C d in Eq. (A) for the pore-mouth poisoned case. A reaction can be considered to be diffusion-free if +G < 0.3.

(A)

For the purpose of using the results obtained so far, a reaction is considered to be diffusion-limited if the generalized modulus defined by:

is greater than 3. The global rates and reactor point effectivenesses are summarized in Table 5.3. These results are used in Chapter 10 for reactor design.

5-7

CATALYST REGENERATION

Deactivated catalyst can be regenerated by simple heating if poisoning is due to reversible chemisorption. If the poisoning species is irreversibly chemisorbed, however, it has to be removed by oxidation or some other chemical means. It is not unusual, when deactivation is a severe problem, to have a guard reactor in which the impurities in the feed are captured before they enter the primary reactor. The subject most studied is the regeneration of coked catalyst because coking is important in many reactions such as cracking, reforming, hydrodesulfurizatian, and various dehydrogenations. The buildup of coke may become significant in less than a minute, as in catalytic cracking, or only over a period of months, as in some catalytic reforming processes. The coke is removed by combustion in an oyxgen-bearing gas stream. The combustion of coke is a complicated phenomenon because of the composition of coke, which can vary from carbon (farmed from relatively high temperature reactions) to a high molecular weight, hydrogen-deficient carbon matrix (relatively low temperature), and because of the manner in which the coke is deposited within the catalyst pellet. The diffusion characteristics of the main and deactivation reactions as well as the mode of the deactivation reaction (parallel versus series) determine the manner in which the coke is deposited. In diffusion-limited reactions, for instance, coke is deposited preferentially at the pore-mouth in parallel coking whereas it is deposited at the core in series coking. Adding complication to these is sintering (Chapter 6) that can take place during regeneration, resulting in reduced surface area. The fact that coke consists of a spectrum of compounds is a complicating factor in determining the intrinsic kinetics of coke combustion, in particular the order of reaction with respect to coke concentration. It is well established, however, that the kinetics can be represented by (Weisz and. Goodwin 1966):

for combustion of monolayer coke deposits when they are formed on inert s or acidic cracking catalysts not containing transition metals. Here, Po, is the partial

pressure of oxygen, and Cc is the coke concentration in moles of coke per unit volume of pellet. The activation energy is 37.6 kcal/mol with the preexponential factor of 1.9 x lo8 sec-I atm-l. The stipulation on coke coverage has to do with the accessibility of coke to oxygen. If each atom of carbon in the coke occupies the same area, a monoatomic layer of carbon on an oxide-type catalyst or would comprise about 5 wt% carbon for each 100 m2/g of surface area: an atom of carbon in graphite occupies about 4A2. Therefore, not all carbon will be accessible to oxygen if this wt% is higher than 5 for each 100 m2/g of surface area. In fact, it has been found that, at higher coke contents, the rate for the first 50 percent or so of the reaction is less than proportional to the amount of carbon present, indicating that some of the carbon atoms are initially inaccessible, but in the latter part of the reaction the rate again becomes first-order with respect to total carbon present. While the combustion of hydrogen contained in coke is also involved in coke combustion, this complication ma!! be neglected for purposes of regeneration analysis and design, since the combustion of carbon is the slow reaction. If the useful life of the catalyst is of the order of a minute or less, very fine particles instead of pellets are used, as practiced in fluidized-beds. In such cases, the coke distribution in the pellet is uniform and the global rate of combustion is essentially equal to the intrinsic rate. If the useful life of the catalyst is of the order of months, the rate of coking would be much smaller than that of the main reaction. Nonetheless, there should exist a nonuniform coke distribution within the pellet if the main reaction is diffusion-limited. A complication involved in treating the combustion of nonuniformly deposited coke is the possible variation of combustion kinetics with coke content. An instructive example of coke burning is provided by Weisz and Goodwin (1966). As one would expect from the discussion in Section 5.4, the mode of coke removal will depend on the combustion temperature. The removal will be uniform at low temperatures, intermediate between uniform and shell-progressive at intermediate temperatures, and strictly shell-progressive at high temperatures. This is illustrated in Figure 5.7. Silica-alumina pellets (beads) can be made transparent by immersing them in a high-refractive index liquid such as carbon tetrachloride. With such immersion the distribution of deposited coke becomes apparent. Photographs of partially regenerated coke catalyst beads obtained by the original authors are represented in Figure 5.7 (Weisz and Goodwin 1966). In the region in which the SPM mode of coke burning prevails ( T > 625OC), they have found that the fraction of coke remaining, 8, can be represented by:

which is Eq. 5.31 for z = 1 (no change in particle size). Here, the subscript B denotes coke, and Cb denotes the bulk concentration of oxygen. A thorough analysis with respect to the effects of bead size, diffusivity, and coke content confirmed the soundness of the above result.

effective diffusivity. In this section, we provide these relationships based on a multilayer kinetic model that s for coke growth. Consider coking taking.place on an energetically uniform catalyst surface. Suppose that a coke precursor, P, can form coke upon deposition onto sites whether the sites are vacant or occupied by coke. This multilayer coking can be represented by the following irreversible elementary surface reactions:

P +S

k, ----+

Initial coke level

P+(2P

-

P . S1

S) L ( 3 P

: monolayer coking

-

S)1

multilayer coking

Figure 5.7 Coke burning as affected by oxidation temperature and the progression of coke concentration: (a) low temperature, (b) intermediatetemperature, (c) high temperature. (Weisz and Goodwin 1963. Reprinted with permission from Journal of Catalysis. Copyright by Academic Press.)

5-8

where it is assumed that the rate constant is the same for all multilayer coking reactions. The first step is monolayer coking; one "molecule" of coke precursor adsorbs onto a vacant site, S, to form monolayer coke. Multilayer coke forms, as shown, when the precursor deposits onto mono- or rnultilayer coke. The net can be written based rate of formation of n-layer (n = 1, . . . , A') coke, R,. on the elementary steps shown above. Thus, we can write, in general,

DEACTIVATION BY COKING

Because of some unique features, coking is perhaps the most challenging problem of chemical deactivation. The activity of coked catalyst does not decrease linearly with coke content, which is typically expressed as weight percent on catalyst pellet, even for structure-insensitive reactions. This is opposed to poisoning in which the activity decreases linearly due to the monolayer nature of the adsorbed poisons. Coking can also cause severe diffusion limitation as the pores get filled with coke. These unique features are a direct result of coke growth, which was discussed earlier. In addition to the uncertainty regarding the chemical nature of coke and coke precursors, there are also complications due to the specificity of coking with respect to catalyst and reaction conditions. The mechanism of coke growth on a nickel catalyst, for instance, is not necessarily the same for different reactions catalyzed by the same catalyst. In spite of these difficulties, one can still deal with coking provided certain relationships are available. One of the required relationships is that between catalytic activity remaining after coking and coke content, since the residual activity can then be determined from the coke content. The other needed relationship is the manner in which the coke content lowers the

where Cnp.,(n = 3, 4, . . . , -V) is the surface concentration of the sites occupied by n-layer coke and not the concentration of n-layer coke. A solution to Eq. 5.80 is

where f is a distribution factor relating the surface concentration of (n - 1) layer coke to that of n-layer coke. This distribution factor is constant over short time intervals, but does change over long intervals. The ratio of the net rates of formation for the mono- and di-layer coke is:

which also follows from the elementary steps. Here, C, is the surface concentration of vacant sites. If we assume that Eq. 5.81 holds for mono- and di-layer coke it follows from Eq. 5.82 that: such that fG.s =

Consider site balances for the determination of the distribution factor, f: A balance for the total sites is:

where Ct is the number of total sites per unit area and Cr ., is the surface concentration of surface intermediate, i, involved in the main reaction and the reaction leading to the formation of the coke precursor. If we restrict attention to coking in which both the main and coking reactions occur on the same sites,

j 1

where

Ep is given by: Ep =

1

KP

(1

+ G)"

(5.90)

It is seen from Eq. 5.89 that f < 1 for y < 1. Now that the distribution is known, the relationship between coke content and the fraction of catalyst deactivated can readily be obtained. Let q be the monolayer coke weight per site per unit pellet weight. Then

where Cc is the coke content in weight per unit pellet weight. The following relationship results when Eq. 5.88 is used in Eq. 5.91:

where Q (= Ctq) is the weight of coke when all sites are covered by monolayer coke. Let N approach infinity and use Eq. 5.89 in 5.92, to obtain where y is the fraction of active sites deactivated. Note that CnP.$is the number of sites occupied by n-layer coke per unit area. Thus, from Qs. 5.84 and 5.85:

As in Eqs. 5.9 and 5.10, we express Z Ct., in of equilibrium constants and concentrations to rewrite Eq. 5.86 as follows:

The coke content due to multilayer coke or coke growth is contained in the second term in the bracket of Eq. 5.93 since Cc/Q = y when only rnonolayer coke is involved. For strictly rnonolayer coking, & approaches infinity, k,, approaches zero, and Eq. 5.93 reduces to that for monolayer coking. For structure-insensitive reactions, the fractional catalytic activity remaining after deactivation, A, is equal to (1 - y). This can be used in Eq. 5.93 to obtain the following relationship:

Since the distribution factor, J is given by Eq. 5.81, it can be used in Eq. 5.85 to give:

provided f is less than unity. Eqs. 5.87 and 5.88 can be used in JZq. 5.83 to arrive at the following expression for the distribution factor for N approaching infinity:

The coke content and the residual activity can be determined experimentally. Therefore, a linear plot of Eq. 5.94 yields the monolayer coke content, and the rate constant ratio Ep. The coking kinetics for the rate at which the active sites disappear follow directly from the rnultilayer model. This rate, R,, is simply the rate of rnonolayer formation. Thus, we have:

With the aid of Eq. 5.87, this can be rewritten as:

As illustrated earlier for poisoning, expressions for G and C, can be derived for

the rate of change of y with time if the elementary steps leading to the formation of the coke precursor and the reaction products are given. Thus, Eqs. 5.93 and 5.96 form the basis for the intrinsic coking kinetics. It will become clear why Eq. 5.93 is necessary when the influence of coking on effective diffusivity is considered. We have so far treated only the case of an infinite layer of coke (N-300). The infinite layer model should be sufficient in most cases since the distribution factor for infinite layers is essentially the same as that for finite layers when y is relatively small. Consider, as an example, the coking data of Dumez and Froment (1976), which were obtained under diffusion-free conditions. Since Eq. 5.93 or 5.94 is applicable when y is relatively small, a plot of Cc/(l - A) versus (1 - A ) / A should still yield a straight line for the initial stages of coking, i-e. for small values of y. Such a plot for the data of Dumez and Froment is shown in Figure 5.8. It is seen that a deviation from linearity occurs at around y = 0.55. The values of Q and Kp, determined from the linear portion of the plot, are also applicable to the finite layer case since the distribution factors for both finite and infinite cases are essentially the same for small values of y. Since the values of Q and & are determined from the linear portion of the plot (Q = 0.015, Ep = 0.96), the value of N is the only unknown in the finite layer case, which can be determined from Eq. 5.92 and the distribution factor given by Figure 5.8 Determination of parameters in Fq. 5.94.

This expression follows from Eqs. 5.83, 5.87, and 5.88. The value of N, determined from Eqs. 5.92 and 5.97 for the data of Dumez and Froment using the stepwise procedures detailed in Klingman and Lee (1984), is 8.4. The comparison between theoretical and experimental results is shown in Figure 5.9 for both infinite and finite cases. It is seen that the infinite layer case gives satisfactory results down to 30% residual activity, indicating that the infinite layer model should be sufficient for all practical purposes since a catalyst usually is regenerated before it loses so much of its activity. The multilayer model is a physical model and therefore the parameters contained have physical meanings. The monolayer coke content, Q determined from. a linear plot of Eq. 5.94 can be compared with estimates obtained by independent means to give credence to the model. For instance, it is known that a monoatomic layer of carbon on an oxide-type catalyst or would comprise about 5 wt

% carbon for each 100 m2/g of surface area (Butt 1972). For coking experiments carried out under constant concentration conditions, Ep is simply a ratio of two rate constants. Thus, the values of K,determined at different temperatures should yield a straight-line Arrhenius plot. This Arrhenius temperature dependence is expected to hold only when the nature of the coke precursor does not change with temperature. The chemical nature of coke precursors, which have been lumped as a single compound, may change with temperature. It is a good indication that the coke precursor changes over the temperature range of interest if the value of Q depends on temperature and the Ep values do not obey the Arrhenius relationship. In such a case, Eq. 5.94 would be valid for segments of the temperature range with a different coke precursor (different Q) for each segment. Consider now the effect of coking on effective diffusivity. As a result of coke accumulation on the catalyst surface, the effective diffusivity will usually

T

: Data by Dumttz and Fromrnt 179781

Infinite Layers

Finite Layers

7'

The change in the pellet porosity due to coking can be related to the corresponding change in the diffusivity through the models already available for effective diffusivity. In a simplified version of Feng and Stewart's model (1 973) (see Chapter 14), the ratio of the diffusivity for the coked catalyst to that for a fresh catalyst is:

where K is the tortuosity factor and the subscript o is for a fresh, uncoked catalyst pellet. The tortuosity factor is a correction for the zigzag nature of the diffusion path along the pores, although constrictions can also play a role. Neglecting any constriction effects, Eq. 5.100 can be written as:

where Eq. 5.99 has been used. Eq. 5.101 has been made dependent on the pellet coordinate x since Eq. 5.99 and Eq. 5.100 are valid at any point in the pellet. If we use the random pore model of Wakao and Smith (1962); on the other hand,

C, (g eokelg catalyst)

Figure 5.9 Comparison of catalytic activity by multilayer technique to experimental data.

In this case, the local effective diffusivity D,(x) is given by:

decrease. The coke occupies pore volume that would otherwise be available for the transport of reacting species. The decrease in the pore volume due to coking is equal to the volume of the deposited coke, as shown by Haldernann and Botty (1959). Thus, we may set the volume occupied by coke deposits equal to the reduction in pore volume:

which follows from Eqs. 5.99 and 5.102. In general, the local effective diffusivity can then be written as:

where EO and E are the catalyst pellet porosities for fresh and coked catalyst, respectively, Vp is the pellet volume per unit weight, C. is the coke content in weight per unit pellet weight, and pc is the coke density. Eq. 5.98 can be rewritten as:

where B is unity for the simple version of the Feng and Stewart model (1973) and is two for the random pore model. If we include the tortuosity factor in the Feng and Stewart model as a power of (c/eo), the value of P will be something other than unity or two. The relationship between coke content and the fraction of catalyst deactivated is applicable to any point in the pellet. Thus, combining Eq. 5.93 with Eq. 5.104, gives:

which is the expression for the local effective diffusivity, where (Cc), is the saturation coke content corresponding to the amount per unit pellet weight required to fill the initial pore volume.

See Chap. 14.

D,(x).

The relationships required for the analysis of a reaction affected by coking can be summarized as:

surface

where De(x) is given by Eq. 5.105 and the expressions for and G follow from the surface elementary steps of the reaction. It is seen that only the fraction of the catalyst deactivated, which is the quantity relevant to catalytic activity rather than coke content, appears in the conservation equations. Thus, the complex problem of dealing with a reaction affected by coking can be treated in a systematic and consistent way based on the relationships surnrna~.izedabove.

Summary

Chemical deactivation by poisoning is well understood and permits a logical representation of the deactivation kinetics in of chemisorption. While our understanding of fouling is not as good, it still permits the representation of the kinetics in the general framework of parallel and series chemisorption. Coking is very difficult to deal with because of the specificity of coking with respect to catalyst and reaction conditions. Nevertheless, the framework of the multilayer kinetic model permits a logical and systematic treatment of the problem. The time dependence of catalytic activity due to deactivation must be taken into for reactor design. The global rates and reactor point effectiveness developed here permit heterogeneous fixed- and fluidized-beds, affected by deactivation and diffusion, to be treated as homogeneous reactors. The single pellet reactor of Hegedus and Petersen (1974), though not treated here, is useful in determining the intrinsic kinetics of deactivation. Two specialized books on catalyst deactivation are available (Delmon and Froment, eds., 1980; Butt and Petersen, in press).

NOTATION

a A 6, m,p

activity of catalyst activity factor defined by Eq. 5.36; area; a constant stoichiometric coefficient ratios

Biot number for mass Biot number for heat concentration of key species pellet center concentration; coke concentration (mole/pellet volume); coke content in weight coke per weight catalyst. coke content in wt coke/wt cat. pellet quantity defined in Eq. 5.99 concentration at the boundary between fresh inner core and cornpletely deactivated outer shell concentration of poisoning species or coke precursor surface concentration of vacant sites total number of sites per unit surface area surface concentration of the sites occupied :by n-layer coke exponents effective diffusivity effective diffusivity of poisoning species Damkohler number activation energy distribution factor in Eq. 5.81 functional dependence of rate of reaction on concentration activity distribution normalized with respect to surface activity of fresh catalyst; G' = dG/dx inhibition term in Eq. 5.1 1 film heat transfer coefficient quantity defined by Eq. 5.47 rate constant for fresh catalyst, main reaction rate constant for the formation of monolayer coke rate constant far deactivation reaction; rate constant for the formation of multilayer coke

L/k, defined in Eq. 5.90 length of the portion of pellet completely deactivated characteristic length of pellet; half-thickness of slab molecular weight number of coke layers concentration of poisoning species; number of moles; number of maximum coke layers coke precursor poison capacity of catalyst (moles poisoning species/pellet volume); monolayer coke content rate of reaction, main reaction rate of deactivation reaction radius at any given time original radius

HETEROCENEOUS REACTOR DESIGN

gas constant global rate of main reaction global rate of deactivation reaction net rate of formation of n-layer coke fraction of vacant active sites active site time temperature temperature at xd pellet volume pellet coordinate as measured from center x at which fresh inner core is separated from completely deactivated outer shell conversion quantity defined in Eq. 5.30

B c d 1 j I

a

M o p s

species B pellet center at the boundary between fresh inner core and completely deactivated outer shell species M fresh catalyst poisoning reaction; pellet at pellet surface conditions

I i

1

Superscript

-

evaluated at surface conditions

PROBLEMS

Creek Letters

constant, quantity defined in Eq. 5.43; quantity defined in Eq. 5.99 quantity defined in Eq. 5.43 fraction of catalyst deactivated quantity defined in Eq. 5.30 Arrhenius number, E,/R, Tb; pellet porosity pellet voidage; (Ea)p/RgTb reactor point effectiveness tortuosity factor effectiveness factor internal effectiveness factor internal effectiveness factor for poisoning reaction, Eq. 5.48 tanh + i n / & ; effectiveness factor given by Eq. 5.57 tanh density molar density coke density usual Thiele modulus generalized Thiele modulus

5 , l . For the following mechanism, show that the kinetics of the main and deactivation reactions can be written in the form of Eqs. 5.4 and 5.5: A + S e A . S A - S e B - S B - S +B S N S -+ N S

+

+

(K*) (Ks) (kd, controlling step)

(b,poisoning)

5.2. For noncatalytic gas-solid reactions, write at least four quantities which are identical to the quantity given below:

+/+

(L- I)(k,/De)x'2 Thiele modulus defined in Eq. 5.67

Assume the

SPM mode of reaction for:

5.3. Show for general kinetics, r, = kf (C), that the activity factor A for uniform poisoning is given by:

Subscripts

A b

species A at bulk conditions

when the main reaction is diffusion-limited. Use the following global rate for a deactivated pellet:

5.6. In the catalytic cracking of gas oil, deactivation is caused by two types of coking: fast and slow. One possible model is: carbon (slow coking)

"d& (fast coking)

First show that:

where qf is the effectiveness factor for fresh pellet. 5.4. Equation 5.41 applies to slab geometry. Obtain the activity factor for spherical geometry. 5.5 Fill the marked corners with the types of catalyst pellet given below and give reasons for the choices. The shaded part of the pellets is inert. Refer to Becker and Wei (1977). For optimal catalyst distribution, see Varghese and Wolf (1980) and Dadyburjor (1982).

I

The mechanism of deactivation is chemisorption for fast coking and simple deposition for slow coking. Assume that all reactions are of pseudo-firstorder. Assume also uniform deactivation. Derive an expression for the time progression of the fraction of the catalyst deactivated. 5.7. It has been shown in Chapter 4 that a reduction in the amount of catalyst used can be realized for diffusion-limited reactions when hollow pellets or pellets with an inert core are used. Determine whether the same is true when the reactions are also affected by deactivation (Lee 1980). 5.8. Write global rates for the main and deactivation reactions in the asymptotic region of strong diffusion effects for the following cases: (1) uniform, (2) pore-mouth, (3) general. Repeat the same when heat effects are negligible. Assume independent poisoning. 5.9. Show for an nth-order reaction represented by:

that the ratio of Eq. 4.102 is:

of 0.7, 0.5, and 0.3, respectively), For surface activities of 0.3, 0.5, and 0.7 determine the value of R for n = 1. To obtain y profiles, first find the times at which 7 becomes equal to 0.3, 0.5, and 0.7. Use the results of Problem 5.8 (part 3) for G. Assume that the Thiele modulus for the deactivation reaction is 6 when y is zero, and that Na/Q = 3.6 X 10d7/4 X loe4 and Dp/L = 10-*cm/sec. 5.10. Suppose that a pellet is deactivated according to the shell-progressive model. Show that the activity factor can be determined from the measurements of external temperature differences:

Severity of poisoning

Severity of poisoning

Figure 5.5P Various t y p s of catalyst pellets.

where (AT.), is the external temperature difference for a fresh pellet. A comparison between A calculated from temperature differences and experimental results is given in the article by Lee (1980a). 5.1 1. A single pellet (Kehoe and Butt 1972) is often used to study various aspects

atm, 525OK, and a space velocity of 30,000 hr-I using 1 cm3 of powdered catalyst particles in an isothermal glass reactor. The feed consisted of 95% N2, 4% HZ,1% CO, and 10 ppm HZS.The nonnal'ized activity and outlet H2S concentrations that they obtained are shown in Figure 5.14P.Would first-order deactivation kinetics fit the experimental results? Give reasons for your conclusion. Note that the relative activity is equal to (1 - y) and that:

of a heterogeneous reaction. For a cylindrical pellet in which a first-order reaction takes place and where deactivation occurs according to the shellprogressive model, show that:

where (7Da)d = LRG/kCbfor a deactivated pellet. Show also that the activity factor A in the asymptotic region of strong diffusion effect is given by: 5.15. A study of coke formation in curnene cracking over silica-alumina has been made by Plank and Nace (1955). They found that the coke is due to an impurity in cumwe feed. They proposed the following for the inhibition of the reaction by the impurity:

= AT, (A Te)f

where (kb)/ is the rate constant for fresh catalyst at the bulk temperature. It is seen that the fraction of catalyst deactivated y can be calculated iteratively from temperature measurements and the observed rate of reaction for a single pellet. Note that Td for 14r can be obtained from Eq. 5.60. 5.12. Derive Eqs. 5.72 and 5.76. 5.13. Show for very fine catalyst particles that: 1

- y = exp (-at)

where S is acidic site. They assumed that the number of vacant sitcs is much smaller than that of occupied sites such that

where a is a constant if the bulk reactant concentration is also constant. 5-14. Bartholomew et al. (1979) studied H2S poisoning of a Ni bimetallic catalyst on alumina for methanation activity. Poisoning tests were conducted at 1

4

St = total sites = (N . S)+ (C .

*

The rate in the absence of the impurity then is:

and the rate in the presence of the impurity is:

RP = ks(C S) Show that

0

4

8

12

16

20

24

Time (h)

Figure 5.14P HzS poisoning of a Ni bimetallic catalyst. (Bartholomew et al. 1979. Reprinted with permission from Journal of Catalysis. Copyright by Academic Press.)

8

s

4

where f = conversion of cumene K = (k4k~)/[kl/(kz k3)] Co = cumene concentration in feed PO= inhibitor concentration in feed

+

j Their experimental results are shown in Figure 5.1 5p.

k

3.0

I

I

I

I

I

I

-

I

cumene hydroperoxide

Figure 5,15P Inhibition of silica-alumina at 8W°F. (Plank and Nace 1955. Reprinted with permission from Industrial and Engineering Chemistry. Copyright by American Chemical Society.)

5.16. Coke burning curves obtained by Weisz and Goodwin (1966) are shown in Figure 5.16P. The oxygen concentration and effective diffusivity are 3 X mol/cm3 and 5 X lod3 cm2/sec. 1. Check whether the criterion of negligible diffusion limitation is valid for their results. 2. Determine the kinetics of coke combustion to the extent you can. 3. Determine the 85% burnoff time for the two cases shown below. Note that about 5 wt% carbon corresponds to monolayer coverage by coke.

I

lo-s

k+

. I

o

E

w

-

I

1

1

1

Figure S.16P Coke burning rates of silica-alumina cracking catalyst, Initial carbon content is 3.4 wt%. (Weisz and Goodwin 1966, Reprinted with permission from Journal of Catalysis. Copyright by Academic Press.)

/-

R, = 0.Olcm

Temperature ( O C )

5.17. Explain for diffusion-limited reactions why the coke content is the highest at the pellet core for series coking whereas it is the highest at the surface for parallel coking. The work by Masamune and Smith (1966) may be useful. REFERENCES Baker, R.T.K., M.A. Barber, P.S.Harris, F.S.Featg and R.J. Waite, J. Catalysis, 26, 5 1 (1972). Baker, R.T.K. and J.J. Chludzinski, Jr., J. Catalysis, 64, 464 (1980). Bartholomew, C.H., G.D. Weatherbee and G.A. Jarvi, J. Catalysis, 60, 257 (1979). Becker, E.R. and J. Wei, J. Catalysis, 46, 372 (1977). Beeckman, J.W. and G.F.Froment, Chem. Eng. Sci., 35. 805 (1980). Beeckman, J.W. and G.F. Froment, Ind. Eng. Chem. Fund., 21, 243 (1982). Bischoff, K.B., Chem. Eng. Sci., 18, 7 11 (1963). Butt, J,B., Adv. Chem. Ser., 109, 259 (19722). Butt, J.B., D.M. Downing and J.W. Lee, Ind. Eng. Chem. Fund,, 16, 270 (1977). Butt, J.B. and E.E. Petersen, Catalyst Deactivation (in press). Carberry, J.J, and R.L. Gorring, J, Catalysis, 5, 529 (1966). Dadyburjor, D,B., AIChE J., 28, 720 (1982), Delmon, D. and G.F. Froment, eds., Catalyst Deactivation, Elsevier Scientific Publishing Co., New York (1980). Dumez, F.J. and G.F. Froment, Ind. Eng. Chem. Prw, Des. Dev., 15, 291 (1976). Feng, C.F. and W.E. Stewart, Ind. Eng. Chem. Fund., 12, 143 (1973). Froment, G.F. and K.B. Bischoff, Chemical Reactor Analysis and Design, p 298, Wiley, New York (1979). Haldemann, R.G.and M.C. Botty, J. Phys. Chem., 63, 489 (1959). Hegedus, L. and E.E. Petersen, Cat. Rev., 9, 245 (1974). Kehoe, P.G.and J.B. Butt, AIChE J., 18, 347 (1972). Khang, S.J. and 0. Levenspiel, Ind, Eng. Chem. Fund., 12, 185 (1973). Klingman, K.J. and H.H. Lee, preprint annual AIChE meeting, San Francisco (1984). Lee, J.W., J.B. Butt and D.M.Downing, AIChE J., 24, 212 (1978). Lee, H.H., Chem. Eng. Sci., 35, 1i49 (1980). Lee, H.H., Chem. Eng. Sci., 35, 905 (1980a). Lee, H.H. and J.B. Butt, AIChE J., 28, 405 (1982). Levinter, M.E.G.M.Panchenkov and M.A. Tanatarov, Intern. Chem. Eng., 7, 23 (1967). Masamune, S. and J.M. Smith, AIChE J., 12. 384 (1966). Maxted, E.B., Adv. Cat., 3, 129 (1951). Mills, G.A., E.R.Boedekker and A.G. Oblad, I. Am. Chem. Soc., 72, 1554 (1950). Petersen, E.E,,Chern. Eng. Sci., 37, 669 (1982). Plank, C.J. and D.M.Nace, Ind. Eng. Chem., 47, 2374 (1955). Supe, S. and 0. Levenspiel, Proc. European Fed. 4th Chem. React. Eng., Brussels, Pergarnon Press (1970). Varghese, P. and E.E. Wolf, AIChE J., 26. 55 (1980). Voorhies, A., Jr,, Ind. Eng. Chem., 37, 318 (1945), Wakao, N. and J.M. Smith, Chem. Eng. Sci., 17, 825 (1962). Weekman, V.W., Jr., AIChE J., 16, 397 (1970). Weisz, P.B. and R.D.Goodwin, J. Catalysis, 6, 226 (1966). Wen, C.Y.,Ind. Eng. Chem., 60, 34 (1968). Wheeler, A., in P.H.Emmett (ed.) Catalysis. Vol. 2, Reinhold, New York (1955). ,

CHAPTER 6 Physical Deactivation and Sintering

6-1

INTRODUCTION

Catalyst deactivation caused by physical processes rather than chemical reactions is termed physical deactivation. Reduction of the exposed surface area of the catalyst is often the result of physical deactivation. This reduction in surface area may be due to the growth of catalyst crystallites dispersed on a , or to the dissolution of metal atoms into the bulk of the . The growth of crystallites leads to a population of larger crystallites resulting in less surface area available for reaction per unit volume of catalyst pellet. When the metal atoms are dissolved into the , the catalyst is in effect lost into the . The exposed surface area of the catalyst is also reduced when particulates present in the feed deposit at the mouth of pores. This mode of physical deactivation is similar in nature to shell-progressive chemical deactivation: the pore mouth is first covered by the deposited particles and then the covered area grows inward with time, and finally the pore mouth is completely blocked, essentially disabling the catalyst since the reactants cannot reach the catalyst inside the pores. There is a fine line to be drawn between this mode of physical deactivation and chemical deactivation caused by the deposition of certain species: the particulates responsible for physical deactivation come with the feed, whereas the species responsible for chemical deactivatian are formed by chemical reactions taking place in the pe1le:t. The physical deactivation processes can be reversed to a certain extent, but only after the damage has been done. Cyclic treatment of cqystallites in an oxidizing atmosphere and then a reducing atmosphere has been found to redisperse large crystallites. Dissolved metal atoms can be made to migrate back to the surface by proper heat treatment, The deposited particulates can be removed either by combustion or by dissolving them with a suitable solvent. Our understanding of the physical deactivation of a catalyst is quite limited. In fact, the physical deactivation due to metal dissolution and particulate deposition has drawn interest only recently. On the other hand, some understanding of crystallite growth has been gained, even though there are still many questions to be resolved. The process of catalyst crystallites growing via the movement of small crystallites or atoms is called sintering (sometimes called coarsening). Large crystal-

PHYSICALDEACTIVATION AND SINTERINC

lites grow at the expense of smaller ones so as to minimize the chemical potential of the system. Since large crystallites have less exposed surface area per unit volume of catalyst, sintering leads to less surface area available for the reaction. For instance, it has been found (Dalla Betta et al. 1976) that the catalysts used in the catalytic converter for automobiles lose 95% of their original activity due to sintering in 50,000 miles of driving. There are also many important industrial reactions in which sintering is a major factor, a typical example being the silver catalyst used for ethylene oxidation. Major emphasis will be placed on sintering in this chapter because it is better understood than the other modes of physical deactivation. Sintering takes place via the movement of either crystallites or atoms (or molecules) detached from the crystallites on the surface of a . For the crystallites or atoms to move on the surface there has to be a source of energy whose level is high enough to overcome the binding energy between the crystallite and the in the case of crystallite migration, or the metal to metal bond energy in that of atom migration. This energy is considered to be derived from the vibration of atoms, the energy of which is proportional to temperature. The presence of certain gaseous species affects the sintering since it can alter the state of the crystallite surface. The presence of oxygen, for instance, can alter the surface of metal crystallites into a metal oxide surface. Therefore, the nature of the interaction between a catalyst and its , for instance, is altered from between a metal and a , to between a metal oxide and a . It is not surprising then that the sintering rate depends on temperature, types of metal and , and the atmosphere in which the sintering takes place in addition to the exposed surface area. Depending on whether the movement of the crystallite or the adatom is responsible for the crystallite growth, sintering is said to take place through the mechanism of crystallite migration (Ruckenstein and Pulvermacher 1973) or atom migration (Flynn and Wanke 1974; Wynblatt and Gjostein 1975). The magnitude of the metal to metal bond energy relative to that of the binding energy between the metal and the determines whether the sintering takes place through the mechanism of crystallite migration or atom migration. It is useful to note that for most of the metals the level of interaction between the metal and the is in the order of alumina > silica > carbon. While the understanding of the other modes of physical deactivation is quite limited, certain useful conclusions can be made on the effect of physical deactivation on the catalyst activity as related to reactor design, provided that the physical deactivation takes place uniformly throughout the pellet. Therefore, this general case will be treated first before proceeding to sintering.

EFFECT O F PHYSICAL DEACTIVATION OF CATALYST

6-2

It has been pointed out that the exposed surface area is reduced in a shell-progressive manner when the deposition of particles causes the physical deactivation. On the

203

other hand, sintering and dissolution of atoms can be considered to take place uniformly within the pellet. The dissolution of the metal atoms into the is largely determined by the density distribution of the atoms on the , which is assumed to be uniform. Therefore, on average, the dissolution would be uniform. As discussed earlier, sintering depends on temperature and the atmosphere in which the catalyst is operating. Since the pellet can be considered isothermal, except when a pellet undergoes shell-progressive chemical deactivation, sintering should take place uniformly provided that the gas composition, in particular the oxygen content, does not change significantly during the course of a reaction. In many reactions of practical interest involving catalyst sintering, oxygen is present in excess, and therefore, sintering can be considered to take place uniformly. It is assumed in what follows that physical deactivation takes place uniformly. Suppose that the exposed surface area of the catalyst can be measured. Given this measurement, an overall pellet effectiveness can be determined for the reactor design, which represents the combined effects of diffusion, chemical deactivation, and physical deactivation on the intrinsic rate of the reaction. The exposed surface area of a metal catalyst is typically measured using the chemisorption method. As pointed out in Chapter 1, however, this method cannot be used for catalysts of metal compounds. Furthermore, the chernisorption method fails to for the whole exposed surface area of metal catalysts if a part of the exposed surface area is irreversibly oxidized or forms a compound with the . Electron microscopy can also be used for the determination of the exposed surface area. While this method is applicable to metal catalysts as well as metal compounds, the size of crystallites counted, based on the micrographs of TEM (transmission electron microscopy), is limited to typically 1 nm for an alumina and to 2 nrn for a silica , which gives a poorer contrast than alumina. SEM (scanning electron microscopy), which can also be used, gives a poorer resolution than TEM. Therefore, small crystallites cannot be detected. A typical SEM micrograph of silver ed on fused alumina is shown in Figure 6.1. The X-ray line broadening method for the average crystallite size does for small crystallites even though its accuracy is rather limited in the nrn size range. In view of these difficulties, it is desirable to use all three methods for the determination of the exposed surface area whenever possible. For metal compounds, selective poisoning reactions can also be used. It is important to recognize that any surface area measurement should not be taken to give an accurate absolute value, but rather to measure the change of surface area, i.e., the ratio of surface area normalized with respect to the initial surface area. Furthermore, smoothed, average data should be used to represent the change of surface area with time. In order to examine the effect of sintering, poisoning, and diffusion, the overall pellet effectiveness E is defined as the ratio of the observed rate (global rate) to the intrinsic rate unaffected by diffusion and catal~ystdeactivation:

PHYSICALDEACTIVATION AND SINTER~NC 205

and so h(S,/So) is simply &/So; for structure-sensitive reactions, the activity has to be correlated to the surface area ratio. Since S, = So for fresh catalyst, and the rate constant k in Eq. 6.3 is for fresh catalyst, it follows that h(1) = 1. Further, it follows from the definition of the overall effectiveness factor (Eq. 6.1 and Eqs. 6.2 and 6.3) that:

In the absence of chemical deactivation, the overall effectiveness factor is simply h(S,/So) since then S ---- St. For diffusion-affected reactions, the generalized internal effectiveness factor , be used to arrive at the following overall pellet effectivefor fresh catalyst, q ~ can ness for uniform chemical deactivation:

Figure 6.1 A SEM micrograph of silver ed on fused alumina.

It is assumed in what follows that sintering is uniform; poisoning can be uniform, shell-progressive, or in between these two extremes. For these various cases, the overall pellet effectiveness will be derived in this section. Since sintering can take place whether the sudace is poisoned or not, it reduces the total surface area St. Furthermore, both the poisoned and the active surface areas sinter the same way under the assumption of uniform sintering. In contrast, poisoning reduces only the active surface area &. In the absence of poisoning, there is no exposed surface area poisoned and so St = &. This distinction of surface areas, between St and &, is important in determining the effects of sintering and poisoning on the rate of reaction. While the heat and mass transfer resistances, both within the catalyst pellet and across the pellet-fluid interface, affect the rate of reaction, it will be assumed that only diffusion within the pellet and heat transfer across the interface are important in accordance with the results obtained in Chapter 4. It is straightforward to obtain the overall effectivenessfactor when the reaction is free of diffusional effects. The global rate in this case is:

where

T C =-

~ ( c b ) = / ~ f ( ~ d c (6.4) ,:c

where L is the characteristic pellet length, D. is the effective diffusivity, Cb is the bulk-fluid concentration of the main reactant, k, is the rate constant evaluated at the pellet surface temperature, and C, is the pellet center concentration. As was the case in previous chapters, the center concentration can be taken as zero when the following modified generalized modulus is greater than 3:

Otherwise, the center concentration is estimated from:

When the poisoning is nonuniform, there exists an activity distribution within the pellet, and the generalized effectiveness factor obtained in Chapter 4 for a nonuniform activity profile can be used to arrive at the following expression: if the intrinsic rate is written as:

where So is the total active surface area for fresh catalyst, and h(&/So) represents the dependence of the catalytic activity on the surface area ratio. For structureinsensitive reactions, the activity is directly proportional to the surface area ratio

where the overbar denotes the quantities evaluated at the pellet surface. This approximate result gives accurate-values - of the pellet effectivenesswhen the surface activity is far from zero, say h(&/So) > 0.5. Another limitation is that it is applicable only to diffusion-limited reactions.


If the rate of poisoning is very high such that shell-progressive poisoning takes place, the surface activity is zero and Eq. 6.9 is not applicable. In order to treat this extreme case of poisoning, one needs to recognize that the poisoned catalyst in the completely deactivated outer shell sinters the same way as the active catalyst in the still active inner core of the pellet, but that only sintering affects this active inner core. At steady state, the rate of reactant transport through the deactivated outer shell should be equal to the rate at which the reactant is consumed by the reaction in the active inner core:

207

where S, is the active surface area that would exist in the absence of sintering. Poisoning reduces only active surface area. Therefore, the total surface area remains at its original area SOin the absence of sintering and the ratio of active surface area to the total surface area is given by &,/SO. Suppose sintering also takes place. Under the assumption of uniform sintering, both active and poisoned surface areas are reduced the same way so that the ratio of active to total surface area remains the same. Since the active surface area in this case is &, and the total surface area is St, then:

where 1 is the length of the deactivated outer shell, and the subscript d denotes the boundary between the deactivated outer shell and the active inner core. Here, the generalized effectiveness factor for the active inner core is given by:

The fraction of catalyst deactivated, y, is simply 1/L for the slab-like pellet. Combining these two equations yields the following expression for the calculation of Cd:

This relationship was used in Eq. 6.14. The ratio E/En has been obtained for various cases (Lee 1981) and these are summarized in Table 6.1. The overall pellet effectiveness in the absence of poisoning can be obtained by simply replacing & by St. In order to use the information on surface area for reactor design, the change of the surface area with time has to be related to the reaction conditions. This is possible only for sintering because of limited understanding of the other modes of physical deactivation. Since surface diffusion plays a pivotal role in sintering, it is considered first for the development of the sintering kinetics to follow.

The definition of pellet effectiveness yields: Table 6.1 Overall Pellet Effectiveness Ratio, E/E,

Note that Cd appearing in Eq. 6.13 can be calculated from Eq. 6.12 and that the activity function h is given in of the total exposed surface area St, rather than the active area S, since no poisoning takes place in the active inner core such that St = &. The overall pellet effectiveness represents the combined effects of diffusion, poisoning, and sintering on the intrinsic rate. A simple multiplication of the intrinsic rate by the pellet effectiveness gives the actual rate at which the reactants are converted, i.e., the global rate. This pellet effectiveness can also be used to determine how much of the lass in catalytic activity is solely due to sintering. For this purpose, it is sufficient to know the pellet effectiveness in the absence of sintering, En, since the ratio of E to E, is a measure of the loss of catalytic activity solely due to sintering. This ratio can readily be obtained from Eq. 6.4 for diffusion-free reactions:

Diffusion-free reaction Diffusion-affected reaction and uniform poisoning Diffusion-affected reaction and general case of poisoning Diffusion-affected reaction and shell-progressive poisoning

h (&/So) h (s/st 1

[ h(g/go)I

112

[h(s)]1'2 so

Cd is given by Eq. 6.12 and (Cd), by:

>

for

/I(

[IC Cr d / ( c ) d c

(Cc )n

f(C)d~]"~


6-3 SURFACE DIFFUSION

A catalyst is usually dispersed on a in a discrete crystallite form as shown in Figure 6.1. These crystallites, which are composites of atoms or molecules, range in size anywhere from below 1 nm to about 100 nm. The crystallites can grow under typical reaction conditions due to the heat liberated by the chemical reaction, which causes the migration of atoms (molecules) or small crystallites. This migration is often aided by the presence of a gas, pa~ticularlyoxygen for platinum group metals and chlorine for base metals. In order for sintering to occur via atom migration, an atom (molecule) has to be detached from the crystallite, forming an ad-atom, which then migrates over the surface to other crystallites. A model of a crystallite is shown in Figure 6.2 (Somorjai 1972). For crystallite migration, on the other hand, a whole crystallite has to migrate over the surface. The question of which mechanism is operative, then, is equivalent to whether the detachment of an atom followed by surface migration is energetically easier to take place than the movement of a crystallite. An understanding of surface diffusion provides a key to the question. Surface diffusion via the movement of atoms is considered to be a multistep process in which atoms break away from their lattice position (e-g., a kink site at a ledge in Figure 6.2) and migrate over the surface until they find their new equilibrium sites. Consider an ad-atom held on the surface by a small binding energy. Let the energy of this interaction be AEd. This energy is the difference between the energy of an ad-atom in its excited state, (i.e., .in between two equilibrium sites) and that at its equilibrium condition. The thermal vibration of the ad-atom about the equilibrium site is the source of the energy for the movement. If the vibration frequency of the ad-atom is vo, the probable number of times that this atom jumps out of the energy well; or the jump frequency is given by:

209

where kB is the Boltzmann constant, and 6 is the number of equivalent neighboring sites. For instance, the 5 value of a (1 11) face of a face-cubic-centered metal is 6. If it is assumed that surface diffusion can occur only via the movement of adatoms, these species have to be created before any appreciable diffusion can occur. The fraction of ad-atoms, C/Co, on the surface is given by:

where A Ef is the energy of formation of an ad-atom, C the concentration of adatoms, and CO the total concentration of surface sites from which the ad-atom can break away. The probabilities multiply and therefore the jump frequency is the product of the two events:

This frequency is for a single jump of an ad-atom to a neighboring equilibrium site. Now, consider the random walk of an atom for long-distance motion. The question is how far a diffusing atom will move in time t and a very large number of jumps. For this net displacement of an atom, a one-dimensional random walk is considered. If the root mean square distance is used as the net displacement, then for the expected value of X2, (X2):

where X is the net distance covered by the atom, nj the number of jumps, and a the distance of one jump, which is taken as the distance between two nearest neighboring sites. If t is the time required to make nj jumps, nj is given by:

which, when combined with Eq. 5.19, yields:

Terrace Monatomic Ledqe

, /

Adatom

It is customary to define the diffusion coefficient as:

where b is the number of coordinate directions in which the jumps may occur with equal probability. For diffusion equally probable in two directions, use of Eq. 6,18 in 6.22 yields: A model of a heterogeneous surface showing different surface sites. (From Principles of Surface Chemistry, Somorjai 1972.)

Figure 6.2

The root mean square distance can be obtained from Eqs. 6.21 and 6.22:

This equation is usually used for the determination of the surface diffusivity 4 from the measurements of t and (X2)lI2.Determination of the activation energy is made in the usual manner using Eq. 6.23. For many cubic metals (Gjostein 1967), AEj is of the order of 40 kcal/mole, and the sum of (A& AEr) can often be approximated at high temperatures by 0.54 AH,, where AHs is the heat of sublimation. The correlations for surface self-diffusion coefficients are shown in Figures 6.3 and 6.4 for face-centered cubic metals and body-centered cubic metals, respectively. These correlations are for the diffusion of metal on the same metal surface. The neighboring site distance "a" is of the order of a few angstrom, and vo is of the order of 1012 sec'l. The surface diffusion of crystallites is more complicated than the diffusion of ad-atoms due to the dependence of surface diffusivity on crystallite size. A review on this subject can be found in the article by Kashchiev (1979). Theoretical models available for crystallite migration predict power or exponential dependence of the diffusivity on the size r (or the number of atoms n, making up the crystallite). The activation energy for the diffusion can be either size-dependent or size-indepen-

+

Figure 6.4 Correlations between surface self-diffusion coefficient and reduced temperature for body-centered-cubic metals. (From Principles of Surface Chemistry, Sornorjai 1972.) I

I

I

1

I

I

I

I

1

I

1

-

-

T,

dent. The surface diffusivity also depends on the mode of crystallite migration. These factors complicate the treatment.

Melting Temperature

-

6 4

-

In this section, the mechanisms ofsintering are treated in detail to arrive at sintering kinetics. The first mechanism to be considered is that of atom migration, sometimes termed Oswald ripening (Lifshitz and Slyozov 1961) by analogy to the growth of particles suspended in a gas or liquid by movement of individual atoms or molecules. The second is the mechanism of crystallite migration, which is the movement of individual crystallites along the surface to cause growth by coalescence, analogous to coagulation by Brownian motion.

-

-

-

I

Figure 6.3 Correlations between surface self-diffusion coefficient and reduced temperature TM/Tfor face-centered-cubicmetals. (From Principles of Surface Chemistry. Somorjai 1972.)

MECHANISMS AND KINETICS OF SlNTERlNC

Mechanism of Atom Migration and Kinetics

Consider individual crystallites dispersed on a flat substrate and assume that only single atoms migrate. Each crystallite will have a concentration of single atoms at the interface (line interface) between the crystallite and the substrate, which depends on the crystallite size. Smaller crystallites will have a higher concentration

than larger crystallites because of their smaller curvature. At some distance away from the crystallites, there exists a "far field" concentration of single atoms (adatoms) on the substrate, representing the "bulk" of ad-atoms for the whole system at any given time. This concentration is higher than the interfacial concentration for the larger crystallites but lower than that for the smaller crystallites. Because of this concentration difference, the ad-atoms will move from the smaller crystallites to the bulk, and then to the larger ones, resulting in the growth of the larger crystallites at the expense of the smaller ones. This mechanism of sintering is termed the mechanism of atom migration or Ostwald ripening. In addition to this ripening, which takes place on a global scale involving the bulk ad-atoms, "direct ripening" (Ruckenstein and Dadyburjor 1978) can also occur locally between crystallites that are sufficiently close to each other, even though the bulk is not supersaturated with respect to the larger crystallites. The atom migration is caused by the difference in the chemical potentials of the crystallites, since the level of concentration of ad-atoms at the interface is determined by the chemical potential. Consider partially wetting crystallites in the form of spherical segments, making a angle 8 with a flat substrate as shown in Figure 6.5 (Wynblatt and Gjostein 1975). The chemical potential p of any crystallite depends a n its radius of curvature r:

5

Volume of the cap = 4 7rr3aar Surface area of the cap = 4wr2a2 a,= (2 3 COS 8 cos3 8)/4 a2 = (1 cos 8)/2

+

-

v

I

L=l(r sin 9)

-

r sin 0

R-

crystallite f errace

is p for an infinite-sized crystallite, a is the interfacial tension between where the crystallite and gas, and V, is the volume of an ad-atom. This relationship can be rewritten in of surface concentrations to give the Gibbs-Thompson relationship:

n

Sites

I

I

Edge Site

Distance

where neq is the surface concentration (number of atoms/area) corresponding to the concentration of an infinite-sized crystallite. Consider the first of the series processes of surface diffusion of ad-atoms and eventual transfer at the interface of the crystallite (Figure 6.5). Taking the center of the crystallite of radius r sin 8 as the coordinate origin, a steady state mass balance in polar coordinate yields:

Figure 6.5 Concentration-distance and energy-distance diagram for the surface diffusion of ad-atoms. (Wynblatt and Gjostein 1975. Reprinted with m i s s i o n from Progress in Solid Store Chemistry. Copyright by Pergamon Press, Inc.)

At the edge of the crystallite (R = r sin B), there exists a certain surface concentration n'. At a "screening" distance L, which is taken as a multiple of the radius ( I sin O), i.e., l(r sin 8 ) where I is the multiple, the surface concentration reaches its bulk value ti. These boundary conditions, when used in the above equation, give:

where n is the local surface concentration of ad-atoms, and Dl is the surface diffusivityof ad-atoms on the substance [ D l = Dlo exp(-AEd/ksT)]. The solution is: The number of atoms attaching per second at the perimeter of the crystallite by surface diffusion is given by:

PHYSICALDEACTIVATION AND SINTERING

I

2f 5

The Gibbs-Thompson relationship of Eq. 6.26 can be used to write n, and fi as

=-2wD1 (fi - n') In I

Consider now transfer at the interface. The rate at which the ad-atoms one interatomic distance a away are attached to the crystallite is equal to the circumferential area (a27rr sin 8) times the rate at which these atoms (n' number of atoms/ area) jump over the energy barrier AEd, i.e., [2wr a sin 8]ntP'where the jump frequency P' is given by:

where n;q and i i e q are the atom surface concentrations on the crystallite edge sites and at the screening distance L, respectively, in equilibrium with an infinitesized crystallite. The,fact that fieq = (P/fl')n$ has been used in Eq. 6.38. When these equations are used in Eq. 6.36, the growth rate expression becomes: dr - VmnFq(P/B') ( 2 r r a P' sin 8)(27~ Dl/ln I ) -

dt

In order for an edge atom to jump out to occupy a site on the substrate, it must AEd, where AEf and AEd are the energy surmount an energy barrier AE' required to detach an atom from the crystallitk and the energy required for an ad-atom to move on the substrate, respectively. Therefore, the jump frequency of these edge atoms is given by:

4 r r2az

27r DJln

I

+ 2 r r a fl' sin 8

+

where vp and vo are the vibrational frequencies on the crystallite edge and substrate sites. Then, the rate of atom detachment is given by [27rr a sin O]n,P, where n, is the surface concentration of atoms on the crystallite edge sites. The net flux across the interface is equal to the difference between these two rates:

This equation essentially defines the critical size i. It is seen that the crystallites of size r' neither grow nor shrink (d?/dt = 0).This means that the surface concentration at the edges of these crystallites is the same as the average surface concentration prevailing at some distance away from the crystallites. This critical size changes with time as the size distribution of crystallites changes with time. The relative importance of interface transfer with respect to surface diffusion can be determined by comparing the relative magnitude of resistances (Wynblatt and Gjostein 1975). If surface diffusion is the controlling step, one should have: 2rD1 In 1

J* = (27~ r a sin 8)(Ptn' - fir) = ( 2 r r a sin B)P'(nl - P/P'nr)

<< 2 w r a /3'

sin 8

The screening distance L is typically 2 to 3 times r sin 8, making (In I) close to unity. The surface diffusivity of ad-atoms on the substrate is given by:

Since J = Js = Ji at steady state, Eqs. 6.30 and 6.33 give:

J=

(27rr a p' sin 8)(2w Dl/ln I) ( f i - fl/P1nr) 27r Dl/ln 1 2 7 ~ar fl' sin 8

+

Therefore, the condition becomes: a

The growth rate (dr/dt) of the crystallite of interest is obtained from:

This condition is always met in physically realistic situations since sin 0 is close to unity and a << r. Thus, surface diffusion is most likely to be the controlling step in realistic situations. This conclusion is a result of the relationship, iiPq = (P/P') n;? When the surface diffusion is the controlling step, Eq. 6.39 reduces to:

which leads to: dr e -

dt

<< r sin 9

V , ( 2 r r a 0'sin 8)(2.rrDJln I) ( f i - P/P'n,) 497 r2al 27r Dr/ln l + 2 7 ~ar P' sin 8

dr - V, n';P@/fl')2r D

dt

4 v r 2 a l In

20

v, - exp k~ Tr

I

The exponential in this equation can be approximated as follows:

in Eq, 6.47. The quantity m can be taken constant if an average temperature is used for the range of temperatures of interest. The growth rate of the critical crystallites can be obtained in a similar manner (Lee 1980):

This approximation is accurate (Lee 1980) when:

Under typical sintering conditions, the value of E ranges from 0.3 to 0.8 for average crystallite sizes ranging from 5 nm to 20 nm. The dimensionless surface energy E is readily calculable for a given system. Substitution of Eq. 6.41 into Eq. 6.40 results in the following expression:

1

where

The sintering kinetics for the mechanism of atom migration can be obtained from the expression for the growth rate (Eq.6.43) and a continuity equation for the size distribution of crystallites:

where the temperature dependence of K, is the same as that of K. This expression can be used to obtairi the time dependence of the average size since in most cases the average crystallite size is very close to the critical size (Wynblatt 1976). The kinetics of sintering for the unlikely case of interface control can also be obtained in a similar manner. It should be noted that the exponential term appearing in the sintering kinetics can be set to unity when the dimensionless surface energy E is less than 0.1. This would be true for a typical platinum system, for instance, if the average size is greater than approximately 60 nm. While sintering can also take place via vapor phase diffusion, the likelihood of this is remote for metal catalysts because of their high melting points (Table 6.2). However, sintering can be considerably aided by the formation of oxides or chlorides. The melting point of metal compounds such as oxides is much lower than that of metals. Since the sublimation energy is a measure of AE,, the energy required to detach an atom from a crystallite is much lower for metal compounds Table 6.2

Metal

+

where f(r,t)dr is the number of crystallites in the size range between r and r dr at any given time t. After a lengthy manipulation, one arrives at the following sintering kinetics (Lee 1980):

where K ( T ) = (constant) AD = (constant) exp[--(A Ef m = (constant)

+ A Ed)&

I 1I

TJ

-----) = (2k"B7

(constant)(y)

Use of Eq. 6.23 in 6.47 results in a rate constant K that is essentially the surface diffusivity D,. A weak temperature dependence of the form of Test2was neglected

1

Properties of Metals (Jones 197 I )

Solid/Vapour Surf4ce Energy at Tm, erg/cm2

Atomic Volume of Solid at T,, crn3/g atom

Melting Molar Surface Point (T,,,), Energy, kcal/g O C atom

Heat of Sublimation at Tm, kcal/g atom

21 8



2 19

Table 6.3 Sintering Kinetics for the Mechanism of Atom Migration

Controlling Step

n

Interface transfer Surface diffusion Surface diffusion (Wynblatt 1976) aided by the formation of oxides or chlorides

3 5

M(s) + (I 01(g)

K,

2 MOW

5

In order to obtain information on Kij, the process of crystallite diffusion must be examined in some detail. A mass balance during a small time interval O0 at or around a given real process time t is given by:

K

AE

- exp(-A E/knT ) - exp(-A Elkn7') - (P,d2 Kt,e x p ( Q / k ~T)*

(AEJ + AEd (AEJ + A&)

Q = AE,

- A Ef

(8)

= equilibrium constant AEJ = energy required to transfer an oxide from the substrate to the vapor phrase

than for metals. The sintering kinetics for various cases are summarized in Table 6.3. Mechanism of Crystallite Migration

According to the mechanism of crystallite migration, crystallites move as such on the substrate surface and then merge into larger ones when they are sufficiently close. Growth is assumed to depend on the rate of surface migration of the crystallites and the nature of their interaction when they are very close to each other. The probable driving force is assumed to be the thermal motion of the atoms at the metal-substrate interface. If the interaction between crystallites is so strong that they merge into a single unit upon within a time period which is short enough compared to the time necessary for the migration, sintering is considered diffusion-controlled.On the other hand, it is considered coalescence-controlled if the merging process is slow compared to the diffusion process. Consider crystallites highly dispersed on a planar substrate (Ruckenstein and Pulvermacher 1973). Let Ck be the surface ,concentration of crystallites of k atoms, i.e., the number of crystallites of k atoms per unit sufiace area of substrate. The concentration Ck increases by the collisions of crystallites composed of i and ( k - i) atoms and decreases by the collisions of crystallites composed of k atoms with any other crystallites. In analogy with chemical kinetics, the rate of collision between crystallites composed of i and j units is written as:

where KG is a rate constant dependent on the mobility of crystallites over the substrate and the nature of the interaction between crystallites. The rate of change of Ck is thus given by:

where Dw is the relative diffusion coefficient of a crystallite k with respect to that of a crystallite j. The cylindrical coordinate system is such that the origin is at the center of the surface of between a crystallite of j units and the substrate. This equation is used during a time interval 00, which must be small compared to the time in which an appreciable modification of the crystallite distribution takes place. Further, the rate of collision should be practically independent of 8 during go. Assuming that the thermal motion of the crystallite of j units is independent of the motion of the crystallite of k units, one can write:

The initial and boundary conditions are:

where ak, is the rate constant for the merging process, and Rkj is the radius of interaction of the two colliding crystallites given by:

When the rate of merging is very high (aM-+ a),the process is diffusion-controlled and Ck = 0 for Rjk = 9 rk. When the rate is very small, on the other hand, the process is coalescence-controlled. The rate of collision +rt is equal to the diffusion flux times the perimeter 2wRw*

+

Since W = KwCb, Ku can be calculated from Eq. 6.57. The expression for Kij thus obtained for large values of (DuO/Ri) or to be more precise, for in (4Qj9/ R t ) >> 1, is (Ruckenstein and Pulvermacher 1973):


22 1

when the growth is diffusion-controlled. When it is coalescence-controlled:

It should be recognized that the diffusion equation (Eq.6.52) ignores the collision between various crystallites. This is appropriate if the time interval needed to achieve the quasi-steady state is an infinitesimal fraction of the process time t. In order to use these results in the rate expression of Eq.6.54 for the derivation of sintering kinetics, Dt3 and a~ have to be expressed in of the crystallite size. These expressions have been suggested by Ruckenstein and Pulvermacher as:

Du = (constant) a ~= j (constant)

;

m 1= 1,2

Using these results in the Kif expressions, one obtains:

Ku =

(constant)(r; (constant)(r:'

n'

+ rJ:"') diffusion-controlled + r y ') coalescence-controlled

where In (4DijO/Rii) is assumed to be relatively constant. With these expressions, the continuous version of Eq. 6.5 1 can be written as:

where m = - n 1 / 3 when sintering is diffusion-controlled, and m = m ' / 3 when sintering is coalescence-cont rolled. These are the sintering kinetics for the mechanism of crystallite migration. According to Eq. 6.61, m' can assume a value of 0, -1, or -2 when the growth is diffusion-controlled, giving an n value of 4, 5, or 6, respectively. When the growth is coalescence-controlled, n can assume a value of 2 or 3. It is seen then that depending on the expression used for the size dependence of Kr3, the value of n can be anywhere from unity to more than ten. In general, however, the value of n appears to be 'smaller far the case of coalescence-control than for diffusion-control. An Arrhenius temperature dependence can be assumed for K. It is of interest at this point to compare the size-dependenceof surface diffusivity for crystallite migration obtainable from the theoretical models with the expressions used by Ruckenstein and Pulverrnacher for the size dependence of the diffusivity [m' of 0,-1, or -23. According to the results of the theoretical models (Kaschiev 1979), rn' can assume a value ranging from 0 to -5. On the other hand, the exponential size dependence of the surface diffusivity suggests that m' can assume a range of values. It is conceivable then that the value of m' and therefore that of n in Eq.6.65 can be wide-ranging when the mechanism is crystallite migration. The mode of crystallite migration dictates the size-dependence of the diffusivity and this size-dependence complicates the a priori determination of the value of n in Eq. 6.65 (Lee and Ruckenstein 1983).

6-5 RELATIVE IMPORTANCE AND S l N T E R l N C BEHAVIOR

where n dv is the number of crystallites per unit substrate surface area having a volume between v and v dv, and K is a constant, independent of size. Equation 6.62 has been used in 6.51 for KW Integrating this equation over all sizes, one obtains:

+

where Nu is the total number of crystallites per unit area. A similarity transfomation of Eq. 6.64 yields the following expression for the rate of change of the total surface area of crystallites per unit substrate area:

The basic question regarding sintering mechanisms is whether the crystallites migrate as such or in the form of ad-atoms. Energetically, the question is whether it takes more energy to detach an atom from the edges of a crystallite than to move a crystallite. A quantitative conclusion regarding the relative importance of the mechanism for a given system can be reached by examining the ratio of diffusivities, Ds/D(r). Because of the dependence of D ( r ) on the mode of crystallite migration, it is difficult to tell exactly under what conditions one mechanism is dominant over the other unless detailed knowledge of the mode of crystallite migration is available for the system under consideration. As the section on surface diffusivity reveals, the activation energy for the surface diffusion of ad-atoms is proportional to the sublimation energy. The activation energy for the surface diffusion of crystallite is dependent on crystallite size. While the mode of crystallite migration dictates the form of this size dependence, it is readily seen that the activation energy increases with increasing size. Therefore, sintering is more likely


to be dominated by crystallite migration when the crystallite size is small. The activation energy for crystallite migration is proportional to the activation energy between an atom and the substrate multiplied by a function of crystallite size, which in turn depends on the mode of crystallite migration. It follows then that crystallite migration, as opposed to atom migration is more likely to take place when the level of interaction between catalyst and substrate is weaker. In fact, it has been observed experimentally that Pt crystallites as large as 20 nm migrate (Chu and Ruckenstein 1977) on a carbon substrate, although undoubtedly the migration was aided by gases evolving from the substrate. Oxides and chlorides have a lower melting point and lower sublimation energy than the metals. Therefore, these compounds are more likely to sinter via the mechanism of atom migration than the metals. It is known that an oxidizing atmosphere is usually more conducive to sintering than in& or reducing atmospheres. This can be explained readily if the species being transported in the oxidizing atmosphere is the oxide, whereas it is the metal in the other atmospheres. It should be recognized that the most substantive assumption made in deriving the sintering kinetics is that the surface is planar. In typical ed catalysts, however, the surface is not planar but rather curved, and there are many crevices and cracks. These nonideal surfaces and boundaries serve as physical barriers against the movement of atoms or crystallites. It has been shown experimentally (Wynblatt 1980) that sintering is localized when the radius of curvature for a curved surface is less than approximately four times the crystallite size. Experiments were carried out on a sinusoidal surface and when the radius of curvature was less than four times the crystallite size, sintering was localized in the valleys of the surface. It can be concluded therefore that the curved surface is essentially planar for crystallites whose size is less than one fourth of the radius of the curvature. Since the typical pore size distribution for ed catalysts is in the range of 1 to 100 nm, undoubtedly some crystallites "see" planar surfaces while others see curved surfaces. Those that see curved surfaces should grow faster in the valleys than those that see planar surfaces, while those in the intermediate range should grow at a rate intermediate between these two extremes. Therefore, the sintering kinetics obtained for planar surface, when applied to ed catalysts, describe sinteing averaged over various nonideal surfaces. &6

223

order to obtain an expression for the rate at which the active surface area is reduced by both chemical deactivation and sintering, the surface area ratio &/ Somust be written as follows:

The sintering kinetics provide the time dependence of (Sc/So):

,(%) d

St

= -K.

(5)

enp[rn'(~t/~d]

atom migration (6.67) crystallite migration

It has been established in Section 6-2, that under the assumption of uniform deactivation, the ratio of active surface area to total surface area should be the same whether or not chemical deactivation also reduces the surface area:

where the term (&/So),denotes the surface area ratio when the surface area is reduced only by chemical deactivation. The rate at which the active surface area is reduced only by chemical deactivation can be obtained from:

where the rate of chemical deactivation rj, can be expressed in of (Z/So),. Here, Q is the poisoning capacity of the pellet in moles of poisoning species per unit pellet volume. Using Eq. 6.68 in Eq. 6.66, one gets:

REACTOR POINT EFFECTIVENESS

Now that the kinetics are known for the change of exposed surface area due to sintering, the results of Section 6-2 can be used to arrive at the reactor point effectiveness. The overall pellet effectiveness derived earlier can be used for this purpose. However, the term &/Sohas to be expressed in of known quantities before it can be used for the reactor point effectiveness. The active surface area can be reduced, not only by chemical deactivation, but also by sintering. While only the active surface area is affected by chemical deactivation, active as well as deactivated surface area is affected by sintering. In

where (St/So) and (&/So), are given by Eqs. 6.67 and 6.69, respectively. Take as an example, independent chemical deactivation due to the chemisorption of poisoning species. The kinetics (Chapter 5) can be represented by:

='

(GIGo ) ~

N {[I

+ G ( ~ ) l n=} +N

{[I

+ G(c)ln

PHYSICAL OEACTIVATION AND

where k, = k;Cb Cto= total number of sites per pellet volume at time zero C, = number of active sites/pellet volume kb = chemisorption rate constant N = concentration of poisoning species C, = number of vacant active sites/pellet volume (= Cr/[l

SINTERINC

Table 6.4 Reactor Point Effectiveness: Uniform Sintering re = kflC)

+ G(c)ln)

Here, the number of vacant sites C;, has been re-expressed in of surface sites per unit volume of pellet, and the subscript p denotes that the active area reduction is solely due to poisoning. The relationship of Eq. 6.70 is also applicable to the case of shell-progressive chemical deactivation since ($/So), = 1 - y, where y is the fraction of catalyst completely deactivated. The value of y is the same under the assumption of uniform sintering whether or not the pellet is also sintered. Here, the time dependence of y can be determined from:

h (&/So) 1.2(-AH)Lksf(Cb)h(&/So) 12h~Rgq

I Diffusion-free reaction

,

11

.

Diffusion-affected reaction and uniform chemical deactivation

I11 Diffusion-limited reaction and general case of chemical deactivation (for $/go > 0.5)

Substitute h($/To) for h(S/So) in the above expression.

IV The reactor point effectivenesscan now be readily determined from the overall pellet effectiveness developed earlier together with the known time dependence of the change of active surface area. Under the assumptions of negligible external mass transfer resistance and an isothermal pellet, the reactor point effectiveness is simply the pellet effectiveness multiplied by ( k / k b ) , where k, and kb are the rate constants evaluated at the pellet surface and bulk-fluid temperatures:

Diffusion-affected reaction and shell-progressive chemical deactivation

(-AH)Dt(l + (Bi)hy)(Cb C d ) B = exp {(A)[ ~ RTb Q h ~ T b L y (1 (Bi)h y)(-AH)De(Cb

+ +

- G)

The following relationships are used for y and the surface area ratios:

The reactor point effectivenesses thus obtained for various cases are summarized in Table 6.4. Take as an example the first entry in Table 6.4, which is for diffusionfree reactions. Since the only transport resistance to consider is in heat transfer across the pellet-bulk fluid interface, the reactor point effectiveness is simply given

'(St/SO)]

by:

crystallite migration For the entry 111, and

Since the ratio of rate constants can be approximated by:

atom migration

s/%= &/So d

-(S/So)p = - -1 Q"

4' = 5 For the entry IV, -

Q and y = 1 dt

225


and the temperature ratio in Eq. 6.75 is given by:

which results from the heat balance across the pellet-bulk fluid interface, use of Eqs. 6.75 and 6.76 in Eq. 6.74 yields the first entry in Table 6.4. While the approximation of Eq. 6.75 has been used for the first three entries in the table, this approximation has not been made for the reactions affected by shell-progressive chemical deactivation. The only difference when compared with the reactor point effectiveness obtained in Chapter 5 for this case is the inclusion of ($/So)1'2 in the expression for (Cb - Cd). The expressions for the time dependence of y and surface area ratios, which are necessary for the calculation of the reactor point effectiveness, are also given in Table 6.4.

Summary

With the possible exception of sintering, the physical deactivation of a catalyst is not well understood. Nevertheless, the effects of physical deactivation can still be quantified provided that the time progression of the change of active surface area is known. Because there is a better understanding of sintering compared with the other types of physical deactivation, emphasis has been placed on sintering. The theory behind the sintering kinetics is perhaps as sound as that behind the chemical kinetics of heterogenwus reactions. The weakness is in the application of the sintering kinetics derived for flat substrate to a porous ed catalyst. A similar weakness in chemical kinetics is the treatment of a heterogeneous surface as homogeneous. In many reactions, sintering is as important as chemical deactivation. For those reactions affected by both sintering and chemical deactivation, reactor point effectivenesses have been developed for use in reactor design, which is treated in Part 111.

NOTATION a

b

bu (Bi)h c b

ct (2

distance for a single jump number of coordinate directions in which jumps may occur rate of collision between crystallites composed of i and j units Biot number for heat bulk fluid concentration of key species pellet center concentration concentration at the boundary between completely deactivated outer region and fresh inner core in pellet

227

crystallites of i, j atoms per surface area fraction of ad-atoms on surface diffusivity of crystallite effective diffusivity of key species in catalyst pellet surface diffusivity of atom or molecule surface diffusivity of ad-atom on substrate pre-exponential factor for the surface diffusivity Dl relative diffusivity of a crystallite i with respect to that of a crystallite j overall pellet effectiveness defined by RG/rc E when sintering is absent binding energy between ad-atom and substrate energy required to detach an atom from crystallite crystallite size distribution jump frequency concentration dependence of intrinsic rate of reaction product inhibition term dependence of catalyst activity on surface area ratio film heat transfer coefficient heat of reaction integral defined in Eq. 6.6 number of atoms crossing the crystallite-substrate interface per time number of atoms attaching per time at crystalliteperimeter by surface diffusion reaction rate constant Boltzrnann constant reaction rate constant at the boundary between completely deactivated outer shell and fresh inner core in a pore-mouth poisoned pellet rate constant for poisoning reaction reaction rate evaluated at pellet surface temperature rate constant defined in Eq. 6.50 screening distance at which n = fi; length of the part of the pellet completely deactivated characteristic dimension of catalyst pellet; screening distance surface concentration of ad-atom; constant n at crystallite edge n for crystallites of c~iticalsize number of atoms making up a crystallite number of jumps concentration of edge atoms associated with a crystallite n in equilibrium with infinite-sized crystallite total number of crystallites poison capacity of catalyst pellet crystallite size critical crystallite size

PHYSICAL DEACTIVATION AND SINTERINC

R R, RG RP

intrinsic rate of chemical reaction intrinsic rate of poisoning reaction distance between two atoms; radial coordinate gas constant global rate (actual rate) global rate of poisoning reaction

Rfi

rf

S

exposed surface area of catalyst crystallite S for fresh catalyst active surface area total exposed surface area at time t time temperature bulk-fluid temperature pellet surface temperature volume of an ad-atom net distance covered by atom pellet coordinate

rc rp

so

& st

t

T Tb

TS vm

X z

s

i 5

Superscript

+ rr

average; at pellet surface 5

8 1

I

f

1

I

i.

i

I

Creek Letters

+ cos3 8)/4 (see Figure 6.5)

a1

shape factor for volume given by (2 - 3 cos 8

P

jump frequency for detachment of atoms from crystallites jump frequency for attachment of atoms over the crystallite-substrate interface fraction of catalyst deactivated quantity defined in Eq. 6.42 generalized effectiveness factor reactor point effectiveness defined in Eq. 6.73 chemical potential p for infinite-sized crystallite vibration frequency at crystallite edge sites vibration frequency on substrate sites fluid density surface energy Thiele modulus Thiele modulus for poisoning reaction quantity defined by Eq. 6.57 generalized modulus in Eq. 6.7 modified +G defined by Eq. 6.7

P' Y E

rlc

A P Elg VP VO

P

u

4) +P

4% 4~ ( 4 )m~

at the boundary between completely deactivated outer shell and fresh inner core in catalyst pellet pellet surface

d I

229

PROBLEMS

6.1. The intrinsic rate of a catalytic reaction measured using fine powders is lo-' rnole/(min volume of powder). These fine powders are compressed into pellets and the observed rate changes to one tenth of the intrinsic rate under identical reaction conditions. After 100 hrs of use, the observed rate is 5 X lowprnole/(min volume). What is the ratio of active catalyst surface area to the initial active surface area? Is the assumption of negligible effects of physical deactivation on the concentration profile justified? If so, what is E/En? 6.2. Show for a structure-insensitive, first-order reaction that the last entry in Table 6.1 reduces to:

!

and Eq. 6.12 to:

when the reaction is diffusion-limited. Show also that E/& approaches unity when the reaction is severely diffusion-limited and the extent of shell-progressive poisoning is significant. Give physical reasons for this behavior. The dominance of shell-progressive chemical deactivation over physical deactivation should be clear from this example. 6.3. If the root mean square distance is used for the distance that a crystallite travels, Eq. 6.24 reduces to:

Subscripts

bulk fluid pellet center

where Dc is the crystallite diffusivity. Suppose that the diffusivity of the crystallite is related to the surface diffusivity of an ad-atom by:

PHYSICALDEACTIVATION AND SINTERINC Dc = 0.3 Di ( r / ~ ) ~ ;a = interatomic spacing, D: = 260, akcallRT (cm2/sec), how far does a 2-nmradius crystallite travel in 1 hour at 770°K? The data are those used by Richardson and Crump (1979) for their study of nickel on silica. 6.4. Chen and Schmidt (1978) studied the sintering of Pt on amorphous silica. The average size was in the range of 5 to 15 nm at 1000°K. Calculate the range of E (Eq. 6.42). Use u of 2100 erg/cm2. 6.5. Show that when E < 0.1, the approximation of Eq. 6.41 is almost the same 1 l/p) in the range of 0.4 < p < 2.0 where p = r/F. The typical as ~ ( size distribution af crystallites belongs to this range of p. 6.6. Sintering data in of average crystallite size obtained at 920°K for Pt on silica are given below:

If D; is given by D[,= 300 e-"

Average Size (nm)

23 1

X' = evaporation-condensation coefficient b E , =energy required to transfer a molecule from the substrate to the vapor phase m' = mass of an oxide or a chloride

Piq = Kw(Pgas) for M ( s )

+

q02

*" MOzq(g)

The quantity (DlP//3') can be written as:

Substituting Eqs. (3) and (4) into (2), one has:

Time (hr) b. Show that Eq. (I), when used in the number baiance equation, yields:

6.8. Show that Eq. 6.64 leads to: If sintering takes place via the mechanism of atom migration and F = rave,,,,/ 1.03, determine the sinteriig kinetics for the average crystallite size. Can you tell the difference between the case of surface-diffusion controlling and that of interface-transfer controlling? 6.7. Derive the result given in Table 6.3 for the case of surface diffusion aided by the formation of oxides or chlorides. The growth rate, when the surface diffusion controls, is given by:

a. Suppose the metal atoms are transported over the substrate in the form of oxide or chloride molecules. The that might be significantly affected by this assumption are (DIP/Pf) and n? The quantity n;q for the molecules can be obtained by equating the vaporization and return fluxes of oxide or chloride molecules:

a. Show that:

since S = bZ where primed quantities refer to the volatile oxides or chlorides and

= v/B(t) and n(v,t) =

Use the transformation b. Show that:

t)dv

v2/3n(v,

where b2 is a geometric shape factor.

h(t)q(r]).

PHYSICALDEACTIVATION AND SINTERING 233

c. Show then that:

where

dS =

- ~ ~ ' - 3 m '

K is a composite of constants.

6.9. Determine the sintering kinetics for the following data: T(OC)

SJSo

d. Repeat the calculation for a Thiele modulus of 1 this time and plot both results on a A versus &/So plane. What can you conclude (note that C, = C ~ / C O($c)m)? S~ 6.12. How does one calculate the reactor point effectiveness for a reaction affected by shell-progressive deactivation (fourth row in Table 6.4) at t = 01 Note that y = 0 at t = 0 and therefore, A becomes infinite at time zero.

Time (hr) REFERENCES

Can you determine whether the sintering takes place via crystallite migration or atom migration? 6.10. Given the sintering kinetics and the kinetics of chemical deactivation, how can one calculate the time dependence of (&/So)? 6.11. Derive the reactor point effectiveness given in the second row of Table 6.4 for a reaction affected by diffusion, uniform chemical deactivation, and sintering.

a. Under the assumptions of an isothermal pellet and negligible external mass transfer resistance, show that the pellet effectiveness is:

b. Use Eqs. 6.73 and 6.75 to arrive at the results. c. Calculate the reactor point effectiveness as a function of &/So for the following conditions:

Assume that the reaction is structure-insensitive.

Chen, M. and L.D.Schmidt, J. Catalysis, 55, 348 (1978). Chu, Y.F. and E. Ruckenstein, Surface Sci., 67, 17 (1977). Dalla Betta, R.A,, R.C. McCune and,J.W. Sprys, Ind. Eng. Chem. Prod. Res. Dev,, 15. 169 (1976). Flynn, P.C. and S.E. Wanke, J. Catalysis, 34. 390 (1974). Gjostein, N.A., in Burke et al. (eds.), Surfaces and Interfaces. Syracuse University Press, Syracuse (1967). Jones, H., Metal Sci. J., 5. 15 (1971). Kashchiev, D., Surface Sci., 86, 14 (1979). Lee, H,H., J. Catalysis, 63. 129 (1980). Lee, H.H., Chern. Eng. Sci., 36, 950 (1981)# Lee, H.H. and E. Ruckenstein, Cat. Rev. (1983). Lifshitz, I.M. and V.V. Slyozov, J. Phys. Chem. Solids, 19, 35 (1961). Richardson, J. T. and J.G. Crump, J. Catalysis, 57, 417 (1979). Ruckenstein, E. and B. Pulvermacher, AIChE J., 19, 356 (1973). Ruckenstein, E. and D.B.Dadyburjor, Thin Solid Films, 55, 89 (1978). Somorjai, G.A., Principles of Surface Chemistry, Prentice-Hall, Englewd Cliffs. N.J. (1972),

Wynblatt, P. and N.A. Gjostein, Prog. Solid State Chem., 9, 21 (1975). Wynblatt, P., Acta Metallurgica, 24, 1175 (1976). Wynblatt, P., in Bourdon (ed.), Growth and Properties of Metal Clusters, p 15, Elsevier Co., Netherlands (1980).

CHAPTER

7 Multiphase Reactions

7-1 INTRODUCTION The reaction between gaseous and liquid reactants catalyzed by a solid is called a multiphase reaction in the context of heterogeneous catalytic reactions. A multiphase reactor is usually used when the liquid reactant is too nonvolatile to vaporize. For instance, a reactor of this type is used to hydrogenate petroleum feeds of wide boiling-point range for desulfurization, cracking, or straight hydrogenation. A fraction of these feeds has too high a boiling point to vaporize. The two most common reactors for multiphase reactions are trickle-bed and slurry reactors. The trickle-bed is a fixed bed packed with catalyst pellets to which both gaseous and liquid reactants are fed downward cocurrently, usually in the downward direction. The gas phase is continuous throughout the bed, whereas the liquid phase is dispersed as the liquid trickles down the bed. While many different configurations and flow regimes can result depending on the mode of feeding (countercurrent or cocurrent, upward or downward) and the feed rates, interest here is with the downward-flowing trickle-bed operated in the trickle-flow regime where the pellets are completely wetted and covered with a thin liquid film. A slurry reactor is one in which catalyst particles are suspended in a liquid reactant by agitation and gas flow, and the gaseous reactant is fed to the reactor through a suitable distributor located at the bottom. The catalyst particles are very small (-100 rnicron) and the heat capacity of the liquid is high so that isothermal conditions usually prevail. The presence of a liquid phase and a liquid-solid interface in multiphase reactors results in added transport resistances. For instance, the effective diffusivity to in liquid-filled pares (of the order of cm2/sec) is much smaller than that in gas-filled pores (of the order of lod2cm2/sec). The solubility of the gaseous reactant is an important factor since the gaseous reactant has to be dissolved into the liquid reactant for the reaction to take place on the catalyst surface. As emphasized in Chapter 4, the Biot number for heat is much larger than the Biot number for mass for liquid-solid systems; the opposite is true for gas-solid systems. Therefore, the major external resistance lies in the mass transport, and the pellet is not necessarily isothermal. In many cases, however, the equilibrium gas concentration in the liquid is quite low and, thus, the heat evolved is small in spite of high heats of reaction. The pellet can be considered isothermal in such a case,

as in the hydrodesulfurization of heavy petroleum feeds. On the other hand, high concentrations and heats of reaction as in the hydrocracking of petroleum fractions can result in nonisothermal pellets. The catalysts used for multiphase reactions are not immune from catalyst deactivation, typically caused by coking and metals deposition. High molecular weight organornetallic compounds present in petroleum feeds, in particular petroleum residues, split off metals (typically Ni and V) upon cracking, eventually leading to pore-blocking. In this chapter, the effects of transport resistances on the rate of reaction will be examined, so that the global rate suitable for direct inclusion in reactor conservation equations can be derived. Because of the important role that the mass transport plays in the overall kinetics, this subject will be taken up first. 7-2

where the function F relates Gg to CfL. If Henry's law applie, this reduces to:

The resistance at the gas-liquid interface is dominated by the liquid side of the interface. When the gaseous reactant is pure, as in hydrogenation, there is no resistance to diffusion from the bulk gas to the gas-liquid interface. Even when the gas phase is a mixture, the gas-side transfer coefficient k, is much larger than k L / H . Also, for slightly soluble gases, the resistance to mass transfer is predominantly on the liquid side. Therefore, Cigcan be set equal to the bulk concentration C, and the above relationship can be written as:

MASS TRANSFER COEFFICIENTS

The mass transfer resistances in a multiphase reactor at the gas-liquid and liquidsolid interfaces are shown in Figure 7.1 (Smith 1981). In the case of a slurry reactor, the continuous gas-phase shown in the figure should be replaced by a gas bubble. If equilibrium exists at the gas-liquid boundary,

-

/

I

1

13

I

I

I

where aL is the gas-liquid interfacial area per unit volume of liquid (bubble-free in the case of slurry reactor), and a, is the reciprocal of the characteristic length of a pellet, E , is the ratio of catalyst volume to liquid volume, and kL and k, are the appropriate mass transfer coefficients. Several correlations (Calderbank and Moo-Young 1961; Misic and Smith 1971; Juvekar and Sharma 1973) for the mass transfer coefficient kL in a slurry reactor are available. One correlation (Calderbank and Moo-Young 1961) that s for the energy dissipation due to agitation by the impeller gives:

Gas-liquid interface

Liquid

Catalyst particle

!

I

Under these conditions the global rate based on catalyst volume can be written as:

where DA = molecular diffusivity of reactant in liquid, cm2/sec v = kinematic viscosity, crn2/s 0 = energy dissipation rate per unit mass of liquid, erg/s0g

where Dl = impeller diameter, cm

N = impeller speed, rps W = mass of liquid in slurry, g Distance

Figure 7.1 Concentration profiles in a multiphase reactor. (After J. M. Smith, Chemical Engineering Kinetics, 3rd ed., @ 1981; with permission of McGraw-Hill Book Company, New York.)

P

Np =(- 10) PLN~D:

4 =a

P = power input, erg/$

correction factor (0

< + < 1) to

for the decrease in the

energy dissipation rate due to gas bubbles. For Q/ND: < 0.035, = 1 - 12.6 Q/ND; where Q is the gas flow rate (crn3/s) p~ = liquid density

+

In the absence of mechanical agitation and for bubbles whose diameter is less than 2.5 mm (the usual size range for slurry reactors), the following correlation (Calderbank 1958) is available:

where Ap = difference in density between liquid phase and. gas bubbles, g/cm3 pL = viscosity of liquid phase, g/cm s g = gravity, cm/s2 pL = density of liquid, g/cm3 This correlation is for bubbles rising through the liquid phase because of a gravitational force. The basis for correlating k, as a function of agitation speed and particle size is Kolrnogoroff's theory (Levins and Glastonbury 1972) of isotropic turbulence. According to this theory, the Reynolds number (Re) is defined in of the energy dissipation rate, given by:

Figure 7.2 Correlation for the mass transfer coefficient, k, (After J. M. Smith, Chemical Engineering Kinetics, 3rd ed., @ 1981; with permission of McGraw-Hill Book Company, New York.)

where DA = molecular diffusivity of the diffusing component, cm2/s GL= superficial mass velocity of the liquid, g/cm2-s p~ = liquid viscosity, g/cm s kLaL= liquid side mass transfer coefficient, l/s a~= a correction factor (-7n~-2(crn)),and n~ = 0.4 for granular particles.

where the eddy size 6 is defined by:

Here, the energy dissipation rate rr is given by Eq. 7.6, and d, is the particle size. The experimental data (Harriott 1962; Brian et al. 1969) for k, are well correlated by plotting the Sherwood number (kcdp/DA) against the Reynolds number given by Eq. 7.8, as shown in Figure 7.2. For trickle-beds, correlations for kLaL are in of the pressure drop in the reactor or the flow velocities. Illustrative of the latter type is the correlation by Goto and Smith (1975):

This correlation applies to the trickle-flow regime; much higher values can be obtained at higher liquid and gas rates. The trickle-flow regime is usually for GL < 3 X 103 g/cm3.h. A j factor correlation (Dharwadkar and Sylvester 1977) in the trickle-flow regime for the liquid-particle mass transfer coefficient k, is:

where jD = -

219

UL= superficial velocity of liquid at = total external area of particles per unit volume of reactor R ~= L dp Uf,p L /pr,

It is notable that both kL and k, in the trickle-flow regime arc independent of the gas flow rate. Sattefield and coworkers (1978) also gave correlations independent of the gas flow rate in the trickle-flow regime. Further details on mass transfer coefficients can be found in the book by Smith (1981), from which the bulk of this section is derived.

7-3 TRANSPORT EFFECTS AND GLOBAL RATES As pointed out earlier, the major external resistance is that of mass transfer, and therefore, the effect of external heat transfer can be neglected. Furthermore, internal (intraparticle) transport effects can be neglected in slurry reactors except under some unusual reaction conditions since the size of the catalyst particles is of the order of 100 microns. In trickle-beds, however, both the internal heat and mass transport effects can be important. Consider first the slurry reactor. Since the internal resistances are negligible, the global rate can be written directly in twms of surface concentrations, and Eq. 7.4 becomes:

where D, is now the effective diffusivity of gas in liquid-filled pores. The global rate can be expressed as:

As in the slurry reactor, these expressions give for the global rate:

where the intrinsic rate is represented by kflC). It follows from Eq. 7.13 that:

where C.is the solution of:

Equation 7.20 can be solved for Cs.This value of C, in turn is used in Eg. 7.19 for the global rate. For a first-order reaction, Eqs. 7.19 and 7.20 can be combined to give: tanh

R'(Ceq)= [ I

Given the intrinsic rate expression kf(C), Eq. 7.15 can be solved for the surface concentration Cs,which upon inserting into Eq. 7.14 yields the global rate in of C e , the rate constant, and the mass transfer coefficients. A pseudo-firstorder rate expression is often adequate when the gaseous reactant dissolves slightly in the liquid reactant. In such a case, the global rate is simply:

which follows from Eqs. 7.14 and 7.15. The relative magnitude of the reciprocals of the rate constant and mass transfer coefficientsdetermines the relative importance of the three resistances. If one of the mass transfer coefficients is much smaller than the rate constant, for instance, the observed rate would appear to be almost independent of temperature, the controlling step for this case being the mass transfer resistance. The existence of internal resistances complicates the analysis of transport effects for trickle-beds since the pellet cannot necmsalily be assumed isothermal. Reactions in which the heat effect is negligible are considered first, and the case of a nonisothermal pellet will be treated in the following section. For arbitrary kinetics kf(C), the internal, isothermal effectiveness factor (Chapter 4) is:

+/+

+ + tanh +/B.L

where the Biot number for mass in of liquid-phase properties is given by:

and the Thiele modulus 4 is L\'k/D,)"*. Here, the characteristic pellet length L is equal to l / ~ .As detailed in Chapter 4, the center concentration Cc in Eq. 7.17 can be set to zero in the asymptotic region of strong diffusion effects, i.e., +c 3 3 where the generalized modulus is given by:

Otherwise, C, is approximated by C ~ / C O+G. S~ While the treatment so far has been based on the assumption of completely wetted pellets, there are cases where the pellets are partially wetted. It has been found (Sattefield 1975; Herskowitz et al. 1979) that at low liquid velocities, a fraction of the pellet surface is not covered by liquid, i.e., partial wetting. This does not mean that the pores are free of liquid. The pores are still filled with liquid due to capillary forces, but the pore-mouth is in direct with gas

rather than liquid. This partial wetting can take place when the Reynolds number is less than about 30. The fraction of the surface defined by Re = L p L U ~ / p L covered by liquid, q, is known to range from 0.6 to 1. It has been found (Tan and Smith 1980) that a linear combination of effectiveness factors gives a satisfactory result for the partially wetted pellet:

where fl = (-AH)DeCL/A Tb, and

E,

is the pellet porosity.

7-4 GLOBAL RATE FOR TRICKLE-BEDS: where the liquid-side overall effectiveness factor r l ~and the gas-side counterpart 9~ are given, for instance, for a first-order reaction by: qr, =

qc =

+/+ tanh +/+

tanh

1 -1-

1

+ tanh +/BmL tanh

+ 4 tanh +/HBmc =-

+

4

, 1

A pellet may not be isothermal when heat effects are significant as in hydrocracking. While complications may arise due to the vaporization of liquid, it will be assumed in what follows that the pellet is completely wetted. Steady-state, one-dimensional balance equations for a slab-like pellet can be written in dimensionless form as:

(7.26)

The gas-side Biot number for mass (HB,c) is usually very large so that its effect can be neglected as shown in Eq. 7.26. The net effect of partial wetting is a higher observed rate than would be expected from completely wetted pellets under identical conditions. Based on this fact, a criterion (Lee and Smith 1982) for partial wetting has been derived for a first-order reaction:

R'd2pp D&L

NONISOTHERMAL PELLETS

du d v with - =- = O dx dx

at x = O

4 tanh 4

+ $ tanh +/BmL

where RA = global rate per unit weight of pellet CI,= bulk concentration of gaseous reactant in liquid pp = pellet density

where u = C/CI.,v = T/Tb,x = z/L,BmI,= K[,L/D, BIIL= hL/h, F(u,v) = rc(C, T)/r,(CI,,Tb) = f ( ~ exp(y(1 ) - I/v)) +2 = ~ ~ ( C L , T ~ ) L ' / fl DC =L (-bH)Dec~/hTb, ~ Y = Ea/RgTb

If the indicated condition is satisfied, the pellets are partially wetted. The criterion, however, cannot say anything about the value of q. There appear to be no accurate correlations available for the prediction of q. An additional complication arises

The notation can be found at the end of this chapter. Here, KL is the overall

when a fraction of the liquid is vaporized due to the heat of reaction, as in hydrocracking of petroleum feeds. In such a case, the pores are filled with both liquid and vapor. This problem has been considered by Dudukovic (1977). The criteria of negligible transport effects are useful for the determination of intrinsic kinetics. Sattefield and coworkers (1969) give the following criterion for negligible mass transfer resistance:

where KI,is the overall liquid-side mass transfer coefficient. A tighter criterion by Lee and Smith (1982) for exothermic reactions is:

(7.34)

mass transfer coefficient defined by:

The major interest here is with an approximate expression for an overall effectiveness factor. Since a general result cannot be obtained for arbitrary kinetics, attention is restricted to a first-order reaction for which F = u exp(y(1 - l/v)). The perturbation method of Pereira and co-workers (1979) is used for this purpose, but with a modification: the boundary condition of Eq. 7.33 for v is retained as such for the liquid-solid system under consideration, for B ~ is L much larger than unity and fl is much smaller than unity (0.1 -- 0.001) such that BhL/P will assume a large value. In the work of Pereira and co-workers, the boundary condition is transformed to (BhL/P)e(l- v ) for the gas-solid system for which BRL/flis finite.

Here, c is a small perturbation parameter equal to 8. Asymptotic expansions of the functions involved yield:

where

and for the

:

where the prime denotes differentiation with respect to x. The zeroth-order solutions of Eqs. 7.40 and 7.41 are:

uo = A cosh +x; and Fu(uo,vo)is aF/au evaluated at (uo,vo), etc. From the definition of effectiveness factor, it follows that:

A =

Brn~

4 siih 4 + BmL C O S ~4

(7.49)

It follows from Eq. 7.39 that: tanh 4

'O

and

+(I

+ 4 tanh + / B r n ~ )

which is the isothermal effectiveness factor. The solutions of Eqs. 7.44 and 7.45 give the first-order corrections ul and vl which are:

V1

When these expansions are substituted into Eqs. 7.30 and 7.3 1, the of equal order in e gives for the EO :

'O

=

+

1 - C O S ~$x/cosh 4 tanh +/BhL 1 4 tanh +/B,L

+

where the constants Ai are defined in Notation. The normalized temperature can be written as:

Eq. 732 gives an estimate of the maximum temperature difference in the pellet:

Av = v - 1 = EAv, or ( Av),,

= fi(Avl),,

V ~ ( X= 1) = ( v ~- )(vl)rntn ~ ~ 1 l/cosh 4 1 -1- 4 tanh +/BmL

( A v I )=~ V~I ( X= 0)

It is seen that the maximum temperature deviation from the bulk-liquid temperature can be approximated by:

(A v

(A T)m, Tb

) =~ ~ =

P(1- 1/~0sh+) 1 tanh +/B,L

++

where (AT),, = Tc - Ti,,and Tc is the pellet center temperature. The fractional maximum temperature rise in the pellet is at most fi according to Eq. 7.55. Therefore, the value of /3 should be sufficient to tell whether the pellet can be treated as isothermal. For instance, if fl is 0.01, the maximum possible temperature rise in the pellet is of the order of 1% of Tb. Typical range of fi is 0.1 to 0.001 for liquid-solid systems. A first-order correction for q can be obtained from:

which follows from Q s . 7.39, 7.44, 7.46, and 7.47. The expression for ul at x = 1 is complex, however, this reduces considerably for 4 3 3 and q l becomes:

P, and that regarding the maximum fractional error in using the isothermal effective-

+

ness factor, which is of the order of &/[2(1 +/BmL)2], should be useful in assessing the suitability of the assumption of an isothermal pellet for complex reactions. Summary

The global rates for multiphase reactions, in which both gaseous and liquid reactants are involved, can be obtained in a straightforward manner for slurry reactors owing to the lack of internal transport resistances. In const rast to gas-phase reactions, the external mass transfer resistance can have a significant effect on the rate of multiphase reactions. In addition, the internal transport resistances can be significant in trickle-beds. For the trickle-beds, similar results as in gas-phase reactions can be used with appropriate redefinition of dimensionless quantities, if the pellet is isothermal. On the other hand, the global rate cannot readily be obtained for a nonisothermal pellet. Nevertheless, an approximate expression for the global rate can be obtained for a first-order reaction. These results provide an order of magnitude estimate of the maximum temperature rise in a pellet and of the error involved in using the isothermal effectiveness factor for rate expressions more complex than first-order. NOTATION

Therefore, the overall effectiveness factor for

-

1 @(I + 4 / B m ~ )[ l

+

py(1/3 +/BhL)] @(@) 2(1+ + / B ~ L ) ~ +

+

4 3 3 can be expressed as for 4 2 3 (7.58)

In view of the fact that BhL 9 BmLfor liquid-solid systems, the maximum fractional error in using the isothermal effectiveness factor is of the order of By/[2(1 B ~ L )for ~ ] 4 a 3. An approximate expression for the global rate of a first-order reaction can be written directly from Eq. 7.58:

+ +/

+

For < 3, Eqs. 7.51 and 7.56 can be used for TI, which together with q o gives a near-accurate value of q, and therefore, of Rc. While specific results have been restricted to a first-order reaction, the result regarding the maximum temperature rise in the pellet, which is of the order of

liquid-solid interfacial area per unit volume of catalyst, reciprocal of characteristic length L gas-liquid interfacial area per unit volume of liquid l/(cosh 4 4 sinh 4/BmL)

+

yA2/2

++

+

+

++

(BmL(A1 A3 sinh A 4 cosh 24) As(sinh 4 cosh 2A4(4 sinh 2+)/A5 +2Ay(cosh 4' 4 sinh 4/BhL)/2 -A i/3 -(4 sinh 4 BmLcosh 4 ) hL/h &LID, KLL/D, concentration of gaseous reactant gas-phase reactant concentration pellet center concentration bulk gaseous reactant concentration in liquid gaseous reactant concentration at liquid-solid interface gas side interfacial concentration liquid side interfacial concentration gaseous reactant concentration in liquid in equilibrium with C,

+ +

4) +

particle size or diameter molecular diffusivity of gaseous reactant in liquid effective diffusivity of gaseous reactant in liquid filled pore activation energy rc(C T)/rc(G, Tb) supeficial mass velocity of liquid Henry's constant in concentration unit rate constant individual mass transfer coefficient at liquid-solid interface gas-side individual mass transfer coefficient at gas-liquid interface liquid-side individual mass transfer coefficient at gas-liquid interface liquid-side overall mass transfer coefficient given in Eq. 7.22 characteristic length of pellet, volume/external surface area fraction of external pellet surface covered by liquid gas constant global rate in volume unit global rate in weight unit Reynolds number defined by Eq. 7.8 Reynolds number defined in Eq. 7.12 temperature liquid bulk temperature pellet center temperature

PP a

pellet density energy dissipation rate given by Eq. 7.6 Thiele modulus for first-order reaction (L(k/De)112) or rc(C+,Tb)L2/DG Thiele modulus for pseudo-first-order reaction generalized modulus given by Eq. 7.23 kinematic viscosity of liquid

4 $1

+G VL

4 2

=

PROBLEMS

7.1. Obtain an expression for the global rate in of C,, and transfer coefficients for a second-order reaction taking place in a slurry reactor. 7.2. Obtain an expression for the global rate in of Ceqand transfer coefficients for a second-order reaction taking place in a trickle-bed. The typical value L from 1 to 10. Suppose that k = 10-5 (s rn0l/crn3)-~, C,, of B ~ ranges = mol/cm3, L = 1 cm, D, = 10-%cm2/s, and the pellet is isothermal. What are the values of the ratio, RG(Ceq)/kC&, for B,,, of 1 and 101 How would the results change if the characteristic length L is changed by a factor of 5 and 0.51 Use Eqs. 7.22 and 7.23 if necessary. 7.3. An estimate of the maximum fractional error ef in using the isothermal effectiveness factor in a trickle-bed is given for a first-order reaction by:

C/C' T/Tb zeroth and first order value of u in asymptotic expansion of u zeroth and first order value of v in asymptotic expansion of v pellet coordinate normalized with respect to L Creek Letters

at

P

a correction factor in Eq. 7.10 (-bIf)DeC~/hTb

T.

Y

Ea / R g

E

a small perturbation parameter equal to /3 ratio of catalyst volume to liquid volume pellet porosity internal effectiveness factor given by E$. 7.17 gas-side effectiveness factor liquid-side effectiveness factor overall effectiveness factor defined by Eq. 7.24 zeroth and first order value of 7 in asymptotic expansion of q effective heat conductivity liquid viscosity liquid density

68

EP

rlr '7c

"7~ 'lo

'70, 7' 1

PL PL

This result can be used for arbitrary kinetics for an approximate estimate if is replaced by given by Eq. 7.23. A typical value of the Arrhenius number y is 30, and the upper bound of /3 is approximately 0.1. From the data given in Problem 7.2, first calculate C, for the second-order reaction, and then calculate +c. For what value of BmL does the fractional error become less than 0. l? 7.4. Derive a criterion of negligible transport effects for a first-order reaction taking place in trickle-bed. Note that one only needs to replace C,, in Eq. 7.21 by CL if k c L is taken as the intrinsic rate. First obtain the criterion for an isothermal pellet and then for a nonisothermal pellet. 7.5. Derive the first-order correction of the effectiveness factor 71 for t# 2 3 (Eq.7.57). Also obtain q l when 4 is less than 3, i.e., when the pellet center concentration is not close to zero.

+

REFERENCES Brian, P.L.T.,H.B.Hales and T.K. Sherwood, AIChE J., IS, 419, 727 (1969). Calderbank, P.H. and M.B.Moo-Young,Chem. Eng. Sci., 16, 39 (1961).

Calderbank, P.H., Trans. Inst. Chem. Eng. (London), 36, 443 (1958). Dharwadkar, A. and N.D. Sylvester, AIChE J., 23, 376 (1977). Dudukovic, M.P., AIChE J., 23, 940 (1977). Goto, S. and J.M. Smith, AIChE J., 21, 706 (1975). Harriott, P., AIChE J., 8, 93 (1962). Herskowitz, M., R.G. Carbonell and J.M. Smith, AIChE J., 25, 272 (1979). Juvekar, V.A. and M.M. Sharma, Chem. Eng. Sci., 28, 825 (1973). Lee, H.H. and J.M. Smith, Chem. Eng. Sci., 37, 223 (1982). Levins, D.M*and J.R. Glastonbury, Chem. Eng. Sci., 27, 537 (1972). Misic, D. and J.M. Smith, Ind. Eng. Chern. Fund., 10, 380 (1971). Pereira, C.J., J.B. Wang and A. Varma, AIChE J., 25, 1036 (1979). Satterfield, C.N., AIChE J., 21, 209 (1975). Satterfield,C.N., M.W. van Eek and G.S. Bliss, AIChE J., 24, 709 (1978). Satterfield, C.N.,A.A. Pelossof and T.K. Sherwood, AIChE J., 15. 227 (1969). Smith, J.M., Chemical Engineering Kinetics, 3rd ed., McGraw-Hill, New York (1981). Tan, C.S. and J.M. Smith, Chem. Eng. Sci., 35, 1601 (1980).

CHAPTER

8 Selectivity and Stability

The selectivity and yield of a desired product is of major interest in multiple reactions. In order to assess the effects of transport processes and catalyst deactivation on the selectivity and yield, discussion shall be confined to simple intrinsic kinetics, for the qualitative behavior of complex reactions is often similar to that of, for instance, first-order reactions. This qualitative behavior of the selectivity and yield as affected by transport resistances and catalyst deactivation will be treated in the first part of this chapter with the understanding that the same approach, when coupled with numerical methods, can lead to quantitative results for any system. An excellent treatment of multiple reactions can be found in the book by Aris (1975). The second part of this chapter deais with the stability of a single pellet in a fluid stream. This stability represents the stability of a reaction at a point in the reactor. This part deals with two types of stability: one in which the stability around known steady states is of concern, and the other in which the knowledge of steady states is not required. For the first type, uniqueness and multiplicity of the solution to the steady state system equations is of interest; for the second, only the stability of the system is of concern. Perhaps, the most difficult part of the stability problem lies in determining whether the equations describing a system adequately represent the physical process. Two different sets of equations describing the same system could give conflicting conclusions on stability. 8-1 SELECTIVITY AFFECTED BY TRANSPORT PROCESSES Consecutive first-order reactions provide a basis for assessing the effects of diffusion on selectivity. For the following isothermal, consecutive reaction network:

steady state mass balances give:

252

f \


SELECTIVITY

AND

STABILITY

253

I

where 4: = L 2 ( k / D i ) for i = A,B, y = De/DA,and the length coordinate is normalized. Here, L is the ratio of volume to external surface area. Boundary conditions are: CA= A, CB= B, on the pellet surface

which is the point yield for homogeneous reactions as represented by the intrinsic rates r* (i = A,B) indicated above. In the other extreme region of strong diffusion limitations, qi+f approaches unity, and Eq. 8.10 reduces to:

(8.3)

The linear equations 8.1 and 8.2 can be solved using the principle of superposition:

A comparison between Eqs. 8.1 1 and 8.12 gives the effect of diffusion on yield. A catalyst would be chosen to maximize the yield of the desired product B. As apparent from Eq. 8.1 1, this implies that s = (kA/ke) would be much greater than unity. For a large value of s, the first term in the right hand side of Eq. 8.12 approaches unity. Therefore, the relative magnitude of yields for this case can be expressed from w s . 8.1 1 and 8.12:

in which ui (i = A,B) is the solution of:

V2uf = +;ui ur = 1 on the pellet surface Substitution of Eqs. 8.4 through 8.7 into Eqs. 8.1 and 8.2 yields:

3 (8.8)

1i I

where s = kA/kB. The point yield in the reactor is given by:

1 I

(8.9)

where n is the outward normal length coordinate. Substitution of Eqs. 8.4 and 8.5 into Q. 8.9 with the known parameters yields:

When the diffusional resistance is negligible, the internal effectiveness factors qt approach unity, and Eq, 8.10 becomes:

which is less than unity for nominal values of the effective diffusivity ratio. It is seen then, that the point yield decreases with increasing diffusion limitations. On the other hand, in rare instances in which s is less than unity, the yield can increase with increasing diffusion limitations. It is noted in this regard that physically -(RC)B/(RC)A 2 0 and therefore, the second terrrk in Eqs. 8.1 1 and 8.12 is less than or equal to the first term. The undersirable effect of diffusion when s(= k ~ / f c>~ 1) is clearly demonstrated in Figure 8.1 for the consecutive dehydrogenation reactions of butene (Voge and Morgan 1972). For a plug-flow reactor in which the effects of external transport resistances are negligible, the point yield is equal to (-dB/dA), where A and B are bulk fluid concentrations. Therefore, an integration of Eq. 8.10 with respect to the dependent variables A and B gives the reactor yield, since -(Rc)s/(Rc)n = (-dB/ dA), and As = A and Bs = B for this plug flow reactor, yielding (Carberry 1966):

1 1

where the subscripts f and L denote the concentrations at the inlet and outlet of ]A the reactor, respectively, and p = Y ~ B + ; / ( ~ 4;). For parallel reactions with a common reactant: A

A

B

(nth order)

C

(m th order)

kc

SELECTIVITY AND STABILITY

255

It is clear from Eq. 8.16 that the relative point selectivity is smaller than unity when n > m and larger than unity when n < m since (CA/A,) 5 1. Therefore, the point selectivity increases with decreasing diffusional resistance when n > m; it increases with increasing diffusional resistance when n < m. The relative point selectivity for the following independent parallel reaction network in which B is the desired product: B

A

kc X ---+ C

(nth order)

(m th order)

can be written as:

The diffusional effect on the selectivity is self-explanatory. It has been shown by Wei (1962) that any complex network of first-order reactions consisting of combinations of consecutive and parallel paths including reversible paths can be decomposed into a system involving only independent parallel reaction paths of irreversible first-order reactions. Let r, be a vector of rate expressions corresponding to a vector of concentrations C for all species. Then:

Particle diameter, mm

Figure 8.1 Selectivity to butadiene at 35% butene conversion for different particle sizes. (Voge and Morgan 1972; reprinted with permission from Industrial and Engineering Chemistry, Process Design and Development. Copyright by American Chemical Society.)

-

the selectivity is of interest only when the orders of the reactions are not the same, for the selectivity would not be affected by diffusion resistance if the orders were the same. For the above reactions, in which species B is the desired product, the point selectivity in a reactor when diffusional effects are absent is:

v~c=K(;c;~,=Kc,K=D-~K/L~ where K is a matrix of rate constants and the spatial coordinate is normalized with respect to the characteristic length L. If A is a diagonal matrix obtained from the matrix by a similarity transformation, one has:

where V is the matrix whose columns are the coordinates of the eigenvectors given by:

When the concentration of species A depends on the pellet coordinate due to diffusion, the point selectivity is:

The eigenvalues are obtained from: det (K where V denotes volume integration. Then, the relative point selectivity defined ~ / ~beAobtained ) directly from Eqs. 8.14 and 8.15: by [ ( R G ) R / ( R G ) A ] / ( ~can

+ h m l )= 0

Then the original balance equation can be written as: A

z

which is now decomposed into independent parallel reaction paths of irreversible first-order reactions. The transfoied concentration Ct is related to the original concentration C through the eigenvectors:


Let us examine the combined effects of these two resistances on sdectivity. A simple way of accomplishing this is to compare the selectivity affected only by diffusion with that affected by both diffusion and external heat transport resistance. Let (Rc)g/(Rc); be the point yield of a consecutive reaction network A B A C based on pellet surface temperature and bulk fluid concentrations. In the absence of external resistance to heat transfer, the surface temperature is equal to the bulk fluid temperature. Therefore, (Rc);/(Rc); is the point yield when only the diffusional resistance is present. For the rate expression r, = kf (composition), the point yield when the reactions are affected by both diffusion and external heat transport, (Rc)s/(Rc)A, is simply:

For such a decomposed system, the gradient at the pellet surface can be written in of the transformed concentrations Ct as:

where q is a diagonal matrix whose elements are the effectiveness factors for the decomposed irreversible first-order reactions. If the global rates are written as RG = K,Ct(l) in which the effective rate constant matrix K, is defined by:

K, = D V A q

257

(8.19)

it follows from Eqs. 8.17 and 8.18 that: where sb and s, are the rate constant ratios evaluated at bulk fluid and pellet surface temperatures, respectively. The rate constant ratio s, can be expressed as:

The point selectivity can readily be obtained from Eq. 8.20. In fact, this technique has been used earlier in examining the effects of diffusion on the selectivity for a first-drder consecutive reaction network. The result obtained by Wei (1962) for a three component system is shown in Figure 8.2. It is seen that the lengths between paths decrease when the reactions are affected by diffusion, indicating that selectivity differences are decreased. This result shows that diffusional effects are usually detrimental to selectivity in the sense that it is more difficult to have products differing in composition. It has been shown in Chapter 4 that major transport resistances under typical operating conditions lie in external heat transport and internal mass transport.

where Ts and Tb are the surface and the bulk temperature and E A and EB are the Arrhenius numbers for kA and kg. Here, the factor 1.2 was introduced for better approximation of the exponential term as discussed in Chapter 4. The temperature ratio is obtained from a heat balance around the external film.

Use of Eqs. 8.22 and 8.23 in Eq. 8.2 1 yields:

I

Figure 8.2 Effect of diffusion on the reaction paths in an integral reactor, equal diffusivities. (Wei 1962; reprinted with permission from Journal of Catalysis. Copyright by Academic Press.)

This ratio for exothermic reactions is larger than unity when EB > E A and smaller when C B < E A ,meaning that the point yield is increased due to the external resistance when E B > E A and decreased when EB < &. It is therefore entirely possible when EB > E A that the combined effects of transport resistances are in fact to increase the point yield. The same conclusions can be made for the point selectivity of the parallel reactions (A --1?-, B, A A C) in which species B is the desired product.

8-2

SELECTIVITY AFFECTED BY DEACTIVATION

The effect of catalyst deactivation on selectivity can be easily assessed for very small catalyst particles such as those used in fluidized-beds, for which the diffusional effect is absent and therefore, deactivation is uniform. A good example is provided by Weekman and Nace (1970) for the catalytic cracking of gas oil (A) into gasoline (G), and dry gas and coke (S) which deactivate the catalyst: A-G

-

- : Predicted : Experimental

-

k1

b.4 S

This lumped version of the network of many reactions involved in the cracking was represented by the following phenomenological rate expressions:

Gas oil conversion (wt.)

where hi (i = 1, 2, 3) is the fraction of catalyst still remaining active after deactivation, and ki is the rate constant for fresh catalyst. Then the point yield can be expressed as:

It is clear from Eq. 8.25 that the point yield is not affected by deactivation if the hi's are the same. This implies that deactivation does not affect the point yield if all the reactions occur on the same sites. On the other hand, deactivation would affect the yield if different reactions take place on different sites. Since the above reaction network is a combination of consecutive and parallel reactions, the same conclusions can be made for simple consecutive and parallel networks. Weekman and Nace (1970) assumed that the hi's are the same, which is represented by:

where a is a constant that would depend on bulk concentration and temperature. The data that they obtained from a moving bed, for which plug flow can be assumed, are compared with the model in Figure 8.3. For plug-flow, - R ~ / R A= -dG/dA and Eq. 8.25 can be integrated to give the predicted curve in the figure. The good agreement shown is an indication that the assumption of hi = h is reasonable. On the other hand, De Pauw and Froment (1975) found for the n-

Figure 8.3 Instantaneous gasoline yield for the catalytic cracking of gas oil: experimental values are for various catalyst residence times. (Weekman and Nace 1970)

pentane isomerization over Pt/A1203 that two side reactions occur on sites different from those involved in the main reaction, resulting in a selectivity that varies with time. While the effect of deactivation alone is rather straightforward to analyze, the combined effect of deactivation and diffusion is not, because of diffusional intrusion into deactivation even when all reactions occur on the same site. Therefore, the selectivity can vary with time regardless of the nature of the active sites. Attention here will be confined to the case in which all reactions occur on the same sites. It has been shown in Chapter 5 that the internal effectiveness factor for a pellet undergoing deactivation is given by:

This equation applies in the asymptotic region of strong diffusion effects, where g is the fraction of external surface activity still remaining active after deactivation, and i, is the intrinsic rate in of surface conditions. This internal effectiveness factor for a deactivated pellet is simply the internal effectiveness factor for a fresh pellet multiplied by Ell2. The restriction placed on the relationship was that the following condition be met:

when all reactions are affected by deactivation. It is also seen that the selectivity for the consecutive network reaches a plateau after poisoning, although it improves slightly with increasing levels of the fraction of catalyst completely deactivated ( y ) or after the plateau is reached. The selectivity for the independent parallel network deteriorates when it is affected by poisoning and decreases with time. The selectivities for any complex reaction networks of first-order reactions can be compactly analyzed using the approach of Wei (1962) discussed earlier. All that is required for the analysis is the expressions for the effectiveness factor vector q, which in this case is the overall effectiveness factor representing both deactivation and diffusion effects. In the domain of transformed concentration Ct, the elements of q, q k are given, for instance, for a slab-like pellet by:

For an nt"order reaction, this condition is equivalent to:

where 4 is the usual Thiele modulus. This condition is usually met in the asymptotic region of strong diffusion effects, unless the fractional surface activity is small. Therefore, in the initial stage of deactivation when g is relatively large, the combined effect of diffusion and deactivation is to reduce the intrinsic rate constant k to an effective rate constant (kg)lI2.It follows then that the point selectivity is independent of time in this initial stage of deactivation. For instance, s in Eq. 8.12 and ks/kc in Eq. 8.15 would remain the same in spite of catalyst deactivation. This time independence cannot hold when the surface activity becomes small or poremouth deactivation takes place. This is clear from Figure 8.4 obtained by Sada and Wen (1967) for simple networks of irreversible first-order reactions affected by independent pore-mouth poisoning. It is seen that the selectivitiesfor consecutive and independent parallel reaction networks are time dependent, whereas the selectivity for the parallel reaction network with a common reactant is time invariant

Figure 8.4 Selectivity behavior of simple multiple reactions affected by poremouth poisoning: equal diffusivities, = 1, kA/ks = 4, CA= Cs.(Sada and Wen 1967. Reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, Inc.)

which was obtained in Chapter 5 for independent pore-mouth poisoning. Results such as those shown in Figure 8.4 can readily be obtained for any complex network of first-order reactions (see Problem 8.6). Finally, it should be recognized that the time independence of selectivity is valid only for the same bulk-fluid conditions, i.e., when the conditions are the same whether or not deactivation takes place. This condition is not satisfied in reactor operation due to changing catalyst activity with time and therefore, the selectivity invariably changes with time except under unusual circumstances. Nevertheless, the time independence or dependence of selectivity is a useful piece of information in laboratory studies in which the bulk-fluid conditions can be held constant.

4

ri

t

i

4

i

THE STABILITY PROBLEM FOR A SINGLE PELLET

8-3

The stability problem for a symmetric pellet in which a single reaction takes place can be stated as follows: Determine the stability of the system described by:

1 5

i

E

when it is subjected to perturbations. The boundary and initial conditions are:

SELECTIVITY

on the boundary and

For one-dimensional transport of heat and mass, the equations can be reduced in dimensionless form:

AND

STABILITY

263

a certain claim on stability requires the resolution of uniqueness or multiplicity of the steady state. A system represented by a set of equations describing a physical system is said to be stable around a steady state if the transients of the system are bounded when the system is subjected to small perturbations from the steady state. The system is said to be asymptotically stable if it is stable and it eventually returns to the steady state. If in addition, the steady state is approached exponentially, it is called exponentially asymptotically stable. If any magnitude of perturbation is allowed, it is said' to be globally stable. The stability problem as related to reactor startup and control is concerned with the following questions (Luss 1977): a) What are the realizable steady states? b) What is the set of initial conditions from which a given steady state is attained? c) Are the steady states stable or unstable? What is the nature of stability if stable?

As indicated earlier, it is rarely possible to fully resolve these questions. In particular, it is extremely difficult to prove global stability. Therefore, in what follows, the main concern will be with asymptotic stability with respect to small perturbations. First, a single wire problem will be considered as an introduction. It brings out essential features of the stability problem and is similar in many respects to the more familiar stability problem with a CSTR.A simple and yet realistic case of negligible internal heat transfer and external mass transfer resistances will also' be treated. The general problem will then be analyzed in a limited form for the case of unit Lewis number (Le) appearing in Eq. 8.27. This considerably simplifies the problem and yet the result can be extended to a certain extent to the general case of arbitrary Lewis numbers. Readers interested in more details on stability can refer to the book by Aris (1975) and the review by Luss (1977), from which the subsequent sections are derived.

with

an the boundary and

where z = C/Cb,y = T/Tb, T = Dt/R2, = r / R , p = (-AH)DCb /A Tb, E = Ea/Rg Tb, +2 = L kd(Cb)/DCb Le = ppD/h, Bi, = &R/D, Bit, = hR/A, f ( z ) =f ( C ) / f ( C , )

(8.29~) A SINGLE WIRE PROBLEM A N D SLOPE CONDITIONS

8-4

Here, the intrinsic rate is represented by r, = k ( T ) f ( C ) , and the subscript b denotes the bulk fluid conditions. The exponent q is a constant depending on pellet geometry: q = 0, 1, and 2 for a slab-like, cylindrical, and spherical pellets, respectively. It is rather difficult to establish stability for the general problem as posed since this requires an analysis of a very wide class of perturbations. It is much easier to establish instability conditions. As a result, most of the studies available are concerned with instability criteria. Steady state operation is invariably concerned with stability since one must question whether or not a system will eventually converge to a steady state when it is subjected to perturbations, i.e., changes in operating conditions. FUI-thermore, steady state solutions do permit certain limited statements to be made about stability. For example, Gavalas (1968) has shown that, given multiple solutions of a steady state equation, there are (2m 1) steady states except under exceptional circumstances, and that at least m of them are unstable. , I t will also be seen that

+

A single catalytic wire on.the surface of which a first-order reaction takes place can be described by;

where the subscripts b and s denote the bulk and surface conditions, k is a rate constant, 6 is a mass transfer coefficient including the volume to surface area ratio, and R is the radius of the wire. At steady state, these equations reduce to:


265

global stability) of a set of nonlinear ordinary differential equations is determined by the stability of the linearized equations about the steady state. Linearization of Eqs.8.30 and 8.3 1 using a Taylor series expansion about the steady state yields: Using Eq. 8.32 in 8.33 yields:

where This relationship is similar to that for a CSTR with cooling: (ZI is the rate of heat removal, and Qh is the rate of heat generation, which is a sigmoidal function of Ts due to the exponential temperature dependence of the rate constant k. A value of T, that satisfies Eq. 8.34 is a steady state solution. This is illustrated in Figure 8.5, which shows (& and lines as a function of Ts. Any intersection of these two Q graphs is a steady state solution. It is seen that three steady states can exist. It has been proved by Lyapounov that the local stability (as opposed to

x T = [As

- Ass,

Ts - T s ]

(8.36)

Here, the second subscript denotes the steady state value. The roots of the quadratic characteristic equation (eigenvalues) of the matrix A determine the stability of the equations: the system will converge exponentially to the steady state if all roots have a negative real part and, therefore, is asymptotically stable. It will show a limit cycle if the roots are imaginary with zero real parts. It is unstable if any of the roots has a positive real part. Since the perturbations will decay asymptotically if and only if all the eigenvalues of the matrix A have a negative real part, it follows that the necessary and sufficient conditions for local stability are:

The first condition is known as the slope condition: a steady state solution is unstable whenever the slope of the heat generation curve exceeds that of the heat removal curve, e.g., the steady state A in Figure 8.5. It is seen from the figure that when three steady states exist, at most two are stable. The second condition has no simple physical meaning and is implied by the slope condition whenever:

For the more realistic case of internal mass transport and external heat transport resistances, the system equations written for a slab-like pellet are: Figure 8.5 Possible steady states for the single wire problem. (Cordoso and Luss 1969; reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press.)

SELECTIVITY

with the usual boundary conditions. Amundson and Raymond (1965) analyzed this problem and obtained the following necessary condition for stability:

This is another form of the slope condition. Since this is a necessary condition, it is not sufficient to guarantee stability. In fact, Lee and coworkers (1972) found that a change in the ratio of Le/B that appears in the dimensionless version of Eq. 8.42 may destabilize a steady state even when Eq. 8.43 is satisfied. However, they found that instabilities exist only for unrealistically low values of Le/P.

8-5 THE CASE OF UNIT LEWIS NUMBER A N D EXTENSIONS

The original problem posed is now analyzed in limited f o r k : first a sufficient condition for instability (Jackson 1973) when external resistances are negligible, and then the global stability for a unique steady state (Luss and Lee 1968) with an additional assumption on the relationship between temperature and concentration. In both cases, the Lewis number is set at unity for simplification. It shall soon be seen that the case of unit Lewis number is an important base case from which the questions raised in Section 8-3 can be resolved. For the nth-order reaction considered by Jackson (1973), the Eqs. 8.26 through 8.29 reduce to:

The distributions yo and zo will give the initial distributions of y and z. However, because one can choose to investigate a very special class of perturbations when wishing to prove instability, it is permissible to insist that E * ( ~ , Tnever ) departs from its steady state value, the constant function E* = 0 for all r 2 0. The reasoning here is that it is sufficient to consider any chosen perturbations to demonstrate instability. Therefore, Eq. 8.49 is reduced to:

and Eq. 8.50 is no longer needed for further consideration. If ys is the steady state under investigation, it satisfies:

The departure from steady state: Y* = ~ ( 5 , t ) YS(O thus satisfies:

+

267

with

with

+

STABILITY

IJtilizing this definition, Eqs. 8.44 through 8.47 can be rewritten for a unit Lewis number as:

with the usual boundary conditions, where:

and with Eq. 8.29 remaining the same. For any steady state, Eqs. 8.44 and 8.45 give y pz = 1 P. Residual enthalpy shall be defined as:

AND

SELECTIVITY A N D

STABILITY

269

+

Figure 8.6 q versus for an exothermic reaction in a spherical pellet. (After R+ Aris 1975)

Here, the quantity F given by:

is the one term approximation of G(()about the steady state for small y*. Assuming a separable solution:

for Eq. 8.58, yields the following eigenvalue problem:

+

at a bifurcation value of the steady state solution y,(E,+) and the effectiveness and: factor are continuously differentiable functions of

and

8(7) = eAr

(8.w

If at least one positive eigenvalue of Eq. 8.62 is found, the steady state will be unstable. The method Jackson (1973) uses to establish the condition for a positive eigenvalue is based on a comparison theorem of Sturrn. Consider:

satisfies, when used in Eq. 8.55:

with

with

dv(O)/de = 0,

+,

v(1) =" 1

(8.66)

This is to be compared with Eq. 8.62. The Sturmian theorem states that if Eq. 8.62 has a positive eigenvalue, then the function v will have at least as many zeros in the interval (0,l) as does v, and in fact the ith zero of w will occur at .a smaller value of 4 than the ith zero of v. Indeed, as Jackson shows, the position of the zero will move continuously as A varies. Hence, if v has at least one zero in the interval (0, l), one will always find a positive eigenvalue for which the solution of Eq. 8.62 satisfies the boundary condition w(1) -- 0. It is therefore a sufficient condition for instability that v should change its sign at least once in the interval (0,1). It is now desirable to relate this criterion to the geometry of the (q,+)-curve. This curve can be separated into branches, as shown in Figure 8.6 which are bounded by branch points (or bifurcation points) at which d q / d $ = m. Except

P(l)=O,

dP(O)/d(=O

But if Eq. 8.68 is multiplied by e 9 , Eq. 8.65 by on integrating over (0,l):

On the other hand:

and so:

eqP,

(8.69)

and subtracted, one has

SELECTIVITY

AND

STABILITY

27 1

where Eq. 8.70 has been used. Since G ( e ) > 0, a sufficient condition for instability will be:

for this can only be the case if v(5) changes its sign at least once in 0 I 5 5 1. The eigenvalues A are displaced continuously as one moves along a branch. Therefore, when a positive eigenvalue crosses the real axis, v(1) = 0 and d(42q)/ d+ + -m. It follows that a shift from instability to stability can occur only at a bifurcation point. Moreover, any branch for which dr]/d+ -m at either one of the branching points is unstable. Shown in Figure 8.6 is an (q,+)-curve having at most five steady states obtained by Aris and Copelowitz (1970) for an exothermic reaction in a spherical pellet. The branches of the solution are OA, AB, BC, CD and DE, the first and last corresponding to the minimal and maximal solutions in the interval cfq, < 4 < + b . Since all solutions on the same branch have the same stability character, any branch for which d($2r])/d+ is negatively infinite (Eq. 8.73) at an endpoint is unstable. If one traces the sign of this derivative, as Figure 8.7, it can be seen that the branches AB, BC, and CD must give unstable steady states. Therefore, only the maximal and minimal solutions may be stable. Luss and Lee (1968) demonstrated that a unique steady state is stable when Le = 1 under the assumption of zero residual enthalpy. This assumption is not too much a restriction since a separable solution for E* (Eq. 8.50) gives:

where the coefficients ai have to be chosen to satisfy the initial distribution Eo(5), and vi(t) are eigenfunctions of:

Figure 8.7

The maximum principle for parabolic equations (Il'in et al. 1962), which is needed for the demonstration, can be stated as follows: if u(r,r) is a continuous function satisfying:

in the domain of interest, and c(r) is bounded from above, then u(r,O) 2 0 in the domain and U ( P , T ) 2 0 on the boundary imply that u'(r,~)2 0 in the domain for all T > 0. Let y, be the unique steady state solution of:

with then: with

v(1) = 0,

dv(O)/de = 0

Slopes for the branches of the solution for the reaction in Figure 8.6. (Aris

1975)

YS

= 1 on the boundary,

Y * = Y(W) - Y&)

(8.80)

(8.76)

Now that E* decays exponentially with time, any time t at which E* is sufficiently close to zero can be chosen as the starting point. Hence, the only equation of interest now is Eq. 8.54, written in the following form:

satisfies: with

(y*), = 0 on the boundary

(8.82)

where with

. where f(y)

y = 1 on the boundary represents the heat generated by the reaction.

(8.78)

Now f '(y*), though unknown, is certainly bounded from above so that the theorem applies to Eq. 8.8 1: if the small perturbation y* is initially one-sided (i.e., either non-negative or non-positive) then it remains one-sided. Moreover, let:

SELECTIVITYAND STABILITY

273

so that partial differentiation of Eq. 8.81 yields:

with

v = 0 on the boundary.

(8.86)

Then the theorem also applies to this equation and v exhibits the same one-sided behavior. Consider the solution of the transient equations for which y(r,O) = 1, and denote the corresponding deviation by y?. Then y? satisfies Eq. 8.8 1, and:

and

v~(r,O)=f(1)

>0

Dimensionlessradial position

Figure 8.8 Approach to the steady state CvJ) of y? and Y,*+~for 0 = 0.2, irreversible first-order reaction in a spherical pellet: - - - - yl*!s, -y?. (Luss and Lee 1968; reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, Inc.) (8.88)

case of unit Lewis number. This allowed a great simplification of the problems. The usual range of Lewis number is 0.5 to 1000 and in most practical applications it is greater than unity. When Le # 1 the transients are described by the two coupled equations 8.44 and 8.45 instead of one, and the stability analysis becomes more difficult. Linearization of these two equations yields:

Thus, according to the maximum principle:

Hence, ~ ~ ( r - rises 7 ) monotonically and with no oscillations to y,(r). An entirely similar argument can be used for y?+~(r,r),the solution of the transient equation whose initial value is constant at (1 + p). It will therefore tend downwards toward the stessdy state. Calculations carried out by Luss and k (1968) for these two extreme cases, given in Figure 8.8, show the way in which these two solutions approach y,(r) in a spherical pellet; it is noticeable that yf+B approaches y, much more rapidly than y?. Finally one can show in the same way that no solution ~ and with initial values y,(r) between 1 and 1 + 6 ever crosses yf or y ~ * +(Luss Lee 1968), which means that:

Assuming separable solutions of the form:

one obtains:

implies:

Y?(~,T) ~ (9 7f)

< Y?+B(~,T)

(8.92)

Hence, any transient y ( r , ~ must ) converge to the unique steady state, and uniqueness implies global stability, when Le = 1. The problems considered so far in this section have been restricted to the

w , = wz = 0

on the boundary.


These non-selfadt equations have an infinite set of corresponding eigenvalues, some of which may be complex. Luss (1977)argues that changes in Lewis number can shift only pairs of eigenvalues across the imaginary axis and therefore, a steady state that has an odd number of positive eigenvalues for Le = 1 will have at least one positive eigenvalue for any Lewis number, implying instability. He also argues that the Lewis number may affect only the stability of those steady states that are candidates for stable steady states as determined by the criterion of Gavalas (1968) discussed earlier. It appears that the Lewis number has a detrimental effect on stability when it is less than unity. The stability analysis becomes quite difficult when both the intra- and interparticle gradients have to be ed for. Only when (Bi)h = (Bi), can the previous analysis be applied to obtain general criteria. The strongest available criterion is that of Gavalas (1968) regarding the stability of (2m 1) steady states.

+

Bs Bib

Ij 1

I I

1

II

1

Bi,

C Ct C CA cs

co

G

D D E* % J

f(C)

Summary

F(5) I? G(5)

It has been seen that transport resistances and catalyst deactivation can have a significant effect on selectivity. It has also been seen that the effect of transport resistances is not necessarily detrimental, even though the selectivity is, in general, adversely affected by diffusion for consecutive reactions. The selectivity is independent of deactivation for structure-insensitive reactions when the same sites are involved for all reactions, provided that the concentration and ternperature do not change with deactivation. This condition is rarely met in reactor operation and therefore, the selectivity changes with the extent of deactivation. The stability associated with a single pellet has been examined in some detail. This stability represents the stability at a point in the reactor. Perhaps the most significant results of the stability analyses are the slope condition regarding the rates of heat removal and heat generation and the criterion due to Gavalas (1968) regarding the number of unstable steady states. The slope condition, which essentially states that the reactor is unstable when the rate of heat generation by the reaction is larger than that of heat removal by the bulk-fluid, is an important criterion for the safe operation of reactors involving highly exothermic reactions.

h

I

NOTATION i = 1 to 4, constants in Eqs. 8.4 and 8.5 bulk fluid concentration of species A reactor inlet concentration of species A reactor outlet concentration of species A pellet surface concentration of species A pellet surface concentration of species A at steady state matrix in Eq. 8.37 bulk fluid concentration of species B reactor inlet concentration of species B reactor outlet concentration of species B

275

pellet surface concentration of species B Biot number for heat defined in Eq. 8 . 2 9 ~ Biot number for mass defined in Eq. 8 . 2 9 ~ concentration vector concentration vector transformed through Eq. 8.18 concentration concentration of species A concentration of species B initial concentration specific heat content effective diffusivity effective diffusivity matrix residual enthalpy defined by Eq. 8.48 activation energy concentration dependence of rate of reaction quantity defined in Eq. 8.60 fractional pellet surface activity remaining after deactivation quantity defined by Eq. 8.56 film heat transfer coefficient; fraction of catalyst still remaining active after deactivation (ht, i = 1 to 3) heat evolved per mole reacted rate constant with subscript denoting the species involved film mass transfer coefficient, k, including volume to surface area ratio rate constant matrix effective rate constant matrix in Eq. 8.19 pellet volume to external surface area ratio Lewis number defined in Q. 8 . 2 9 ~ order of reaction outward normal vector YTB4; /(VA4; ) quantity defined in Eq. 8.67 0 for slab, 1 for cylinder, 2 for sphere rate of heat removal given in Eq. 8.34 rate of heat generation given in Eq. 8.34 spatial coordinate vector intrinsic rate of reaction given by r, = kf(C) r, evaluated at pellet surface intrinsic rate of reaction for species A and B. respectively length of spatial coordinate global rate of reaction global rate of reaction for species A and B, respectively (Rc)A and (Rc)B evaluated at pellet surface temperature global rate for ith reaction path

k~/ k ~ s evaluated at bulk fluid temperature .

SELECT~V~TY AND STABILITY

external surface area time temperature bulk-fluid temperature pellet surface temperature initial temperature pellet surface temperature at steady state transformed concentration given in Eq. 8.4 columns of V pellet volume matrix in Eq. 8.18 variable given in Eq. 8.61 vector in Eq. 8.35 pellet surface concentration of species X T/Tb variable defined in Eq. 8.57

277

PROBLEMS

8.1. Consider first-order consecutive reactions A B C Show that the point selectivity - ( R c ) B / ( R c ) ~can be expressed as follows:

~ )a~ function of q s / q * for s = 0.5, 2, when y = 1. Plot I ( R [ ; ) ~ / ( R as and 10, with (-rB/rA) set to 0.5. What can you conclude about the effect of diffusion? Note that when s > 1, q ~ / > q 1.~ 8.2. Consider the following reaction network:

C/Cb Greek Letters

constant used in exponential deactivation function constant defined in Eq. 8 . 2 9 ~ fraction of catalyst deactivated; Ds/DA diagonal matrix in Eq. 8.19 Ea /R,T b

for r~ and r ~ respectively , effective thermal conductivity eigenvalue r/R effectiveness factor internal effectiveness factor z - 2, Y - Ys q for r ~ r, ~ and , r,, respectively pellet density Dt/R pC, for single wire Thiele modulus defined in Eq. 8 . 2 9 ~ for r ~and r g , respectively E

+

Find an expression for the point selectivity -(RG)BI'(RG)A when all reactions D is second-order, the network are first-order. When the reaction path A ---+ is known as a van de Vusse network. For the van de Vusse network, would you expect the yield to increase when kd is decreased? What if the reactions are severely diffusion-limited? For detailed results, refer to Carberry (1976). 8.3. For the following reactions: A

A B (nth order)

A

kc ---* C (m th order)

in which B is the desired product, Roberts (1972) has shown that:

in the asymptotic region of strong diffusion effect if:

Subscripts

bulk-fluid species i pellet surface; steady state pellet s evaluated at pellet surface temperature

Derive this result. 8.4. Use the diagonalization technique to write the global rates as in Eq. 8.20 for the following first-order reactions:


8.5. Show for the case of uniform deactivation that the point selectivity is not dependent on time and therefore, is independent of deactivation, provided that all reactions are first-order and take place on the same sites. Is the statement correct if the kinetics for some reactions are not first-order? 8.6. Use the approach of Wei (1962) to obtain global rate expressions Rc for a complex first-order reaction network when the reactions are also affected by catalyst deactivation. Assume pore-mouth poisoning and slab geometry. Note that q k for each reaction path is given by:

awl ae

ayl Le---

ae

where g6 = b.

Based on the expressions obtained, construct the selectivity curve given in Figure 8.4 for the case of pore-mouth poisoning. 8.7. Show that Eq. 8.43 is a necessary condition for the stability of the system described by Eqs. 8.41 and 8.42. What is the physical meaning of the necessary condition. Why can the system become unstable even when Eq. 8.43 is satisfied? Give physical reasons. 8.8. For this problem, follow the steps given below to obtain an instability criterion.

azwl - -(Uogbw1+ az2

279

UAgoy1) for c = 1

azyl = p(uoggwl+ Ubgoy1) az2

I

d w w=w,

and U;= -

Assume separable solutions for w l and yl:

so that Eqs. (7) and (8) can be written as:

Dimensionless mass and heat balances for a slab-like pellet in which a single reaction takes place can be written as follows: Define Y1 by:

y=- -B Le where w and y are the dimensionless concentration and temperature, respectively. Here, the rate expression is given as a product of a temperature dependent function U(y)and a concentration-dependent function g(w). Le and fl are Lewis number and Prater number, respectively. a. Expand the equations about a steady state using:

so that upon substituting Eqs. (3) and (4) into (1) and (2), and collecting of e n of same order for n = 0 and n = 1, there result:

w + (Le- 1) Yl

and show that Eqs. (12) and (13) can be transformed into:

Note that one is interested in the sign of h for stability. Also note that now Eqs. (15) and (16) are sufficient for considerations of stability. c. Seek a series solution for W in of an orthogonal set fn over @,I), 1.e.:


Substitute Eq. (17) into (15) and (16) and integrate over the interval to obtain the following:

where B =

Pmn

=

1;

fidz

fmh

( uo& -

P

u&o)d~

d. For a nontrivial solution of Eq. (18) for Am's, the determinant of the matrix given by the equation must be zero, which will yield solution(s) for h. For stability, A should be positive. Equation (18) is an infinitedimensional matrix for the solution of Am's. However, 1 X 1 (n = rn = 0) case can be considered for an instability criterion. Obtain the instability criterion for the 1 X 1 case.

REFERENCES Arnundson, N.R. and L.R.Raymond, AiChE J., 11, 339 (1965). Aris, R. and J. Copelowitz, Chern. Eng, Sci., 25, 909 (1970). Axis, R,,Mathematical Theory of Diffusion and Reaction in Permeable Catalyst, Vol. I and 11, Oxford University Press,'London (1975). Carberry, J.J., Chem. Eng. Sci., 21, 275 (1966). Carberry, J.J., Chemical and Catalytic Reaction Engineering, pp 48-90, McGraw-Hill, New York (1976). De Pauw, R. and G.F.Froment, Chem. Eng. Sci., 30, 789 (1975). Gavalas, G.R., Nonlinear Diflerential Equations of Chemically Reacting System. SpringerVerlag, Heidelberg (1 968). Il'in, A.M,, A.S. Kalashnikov and O.A. Oleinik, Russ. Math. Sum., 17. (31, 1 (1962). Jackson, R., Chem. Eng. Sci., 28, 1335 (1973). Lee, J.C.M., L. Penabhan and L. Lapidus, Ind. Eng. Chem. Fund., 11, 117 (1972). Luss, D., in Lapidus and Amundson, ds., Chap. 4, Prentice-Hall, Englewood Cliffs, N.J. (1977). Luss, D. and J.C.M.Lee, Chem. Eng. Sci., 23, 1237 (1968).

Roberts, G., Chem. Eng. Sci., 27, 1409 (1972). Sada, E. and C.Y.Wen, Chem. Eng. Sci., 22, 559 (1967)). Voge, H.H. and C.Z. Morgan, Ind. Eng. Chem. Proc. Des. Dev., 11, 454 (1972). Weekman, V.W., Jr. and D.M.Nace, AIChE J., 16, 397 (19713). Wei, J., J. Catalysis, 1. 526, 538 (1962).

281

PART

III Reactor Design and Analysis

CHAPTER 9 Fixed-Bed Reactors

9-1

INTRODUCTION

As evident from the list of typical catalytic processes given in Table 1.1, most catalytic processes are carried out in fixed-bed reactors. These reactors are usually large capacity units, reaching in some cases, as in ammonia synthesis, capacities of half a million tons per year. The reactors are not single packed tubes. With all the auxiliary equipment for gas processing such as feeding, compressing, and heating or cooling, and the units, the reactors are indeed complicated. Large capacity reactors can have tens of thousands of tubes operating in parallel. At the heart of these reactors is still a single tube packed with a ed catalyst in one form or another. Many considerations go into the design of a fixed-bed reactor. These include the mode of heat exchange, bed pressure drop, safe operating temperature range, the mode of catalyst packing, and so on. Sizing a reactor, which is usually of primary concern, may turn out to be of secondary importance in the overall scheme of the design when economics and operability are considered. Rapid temperature rises in the bed caused by highly exothermic reactions may have a significant unfavorable effect on selectivity and equilibrium conversion. Or it may be altogether unsafe to operate the reactor under certain feed conditions. When effluent gases have to be recycled, the pressure drop in the bed has to be kept as low as possible to minimize the costs of recycling. It may be desirable to pack different parts of a reactor with catalyst particles of different size. These design considerations place limits within which a designer is free to choose the size and operating conditions of a reactor. Conservation of mass, energy, and momentum forms the basis on which these limits can be specified, as well as the size and operating conditions of the reactor. Certain assumptions have to be made to write realistic and yet useful conservation equations for a fixed-bed reactor. Different models result depending on the assumptions made. The simplest model is the pseudo-homogeneous model in which the solid phase (catalyst particles) and the fluid phase as a whole are treated as homogeneous. In such cases, the usual conservation equations for homogeneous reactors, in which only one phase is involved, apply. In reality, however, there are certain transport resistances within the catalyst particles and also across the interface between solid and fluid phases. The reaction takes place within the catalyst

particles (solid phase) and the fluid phase is simply a medium that suppliis the reactants and carries off the products. Heterogeneous models precisely represent this fact. The fluid and solid phases are treated separately, and the linkage between the two phases is the interfacial transport processes. Heterogeneous models are treated in this chapter. The treatment of homogeneous reactors (or equivalently pseudo-homogeneous models) can be found in a number of textbooks (e.g., Levenspiel 1972). It should be recognized that heterogeneous models reduce to pseudohomogeneous models when all transport resistances are absent. A rather complete treatment of heterogeneous models is presented first. It should become immediately evident that the conservation equations need not be solved in their entirety because of the simplifications that can be made for the design and analysis of fixed-bed reactors. The model used for the design and analysis should be as simple as the particulars of the reaction system under consideration will allow. In this regard, it is noteworthy that it is possible these days to tailor the design of a reactor according to the conservation equations. This capability, when fully utilized, should eventually lead to new types of reactors and to direct scale-up from experimental data to full-size plant reactors. Before this can be a reality, however, uncertainties regarding the accuracy of correlations for the transport properties (Chapter 14) have to be resolved, especially for highly exothermic reactions. In this chapter, the classical approach to the design and analysis of fixedbeds is considered. Because of the complexity involved in using the classical approach, attention here is limited to reactions affected by diffusion, and the treatment of reactions affected by simultaneous diffusion and deactivation are postponed to Chapter 10, where the concept of reactor point effectiveness is used.

9-2

CONSERVATION EQUATIONS

molar flux Jt can be due to gradients of concentration, pressure, and temperature. Here, however, the effective dispersion coefficient Dt is defined by the following relationship:

b

I

Far fixed-beds, the cylindrical coordinate system can be used. If only the velocity in the direction of flow (2) is taken, Eq. 9.2 can be rewritten with the aid of Eq. 9.3 as:

i

I1 i

ac,

,zg-=--

at

a a~ +

a {

+

{

D

~

}

-

-(9.4) ~

where Da and D, are the axial and radial effective dispersion coefficients, respectively, based on the total area including void space, v is the superficial velocity, and EB is the bed voidage. Here, the global rate RGt is multiplied by the factor (1 - ~ b since ) R G ~is based on pellet volume. For constant velocity and dispersion coefficients, this equation is further reduced to:

3:

I

The energy conservation equation can be written directly from Eq. 9.4 by simply replacing Ci by p q T , D by the effective thermal conductivity PC&, and RGi by 2(-AHi)Rci, and by neglecting the kinetic and potential energy . For a single reaction taking place in the fixed-bed, the conservation equations for constant v and p G are:

The conservation of mass is considered first using fixed coordinates. The equation of continuity (Bird et al. 1960) for a multicomponent system can be written for each species i as:

;, This equation can be rewritten in of concentration by dividing by the molecular weight Mi:

1

i I

where the subscript i has been dropped for the single reaction, and Ka and K, are the axial and radial effective thermal dispersion coefficients,respectively. Here, the subscript b has been used for C and T to denote that these are bulk-fluid quantities. The radial boundary conditions are: at the center

where the observed rate of formation of species i due to chemical reaction, Ri, which is based on fluid volume, is now expressed in of concentration. The

aCb

- =arO ,

hw(Tb- Tc)=-pqKrdraTb

at the wall

(9.8)

The first set of boundary conditions results from symmetry about the centerline of the tube. The second set results from the fact that no reactant transport takes place across the wall, and that heat transfer to the coolant, whose temperature is TC.is equal to the heat conducted at the wall. The axial boundary conditions (Wehner and Wilhelm 1956) are:

at the inlet

where (-AH) = heat of reaction h = film heat transfer coefficient Ts = pellet surface temperature A, = effective thermal conductivity In order to relate the intrinsic rate of reaction to the global rate, the effectiveness factor, or equivalently, the gradient at the pellet surface has to be calculated. This is obtained by solving the conservation equations for the pellet:

at the outlet These boundary conditions are based on continuity of flux across a boundary, i.e., at the inlet and outlet of the bed. At the inlet continuity of flux requires that:

If the feed is well mixed such that the dispersion term is zero, the boundary + , (dC/dz),+ = condition of Eq. 9.10a results. At the outlet (vC),- = ( V C ) ~ and 0 since no reaction takes place. These conditions lead to the boundary condition of Eq. 9.1 1. The initial conditions are the steady state solutions of Eqs. 9.6 and 9.7. The left hand side of Eqs. 9.6 and 9.7 represents the changes taking place in the fluid phase as a result of the reaction in the solid phase (catalyst pellets), which is given by the right hand side. The global rate RG is equal to the rate at which mass is transferred across the interface between fluid and solid phases, which in turn is related to the flux at the catalyst surface:

where k, = film mass transfer coefficient L = characteristic length of catalyst pellet, volume/external surface; surface D, = effective diffusivity C.= fluid concentration at pellet surface q = effectiveness factor r, = intrinsic rate of reaction where the subscript p is for the pellet. In a similar manner, one gets:

where D, and A, have been assumed constant, and is the pellet voidage. The usual boundary conditions at the center and the surface of pellet apply. When Eqs. 9.6 and 9.7 are made dimensionless, key reactor parameters emerge. Letting:

where d, = pellet diameter R = radius of fixed-bed Z = length of fixed bed Gin, Tin = reactor inlet concentration and temperature the reactor conservation equations become:

P The axial Peclet number for mass dispersion , P,, are given by:

and the radial Peclet number

The Peclet numbers for heat dispersion are given by:

It is seen then that the Peclet numbers and aspect ratios n, m, and A determine the extent to which the dispersion affects the reactor behavior. Solving Eqs. 9.6 through 9.15 for the design and analysis of a fixed-bed reactor is not a simple matter by any means and may be a formidable numerical problem. The steady-state versions of these equations are used for design purposes. It shall soon be seen that considerable simplifications can be made for tvnical -- operating conditions.

9-3 SIMPLIFICATIONS The nature of a reaction and the mode of reactor operation immediately suggest the kind of simplifications that can be made. If the heat of reaction is negligible so that a reactor is ntn isothermally, all heat balance equations can be eliminated. When the heat of reaction is not very high, the reactor is usually run as an adiabatic reactor, as many reactors are in practice. In such cases, radial gradients are negligible since only the velocity profile, which can be assumed to be flat, can give rise to gradients. Therefore, a one dimensional treatment of the conservation equations is sufficient for adiabatic reactors. For very exothermic reactions, however, external cooling is required to preserve the integrity of the catalyst pellets and to ensure the safety of reactor operation. In such cases, the whole equations as posed may have to be solved. As indicated earlier, the Peclet numbers and aspect ratios determine the relative importance of dispersion . It is apparent from Ea. 9.16 that axial dispersion can be neglected compared to the convection term when (nP,.) - --islarge. The axial Peclet number for mass, P,., is about 2 for gases flowing through packed beds for Reynolds numbers (based on d p )greater then 10. Since the axial for typical reactors is quite large and aspect ratio n (= Z / d p ) is large, (nP,) the axial mass dispersion tern can be neglected. Much less is known about the axial and radial Peclet numbers for heat. However, Pha is in general smaller than Pma, as evidenced (Price and Butt 1977) by the necessity of including the axial heat dispersion term even when the axial mass dispersion term is negligible. Details on effective dispersion coefficients can be found in Chapter 14. Based on the mixing.. .. cell dispersion analogy, in which the bed is considered to consist of n CSTRs in series, Carberry (1958) suggests that all axial dispersion can be neglected when n is greater than 150, except for unusual reactor confiaurations. While a large value of (nP)for axial dispersion results inWnegligible dispersion effects compared to convection, the inverse is true for radial dispersion simply because the only transfer mechanism is conduction and a larger value of radial ....

W

.. .

-

Peclet number (a smaller value of K) m a n s a higher resistance to conduction. The radial Peclet number for mass is typically 11 for Reynolds number greater than 30. The corresponding value for heat is generally smaller. On the other hand, the combination of radial and reactor aspect ratios m/A (= R*/dpZ) is quite It is readily seen then that the radial dispersion cannot readily - be small. - -neglected unless the bed is adiabatic. If the tube radius is of the order of one pellet diameter and heat exchange is only at the tube wall, on the other hand, the radial gradients can be neglected since then the whole cross-section of the tube can be considered to be well-mixed. In general, the radial gradients can be neglected for all practical purposes if the radial aspect ratio m (= R / d p ) is 3 or 4 as suggested by Carberry (1976). As discussed earlier, the radial gradients exist when the bed is cooled externally, and this cooling is done only for very exothermic reactions: otherwise, the reactor is run adiabatically. Since the radial thermal gradient is caused by the external cooling and this in turn causes the radial concentration gradient, the major dispersion effect is in the radial direction, and axial dispersion can be neglected in comparison to radial dispersion. The conservation equations in such a case can be written as given in Table 9.1, and involve only axial convection and radial dispersion. Furthermore, the very exothermic nature of the reaction in such cases dictates that the radial temperature gradient be minimized to ensure safety of operation and to preserve the integrity of the catalyst pellets. Therefore, typical industrial reactors for highly exothermic reactions are designed to have a small radial aspect ratio, as practiced in the oxidation of naphthalene. In such cases, it is sometimes possible to neglect both axial and radial dispersion , yielding plug-flow conservation equations. To sum up, it is sufficient to treat an adiabatic reactor as a plug-flow reactor. If the axial dispersion effect is to be included, only the heat dispersion tern needs to be added. In the case of nonadiabatic, nonisothemal reactors, axial dispersion -. . can be neglected in comparison to the radial dispersion . In add~tlon, the radial dispersion can often be neglected if the radial . aspect . - ratio - . is small. The conservation equations for various cases are summarized in 17ableY. 1. The aspect ratios appearing in the conservation equations deserve further comment. The quantities appearing in the aspect ratios are important design parameters. The pellet size dp is an important parameter determining the effects of interand intraphase transport, the tube radius is a major factor in heat transfer at the wall (in the form of 2 / R in the overall heat transfer coefficient [I), and the ,tor length detennina the time or the conversion. These parameters may then be determined a priori based on certain reactor requirements. This does not necessarily mean that the aspect ratios cannot be manipulated to achieve a desired design objective. If, for instance, it is desired to minimize radial gradients for a highly exothermic reaction without affecting the effectiveness factor and the conversion, R may be reduced, which results in a reduction of m and an increase in Upboth of which reduce the radial gradients. As discussed fully in Chapter 4, the Biot number is large enough under typical reaction conditions to allow isothermal treatment of catalyst pellets. Furthermore, the external mass transfer resistance (interphase mass transfer) can also be --

- -

.

-...

.

.

A

292


t

\

Table 9.1 Conservation Equations for Design Reactor Conditions

i

Left Hand Side of Eqs. 9.6 and 9.7

where A P = pressure drop GS= fluid mass rate f = friction factor p~ = fluid density

Otherwise, pqv- a Tb

az

Nonadiabatic, nonisothermal

Y-

acb

az

OTHER CONSIDERATIONS

The pressure drop in a fixed-bed due to packed particles is an important design consideration, especially when a large amount of gas needs to be recycled. For very large capacity units, in fact, as in catalytic reforming and ammonia synthesis, a radial flow arrangement (Finneran et al. 1972) is used to reduce the pressure drop and thus enhance the recycle compressor capacity. The Fanning equation may be used for the estimation of the pressure drop:

Adiabatic

Isothermal

9-4

i

i

B

which can be obtained from the Bernoulli equation by neglecting the kinetic and potential . The friction factor f is given by the Ergun relationship:

R/dp > 4

Here, the Reynolds number is based on the particle diameter d,. For a radial aspect ratio R/dp less than 25, Mehta and Hawley (1969) suggest the following modified Ergun correlation, which includes wall effects:

Pellet can be considered isothermal if k, h,/D,h >> 1. Pellet surface concentration can be considered to be equal to bulk fluid concentration when +/(Bi), < 0.05. --

neglected under typical reaction conditions. Therefore, the heat balance of Eq. 9.15 can be eliminated and the pellet surface concentration can be taken to be equal to the bulk fluid concentration. These results ate also summarized in Table 9.1. It is to be noted that the results given in Table 9.1 for the conservation equations are for guideline purposes only and that exceptions can always be found. Therefore, the results should be used with caution. The numbers used for pellet conditions are somewhat arbitrary. The condition of +/(Bi), < 0.05, for instance, ensures for a first-order reaction that the error in not including the interphase mass transfer resistance in the overall effectiveness factor will be less than 5% for an isothermal pellet,

:

I

1

Hicks (1970) suggests the use of the following correlation for smooth spheres in the range of 300 < Re/(l - e B ) < 60,000:

The conservation equations of 9.6 and 9.7 have been written with the assumption of constant fluid velocity. If there is a net change in the number of moles due to the reaction, or if there is a significant change in the reactor temperature or pressure, the velocity cannot be considered constant. This change in velocity can be ed for with the aid of the ideal gas law and the reaction stoichiometry:

where cr = 8 WArn 8 = change in the number of moles per one mole of species of interest, A WAfn = mole fraction of species A in feed , y~ = conversion of species A This change has to be included in the conservation equations with the velocity appearing inside the differential operator as in Eq. 9.4. In addition, a momentum balance has to be added unless the bed pressure drop is negligible. In such a case, the momentum balance may take the form of the Bernoulli equation:

where F is given by the right hand side of Eq. 9.20. Use of Eq. 9.24 in 9.25 then yields an expression for dP/dz in of dT/dz and dyA/dz. This expressiion may be solved with the mass and energy balance equations simultaneously for the design and analysis of fixed-beds. It should be recognized, on the other hand, that the effect of changing velocity is usually negligible in practice, even for highly exothermic reactions unless the reaction results in a significant change in the number of moles due to the reaction. For such reactions, the feed is usually quite diluted with inert gas for the reasons discussed below, and no excessive temperature rise should occur.

Figure 9.1 Runaway criteria for first-order reactions obtained by various authors. (1: Barkelew 1959; 2: Dente and Collina 1964; 3: Hlavecek et al. 1969; 4: Van Welsenaere and Froment 1970.) (After Froment and Bischoff, Chemical Reactor Analysis and Design, @ 1979, Wiley. ~ e ~ r i h t ebyd permission of John Wiley & Sons, Inc.)

Design of a fixed-bed for highly exothermic reactions requires additional consideration with regard to the integrity of the pellet and the safety of reactor operation. When a reaction is highly exothermic, a slight change in operating conditions such as the inlet temperature can lead to a situation where the reactor can literally explode. This is called runaway or parametric sensitivity. Several criteria have been developed that permit a selection of operating conditions and reactor dimensions for safe reactor operation. These criteria, shown in Figure 9.1, have been derived based on the pseudo-homogeneous plug-flow model for a firstorder reaction. Runaway criteria for an n th-order reaction can be found in Morbidelli and Varrna (1982). The dimensionless quantities involved are the dimensionless adiabatic temperature rise S given by:

S=

Tad

(Tb)in- (-AH)(Cb)in (Tb)in ~ q ( T b ) i n

and N given by:

where k is the volumetric rate constant. A general treatment for the parametric sensitivity problem is given in Section 9-9. The sensitivity problem considered above places constraints on the choice of operating parameters and reactor dimensions. The same is true when the bed pressure drop is significant, the constraint in this case being on the pellet size. The pellet size is also an important factor affecting inter- and intraphase transport. A smaller size reduces diffusional effects but causes a higher pressure drop and lower external transfer rates, resulting in poorer heat transfer. Therefore, a compromise has to be made regarding the choice of pellet size unless one of the factors is overriding. A similar situation occurs regarding the choice of reactor tube diameter. A smaller diameter ensures better heat transfer and better temperature control but results in a lower capacity per tube. The pellet size and reactor diameter are usually chosen before the reactor is sized. For highly exothermic reactions, not only is the feed diluted with inert gas but the bed packing is also diluted with inert pellets to reduce the heat liberated by the reaction on a unit volume basis. In addition, a cooling medium that gives a favorable reactor temperature profile can be chosen. The design of fixed-bed reactors eventually involves solving the conservation equations (Eqs. 9.6 through 9.15). It is rare that analytical solutions are obtained because of the nonlinearity involving the Arrhenius expression and intrinsic kinetics. Therefore, the design invariably requires numerical solutions of the equations. This is dealt with in the next section.

9-5

NUMERICAL METHODS

As discussed in Section 9-3, the following conservation equations are sufficient for most practical reactor designs: Reactor fluid side:

with

say, the one-dimensional reactor (e.g., adiabatic reactor) is a special case of the general case given above. Numerical solutions of these equations start at z = 0,at which Tb and Cb art: known as a function of r. For the global rate RG at each grid point in r, Eqs. 9.33 through 9.36 can be solved iteratively for T,(O,r) and RG(O,r).For the first iteration Ts is set equal to Tb, which allows the solution of Eq. 9.33 for the global rate Rc through Eq. 9.35. This Rc is used in Eq. 9.36 for the calculation of T,. Based on this calculated G, the whole procedure is repeated until the new value of Ts is close to the value of T, in the previous iteration. Based on all information at z = 0 now available, the radial profiles of Tb and Cb at the next grid point in z are obtained using a finite difference version of Eqs. 9.27 and 9.28. Using these newly calculated T and C profiles, the procedures described above at z = 0 are repeated to obtain Rc and T, at this new grid point in z. These procedures can be repeated until z = Z is reached. If an analytical expression for the internal effectiveness factor q is available, the whole procedure is greatly simplified since then all equations for the pellet and the interface can be replaced by:

-acb TO ar

and

h,(Tb-Tc)=--pC,K,--

a Tb ar

at r = R

(9.32)

Pellet side:

with

-= 0

ax

= Cb

at the center

at the surface ( x = 1;)

Interface between pellet and reactor:

These equations are the steady-state version of Eqs. 9.6 through 9.15 with the assumptions of negligible axial dispersion, negligible external mass transfer resistance, isothermal pellets, and constant v and physical properties. Note that the external heat transfer resistance is still present as given by Eq. 9.36. Needless to

The nonlinear algebraic equation 9.37 can be solved for T,, which in turn gives RG in of the bulk fluid quantities Cb and Tb. No iteration procedures are necessary for the calculation of Ts in this case. There are explicit and implicit numerical methods for the finite-difference version of Eqs. 9.27 and 9.28. There are also approximaire numerical methods in which the radial derivatives are replaced by the functions obtained by differentiating assumed trial functions for the radial profiles, essentially transforming the equations into ordinary differential equations in z. The cell model (Hlavacek and Votruba, 1977) can also be used for the solution. The explicit method gives the radial profiles at the new grid point in z explicitly, whereas the implicit method involves simultaneous solution of algebraic equations for the new radial profiles. The former replaces the axial derivative by a forward differenceformula and the radial derivative by a central difference formula. The latter uses a backward difference formula for the axial derivative. Perhaps, the most important difference between the two is the stability of the numerical methods: the explicit method needs to satisfy much more stringent stability conditions than the implicit method. The grid system for explicit methods is shown in Figure 9.2. The dark circles represent the grid points from which the information for the open circle is obtained. The finite-difference formulas far the grid system of Figure 9.2 are given in Table 9.2. The finite-diffeence method corresponding

b. Explicit method of Figure 9.26 (Liu 1970) L-method ( j = 2,. . . ,J - 1)

:

- = - [ y ( i+ l d ) - y(i.j)]

Figure 9.2a

Grid system for explicit method. (Hlavacek and Votruba 1977)

1,. R-method (j= same

+ 3y(i + l j ) + y(i,j + 1) - y(i,j - I)]

- 3y(i + 1,j) + y(i,j + 1) - y(i,j - I)]

+ 3y(i,j + 1) - y(i + l j - 2 )

- 3y(i + 1 j ) + y(i,j + 1 ) - 4y(i,j)

+ 4y(i + 1

J

- 1)

- 3y(i + Lj)]

+

At j = 1, a hypothetical point, ( i 1,-I), is necessary for the derivatives. For y(i 1,-1) use:

+ 3y(iPj- 111 At j = J - 1, a hypothetical point, ( i + 1, J

+

and other derivatives in the conservation equation, and then solve the resulting equation for y(i l ,-I). Figure 9.2b Liu's grid system for explicit method. (Liu 1970)

. . ,J - 2 )

+

+ 1) is necessary for the derivatives.

For this point, use:

and other derivatives in the conservation equation and then solve the resulting equation for y ( i l,J 1).

+ +

c. Implicit method of Figure 9.3

avaz Table 9.2

=+!

r [ y ( i + l j ) - y(i,j)

0

<8 S 1

Finite-Difference Formulas for Various Methods O ~ 2 Zf , O < r 5 1 Y" Cb/(Cb)in Or T b / ( T b ) l n ; J: at the tube wall (= ( l / m ) )

a. Explicit method of Figure 9.2a

0'=1,. . . ,J-1) to Figure 9.2a is stable if the grid sizes are chosen in such a manner that:

':-i2+

--ar

[y(i,j

1) - 2y(i,j)

+ y(i,j - I)]

a=

ZD, for mass R2v . .

.

ZKr R2v for heat

Figure 9.3 Grid system for implicit method.

Upon substituting Eqs. 9.39 and 9.40 into 9.38, the equation becomes:

+

where 1 and m are the step sizes of integration in the z and r directions, respectively ( I = 1/1, m = 1/J). While this stability condition for numerical solution has been derived for linear equations, it can be used for nonlinear equations since the stability is mainly determined by the manner in which the derivatives are approximated by the finite-difference. While the scheme of Figure 9.2a has the stability limitation, it is by far the simplest method, involving straightforward marching in the z direction based on the information at previous grid points, i.e., at z = li for the calculations at z = I(i I). The finite-difference version for this scheme given in Table 9.2 can be applied for all j from j = 1 to j = J - 1 at any z, i.e., at any (i l), where J (= l / m ) corresponds to the tube wall and j = 0 corresponds to the tube center. At j = 0 and j = J , the boundary conditions can be used to obtain Tb and Cb at the center and at the wall. The procedures for obtaining the values of Tb and Cb at these points can best be illustrated by re-writing the conservation equations as follows:

+

+

where y can be either Tb/(Tb)i, Or Cb/(Cb)fn, and y and are constants. For the value of ~ ( fi 1.01, y may be expanded around r = 0 in a Taylor series to give:

This equation can be sblved for y(i 1,O) since it is the only unknown quantity. The basic idea here is that the differential equation has to be satisfied everywhere on the z-r domain including the boundary. Similar proct:dures can be followed at j = J for y(i 1,J). A five point numerical scheme was devised by Liu '(1970) to alleviate the stability problem. Depending on whether the integration in r proceeds from left to right or from right to left, the scheme was termed L-method or the R-method. Here again, the integration procedures are straightforward with the finite-difference formulas given in Table 9.2 for j = 2 to j = J - 1 for the L-method and for j = 1 to j = J - 2 for the R-method. The calculation procedures at j = 1 for the L-method and at j = J - 1 for the R-method are detailed in the table. It is apparent that trial and error iterations are necessary to obtain the values at j = 0 and 1 (or j = J - 1 and J for the R-method), using the boundary condition at j = 0 and the suggested procedure for j = I given in the table. It is suggested by Liu (1970) that RG evaluated at z = i be used for the L-method and once the tube wall is reached the R-method be used with the value of Rc at z = i 1, which is now available from the integration by the L-method, so that the average of the values obtained by the two methods can be used for the solution at each grid point. The implicit method, while it requires additional computation, is better than the explicit method as evident from Table 9.3. The implicit method given in Table 9.2 i s called the Crank-Nichobon method when the weighting factor 8 is 0.5. As discussed earlier, this method requires simultaneous solution of algebraic equations at each i When the expressions for the derivatives in Table 9.2 are used in Eq. 9.38, for instance, it is transformed into:

+

+

Table - 9.3 Comparison of Numerical Methods (Carnahan et al. 1969)

which yields, when solved for @y/ar2: Methods

Stability or Convergence Condition for Linear Equations

Order of Magnitude of Error

Explicit (Figure 9.2a)

The first derivative disappears due to the boundary condition of Eq. 9.31. The term (l/r) (ay/ar) is indefinite at r = 0.Applying the I'Hopital's rule, it becomes:

Explicit (Figure 9.2b) Implicit

none none

-(I 12

+ m2) +m2

The right hand side of Eq. 9.41 is a constant since all quantities are at z = il, which are known. This equation can be written ( J - 2) times and the number of unknowns is ( J - 2). Written in full for j, these equations form a tridizigonal matrix equation in the form of:

+

where A is a tridiagonal matrix of dimension of ( J 1) including the points at 1) and the center and the wall, and yi+l and bi are vectors for y at z = l(i the right hand side of Eq. 9.41 evaluated at z = li, respectively. The Thomas method (Lapidus 1962) can be used to solve for y i + ~ Similar . procedures as in the explicit method can be used at the boundaries for y at j = 0 and j = J. The matrix equation is solved at each i until the reactor outlet is reached. Mihail and Iordache (1976) compared various numerical techniques and found Liu's scheme to be sensitive to step size and to lead to stability and convergence problems for severe operating conditions. The method of lines lies midway between analytical and grid methods, and can also be used to solve the conservation equations. This method involves substituting finite differences for the derivatives with respect to one independent variable and retaining the derivatives with respect to the remaining variables. This approach replaces a given differential equation by a system of differential equations with a smaller number of independent variables, typically reducing a partial differential equation to a set of ordinary differential equations. Consider Eq. 9.38. Suppose lines are drawn parallel to the r axis, assuming that the distance between two adjacent lines is constant and equal to h. Suppose that the z-r region is intersected by the lines z = zo kh = zk (k = 0, 1, 2, . . . , n). Since y(0,r) is known, the z derivative can be replaced by a difference formula. For example, one can set:

+

The set of differential equations resulting from the method of lines is solved analytically (Mkhlin and Smolitskiy 1967) or numerically (Carver 1976). Since the set of ordinary differential equations resulting from the application of the method of lines to Eq. 9.38 is not amenable to analytical solution, the set has to be solved numerically. Thus, the application of the method of lines leads to another grid method. There are two useful approximate methods for the numerical integration of the conservation equations, approximate in the sense that radial profiles are approximated by trial functions. One of them is due to Villadsen and Stewart (1967), called the orthogonal collocation method, and the other is due to Ahmed and Fahien (1980) involving a three parameter function. It is instructive to examine typical radial profiles shown in Figure 9.4 since these methods involve an approximation of radial profiles. Because of the heat sink at the tube wall, the temperature gradient is steep at the wall. On the other hand, the concentration gradient at the wall is flat since no transport takes place there. The essence of these methods lies in approximating the profiles with suitable trial functions and then replacing the radial derivatives with the results obtained by differentiating the trial functions. The collocation method is one of a general class of approximate methods known as the method of weighted residuals (Ames 1965). The method involves expanding the temperature and concentration in a series Zai(z)Pi(r) of known functions of radius, Pi(r), multiplied by unknown functions ai of z The trial functions are substituted into the partial differential equations that are satisfied at discrete radial collocation points, 0. This gives a set of ordinary differential equations governing a&). The trial functions are orthogonal polynomials, e.g.:.

+

This leads to the following sequence of equations: center

r

wall

center

r

Figure 9.4 Typical radial profiles of temperature and conversion.

wall

such that a minimization condition of the least-squared error integrated over f can be met. For the reactor simulation, a set of ordinary differential equations corresponding to the original conservation equations is obtained at each collocation point upon substitution, which are in turn solved for the quantities of interest. This application of the collocation method to reactor simulation is detailed by Finlayson (197 1). The averaging method of Ahmed and Fahien utilizes three-parameter radial profiles (1980) given by:

Use of boundary conditions leads to the expressions for the parameters a through c, which depend on centerline conversion and temperature, average conversion and temperature, and the conversion at the wall. Averaging over the cross section of the original conservation equations and evaluation of these equations at the centerline and wall gives five ordinary differential equations for the five quantities given above. This method is useful for the calculation of the hot-spot timperature. The solution of the pellet-side conservation equation (Eq. 9.33) is a twopaint boundary value problem but is not difficult to solve numerically so long as the intrinsic kinetics are simple and the reaction is moderately diffusion-limited. The problem becomes difficult to solve because of numerical sensitivity when the reaction is severely diffusion-limited and the accuracy of the result is not assured. However, an analytical expression can be obtained in this case for arbitrary intrinsic kinetics as shown in Chapter 4. 9-6

EXAMPLES AND PAITERNS

The simulation of an adiabatic reactor for the oxidation of SOz was carried out by Minhas and Carberry (1969). With the assumptions of negligible axial dispersion and constant fluid velocity, the conservation equations (Eqs. 9;27 and 9.28) reduce to:

The boundary conditions are reactor inlet conditions. The intrinsic kinetics over a platinum catalyst are represented by:

Instead of solving the pellet conservation equations, they used a definition of the Thiele modulus in of the limiting reactant SO2 and the equilibrium nature of the reaction SO2 1/2 O2 = SOs to obtain:

+

where the pellet was assumed to be isothermal. Simulation results are shown in Figure 9.5. These results are fairly close to actual plant conditions. For instance, a typical q value for the industrial SO2 reactor is around 0.1. The patterns shown in the figure are typical of an adiabatic reactor for exothermic reactions. The effectiveness factor generally decreases with reactor length, but more rapidly than shown in the figure. Because of the reaction equilibrium, the decrease is much more moderate in this case. The conversion and temperature usually increase rapidly near the inlet, but these increases are moderate toward the outlet, reaching plateaus as the reactant is depleted. For a nonadiabatic reactor, consider the analysis by Carberry and White For the simplified network (1969) for the oxidation of naphthalene over VzO5. of first-order reactions: kl

+

k2

naphthalene ---+ anhydride ------t C02 H 2 0 (1) RG, (2) '~r the conservation equations are:

V

ac, az - D

(

+ -1r )ac ar

=- 1

)

-

-B

for naphthalene

= -( 1 - C B )(Rc2 - RC I)

for anhydride

with the same boundary conditions as in Section 9-5. The pellet conservation equations were not solved. Instead, an analytical expression was used for the isothermal effectiveness factor for naphthalene. They found that the effect of radial mass dispersion was negligible, which makes it possible to write a reactor point yield ratio d C l / d C 2 directly from the reactor conservation equations for naphthalene and anhydride. Results are shown in Figures 9.6 through 9.9. It is seen in Figure 9.6 that the centerline temperature goes through a maximulm at a fractional length of about 0.1. This point is called a hot spot. For a given cross-section, the maximum temperature occurs at the centerline with shell-side coo1,ing. The presence of a hot spot is a characteristic of nonadiabatic reactors with shell-side cooling. At the hot spot, the heat generated by the reaction is equal to the heat transferred

Tat Tin

Figure 9.5 Simulation results for SO2 oxidation. (Minhas and Carberry 1969)

I 1.6

Tin Tat

1.o

0.1 0.3 0.5 0.7 REDUCED A X I A L DISTANCE

Figure 9.7 Radial catalyst temperature profiles: -for P h r = 10, -- for Phr = 5. (Carberry and White 1969; reprinted with permission from Industrial and Engineering Chemistry. Copyright by American Chemical Society.)

to the coolant through dispersion and heat conduction at the wall. The heat generation rate dominates the heat removal rate in the region close to the reactor entrance. Once the hot spot is ed, the heat removal rate dominates. This hot-spot temperature places a constraint on the reactor design since the viability of the catalyst is lost if the temperature exceeds a certain level: the catalyst as well as the can be transformed into a different material at high temperatures. In industrial reactors, the hot spot migrates toward the outlet with the time on stream as the activity of the catalyst decreases due to deactivation. As expected, the hot-spot temperature as well as the catalyst temperature profile is lower for a smaller Peclet number for heat (better heat dispersion). This is also shown in Figure 9.7, in which the radial temperature profiles are given. The temperature patterns shown in these figures are typical of nonadiabatic reactors with shell-side cooling. The importance of good radial heat dispersion (approaching plug-flow behavior) is exemplified in Figure 9.9 where the yield of the desired intermediate product is given. The average yield is obtained from the mixing-cup average concentration: (Cn(z)) =

1 0

Figure 9.6 Normalized catalyst temperature profile at bed centerline: effect of radial Peclet , = 10. (Carberry and White 1969; reprinted with pennisnumber for heat on the profile, P sion from Industrial and Engineering Chemistry. Copyright by American Chemical Society.)

rC2(r.z)dr

/1

rdr

0

The overall effectiveness factor for the first reaction shows some interesting features (Figure 9.8). It is seen that the temperature effect dominates the overall effectiveness factor, causing a minimum near the hot spot. It should be recognized that the

effective rate constant at the hot spot is still quite high despite the very low overall effectiveness factor there (10-3 -This is due to the fact that the effective rate constant is proportional to q t k l (-kl1I2) and k, is quite high at high temperatures. A successful application of the plug-flow heterogeneous model has been reported by Cappelli and coworkers (1972). Their simulation results show excellent agreement between the model predictions and the data that they obtained from a nonadiabatic industrial reactor for the synthesis of methanol from CO and Hz. The work by Durn& and Froment (1976) is another example of the application of the plug-flow heterogeneous model to an industrial reactor.

9-7 REACTORIHEAT EXCHANGER COUPLING: AUTOTHERMIC REACTORS

REDUCED REACTOR LENGTH

Figure 9.8 Effectiveness factor at reactor wall and centerline: -for Phr = 10, ---- for Phr = 5. (Carberry and White 1969; reprinted with permission from Industrial and Engineering Chemistry. Copyright by American Chemical Society.)

When an exothermic reaction requires a high temperature, the heat of reaction can be used to preheat the feed. The coupling of the reactor with the heat exchanger through the use of reactor effluent fluid for the purpose of self-ing the preheating requirement of the feed is termed autothermic operation (Van Heerden 1935). This can be accomplished externally, internally, or through recycling. These arrangements are shown in Figure 9.10. Ammonia synthesis reactors (Baddour et al. 1965) of the Nitrogen Engineering Co. and the TVA (Tennessee Valley Authority) are examples of the autothermic reactors having external heat exchange. Consider the external arrangement. Since the reactor is adiabatic, it can be represented by the plug-flow model. Referring to Figure 9.10, the conservation equations are:

dv

Fin = (1 - B )Rc dz Product Reactor '-

1

Reactor

5

6 7 8 9 1 0 RADIAL PECLET NUMBER

Figure 9.9 Effect of radial Peclet number on overall yield of anhydride, conversion = 99%. (Carberry and White 1969; reprinted with permission from Industrial and Engineering Chemistry. Copyright by American

Chemical Society.)

I

Tin Feed

~ T ~ ( o ) Product

External

1

Feed

Internal

Recycle

Feed

Figure 9.10 Autothermic reactor arrangements. (After I. M.Smith, Chemical Engineering Kinetics, 3rd ed., @ 1981; with permission of McGraw-Hill Book Company, New York.)

with C = Gin, T = TI(Z1) Td0) = Tin T 2 ( Z 1= ) T(Z)

On a y-T diagram this equation is a straight line with a slope of f . This is shown as dotted lines in Figure 9.11. A higher Ti, shifts the line more to the right of the diagram. Another relationship between y and T can be obtained by inserting Eq. 9.46 into the right hand sides of Eqs. 9.42 and 9.43: Eq. 9.42 then gives y ( Z ) = f l [ c n J and Eq. 9.43 gives T ( Z ) = fz[Z,], combination of which yields a relationship between y ( Z ) and T ( Z ) . For a given reaction system, y ( Z ) is a function solely of T ( Z ) if yin is zero as Tin is eliminated in the process of linking the function fxto f2 for the relationship. Because of the Arrhenius temperature dependence, this relationship gives a sigrnoidal-shaped curve (solid line) as shown in Figure 9.1 1. Since the relationship is independent of T i n Q i n = O), there exists only one curve for a given reaction system. The reaction system, on the other hand, has to satisfy not only this relationship, but also Eq. 9.5 1, meaning that the intersection points between the two curves are the steady state operating conditions. If the choice of 5 and Ti, is such that Eq. 9.51 crosses the sigmoidal curve at three points as in curve b, there exist three steady states, bl, b2 and b3. Outside the region defined by lines a and c, there is only one steady state. These lines a and c themselves touch the sigmoidal curve at a tangent and two steady state operating points exist. For the line b, the points bx and b3 represent stable operating points. The middle point, b2, is unstable in the sense that a slight perturbation in operating conditions moves the operating point to either bl or b,. For instance, a slight increase in T ( Z ) at b2 will cause a shift to bl since this increase has a

at z = 0 at z' = O at z = Z or z' = 2'

where A, = heat exchanger surface area per unit volume (2/radius) Fin= inlet molar flow rate y = conversion U = overall heat transfer coefficient Z = reactor outlet 2'= heat exchanger outlet Combining Eqs. 9.42 and 9.43 gives, when integrated from z = 0 to Z

Assuming

1I

= (vp)=, Eq. 9.45 gives:

I

a

when evaluated at z' = 2'.Equation 9.47 shows that (T2 - TI) in Eq. 9.44 is constant. Therefore, Eq. 9.44 gives, after integration:

UACZ1

Tl(Z') - Tin = -[Tz(O)- Tin ] (vP)~ Adding T ( Z ) to bath sides of this equation and using Eq. 9.48 gives: T ( z ) - f i n = (1

+ TV P) G ) ~[ T ( z ) - T1(z1)] UACZf

I

(9.50)

This equation together with Eq. 9.46 yields: ~ ( 2-) in =

T ( z ) - Tin

Fin ( - A H )

i

5 [ T ( Z )- Tin]

(9.5 1)

j i 1

"r ( Z )

Figure 9.11 Possible steady state operating points for autothermic reactors.

much larger effect on the heat generated than the heat exchanged (the sigmoidal curve has a much larger slope at around b2 than the line 6). This is very similar to the phenomenon found for stirred-tank homogeneous reactors (e.g., Smith 1981). It is often undesirable to operate near the extinction point (where the heat of reaction cannot the heating requirement) since a perturbation may lead to the quenching of the reaction (blowout). Nevertheless, the TVA ammonia reactors are operated near the extinction point to maximize throughput. Similar results can be obtained for the other arrangements shown.in Figure 9.10. Results for the recycle arrangement are given by Reilly and Schmitz (1966).

9-8

Figure 9.12 Radial flow reactor and idealized flow patterns. (Hlavacek and Votruba in Chemical Reactor Theory: A Review, ed. Lapidus and Amundson, Q 1977. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.)

VARIATIONS OF FIXED BED REACTORS

Radial flow reactors are used in large capacity units where the pressure drop can become a limiting factor. Radial reactors are used in catalytic reforming and ammonia synthesis to eliminate problems of pressure drop associated with high throughput. A diagram for a radial flow reactor is given in Figure 9.12. Also given are idealized flow patterns [(a) and (b) in the figure] for a section (section A in the figure) of the catalyst bed. As shown in the figure, the fluid flow through the catalyst bed is either from a center pipe toward the wall or from the annulus to the center pipe. The void space in the vessel is for transportation of fluid and no reaction takes place. According to the idealized flow patterns (Hlavacek and Kubicek, 1972), the slice of the bed cut in the radial direction can be considered as a fixed-bed with the inlet condition at r = R 1 and the outlet condition at r = Rz. Conservation equations written for this slice should be applicable to the whole length of the bed. Therefore, the conservation equations of Eqs. 9.27 through 9.36 are applicable and aT/az replaced by to radial flow reactors with the convection ac/ar and a T / a r . This model is then identical to the one-dimensional model with axial dispersion as for the usual fixed-bed reactor. The same boundary conditions as those of Eqs. 9.10 and 9.1 1 apply, but this time Eq. 9.10 applies at r = R and Eq. 9.1 1 at r = R2,with z replaced by r for the flow pattern (a) in Figure 9.12. Unless the bed depth is quite shallow, the dispersion term can be neglected, resulting in a plug-flow model in the radial direction. With the advent of automobile catalytic converters, monolithic catalyst beds have been developed to convert unburned hydrocarbons and carbon monoxide to combustion products. These oxidation reactions are highly exothermic. Thus, the catalyst is deposited only on the surface layers of the . Other design considerations are a mechanically strong bed to withstand mechanical and thermal shock, and low pressure drop. These requirements led to monolithic catalyst beds of various configurations. At the heart of the various configurations is an open channel, the inside wall of which is "coated" with the catalyst. As fluid es through the open channel, it reacts on the catalyst coated on the inside wall. For the reactant i in the channel, one has:

where Ac is the cross sectional area of the channel, I the circumference of the channel, (C,)*the concentration of species i at the catalyst surface, and (kg)i

the film mass transfer coefficient for species i. Since the mass transfer rate should be equal to the reaction rate at the surface at steady state:

ignition are clearly delineated in of the system parameters. Utilization of this phase diagram for the design is then examined.

where the global rate R b is based on the external surface area of the catalyst. Similarly, the heat balance for the reacting fluid is:

designed to have a small ratio of pellet size to reactor radius so as to avoid undesirable effects of dispersion. It is therefore adequate to use a plug-flow model for such reactors. I f f , is the global rate expressed in of conversion, the conservation equations are:

As discussed earlier, reactors for highly exothermic reactions are usually

On the other hand, the heat balance for the solid wall, which is assumed to be at T,,is:

where 6 is the wall thickness, which is assumed to be small enough to give the area perpendicular to the heat flux as 81. The boundary conditions for the fluid are the channel inlet conditions. The boundary conditions for Eq. 9.55 are:

d T'. =0 dz

at the inlet and outlet

Equations 9.52 through 9.56 represent the basic model for monolithic catalyst beds. More detailed modeling of these beds can be found in the articles by Hegedus (1975) and Votruba et al. (1975). Design of a fixed-bed for highly exothermic reactions is perhaps the most demanding because of the sensitivity problem that these reactions can cause. In the next section the design of a countercurrent reactorheat exchanger for highly exothermic reactions is considered.

where G, = mass specific heat of coolant r= mass specific heat of reacting fluid fa = (1 - EB)RG F = molar flow rate (mol/time) of key species m = mass flow rate of coolant M = mass flow rate of reacting fluid Tc = coolant temperature y = conversion Assuming that mass flow rates, specific heats, and the overall heat transfer coefficient U are constant, the above equations reduce to:

9-9 DESIGN O F A COUNTERCURRENT REACTORIHEAT EXCHANGER W I T H PARAMETRIC SENSITIVITY

where y = ( f i n - F)/F,,, The parametric sensitivity problem associated with a countercurrent reactodheat exchanger in which a highly exothermic reaction occurs has been discussed in Section 9-5. The overriding design concern is that no runaway situation should occur so that the reactor can be operated safely. Criteria developed for a firstorder reaction have been presented in Figure 9.1 for the safe operation of a homogeneous reactor. This section develops a phase diagram of inlet reactant and coolant temperatures on which the regions of safe operation, negligible reaction, and possible

f = z/Z

fin = inlet molal flow rate of key species

Given a reaction, the design problem is to choose the parameters involved in a l through a4 and the reactant and coolant inlet temperatures that give the best

possible performance of the reactor within the constraint of a no runaway situation. For a given set of parameters, certain choices of inlet temperatures will result in ignition of the reaction and possible explosion. On the other hand, some other choices may be such that no appreciable amount of reaction can take place. The question at hand for the phase diagram is then to find "workable" or safe ranges of reactant (Ti,) and coolant (T,,,) inlet temperatures, which lie between these two extremes, on a Tin-Tcinplane, as shown in Figure 9.13. The phase diagram is therefore divided into three regions: the region of safe operation, that of negligible reaction, and the ignition region, in which the runaway situation can occur. The line separating the safe region from the region of negligible reaction is to be called the quenching line. The line separating the safe region from the ignition region is to be called the ignition line. Referring to Figure 9.13, the ignition line in the region En 2 Tcin is to be called the upper ignition line and that in the region En < Tcin is called the lateral ignition line. Approximate relationships for these lines have been derived from simple runaway criteria derivable from the system equations (Akella and Lee 1983). Procedures for constructing a phase diagram such as the one in Figure 9.13 have been developed based on the approximate relationships, summarized in Table 9.4. Consider a realistic problem involving a general hydrocarbon oxidation process. The reaction system given in Table 9.5 is similar to the one treated by Van Welsenaere and Froment (1970). A gaseous hydrocarbon mixed with oxygen and inerts is fed to a fixed-bed with a countercurrent heat exchanger. The reactor consists of 2500 tubes, 3 m long and 2.5 cm in diameter. The hydrocarbon concentration in the feed is 1.5 mol% in excess oxygen. Under these conditions a single pseudofirst-order irreversible reaction is assumed to take place:

where M and (MW) are the mass velocity and average molecular weight of the reacting fluid, respectively, and N is the number of tubes. In order to construct the phase diagram of Figure 9.13, one has to know the relationships for various lines. According to step (7) in Table 9.4, the quenching line is defined by:

+ ( Tin - Tcin)exp(a33.6)

TI = Tci, and

1.956(109)exp(- l3,636/ TI) = 0.01

(9.68)

when the parameters in Table 9.5 are used. Since e-33.6 is negligibly small, TI = Tc,, practically, and hence, the quenching line is given by a straight line parallel to the ordinate at Tcfn= 525°K.Consequently,' the quenching line for the upper operating zone is given by Tcin = 5 2 5 O K - The quenching line for the lower operating

I Upper Operating Zone

0

lgnition Region

ol,

-

where p is the partial pressure of the hydrocarbon. Since the concentration of the key reactant is very low, the total number of moles in the gas stream can be assumed constant to give:

I

Simulation Data Ignition Points 0Quenching Points

Negligible Reaction Region

520

It follows then that the function f, is:

/

500

I)

Quenching Line I

I

I

where P is the total pressure and Winis the feed mole fraction of the hydrocarbon. Since the reactor consists of 2500 tubes, a l through a4 in Eqs. 9.60 through 9.62 should be rewritten appropriately for the system:

I I

300

500

1

I

I

520 540 560 580

1

I

.

600 620 640

Coolant Inlet Temperature

I

1

1

660 680 700

I

720

K (Tc+ ~n)

Figure 9.13 Phase diagram of inlet temperatures for a hydrocarbon oxidation reaction. (Akella and Lee 1983)

Table 9.4 Procedures for Constructing Phase Diagram 1

Global rate f, =f,(Tt,,j~)

of John Wiley & Sons, Inc.)

- Tc,, plane with

(1)

The phase diagram is plotted on the Tin En as the ordinate.

(2)

The phase diagram is divided diagonally along the En = Tci, line. The upper section where Tt, 2 Tc,, has an upper operating zone and the lower section where Z n < Tcinhas a lower operating zone. The upper operating zone is bounded on the left by the quenching line, on the top by the upper ignition line, and by the diagonal (see Figure 9.13). The lower operating zone is bounded on the right by the lateral ignition line, at the bottom by the quenching line, and by the diagonal, The upper ignition line is defined as the locus of the lowest values of (Ti,,T,,,) satisfying the conditions

(3) (4)

(5)

with (6)

Tc (;=o = Kin

+ CI $- Cz(Ttn - Gin);

(7)

Tcin as the abscissa and

= Tc [;=I

The lateral ignition line is the vertical line at Tci, defined by:

-dTctn --0.

such that:

Table 9.5 Reaction System of Hydrocarbon Oxidation (After Froment and Bischoff, Chemical Reactor Analysis and Design, @ 1979, Wiley. Reprinted by permission Reactor: 2500 packed tubes of 3 rn long and 2.5 crn diameter Parameters .--

Reactor length (2) Reactor radius (R) Mole fraction of hydrocarbons in feed (Win) Molecular weight of feed (M W) Feed mass velocity or mass flow rate per area (M/NwR2)1 Number of tubes (N) Coolant mass rate ( m ) Heat of reaction (-AH) Average specific heat of gas mixture (q,) Average specific heat of coolant (&) Overall heat transfer coefficient (U) Pre-exponential factor (ko) Activation energy divided by gas constant Catalyst bulk density ( p ~ ) Total pressure (P) Pellet diameter (d,) Feed temperature ( E n ) Coolant inlet temperature (Tci,)

3m 2.5 cm

0.015 29.5 kg/kmol 5000 kg/rn2hr 2500 1.q10s) kg/hr 1.28(10s) kJ/kmol 1.0 kJ/kg°K 4.0 kJ/kg°K 3.5(102)M/m2hr0K 8.5(107) krnol/kg hr 1.3636(1(r)"K 1.3(105) kg/rn3 1 atm 3 mm variable (3300°K) variable

dTm

The quenching line is defined by:

with

T1 = Tci, +(Tin - TC,,)e-a3;

where T,,, = maximum temperature Cl = Fin(-AH)/mCw C2 = M G ~ / ( ~ c ~

T,= T,I;=,

)

ol through a, are given by Eq.9.63 For f,(Tm,O) in Eq. 9.66 above, simply set TL, = Tm. and y = 0 in the global rate expression,

h( T b n ~ ) .

zone is given when Tin is set to the minimum available feed temperature. In this case it is assumed that G,,= 300°K is the quenching line for the lower operating zone. These lines are shown in Figure 9.13. According to step (5) in Table 9.4, the upper ignition line is obtained from:

and

TcIi=O

+

= Tcin 13.31

+ 0.015 (Zn- Tcin)

(9.69~)

For a given value of Tcin, Eq. 9 . 6 9 ~gives Tqn in of Ti,, which upon insertion into Eqs. 9.69a and 9.69b yields En values. The lower value of Tf, (one from 9.69a and the other from 9.69b) is then chosen for the value of TCi,. Since the quenching line is given by Tc,, = 525 K, pairs of (Tin - Tc,,) corresponding to Tc,, = 525, 550, 575, 600, etc. have been calculated. A continuous curve drawn through these points is shown in Figure 9.13. According to step (6) in Table 9.4, the lateral ignition line is given by: Tcin = T m 3.79 (loL0)~xP(-13,636/Tm) (9.70) such that dTcin/dTm= 0. Therefore, the lateral ignition line is the vertical line at TCin= 621°K.This line is also shown in Figure 9.13. Using these relationships, the procedures of Table 9.4 were followed to construct the phase diagram of Figure 9.13 for the present problem. The dark and open circles in the figure are the temperatures corresponding to the onset of ignition and to the quenching region

numerically calculated using the original system equations for comparison. It is seen that the phase diagram predicts the true nature of the reactor behavior very closely. It is evident from Figure 9.13 that the phase diagram not only tells whether a set of inlet temperatures, which are most susceptible to unexpected fluctuations, would guarantee safe operation, but also shows the margin of safety. The boundaries surrounding the safe operation region are defined uniquely by the system parameters. It is thus possible to expand the boundaries of the safe operating region by proper selection of the system parameters, which in turn simplifies the design problem considerably. The design problem for a reactor/heat exchanger is to choose reactor length (Z), reactor diameter (R), overall heat transfer coefficient (U), coolant and feed rates ( m and @), feed composition ( W i n ) ,and inlet temperatures (Ti, and Tci,) in such a way that design objectives are achieved, yet the reactor is safe from any possible runaway due to uncontrollable fluctuations in the operating variables. Due to the large number of design and operation parameters involved, the design problem is generally complicated. Since all these parameters are lumped into the coefficients a l through a4 of the system equations and these coefficients in turn define the boundaries of the safe operating region, the design problem can be considered from the viewpoint of the phase diagram. One broad objective of the design is the proper choice of parameters such that the reactor can be operated safely over a wide range of temperature, since higher temperatures will result in higher conversions. Thus, the design problem can be condensed into a single problem of expanding the boundaries of the safe operation region by moving the quenching and ignition lines outward or upward, subject to the requirements of throughput. Consider the example of hydrocarbon oxidation. The quenching line given by Eq. 9.67 is exclusively determined by a l . This line is to be pushed to the left in Figure 9.13, allowing high conversion even at low coolant temperatures. In accordance with Eq. 9.65, this can be done by increasing 2, or decreasing M and Wt,, which results in either a larger reactor or lower throughput. There should be an optimum search. In addition, the effect of changing these variables on other system parameters should also be taken into . Nevertheless, since Wi, is not involved in any other system parameters, in principle a1 can be varied independently. The second quenching line does not pose a design problem since it is always represented by the lowest allowable feed temperature. The upper ignition line, on the other hand, is to be elevated in Figure 9.13. From Eq. 9.69, it is not possible to make any a priori comments as to the effect of a l through a4. Nevertheless, a search can be made using Eq. 9.69 for moving the line upward. The movement of the lateral ignition line on the phase diagram is mainly controlled by the ratio of a2/a3. This line, given by Eq. 9.66, can be moved to the right of Figure 9.13 by decreasing a2/a3 or by decreasing the ratio R/U since a 2 / a 3 = (-AH) R / U . While decreasing R causes a reduction in throughput, increasing U (choice of material of construction and coolant) does not have any adverse effect on throughput. One interesting observation emerges from this analysis: fewer design variables are involved for reactor operation in the lower operating zone.

The phase diagram of Figure 9.13 reveals an important point. For a given coolant inlet temperature, any point in the upper operating zone is more sensitive to ignition than a point in the lower operating zone. This sensitivity is shown in Figure 9.14. It is seen that a slight increase in feed temperature in the upper operating zone can lead to ignition, whereas the sensitivity is not as severe in the lower operating zone. Therefore, from the sensitivity viewpoint, it is always better to operate the reactor in the lower operating zone than the upper. This fact leads to an interesting design alternative: a countercurrent reactor/heat exchanger without preheater, as opposed to the usual arrangement of a preheater followed by a reactor/heat exchanger. This alternative eliminates the sensitivity problem with respect to changes in the reactant inlet temperature and allows elimination of the preheater, reducing fixed costs. The additional reactor length required of this design for the same conversion is minimal when compared with the usual design with preheater (Akella and Lee 1983). The sensitivity with respect to coolant inlet temperature remains almost the same for both types of reactor configuratian, as shown in Figure 9.15. This sensitivity problem is much less severe than that with respect to feed temperature since there is much better control over the coolant inlet temperature. However, this precludes the attainment of exit conversions in excess of a certain level, for instance, 60% in Figure 9.15, since around this region even a slight increase in the coolant temperature leads to runaway, as shown in Figure 9.13, no matter what the feed temperature is. One way of circumventing this predicament is to rkalize that faster heat removal from the reactor flattens the reactor temperature profile, and thus, the reactor becomes operable even at higher coolant temperatures. This can be accomplished by increasing the value of a3 in Eq. 9.61, which can he varied independently of the other a's. The results when a3 is increased by 50% to 150% are shown in

l OWER QPFRATING RANGE

90

300

4 y

-

350

400

450

UPPER OPERATING RANOE

500

FEED TEMPERATURE

550

600

650

700

K (T,,,)

Figure 9.14 Sensitivity of conversion with respect to feed temperature.

Summary

COOLANT INLET TEMPERATURE 'K

Figure 9.15 Sensitivity of conversion with respect to coolant inlet temperature.

Figure 9.16 for the reactor configuration without a preheater; slightly worse results are obtained for the configuration with a preheater. In view of the advantages discussed, the reactor configuration with no preheater should be considered as a viable alternative to the usual configuration for highly exothermic reactions. The results obtained in this section are also applicable to the case of constant wall temperature, in which case the parameter a4 can be set to zero for the construction of the phase diagram. Since a plug-flow model was used for the phase diagram, radial dispersion should have the effect of reducing the range of safe operating conditions; axial dispersion should tend to broaden the range. Figure 9.16 Effect of increasing the parameter a3 on reactor sensitivity.

L 575

I

I

I

I

600

625

650

675

COOLANT INLET TEMPERATURE K (T

1 700

Design and analysis of heterogeneous fixed-bed reactors requires solving conservation equations of mass and energy for both the fluid phase (reactor) and the solid phase (pellet). These equations in their full form are extremely difficult to solve. The mode of reactor operation and the nature of the reaction, however, allow considerable simplification. Even with the simplifications that can be made, numerical solutions are not easy to obtain. In addition, there are design requirements that have to be satisfied. There are usually limitations placed on the pressure drop and operating conditions for safe reactor operation. Special types of fixedbeds are sometimes necessary as in radial flow reactors and monolithic catalyst beds. The desire to utilize the heat of reaction motivates the coupling of the reactor with a heat exchanger. If the reaction is highly exothermic, the reactor becomes quite sensitive to changes in operating conditions. For such cases, the phase plane diagram and the phase plane analysis can be utilized for the design. When axial dispersion is significant, the question of multiple steady states (Luss and Arnundson 1967) becomes important for the reactor start-up. Unless the bed is quite shallow, however, this should not pose a problem. Not considered in this chapter was the design and analysis of fixed-beds under conditions of catalyst deactivation. This is treated in the following chapter based on the approach of reactor point effectiveness. NOTATION

parameters defined in Eq. 9.63 (or in Eq. 9.65) parameters in orthogonal polynomial z-dependent parameters for radial concentration profiles z-dependent parameters for radial temperature profiles Biot number for mass Z/R heat exchange surface area per unit voluime concentration bulk fluid concentration concentration of species i Cb/(Cb)fn constants defined in Table 9.4 specific heat mass specific heat of coolant mass specific heat of reaction fluid pellet size (pellet diameter) dispersion coefficient for mass axial dispersion coefficient effective diffusivity radial dispersion coefficient activation energy

T1 Tad Tb

Tc Tin

Tcin

friction factor global rate in of bulk fluid temperature and conversion molar flow rate of key species mass rate of fluid/area film heat transfer coefficient wall heat transfer coefficient for two-dimensional model heat of reaction grid point in z-direction grid point in r-direction mass flux vector for species i molar flux vector for species i rate constant film mass transfer coefficient heat dispersion coefficient (pK = h) axial dispersion coefficient for heat radial dispersion coefficient for heat grid interval in z-direction; circumference of an open channel characteristic pellet length grid interval in r-direction; R / d p mass flow rate of coolant mass flow rate of reaction fluid Z/d, 2-dependent parameters for radial profiles of concentration and temperature, respectively parameter defined by Eq. 9,26; number of tubes partial pressure and total pressure, respectively axial and radial Peclet numbers for heat, defined in Eq. 9.19 axial and radial Peclet numbers of mass, defined in Eq. 9.18 radial coordinate (0at tube center) rate of formation for species i in mass unit per unit volume of fluid intrinsic rate in moles per unit volume of pellet radius of tube ri in mole unit global rate of formation by key species global rate based on external surface area Reynolds number based on d, parameter defined by 9.25 time temperature reactor outlet temperature used in Eq. 9.67 adiabatic temperature at reactor outlet bulk-fluid temperature coolant temperature Tb .at reactor inlet (at z = 0) coolant inlet temperature (at z = Z)

maximum temperature in reactor Tb/(Tb)fn fluid velocity vector overall heat transfer coefficient based on average temperature superficial fluid velocity mole fraction pellet coordinate conversion of key species; C b or Tb in difference equations (Eq. 9.38) axial coordinate (z = 0 at reactor inlet) z/Z heat exchanger coordinate reactor length heat exchanger length 1

Greek Letters

ZDr/R2v for mass, and zKr/R2v for heat constants in Eq. 9.38 quantity defined in Eq. 9.26; wall thickness bed porosity effectiveness factor Z/v; weighting factor for implicit numerical method in Table 9,2 thermal conductivity effective thermal conductivity catalyst bulk density mass density of species i quantity defined in Eq. 9.24 Thiele modulus Subscripts

b c

E 1

f i

in s I

P

3

; I

1

bulk-fluid coolant fluid species i reactor inlet pellet surface pellet

PROBLEMS

9.1. List the assumptions made in writing Eq. 9.5 from 1Eq. 9.1. 9.2. Suppose that the pellet size is such that the diffusional effects are negligible. What equations can be deleted from Eqs. 9.6 through 9.15? Does this necersar-

ily mean that the global rate ( R c ) is equal to the intrinsic rate (rc)? Under what conditions is Rc equal to r,? If RG # rc, what additional conditions are neeeded in addition to Eqs. 9.6 and 9.71 9.3. According to a dispersion model (Carberry 1958), the dispersion coefficient is given by:

Da =-

l2

20D

1: mixing length .s l/v

80

If the axial mixing length 1 in a packed bed is defined as the bed length divided by the number of perfect mixers n, 1 = Z/n. Therefore, a reactor with axial dispersion may be considered equivalent to n CSTR's in series where:

Show then that a packed bed is equivalent to Z/d, perfectly mixed, stirred tanks in series. Use the typical P,, value of 2. By extending this analogy to radial dispersion, show that the number of perfectly mixed cells in the radial direction is 5(R/dp) when P,, = 10. 9.4. Consider a bed packed with spherica.1 pellets. Show that the relative pressure drop is given by:

when the wall temperature is constant at T,. The subscript 0 is for the centerline quantities. The cup-mixing quantities are denoted by the overbar. From the expressions for y and c,, show that:

The value of n~ can be found from the fact that Tw is constant: dTW/dz= 0. Obtain 5 ordinary differential equations from Eqs. 9.27 and 9.28 for yo, To, yw, j, and T. Note that the equations for the centerline quantities can be obtained by evaluating the radial derivatives at r' = 0 and using the assumed profiles. 9.7. Determine the length of a reactor for 35% yield (B f~med/Ai.) of phthalic anhydride by partial oxidation of naphthalene in air over VzO5. Assume each reaction to be of first-order and that the coolant temperature is constant.

P ,

=8 =6 P = 1 atm

Phr

Ti, = 320°C Re=184 R=4cm Tc = 310°C d, = 0.5 crn yt, = 0.5% D, = 0.1 crn2/sec EB = 0.5

napthalene

-AH,

-A Ho

anhydride ---+ C02and HzO kn

kl

for an isothermal reactor. This relationship should be applicable to the radial flow reactor with the idealization discussed in Section 9-8. If a fixed-bed with a length to depth ratio of 50 is operated as a radial flow reactor, how much of an increase in throughput would you expect if the relative pressure drop is the same for both modes of operation? Assume P to be the same for both modes. 9.5. Write the finite-differenceversion of Eqs. 9.27 through 9.32 using the explicit method of Figure 9.2a and the implicit method of Figure 9.3. Draw flow sheet diagrams for both methods. Make the equations dimensionless first. 9.6. Show for the averaging method of Ahmed and Fahien (1980) that the use of boundary conditions leads to:

Use the kinetic viscosity of air for the reaction mixture. Use the two-dimensional model. Assume a spherical, isothermal pellet. 9.8. For the following reaction carried out in an adiabatic reactor, calculate the reactor length required for 40% conversion.

A

k ---+

products

REFERENCES

L = 0.5 cm, bed cross-sectional area = lo4 cm2 E B = 0.5, P = 1 atm, yin (mole fraction) = 0.06 Molar flow rate = 1.5 mole/hr cm2, D, = 0.05 cm2/sec (-AH) = 20 kcal/mol, Tin = 580°C

Use the physical properties of air for the mixture. Also use the same j factor correlations as given in Problem 9.7. 9.9. Construct a diagram similar to the one given in Figure 9.11 for the following conditions: in Eq. 9.51 = 125OK-I Rc = 1017 exp(-18,000/RT) Reactor length = 7 rn

mol/hr

. m3

m2 vp/(-AH)Fi, = 2.5 X lom3(l/K)

fin = 1.5 X lo4 mole/hr Ee = 0.5

Obtain the extinction temperature, i.e., the Tinthat results in extinction of the reaction. Alsa calculate the Tinthat gives a conversion of 0.5. Expand the Arrhenius exponential around T ( Z ) to rewrite it as A exp(bT). 9.10. For the reaction system of Table 9.5, calculate the reactor lengths for 29.5% conversion, first for a reactor with a preheater (Tin= 600°K)and then for the reactor without a preheater ( F , = 3006K).Use (Tc)inof 600°K.Also calculate the preheater length for the reactor with a preheater by assuming that the exit stream from the reactor is used to exchange heat with the cold feed in a preheater geometrically and thermally similar to the main reactor. Assume that the main temperature difference across the heat exchanger is 10°K. Reactor Length (rn) With preheater Without preheater

3

3.25

Preheater Length (m) 2.65 0

9.1 1. Construct a phase diagram of inlet temperatures for the following reaction system:

All quantities are in cgs unit.

Akella, L.M. and H.H. Lee, AIChE J., 28, 87 (1983). Ahmed, M. and R.W. Fahien, Chem. Eng. Sci., 35, 897 (1980). Anes, W.F., Nonlinear Partial Differential Equations in Engineering, Academic Press, New York (1965). Baddour, R.F., P,L.T. Brian, B.A. Logeais and J.P. Eymery, Chem. Eng. Sci., 20, 28 1 (1965). Barkelew, C.R., Chern. Eng. Progr. Symp. Ser., 55(25), 38 (1959). Bird, B.R.,W.E. Stewart and E.N. Lightfoot, Transport Phenomena. Wiley, New York ( 1 960). Carver, M.f3., in Numerical Methods fir Diflercntial Systems, p 243, eds. Lapidus and Schiesser, Academic Press, New York (1976). Cappelli, A., A. Collina and M. Dente, Ind. Eng. Chem. Proc. Des. Dev., 11, 184 (1972). Carberry, J.J. and D. White, Ind. Eng. Chem., 61. 27 (1969). Carberry, J.J., Chemical and Catalytic Reaction Engineering, McGraw-Hill, New York (1976). Carberry, J.J., Can. J. Chem. Eng., 36, 207 (1958). Carnahan, B., H.A. Luther and J.O.Wilkes, Applied Numerical Methods, Chap. 7, Wiley, New York (1969). Dente, M., and A. Collina, Chim. et Industrie, 46, 752 (1964). Dumez, F.J. and G.F. Froment, Ind. Eng. Chem. Proc. Ds.Dev., 15, 291 (1976). Finlayson, B.A., Chem. Eng. Sci., 26, 1081 (1971). Finneran, J.A., L.J. Buividas and N. Walen, Hydrocarbon Processing, 51. 127 (1972). Froment, G.F. and K.B.Bischoff, Chemical Reactor Analysis and Design, Wiley, New York (1979). Hegedus, L.L., AIChE J., 21, 849 (1975). Hicks, R.E.,Ind. Eng. Chem. Fund., 9, 500 (1970). Hlavacek, V. and J. Votruba in Chemical Reactor Theory, Chap. 6, eds. Lapidus and Amundson, McGraw-Hill, New York (1977). Hlavacek, V. and M. Kubicek, Chern. Eng, Sci., 27, 177 (1972), Hlavacek, V., M. Marek and T.M. John, Coll. Czrhoslov. C h m . Comm., 34 3868 (1969). Lapidus, L., Digital Computation .for Chemical Engineers, McGraw-Hill, New York (1962). Levenspiel, O., Chemical Reaction Engineering, Wiley, New York (1972). Liu, S.L., AIChE J., 16, 501 (1970). Luss, D. and N.R.Amundson, Chem. Eng. Sci., 22, 253 (1967). Mehta, D. and M.C. Hawley, Ind. Eng. Chem. Proc. Des. Dev., 8, 280 (1969). Mihail, R. and C. Iordache, Ind. Eng. Chem. Proc. Des. Dev., 31, 83 (1976). Minhas, S. and J.J. Carberry, Br. Chem. Eng., 14, 799 (1969). Mkhlin, S.G.and K.C. Srnolitskiy, Modern Analytic and Computational Methods in Science and Mathematics, eds. Bellman and Kalaba, American Elsevier Co., New York (1967). Morbidelli, M. and A. Varma, AIChE J., 28, 705 (1982) Price, T.H. and J.B. Butt, Chem. Eng. Sci,, 32. 393 (1977). Reilly, M.H. and R.A. Schmitz, AIChE J., 12. 153 (1966). Smith, J.M., Chemical Engineering Kinetics, 3rd ed., Chap. 5, McGraw-Hill (1981). Van Heerden, C., Ind. Eng. Chem., 45, 1242 (1935). Van Welsenaere, R.J. and G.F. Froment, Chem. Eng. Sci., 25, 1503 (1970). Villadsen, J.V. and W.E.Stewart, Chem. Eng. Sci., 22, 1483 (1967). Votruba, I., 0.Mikus, K. Nguen, V. Hlavacek and J. Skrivanek, Chem. Eng. Sci., 30, 117, 201 (1975). Wehner, J.F. and R.H. Wilhelm, Chem. Eng. Sci. 6, 89 (1956).

CHAPTER

10

Design and Analysis of FixedBeds: Approach of Reactor Point Effectiveness

10-1

INTRODUCTION

Rigorous procedures for the design and analysis of fixed-bed reactors based on heterogeneous models have been treated in Chapter 9. While such procedures can be used to solve the design and analysis problem in principle, the time and effort involved is often too costly. This is particularly so when the kinetics are too complex to yield an analytical expression far the effectiveness factor. The stability of the numerical scheme may also pose a problem. This chapter describes the use of reactor point effectiveness as an alternative to the rigorous design procedures of Chapter 9. For a fixed-bed reactor, the reactor conservation equations

are: Lc(Cb)= -Rc(pellet and film conditions)

&(Tb) = (-AH) Rc (pellet and film conditions) where the linear operators LC and &- operate on the bulk concentration and temperature, respectively. The linear operators take on different forms depending on whether a plug-flow model or dispersion model is used. The conservation equations for the pellet in which the reaction takes place are given by:

where the linear operators and operate on the pellet concentration and temperature, respectively. The intrinsic rate r, is related to the global rate Rc through the transport processes across the pellet-bulk fluid interface. For the design and analysis of fixed-beds, therefore, the pellet conservation equations have to be solved first, which involves solving nonlinear two-point boundary value problems. These solutions are then related to the bulk conditions through heat and mass

DESIGN AND ANALYSISOF FIXED-BEDS 333 transfer across the pellet-bulk fluid interface. The reactor conservation equations can then be solved for the reactor concentration ( C b )and temperature (Tb). This heterogeneous reactor problem can be transformed into a homogeneous one, in which only the reactor conservation equations are involved, through the use of the reactor point effectiveness. The reactor point effectiveness developed in Part I1 enables the global rate to be expressed in of the bulk-fluid conditions. Therefore, the design and analysis problem reduces to solving:

tionship. This is quite adequate since the ratio of the bulk-fluid to the pellet surface temperature is close to unity. This approximation can be eliminated if desired, as will be seen in the treatment of reactors affected by catalyst deactivation. This leaves the third approximation regarding the pellet center concentration. For the intrinsic kinetics, rc = kf(C), the internal effectivenessfactor (Chapter 4) is:

which are equivalent to the conservation equations for homogeneous reactors. The usefulness of this approach will be fully appreciated when reactions affected by catalyst deactivation are considered. The approach of reactor point effectiveness, however, does require some approximations. The adequacy of these approximations will be examined in detail. This will be followed by a section on the equivalence between the plug-flow model and the axial dispersion model, and that between the plug-flow and the radial dispersion model. With these equivalences established, it is possible to concentrate on the plug-flow model. Using this model, the design and analysis problem for reactions affected by diffusion, those affected by both diffusion and chemical deactivation, and those affected by catalyst sintering will be treated in detail. Detailed design and analysis procedures result from this treatment. The simplification made possible by the use of the reactor point effectiveness allows a close examination of the design of a reactor affected by catalyst deactivation, the characteristic of which is time-dependence. This leads to the optimal design for a reactor whose performance is time-dependent. Finally, reactor design involving multiple reactions is considered, using essentially the approach of reactor point effectiveness.

and the generalized modulus is:

10-2

ADEQUACY OF APPROXIMATIONS

The basic approximations made in arriving at the reactor point effectivenessare: (1) isothermal pellet, (2) negligible external mass transfer resistance, and (3) estimation of the pellet center concentration by a simple relationship when the reaction is not severely diffusion-limited. The first two approximations are quite adequate in view of the fact that the mass Biot number is of the order of hundreds under realistic reaction conditions. Both theoretical and experimental justifications for these approximations have been given in Chapter 4. The first approximation will be relaxed when reactions affected by pore-mouth poisoning are considered since a definite temperature gradient then exists within the pellet. An additional approximation is the representation of the difference between the Atrhenius exponentials evaluated at the pellet surface and the bulk-fluid temperatures by a linear rela-

where the pellet surface concentration has been replaced by the bulk concentration in view of the second approximation. As was shown in Chapter 4, the pellet center concentration Cc can be set to: 0

Cc/Cb=

I ,

I'

cosh #jc

for +c 3 J for 0.3 < +c

<3

The approximation for the pellet center concentration when 0.3 < +c < 3 is based on the fact that the value of the integral in Eq. 4.73 is quite insensitive to an error in the estimated value of Cc. Because of this tolerance, the approximation is adequate in the region of moderate diffusion effect (0.3 < +c < 3). On the other hand, the error in the calculated value of qi can be significant as +c approaches zero. This should not pose any problem since the reaction can be considis small, say ered diffusion-free for all practical purposes when the value of less than 0.3. As discussed in some detail in Chapter 4, the approach of reactor point effectiveness cannot be used when the intrinsic kinetics exhibit an apparent negativeorder behavior.

10-3

EQUIVALENCE OF FIXED-BED MODELS

The reactor conservation equations given in Chapter 9 contain dispersion . Depending on the relative importance of dispersion effects, there results a twodimensional model with radial dispersion, a one-dimensional model with axial dispersion, or a plug-flow model. If plug-flow is taken as the basic model, the dispersion models can be viewed as corrections to the plug-flow model, resulting

i.e., T R =~ F(Cb), Burghardt and Zaleski (1968) obtained an approximate solution of Eq. 10.1 using a perturbation technique:

from mixing effects. If the deviation from the plug-flow model due to axial dispersion is small, which is usually the case, this deviation can be ed for by a firstorder correction term added to the solution of the plug-flow model. A perturbation technique (Cole 1968) will be used to show the equivalence between the plugflow model and the one-dimensional model with axial dispersion. When there exists a radial gradient, the process of averaging over the cross-section usually leads to simpler results. In fact, the quantities of interest are the average concentration and temperature rather than the local quantities. It will be shown that the averaging process leads to an equivalence between the plug-flow model and the two-dimensional model with radial dispersion under certain simplifying conditions. For adiabatic fixed-bed reactors with axial dispersion, the steady state conservation equations are:

where Co= solution for the plug flow model, F = rRC(CO) 4 = F evaluated at reactor inlet

6

=0

This result can be used directly in Eq. 10.1 if a relationship between Tb and Cb can be found such that rRG(Tb,Cb)is expressible solely in of concentration. This relationship has been obtained by Lee (1981) using a perturbation technique:

-AH

-AH

(10.6)

P

where dCo/dy is the concentration gradient obtained from the solution of the plug-flow model. It can be seen that the second term in the right hand side of Eq. 10.6 is a first-order correction for the effect of dispersion on the relationship between Tb and Cb, since the first term represents the temperature change for a plug-flow reactor. Now that Eq. 10.6 gives Tb in of the solution of the plug-flow model and Cb, use of this relationship in Eq. 10.1 allows an expression for r R c in of Cb and known quantities. Therefore, the solutions for the reactor with axial dispersion follow from Q s . 10.5 and 10.6. It is seen that the plug-flow model leads directly to near-accurate solutions of the axial dispersion model. Details of using Eqs. 10.5 and 10.6 will be given when design and analysis are considered in the following section. Consider now the radial dispersion model. The steady state conservation equations are:

where E: = l/Pm = l/(Zv/Da) fi = l/Ph = 1/(Zv/KU) 7 = Z(1 - E ~ ) / v E B = bed porosity p G = molar heat content RG= global rate y = normalized reactor coordinate Z = reactor length v = superficial fluid velocity D, = axial dispersion coefficient for mass K, = axial dispersion coefficient for heat The dimensionless quantities E and are much smaller than unity since the Peclet number for heat (Ph) and the Peclet number for mass (P,) are large for most reactors of interest. The boundary conditions are:

Zi

Nonadiabatic reactors in which heat exchange occurs with the environment are not considered here, since then radial dispersion would dominate over axial dispersion. For an isothermal reactor for which T R is~a function only of concentration,

with the boundary conditions:

a Cb -=o, ar

aTb --=ar

h,Rt pKr (

T

)

atr=l

DESIGN AND ANALYSIS OF FIXED-BEDS 337 where T~ = Z/v D, = radial dispersion coefficient for mass K, = radial dispersion coefficient for heat r = normalized radial. distance, r = 0 at the center R, = tube radius h, = overall heat transfer coefficient based on ( T I - Tc) Tc = coolant temperature T I = Tb at r = 1

ratio of tube diameter to pellet size such that the undesirable radial gradients can be eliminated. Therefore, the design and analysis of reactors can be carried out in most cases based on the plug-flow model. 10-4 DESIGN A N D ANALYSIS OF FIXED-BED REACTORS

Multiplying Eqs. 10.7 and 10.8 by r and integrating the results with respect to r from zero to unity for the average values gives:

The approach of reactor point effectiveness is used in this section to obtain detailed design and analysis procedures for reactions affected by diffusion, those affected by both diffusion and chemical deactivation, and those affected, in addition, by catalyst sintering. Results obtained in Chapters 4 through 6 are used to arrive at the procedures.

10-4-1

where the boundary conditions have been used in the integration, and TI is Tb at r -- 1. The mixing-cup averages are:

C=

1: / 11 1: rcbdr

rdr = 2

rcbdr

rR~dr

rdr = RG 1 T b=T,cb =r

The steady state conservation equations for a plug-flow reactor with countercurrent cooling are:

(10.13)

A comparison between Eqs. 10.11 and 10.12 and the conservation equations for the plug-flow model reveals that the radial dispersion model is the same as the plug-flow model in of average concentration and temperature if:

1; /1j

Reactions Affected by Diffusion

(10.15)

and if the overall heat transfer coefficient is based on the average temperature (U) instead of the fluid temperature at the wall (h,) such that the heat transferred from the reactor t o the coolant is written as 2 u(T - Tc)/R, instead of 2h, (TI - Tc)/R,. In general, the identity of Eq. 10.15 does not hold. The goodness of the plug-flow model, therefore, hinges on a proper choice of the value of U when radial gradients exist. It has been shown that the solutions of the plug-flow model lead directly to near-accurate solutions of the axial dispersion model and that the plug-flow model is equivalent to the radial dispersion model with some simplifying assumptions. Radial gradients usually exist when there is a significant heat effect (Chapter 9). For highly exothermic reactions, the usual design practice is to select a small

where vc = coolant velocity (PC,), = heat content of coolant T c = z/vc The boundary conditions are reactor inlet conditions (at y = 0) for Cb and Tb and the coolant inlet temperature (at y = 1) for Tc. It has been assumed that the velocity of the reacting fluid is constant. The case of a variable velocity will be treated later. The global rate is given by:

which can be rewritten as:

Table 10.1 Design Equations for Reactions Affected by Diffusion

where the expression for the reactor point effectiveness developed in Chapter 4 (Eq. 4.83) has been used. For the reactor design, Eq. 10.16 can be integrated from the inlet to the outlet of the reactor with RG given above to yield:

a. Nonadiabatic, nonisothermal reactor

The right hand side of this equation can be integrated analytically or numerically to give the reactor size (T), given the desired conversion (Gout). In order to integrate this equation with respect to Cb, however, Tb has to be expressed in of Cb.Combining Eqs. 10.16 and 10.17,and integrating the resulting equation from the reactor inlet to any point in the reactor gives: b. Adiabatic, nonisothermal reactor

where Tinis Tb at y = 0 and (Te),,tis Tcat y = 0: This equation provides the relationship necessary for the integration of Eq. 10.21.Because of the split boundary conditions on Tb and Tc, however, iterations on Tc are necessary as detailed below. Design equations for this general case are summarized in Table 10.1. The design of an adiabatic reactor is quite simple since then Eq. 10.22 reduces to: ~b

= Tin -

(-)-AH P

(cb - tin)

-

c. Isothermal reactor

When Eq. 10.23 is used in Eq. 10.21, there results a simple design equation for

adiabatic reactors:

-I

-

p":)"' u~fialda]'12

with I f +c 3 3, C, = 0 for Eqs. (A) through (G). If 0.3 < € 3. obtain Cr by:

where ko is the preexponential factor. It is seen that a straightforward integration as a function of Cb gives the reactor volume for adiabatic reactors. A simpler design equation results for isothermal reactors. These results are summarized in

I f +c

1.2LEo(-AH)

2hRg[Trn - (-AH/pG)(C.b

< 0.3, the reaction is diffusion-free.

.

- Cin)I2

dCb = T

(E)

DESIGN AND ANALYSISOF FIXED-BEDS 341

Cdculate

Table 10.1. While the design procedures for isothermal reactors and adiabatic reactors are straightforward to use, those for nonadiabatic reactors are complicated by the split boundary conditions on temperature. Procedures for nonadiabatic reactors are summarized in Figure 10.1 in the form of a flow chart. Note that Eq. (D) in Table 10.1 is the Euler version of Eq. 10.18 for numerical integration. Using the procedures given in Figure 10.1, a table of T versus Gout can be generated, from which the value of r corresponding to the desired conversion (COuI) can be selected. In order to illuistrate the use of the design and analysis procedures, the reaction system detailed in Table 10.2 has been completely analyzed. For the adiabatic reactor under consideration, the design equations (Eqs. (E) and (F) in Table 10.1) reduce to:

vc)J urlng Eq. (0)In Tabla 10.1.

I1 Obtain Tj

t(Cj) using Eq. (C1.

-

Use the condltlons at I 1 grld polnt from Eq. tJ1. to cmlculate *

Uas Eq. IKl tor Cc

I

with

I

Determine Clcorrsapondlngto yj by adsctlng a value of S that yield# the known value 01 (-ry1) when thla'valur of Cj I8 unad tor the Integratlori.

I

Tb =

En + 4 x

107 (tin - cb)

In view of the fact that +G at the reactor inlet is 5.6, the pellet center concentration was set equal to zero. Eq. (J) in Table 10.1 was used to evaluate the value of +c- The bulk temperature was used for the evaluation of the equilibrium constant 5. The relationship between conversion and residence time (equivalently the reactor

I

Calculate A mnd T./T. from Eqa (I) and (H).

Table 10.2

Parameters for the Example Reaction (Lee and Butt 1982)

I

k = exp

Obtain new

*

(-

T

(Bi), = 2500 (0, = 1 X

+ 14.6

(l/sec).

cm2/sec, L = 1 cm, k,,, = 2.5 crn/sec)

Obtaln Cout and reactor profilea.

Figure 10.1 Design and analysis procedures for nonadiabatic, nonisothermal reactors for a given value of T .

+G = 5.63 at

& = 23.76 kcaI/rnole

the reactor inlet

DESIGN AND ANA.LYSIS OF FIXED-BEDS 343 0.24

1.12

NORMALIZED REACTOR LENGTH

Figure 10.3 Reactor profiles of reactor point effectiveness and &/Tb. RESIDENCE TIME (SEC.)

f

Figure 10.2 Conversion versus residence time for the example reaction of Table 10.2.

size) shown in Figure 10.2 was obtained by integrating Eq. 10.26. Simpson's method was used for the integration. It is seen that there exists a critical size of reactor where an incremental increase in reactor size significantly increases the conversion. This size corresponds to a residence time of approximately 17 seconds. Detailed calculations have been carried out for this case. The ratio of pellet surface temperature to bulk fluid temperature calculated from Eq. (H) in Table 10.1 is shown in Figure 10.3 along with the reactor point effectiveness calculated from Eq. (I). Reactor profiles of bulk fluid concentration and temperature are shown in Figure 10.4. It is seen from these figures that the ratio (Ts/Tb)increases rather rapidly as the conversion increases, whereas the reactor point effectiveness decreases monotonically. This example clearly demonstrates how simple it is to design and analyze a fixed-bed reactor using the approach of reactor point effectiveness, in sharp contrast with the usual approach given in Chapter 9. Consider another example in which there are more than one species involved in the reaction:

A

+ 2B

4

30

I

Suppose further that the effective diffusivities are 10-2, 4 X 10-1, and 2 X 10-1 crn2/s for the species A, B, and D, respectively, and that the reactor inlet concentrations are mol/cm3 for A and 4 x lod5 rnol/cm3 for B with no products in the feed. Here, the rate r~ is sought, expressed solely in of CA such that it

I

1

+ other products

Suppose that the intrinsic rate based on species A is given by:

0

0.2

0.4

0.6

0.8

1 .O


Figure 10.4 Reactor profiles of bulk-fluid conceniration and temperature.

DESIGN AND

ANALYSISOF FIXED-BEDS

345

can be used for the concentration dependency f(C)in Table 10.1. As detailed in Chapter 4, the relationships between the key species and the other species can be obtained by combining the pellet consewation equations for the species involved, which in this example yields:

where the Di's (i = A, B, D ) are the effective diffusivities, and the subscript b denotes the bulk fluid value. If the reactor dispersion coefficients are for the fluid mixture, which is usually the case, the stoichiometric relationship holds and in addition one has the following relationships:

where the subscript in denotes the reactor inlet value. The pellet local concentrations CB and CD then become:

since (CD)in = 0. Therefore, the rate expression for the key species A is:

Since the concentration dependency f ( C ) for the rate expression in Table 10.1 is for the key species, which is A in this example, one has:

where (Cb), = concentration evaluated at reference temperature and pressure r r = T evaluated at reference velocity T, = reference temperature 6 = mole fraction times the change in the number of moles due to reaction per mole af species of interest x b = conversion of species of interest The subscript r is for the reference conditions, which can be taken as the inlet conditions if so desired. The ideal gas law was used to rewrite Eqs. 10.16 and 10.17. Note that these equations are in the same form as Eqs. 10.28 and 10.29, Cb being replaced by the quantity within the bracket in the function RG. Since these equations are in the same functional form as Eqs. 10.16 and 10.17, the same design and analysis procedures developed earlier can be applied. Combining J2qs. 10.18, 10.28, and 10.29, a relationship similar to Eq. 10.22 results. It has been shown in Section 10.3 that the solution of the plug-flow model leads directly to that of the one-dimensional model with axial dispersion. These results can be applied to reactions affected by diffusion. Use of the results leads to the design equations for the fixed-beds with axial dispersion given in Table 10.3. Note that Co is the solution for the plug-flow model and that Eq. (E) in Table 10.3 is used for kb in Eq. (B), whereas Eq. (G) is in Eq. (F) (see Problem 10.9 for the case of E = 0).

10-4-2 Reactions Affected by Diffusion and Chemical Deactivation

Design and analysis are treated separately in this section, for the analysis procedures do .not lead directly to the design because of the time dependence of conversion caused by catalyst deactivation. The analysis procedures are treated first. This expression can be used directly in the design equations in Table 10.1 since the integration of f(C) in Eq. (E) in Table 10.1, for instance, is with respect to CAPand thus (CA)bis constant for the integration of f(C).This example clearly demonstrates the use of the design equations for a single reaction involving many species. As was the case in the earlier example, the bulk fluid temperature can be used for the value of the equilibrium constant K. When the temperature change in the reactor and the volume change due to the reaction are significant, Eqs. 10.16 and 10.17 need to be rewritten to for these changes. These are:

Analysis Procedures

It has been shown in Chapter 5 that the chemical deactivation of a catalyst can be uniform or shell-progressive(pore-mouth deactivation) depending on the relative speeds of the reaction and the transport of the species causing deactivation. Detailed analysis procedures are developed here. While deactivation reactions can cause shell-progressive or pseudo-shell-progressivedeactivation in many cases, the activity profile of a pellet undergoing chemical deactivation can also exhibit a certain gra-

Table 10.3 Design Equations for Adiabatic Fixed-Beds with Axial

Dispersion

The boundary conditions are the usual ones at the pellet center and surface. Here, the fraction of catalyst deactivated, y, is dependent on the pellet coordinate z. Without loss of generality, it may be assumed that the reaction is structure-insensitive, as indicated by the effective rate constant k(l - y) in Eqs. 10.31 and 10.32. The rate of deactivation is much slower than the rate at which concentration and temperature reach their steady state values. This condition leads to the pseudosteady state assumption with respect to the reactor conservation equations. With this assumption, the fraction of catalyst deactivated can be expressed (Chapter 5) as:

where the intrinsic rate of deactivation can depend on concentrations of the main reactant and the poisoning species. For independent deactivation, this reduces (Chapter 5) to: (b)sNs

dv

Q

-y )

uniform shell-progressive

b = o P { - / ~ / [ n - ( ) c - G n ) ] }

(G)

dient that cannot be well approximated by the shell-progressive model. This general case of deactivation with an arbitrary profile is treated later. For a plug-flow reactor in which the catalyst undergoes either uniform or shell-progressive deactivation, an additional mass balance for the species causing deactivation is required along with Eqs. 10.16 through 10.18:

where k, = rate constant for deactivation reaction Q = deactivation capacity of catalyst in moles of the species causing deactivation per apparent pellet volume ( h ) d = L[(kp)d/DpI1l2 Dp = effective diffusivity of the species causing deactivation s = subscript denoting pellet surface d = subscript denoting the boundary between the fresh inner core and the completely deactivated pellet N = concentration of the species causing deactivation The pellet conservation equations are related to their reactor counterparts through transport across the pellet-bulk fluid interface:

where Rp is the global rate of the deactivation reaction and Nb is the bulk concentration of the species causing deactivation. The steady state pellet conservation equations are: where (b),is the mass transfer coefficient for the species causing deactivation. The global rate of deactivation is given by:

DESIGN AND ANAI.YSISOF FIXED-BEDS 349 where q, is the internal effectiveness factor for the deactivation reaction. independent and uniform deactivation this reduces to:

For

In the case of shell-progressive deactivation, the global rate is given by the solution (Chapter 5) to: Equations 10.16, 10.34, and 10.43 then form a set of equations for the analysis procedures along with Eqs. 10.22 and 10.44:

Under the conditions of negligible external mass transfer resistance and an isothermal pellet, Eqs. 10.32, 10.37, and 10.38 can be removed from consideration for the case of uniform deactivation. However, Eq. 10.32 still needs to be retained in a limited form to for the temperature drop in the deactivated outer shell in the case of shell-progressive deactivation. The problem posed by Eqs. 10.16 through 10.18 and Eqs. 10.30 through 10.41 can be solved in principle for the analysis of a fixed-bed reactor. However, it is quite complicated and the numerical problem involved is not simple by any means, especially when it involves the design ing for the time dependence of conversion. The reactor point effectiveness in Chapter 5 is now used to reduce this problem to a simpler one. Uniform, Independent Deactivation. Since dependent deactivation can be treated in conjunction with the general case of deactivation considered later, attention here is restricted to independent deactivation. For a uniformly deactivated pellet, Eq. 10.30 can be combined with 10.40 to give:

The external mass transfer resistance has been neglected such that N, is set to Nb. This equation can be integrated with the pseudo steady state assumption to give:

where (Nb)in is the reactor inlet concentration of the species causing deactivation. The global rate (Chapter 5, Table 5.3) is:

Nb

= (Nb)in exp

{-I:

(kph~(l y)dy}

(10.43)

]Equation 10.45 has been obtained by combining Eq. 10.43 with Eq. 10.16 while Eq. 10.46 is the Euler version of Eq. 10.34 in time for numerical calculation. Equation 10.18 is also required for T,. Let us examine in some detail the analysis procedures for an adiabatic reactor since the procedures are similar for nonadiabatic reactors. At time t = O+ (y = 0), an integration of Eq. 10.45 yields all information on the reactor since the left hand side of this equation is a function only of Cb when Eq. 10.22 is used in 10.45. The last term in Eq. 10.22 disappears for an adiabatic reactor. The pellet surface temperature is then calculated for the evaluation The initial profile (t = 0+) of Nb is calculated from Eq. 10.43. The of (h)s. reactor profile of y is obtained from Eq. 10.46 for the next time interval. This new y profile (yj) is used for the integration of Eq. 10.45 for Cb corraponding to the new time, i.e., at tj = tj-r f At. These procedures are repeated until the final time of interest is reached. Equations necessary for thie analysis are summarized in Table 10.4 for various modes of operation. The equation equivalent to Eq. 10.45 for an isothermal reactor is obtained by simply, setting (-AH) equal to zero. The analysis procedures for an adiabatic reactor based on these equations are given in Figure 10.5 in the form of a flow chart. A similar chart can be obtained for nonadiabatic reactors. Simulation results for an adiabatic reactor undergoing uniform deactivation (Lee and Butt 1982) are shown in Figures 10.6 and 10:7 for the reaction system given in Table 10.5. These results were obtained accarrling to the procedures of

DESIGN AND ANALYSIS OF FIXED-BEDS

Table 10.4 Equations for Analysis of Plug-Flow Reactors: Uniform Deactivation and Diffusion Limitation

0

rc.,(c.T3 = kf(C)

START

a. Adiabatic, nonisothermal reactors

'I

Obtain Cb(y) using Eq. (A) with the aid of Eq. (B)

Obtmln A (y) and T,(y)

using Eqs. (Eland (F)

Obtain Nb(y) using Eq. (C) with the aid of f ,(y) above and (kpIs [Eq. (GI1

Increment t by O t

b. Nonadiabatic, nonisothermal reactors

Tb = T i n -

-AH (=) (Cb - G n )

c-+ VP

G

I

[Tc - (Tc)out]

I

Obtain y ( y ) for t

-

tf At

using Eq. ( 0 )

[Equations (A) through (H) with (B) replaced by (B')] c. Isothermal reactors

Figure 10.5 Calculation procedures for adiabatic, plug-flow reactors affected by diffusion and uniform poisoning* Ce is calculated as given in Table 10.1.

351

DESIGN AND ANALYSISOF FIXED-BEDS 353 E

I

I

1

I

t=O sec

-

0. Normalized Reactor Length

0.2

0.4

0.6

0.8


Figure 10.6 Reactor profiles of A and fraction of catalyst deactivated for the model system of Table 10.5: uniform poisoning and r = 17s.

Figure 10.7 Profiles of concentration and temperature for the model system of Table 10.5: uniform poisoning and T = 17s.

Figure 10.5 for a residence time of 17 seconds. The reactor profiles for the fraction of catalyst deactivated, shown in Figure 10.6, reveal that the fraction deactivated increases rather sharply as the reactor outlet is approached, and that the time progression slows down with time. The concentration Nb turns out to remain relatively constant up to 12 X lo4 seconds. The reactor profiles for y are expected to show the trend in Figure 10.6 in view of the relatively constant concentration of the species causing deactivation ( N b ) throughout the reactor, and in view of the fact that the maximum reactor temperature is usually reached at the outlet for an adiabatic reactor. The reactor point effectiveness, also shown in Figure. 10.6, does not depend on time as much as y. This can be attributed to the compensating effect between the fraction of catalyst deactivated and the internal effectiveness factor: a larger fraction of catalyst deactivated results in a smaller catalyst activity while an increase in y, on the other hand, causes an increase in the internal effectiveness factor. The outlet concentrations shown in Figure 10.7 reveal that the conversion decreases rather sharply initially, but then moderately with the time on stream. As expected, the reactor temperature decreases with increasing level of catalyst deactivation as shown in Figure 10.7.

In view of the fact that the reactor profile for Nt, appears to have a strong effect on the other profiles, a T value of 20 seconds has been chosen to examine the effect of a change in the h$ profile. The simulation results are shown in Figures 10.8 through 10.10. All the other parameters are the same as those given in Table 10.5, the only difference being the change of r froni 17 to 20 seconds. It is seen in Figure 10.8 that the species causing deactivation is almost completely depleted initially at the reactor outlet. This condition leads to reactor profiles for y with the maximum near y = 0.8, as shown in Figure 10.9. As discussed earlier, the fraction of the catalyst deactivated is determined by theconcentration of the species causing deactivation ( N b ) and the local temperature. Initially, the species causing deactivation is nearly completely depleted at the outlet. Because of this depletion, the catalyst is not deactivated the most at the outlet even though the temperature is the highest there. As the time on stream increases, less of the species causing deactivation is consumed in the front part of the reactor because of the presence of partially deactivated pellets, making more of the poisoning species available toward the outlet. This results in an increase in Nb toward the outlet as shown in Figure 10.8, eventually resulting in a flat profile. The existence of a maximum

DESIGN AND ANALYSISOF FIXED-BEDS

355

Table 10.5 Parameters for a Model Reaction System (Lee and Butt 1982)

t=19xlo4 sec t=16x104

k = exp

(-

f = exp

(y+

+ 14.6) (l/sec),

& = 23.76 kcal/mol

)

3.86 (cm3/mol)

Ti, = 673OK Ct, = 1.81 X 10-5 mole/cm3

B, = 2500, D, = 10-3 cm2/sec, L = 1 cm,

(2) =4

X

10'

+G

( - A H ) = 24. (.1 kcaVrnol

OK crn3/rnole;

h = 3.245 X

= 2.5 cm/sec

cal/sec cm2OK

= 5.63 at the reactor inlet

(Nb)rn = 5 X

rnole/cm3

Q = lo-' mole/cm3 cat, pellet

k, = e x p ( - y

+3.27)*

&!I

= 13.86keal/mol

1


Figure 10.8 Reactor profiles of the concentration of the species causing deactivation: uniform deactivation and r = 20s.

t

i

bulk-fluid temperature decreases with time, the catalyst temperature can increase with increasing time despite the fact that the bulk-fluid temperature decreases with increasing time (Eq. 10.36). It is seen in Figure 10.10 that the local global rate (at y = 1) does increase with increasing time initially. The outlet concentration, on the other hand, decreases with increasing time, resulting in the monotonic decrease of the bulk-fluid temperature as shown in the figure.

ax smx paint w i k ha9r nsc-xwwrk to wae unusual behkshior as shown in Figure 10.10, in which the time p r o p s i o n of &sewed rate, catalyst temperature, and bulk temperature at the outlet are shown. The most important fact revealed in this figure is that the maximum catalyst temperature can increase with increasing time. This fact has also been observed experimentally by Price and Butt (1977). This finding is important in view of the fact that adiabatic reactors are often designed with a constraint on the maximum allowable temperature. A design that disregards the effect of deactivation may not adequately meet the constraint. It is interesting to observe in Figure 10.10 that the bulk-fluid temperature decreases monotonically with time, in sharp contrast with the behavior of the catalyst temperature. This difference can be explained by the way in which the temperatures are determined: the bulk-fluid temperature is determined by the amount of reactant consumed up to the point of interest in the reactor (Eq. 10.23), whereas the catalyst temperature is determined by the local conditions (Eq. 10.36) of RG and Tb. If the local global rate increases faster than the rate at which the

Shell-Progressive, Independent Deactivation. Consider the analysis procedures for reactors undergoing shell-progressive deactivation. The profile of Nb can be obtained by combining Eqs. 10.30 and 10.41 and then solving the resulting equation: )

The reactor point effectiveness can be expressed (Chapter 5) as:

DESIGN A N D ANALYSIS OF FIXED-BEDS 357

Oa5


5 1500

Figure 10.10 Time dependence of global rate, catalyst and bulk-fluid temperatures at the reactor outlet: uniform deactivation and r = 20s.

!

Figure 10.9 Reactor profiles of the fraction of catalyst deactivated: uniform deactivation and 7 = 20s.

When this is combined with Eq. 10.16, there reults:

At time zero, the value of y is zero. Therefore,

Eq. 10.45 should be used at time x r o in place of q.10.49 with y set equal to zero. A set of ~ u a t i o mn W for the analysis of an adiabatic reactor can then be written as follows: (cb) = (Cb)i-la-

.rDe h y {(Cb)i-1 L2y

- (Cd)i -1)

(10.50)

*. -

{(*b)[h

(-AH)oe(l + ( B i ) h ~ ) ( ~-b r b y (1 (Bi)h'Y)(cb - cd)(-AH)De

+ +

]}

(10.51)

Equation 10.51 has been obtained in Chapter 5 by equating the diffusion-limited transport rate through the completely deactivated outer shell to the diffusion-limited reaction within the fresh core of the pellet. The temperature drop in the deactivated outer shell has been ed for in of the heat Biot number (Bi)n and y. Equation 10.50 is the Euler version of Eq. 10.49 while Eq. 10.52 is that of Eq. 10.35 for numerical calculation. For nonadiabatic reactors, Eq. 10.18 needs to be added to the above set of equations with Eq. 10.22 in place of Eq. 10.23. These equations are summarized in Table 10.6 for various modes of operation while detailed analysis procedures for adiabatic reactors are given in Figure 10.11 in the form of a flow chart. The structure of the procedures shown in Figure

358


Table 10.6 Analysis Equations for Plug-Flow Reactors: Shell-Progressive, Independent

Deactivation, and Diffusion Limitation


I

Table 10.6 (cont.)

a. Adiabatic, nonisothermal reactors Use Eqs. (A), (B), (E), and (F) In Table 10.4 for the profiles of Cb, Tb, A, and Ts at t = 0 (y = 0). Based on these initial profiles, proceed with the equations given below.

f

0

for

10.11 is easy to understand if it is recognized that l2q. 10.23 is an auxiliary equation for Eq. 10.5 1 while Eq. 10.47 is the same for Eq. 10.52,. For a given time then, simultaneous solutions of Eqs. 10.50 and 10.51 yield the reactor concentration profile. To proceed to the next time interval, Eq. 10.52 is solved for the y profile for use in Eqs. 10.50 and 10.51, which yields the reactor concentration profile at time ( t At). In order to initiate the numerical calculations, the initial concentration and temperature profiles are obtained using Eq. 10.45. Based on these profiles at t = 0+, the No profile is obtained from Eq. 10.47 which in turn is used in Eq. 10.52 for the y profile at time (0+ At). Given the y profile, Eqs. 10.50 and 10.51 are solved simultaneously. These procedures can be repeated at each successive time interval with the aid of Eq. 10.52 until the final time of interest is reached. The design equation (K) in Table 10.6 can be obtained directly from Eq. (A) by setting (-AH) equal to zero. Equation (N) has been obtained by integrating Eq. 10.35. Simulation results for an adiabatic reactor undergoing shell-progressivedeactivation (Lee and Butt 1982) are shown in Figures 10.12 through 10.17 for the model reaction system in Table 10.5. The procedures of Figure 10.1 1 have been followed to obtain these results for T of 20 seconds. The relative speeds of transport and reaction of the species causing deactivation determine whether the catalyst is uniformly or shell-progressively deactivated. Therefore, the intrinsic rate of deactivation given in Table 10.5 should have been changed for the shell-progressive deactivation being considered. However, the same conditions have been used in the simulation to contrast the results with those obtained for uniform deactivation. The most striking thing is that the fate of the reactor is literally determined in the initial stage of deactivation, although this is not unexpected considering the nature of shell-progressive deactivation, This is most evident in Figure 10.12 in which the outlet concentration is shown to go through a drastic change initially with time, in sharp contrast with the concentration behavior shown in Figure

+

+


[]Equations(A) through (J) with (C) replaced by (C')] c. Isothermal reactors

Use Eqs. (I) and (L) in Table 10.4 for the profiles of Cb and A at t = 0. Based on these initial profiles, proceed with the equations given below.

23

DESIGN AND ANALYSIS OF FIXED-BEDS 361

0 START

t=20x104 sec t=ix104 t=2,5x1o3

Obtain initial profiles of Cb. Tb. A and T,, uslng Eqa (A). (B). IE) and IF)

I

Obtain Nb(y) using Eq (D) with the aid of TSabove for (kp)d and ( bp)d.

A Increment t by At.

t=O

0 0

I

Solve Eq. (A) for Cd with the aid of Eq. IC).

I

0.2

0 -4

0.6

0.8

1.O

NORMALIZED REACTOR LENGTH Figure 10.12 Profiles of the bulk-fluid concentration: shell-progressive deactivation.

using Eq. 18).

I

no Obtain A (y). f,lyl. Tdly) using Eqs. (F),(GI and (HI.

I I

" 6

Obtain Nb(y) using Eq. (D) with the aid of Td above for (kp)d and ( 9p)d IEq. l1I1.

,

Figure 10.11 Calculation procedures for adiabatic, plug-flow reactors using the equations in Table 10.6.

10.7 for uniform deactivation under identical conditions. The y profiles again go through a maximum as shown in Figure 10.14. However, the peaks are sharper than those for uniform deactivation. The species causing deactivation is also depleted at the outlet initially as was the case with uniform deactivation (Figure 10.13). Similar reasoning as for uniform deactivation applies for the existence of a maximum in the y profile. The existence of a maximum causes aberrations in the profiles of the reactor point effectiveness and the concentration at the forefront of the fresh inner core, apparent in Figures 10.15 and 10.16. The profiles of reactor point effectiveness shown in Figure 10.15 are quite different from those obtained for uniform deactivation (Figure 10.6). It is seen that the reactor paint effectiveness does decrease more rapidly with time than for uniform deactivation. This example clearly shows the devastating effect of shell-progressive deactivation on reactor performance as evident from a comparison of Figures 10.7 and 10.12 for the outlet concentration. If the rate of deactivation is higher than the rate given in Table 10.5, which should be the case for shell-progressive deactivation, the effect will be even more devastating than shown. This fact has an important bearing on the design of a reactor undergoing shell-progressivedeactivation, as will be seen shortly.

362

i


DESIGN AND ANALYSIS OF FIXED-BEDS 363

t=10x10~sec t=ix104 t=2.5x1 o3


Figure 10.13 Profiles of the concentration of the species causing deactivation: shell-progressive deactivation.

General Case of Deactivation. As discussed earlier, the activity profile of a pellet undergoing chemical deactivation cannot always be adequately represented by either uniform or shell-progressive deactivation. Deactivation intermediate between uniform and shell-progressive deactivation results in a smooth activity profile within the pellet with its activity decreasing toward the pellet surface. Therefore, the result obtained in Chapter 4 for a pellet with a nonuniform activity distribution can be used for this general case. Utilizing the global rates derived in Chapter 5 for this case (Table 5.3), for independent deactivation:

0

0.4 0.6 0 -8 NORMALIZED REACTOR LENGTH

0.2

1.O

Figure 10.14 Profiles of the fraction of catalyst deactivated: shell-progressive deactivation.

[l

+

where 7 = fraction of catalyst deactivated at pellet surface ep = Arrhenius number for deactivation reaction, (Ep)/RgTb G(Cb)] = product inhibition term.

For dependent deactivation:


t=2.5x103 sec

t=2.5x103 sec

0

0.2

0.6

0 -4

0.8

1.O


Figure 10.15 Profiles of reactor point effectiveness: shell-progressive deactivation.

NORMALIZED REACTOR LENGTH Figure 10.16 Profiles of concentration at the forefront of fresh inner core: shell-progressive deactivation.

(kp)s(C~)b(l 1

7)

+ 2 mi]"bulk

parallel

1

( b ) s ( c ~ ) b (l 9) 1 (fCi~)mfln

+ Ci

series

In Eq. 10.54, the pellet center concentration has been set to zero for the diffusionlimited case being considered. As pointed out in Chapter 5, the global rates are usually valid for 7 less than 0.5 (refer to Sections 4-8 and 5-6 for details). The analysis equations and procedures given in Table 10.4 and Figure 10.6 for uniform deactivation can be utilized for independent deactivation provided that y is replaced by 7, and Eqs. (C) and (J) in Table 10.4 are replaced by:

bulk

where CAis the concentration of the main ractant, and CBis that of the intemediate prduct in the dependent deactivation mdels: Parallel: A

+ S --k--B, + S

A+s--L - SA Series: A

+S

k

---+ B

+S

B+SAB.S

i = grid point in y

which is the Euler version of the reactor conservation equation:


367

Eq. 10.58 can be replaced by:

and the same analysis procedures discussed above can be applied. One assumption made in deriving the global rates for the case of dependent deactivation is that the rate of the deactivation reaction is much smaller than that of the main reaction. This must be true fordependent deactivation in which the main reactant or product itself is the species causing deactivation. If not, the rate of deactivation reaction becomes of the same order of magnitude as the rate of the main reaction, and such a catalyst cannot be useful in the form of a pellet. This is precisely the reason why small particles, instead of pellets, of silica-alumina catalyst are used in a fluidized-bed for the catalytic cracking of hydrocarbons, for which the coking rate is high. Design

NORMALIZED REACTOR LENGTH Figure 10.17 Profiles of temperature at the forefront of fresh inner core: shell-progressive deactivation.

The same equations and analysis procedures are also applicable to dependent deactivation provided that y is again replaced by 7, and Eqs. (C) and (J) in Table 10.4 are eliminated. It is seen that the analysis procedures for the general case are straightforward for both dependent and independent deactivation. On the other hand, their utility is limited to small values of 7, say less than 0.5. More specifically, the value of R given by Eq. 4.102 (Section 4-8) should be much less than unity for accurate results. Further, +G has to be much larger than unity. While closedform equations for global rates cannot be written when these restrictions are not met, similar analysis procedures as for 7 less than 0.5 can be developed by adding more correction to the calculation of the flux at the pellet surface (Section 6 1 0 ) and relating this flux to the global rate. While specific kinetics have been specified for dependent deactivation according to the model given above, arbitrary deactivation kinetics can be handled without problem. For instance, if the dependent deactivation kinetics are represented by:

The time dependence of conversion can complicate the design of a reactor affected by catalyst deactivation. In practice, however, this complication is avoided: the usual practice is to design the reactor as if there. were no deactivation taking place and then operate the reactor at the desired conversion by constantly raising the reactor temperature to compensate for the declining catalytic activity. When the conversion drops to an unacceptable level despite adjustments in reactor temperature or the temperature reaches too high a value, the catalyst is regenerated. Therefore, a typical operating cycle consists of start-up, operation, and catalyst regeneration. An obvious problem with this design approach is that it does not take advantage of the knowledge of catalyst deactivation during the design phase. Consider the choice of reactor size T when the reactor is subject to catalyst deactivation. A measure of the performance of the reactor is the average conversion per unit volume of reactor in a given time period or some reference time period t,. If is defined as the duration of the reaction phase of the cycle, and b as the time required for catalyst regeneration, the number of cycles is:

There are additional costs incurred due to the reactor shut-downs caused by deactivation, which can be represented by a constant M. A performance index suitable for the choice of reactor size T can then be written as:


In essence, this index represents the net value of the product produced per unit volume of reactor, averaged over a reference time period t,. This index, when Eq. 10.60 is utilized, reduces to: Max r J(T) = Max .r

{7 ( t /'+ b )[ ~ ot f ( 2 n - C o u c ) d t - M11

(10.62)

where (Gout), is the reactor outlet concentration corresponding to the conversion below which the catalyst is regenerated. When the duration of the reaction phase is relatively short due to rapid deactivation, the major concern has to be the longevity of the operation. In such cases, a suitable index for the choice of T may be written as

perature (Tin)solely determines the maximum catalyst temperature. The inlet temperatures that satisfy the constraint on the maximum catalyst temperature are given in Table 10.7 along with the duration of operation Q. It is clear from the table that the duration of the operation before catalyst regeneration is relatively short. On this basis alone, one may decide on the reactor size choosing, say r of 30 seconds to have a relatively long reaction phase. It is nevertheless useful to determine how effective the reactor is on a unit volume basis, using the criterion of Eq. 10.63. A plot of J (Eq. 10.63) versus r is given in Figure 10.18. The J values in the figure are normalized with respect to the S value corresponding to T of 15 seconds. Although no maximum is reached under the conditions specified, it is clear that a larger reactor should be favored over a smaller reactor for higher conversion per unit reactor volume. Also given in the figure as the dotted line is

This index represents the amount of the product produced per one cycle on a unit reactor volume basis. The specification of T based on the criteria of Eq. 10.62 or 10.63 is relatively simple with the analysis procedures developed in the previous section. A value of T can be chosen to calculate J(T), subject to any additional constraints placed. From a plot of T versus J, the value of T that gives the maximum J can be chosen. We consider here the use of Eq. 10.63 for the specification of T and then that of Eq. 10.62 in the next section. For the reactor system in Table 10.5, values of J versus T have been calculated for the case of uniform poisoning. Constraints are the maximum allowable catalyst temperature (or the maximum reactor inlet temperature) and the minimum allowable conversion. For the adiabatic reactor being considered, the reactor inlet tem-

Table 10.7 Performance of the Model Reactor in Table 10.5

(8.1 days)

(2 1.8 days)

Figure 10.18 Performance index of Eq. 10.63 for a reactor affected by uniform deactivation.

DESIGN AND ANALYSISOF FIXED-BEDS 371 the value of J (J, in the figure) when catalyst deactivation is not taken into consideration, i.e., the usual design practice. This can be obtained by simply dividing the conversion at t = 0 by T . It is seen that the usual design practice leads to a completely wrong conclusion that a smaller reactor should be favored over a larger one. Another pitfall that one can come across when catalyst deactivation is not taken into consideration has already been pointed out in a previous section: the fact that the maximum catalyst temperature can increase rather than decrease with an increasing level of deactivation, which will not be ed for if the usual design practice is followed. Similar conclusions can be made for shell-progressive deactivation regarding the choice of T . Some important design considerations also emerge from the dominant feature of shell-progressivedeactivation, i.e., the rapid initial change of conversion with time. This is shown in Figure 10.19, which has been obtained with a constraint on the maximum allowable catalyst temperature. It is seen that the conversion decreases exponentially with time initially, followed by an almost linear decrease with time. It is also seen that the level of conversion in this linear range is largely determined by the reactor size T . This dominant role of the reactor size in determining the level of conversion in the linear range is also apparent in Figure 10.20, where the conversion for T of 20 seconds is given as a function of time when the constraint on the maximum catalyst temperature is removed. Even with a change in the reactor inlet temperature from 662% to 673OK,which can double: the rate of reaction as indicated by the conversions at time zero in the

CON". A T t=O

9

(

0.470 20 sec 661.7 K 0.448 17 rec 673.0 K 0.425 15 roc 681.6 K

-

0.2

-

(I)

a

u Z

g

0.1 -

---- --

T

=20 sec, T=673 O K

r =20 sec, T=661"7"K

Figure 10.20 Time dependence of conversion: shell-progressive deactivation without the constraint on the maximum catalyst temperature.

0.3

>

--

.

=20 SeC 17 sec 15 sec

Figure 10.19 Time dependence of conversion: shell-progressive deactivation with a constraint on the maximum catalyst temperature.

figure, the conversions converge to the same level in the linear range. A few canclusions can be made from these observations on a reactor undergoing shell-progressive deactivation. The reactor may be sized according to a desired conversion in the linear range. Furthermore, the initial inlet temperature may be set as high as the constraint on the maximum catalyst temperature allows. Given the desired conversion a,then, the value of 7 that yields xd in the linear range can be found using the analysis procedures developed for the case of shell-progressive deactivation. A more sophisticated design criterion may be used for the choice of T . For instance, the choice of T based on Eq. 10.62 or 10.63 may be subject to an optimal temperature policy during the operation so that J is maximized with regard to both r and the temperature policy. This optimization problem will be treated in Chapter 13. The point here is that much better reactor pedormance can be attained by including the effects of catalyst deactivation in the design, as will be exemplified in the next section, and that such complicated design problems, whatever the performance index of the design may be, can be. handled with relative ease using the analysis procedures developed in the previous sections.

DESIGN AND

104-3

+

Design and analysis procedures based on reactor point effectiveness are developed in this section for reactions affected by diffusion, chemical deactivation, and catalyst sintering. Since these procedures closely parallel those developed in the previous section, only the general case of catalyst deactivation will be treated in detail. For an adiabatic plug-flow reactor, the conservation equations are:

where the external mass transfer resistance has been neglected by setting C, = Cb in Eq. 10.69. Therefore, the mass balance of Eq. 10.41 can be rewritten as:

where the reactor point effectiveness is given by:

I

- -

where 7 is related to Sr/Soby:

Using the pseudo-steady state assumption, the rate at which the active surface area changes with time is obtained from Table 6.4 as: .

-K. ($/%nexp[rn'(S/go)]

i

For the species causing independent chemical deactivation:

The pellet mass balance for this species is:

373

The factor (1 - y)/[l G(C)Inis the fraction of active sites that are vacant, as shown in Chapter 5. This factor represents the profile of sites within the pellet that are available for both the main and deactivation reactions. If this profile is treated as the activity profile for the rate constant kp, the result obtained in Sectian 6 8 for the internal effectiveness factor for a pellet with a nonuniform activity distribution can be applied (Section 5-6):

Reactions Affected by Sintering

Here, Sr and So are the active surface area and total surface area, respectively, for fresh catalyst, and the overbar denotes evaluation at the pellet surface. Without loss of generality, the functional dependence of the rate constant on surface area h ( S / S o ) can be set to $/So for structure-insensitive reactions. This reactor point effectiveness for a certain smooth activity profile has been developed in Chapter 6 and is given in Table 6.5 along with the reactor point effectiveness- for the other cases. Combining Eqs. 10.64 and 10.65 and rearranging, there results:


atom migratipn

-K,(g/F0) nc with (%/TO) = ($/%) (s/go)

crystallite migration

I

where is the total catalyst surface area at time t at the pellet surface, and K. and Kc are the rate constants for sintering. Here, n, h,and m' are the constants for the sintering kinetics. For the design and analysis of the reactor affected by diffusion, chemical deactivation, and sintering, then, Eqs. 10.66 and 10.70 need to be solved with the aid of Eqs. 10.23, 10.68, and 10.71 through 10.73. The solution of Eq. 10.70 is: Nb kt]= ( N b ) i n

(S;/%) tll~lh(4p). [l+G(Cb)ln(Cbp)a dy

T(ILI1)S

exp 0

}

(10.74)

DESIGN AND ANALYSISOF FIXED-BEDS 375 Table 10.8 Design Equatians for Plug-Flow Reactors: Diffusion-Limitation and General Case of Independent Catalyst Deactivation (Reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)

Table 10.8 (cont.) c. lsotherrnal reactors

a. Adiabatic, nonisothermal reactors

Nb [Y;t] = (Nb)tn

eXp 0

(+

-

p a -

~(kp)s SO tanh ( + p ) a 11 + G(Cb)l " ( + p ) a

{[(b)s @r/%) }lt2 1 + GCb)]"Dp

+

- I(~~)~(S/%~)~exp[rn'(%/~~)]l-~~t (s/.So)j-I

( S / ~ O )= ,

atom migration - ((Kc),(S/~o)n~}j -IAt + (S/%)j -1 crystallite migration

s/go = (S/S)(S/%.)

where Eq. 10.71 has been used. This solution, along with the solutions of Eqs. 10.72 and 10.73, is required for the integration of the reactor conservation equation, Eq. 10.66. The complete set of equations necessary for design and analysis is given in Table 10.8. Equations (F) and (G) in the table are the Euler versions of Eqs. 10.72 and 10.73 for numerical calculations. Design and analysis procedures for adiabatic reactors are summarized in Figure 10.21 in the form of a flow chart. - At time zero, the catalyst is free from deactivation and therefore &/So = 1. Integration of Eq. (A) in the table yields C a b ) at time zero when Eqs. (B) and (C) are used in Eg. (A). The temperature profile at time zero then follows directly from Eq. (C). The initial pellet surface temperature T , Q follows from Eqs. (I) and (J). With this TsCy), the initial profile of N b b ) can be obtained from Eqs. (D) and (E). Having obtained all necessary information at time zero, the new surface area ratio for the next time interval (t = 0+ At, or next j) can be obtained from Eqs. (F) and (G) using the calculated TsCy) and Eqs. (B) and (H), which in turn are used for the integration of Eq. (A) for the next time interval. These procedures can be repeated until the final time of interest (Q) is reached. Design and analysis procedures for the limiting cases* of uniform and shellprogressive chemical deactivation coupled with sintering closely parallel those given for the general case and arc very similar to those already developed in the previous section on reactions affected by diffusion and chemical deactivation. Likewise, the

+


[Equations (A) through (J) with (C) replaced by (C'] See problems at the end

of this chapter

0


START

case of dependent deactivation can be handled in a similar way (see Section 1-2)When the diffusional effect is totally absent, the reactor conservation equation reduces to:

Integrate Eq. (A) for Cb(y) with the aid of Eq. (C).

Obtain Nb(y) using Eqs. ( 0 )and (El with the aid of T,(y\above. (k,), [Eq. (B)1 and 8,/% profile calculated.

--

Increment t by At chosen, j = j

+

1

I Calculate reactor profile of ($/Soli (&/% for t = t f. A t) uslng Eq. (G).

where the reactor point effectivenessgiven in Table 6.4 has been used. The conservation equation for the poisoning species is:

The design and analysis procedures based on Eqs. 10.77 and 10.78, and Eqs. 10.72 and 10.73 with the overbar removed can readily be obtained. In the absence of poisoning, all that is required is to set (Nb)in equal to zero in the flow chart given in Figure 10.21. As an example, consider the design of an adiabatic reactor for the following reaction:

I Calculate reactor profile of ( S r / ~ o ) j ( z , ~ S , f ~t = t + A t) using Eq. (F) and (HI,and (St/So)j calculated above. Use Eq. (B) for (KBISand (kpIs

A +SeA

. SS

. S(KA)

B - S (k,', controlling step) B S+B+S(Ke) N S + N . S (Kp poisoning) A

+

New S,/So for next

Figure 10.21 Design and analysis procedures for adiabatic, nonisothermal plug-flow reactors affected by diffusion, chemical deactivation, and sintering. (Reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)

where S represents active sites. These elementary steps lead to:

where &/SO = G/Ct,. Here, Cr and Cto are surface concentrations of active sites and initial total sites, respectively. For this independent poisoning, the inhibition term (1 G)nis:

+

DESIGN AND ANAL'YSIS OF FIXED-BEDS 379 where the effective diffusivities are assumed to be the same for both A and B and C A , is the reactor inlet concentration of species A. In Table 10.9, C = C A I and Cb = Ci,for the key species A. It is also assumed that the reaction is facile such that h(Sr/&) = &/So, and that the catalyst crystallite size is sufficiently large that sintering takes place via atom migration. For large metal crystallites, surface diffusion is the controlling step and thus the value of n in the sintering

kinetics given in Table 10.9 has been set to 5. Other pertinent parameters and conditions are also given in the table. The problem is to choose the reactor size T that maximizes the performance index J of Eq. 10.62, subject to:

(Teat),,S 853OK and (Gout), 5 0.75 X

In order to calculate J as a function of T , the Z, that yields the maximum catalyst temperature ,allowed at t = 0 for each of T values selected was first calculated. This allows calculation of J with the procedures in Figure 10.21. The smallest value of +G was 2.5, and therefore, C, can be set to zero. Values of the performance index J were then calculated using the procedures of Figure 10.21 for the selected T values. The values of J normalized with respect to that of J for T of 20 seconds are plotted in Figure 10.22 for b = 2 X lo3 rnin (1.4 days) and various values

--

Table 10.9 A Model System of the Reaction A B (reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)

Sintering:

d (&/So) = -Ka(St/So)%xp(rn dt

'(St/So)]

12,000 (- + 14.6 (cm3/seccm3 pellet volume); T

k

= cxp

k,

= exp(-50001' T

KA = exp(3600/ T Kg = exp(2800/ T K = exp(6400/ T

rnole/crn3 [conversion > 58.3% J

E. = 23.76 kcaVmole

+ 3.27) + 3.86)(cm9/mol) + 3,70) + 7.56)

En = 673OK; D, = Dp = 5 X 10-3 cm2/sec h = 3.25 X 10-4 cal/sec crnz0K Ci, = CAO = 1.8 X lo-= mol/cm3; (-AH/p) = 1.1 X 1070Kcm3/rnol Ni,,= 5 X mol/cmg; L = 1 crn; Q = lQ"4mol/cm3 cat. pellet; (-AH) = 30 kcal/rnol

Maximum catalyst temperature allowed = 853OK (Gout), = 0.75 X 10-5 mol/cms

A t = 105 (min);

Ay = 0.005;

6 T: varied

I

Figure 10.22 Performance index of Eq. 10.62 for a reaction affected by both sintering and poisoning. (Reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)


I

of M. It is seen from the figure that a maximum in J exists between T of 20 and 30 seconds, and that this maximum point moves to higher values of T as M increases. An M value of 0.05 for a reactor with T of 30 seconds means that the costs incurred due to catalyst regeneration are approximately 1% of the value of the product. Perhaps, more important than the specific values shown in the figure is the fact that the performance of a reactor undergoing catalyst deactivation can be significantly enhanced by properly sizing the reactor, and that this enhancement

Figure 10.24 Effect of poisoning on conversion. (Reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)

-

poisoning and sintering

0

I

I

I

40

80

120

160

t x 10-3 (rnin)

Figure 10.23 Reactor profiles of surface area ratios for the reaction of Table 10.9.(Reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)

can only be realized by taking the effects of catalyst deactivation into consideration during the design phase. Some interesting behavior emerges from the simulation results obtained in the process of calculating the performance index. Shown in Figures 10.23 through 10.25 are the simulation results for a reactor with 7 of 25 seconds. The surface area ratios shown in Figure 10.23 for the active and total surface areas indicate that deactivation is dominated by catalyst sintex-ing: the reduction of active surface area is almost entirely due to the sintering except for the entrance region. The poisoning species is almost entirely consumed in the entrance region so that its concentration is too small in the rest of the reactor to cause appreciable poisoning. On the other hand, a higher temperature toward the reactor outlet causes a higher rate of sintering, resulting in the decrease of the active surface area toward the outlet shown in the figure. Most of the conversion occurs near the outlet in an adiabatic reactor, and therefore, it is expected that the reduction of active surface area due to poisoning in the entrance region will not have a significant impact on the conversion, as verified by the results shown in Figure 10.24. It is s e n from this figure that poisoning does not affect the conversion significantly at low times on stream. As the time on stream increases, however, the catalyst is deactivated too much near the outlet, and the reactor is eventually affected by the poisoning in the entrance region. This is reflected in the difference in the conversions shown in Figure 10.24 for t > 40 X 103 rnin. The conversion corresponding to

DESIGN AND ANALYSIS OF FIXED-BEDS 383 10-5

MULTIPLE REACTIONS

The reactor point effectiveness for multiple reactions cannot be expressed explicitly in of bulk concentrations and temperature. Nevertheless, the approach of using the reactor point effectiveness to obtain the global rates in of concentrations still applies as detailed in Section 4-9. Consider multiple reactions taking place in an adiabatic reactor. If A . is designated as the main reactant from which all key species At (i = 1,2, . . . , N)at each node shown in Figure 10.26 originate, the reactor conservation equations are:

~ the net global rate for the key species Ai such that: where ( R G ) is

A few comments are in order at this time. It is understood in Figure 10.26 that there can be more than one species at each node. However, only the key species is required at each node as shown since the other species can be related to the key species through the diffusivity ratios and staichiometric coefficients as detailed in Chapter 4. The use of the net global rate in place of the sum of individual global rates for all reaction paths involving the key species At necessitates the use of the heat of formation (AHf)t in Eq. 10.83 in place of the heat of reaction. Figure 10.25 Profiles of reactor point effectiveness for the reaction system of Table 10.9. (Reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)

the case of no poisoning was obtained by setting (Na)i, q u a 1 to zero in the procedures given in Figure 10-21. The profiles of the reactor point effectiveness shown in Figure 10.25 reveal that a maximum exists, which is due to the rapid change of the active surface area caused by poisoning in the entrance region. Based on the simulation results and the reasoning given earlier, it can be concluded for adiabatic reactors that the conversion is mainly affected by sintering initially, but then poisoning dominates as time increases. This will not be true, however, when takes place in a shell-progressive manner.

1

8

f

-Am

Figure 10.26 A reaction network for multiple reactions.

DESIGN AND ANALYSIS OF FIXED-BEDS 385 The pellet conservation equations are:

For a selected increment AAo, Eq. 10.90 represents N algebraic equations for N unknown increments AAi at each grid point j in the pellet. The concentration increments ace given by:

where

Here, ril is the intrinsic rate for the reaction path from species Ai to Al, and Yir is the stoichiometric coefficient per mole of Ai with the usual sign convention that Ri is the net intrinsic rate of consumption of species Ai. The intrinsic rate ril may be expressed as:

where k t l is the rate constant for the reaction path between species Ai and Al. With the assumptions of an isothermal pellet and negligible film mass transfer resistance that have been used throughout this chapter, one only needs to consider the film heat transfer to relate the reactions in the pellet to mass and heat transport in the bulk fluid:

According to the results obtained in Chapter 4, the net global rates are given by:

where ( A A i ) is negative for i = 0 and positive for i > 0 in view of the fact that A . is "reactant" and the Ai's are "products." If ( A A o ) j + l is chosen to be the same for all intervals, (AAo)j+1 may be replaced by the constant increment AAo, but ( A A i ) j + l will still depend on the position j. The design and analysis procedures follow from Eqs. 10.81 through 10.83 and Eqs. 10.87 through 10.91. At the reactor inlet at which (Ao)b = Ain and ( A i ) b = 0, which are the values of Ai at the pellet surface ( j = 0) in Eq. 10.90, Eq. 10.90 can be solved for the pellet concentration profiles Ai (i = 1,2, . . . , IV) with the aid of Eq. 10.91 using the chosen value of AAo. As indicated earlier, (AAo)j may be chosen to vary with each increment. These concentration profiles are used in Eqs. 10.88 and 10.89 for the global rates. In order to solve Eq. 10.90, however, the pellet surface temperature, Ts,has to be used to calculate the rate constants in Ri. This is unknown, but by trial and error solution of Eqs. 10.87 and 10.90 the correct value of Ts can be calculated. Based on the correct global rates obtained from the trial and error solution, Eqs. 10.81 through 10.83 can then be used to numerically calculate ( A o ) ~(,A i ) b , and Tb for the next grid point in the reactor. These procedures can be repeated until the reactor outlet is reached. In order to calculate the global rates from Eqs. 10.88 and 10.89, the pellet center concentrations ( A & and ( A i ) , have to be known. As detailed in Chapter 4, ( A & is estimated from:

where

where ( A i ) , is the pellet center concentration of species Ai. The relationships between A. and At in the pellet, which are required for the numerical integration of Rt in Eqs. 10.88 and 10.89, are obtained in the process of solving the following equations at each grid point j in the pellet:

i=1,2,. . . , N j = 0 at the pellet surface

This means that Eq. 10.90 is solved until (Ao)j reaches zero for use in Eq. 10.93. The grid index j corresponding to the value of (Ao), calculated from Eq. 10.92 can be located for the values of corresponding to this j, which are the approximate pellet center concentrations. The procedures outlined above can be quite time-consuming due to the iterations involving Eqs. 10.87 and 10.90. One way of overcoming the trial and error iteration problem is to combine Eq. 10.83 with 10.87:

DESIGN AND ANALYSIS OF FIXED-BEDS 387 This can be integrated from any point y, to a point Ay away

to give:

It follows then that the net intrinsic rates for species Ao, AI, and A* are: If the trapezoid rule is used for the integration of the right hand side of Eq, 10.95, one has:

The pellet conservation equations (Eq. 10.85) for all the involved species can be combined two at a time and integrated twice to express the non-key-concentrations in of the key concentrations: 1 D2 (BO)b- B 0 = - - [(A 2

Here, m is the index for the grid point in y, and Ay is the grid interval. Consider the use of Eq. 10.97 in place of 10.87. If the pellet surface temperature is known at the reactor inlet, Eqs. 10.81 through 10.83 allow the calculation of Tb and (At)b at the next grid point in y, i.e., m = 1. Then, the only unknown in Eq. 10.96 is (T,)l, which is the pellet surface temperature at the next grid point. This value of (Ts)l is then used in Eq. 10.90 for the global rates at m = 1, thereby eliminating trial and error iterations for all m except at the reactor inlet since the same procedures as those for m = 1 apply to all m due to the recursive relationship of Eq. 10.97. It is seen that the use of Eq. 10.97 in place of Eq. 10.87 results in a numerical scheme that requires iteration only at the reactor inlet. As discussed in Chapter 4, the rectilinear rule can be used in place of the trapezoid rule in deriving Eq. 10.90 with the following result:

a90

Dl

- A 21 - [(A I)* &I 0

A 11

(10.101)

where the subscript b denotes the bulk-fluid concentration and the D's are effective diffusivities. The bulk-fluid concentrations can be related to reactor-inlet concentrations by combining the reactor conservation equations (Eq. 10.82) written for all the involved species two at a time and then integrating once. Equations 10.81 and 10.82 with the assumption of no products in the feed, yield: 1

( B o )= ~ (Bob - 2 ( A 2)s - (A i)b (B1)b = ( A d i n - (A0)b

1

-5

(10.102) (Adb

Here the subscript in denotes the reactor inlet concentration. Equations 10.101 and 10.102 can be used in Eq. 10.100 to give:

The use of Eq. 10.98 requires solving linear algebraic equations as opposed to the nonlinear algebraic equations of Eq. 10.90 and thus considerably simplifies the numerical problem. Furthermore, accurate results can be obtained using a predictor-corrector method, as detailed in Section 4.9. Take as an example the following multiple reactions, which are the basic unit. in any complex reaction network:

Suppose that the reaction stoichiometry and rates are given as follows:

I -

I

where the effectiveness diffusivities have been assumed the same for all species for illustration purposes.

I

Exprau m t Intrlndc ratem [Eq. (lO.MI)] in of key species concsntratlons.

I

DESIGN AND


389

It is seen that the net intrinsic rates are now expressed solely in of the concentrations of the key species. Upon substituting Eqs. 10.91 and 10.103 into 10.90, the nonlinear algebraic equations of Eq. 10.90 become:

Uae the reactor inlet conditions to obtain tha psllat concsntrution profiles from Ep. (10.90) wlth m nsrurnsd value of f ,.

I

I

Iterate urlng Eqa (10.87) and (10.90) untll Eq. (10.87) I8 aatldled.

I

Obtain h and (&Ic from E q r (10.92) and (10.93).

I

1 + .

. .

Uss the index j corrsspondingto (Ao), to obtain (Aile.

Obtain (R,), from Eqk (10.88) and (10.119) u8ing HI* w l k t

concsntrntlonprofiles.

I

Obtain (Ailb and Tb for the next grid point in y uming dlfferencs vsrrions of Eqa. (10.H), (10.82) and (10.83).

I

I

'

Use the calculated values of (Ai)b end T, to solva Eq. (10.90) or Eq. (10.98) for ma pellst concrntratlon profilss until A 0 bscomes zero.

Use the valuer of (Ai)b and f b to obtain T, for the next grid point in y ullng Eq. (10.96)/(10.97).

A

no

w

( END

Figure 10.27

1

Design and analysis procedures for multiple reactions in an adiabatic reactor.

where F1 and F2 are functions of the unknowns, (AAllj+l and (AAz)j+l, and f l and fi are constants for a selected value of (AAolj+l. These equations can be solved for the two unknowns at each grid point j to give the pellet concentration profiles of A 1 and A 2 versus Ao. In order to generate the pellet concentration profiles, a value for (AAo)j+l, which can be a constant or a variable for each grid point, is first chosen. The concentrations at j = 1 can then be calculated from Eq. 10.104 with (Ai)o = (Ai)* at the pellet surface. The calculated values (A1)l, and of (AA1)l and (AA& are used in Eq. 10.91 to obtain values for (AO)~, (AZ)~,which in turn are used in Eq. 10.104 to obtain (dAAl)2and (AA& with a chosen value of (AAo)z.These procedures can be repeated until A . reaches a value of zero, at which point the' generalized modulus +G (Ecl. 10.93) is calculated for the pellet center concentration of A . (Eq. 10.92). The index j corresponding to (Ao), is then located to obtain the center concentrations (Al), and (A&. Now that all quantities in Eqs. 10.88 and 10.89 are known, the net global rates can be calculated. The above example is just for the cafculations of n!et global rates at a point in the reactor. These global rates have to be used in conjunction with the reactor conservation equations for design and analysis. The complete design and analysis procedures for the reactor are outlined in Figure 10.27. Even with the simplifications made possible using the reactor point effectiveness, the design and analysis problem is still not simple. It should be clear, however, that the alternative of solving nonlinear differential equations with split boundary conditions on each is much less attractive.

Summary

The approach of reactor point effectiveness allows a general treatment of reactor design, whether the reaction is affected by transport processes, catalyst deactivation, or bath. Relatively simple procedures have h e n developed for the design and analysis of heterogeneous reactors. These procedures, applicable to general forms of rate expressions for the main reaction, poisoning, and sintering, allow a rather straightforward treatment of complex design and analysis problems for reactors affected by catalyst deactivation. Including the effects of catalyst deactivation during the design phase has been shown to deliver better reactor performance. Design and analysis for multiple reactions is a very complicated problem. Use of the reactor point effectiveness, however, considerably simplifies it.

DESIGN A N D ANALYSIS OF FIXED-BEDS 391

NOTATION cancentration of species Af Biot number for heat Biot number for mass concentration bulk concentration for a species of interest Cb evaluated at reference conditions pellet center concentration cup-mixing concentration concentration of species i reactor inlet concentration for a species of interest solution for plug-flow reactor in perturbation met hod reactor outlet concentration concentration of the species causing deactivation; specific heat content of reaction fluid pellet surface concentration axial dispersion coefficientfor mass effective diffusivity of main reactant effective diffusivity for species At radial dispersion coefficient for mass effective diffusivity of the species causing deactivation activation energy for sintering activation energy for the main reaction activation energy for the deactivation reaction concentration dependence of intrinsic rate of the main reaction concentration dependence of intrinsic rate of the deactivation reaction rRc(Co) in Eq. 10.5 F evaluated at reactor inlet product inhibition term in the deactivation kinetics film heat transfer coefficient wall heat transfer coefficient for two-dimensional model heat of reaction ([1 G(Cb)] " [l G(C = O)]" ) / 2 heat of formation grid point in space integral defined in Eq. 10.44 grid pointin time; grid point in z integral defined by Eq. 10.54;integral defined in Eq. 10.62 rate constant for main reaction k for pseudo-first-order reaction k evaluated at Tb film mass transfer coefficient k, for the species causing deactivation

+

+ +

rate constant for deactivation reaction k, evaluated at Td preexponential factor for k preexponential factor for k, axial dispersion coefficient for heat; rate constant for sintering kinetics in Eq. 10.73 rate constant for sintering kinetics in Eq. 10.73 radial dispersion coefficient for heat equilibrium constant for the surface reaction in Eq. 10.79 characteristic pellet size constant for sintering kinetics in Eq. 10.73 exponents for sintering kinetics in Eq. 10.73 concentration of the species causing deactivation bulk-fluid value of N N at reactor inlet Peclet number for heat Peclet number for mass deactivation capacity of catalyst in moles of the poisoning species per pellet volume normalized radial coordinate intrinsic rate of the main reaction [r, = kf(C)] intrinsic rate of the deactivation reaction [r, = W p ( C ) ] intrinsic rate for the reaction path between species At and Al gas constant net intrinsic rate of consumption of species At global rate of the main reaction net global rate for species At global rate of deactivation reaction tube radius surface site total catalyst surface area at time zero or for fresh catalyst active catalyst surface area at time t total catalyst surface area at time t including the surface area made inactive by chemical deactivation time time at which catalyst is regenerated temperature bulk fluid temperature coolant temperature Tc at y = 1 temperature at the boundary separating fresh inner core from the completely deactivated outer shell of a pellet reference temperature pellet surface temperature Tb at tube wall


T

at the boundary between fresh inner core and completely deactivated outer shell of a pellet deactivation reference condition pellet surface reactor inlet

cup-mixing temperature reactor inlet temperature interstitial fluid velocity overall heat transfer coefficient based on superficial fluid velocity coolant velocity conversion conversion of main reactant in bulk fluid normalized reactor coordinate pellet coordinate reactor length

Tin

U

U v

Greek letters

393

Superscript

evaluated at pellet surface; cup-mixing average

k !

I /Ph ; h ~ / ( pL) fraction of catalyst pellet deactivated reactor point effectiveness mole fraction times the change in number of moles due to reaction per mole of species of interest Arrhenius number (EJR, Tb);l/Pm bed porosity E for deactivation reaction internal effectiveness factor qr for fresh pellet internal effectiveness factor for deactivation reaction effectiveness factor given in Eq. 10.68 vt for the fresh inner core of a shell-progressively deactivated catalyst pellet effective thermal conductivity equilibrium constant molar density Z(1 - E B ) / V

PROBLEMS 10.1. Consider an isothermal reactor for which:

For a large mass Peclet number, the above equation can be rewritten as:

where

E =

l / P m and g ( C ) = TRC( C ) .

The boundary conditions are:

z/v

z/vc calculated at reference velocity Thiele modulus defined by Eq. 4.44 generalized modulus defined by Eq. 4.74 L(kp/DP)'l2 Thiele modulus defined by Eq. 10.69 L[(kp)d/D~l"~ T

Using perturbation, with E as a small parameter, C(y,e) for E f 0 can be expressed as a superposition of Co (solution for C when E = 0) and some perturbation :

1

Subscripts

bulk fluid pellet center; coolant

+

Normally, a first-order correction or C(y,E ) = Co E C is~accurate enough.

a. Show that g(C), when expanded in a Taylor series, can be expressed as:

f. Finally, show that:

10.5. Determine whether the equality of Eq. 10.15 holds for a first-order reaction. Expand Rc in a Taylor series around T and 10.6. For a first-order reaction taking place in an isothermal, plug-flow reactor, calculate T : a. assuming that no diffusional resistance exists b. taking diffusional effects into .

Use 5 as the Thiele modulus. CoUt/Ci,= 0.2. Compare the two T values to recognize the importance of diffusional effects in sizing a reactor properly. g. Show that for a first-order reaction with k r = 1, use of the first-order , > 30. term is sufficiently accurate for P 10.2. Derive Eq. 10.6 by letting:

10.7. Calculate the length of an adiabatic reactor required to convert 65% of SO2 to SOs. Obtain the reactor profiles of Tb, Cb, Ts, and A.

Ti,= 643OK. Feed at atmospheric pressure (7.8 mole % SOz, 10.8% 0 2 and 81.4% Nz)at 5700 kg/hr m2. (-AH) = 2 1.4 kcal/mole, mean specific heat = 0.22 1 kcal/kg°C.

r,(kmole SOz/kg cat - hr) =

and using the fact that:

-

22.414 (1

-&

-

+ KzPso2+ K ~ P s o , ) ~

Ee = 0.5

First arrive at the following result:

.. .

+ +

.

10.3. It is desired to examine the adequacy of thlug-flow model for an adiabatic reactor that is in operation. This is to be done based on temperature and concentration measurements at the inlet and outlet of the reactor. The following are known: = 673OK ci, = 5 x gmol/cm3 -A H/p C, = lo7 (OK cm3/rnol) T = 10 sec

L = 0.8 crn k = exp(12.16 - 5473/ T ) Kz = exp(-9.953 8619/T) K 3 = exp(-72.745 52596/7) Kp = exp(l1300/ T - 10.68) h = 3.3 X cal/sec cm2 O p, = 1.5 g/cm3 De = 0.5 X cm2/sec

To,, = 1073OK tout = 0.5 x 1ok5gmol/crn3 Pm=lOO Ph=50 Rc I =I = 5 X gmol/(cm3sec)

What can you conclude on the adequacy of PFR model? What is the outlet concentration difference between the value calculated from the PFR model and that from the axial dispersion model if the observed rate at the inlet is 10 times the rate at the outlet? 10.4. Show that the inclusion of the radial velocity profile in Eqs. 10.7 and 10.8 also leads to Eqs. 10.11 and 10.12 if the cup-mixing concentration and temperature are used for and T (Ahmed and Fahien 1980).

K

Use bulk temperature for K's. Use the equations given in Table 10.1 and the design procedures in Figure 10.1. Express r, in of Pso2 only using the stoichiornetry and inlet conditions. The solution obtained using a hand-held calculator for Ti,, = 643°K is 145 m. Note that small intervals have to be used for integration in the entrance region because of very slow conversion in that region. Redo the problem with Tin= 720°K.Assume that the effective diffusivities are the same for all species. 10.8. Derive Eqs. 10.28 and 10.29 from Eqs. 10.16 and 10.17. Use these equations in Problem 10.7 and resolve it. Show that no significant difference can be found for the exit conversion. Discuss the reasons. 10.9. Consider an adiabatic reactor for which P, = 200 and Ph = 20. The axial Peclet number for mass dispersion is typically in the range of 200 500. The Peclet number for heat dispersion is in general smaller than that

-

DESIGN AND ANALYSISOF FIXED-BEDS for mass dispersion. Even when mass dispersion can be neglected, heat dispersion may be significant. Obtain the reactor profiles of temperature and concentration for the following conditions: Ci, = r,

rnole/cm3

En = 620°K

=kC

cm2/sec -AH = 20 kcal/mole h =4 X cal/sec cm2 OK r = 10 sec.

k = exp(--12,00O/T L=lcm

13, =

pC, = 3.8 X

+ 14.6) (l/sec)

In addition, obtain Cd and Td profiles.

cal/cm3 OK

What can you conclude from the results? Neglecting the mass dispersion does not mean that Co = Cb. Rather, Eq. 10.1 should be solved as such instead of Eq. (A) in Table 10.3 with the aid of Eq. (E) in Table 10.3 but with E set to zero for both Eqs. 10.1 and (A). 10.10. For the reaction system given in Table 10.5, calculate the generalized modulus +c at the reactor inlet. 10.11. For the reaction system given in Table 10.5, plot the reactor point effectiveness against y at the reactor inlet. 10.12. Consider a first-order reaction taking place in an isothermal reactor. If the packed pellets are uniformly poisoned and the concentration of the poisoning species is constant throughout the reactor, what is the outlet concentration at t = lo7 sec? L=lcm D, = 0.1 cm2/s

= 10 seconds sec. Nb = g mole/cm3 cat. vol. Q= T

(b)=

10.14. Repeat Problem 10.13 for the case of shell-progressive deactivation. Use the following rate constant for the species causing deactivation.

10.15. Repeat Problem 10.12 for the case of shell-progressive deactivation. Use the following effective diffusivity:

Plot the results for the following eases: a. Assume plug-flow, neglecting dispersion of both heat and mass. b. Include heat dispersion. c. Include both mass and heat dispersion. d. Repeat for Ph = 2.

L(k/De)"*= 3 Ci, = 10-5g mole/cm3

399

g moe/crn3

Suppose that the catalyst is regenerated when the outlet concentration (C,,[) becomes 50% higher than Gout calculated above for r = 10 sec. Specify T based on Eq. 10.63. 10.13. For the reaction system of Table 10.5, obtain reactor profiles of Tb, Cb, T,, A, y, and Nb as a function of time on stream for the case of uniform chemical deactivation. Use the following rate constant:

10.16. Consider a pellet undergoing uniform chemical deactivation and sintering. If the main reaction is diffusion-free, the 'global rate is:

and the observed rate of the deactivation reaction is given by:

For an adiabatic plug-flow reactor, then, the conservation equations are:

Based on Eqs. (c) and (d), and Eqs. (F) and (G) in Table 10.8, procedures very similar to those in Figure 10.21 can be obtained. - results For the case of a diffusion-limited main reaction, show that the given in Table 10.8 can be used as such by simply replacing (Sr/So) by %/SO and S/Soby St/So. Also note (Chapter 4) that [ l G(Cb)] is close to unity (or G(Cb << 1) for a uniformly deactivated, diffusion-limited reaction. Therefore, G(Cb)in the equations of Table 10.8 can be set to zero. Further, show that Eq. (D) in Table 10.8 becomes for this case:

+

10.17. Consider a pellet undergoing shell-progressive chemical deactivation and uniform sintering. Show that the design equations given in Table 10.6 apply to this case. The only change is in Eq. (A) of Table 10.6:

CHAPTER I 1 Fluidized-Bed Reactors

for which the sintering kinetics given in Table 6.4 can be used for (St/ a ,

10.18. Derive Eqs. 10.101 and 10.102. 11-1

INTRODUCTION

REFERENCES Ahmed, M.and R.W.Fahien, Chem. Eng. Sci., 35, 897 (1980). Burghardt, A. and T. Zaleski, Chem, Eng. Sci., 23, 575 (1968). Cole, J.D., Perturbation Methods in Applied Mathematics, Blaisdell, New York (1968). Lee, H.H., AIChE J., 27, 557 (1981). Lee, H.H. and J.B. Butt, AIChE J., 28, 410 (1982). Lee, H.H., and E. Ruckenstein, Cat. Rev., 25, 475 (1983). Price, T.H.and J.B. Butt, Chem. Eng. Sci., 32, 393 (1977).

A fluidiked-bed for catalytic reactions is one in which small catalyst particles (typically in the range of 20 to 300 pm) are suspended by fluid reactants. The earliest application of fluidization was for the catalytic cracking of gas oil into gasoline. Rapid deactivation of the catalyst due to coking necessitated the change from fixed-beds to fluidized-beds for the cracking reactions. In the continuous reactorregenerator system (Froment and Bischoff 1979), show11 in Figure 11.1, the oil is fed at the bottom of the reactor through a perforated plate distributor and the reaction products are removed at the top. Fines entrained in the product stream are separated in cyclones and are then fed back to the reactor. Catalyst particles are allowed to leave the reactor through a bottom standpipe in which they are steam-stripped to remove adsorbed hydrocarbons. They then plow under a static head through the transfer line to the regenerator. The lower static pressure in the riser leading to the regenerator is attained by aeration of particles with the air required for burning off the coke. The regenerated catalyst flows into a downcomer and back to the reactor. The pressure difference required for this transport is attained by injection of the oil into the riser of the reactor. The heat produced by the regeneration is carried to the reactor by the catalyst particles and there they evaporate, heat, and crack the oil. With the advent of zeolite catalysts, the traditional fluidized-bed has been modified to the riser cracker. These zeolite catalysts are so active that the cracking takes place entirely in the riser, leaving the traditional reactor vessel only for housing cyclones for rapid separation of the catalyst from the product gas and a stripper. Significant catalyst losses by entrainment and attrition can occur because of the fluidization of small particles. On the other hand, the high turbulence created by fluidization leads to much better heat and mass transfer than is possible in a fixed-bed. In fact, fluidized-beds are now used for highly exothermic reactions for which close temperature control is important since the fluidized-bed can be operated isothermally. Fluidized-beds are an alternative to fixed-beds for highly exothermic reactions and when frequent regeneration of the catalyst is required. A quantitative description of a bubbling fluidized-bed is complicated because of the presence of gas bubbles that form at the feed distributor, rise, and then rupture when they reach the free surface at the top of the bed, releasing downward the particles drawn into the wake of the bubbles as they rise. Despite the complexity 40 1

Figure 11.1 Reactor-regenerator system for catalytic cracking of gas oil. (Zenz and Othmer 1960)

Regenerator

I

Air

Gas oil

involved, simple-minded approaches have been used: one of them is to treat the bed as a continuously-stirred tank reactor. In an analysis of a fluidized-bed for the regeneration of coked catalyst, Pansing (1956) assumed the coked catalyst to be well-rnixed but for the gas phase to be in plug-flow. His work contains the essential elements of two-phase models to evolve later: the presence of two distinct phases, gas-rich and solid-rich, and the degree of mixing and ing within and between the phases. For coke burning, the rate of reaction is equated to the mass transfer between the gas and the catalyst particles:

where Cc = concentration of carbon on catalyst pi = oxygen partial pressure at catalyst surface p = bulk gas-phase oxygen partial pressure k, K = rate and mass transfer constants R;;= observed rate based on catalyst weight A mass balance on oxygen in the plug-flow gas-phase gives:

which can be integrated with the aid of Eq. 11.1 to yield: --PC, =-c,

Slnf

K

+-k1

where f = N / N o , S = FNo/w, and P is total pressure. Here, F is the volumetric feed rate, N is oxygen concentration, No is inlet oxygen concentration, and w is the weight of the catalyst. Pansing correlated K to the mass flow rate and particle size (K a G m / d p )to' fit pilot plant date and found the activation energy of k to be 35 kcal/mol, which is close to the 36 kcal/mol value found by Weisz and Goodwin (1966) for coke combustion. Pansing's concept of separating the solidrich phase (emulsion phase) from the gas-rich phase (bubble phase) forms the basis for many two-phase models. The understanding of fluidized-beds is far from satisfactory, particularly regarding the fluid mechanics. The value of various models for fluidized-beds, however, lies in providing the framework within which each specific application can be considered. This chapter examines fluidization characteristics, the role that gas bubbles play, and the rationale behind the two-phase theory, which naturally leads to the models based on the two-phase theory. A full section will be devoted to the catalytic cracker, in particular the riser cracker, since it represents the most important application of fluidized-beds to catalytic reactions. This chapter starts with the understanding that the intrinsic rate is essentially the same as the global rate in fluidized-beds, unless the catalyst is deactivated.

11-2

FLUIDIZATION

The flow regimes that can exist when a mass of fine particles is subjected to a fluid flowing upward are shown in Figure 11.2, along with a diagram of pressure drop versus superficial fluid velocity. When the superficial fluid velocity is just high enough to the solid weight of the particles, incipient fluidization occurs and this velocity is called the incipient fluidization velocity or minimum fluidization velocity Uw. This situation is represented by the flow regime (a). The incipient velocity is usually determined by drawing the lines shown in the lower diagram and finding their point of intersection. The voidage at minimum fluidization is calculated from:

where A PbOm= pressure drop at Uw p, = solid density p = fluid density EN = vaidage at incipient fluidization HmJ = bed height at UmS g = gravity

appear as shown in the slugging regime (c) of Figure 11.2. In this regime, the bubble size is comparable to the diameter of the bed and the bubble motion is restricted by the walls. At a still higher pressure, slugs occupy the whole cross section and the particles rain through the low-voidage regions. This extreme case is often found in small diameter beds and is really a breakdown of proper fluidization. As the velocity exceeds the terminal velocity, solid particles are carried over, approaching the behavior of a transport reactor. The pressure drop remains relatively independent of the velocity until entrainment takes place. The terminal velocity can be estimated from:

.

I

Incipient

~ubbling

Slug

Carry-over

(a)

(b)

(c)

(dl

Superficialvelocity

Figure 11.2 Flow regimes and pressure drop versus gas velocity for fluidized-beds (Kunii and Levenspiel 1969). (Reprinted from Fluidization Engineering, @ Wiley, by permission of John Wiley & Sons, Inc.)

In the absence of experimental data, it is a good approximation to take ~d = 0.4 (Davidson et al. 1977) for uniform spheres. The minimum fluidization velocity can be estimated by combining the Ergun equation with Eq. 11.4:

which is an implicit expression for Uw. Here, d, is the particle size, and p is the fluid viscosity. While the fixed-bed Ergun equation has been used, it should give a reasonable approximation of Ud. In fact, the pressure drop through a fixed-bed has been shown (Richardson 1971) to be slightly more than through a fluidized-bed with the same particles and voidage fraction. As the velocity is increased beyond U,,,J, gas bubbles start to appear, as shown in Row regime (b) of Figure 11.2. This bubbling regime, in which the larget bubble are smaller than the bcd diameter D, h the one rnrfit w,rnrnonly found in industrial bubbling beds. When the velocity is further i n c r e a d , slugs

As seen from the lower diagram of Figure 11.2, a certain hysteresis exists on the bubbling decreasing the velocity. In the range of practical interest (U> 2 Urn/), regime prevails since industrial bubbling beds often have a height to diameter ratio close to unity, and thus, the bubble diameter is likely to be less than D. For a bubbling fluidized-bed, the "excess flow" above the value needed far incipient fluidization is (U - UmJ) (area). The two-phase theory is based on the supposition that much of the excess flow es through the bed as voids or bubbles. In this way the bed is divided into two phases, a bubble phase and an emulsion phase, which is like the bed at incipient fluidization. The emulsion phase is assumed to have the same voidage as at incipient fluidization. The presence of bubbles, shown in Figure 11.3, in the bubbling regime then leads to the two-phase theory. The essence of this theory is the hypothesis that the bubbles behave as if they were in a liquid. This analogy is often used to obtain an insight into the fluid mechanics of bubbles. In the absence of bubbles, a fluidized-bed can be treated as a simple CSTR, or a PFR. Fluidization behavior unique to fine particles will be discussed in Section 11-4, which involves nonbubbling beds at high gas velocities. The operating range of superficial gas velocity is bounded by Uw < U <

Particle Density

"O

Particle Diameter (mm)

F

a Air

Air (1.0 2S°C, 1 atm 10 (glcm3) X

Water 1.5 25OC ( 1 0

0.1

1 .o

Figure 11.4 Ratio of terminal velocity to incipient fluidizing velocity with (Richardson 1971)

EW

= 0.4.

U,.This range has been plotted as a function of particle size by Richardson (1971) for uniform spheres with E,,,J = 0.4 (Eq. 11.5) as shown in Figure 11.4. It is seen that for large particles (> -1 mm), the allowable range of U is only about 10 U,,,J.For beds of mixed particle sizes, U is often larger than Ut for the smallest particles, so there is continuous elutriation and a cyclone recycle system is used for particle recycle. The rate of elutriation of fines (Kunii and Levenspiel 1969) at unsteady state is often expressed as:

where k, = elutriation rate coefficient (wt/area . time) W = total weight of solids in the bed W, = weight of the amount of particles of diameter d, At = cross-sectional area of bed surface Correlations compiled by Leva (1960) for the elutriation rate coefficient are shown in Figure 11.5. The elutriation constant is obtained from the definition of k, and data from steady state runs (see Problem 11.2):

k=

Air

rate of removal of solids of size d, per area of bed surface fraction of bed consisting of size d,

(1 1.7a)

Figure 11.5 Correlation of elutriation rate coefficients for two-component systems. (Leva 1960)

11-3

TWO-PHASE MODELS

Two-phase models for bubbling beds focus on a single bubble, which is representative of all bubbles. This necessarily requires the assumptions that each bubble rises without coalescence through the bed and that all bubbles have the same diameter and behave the same as the representative bubble. Bubbles do grow as they rise and do coalesce. Therefore, these assumptions are oversimplifications and yet they bring out the most important characteristic of the bubbling bed, namely bying of the catalyst due to bubbles. This recognition of bying is the main improvement over earlier models (up to around 1960) that could not explain observed conversions below those for complete mixing. Since bubbles are considered to be free from catalyst particles and no reaction takes place in them, the gaseous reactant in the bubbles is depleted only through mass transfer from the bubble to its surroundings, which is the emulsion phase, and possible crossflow between the bubble and emulsion phases. Various two-phase models (Davidson and Harrison 1963; Partridge and Rowe 1966; Kunii and Levenspiel 1969) all

(sometimes, it is treated as a PFR). Therefore, Eq. 11.8 can be integrated directly to give:

Figure 11.6 Two-phase model of Davidson and Harrison. (Davidson, Harrison, Darton, LaNauze, in Chemical Reactor Theory: A Review, ed, Lapidus and Amundson, Q 1979. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.)

+

where Q = q k . S . Since the basis of the two-phase theory is that the excess flow (flow in excess of the value needed for incipient fluidization) represented by MU, - Uw) (area)] es through the bed as bubbles and the gas flow in the emulsion phase is at its fluidization velocity Uw, a mass balance at the top of the bed gives:

f '

t t t

1

where the subscript h refers to the value at z = H. Since C. is constant at Ceh, Eq. 11.10 can be rewritten with the aid of Eq. 11.9:

!

Ci,'Ji

have these basic ingredients, but they differ in treating the interchange of mass between the two-phases. The models due to Davidson and Harrison, and Kunii and Levenspiel will be examined in some detail here. The two-phase model of Davidson and Harrison is shown in Figure 11.6. The bubbles are assumed devoid of catalyst particles and to maintain representative characteristics such as diameter as they rise. Thus a bubble acts as a plug rising through the bed and the concentration within the plug changes only through interacis an overall mass transfer coefficient for the tion with the emulsion phase. If mass transport across the bubble film, and q is a volumetric rate of the crossflow, a steady-state mass balance for a single bubble is:

where

kc

= overall film mass transfer coefficient q = volumetric rate of cross-flow S = surface area of a bubble/volurne Ce = concentration of key species in emulsion phase Cb = concentration of key species in the bubble UA = absolute bubble velocity Vb = volume of the bubble

As indicated earlier, representative bubble characteristics such as volume and surface area are assumed to remain constant, i-e., independent of I. Because of very good mixing in the emulsion phase, this phase can be treated as a CSTR

A mass balance around the bed gives:

where E is the bed voidage inclliding bubbles and the interstitial gas in the emulsion phase, and S is the bed cross-sectional area. In this regard, it is noted that R,; is based on the apparent catalyst volume and that the global rate is equal to the intrinsic rate in most cases where catalyst deactivation is unimportant. Given the global rate of the reaction, Eqs. 11.1 1 and 1 1.12 can be solved simultaneously for C, and Ch. It is instructive to examine the conversion behavior of a first-order reaction for which RG[Ce]= r, [C,]= kc,. Equations 11.1 1 and 1 1.12 yield:

where Z = @emu = ( Ui - Uw)e-a/Ui, and K = Hk(l - E)/ Ui. At incipient fluidization, Ui = Uw, Z = 0, and there is no bubble flow. Therefore, Eq. 1 1.13 reduces to Ch/Ci = 1/(1 K), which is the conversion behavior of a CSTR.

+

An actual bed at incipient fluidization would operate more nearly in plug-flow; the results obtained are for a vigorously bubbling fluidized-bed. Two-phase models have been credited for ing for the conversion behavior of a bubbling bed, which gives a conversion lower than that for a CSTR. This can be clearly seen by subt-&ting from Eq. 11.13 the concentration ratio for a CSTR [(Ch/C1),= l/(l

+ K)]:

z-(z)

=

K2Z (1+K-Z)(l+K)

(1 1.14)

Since 0 < Z < 1, the right hand side of this equation is always positive. Therefore, the conversion with a bubbling bed is always poorer than with a CSTR,due to bying of gas through the bubbles. It is seen, on the other hand, that the bed approaches the behavior of a CSTR when the value of a is large since then Z approaches zero. According to the definition of a,a large value of a corresponds to slowly rising bubbles, whose velocity is proportional to (Ui - Uw), a large mass transfer rate from the bubble to the emulsion phase, or both. One would expect the bed to behave as a CSTR in both cases. When (Ui - Uw) is small, bubbles are also small and it is well recognized that the bed behaves as a CSTR, and when the mass transfer rate is high, bubbles essentially act as a source of reactant and bying of reactant is minimized. General conversion behavior of a fluidized-bed as affected by increasing velocity emerges from the foregoing discussion. A fluidized-bed behaves more like a PFR at incipient velocity. As the velocity is increased, small bubbles appear which causes good mixing, leading to the behavior of a CSTR. As the velocity is further increased, bubbles become larger causing the bying of reactant, leading to the conversion behavior of Eq. 11.13, which gives a conversion smaller than that for a CSTR. It is seen then that the conversion attainable decreases with increasing velocity. Therefore, typical industrial beds operating in the bubbling regime result from a compromise between conversion and the high throughput required of an industrial bed. Controllability of conversion for highly exothermic reactions also emerges from Eq. 11.13. When K is very large, Eq. 11.13 approaches 2. The model of Kunii and Levenspiel also focuses on a single bubble, as shown Figure 11.7 Single-bubble model of Kunii and Levenspiel (1969). (Reprinted from Fluidization Engineering, @ Wiley, by permission of John Wiley & Sons, Inc.)

i

in Figure 11.7. Here, y is the ratio of solid volume to bubble volume. A representative bubble with its fixed diameter and other bubble properties is treated as a batch reactor ascending through the bed:

The events of the reaction in each phase and the mass transfer between phases are traced for the determination of the overall rate of the reaction: transfer to cloud and wake. Overall disappearance = reaction in bubble Transfer to cloud and wake = reaction in cloud and wake transfer to emulsion. Transfer to emulsion = reaction in emulsion.

+

+

where the KU's are coefficients of mass transfer from phase i to phase j. Equations 1 1.16, 11.17, and 1 1.18 can be used to eliminate the unknowns, C, and C,. For a first-order reaction RG = rc = kC and Eq. 11.16 can be expressed as:

where the overall reaction-mass transfer coefficient KO is obtained from Eqs. 11.16 through 11.18:

Then, the conversion x is simply given by:

While the solution is straightforward for a first-order reaction, trial and error calculations have to be carried out for Cc and C, for more general rate expressions. In general y, > y, >> y b . and ya can be set to zero in Eq. 11.19.

The model of Partridge and Rowe (1966) takes into the bubble clouds, while the bubble wakes are treated as a part of the emulsion phase. While the two-phase models are superior to earlier simple models, there are ample opportunities for further improvements. These have to do on one hand with better ways of determining a priori the bubble properties appearing in the models. On the other hand, bubble growth and its effect on the performance of the fluidized-bed has to be ed for. All the models discussed above assume a constant effective bubble diameter. The correlations for the parameters in the two-phase models are given in Section 11-5. It is a well-known experimental fact that the bubbles grow as they ascend through the bed. This growth can be ascribed to expansion, coalescence, and gas transfer from the emulsion phase. On the other hand, bubbles also tend to split. Therefore, there will be a broad distribution of bubble sizes at any point in the bed and this distribution will change with the bed height. This section concludes with the comparison of various models by Pyle (1972), shown in Figure 11.8. For the first-order reaction considered, the differencesbetween the models are particularly noticeable near the bottom of the bed where the effect of the small bubbles used by Partridge and Rowe is in evidence. Over much of the range covered, moreover, the predictions of the models differ by up to 100% in either concentration or bed height. What is obvious from this comparison is a need for more reliable and general correlations for bubble properties.

- - . .

Kunii and Levenspiel model, 0, = 5cm Kunii and Levenspiel model, D, = 1Ocm Davidson and Harrison model, D, = IOcrn Davidson and Harrison model, 0, = 5cm Partridge and Rowe, D, = 5cm

11-4

r

I

I

FURTHER CONSIDERATIONS

The pressure drop versus superficial velocity diagram shown in Figure 11.2 for various flow regimes is for a particular particle size. A different diagram results for a different particle size. Geldart (1973) made a useful classification of solid particles into four characteristic groups, as shown in Figure 11.9, in order to identify the modes of fluidization according to the size arid density of the particles. Bubble-correlations are often based on particles belonging to group B (Horio and Wen 1977). These correlations, when extended to group A particles, for instance, could cause the model behavior to deviate from experimental observation unless care is taken. The classification of Geldart is useful in recognizing the dependence of bubbling behavior on the particle size. When fine particles are fluidized, there is a region in which the bed expands without bubble formation. This bubble-free region disappears when larger particles are fluidized, as shown in Figure 11.10 (DeJong and Nomden 1974). For large particles, bubbling will start when the gas velocity reaches the minimum bubbling velocity. Fluidization has become virtually synonymous with operation in the bubbling regime, corresponding typically to gas velocities of around 1 ft/s. An underlying function of fluidization has often been to afford between a gas and a large inventory of solid surface per unit bed volume. A question that can be raised is whether the same end can be achieved at gas velocities beyond the bubbling regime. The point advanced by Yerushalmi and co-workers (1976) is that for particles belonging to group A in Figure 11.9, fluidization can be achieved at gas velocities much higher than 1 ft/s. Since the solid feed rate determines the state of fluidization,

-

-----

4

k = 10s-l

I F

Particle diameter = 1 1Oum

0

50

100

150

200

Bed height (ern)

Figure 11.8 Comparison of various models and their dependence on effective bubble diameter. (Pyle 1972; reprinted with permission from Advances in Chemistry Series. Copyright 1972 American Chemical Society.)

0.2

0.1

---

, Extremely difficult to &"

10

fluidize I

20

-

1

1

t

I

I

50

100

200

500

I 1 1000

*

Figure 11.9 Four groups of particles proposed by Geldart (1973).

) .

I

I

Bubbling line

\

pp = 0.82 - 0.92 9~~

-

-+

I

sizes undergoing growth and coalescence as they rise. The inadequacy becomes acute when the rate of reaction is fast and the phenomena near the distributor region dominate (Horio and Wen 1977). A logical alternative is to allow the bubble size to vary along the bed axis. Another problem with the approach of a constant bubble size is its inability to deal with the flow reversal of gas in the emulsion phase, which occurs when the gas velocity is much higher than the minimum fluidization velocity. For this flow reversal, first verified experimentally by Kunii and co-workers (1967), Fryer and Potter (1972) proposed the following equation for estimating the critical superficial gas velocity U,, above which the flow reversal will occur:

1

Figure 11.10 Region of bubble-free operation, (Dejong and Nomden 1974)

for satisfactory fast fluidization the solid rate has to be high. In this "fast fluidization" regime, there is no gas-bying since there are no bubbles. A relatively dilute phase (low solid concentration) results from the fluidization. The fast fluidization, already practiced commercially (Reh 1971), needs further systematic studies. Another important state of fluidization for fine particles is "turbulent fluidization" (Kehoe and Davidson 1971; Massimilla 1973). As the gas velocity is increased, a breakdown of the slugging regime into a state of continuous coalescence occurs before the carryover of particles takes place. Thus, industrial beds of fine particles, which often operate at velocities around 2 to 3 ft/s, might very well be operating in the turbulent regime. Such a bed would operate as a nonbubbling bed at high velocities and could be modeled as a simple CSTR,plug-flow reactor, or a combination of both, in which the gas phase is treated as plug-flow and the solid phase as well-mixed flow. A model based on the latter approach is detailed later for a riser cracker. The two-phase theory originally proposed by Toomey and Johnstone (1952) made a significant contribution in the development of models for bubbling fluidized-beds. Although the general concept of gas bying in the form of bubbles is applicable to any bubbling bed, the interpretation of the concept as applied to the specifics of modeling is diverse. This diversity has led to a number of bubbling fluidized-bed models that have varying degrees of complexity. The two-phase models considered earlier use the gas bubbles as the means of relating the model parameters to operating conditions. Thus, an average bubble size, which is often estimated at the middle point in the bed, becomes the key parameter. The bubble size is often considered as an adjustable parameter in the absence of bubble size information. Although this approach could be used successfully in some cases, there are a number of problems associated with it. In the first place, use of an average bubble size cannot adequately describe the bed since bubbles are present in varying

where fw is the ratio of wake volume to bubble volume. Under flow reversal conditions, the average gas concentration goes through a minimum in the bed. In the following sections, a model for a bubbling fluidized-bed that allows for the variation of bubble properties along the bed axis will be considered, together with a model for a nonbubbling fluidized-bed at high velocities.

11-5 A MULTIPHASE, VARIABLE BUBBLE PROPERTY MODEL

Peters and co-workers (1982) combined the bubble assemblage concept of Kato and Wen (1969) and the three-phase model of Fryer and Potter (1972) to advance a cell (compartment) model in which an equivalent spherical diameter for a bubble is the cell size (height). The equivalent diameter is allowed to vary along the bed axis and each cell, consisting of bubble, cloud, and emulsion phases, is considered to be perfectly mixed in each phase, as shown in Figure 11.11. For each phase in a cell, a mass balance can be written, ing for the convective mass flow into and out of the cell, and for the mass transport between phases in the same cell. A solution for the axial concentration profiles is obtained from the bottom up, starting with the inlet conditions that specify the size (height) of the first cell. The solution for the first cell is then used to calculate the next cell size (equivalently the bubble diameter) which allows solution of the mass balance equations for the second cell. These procedures are repeated until the cumulative cell size reaches the expanded bed height. Consider the nth cell in Figure 11.1 1. According to the model, the mass balance for the bubble phase is:

CoutnUi

Figure 11.11 Cell model of Peters et al. (1982). (Reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, Inc.)

t H

where the subscript e denotes the emulsion phase. The mass balance for the emulsian phase has to be written for three different regions of the bed when flow reversal occurs at one of the cells, say n,. The cell number rt, at which flow reversal starts is found in the cell calculations when the superficial gas velocity in the emulsion phase, calculated according to the relationship given below, becomes negative:

where the absolute bubble velocity for the nth cell, UAn,is determined (Davidson and Harrison 1963) from:

where the first subscript b is for the bubble phase, the second subscript n is for the cell number, and the subscript c is for the cloud phase. Here, U is the superficial gas velocity such that Ubn is the superficial gas velocity of the bubble phase out of the nth cell, S is the bed cross-sectional area, C is the gas concentration, V is the volume of the phase such that Vbn is the volume of bubble phase in the n t h cell, Pbc is the gas cross-flow between bubble and cloud, Kbc is the gas interchange coefficient from the bubble to the cloud phase, and N is the total number of cells. The volume fraction of gas in the bubble, ~ b is, assumed constant at unity; the corresponding quantities for the cloud (eC) and emulsion phases (E,)are assumed to be constant at E N .Since ~b is unity by assumption, there are no catalyst particles in the bubble, and thus, no consumption term due to reaction appears in Eq. 11.22. The first two in the right hand side of Eq. 11.22 are the convective mass flow into and out of the nth cell. Although the third term is called mass exchange due to "cross-flow," it is not the cross-flow referred to in the Davidson and Harrison model (Eq. 11.8), but rather, it is included to satisfy the overall mass balance for the cell. This cross-flow is included in the mass balance for the emulsion phase to for the emulsion phase degasing. The degasing coefficient for the emulsion phase, (PC,),, may be evaluated by:

Here, g is the acceleration due to gravity, De is the effective bubble diameter, and Sin is the volume fraction of the bed in the nth cell occupied by phase i. Thus, for the emulsion phase:

E~

dcen Ven= G , , - ~ s C e , n -, UenSCen lSn
Cell by cell calculations can be carried out for given initial conditions and the following relationship:

Cbo=Cco=o=Ct C,,,

The fourth term in Eq. 11.22 represents the mass transfer at the bubble-cloud interface. The mass balance for the cloud phase is:

attheinlet

(1 1.31)

The gas concentration averaged over the cross-sectional area of the bed, is given by:

41 8

HETEROGENEOUS REACTOR DESIGN 1.0 -

0.8

-

I

I

1 0

I

I

1

I

Experiment

-

- Model Prediction

6 0 0 i.

Figure 11.12 A comparison between cell model predictions and experimental data. (Peters et al. 1982; reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, Inc.)

tions at the minimum fluidization (Uw, E , and Hw), reaction kinetic parameters, the molecular diffusivity of the gas (D,),the bed diameter (D), and the number of orifice openings on the distributor ( N D ) . It is seen that the model describes the experimental data reasonably well. Perhaps more important is the ability of the model to predict the minimum concentration in the bed caused by flow reversal in the emulsion phase. In order to use Eqs. 11.22-1 1.30, the model parameters in these equations have to be evaluated from the basic parameters. The relationships for the model parameters, which are given by Peters and co-workers, are summarized in Table 11.2 in the order in which they will be calculated. As indicated earlier, these calculations have to be carried out cell by cell. Since the e:fKectivebubble diameter D, required for specifying the cell height is implicit, an iterative procedure involving

Table 11.2 Relationships for Model Parameters (Adapted from Peters et al. 1982 by permission of Pergamon Press, Inc.) -

The average outlet concentration, when flow reversal occurs, is obtained from:

De (Mori and Wen 1975):

D m = 0.652[S(Ui - Uw)]2/5 The inlet concentration to the Nth cell in Eq. 11.30, i.e., Ce.N+l,is given by C,,,. If flow reversal does not occur, only Eq. 11.28 is required for the emulsion phase and the outlet concentration is given by:

(Ddo = 0.347[S( Ui

--UUw)2: w ) / N ~ ] 2 / 5perforated : distribution plates porous distribution plates

0.00376 (&

Range of validity:

< U,,,J< 20 crn/s; Uf-UmS<48crn/s;

0.006 < dp < 0.045 crn DC130cm

0.5

The steady state version of Eqs. 1 1.22 through 1 1.30 can be used for design purpose. A comparison between cell model predictions and the experimental data of Fryer and Potter (1975) is shown in Figure 11.12 (Peters et al. 1982). The basic parameters used to calculate the model parameters for the prediction in Figure 11.12 are given in Table 11.1. These parameters are the inlet conditions, the condiTable 11.1 Basic Parameters for the Simulation in Figure 1 1 12 (Peters et al. 1982)

S c / S p (Murray 1965):

-8,= Sb

E

1 E~UA/U,,,J

(Kato and Wen 1969): HnLr 7 (1 -%"Y)

for h

1-E=

Hd

[

-€ 1

-

( I ] h-Hw

< Hw

for h 3 Hw

Table 11.2 (continued)

Nb (Kato and Wen 1969):

sj(j = 6, c, e): Sj = V,/SD,

vj(j = 6, c, e): Ub=UASb~b;

6c

Uc=--

~c

Sb Eb

Ub;

We given by Eq. 11.26

KU (Murray 1965; Sit and Grace 1981): i i

1

Pu: Eqs.

Figure 11.13 Interphase transfer factors in Eq. 11.35. (Davidson, Harrison, Darton LaNauze, in Chemical Reactor Theory: A Review, ed. Lapidus and Amundson Q 1977. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ)

11.23 and 1 1.24

1

improvement of D, calculated from the previous cell information is required. In arriving at the superficial gas velocity in the emulsion phase, the gas flow belonging to the wake phase was neglected (Yacono et al. 1979) such that the superficial gas velocity, which is constant at the inlet velocity (Ir, is the sum of (Ib, U,, and U,. The bubble correlations given in this section are also applicable to the twophase models. The best results are often obtained when these properties, particularly the bubble size, are evaluated at the conditions halfway up the bed. Since the bubble correlations are given in Table 11.2, only the parameters in the two-phase models are given here, which are different from those in Table 11.2. The factor a (Eq. 11.11) in the Davidson and Harrison (1977) model (Eq. 11.8) is given by:

4

i,

I

For the Kunii and Levenspiel model, Kc, is that given in Table 11.2, but they use the following for Kbc:

For the volume fraction y, they give:

i

+ + ye) = (1 - E N I J ) ( ~ - c b )

~ b ( ~ Yr b

+

where the factors F, and Fdt which depend on DJD, are given in Figure 11.13.

where is estimated from Figure 11.14. The value of since it is in the range of 0.001 to 0.01.

ra

is usually set to zero

Figure 11.14 Value of 4 in Eq. 1 1-38. (Kunii and Levenspiel 1969; reprinted from Fluidization Engineering, @ Wiley, by permission of John Wiley & Sons, Inc.)

where m is the ratio of the product molecular weight to the reactant molecular weight. Combining Eqs. 11.40 and 11.42, and rendering the result dimensionless, one has:

where 8 = t/t,, x = z/z0, and t, is the catalyst residence time. If V, is the reactor volume and- E the bed voidage, vopo/zo = Fo/Vre, FO being the mass rate of feed. Then, Eq. 1 1.43 can be rewritten as:

oil transit time m poE: pSt,k -I- m(1 - fm)12 catalyst decay time (t,) where B = 11-6

NONBUBBllNG BEDS AT HIGH VELOCITIES

As indicated earlier, industrial beds of fine particles operating at high gas velocities may be in the turbulent regime or may operate as a fast fluidized-bed. For such a bed, no bubbles are present, and thus, the bed could be modeled as a simple CSTR,as a plug-flow reactor, or as a combination of both in which the gas phase is treated as plug-flow and the emulsion phase as well-mixed. Here, the latter approach taken by Weekman (1968) is considered for a riser cracker since the same approach can I x applied to nonbubbling beds operating at high velocities. Weekman's model is based on assumptions of plug-flow for the gas and of CSTR behavior for the emulsion phase with a CSTR internal age distribution of particle. In essence, this model treats the fluidized-bed as a PFR with a CSTR age distribution of the catalyst being deactivated in the bed. If the weight fraction of gas oil in the riser is fmr the continuity equation is:

S = Fo/Vrj5;

= liquid density of feed before the feed

(1 1.45b)

enters the riser

Here, S is the space velocity based on the liquid feed, which vaporizes upon entering the riser. The catalyst decay time is much longer than the vapor residence time. Therefore, Eq. 11.44 can be approximated by:

An exponential decay model was used to represent the decay of activity due to coking, such that:

where A is a constant. The internal age distribution of the catalyst according to an ideal CSTR model (Levenspiel 1972) is: where po and vo are the density and velocity at the inlet, respectively. Here, the following relationship has been used: Then, the average activity of the catalyst particles in the bed can be calculated from: The density can be related to the weight fraction by:

1;

K= k

= k/(l

r(e)exp(-he)de

+ A)

(1 1.49)

Hence,

Given the concentration dependence of the rate of the reaction fCfn,), this equation can be integrated for the conversion. Second-order kinetics were used by Weekrnan (1968) for fCf,).

Summary

The state of fluidization is perhaps the most important factor in the design of a fluidized-bed. Particle size is a dominant factor in determining the state of fluidization. Thus, for industrial beds of fine particles belonging to group A in Figure 1 1.9, bubbling fluidization would be rarely found since turbulent and fast fluidization would allow better between the catalyst particles and the gas at much higher throughputs than bubbling fluidization could allow. For larger particles, however, bubbling fluidization would be used. For nonbubbling beds operating at high velocities, simple models, in which CSTR behavior for the emulsion phase and PFR behavior for the gas phase are assumed, seem to describe the conversion behavior adequately, as the studies of Pansing (1956) and Weekman (1968) indicate. For bubbling beds, however, multiphase treatment involving bubbles is required. The two-phase theory for bubbling beds is based on a single bubble with a fixed diameter and associated bubble properties that are assumed representative of all bubbles in the bed, While the two-phase models based on this theory for gas bying, and the bubble properties can be estimated a priori, the model predictions are often in error. A better understanding of bubble properties is certainly warranted. The cell model, which allows the parameters to vary along the bed height, is an improvement over the two-phase model and can for flow reversal in the emulsion phase. While the accuracy of the fluidized-bed reactor models, in particular the models for bubbling beds, is still insufficient for general use in the design and scale-up of the beds, these models do provide a frame within which each particular application can be formulated.

I

I

I

I

j

i

i 1

1f I

1 I

!

II y

1 I

i NOTATION A

C cb

CC

ce c h

ci

constant defined in Eq. 11.45~ concentration of key species bubble-phase gas concentration cloud-phase concentration; coke concentration emulsion-phase gas concentration C at the top of bed inlet concentration

t

concentration of gas in phase j ( j = 6, c, e) in the nth cell outlet concentration particle diameter bed diameter effective bubble diameter molecular diffusivity I;. = kf(c) weight fraction of gas oil in feed ratio of wake volume to bubble volume volumetric flow rate, mass rate in Eq. 11.45b constants in Eq. 11.35 acceleration due to gravity mass rate per area height from distributor plate bed height occupied by fluidized particles H at incipient fluidization intrinsic rate constants elutriation constant defined in Eq. 11.7a gas-phase film mass transfer coefficient in Eq. 11.8 rate constant in Eq. 11.19 film mass transfer coefficient in Eq. 11.1; constant defined in Eq. 1 1.13 mass transfer coefficient from phase i to phase j overall reaction-mass transfer coefficient defined by Eq. 11.19 ratio of product molecular weight to reactant molecular weight number of cells as counted from bed inlet; moles of main reactant in Eq. 11.5 n at which flow reversal occurs in the emulsion phase flux, oxygen concentration; last cell number of bubbles in a cell number of distributor holes bulk pressure pressure drop across bed quantities given by Eqs. 11.23 and 11.24 interfacial pressure cross-flow rate in Eq. 11.8 q+kS global rate based on particle volume global rate based on pellet weight F N J W in Eq. 11.3; space velocity defined by Eq. 11.45b; bed crosssectional area time transit time of gas oil superficial velocity superficial absolute bubble velocity given by Eq. 11.27 average gas velocity in emulsion phase in Figure 11.9 superficial velocity defined in Eq. 11.21

inlet supeficial velocity supeficial velocity of gas in phase j ( j = b, c, e) in the nth cell incipient superficial velocity terminal velocity velocity of gas oil vapor bubble volume volume of phase j in a cell ( j = 6,c, e) in the nth cell catalyst weight total weight of solids in bed weight of solid particles of diameter d, conversion; z/z, distance from the bottom of the riser of catalytic cracker height of the riser of catalytic cracker constant defined in Eq. 11.13

Creek Letters QH/UAVb ( Ui - UW)/Ui solid volurne/bubble volume volume fraction of bed occupied by phase j bed voidage volume fraction of gas in phase j ( j= 6, c, e) E at incipient fluidization constant in Eq. 11.47 fluid viscosity fluid density bulk density of particulate phase solid density constant in Eq. 11.38 t/t, Subscripts

b c

e i

n

bubble-phase cloud-phase emulsion-phase inlet or feed cell number as counted from bottom

PROBLEMS

1 1.1. Plot incipient and terminal velocities as a function of particle size for particle sizes in the range of 10 to 300 microns. Use the properties of air for the fluid (p, = 0.5 g/crn3). What can you conclude regarding the range of veloci-

ties for fluidization (UM iU < Ut)as affected by particle size? Assume EN = 0.4. 11.2. Since any particle whose terminal velocity is smaller than the operating velocity is elutriated, the elutriation constant in Eq. 11.7 depends on particle size. When particles of wide size distribution Po(dp) are fed continuously into a bed as shown in Figure 11.2P,fine particles are entrained by the gas while the remainder are discharged through an overflow pipe. Suppose that the bed weight W, the feed properties, Ff (mass rate) and Pf(dP), and elutriation constant k ( d , ) are known and that one wants to find the exit conditions: F , , Fz, Pl(dp), P2(dv).For steady state operation, the definition of k. (Eq. 11.7a) gives:

where the size distribution in the bed Pb is the same as P I under the assumption of perfect mixing. a. Show that

Use the fact that FOPo= FlP I

+ FzP2 Figure 11.2P Fluidized-bed at steady state for the determination of size distributions. (Kunii and Levenspiel 1969; re-

printed from Fluidization Engineering, @ Wiley, by permission of John Wiley & Sons, Inc.)

b. Noting that jsizI,P(R)dR= 1, integrate Eq. p11.2 to obtain

1 II I

c. Write procedures for determining Pl(dp) and Pz(dp). Refer to Kunii and Levenspiel (1 969).

+

11.3. For the simple Langmiur-Hinshelwood kinetics of r, = kC/(l K,C), obtain the conversion using the Davidson-Harrison model. Only the parameters K,, Z(= fieha) and K[= Hk(1 - F)/ U i ] should appear in the expression. Plot Ch/Ci as a function of Z and K for K, = lo5 and K, = lo4. Use Ci of 5 X low5g rnol/cm3. 1 1.4. Consider a deactivation-affected reaction of first-order. Since the particle size is very small, uniform deactivation can be assumed. Further, assume that a pseudo-steady state is applicable with respect to deactivation.

a. Show that the effective rate constant at any given time can be expressed as

and therefore, RG = koCexp(-yt). Here, y is a constant depending on concentrations and temperature, and ko is the rate constant for fresh catalyst. At steady state with a continuous flow of catalyst particles, y remains constant. b.

Show that the average activity of a catalyst (average effective rate constant) is given by:

if one assumes complete mixing for the internal age distribution of the catalyst particles.

c.

Use the Davidson-Harrison model to obtain an expression similar to Eq. 11.13. Compare the Ch/Ci ratio between deactivating and nondeactivating catalysts for y of 5.

11.5. Write the equations necessary for the calculation of conversion according to the Kunii-Levenspiel model for a second-order reaction, for which Re - r, = kc2.Note that an overall reaction-mass transfer coefficient such as the one in Eq. 11.19 cannot be expressed explicitly. 11.6. Use the data given in Table 11.1 to obtain the results shown in Figure 11.12

based on the cell model.

i iI

I 1

i

fi

i\

!

REFERENCES Davidson, J.F. and D. Harrison, Fluidized Particles, Cambridge University Press, London (1963). Davidson, J.F., D. Harrison, R.C. Darton and R,D. LaNauze, in Chemical Reactor Theory, Chap. 10, pp. 625, 630, eds. Lapida and Amundson, Prentice-Hall, Englewood Cliffs, N.J. (1977). DeJong, J.A.H. and J.F. Nornden, Power Technol., 9, 91 (1974). Froment, G.F.,and K.B. Bischoff, Chemical Reactor Analysis and Design, p. 664, Wiley, New York (1979). Fryer, C. and O.E. Potter, Ind. Eng. Chem. Fund., 11, 338 (1972). Fryer, C. and O.E. Potter, International Fluidization Conference, Preprint 111-1, Pacific Grove (1975). Geldart, D., Powder Technol., 7, 285 (1973). Horio, M. and C.Y. Wen, AIChE Symp. Ser., 73, No. 161, p 9 (1977). Kato, K.and C.Y. Wen, Chem. Eng. Sci. 24, 1351 (1969). Kehoe, P.W.K. and J.F. Davidson, Inst. Chem. Eng. (London) Symp. Ser., 33. 97 (1971). Kunii, D. and 0. Levenspiel, Fluidization Engineering. Wiley, New York (1969). Kunii, D., K. Yoshida and I. Hiraki, Proc. Int. Symp. Fluidization, p 243, Netherlands Univ. Press (1967). Leva, M., Fluidization, McGraw-Hill, New York (1960). Levenspiel, O., Chemical Reaction Engineering, 2nd ed., Wiley, New York (1972). Massirnilla, L.,AIChE Symp. Ser., 69, No. 128, 11 (1973). Mori, S. and C.Y. Wen, AIChE J., 21, 109 (1975). Murray, J.D., J. Fluid Mech., 21, 465 (1965). Pansing, W.F*, AIChE J., 2. 71 (1956). Partridge, B.A. and P.N. Rowe, Trans. Inst. Chem. Eng. (London), 44, T335, T349 (1966). Peters, M.H., L.S. Fan and T.L. Sweeney, Chem. Eng. Sci,, 37. 553 (1982). Pyle, D.L., Adv. in Chem. Ser., 109, 106, Am. Chem. Soc., Washington (1972). Reh, L., Chem. Eng. Prog., 67 (2), 58 (1971). Richardson, J.F., in Fluidization, p 25, eds. Davidson and Harrison, Academic Press,London (1971). Rowe, P.N., in Fluidization, edr. Davidson and Harrison, Academic Press, London (1971). Sit, S.P. and J.R. Grace, Chem. Eng. Sci., 36, 327 (1981). Toomey, R.D. and H.F. Johnstone, Chem. Eng. Prog., 48, 220 (1952). Weekman, V.W.,Jr., Ind. Eng. Chem. Proc. Des. Dev., 7, 90 (1968). Weisz, P.B. and R.D. Goodwin, J. Catalysis, 6, 226 (1966). Yacono, C., P.N.Rowe and H. Angelino, Chem. Eng, Sci., 34, 171 (1979). Yerushalmi, J., D.H. Turner and A.M. Squires, Ind. Eng. Chem. Proc. Des. Dev., 15. 47 (1 976). Zenz, F.A. and D.F. Othmer, Fluidization and Fluid-Particle Systems, Reinhald Pub. Co.,

CHAPTER 12 Multiphase Reactors

12-1 INTRODUCTION

Gas-liquid reactions catalyzed by a solid catalyst are carried out in multiphase reactors. These reactors are usually used when the gaseous reactant is too volatile to liquefy or when the liquid reactant is too nonvolatile to vaporize. Typical examples are hydrogenation, desulfurization, and hydrocracking of petroleum feeds. Many different reactor configurations can result depending on the mode of feeding and the manner in which the catalyst is placed in the reactor. The two most common multiphase reactors are slurry and trickle-bed reactors, which have been described in some detail in Chapter 7. Since other reactor configurations can be analyzed in a similar manner as for slurry and trickle-bed reactors, attention here will be focused on these reactors. There are some similarities between fluidized-beds and slurry reactors. However, there is a distinct difference to be recognized with respect to the relative movement between particles and fluid. Unlike the fluidized-bed, there is little relative movement between particles and fluid in a slurry reactor, even though the liquid is agitated mechanically. The particles tend to move with the liquid, which considerably simplifies the modeling. Because of the high heat capacity and relatively high thermal conductivity of the liquid, uniform temperature control is easy to achieve for highly exothermic reactions. Therefore, slurry reactors should be considered as an alternative to fixed-beds for highly exothermic reactions. Because of the very small size of the catalyst particles, diffusional effects are often negligible, even though the intraparticle effective diffusivity in liquid-filled pores is of the -5 order of 10 cm2/s. The reactor can be run in a batch or continuous mode with respect to the catalyst. The catalyst particles are retained in the vessel in the batch mode. The most serious disadvantage of slurry reactors is the difficulty in handling the catalyst particles in the vessel. There is a ,problem of retaining the catalyst when operating in a batch mode and of separating the catalyst from the product when operating in a continuous mode. For instance, screens placed in the outlet lines tend to clog. Gaseous and liquid reactants can be fed to a fixed-bed packed with catalyst pellets cocurrently or countercurrently and in the case of cocurrent feeding, the flow may be either upward or downward. Typical industrial practice is to feed both reactants downward in a cacurrent manner. A fixed-bed with this type of

for which the intrinsic rate based on the species A is represented by:

Spray flow

Here, CA is the liquid-phase concentration of species A, and Cg is the liquidphase concentration of species B. The basis for r~ is unit volume of the catalyst.

12-2 Trickle flow

Liquid mass rate (glcm2h)

SLURRY REACTORS

With good agitation and little relative movement between particles and liquid, it is a good assumption that the liquid phase is well-mixed. The situation with respect to the gas phase is less certain. With very strong agitation, the gas bubbles may coalesce and redisperse, and may have a residence time approaching that of a completely mixed fluid. With less agitation, the bubbles approach plug-flow as they rise through the liquid. Fortunately, it is only for cases of intermediate solubility of the gaseous reactant in the liquid that the residence time distribution of the bubbles is important. It has been shown (Niiyana and Smith 1976) that for either very soluble or slightly soluble gases, the residence time distribution of the gas bubbles in the liquid has no effect on reactor performance. Since most applications of importance involve either very soluble or slightly soluble gases, the gas phase will be treated as being in plug-fiow. For the slurry reactor shown in Figure 12.2, a mass balance for the gas phase (bubbles) can be written as:

Figure 12.1 Flow regimes in downflow trickle-beds. (After I. M. Smith, Chemical Engineering Kinetics, @ 1980; with permission of McGraw-Hill Book Co., NY,)

feeding is called a trickle-bed. The advantages are that there is no limitation on the flow rates imposed by flooding and the liquid is much more thinly and evenly distributed than with upward flow. One important characteristic of a trickle-bed is that the gas-phase is continuous throughout the bed. Downflow packed beds can have different flow regimes depending on the respective flow rates of gas and liquid as shown in Figure 12.1 (Smith 1981). The trickle-flow regime results when both flow rates are relatively low, in which the gas phase is continuous and the liquid phase is dispersed. Increasing the gas flow rate leads to pulsed flow. For high liquid throughputs, the liquid phase may be continuous and the gas phase dispersed: this is called bubble flow. The lines separating different flow regimes are approximate boundaries, and thus, should not be taken too rigorously. With the global rates developed in Chapter 7 for slurry and trickle-bed reactors, it is now possible to proceed with the design of these reactors. Reactions of interest are:

A(g)

+ bB(1)

products

where CAg= bulk concentraticn of species A in the gas phase vb = bubble velocity kL = liquid-side mass transfer coefficient al, = interfacial surface area of bubbles per unit liquid volume ~b = fraction of the liquid volume occupied by bubbles es = fraction of the liquid volume occupied by catalyst particles CAL= bulk concentration of species A in liquid phase Here, it has been assumed that the resistance to mass transfer is dominated by the liquid-side of the interface so that the gas-side interfacial concentration Cig is equal to the bulk concentration C, for the gaseous species A. It has also been assumed that Henry's law holds so that CAg= HCeq where H is the Henry's law constant and C,, is the liquid-phase concentration in equilibrium with CAS. For slightly soluble gases, the bubble velocity vb can be assumed constant, as implied by Eq. 12.2. For very soluble gases, vb is not constant, but it will be

where Eq. 7.14 has been used for RG.Here, a, is the external area of a catalyst particle per unit volume of catalyst. The pellet surface concentration of species A, CAs,is determined from:

Gas feed

-

I

-

---

Liquid feed

--

Gas bubble

b Liquid product 0

0

.

. 0

0

1

. 0

1

0

0

- * * .

1 1 .

I

The assumptions involved are that the reaction is diffusion-free and that the concentration of liquid reactant B remains constant. Given the intrinsic kinetics, Eqs. 12.5 through 12.7 can be used to design the reactor. For a first-order reaction ( r A = kcA), Eq. 12.6 can be solved far CAs,which upon inserting into Eq. 12.5 gives:

Catalyst '

particle

Figure 12.2 Slurry reactor. (After I. M. Smith, Chemical Engineering Kinetics, @ 1980; with permission of McGraw-Hill Book Co., NY.)

seen that this balance equation is not necessary. Mass balances for the well-mixed liquid phase are:

i

When the liquid reactant is the species of primary interest, Eq. 12.2 need not be included in the design if the gaseous species is present in excess and the liquid is saturated. For instance, hydrogen used for the treatment of petroleum feeds is usually present in excess and readily soluble in the petroleum. In such a case, only Eq. 12.3 is involved in the design, which can be rewritten as:

where V = liquid volume of the reactor Q = volumetric flow rate of liquid CB1,=bulk concentration of liquid reactant B b = stoichiometric coefficient in Eq. 12.1 CtlLin= Csr,at the reactor inlet Z = length of the reactor Equation 12.3 is the mass balance of liquid reactant B, which is well-mixed. In Eq. 12.4, the inlet concentration of A in the liquid phase is zero, and therefore, the total mass of A transferred from the gas to the liquid phase should be equal to the sum of species A leaving the reactor and the amount of this species consumed by the reaction. Since the liquid phase is well-mixed, both CAr.and CBLare uniform throughout the liquid phase. Equations 12.2 through 12.4 completely describe the slurry reactor provided that the global rate is expressed in of bulk-liquid CBL]. concentrations, i.e., (Rc)A[CAI,# Two types of reactions can be considered: one in which the removal of species A in the gas stream is of major interest, as in the removal of pollutants, and the other in which the treatment of liquid reactant B is of major interest, as in the desulfurization of petroleum feeds. For the former, only Eq. 12.2 is of concern. The global rate developed in Chapter 7 can be used to rewrite the equation:

The particle surface concentration CBS is obtained from:

It is seen that Eqs. 12.9 through 12.11 are sufficient to design the reactor. The general design problem posed by Eqs. 12.2 through 12.4 can be solved without difficulty. The solution of Eq. 12.2 yields:

I

i i

where A = kLaL/Hebvb (CAg)o = C A at ~ reactor outlet ( C A g ) i n = CAgat reactor inlet This equation has to be solved together with the following algebraic equations:

the reactor is isothermal. Even with a very small particle size (-100 micron), the diffusion resistance can become important for some fast reactions since the effective diffusivity in liquid-filled pores is of the order of 10-5 cm2/sec. In such cases, the rate constant k should be replaced by the effective rate constant kq, where q is the internal effectiveness factor. Equations 12.13 and 12.14 follow from Eqs. 12.3 and 12.4. The integral in Eq. 12.4 can be evaluated using Eq, 12.2, as given in Eq. 12.14. Take as an example a simple reaction for which r~ = kf (CA,CB)= kCACB. Assume, for brevity, that the mass transfer coefficients are the same, i.e., It follows from Eqs. 12.15 and 12.16 that: (kc), =

12-3 j

I I

'ie This, upon inserting into Eq. 12.15, yields:

This can be solved for CASto give: I

r

It has been assumed that the root of Eq. 12.18, given by Eq. 12.19, yields CAs, which is positive. Now that CAsand Cssare expressed in of CAI.and CBL, these can be used in Eqs. 12.13 and 12.14 to obtain CB1, and (CAg)Owith the aid of Eq. 12.12. If the problem is to specify the liquid volume (reactor volume) V far a given desired conversion, say 60% conversion of B, Eq. 12.13 yields:

1 r

kl

TRICKLE-BED REACTORS

The design of trickle-beds is similar to that of fixed-beds, although it is more complex. In trickle-beds, reactions take place in liquid-filled pores, and therefore, the presence of both gas and liquid phases increases the transport resistances. Many complex situations can arise depending on the nature of the reaction, the distribution of liquid flow, and wetting. For extremely exothermic reactions, the pellet is not necessarily isothermal. Further, vaporization of the liquid reactant can occur. The gas can be assumed to be uniformly distributed across the diameter, but the liquid tends to flow toward the reactor wall. A reactor-to-pellet diameter ratio of 18 or more, however, ensures uniform liquid distribution (Herskowitz and Smith 1978). When the liquid velocity is relatively low, the catalyst pellets are not completely wetted with liquid and partial wetting occurs. Further, axial dispersion cannot be neglected at low velocities. Trickle-beds play an important role in petroleum processing and several reviews are available by Sattefield (1975), Goto et al. (1977), Hofmann (1978), and Herskowitz and Smith (1983). A book by Shah (1979) is also available. In the first part of this section, attention will be restricted to the simple conditions of uniform gas and liquid distributions, a nonvolatile liquid, trickling flow, isothermal pellets, and complete wetting of the pellets by the liquid. Nonisothermal pellets will also be treated for the case of first-order reactions. In the second part, a more complex case at low liquid velocities is examined.

1 12-3-1

This can be solved for CAI,,which upon inserting inta Eq. 12.14 gives a relationship containing V as the only unknown when Eq. 12.12 is used for (CAy)o.It is seen that even for relatively simple kinetics, the design requires trial and error calculations for this general case. It has been assumed that the intraparticle resistances are negligible and that

Simple ~rkkle-6eds

Consider a simple trickle-bed for which one can assume uniform gas and liquid distributions, a nonvolatile liquid, trickling flow, isothermal pellets, and complete wetting. For an isothermal reactor with a uniform distribution of the gas and liquid phases, there will be no radial gradients of concentration. Although axial dispersion is more important in trickle-beds than in fixed-beds because of relatively low fluid velocities, we will assume that the dispersion is negligible. Under the assumptions, one can write for the gas phase (refer to Figure 12.3):

1

1

4

-

Liquid feed

where CA and Cg are the concentrations of A and B in the pellet, and DA and DB are the effective diffusivities of A and B in the liquid-filled pores. Integration of this equation using the usual boundary conditions yields:

Therefore, the intrinsic rate can be written as:

1-

Liquid product

Figure 12.3 Trickle-bed reactor. (After J. M. Smith, Chemical Engineering Kinetics, @ 1980; with permission of McGraw-Hill Book Co., NY.)

It follows that a generalized effectiveness factor similar to Eq. 7.17 in Chapter 7 can be written as:

where v, is the superficial velocity of the gas. For reactant A in the liquid phase, the conservation equation is:

where vl is the superficial velocity of the liquid. Here, the gaseous species A in the liquid phase is replenished by gas-liquid mass transfer and depleted by reaction through liquid-solid mass transfer. For the liquid reactant B, the conservation equation is:

Boundary conditions are the feed concentrations. The relationships between the surface and bulk-liquid concentrations are obtained by equating the mass transfer rate at the solid interface to the global rate:

where C i is the pellet center concentration of species A, and fi is defined in Eq. 12.27. As in Chapter 10, the pellet center concentration C i can be approximated by C*,/COS~ +(;, where &: is the generalized modulus. Equations 12.23, 12.24, and 12.28 can be solved for CAsand Cns in of CAI, and CBL.Then the balances of Eqs. 12.20 through 12.22 are solely in of CAY.C A I , , and CHI,, which can readily be solved. It should be recognized in Eqs. 12.25 through 12.27 that the reaction stoichiometry is not sufficient to relate Cgs to CASor CAI,to CHLsince CA is determined not only by the reaction stoichiometry but also by the mass transport rate. Take as an example a reaction which is first-order with respect to both A and B: rA = kf(CA.C,j) = kcnCH.The intrinsic rate can be expressed (Eq. 12.27) as:

If it is assumed that the reaction is diffusion-limited so that C i = 0, Eq. 12.28 becomes: where r] is the internal effectiveness factor excluding the effects of external resistances. Given the intrinsic rate kf(CA,CB), Eqs. 12.20 through 12.24 can be solved far the design. Since only one species was involved in Chapter 7 for the effectiveness factor, here the effectiveness factor for two species is examined. The pellet conservation equations for species A and B, when combined, give:

*3 where 4 = L ( ~ / D B ) ' '> B m ,=L(~)B/DH Bhl. = Lh/A P = (-AH)DBCBL/A TL y = &/Rg TL

Then, Eqs. 12.23 and 12.24, for instance, become:

The three balance equations (Eqs. 12.30 through 12.32), need to be solved for the design with the aid of Eq. 12.33 for the global rate. While relatively simple results have been obtained for first-order reactions, full pellet-side conservation equations have to be solved for rate expressions more complex than first-order.

These two equations can be solved with Eqs. 12.20 through 12.22. It is seen that even for the simplest form of rate expression, a numerical method has to be used. Considerable simplification results when the rate of the reaction depends only on species A. In such cases, the global rate developed in Chapter 7 can be used directly for the design. As indicated earlier, the pellet can be nonisothermal for some reactions such as hydrocracking. Here, a simple case in which the gaseous reactant is present in excess and the liquid reactant is saturated with the gaseous reactant will be considered. Under these conditions, the equilibrium concentration Cr,(= Cns/H) is the same everywhere in the reactor, and C A I , is also constant. Further, it is assumed that the reaction is first-order with respect to the liquid reactant. Then the rate of the reaction is represented by:

where the apparent intrinsic rate constant k, is now constant with respect to CAL since gl(CAL) is constant throughout the reactor. Thus, only the conservation equations concerning species B are necessary for the design. For an adiabatic reactor, these are:

12-3-2

Trickle-Beds at Low Liquid Velocities

As indicated earlier, the catalyst pellets are not necessarily wetted completely when

'

the liquid velocity is low. According to the literature data summarized by Sicardi and co-workers (1980) partial wetting can occur when the liquid mass flow rate is less than approximately 10 kg/rn%. Their results show that partial wetting always occurs when the liquid rate is less than 2 kg/m%. Although internal partial wetting in which the pores are not completely filled with liquid can occur, the term partial wetting normally refers to the condition where the pores are full, but the external surface of the catalyst is only partially covered by liquid. At low liquid velocities, dispersion in the liquid phase cannot be neglected. Thus, even with the assumptions of a nonvolatile liquid, isothermal pellets, trickling flovv, and uniform gas and liquid distributions, one has to deal with axial dispersion and partial wetting at low liquid velocities. The fraction of the pellet surface covered by liquid, q, which is sometimes termed the wetting efficiency, has been correlated by Mills and Dudukovic (1981) as follows:

vldp pi

where Rer = Pl

where TL is the bulk-liquid temperature, T, the bulk-gas temperature, and U the overall gas-liquid heat transfer coefficient. The approximate global rate developed in Chapter 7 gives: Here, d, is the pellet diameter, pi the liquid density, p~ the liquid viscosity, at the external surface area of the pellet per unit volume of the bed, L, the mass

superficial velocity per unit area, g the acceleration due to gravity, 6! the liquid surface tension, and E B the bed porosity. This correlation is based on very few data points and is less accurate at lower liquid velocities. As indicated in Chapter 7, the usual range of q is between 0.6 and 1.0. An exact value of q is not often critical in assessing the effect of the wetting efficiency on the overall effectiveness factor (Mills and Dudukovic 1979) unless the mass transfer resistances are significant. This can be readily seen from the following approximate relationship for the overall effectiveness factor for first order kinetics given in Chapter 7:

In fact, correlations for Peclet numbers required for reactor modeling are available almost exclusively for the liquid phase. Bufiam and Rathor (1978) give the following correlation:

where

where the effectiveness factors for a pellet completely covered by liquid ( q ~ and ) by gas ( q ~ are ) given by 71-=

qc =

Here, E is the axial dispersion coefficient, and he is the dimensionless external holdup, which is a sum of the dynamic holdup and the external static holdup. Correlations for the holdup can be found in Shah (1979) and Gianetto et al. (1978). Consider the design of a trickle-bed operating at low liquid velocities for the reaction given by Eq. 12.1. For the gas phase, one has:

+/+

tanh 1 + 4 tanh + / B m L tanh

1

+/+

+ + tanh +/HB,I:

If the mass transfer resistances are small such that Bml, and Bmc: are large, q L is approximately the same as q ~ and , thus, q, is independent of q as apparent from Eq. 7.24. However, an accurate value of q would be required for the overall effectiveness factor if the mass transfer resistances are significant. In light of the fact that BULLis usually small, although HB,c is large, better correlations than Eq. 12.34 are necessary to describe a trickle-bed at low liquid velocities. In general, partial wetting can be detrimental or beneficial depending on which phase contains the limiting reactants (Herskowitz and Smith 1983). If the limiting reactant is liquid as in hydroprocessing reactions, partial wetting reduces the rate of the reaction due to the decrease in liquid-solid mass transfer area. If the gaseous reactant is the limiting reactant as in hydrogenation reactions, partial wetting enhances the rate of the reaction because of the absence of gas-liquid mass transfer resistance for the nonwetted part of the pellets. This can readily be seen from Eqs. 7.24 through 7.26 above since qc > ql,(BmI,< HB,,c), and thus, the overall effectiveness factor increases with decreasing q. The approximate overall effectiveness of Eq. 7.24 above can be used for reactor design for first-order reactions. For nonlinear kinetics, however, it is difficult to obtain an analytical expression for the overall effectiveness factor that is accurate for partially wetted pellets. Even when a relatively accurate effectiveness factor is available, the presence of different bulk and interfacial concentrations in bath phases reduces the utility of the overall effectiveness factor except for limiting cases. Therefore, an internal effectiveness factor applicable to both wetted and nonwetted parts of the pellet is used here instead. For the low liquid velocities being considered, only dispersion in the liquid phase need be considered, and the gas phase may be treated as though it were in plug-flow (Levec and Smith 1976).

since a fraction, q, of the pellet surface is covered by liquid and a fraction, (1 - q), is covered by gas. Here, CA, is the concentration and k, is the mass transfer coefficient at the gas-solid interface. For reactant A in the liquid phase, the conservation equation is:

I

!

For the liquid reactant B, one has:

/

The boundary conditions are:

where the subscript f denotes feed conditions. The relationships for the interfacial concentrations are obtained by equating the mass transfer rate at the interface to the global rate. Unlike the simple trickle-bed, however, one has to consider wetted and nonwetted parts separately. Thus, for reactant A: ( k a c ) (CAL ~ - C A ~=) ( R c ) A = ~ k q d ( C ~ , Cm): . ( ~ C (CAG ) A - CA~= S )( R c ) A = ~ kq&(C~gs/H,CHs):

wetted part

Summary

In this chapter, the design of the two most common multiphase reactors has been treated. While the design of slurry reactors is relatively straightforward, that of trickle-beds can be complicated when the reaction is highly exothermic. For the nonisothermal case, only a very simple case where the liquid is nonvolatile has been treated to illustrate the design procedures. Further complications can arise when liquid velocity is low. At low liquid velocities, the catalyst particles may become partially wetted and axial disperion may be significant, at least for the liquid phase. One distinct feature of multiphase reactors, when compared to fixedbed reactors, is that the conservation equations have to be written for the gaseous reactant for both the gas and the liquid phases. The other feature is the important role of mass transfer resistances due to the presence of three phases, in particular the liquid phase. Modeling of trickle-beds in which the liquid evaporates and the pellet surface is partially wetted at low liquid velocities requires further work.

(12.43)

dry part (12.44)

I

where q, and qd are the internal effectiveness factors based on CAsand C*p/H, respectively. Here, it is assumed for qd that the concentration of A in the liquid at the mouth of the liquid-filled pore, which is exposed to gas, is at CAw/H. It is also assumed that the concentration of B at the pore mouth is at Cs,. For the reactant B, one has:

Here, q, is given by Eq. 12.28, and 176 is simply q given by Eq. 12.18 with CAs replaced by CA,/H. The pellet center concentrations necessary for q, and 116 can be obtained from the following approximate relations:

where ( + G ) and ~

(+~)d

are given by:

For (&)LO. fl(CA) is given by Eq. 12.27; for ( + ~ ) d .fl(C.) is given by Eq. 12.27 with C A replaced ~ by CAw/H. Design of a trickle-bed at low liquid velocities requires solving Eqs. 12.38 through 12.45. Considerable simplification results if the wetting efficiency is unity, in which case q = 1, 11, is q given by Eq. 12.28, and the equations for the dry part of the pellet disappear. While the design can be carried out using the conservation equations, the accuracy with which the Peclet number and the wetting effiency can be estimated would limit the utility of the result.

NOTATION

1

external area of catalyst particles per unit volume of catalyst gas-liquid interfacial area per unit volume of liquid external surface area of pellets per unit volume of bed stoichiometric coefficient for the liquid reactant in Eq. 12.1 Lh/h &L/D kt,L/D; L ( k ) s / D B in Eq. 12.33 concentration of species A concentration of species B molar specific heat gas-phase bulk concentration of species A liquid-phase bulk concentration of species A pellet center concentration of species A C; for dry part of pellet pellet surface concentration of species A liquid-phase bulk concentration of species B pellet surface concentration of species B concentration of A at gas-solid interface pellet (particle) diameter effective diffusivity in liquid-filled pores effective diffusivity of species A in liquid-filled pores effective diffusivity of species B in liquid-filled pores axial dispersion coefficient (rn2/s) for mass dependence of rate on concentrations gravity gas-solid film heat transfer coefficient external holdup Henry's constant

rate constant apparent rate constant mass transfer coefficient at the interface between catalyst and liquid mass transfer coefficient at the gas-solid interfalce mass transfer coefficient at.the interface between gas and liquid characteristic length of pellet liquid mass rate per unit area fraction of pellet surface covered by liquid (wetting efficiency) volumetric flow rate of liquid intrinsic rate of reaction based on species A intrinsic rate of reaction based on species B global rate for species A global rate for species B gas temperature liquid temperature overall heat transfer coefficient between gas and liquid bubble velocity superficial velocity of gas superficial velocity of liquid liquid volume axial coordinate reactor length Creek Letters

quantities defined in Eq. 12.33 fraction of liquid volume occupied by bubbles bed porosity fraction of liquid volume occupied by catalyst particles internal effectiveness factor r] given by Eq. 12.28 with CAsreplaced by CA,,I/I 7 given by Eq. 12.28 Thiele modulus, L(k/D)lI2 generalized modulus given by Eq. 12.48 or 12.49 molar density effective thermal conductivity viscosity surface tension Subscripts

A b

B d

species A bubble species B nonwetted part of pellet

f G < I

1 I, w

o in

feed gas phase liquid liquid phase wetted part of pellet reactor outlet reactor inlet

PROBLEMS

12.1. Specify the liquid volume of a slurry reactor required to convert 60% of the liquid reactant at 52S°K for the following reaction:

A (g)

+ 3B ( l )

products

The gaseous reactant is present in excess for saturation at lo-' mal/cm3 with a Henry's constant of 3 (mol/crn3 gas)/(rnol/cm3 liquid). The inlet mol/crn3 with a voluconcentration of species B in aqueous solution is metric flow rate of 1 cm3/s. The liquid-solid mass transfer coefficient including the surface area to volume ratio (&%) is 2 sVX for both A and B. The intrinsic rate is:

r~ = loLoexp(-20,000/Rg 7')CsCA mol/'(s cm3 catalyst) Assume that the fraction of the liquid volume occupied by the caialyst is 0.02. 12.2. Resolve Problem 12.1 when the reaction is affected by dependent deactivation. The rate of deactivation is given by:

Assume that the catalyst particles are continuously removed from the vessel, but that the internal age distribution of the particles is that of an ideal CSTR:

12.3. Show for an adiabatic trickle-bed that the conservation equations can be written as:

If axial dispersion effects are included, the conservation equations become:

CHAPTER

13 - - .

Optimization

Using the results obtained in Chapter 9 regarding the equivalence between plug-flow and axial dispersion models, determine whether the plug-flow conservation equations given above are sufficient as a first-order approximation of the dispersion model. 12.4. Suppose that the reaction in Problem 12.1 is carried out in an adiabatic trickle-bed. What is the catalyst volume required for the desired conversion? Assume that the effective diffusivities are the same at 10-~cm2/s and the spherical pellet diameter is 1 cm. Use the following data:

k, = 0.02 cm/s for both A and B

12.5. Explain why Eqs. 12.30 and 12.33 are not sufficient to describe the tricklebed when in fact Eq. 12.33 contains only CBL.

REFERENCES

Buffham, B.A. and M.N. Rathor, Trans. Inst. Chem. Eng., 56. 266 (1978). Gianetto, A., G. Baldi, V, Specchia and S. Sicardi, AIChE J., 24. 1087 (1978). Goto, S., J. Levec and J.M. Smith, Cat. Rev., 15, 187 (1977). Herskowitz, M. and J.M. Smith, AIChE J., 24, 439 (1978). Herskowitz, M.and J.M. Smith, AIChE J., 29, 1 (1983). Hofmann, H.P., Cat. Rev., 17, 21 (1978). Levec, J. and J.M. Smith, AIChE J., 22, 159 (1976), Mills, P.L. and M.P. Dudukovic, Ind. Eng. Chem. Fund., 18, 139 (1979). Mills, P.L. and M.P. Dudukovic, AIChE J., 27, 893 (1981). Niiyana, H. and J.M. Smith, AIChE J., 22, 961 (1976). Satterfield, C.N., AIChE J., 21. 209 (1975). Shah, Y.T., Gas-Liquid-Solid Reactor Design. McGraw-Hill, New York (1979). Sicardi, S., G,Baldi, A, Gianetto and V. Specchia, Chcm. Eng. Sci., 35. 67 (1980). Smith, J.M., Chemical Engineering Kinetics. 3rd ed., McGraw-Hill, New Yotk (1981).

13-1

INTRODUCTION

There are t w major ~ categories of reactor optimization. The first type, which is usually the most important for single reactions, is that of maximizing conversion for a given reactor size by properly manipulating the operating conditions, typically temperature. This problem is equivalent to minimizing reactor size for a given conversion. The second type, which is relevant to multiple reactions, is that of maximizing the yield of the desired product. The overriding factor in the latter type is the relative magnitude of activation energies associated with each reaction path (Denbigh and Turner 1971; Millman and Katz 1967). In order for an optimization problem to exist, there must be conflicting factors. These are, for instance, the rates of formation and disappearance of the species of interest in multiple reactions. If there are no constraints, there are essentially no optirnization problems for irreversible single reactions since the highest possible temperature will always lead to maximum conversion and minimum reactor size, except under some unusual circumstances. Even for reversible single reactions, the highest possible temperature will lead to the maximum conversion if the reactions are endothermic since the equilibrium conversion should increase with temperature. For single reactions, therefore, only exothermic reversible reactions are of interest since then the equilibrium conversion becomes the conflicting factor against the rate constant, which increases with increasing temperature. For such single reactions, the conversion is constrained by the equilibrium conversion. On the other hand, any reaction is of interest for optimizatioil if the reaction is affected by catalyst deactivation since a temperature increase will not only increase conversion, but will also accelerate catalyst deactivation. Thus, the conversion averaged over a period of time can be maximized only when the temperature is properly manipulated. In the sections to follow consideration will be given to single or multiple reactions constrained by equilibrium conversion and catalyst deactivation. There are many techniques (Beveridge and Schechter 1970; Ray and Szekely 1973) available that can be used for reactor optimization. Some that are used here include simple maximization, the calculus of variations, dynamic programming, and the maximum principle. Use of these techniques leads to some very powerful optimal policies in simple cases. These policies, however, are not always applicable to realistic heterogeneous reactors since the restrictions placed on the

optimal policies are not always compatible with heterogeneous reactions inherently affected by transport resistances. When an optimization problem is posed, the assumption is that the system is characterized well enough for quantitative description. For reactions affected by catalyst deactivation, however, this is not always true. Therefore, there is a need to estimate the extent of deactivation from process measurements for the optimization. This problem will be treated in some detail. Undoubtedly, much can be gained from the optimization of reactor operating conditions. Often neglected, however, is the fact that much more can be gained by optimizing the reactor size at the same time. This fact will be pointed out and illustrated whenever appropriate to emphasize the importance of optimizing with respect to both reactor size and operating conditions.

13-2

of the main reactant concentration, stoichiometric coeficients, and diffusivity ratios. Therefore, the net int+nsic rate can be expressed as:

where gr and gb are the concentration dependences of the forward and backward rates, and k and k d l are the corresponding rate constants. Thus, the expression for f is:

EQUILIBRIUM CONSTRAINTS

where K can be expressed as: Consider a single reaction constrained by an equilibrium conversion. An optimization problem for a given throughput is that of choosing the inlet temperature or the temperature trajectory along the reactor that will minimize the reactor size for the desired conversion. If the maximum possible conversion is the overriding concern such that a multibed arrangement is used, one can have another optimization problem of allocating the extent of reaction to each bed so as to maximize the final conversion with certain constraints. In general, an endothermic reaction in which the equilibrium conversion increases with increasing temperature is of no interest since the highest allowable temperature is the best policy. On the other hand, one can pose an optimization problem for an exothermic reaction since an increase in temperature that results in an increase in the rate also causes a decrease in equilibrium conversion. Suppose that an exothermic reaction is carried out in an adiabatic fixedbed. The design equations given in Table 10.1 in Chapter 10 for the adiabatic reactor are:

Here, Eb is the activation energy for the backward rate constant kWl.Consider now the optimization problem of minimizing the reactor size, or equivalently T , by properly choosing the inlet temperature (Tin) for a given throughput, desired conversion, and inlet concentration. It should be clear that this problem is equivalent to that of choosing the optimal temperature for the maximum conversion for a given reactor size T . Since r is a function only of Tin when Eqs. 13.2, 13.4, and 13.5 are used in Eq. 13.1, the optimization is simply a one-parameter problem, which can be solved using the usual conditions for a minimum. That is, the optimal Tin is that which satisfies the following conditions:

-37- - 0

aT~,!

and

a27

-> O a2

zn

A simple search can also be used in which various values of T are calculated for assumed values of Tin until the minimum is found. Here again, the pellet center concentration, C,, is obtained from:

One can readily show that these equations are also applicable to a single reversible reaction provided that C is the concentration of one of the reactants, the heat of reaction (-AH) is based on a unit mole of the main reactant, and the concentration dependence of the intrinsic rate f is properly expressed (see Problem 13.1). As detailed in Chapter 10, any forward and backward rates can be expressed in

where f is given by Eq. 13.4, It is always desirable to know the best possible performance even when such a performance is difficult to attain due to implementation difficulties. For the optirni-

zation being considered, the question is what the temperature trajectory along the reactor should be to minimize the reactor size. For a single reaction, the optimal temperature trajectory is that satisfying the following condition:

where v is the superficial fluid velocity, A, the cross-sectional area, V the reactor volume, and C, and C n - l are the inlet concentrations to the ( n and nth beds. The net profit function Qn can be reduced to the following performance index:

+

where Rc for use in the above equation is: since v is constant for a given throughput, and therefore, PI, is equivalent to Ql,. Here, Zn is the reactor length for the nth bed. The problem is then to maximize:

This result follows directly from the results of Aris (1965) obtained for reactions free from transport effects. Obviously, the optimal temperature depends on concentrations, and therefore, the implementation of the optimal policy requires concentration measurements at every point along the reactor. Furthermore, the optimal policy at some points along the reactor may require a negative or infinite temperature (Fournier and Groves 1970) at low conversions. An alternative to this ideal policy is a multibed arrangement. The discrete, stagewise arrangement of multibed adiabatic reactors can be optimized using the techniques of dynamic programming (Bellman 1957; Roberts 196.4). Dynamic programming is based on the principle ofoptimulity, which states that the optimal policy for an N-stage system must be such that the decision for 1 - n) stages of the the ntQtage ( n = 1, . . . , N ) is optimal for the last ( N N-stage system for any input to the nth stage. Obviously, the search for the optimal policy should start with the last bed for the multibed arrangement ( N number of reactors) since the optimization for n = N involves only one stage, which in turn involves the optimal choice of inlet conversion and temperature to the last bed. The principle of optimality requires that the choice of inlet conversion and bed should be optimal for the last two beds, but temperature to the ( N subject to the optimal policy already made for the last bed. The same is true for the last three reactors, but this time subject to the optimal policy already known for the last two. This stage by stage optimization can be carried out until the first bed is reached. The principle of optimality, therefore, considerably simplifies the optimization problem. Perhaps as important as this simplification are the relationships that result from the application of the principle. Suppose that the optimization problem is to maximize a measure of the profit. If a is the price of product and y is the cost incurred in producing the product per unit volume of reactor per unit time, a measure of the net profit for each bed Qn is:

I

by properly selecting the inlet concentrations and temperatures to N beds. In order to change the inlet temperatures, there has to be a heat exchanger between two adjacent adiabatic reactors. This multibed arangement is shown in Figure 13.1. The optimization problem is to be solved subject to the following system equations:

1

where E H is the bed voidage, (Cb),, and ( ' f b ) , , are the bulk-fluid corlcentratiori and temperature in the nth bed, and the global rate RG is that given by Eq, 13.9. For the adiabatic reactor under consideration, (Tb), is related to (Cb)n by:

+

A

Y

which results when Eqs. 13.14 and 13.15 are combined. Therefore, (RG')n is a function solely of (Cb)n, which allows one to rewrite Eq, 13.14 in the form of Eq. 13,l for r,, i.e.:

t

Figure 13.1 Multistage reactor/heat exchanger arrangement.

since the integrand of Eq. 13.1 is simply -l/(Rc), for the nth bed P, can be rewritten as:

. Thus, the performance index

The last part of the integrand follows from the fact that vZ,/v = v ~ , / ( 1- E B ) . As discussed earlier, application of the principle of optimality begins with the last bed for the overall optimization, which can be stated as:

It should be noted that the feed concentration Cois specified such that the decision variables for the first bed are To and C 1 since T I is fixed by the relationship of Eq. 13.16 applied at the outlet of the first bed. This means that the decision variables for the Nth reactor are TN-Xand CN. It should also be noted that Tn # Tn because of the heat exchanger between the two adjacent beds. Thus, TN-Iand CN have to be chosen such that:

i I

The first condition states that:

(Tinin Eqs. 13.1 and 13.2) as the The integrand in Eq. 13.21 contains only unknown parameter, which can be differentiated with respect to TN-1 for ~ in Eq. 13.1). Simultaneous the indicated integration with respect to ( C b ) (Cb solution of Eqs. 13.21 and 13.22 is illustrated in Figure 13.2. A curve satisfying Eq. 13.22 is first drawn as shown in the figure (AN curve), which can be generated by calculating the CN values corresponding to assumed values of T N .For every point on the AN curve, there is a pair of C N - Iand TN-1satisfying the adiabatic line of Eq. 13.16, i.e.:

Figure 13.2 Optimal solution paths for threestage reactor/heat exchanger on C-T plane.

1

i

is expressed in of T , Eq. 13.21 can be solved for T N - -as ~ the only unknown. The pairs of CN- I and T , corresponding to the pairs of CN and TN on the A, curve can be plotted as shown in Figure 13.2 (BN curve). The optimal values of the inlet temperature and concentration to the Nth bed should be on this curve BN since any pair of C N - Iand on this curve satisfies both Eqs. 13.21 and 13.22 due to the adiabatic relationship of Eq. 13.23. Consider now the last two beds. According to the principle of optimality, TN-2and CN-Ihave to be chosen in such a way that:

rN-l

In order to find the pair, C N - Iand T , - l , corresponding to a point ( C N , T Non ) the AN curve, the values of CN and TN satisfying Eq. 13.22 are used in the above relationship to obtain CN-l in of T N - l .NOWthat CN is known and C N - l

-Max

The optimal condition states that the rate at the outlet of the ( N bed must equal that at the inlet of the last ( N t h ) bed. Thu:;, this condition specifies

(PN+PN-l)=Max[PN-l+Max P N ]

(TN-2*CN-l)

= Max

[ICN-' + -) lCN + l) (-1

CN - 2

+~

a r

C N -1

A

R~~ -I

d ( ~ a ) ~ ]

(-1

the heat exchanger load (refer to Figure 13.2). Here again, a pair of CN-l and TN-I values corresponding to a point on the curve BN (left hand side of Eq. 13.29) can be calculated from Eq. 13.29. The locus of CN-, and TN-1satisfying the equation is shown in Figure 13.2 as the curve AN-^. The second optimal condition of Eq. 13.26 leads to:

~(CD)N-I (13.24)

R ~ N

The maximization requires the following conditions to be satisfied:

- conditions already established for the last bed. Since PN subject to the optimal is dependent on CN, TN and CN Eq. 13.25 becomes:

i

I

The first and third vanish because of the optimal conditions (Eqs. 13.20 and 13.21) already determined for the last bed. It follows then that: I

which reduces to:

This relationship is identical to that of Fq. 13.21 with the exception of the stage subscripts. Procedures identical to those used to generate the BN curve can be followed to obtain the locus of CN-Zand TN-2 satisfying Eqs. 13.29 and 13.30, which is the curve B N - l in Figure 13.2. This curve represents the optimal inlet temperature and concentration to the (N - l)th bed. The procedures discussed so far can be repeated for the optimization until the first bed is reached. The optimal solution for a three-bed arrangement ( N = 3) is illustrated in Figure 13.2. First, AN and BN curves representing the optimal conditions for each bed and heat exchanger are constructed as detailed above and shown in Figure 13.2. Once these curves are available, the optimal solution is straightforward if it is recognized that the AN and BN curves are related to each other by the adiabatic operating line with a slope of (-B) as evident from Eq. 13.23 and the AN-1 and BN curves are related to each other by the constant concentration line for the heat exchanger (Eq. 13.29). All that is needed for the solution is to locate one point on any of the curves. Since the feed concentration Co is specified, one can locate the point corresponding to Co on 131 ( B N - z in Figure 13.2 for Co versus To) designated as S. From S, the adiabatic line is drawn to the A N - 2 curve as shown by the dotted line in *he figure; the constant concentration line is then followed to the BN-1 curve, and the adiabatic line is drawn again to the AN-I curve followed by the constant concentration line to the BN and then to the AN curve via the adiabatic line. Thus, the points on the AN and BN curves at which these meet with the dotted lines represent the optimal inlet conditions to the three beds, corresponding to the specified feed concentration Co. It is seen that the inlet temperature is the highest for the first bed and becomes progressively lower for subsequent beds. Further, the outlet stream from each bed should be cooled before it is fed to the next bed. The problem of including cooling costs in the performance index has been treated by Lee and Aris (1963). It should be recognized that a shell and tube reactor/heat exchanger in which the sections of the heat exchanger are maintained at different temperatures to obtain optimal operation is entirely analogous to the rnultibed arrangement just considered. The same procedures should apply even though the calculations are

The pseudo-steady state assumption has been made in writing Eq. 13.32 for a slowly deactivating catalyst. Equation 13.31 can be solved for T to give:

much more complex since the adiabatic relationship no longer holds and a term for cooling has to be added to Eq* 13.15. The application of dynamic programming invariably involves numerical calculations. Nevertheless, some very useful relationships emerge, such as in Eq. 13.29 in the process of applying the principle of optimality.

Thus, the conversion can be expressed as: 13-3

CATALYST DEACTIVATION

Many optimization problems arise as a natural consequence of the fact that the catalytic activity decreases with time when the catalyst is deactivated. Given this fact, the question is how to optimize the reactor operation so as to maximize yield or a performance index that is a measure of profitability. The optimization can also involve sequences of beds to allow reaction and regeneration in a swingcycle. Often neglected, but just as important as the optimization of operating conditions, is optimization with respect to reactor size, which was treated to some extent in Chapter 10. The usual industrial practice of operating a reactor affected by catalyst deactivation is to manipulate the reactor inlet temperature so as to maintain a constant conversion at the outlet. The reactor throughput is also rnanipulated to compensate for the declining catalytic activity, but usually only when the temperature cannot be raised any further due to the limitation on the maximum temperature allowed. When a shell and tube reactor/heat exchanger is used, the coolant temperature is also manipulated. The usual practice of maintaining a constant conversion is optimal wirh respect to the maximum possible yield under some limiting conditions (Szepe and Levenspiel 1968; Chou et al. 1967). Consider an isothermal, plug-flow reactor. Suppose the deactivation kinetics are such that the fractional catalytic activity A can be described by:

where Eq. 13.34 has been used for 13.33 as:

According to a basic theorem of the calculus of variations (Bliss 1961), the condition for an extremum of the function given in the form of Eq. 13.36 is: x ( A , A t ) - A ' x n , ( A , A t )= constant;

P

=ltf

xdt

0

ax

xA, E aA'

(13.37)

It is seen that the optimal policy would be that of maintaining the conversion constant if A t x A , is zero or a function solely of x since then the left hand side of Eq. 13.37 should be constant, which implies constant x. It can be shown (Chou et al. 1967) that Eq. 13.37 leads to the optimal policy of maintaining a constant conversion for an irreversible reaction in which only one rate constant and temperature dependence is involved, and that the policy is also applicable to nonisothermal reactors, The limitations under which the policy of a constant conversion is valid are embodied in Eq. 13.37: x should be a function of A and A' and not dependent explicitly on t, and A ' x * . should be a function solely of x. The policy gives the maximum of P when an optimal temperature is chosen and hence requires a search for the optimal temperature. The problem of maximizing the performance index of Eq. 13.33 can also be solved using the maximum principle, which will be discussed shortly. For the deactivation kinetics given by:

Suppose further that transport resistances are negligible so that the mass balance for a single reaction in of conversion x can be written as:

If the objective is to maximize the yield, Le., outlet conversion, over the catalyst life tf, which is fixed, the performance index P becomes:

T. Equation 13.35 allows one to write Eq.

! I

the optimal policy (Crowe and Therien 1974; Gruyaert a,nd Crowe 1976) obtained from the maximum principle is again to maintain a constant exit conversion when

the inlet temperature is manipulated without any constraint, provided that the temperature dependence in Eq. 13.32 is separable such that:

Even for this simplest case of deactivation, the condition is not met because the rate constant kt,, for instance, is not separable as apparent from Eq. 13.42, not Nb, and y. to mention the separability of the temperature dependence of (b),, This does not mean that realistic problems cannot be solved. They can still be solved by the maximum principle, for instance, although the results are not as elegant as the policy of constant conversion. The results discussed earlier for concentration dependent deactivation kinetics were based on Pontryagin's maximum principle (e.g., Koppel 1972) as extended to pseudo-steady state systems (Sirazetdinov and Degtyarev 1967). Here, a weak but more general maximum principle is considered due to Ogunye and Ray (1971). Let the concentrations (conversions) and temperatures be denoted by state variables x and the control variables such as temperatures (which are not state variables) by U and the catalytic activities by A. In general, under the pseudo-steady state assumption one has:

When unconstrained temperature control is distributed over the entire reactor, the optimal policy is to maintain a constant exit conversion if the activation energy for the deactivation reaction is the same as that of the main reaction; if not, it is constant conversion over the entire reactor. It should be pointed out here that the optimal policy based on the pseudo-steady state assumption regarding Eqs. 13.32 and 13.38 is valid (Gruyaert and Crowe 1976) only when the change of temperature is felt at the outlet before another change is made at the inlet. The optimal policies obtained so far are very powerful. However, the conditions under which the policies are valid are restrictive and the deactivation kinetics are somewhat unrealistic. For instance, the temperature dependence in rate expressions cannot be completely factored out for heterogeneous reactions, which are inherently affected by transport resistances. Consider independent deactivation. For an adiabatic reactor (Chapter 10):

dAi -

do -&.(x,A,U);

Aj(zPO)=wr(z)

J

,

,

.

,

(13.47)

3

4

If the deactivation is due to uniform poisoning, the global rates for the single

i

Suppose that the problem is to maximize the following performance index:

main and deactivation reactions are: \

i

i 4 2 where (b), is the value of k, evaluated at the pellet surface temperature K. The rate at which the fraction of catalyst deactivated y changes with time is given by:

If the reaction is structure-insensitive, the activity A is simply:

It is immediately seen that the condition of separable temperature dependence, which is necessary for the optimal policy of constant conversion, cannot be satisfied.

1

1

subject to Eqs. 13.46 and 13.47 and the following upper and lower limits on the distributed (u and w) and boundary (v) control variables: v*=Sv
pj

(13.49) for Eqs. 13.46

4

it is necessary for optimal controls to satisfy the conditions in Table 13.1. The adt variables are then defined by:

462

HETEROGENEOUS REACTORDESIGN

Table 13.1 Necessary Conditions for Optimal Controls (Ogunye and Ray 1971) Category

UCv. 6 )

the conversion with the feed of only B1 can be described by:

Necessary conditions

aH = 0 for all

Uk and H be a maximum with respect to Uk at a constraint on ZI*.

auk

1:

HdB be a maximum with respect to Uk at a constraint on Uk,

aw dy = 0 and

I:

where U 1is the dimensionless temperature R,T/El, and & are constants involving reactor length and preexponential factors, p is the ratio of activation energy for reaction 1 to that for the deactivation reaction, and p1 is the ratio of activation energy for reaction 2 to that for reaction 1. Suppose that the objective is to maximize the production of B2,i.e.:

Hdy be a maximum at a constraint on Uk.

2 =I1

dy

=

HI

O

Gdy

P

+ hi(0,B)= 0 and

+ hr(0,6)vj(B)be a maximum with respect to

constraint on

=!:I

(fi -fz)dyd@

(13.56)

subject to the following constraint: vj

at a

vj.

constant v

1:

The Hamiltonian is then given by: H1d0 be a maximum with respect to vi at a constraint on vi

=I'

H2

O

constant W

and the necessary condition, which is for U(8) in Table 13.1, is:

Gd0 + pi(y,0 ) w i b ) be a maximum with respect to a constraint on w,.

Wj

at

11%

aH dy = 0 and H be a maximum at a constraint

on U1

(13.59)

dy = 0 and

1;

where a H / a Ul is given by:

Hzdy be a maximum with respect to wj at a constraint on wj.

Take as a simple example an isothermal reactor in which consecutive reactions (B1 B , 1, Ba) take place. Suppose that the isothermal temperature is the control variable. If the catalytic activity A is the same for both reactions, which is given by:

Since the necessary condition does not yield any analytic solution, and thus, requires a numerical search for a maximum starting with an initial guess for U1(0), the condition for the numerical search can be written as:

of catalyst deactivation can be estimated from process measurements and, if so, how control can be realized with the measurements. A related question

where E is a small, positive scaling number for convergence to the optimal U1(8). The last condition simply means that at every ascent to a local maximum, 8U1(0) should move in the direction of the integral to reach the local maximum. When a constraint is reached, U1(8) should follow U,". The numerical search involves solving the equations for A1, X I , and xz with an initial guess of U1(8)and then the equations for the adt variables, which are in turn used to calculate a H / aUl (Eq. 13.60) for the increment aU1(8), obtaining the improved Ul(8) for the next iteration, and repeating the same procedures until 6 U1 or SP/P reaches a small number. The adt equations for the example can be written from Eqs. 13.51 and 13.52. For instance, the adt variable hl is defined by:

+ Ax&(-

31

1 4- hz) exp - - ;

(

hl(l ,8) = 0

(1 3.63)

The weak maximum principle can certainly be used to maximize the performance index of Eq. 13.33, for instance, for the independent deactivation problem given by Eqs. 13.40 through 13.45 with the addition of a heat balance at the pellet-bulk fluid interface for the pellet surface temperature [(&),I. Nevertheless, the optimal inlet temperature will be a function of time. Thus, the inlet temperature is to be manipulated in a prescribed manner in time without any regard to what happens to the reactor. This open-loop control is rarely practiced in actual applications because of the uncertainties regarding the model, measurements, and disturbances. Rather, closed-loop control using from the process is usually practiced. This fact should not discourage one from using the maximum principle for optimization, for it will at least indicate what the best possible performance is in a relative sense. At the same time it can yield in some simple cases a very powerful optimal policy such as the constant conversion policy, which can be implemented by a control scheme. It is evident from the foregoing discussion that optimal policies ace difficult to obtain for realistic heterogeneous reactors undergoing catalyst deactivation. Further, it is clear that some information needs to be used for the implementation of a control scheme. While it has been assumed so far that the deactivation kinetics are known, detailed knowledge of deactivation is rarely available. This problem is particularly acute for structure-sensitive reactions in which the activity of the catalyst changes irregularly with the extent of deactivation. A number of questions can be raised, therefore, regarding optimal control of a heterogeneous reactor undergoing catalyst deactivation. A central question is whether a measure

that was alluded to earlier is the interrelationship between reactor design and control and its use to achieve the best possible reactor performance. These questions will be considered one at a time. Consider a single reaction in an adiabatic reactor. Assume that the global rate for fresh catalyst, which is Eq. 13.9, is known and let the "local activity factor" h be defined by:

where R G and ~ Rc, are the global rates for fresh and deactivated catalyst, respectively. Now the intrinsic rate for Eq. 13.9 is expressed as:

where K denotes equilibrium constants that are evaluated at the bulk-fluid temperature Tb when used in Eq. 13.9. Further, Tb can be expressed in of the bulk-fluid concentration of the main reactant Cb and the inlet temperature and concentration for the adiabatic reactor under consideration. Therefore, Rcf given by Eq. 13.9 is a function solely of Cb.Thus, for the deactivated catalyst:

If this equation is integrated from the reactor inlet to the outlet, one has:

where Gout and Ct, are the reactor outlet and inlet concentrations, at any given time. By definition, H[O] = 1 for fresh catalyst since h = 1 throughout the reactor at t = 0.The overall activity factor H is an overall measure of catalyst deactivation for the reactor at any given time and can be calculated directly from Eq. 13.67 by measuring the outlet concentration. Further, the overall activity factor can be calculated whether the deactivation is independent, dependent, or for that matter, of any kind since by definition h is simply the ratio of Rcf to Rod regardless of what form RG, may take. The overall activity factor can be used in a straightforward manner for a piecewise control. Suppose that the control policy is to maintain a constant outlet conversion. Suppose further that the inlet temperature is manipulated intermittently, as in the usual operation of an adiabatic reactor, so as to maintain the constant outlet conversion. If one lets the subscript c denote the current quantities

and n the new ones resulting from a change in the inlet temperature, Eq. 13.67 can be written twice to give:

cd,

where is the adjusted Cd when a step change in the inlet temperature is made at the ith step, P is a proportionality constant with a value between zero and unity, and C, is the measured outlet concentration. For a given bandwidth for the allowed decrease in the outlet concentration that will trigger the temperature adjustment, the piecewise control algorithm can be stated as follows:

where Cd is the desired outlet concentration corresponding to the desired conversion, which will be attained by changing the current inlet temperature (Ti,), to the new inlet temperature (Z,),. As shall soon be seen, the local activity factors h, and hn depend mainly on the fraction of the catalyst deactivated and slightly on the temperature difference between the bulk fluid and the pellet surface. If the change in the temperature difference due to a change in Zn is neglected for the time being, one can set:

for the purpose of calculating the new inlet temperature that yields the desired outlet concentration. The justification here is that the fraction of the catalyst deactivated changes negligibly while the inlet temperature is adjusted from the current value to a new one. Then, Eqs. 13.68 through 13.70 can be used to obtain:

where Hc is the known current overall activity factor calculable from Eq. 13.68 just prior to the change in the inlet temperature based on the measured current outlet concentration Gout and inlet temperature (Tin),. muation 13.71 can be solved for (En), by a trial and error method since the left hand side of the equation can be evaluated for an assumed value of (q,),.The value of (En), that satisfies Eq. 13.71 will yield the desired conversion, and therefore, the inlet temperature can be changed to (En),to maintain the conversion at the desired level. Since the control is piecewise, it will maintain the conversion at the desired level only for a short period of time. The conversion thereafter will gradually decrease with time until the inlet temperature is raised again. A bandwidth for the allowed decrease in conversion can be used to trigger the adjustment of the inlet temperature. The control algorithm of Eqs. 13.68 and 13.71 is based on the assumption that the temperature difference between the bulk fluid and the pellet surface is negligible. When the temperature difference is not negligible, the discrepancy between h, and h, caused by the difference can be ed for by adjusting Cd as follows:

where (H,)i -1 is calculated from:

with the measured outlet concentration and inlet temperature, and Giis given by Eq. 13.72. Each time the outlet concentration decreases to the allowed bandwidth, the new inlet temperature is calculated for the manipulation of the inlet temperature. The reactor behavior (Hong and Lee 1983) that results when the adiabatic reactor detailed in Table 10.5 is controlled according to the algorithm of Eqs. 13.72 through 13.74 is shown in Figure 13.3. The reactor response for T = 20s, which was simulated according to the procedures in Chapter 10 for the reaction affected by uniform independent deactivation, was used as the process response for the manipulation of the inlet temperature. The bandwidth was set so that temperature adjustments were made when percent conversion fell to 60% from the initial conversion of 70%. The final time was set as the time at which the final bandwidth becomes one-tenth of the initial bandwidth. As shown in Figure 13.3, the outlet temperature, which is the maximum reactor temperature in this case, is well below the maximum temperature allowed. Due to the deactivation, both the conversion and the outlet temperature decrease with time and the conversion eventually reaches the lower bound given by the bandwidth, triggering an adjustment of the inlet temperature. The new inlet temperature (Tin)l calculated from the algorithm is seen to bring the conversion back to the desired level. When the outlet temperature reaches the allowed maximum, which occurs at around t = 2 x lo%, the control policy is simply to increase Tin such that Tout = T ,, since the conversion can no longer be brought back to the desired level due to the temperature constraint. Consequently, the bandwidth decreases with time and eventually becomes less than one-tenth of the original bandwidth, resulting in the reactor shutdown for catalyst regeneration according to the constraint imposed. In practice, however, the conversion may be allowed to decrease below 60% when the maximum temperature is reached. In the example, the time average conversion is at the middle of the bandwidth, which is 65%. The control algorithm of Eqs. 13.72 through 13.74 does not require any

Max ~ = l l ' ~ x d r ( r p

Tin( t)J

T

subject to mass and heat balances for the reactor and T,, S T ,;(),i with a fixed 9. This two-parameter optimization problem can be solved one at a time by first searching for the maximum for a given T and then finding the value of 7 that yields the maximum of F. The results obtained for a constant conversion policy for the optimal Ti,(t) are shown in Figure 13.4. The algorithm of Eqs. 13.72 through 13.74 was used for the policy of constant conversion, which depends on T and initial Ti,. Under the constant conversion policy, therefore, the problem for a given T is that of finding an initial conversion (or initial inlet temperature) that yields a maximum. As evident from Figure 13.4, much better reactor performance is attained when the reactor is optimally sized, T of 22s yielding the rnaximum. Any other choice of T would have resulted in poorer perfomance even if the reactor were operated optimally. This example clearly demonstrates the need for optimizing both reactor size and operating conditions. So far optimization problems have been considered in which the regeneration

Figure 13.3 Reactor behavior when subjected to piecewise control for constant

conversion. knowledge of catalyst deactivation and is applicable to any type of deactivation. It can also be extended (Hong and Lee 1983) to multiple reactions and nonadiabatic reactors, although the application to nonadiabatic reactors requires measurements of coolant temperature along the reactor. It should be noted here that reactor transients can exhibit "wrong way7*behavior (Crider and Frost 1966) in which the transients initially move away from the "right" steady state, but eventually reach the right steady state in a few residence times. Since the control algorithm is piecewise and intermittently used for the temperature manipulation in intervals of hours rather than seconds due to the time scale of deactivation, which is much larger than the residence time under the assumption of pseudo-steady state, the right steady state v a l u s would be used in the control alogrithm by the time the temperature change is made. Further, wrong way behavior is caused by a decrease in temperature rather than an increase (Mehta et al. 1981). As indicated earlier, the usual optimization practice is to design first without due consideration of catalyst deactivation and then optimize the reactor pedormance. It is intuitively clear that much better reactor performance can be attained if the reactor size and operating conditions are both optimized during the design phase. Suppose that one wishes to maximize the yield of a single reaction per unit reactor volume by properly specifying the reactor size T and inlet temperature Tin(t). Then the optimization problem is:

Figure 13.4 Maximum values of the performance index of Eq. 13.75 as a function of reactor size.

time is negligible compared to the operation time as in, for example, reforming and hydrocracking reactions. If the regeneration and operation times are of similar magnitude, there would be a unique operation/regeneration cycle optimal for a single fixed-bed or a train of fixed-beds. This optimization problem was considered by Weekrnan (1968) for the fixed-bed catalytic cracking of gas oil. The simplified model for catalytic cracking discussed in Chapters 8 and 11 was used:

in which F is gas oil, G is gasoline, and C is coke. As detailed in Chapter 8, the rate affected by the coking is expressed as:

where A is the 'fractional activity remaining after coking, and yl is the weight fraction of gas oil. The fractional activity is correlated as follows:

where Fo is the feed rate, and tt is the total time for N cycles of reaction of length to and regeneration of length t,. Thus, the reactor efficiency becomes:

where Eq. 13.78 has been used in 13.81. Two different cases can be considered for the optimal operation: regeneration independent of operation and regeneration dependent on operation. For the first case, the reactor efficiency is simply a function solely of to. An inspection of the expression of Eq. 13.82 should reveal that there is u maximum for a given value of t,. The dependence of E, on to is shown in Figure 13.5 for various regeneration times. It is seen that the maxima are not materially affected by the regeneration time, although the optimal cycle length increases with increasing regeneration time as would be expected. When regeneration and operation times are dependent on each other, the

where a is a decay constant dependent on the catalyst and feedstock, and t is time on stream. For an isothermal plug-flow reactor in which the rates are affected only by coking, Eqs. 13.76 and 13.77 can be used in the reactor conservation equation to obtain the following time-averaged conversion (Weekman 1968) of gas oil:

where

Here, S is the liquid hourly space velocity (vol/vol time), po the initial vapor density, p~ the liquid charge density at room temperature, and to the duration of reactor operation. The performance criterion used was a measure of the approach to operation at 100% conversion without coking, termed the reactor efficiency and given by:

NFot,F &=-Fotl

- total actual product total ideal product

(13.81)

Operation Cycle Time, Hrs.

Figure 13.5 Reactor efficiency for regeneration independent of operation. (Weekman 1968; reprinted with permission from Industrial and Engineering Chemistry, Process Design and Development. Copyright by American Chemical Society.)

manner in which the regeneration is carried out has to be determined for the interrelationship between regeneration and operation. As shown by Voorhies (1945), coke formation in catalytic cracking can be well correlated by a negative order, i-e., self-inhibiting reaction:

r

I

I

I

I

I

-

k, = 20 hr"

-

b = 0.5

--..

.

m = l

I

pace Velocity

which, upon integration with C, = 0 at t = 0, yields the Voorhies relationship: Cc = at:

(13.84)

where

and

0

Further, if the coke burning is diffusion-free, one has:

.1

.2

.3

.4

.5

.6

Operation Cycle Time, Hrs,

Figure 13.6 Reactor efficiency with regeneration dependent on operation. (Weekrnan 1968; reprinted with permission from Industrial and Engineering Chemistry,Process Design and Development. Copyright by American Chemical Society.)

where m is unity for monolayer coke and less than unity for multilayer coke as detailed in Chapter 5. Solving Eq. 13.87 for constant Co, with C, = at: at tr = 0 and with Cc = 0 as tr + yields:

It can be observed from Eq. 13.82 that if E, + 0 as to -4 0,an optimum E, will be found as to is increased since as to + m, E, + 0.Substituting Eq. 13.88 into 13.82 and taking the limit as to -+ 0 by l'H8pital's rule gives:

For the reactor efficiency to approach zero as to --+ 0, the exponent on to must be negative. Thus, if n > 0 and m 2 0, an optimum will exist in & as to is increased. This result shows that for zero or any positive order coke burning an optimum operation/regeneration cycle will exist, provided the Voorhies parameter b is between zero and unity. Shown in Figure 13.6 is such a relationship, illustrated

for m = 1, b = 0.5, and a final coke content at the end of regeneration of 0.03 wt% coke on catalyst. Considerable improvement is seen to result from the optimization. Here again, the reactor size (inverse of space velocity) is seen to play an important role. Summary

Equilibrium constraints and catalyst deactivation lead naturally to optimization problems. Simple maximization, dynamic programming, maximum principles, and other techniques can be used to solve the optimization problems. Optimization is usually carried out with respect to operating conditions. As pointed out repeatedly in this chapter, it should also be done with respect to reactor size. Perhaps the most powerful optimal policy for a reactor affected by deactivation is that of a constant conversion. However, it is not usually applicable to realistic systems and more work is needed for simple and yet general policies applicable to realistic systems. Means for estimating the extent of deactivation from process measurements and their use for optimization is another area that needs further work since detailed knowledge of deactivation is usually unavailable. An extensive review of reactor

optimization in the presence of catalyst deactivation is given by Kovarik and Butt (1 982). NOTATION

constant given by Eq. 13.85 fraction of catalyst activity remaining after deactivation cross-sectional area quantity defined in Eq. 13.79 dA/dt vector of A constant given by Eq. 13.86 concentration of main reactant bulk-fluid concentration of main reactant Cb in nth bed coke concentration; pellet center concentration given in Eq. 13.7 desired concentration of main reactant reactor outlet concentration concentration of main reactant to the nth stage feed concentration of main reactant concentration of poisoning species C d at time zero modified Cd at the it" control action, given by Eq. 13.72 inlet concentration of main reactant outlet concentration of main reactant effective diffusivity activation energy for key reaction, forward reaction activation energy for backward reaction activation energy for poisoning reaction reactor efficiency defined by Eq. 13.8 1 concentration dependence of the rate of reaction function given in Q. 13.46 performance index defined in Eq. 13.75 feed rate concentration dependence of rate of the backward reaction concentration dependence of rate of the forward reaction function given in Q. 13.47 functions given in Eqs. 13.53, 13.38, and 13.39, respectively function given in Eq. 13.48 local activity factor defined by Eq. 13.64, film heat transfer coefficient overall activity factor defined by Eq. 13.67; Hamiltonian defined by Eq. 13.50 current H value given by Eq. 13.68 value of I& calculated at the ith control, given by Eq. 13.73 new H value given by Eq. 13.69 heat of reaction per mole of main reactant

I

I 11

i d

4

I

1 i

1:

5

If i

I ;a

tr to

tr

quantity given in Eq. 13.42 rate constant of main reaction k elevated at Tb rate constant for coking reaction preexponential factor of k rate constant for poisoning reaction preexponential factor for k, rate constant for coke regeneration reaction k, evaluated at pellet surf- temperature rate constant in Eq. 13.76 equilibrium constant; k/k -1 preexponential factor of K; constant given in Eq. 13.79 vector of K pellet characteristic length exponent in Eq. 13.87 negative order of coking reaction total number of beds or stages; number of operatio_n/regeneration cycles bulk-fluid concentration of poisoning reaction Ea /Ep performance index given by Eq. 13.33; performance index given by Eq. 13.48 performance index for nth stage given by Eq. 13.18 poison capacity of catalyst in moles poisoning species per pellet volume; Eb - Ea measure of net profit for nth stage given by Eq. 13.12 intrinsic rate of main reaction rate given by Eq. 13.76 gas constant global rate of deactivation global rate of main reaction global rate of deactivated catalyst global rate of fresh catalyst Rc for nth stage space velocity, vol/(vol time) time final time at which catalyst is regenerated; catalyst life duration of operation in operation/regeneration cycle length of time for catalyst regeneration

+ tr

!j

4

to

S

T

temperature bulk-fluid temperature Tb in n t h bed outlet temperature for nth stage inlet temperature to nth stage after cooling reactor inlet temperature

t

Tb

(Ta)n

f

Tn T R -1

Tin $\

current value of En new value of Z, R,T/El in Eq. 13.53 vector of control variables maximum value of control variable minimum value of control variable superficial fluid velocity inlet value of state variable xi maximum value of vi minimum value of vi reactor volume initial distribution of activity maximum value of wj minimum value of wj conversion ith state variable vector of state variables average conversion

n

new; nth stage Nth stage pellet surface reactor inlet reactor outlet

N s in out PROBLEMS

13.1. Consider the following second-order reversible reaction:

where C is the concentration of the main reactant A. Show that combining the pellet conservation equations for A and B yields:

ax/aA

dimensionless reactor length weight fraction of gas oil total reactor length Creek Letters

a

P Pl, P 2

Y E EB

8

h hr Pj

P Po

PL V

7

4%

product price per mole; decay constant in Eq. 13.77 (-A H)/p G; proportionality constant in Eq. 13.72 constants in Eqs. 13.54 and 13.55 costs per unit volume of reactor; fraction of catalyst deactivated small nonnegative constant in Eq. 13.62 bed voidage dimensionless time v/(l - E B ) ; ato adt variable in Eq. 13.51 adt variable in Eq. 13.52 constant in Eq. 13.53; density initial vapor density liquid charge density at room temperature ?'/a (1 - E B ) ~ / V generalized Thiele modulus given by Eq. 13.8

if one assumes equal diffusivities. Show further for a plug-flow reactor that:

if one assumes no B in the feed, where Cin is the inlet concentration of A, and the subscript b denotes the bulk-fluid concentration. Show that f(C) in Eq. 13.4 for this problem is:

Obtain an expression that can be solved for the optimal Tinwhich maximizes Gout using Eq. 13.1, given Cf,and T. Assume that the pellet center concentration C, is negligible. 13.2. Construct a diagram corresponding to Figure 13.2 for a two-bed adiabatic reactor and specify optimal inlet conditions to each bed using dynamic programming. For the SOzoxidation under consideration, assume that the global rate is the same as the intrinsic rate given (Collina et al. 1971) by:

Subscripts

b c

bulk fluid current

where r, = mol S02/g cat hr k = exp(12.160 - 5473/ T )

REFERENCES

Use the following conditions of Lee and Aris (1963): Feed: 7.8 mole % SOz, 10.8 mole % 02,8 1.4 mole % N2a t 1 atm and 37OC Mean specific heat = 0.22 1 cal/g°C, (-AH) = 2 1.4 kcal/rnole, )I = 0.00068 mol/g cat h r 13.3. Obtain the temperature trajectory, T(t), which is optimal with respect to the performance criterion of Eq. 13.33. Assume for an isothermal plug-flow reactor that the conservation equation is given by:

I

I

and the activity A is given by:

Show first that under the condition of constant conversion, the activity is given by:

where p is the ratio of activation energy for k to that for constant. Assume that the rate constants are given by:

b,and fi

is a

I 1

Find an expression for the optimal constant conversion. 13.4. For the example problem in the text (Eqs. 13.53 through 13.57), write the complete equations necessary for the optimization. Based on these equations, give detailed procedures readily usable for computer programming. 13.5. Extend the method of estimating the extent of deactivation to simple parallel and consecutive reactions. Specifically, derive a n expression for Hc (Eq. 13.68) when the local activity factor h is the same for both reactions. Repeat the case in which h is different for different reaction paths. 13.6. Construct a diagram similar to Figure 13.6 using the following conditions:

Assume that the final coke weight after burnoff is 0.03% and that the initial coke content is 5%.

Aris, R., Introduction to the Analysis of Chemical Reactors, Prentice-Hall, Englewood Cliffs, N.J. (1965). Bellman, R., Dynamic Programming, Princeton University Press, Princeton, N.J. (1957). Beveridge, G.S.B. and R.S. Schechter, Optimization Theory and Practice. McGraw-Hill, New York (1970), Bliss, G.A., Lectures on the Calculus of Variations, University of Chicago Press, Chicago (1961). Chou, A., W.H. Ray and R. Aris, Trans. Inst. Chem. Eng., 45, 53 (1967): Collina, A.,D.Corbetta and A* Cappelli, Eur. Symp., Use of Computers in the Design of Chemical Plants, Firenze (197I), Crider, J.E. and A. S. Frost, AIChE J., 12, 514 (1966). Crowe, C.M. and N. Therien, Can. J. of Chem. Eng., 52, 822 (1974). Denbigh, K.G.and J.C,R. Turner, Chemical Reactor Theory, 2nd ed., Cambridge University Press (1971). Fournier, C.D. and F.R.Groves, Chern. Eng., 77, 157 (197O), Gruyaert, F. and C.M. Crowe, AIChE J., 22, 985 (1976). Hong, J.C. and H.H. Lee, AIChE J. (in press), Koppel, L.B., Introduction to Control Theory with Application to Process Control. PrenticeHall, Englewood Cliffs, N.J. (1972). Kovarik, F.S. and J.B, Butt, Cat. Rev., 24, 441 (1982). Lee, K.Y. and R. Aris, Ind. Eng. Chern. Proc. Des. Dev., 2, 300 (1963). Mehta, P.S.,W.N. Sams and D.Luss, AIChE J., 27, 234 (1981). Millman, M.C. and S. Katz, Ind. Eng. Chem. Proc. Des. Dev., 6, 447 (1967). Ogunye, A.F.and W.H. Ray, AIChE J., 17, 43 (1971). Ray, W.H.and J. Szekely, Process Optimization, Wiley, New York (1973). Roberts, S.M.,Dynamic Programming in Chemical Engineering and Process Control, Academic Press, New York (1964). Sirazetdinov, T.K. and G.L. Degtyarev, Automation ~ e m o t c ~ o n t r o28, l , 1642 (1967). Szepe, S. and D. Levenspiel, Chern. Eng. Sic., 23. 881 (1968). Voorhies, A.,Ind. Eng. Chem., 37, 3 18 (1945). Weekrnan, V.W., Jr., Ind. Eng. Chem. Proc. Des. Dev., 7, 90, 253 (1968).

PART

IV Transport Properties

CHAPTER

14 --

Transport Properties and Experimental Methods

Throughout this book various transport properties and transfer coefficients have been used. These include effective diffusivity and thermal conductivity for mass and heat transport in catalyst pellets, film transfer coefficients for mass and heat transfer across the pellet-bulk fluid interface, transport propertis for the degree of dispersion of mass and heat in the reactor, and heat transfer coefficients for heat exchange between the cooling medium and the reactor. In this chapter these transport properties and transfer coefficients are treated in detail, including experimental methods for obtaining these properties. 14-1

PELLET TRANSPORT PROPERTIES

This section considers the effective diffusivity and thermal conductivity for the transport of mass and heat in the pellets. In Chapter 4, the effective diffusivity for a key species was defined by the following flux relationship:

where the subscript 1 in Eq. 4.7 is now eliminated for the key species. An explicit expression for D, is impossible to obtain from the general flux expressions of Eq. 4.13 for a rnulticomponent system even when the pellet is assumed to be isobaric and surface diffusion is neglected, as has been assumed in all treatments so far. It is often satisfactory, however, to calculate a theoretical value of the diffusivity based on a pseudo-binary system in which the key species is one species and the test of the system is another. This theoretical estimate of the diffusivity may be used for comparison with the diffusivity obtained experimentally using the definition of Eq. 14.1. For a binary system, the effective diffusivity D, can be expressed as:

which is Eq. 4.16 for the key species. Here, and D , I are effective molecular and Knudsen diffusivities, x is the mole fraction of the key species, and m is the ratio of the molecular weight of the key species to that of the other species. The effective diffusivities, which for the obstruction of movement of gas molecules in a porous medium, are related to the obstruction-free molecular and Knudsen diffusivities by:

where ~ ( E / Kis) a correction factor for the effective values of the diffusivities in the porous medium. Here, E and K are the porosity and tortuosity of the medium. ,2. the ChapmanFor an accurate calculation of the molecular diffusivity Dm, Enskog formula (Bird et a]. 1960) can be used:

{ [ T.(&+ &)]

112

Dm,12 = 0.0018583

[Pv:.

a12]

(cm2/sec)

(14.5)

where the total pressure P is in atm, T is in OK, and alzis in A and ILlz are the parameters calculable from Lennard-Jones parameters: all = (al a2)/2 and e12 = ( ~ l e ~ ) These " ~ . parameters are given in Tables 14.1 and 14.2. When the pore radius is very small, collisions will occur primarily between gas molecules and the pore wall, rather than between molecules. For such Knudsen diffusion in a long capillary, the diffusivity obtainable from the simple kinetic theory of gases is:

+

c - m

O

W

O

~

O

Fm mq Tn l en gmFmEm?bq mF .e- ;vg*

where i is the pore radius (capillary radius), and 5 is the mean molecular velocity given by:

where M is the molecular weight, and R, is the gas constant. The Knudsen diffusivity, therefore, is given by:

For Knudsen diffusion in a porous medium, an average pore radius may be used for i (in cm) in Eq. 14.7. It is instructive in this regard to compare the relative magnitude of molecular and Knudsen diffusivities with the help of an approximate expression for the former based on the simple kinetic theory, given by:

~

~

~

I Table 14.2 Functions for Prediction of Transport Properties of Gases at Low Densities (after Bird, Stewart, and Lightfoot, Transport Phenomena, @ 1960, Wiley. Reprinted by permission of John Wiley & Sons, Inc.)

TRANSPORT PROPERTIES AND EXPERIMENTAL METHODS487

where h is the mean free path. This relationship is for rigid spheres of equal mass and radius for the two species. Taking the ratio of the two diffusivities, one gets from Eqs. 14.6 and 14.8:

n,

n,

= nk

(For

k TIE or

(For Viscosity and Thermal

flu. 12 (For Mass

k ?'/€AH

Conductivity)

Diflusivity)

2.785 2.628 2.492

-

= n,

kT/E or k T/rAtl

fla

12

and Thermal

(For Mass

Conductivity)

Diflusivity)

0.9996 0.9878 0,9770 0.9672 0.9576

0.30 0.35 0.40 0.45

2.368

2.662 2.476 2.318 2.184

2.90

1.093 1.081 1.069 1 .OS8 1.048

0.50 0.55 0.60 0.65 0.70

2.257 2.156 2.065 1.982 1.908

2.066 1.966 1.877 1.798 1.729

3.00 3.10 3.20 3.30 3.40

1.039 1.030 1.022 1.014 1.007

0.9490 0.9406 0.9328 0.9256 0.9186

0.75 0.80 0.85 0.90 0.95

1.841 1.780 1.725 1.675 1.629

1.667 1.612 1.562 1.517 1.476

3.50 3.60 3.70 3.80 3.90

0.9999 0.9932 0.9870 0.981 1 0.9755

0.9120 0.9058 0.8998 0.8942 0.8888

1 .OO

1.587 1.549 1.514 1.482 1.452

1.439 1.406 1.375 1.346 1.320

4.00 4.10 4.20 4.30 4.40

0.9700 0.9649 0.9600 0.9553 0.9507

0.8836 0.8788 0.8740 0.8694 0.8652

1.25 1.30 1.35 1.40 1.45

!.424

1.296 1.273 1.253 1.233 1.215

4.50 4.60 4.70 4.80 4.90

0,9464 0.9422 0.9382 0.9343 0.9305

0.8610 0.8568 0.8530 0.8492 0.8456

1.50 1.55 1.60 1.65 1.70

1.314

1.198 1.182 1.167 1.153 1.140

5.0

1.296 1.279 1.264 1.248

6.0 7.0 8.0 9.0

0.9269 0.8963 0.8727 0.8538 0.8379

0.8422 0.8124 0.7896 0.7712 0.7556

1.75 1.80 1.85 1.90 1.95

1.234 1.221 1.209 1.197 1.186

1.128 1.116 1.105 1.094 1.084

10.0 20.0 30.0 40.0 50.0

0.8242 0.7432 0.7005 0.6718 0.6504

0.7424 0.6640 0.6232 0.5960 0.5756

2.00 2.10 2.20 2.30

1.175 1.156 1.138 1.122 1.107

1.075 1.056 1.041 1.026 1.012

60.0 70.0 80.0 90.0 100.0

0.6335 0.6194 0.6076 0.5973 0.5882

0.5596 0.5464 0.5352 0.5256 0.5170

1.05 1.10 1.15 1.20

1.399 1.375 1.353 1.333

2.40

-

--

2.50 2.60 2.70 2.80

Viscosity

-"

It is seen from Eq. 14.9 that the magnitude of the average pore radius relative to the mean free path determines the predominant mechanism of diffusion. Since h is of the order of 1000 A for gases at atmospheric pressure, diffusion in a porous medium will be predominantly by the Knudsen mechanism if the average pore radius F is much less than 1000 A. Theoretical calculations for the effective molecular and Knudsen diffusivities, and therefore, of the effective diffusivity D, can be made from Eqs. 14.5 and 14.7 provided that the factor S, representing the pore characteristics is known. Various models have been developed for the calculation off,. Two such models are considered here. The first is that due to Feng and Stewart (1973). The simplest form o f f , ( E / K ) that they obtained is:

Assumptions made are isotropic pores with no dead-ended branches, independence of e and K with pore size, and lumping of an integral over the entire pore structure into a single term. The second model to consider is the random-pore model due to Wakao and Smith (1962), shown in Figure 14.1. This model was developed for a bidisperse porous medium, but is equally applicable to a monodisperse medium. The nature of the interconnections of macro and micro void regions for a bidisperse pellet shown in the figure is the essence of the model. Transport in the pellet is assumed to occur by a combination of diffusion through the macro regions (of , micro regions (of void fraction E,), and a series contribution void fraction E ~ )the involving both regions but dominated by the diffusion through the micro regions. A probabilistic argument is used to assign the probability of access of gaseous molecules to these three regions equal to respective available area, which are E& for the macro region, (1 - E ~ for) the ~ micro region, and 2 ~ ~ -( clM ) for the series region. Since the micropore diffusivity D, is based on the micro void, the ratio of eV(1 - E ~is used ) ~ for the (micro void/particle) area ratio, yielding:

where DM and D , are the macropore and micropore diffusivities, respectively. For a monodisperse pellet, the above relationship reduces to either E L D ~or L ~

D

~

TRANSPORTPROPERTIES AND EXPERIMENTAL METHODS489 Direction of diffusion

M

I

-

Macropores ,Mir~oporous

1

Figure 14.2 Constant-pressure apparatus for measuring diffusion rates in porous catalysts. A: mixing chambers; B: detector for composition of Nz in Hz stream; C: detector for composition of Hz in N2 stream; D: pressure equalization gauge; E: catalyst pellet. (After J.M. Smith, Chemical Engineering Kinetics, @ 1980; with permission of McGraw-Hill Book Co., NY.)

Figure 14.1 Random-pare model. (Wakao and Smith 1962; reprinted with permission from C'hemicul Engineering Science. Copyright by Pzrgamon Press, Inc.)

Therefore, the factor & ( E / K ) for the random pore model is:

Theoretical values of effective diffusivities can now be calculated with Eqs. 14.5 and 14.7 along with the factor f, given by either Eq. 14.10 or Eq. 14.12. It must be pointed out, however, that the diffusivities thus calculated do not often compare satisfactorily with experimental values. Often the tortuosity is adjusted in such cases to force the theoretical values to conform to the experimental values. Detailed treatment of the tortuosity factor can be found in the book by Sattefield (1970). Experimental determination of the effective diffusivity D, is usually carried out in an apparatus, shown in Figure 14.2, known as a Wicke and Kallenbach chamber (1941). The chamber is operated at steady state and nearly constant pressure as is usually the case in reactors. Rewriting the definition of Eq. 14,l for the flux, one has:

which, upon integration, yields:

where y is the mole fraction of the key species, and Az is the length of the pellet. If the flow rates and concentrations are measured, N can be calculated for the

Ilk'

key species, say H,,(see Figure 14.2), in which case the subscript 2 is for the bottom of the chamber and 1 for the top. Then this flux, the measured concentrations and pellet length are substituted in Eq. 14.13 to obtain an experimental value of the effective diffusivity. It is noted that the effective diffusivities determined from the Hz-N2 pair, for instance, can be used to calculate the diffusivitites for any other pairs for the same pellet (see Problem 14.1). The dynamic method (Dogu and Smith 1975) has also been used for the measurement of D, using an apparatus similar to the Wicke-Kallenbach chamber. Their method involves introducing a pulse of one of the gases and analyzing the measured response based on a moment analysis. The dead-ended pores can affect the diffusion in this method. Experimental results show that the dynamic method yields similar (Baiker et al. 1982) or smaller values of D, (McGreavy and Siddiqui 1980) when compared with the results of the static method. The latter discrepancy was attributed to pore size distribution. A flow-through diffusion cell has been proposed by Frost (1981). In contrast to the Wicke-Kallenbach cell, it can be operated at high temperature and pressure and does not require special sample preparation. The effective diffusivity can also be determined from reaction conditions provided that the intrinsic kinetics are known. In the region of strong diffusion effects, the product of the effectiveness factor and the generalized Thiele modulus is unity, i.e., q+c = 1, where +(; is given by:

Here, L is the characteristic length of the pellet, and r, is the intrinsic rate of the reaction. Since 7 can be determined from the ratio of Rc/rc where RG is the observed rate, one has from Eq. 14.14:

TRANSPORTPROPERTIES AND EXPERIMENTAL METHODS491

It is seen that given the intrinsic kinetics, the actual rate of the reaction in the diffusion-limited region can be used to determine the effective diffusivity. The theory of diffusion in liquids is not as well developed as that in gases. In fact, no satisfactory theory appears to exist. The main interest here is the diffusion of a dissolved gaseous reactant in the pores of catalysts filled with liquid reactant, as in a slurry or trickle-bed reactor. The mean free path in liquids is so small that Knudsen diffusion is not significant. Thus, the diffusion rate is unaffected by the pore diameter and pressure* The effective diffusivity may be defined in the same manner as in J2q. 14.1 with the diffusivity DAB,for instance, representing the effective diffusivity of species A of interest in pores filled with a solution of A and B. As expected, values of DABare much less than those for gases and are of the order of 1 X low5cm2/sec. For gases dissolved in liquids, the diffusivity can be an order of magnitude larger. Several correlations (Reid et al. 1977) are available for estimating diffusivities in liquids at low concentrations (infinite dilution), but these should be used with caution.

where the values of u and Slk are given in Tables 14,l and 14.2. For a polyatomic gas, one can use the following:

where is the mass heat capacity, and p is the viscosity. For the conductivity of gas mixtures, the following can be used:

Effective Thermal Conductivity

It is not surprising, in view of the minor role played by internal heat transport, that relatively little attention has been paid to the effective thermal conductivity inside pellets. In general, a solid is a much better conducting material than a gas. It is natural, therefore, that the conductivity of gas is the controlling factor in the overall (effective) thermal conductivity. This manifests itself in the strong dependence of the effective thermal conductivity on the void fraction of the pellets. The thermal conductivity of solid alumina is of the order of lod3 (cal crn-hec-I OK-') while that of helium, which has the highest thermal conductivity of gases, is of the order of (cal cm-l sec- OK-'). In the range of typical reaction temperatures and pressures, the experimental data obtained by Masarnune and Smith (1963) and Sehr (1958) indicate that the effective thermal conductivities fall in a very small band around 3 X lod4 (cal cm-l secdl OK-'). It is thus seen that the gas conductivity is indeed the controlling factor and that a simple-minded approach may be sufficient for theoretical calculations of the effective thermal conductivity. Such an approach by Woodside and Messmer (1961) yielded the following relationship:

where A,, A,, and A, are the effective thermal conductivity of the pellet, and the thermal conductivities of solid and gas, respectively, and E is the fractional void. A more rigorous approach can be found in the article by Butt (1965). The theory (Bird et al. 1960) of gas conductivity is well developed and the Chapman-Enskog theory for a monoatomic gas at low density gives:

14-2

MASS AND HEAT TRANSFER COEFFICIENTS

Average transport coefficients for transfer between the bulk-fluid and particle surface can be correlated in of dimensionless groups that characterize the flow conditions. It is common practice to correlate experimental data in of j-factors. Usually, the mass transfer coefficient is obtained from the j factor for mass; the heat transfer coefficient is obtained from j factor analogy. There have been many experimental studies of mass transfer in fixed-beds and summaries and analyses of the results are available (Whitaker 1972; Dwivedi and Upadhay 1977). For Reynolds numbers greater than 10, the following relationship,(Dwivedi and Upadhay 1977) between JD and the Reynolds number represents available data: 0.458 ( d ; 9 - 0 . 4 0 7 j,, = - EH

! !

where G = mass (superticial) velocity based on the cross-sectional area of the empty reactor (= vp) p = viscosity of fluid E B = void fraction of the bed d, = diameter of a spherical catalyst pellet (For other shapes, an approximate value of d, is that of a sphere with the same external area as the nonspherical pellet.) p == fluid density v = superficial velocity

492


TRANSPORT PROPERTIESAND EXPERIMENTAL METHODS 493

The j-factor for mass is defined as:

where Rc = measured rate of reaction L = pellet volume/external surface area T, = pellet surface ternperature Tb = bulk-fluid temperature (-AH) = heat liberated per mole reactant

where D is the molecular diffusivity of the component being transferred, and k, is the mass transfer coefficient. For a Reynolds number less than 10, a correlation based on a boundary layer model (Carberry 1960) may be used:

If the temperatures are measured, then the heat transfer coefficient can be determined from: h=

In view of the good agreement obtained between measured and theoretical temperature differences across the pellet-bulk fluid film (Butt et al. 1977), the temperature measurements may give accurate values of h under reaction conditions.

where v is the kinematic viscosity. This correlation may be used down to a Reynolds number of 0.5. As the Reynolds number gets smaller, the applicability of j-factors becomes increasingly uncertain. While literature correlations sometimes make a distinction between jtf and ju, the validity of the difference is uncertain in the absence of radiation. Hence, the original j-factor analogy may be used to obtain heat transfer coefficients:

14-3

REACTOR TRANSPORT PROPERTIES

In this section, consideration is given to effective thermal conductivities and diffusivities, in both the radial and axial directions, for the dispersion of mass and heat in a fixed-bed. The flux relationships for the mass and heat transport in the bed may be considered to define these transport properties. Also considered are the heat transfer coefficients at the tube wall.

where h is the thermal conductivity of fluid, and h is the heat transfer coefficient. Proper values of the mass and heat transfer coefficients that should be used when axial dispersion is included in the reactor model are given by Wakao and Funazkri (1978) and Wakao et al. (1979). For fluidized-beds, Kunii and Levenspiel (1969) have summarized available mass and heat transfer data in the form of graphs of Sherwood and Nusselt numbers versus Reynolds number. Typical of the results are those of Chu et al. (1953) which may be expressed in of j-factors as:

Effective Thermal Conductivities (pC,K)

I

) 5000. The transfer coefficients for multifor the range 30 < dpG/p(l - E ~ < phase reactors have already been treated in Chapter 7. Of the two transfer coefficients for gas-phase catalytic reactions, the heat transfer coefficient is much more important than the mass transfer coefficient since the external mass transfer resistance can be neglected under realistic conditions. The heat transfer coefficient under reaction conditions may be determined using a single-pellet reactor. At steady state, one has:

(-AH)RcL Ts - Tb

1

Many models and correlations have been develowd for the radial effective thermal conductivity (see the review by Kulkarani and Doraiswamy 1980). The thermal dispersion coefficient K used in Chapter 9 is related to the effective conductivity by A = pK. For nanacliabatic fixed-beds, the main heat conduction is in the radial direction, and thus, the radial conductivity is much more important than the axial conductivity. The axial conductivity represents the conduction supel-irnposed on the bulk flow, which is quite small relative to the heat transport by the bulk flow. Therefore, most work has been directed to the radial conductivity. The radial heat transport is complex, involving conduction, convection, and radiation between voids and solid and between solid particles. Possible modes of heat transfer in the radial direction are shown in Figure 14.3. Different physical models result depending on whether various resistances to the heat transport are in series, parallel, or a combination of both. Here an additive model is considered, which assumes that the radial effective thermal conductivity consists of static (conduction and radiation) and dynamic contributions, the latter caused by fluid motion. These two contributions are considered to be additive:

Heat Flux

Through Fluid

f

& donduction

n f - \ LAJ -

~etwesn Solid Surfacer Conduction Through Solid

Convection Due To Fluid Flow

Fluid

t

Flow

Figure 14.3 Various modes of heat transport in a packed bed. (Reprinted from Kulkarani and Doraiswamy 1980, by courtesy of Marcel Dekker, Inc.)

where h,O and A,O are the static and dynamic contributions. The static contribution is considered to consist of conduction and radiation, which are again additive:

A," = h,E + A;

(14.27)

where A: and :A are the contributions due to conduction and radiation, respectively. The Kunii-Smith model (1960), when the radiation term is neglected, reduces to:

A

31

A,

Figure 14.4 Correlation for S in Eq. 14.28. (Kunii and Smith 1960)

where d, is the particle diameter and the radiant heat transfer coefficient hb is given by:

hb = 1.952 X loF7gT3

kcal/hr m2"C

(14.30)

Here, g can be assumed to be equal to the ernissivity e. For the dynamic contribution, the relationship suggested by DeWasch and Froment (1972) may be used:

where A, = fluid thermal conductivity A, = solid thermal conductivity en = bed porosity 0 = ratio of the effective distance between particles to particle diameter (= L / d p ) 6 = stagnant fluid thickness Here, the thickness 6 can be read from curve A of Figure 14.4 for dense packing with @ of 0.895, and from curve B for loose packing with 6 of 1. An expression for the radiant contribution (Kulkarani and Doraiswamy 1980), which is equivalent to the radiation part of the Kunii-Smith model, is:

where dt is tube diameter. Use of Eqs. 14.28, 14.29, and 14.31 in Eq. 14.26 then yields the effective thermal conductivity in the radial direction. The effective thermal conductivity increases almost linearly with increasing particle Reynolds number (dpG/p) and ranges from 0.2 to 2 kcal/hr m°K for Reynolds number ranging from almost zero to 800. Because of the very minor role that conduction has in axial heat transport, not much work has been done and a relatively simple correlation should suffice for the purpose of estimating the approximate magnitude of the axial effective thermal conductivity. For this purpose, the correlation of Yagi and co-workers (1960) may be used:


where A,, = axial effective thermal conductivity h," = static contribution, which is the saxlie as that for radial thermal conductivity (Eq. 14.27) Re = dpG/p a = factor varying with solid thermal conductivity, 0.7 for steel and 0.8 for glass Pr = Prandtl number of fluid Here again, the static and dynamic contributions have been assumed to be additive.

where the static contribution h g is given by:

The overall heat transfer coefficient is based on an average bed temperature. If there exists a significant radial temperature profile and yet the one-dimensional model is used through a choice of the overall heat transfer coefficient, the following equation can be used to for the effect of the radial profile (Crider and Foss 1965):

Wall Heat Transfer Coefficients In Chapters 9 and 10, two heat transfer coefficients were used: the overall heat transfer coefficient for a one-dimensional model (U) and the wall heat transfer coefficient for two-dimensional model (h,). The averall heat transfer coefficient can be written as:

where h, is the heat transfer coefficient on the bed side, and h, is that on the cooling or heating medium side. Here, the conduction resistance at the wall was neglected. The surface area ratios were also set to unity since, in general, the wall thickness is so small that the ratios are close to unity. The correlations for hc can be found in any book on heat transfer. Here again, additive models for the heat transfer coefficients will be used. For the heat transfer coefficient he, the correlation of DeWasch and Froment (1972) may be used for an accurate estimate:

Assuming a parabolic profile, for instance, for the temperature, the factor appearing in the denominator would be 8 instead of 6.2. This relationship also gives an indication as to whether the one-dimensional model is adequate. If (Bi)h is small, say less than 1, one may conclude that the one-dimensional model is adequate since then he is quite close to h,. It has long been recognized that the wall heat transfer coefficients (he and h,) as well as the effective thermal conductivity are not constant but depend on the reactor length. In general, these coefficients decrease with increasing reactor length. On the other hand, it is extremely difficult to determine these coefficients as a function of reactor length, and therefore, constant values are used for design purposes. Most of the correlations presented so far are based on the data obtained from experimental reactors in the absence of a reaction for which the ratio of reactor length to tube radius is relatively small. If this ratio is large enough, (100 200) which is true for typical industrial reactors, these coefficients will approach asymptotic values, which are independent of the reactor length. Li and Finlayson (1977) obtained these asymptotic values based on the available literature data. Their results are summarized in Table 14.3.

-

where (Bi)h = hedp/h, (Bi)g = hzdp/h, Effective Diffusivities

For the static contribution (h,D), the following can be used:

DeWasch and Froment also give the following relationship for the wall heat transfer coefficient for the two-dimensional model (h,):

Unlike the diffusion in catalyst pellets, molecular as well as eddy (turbulent) diffusion causes the mass dispersion in fixed-beds. The effective molecular diffusivity may be obtained by simply multiplying the gas molecular diffusivity by the factor E ~ / K ,where the tortuosity factor K is often taken as 1.5. The theory on eddy diffusivity is not well established. Therefore, the effective diffusivities are often correlated in the following form:

Table 14.3 Asymptotic Values of Coefficients (Li and Finlayson 1977; reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, inc.)

Table 14.4 Approximate Ranges of Transport Properties and Coefficients (Reprinted from Kulkarani and Doraiswamy 1980, by courtesy of Marcel Dekker, Inc.) 1-10 (kcal/m2hr0C) for 0 < Re < 800

Radial effect thermal conductivity (her) Overall Heat Transfer Coeficienr

Wall Heat Transfer Coeficient

-------

Spherical packing 20

d G

< 7600 and 0.05 < dp < 0.3 dl

P

Cylindrical packing 20 <*< P

0.2-2

Axial effective thermal conductivity Gas thermal conductivity Heat transfer coefficient for a one-dimensional model (he) Wall coefficient for a two-dimensional model (h,) Radial Peclet number for mass dispersion ((Pe)r) Axial Peclet number for mass dispersion ((Pc),) -

-----

.

dt

1 ---

(Pe),

where (Pe), and (Pe)[ are the Peclet numbers based on particle diameter for molecular and turbulent (eddy) diffusion, respectively, and Pe(= dpv/D,) is the Peclet number for the combined mass dispersion. In of Reynolds and Schmidt numbers, the Peclet number for molecular diffusion can be written as:

( )

Re=- d p G ,

I-L

P D ~

+

Ei9

1.5ReSc

1 + ll[1+19.4(dp/dt)2J'

(pe)r=Ydp Der

(14.41)

EB

1.5ReSc

+ -21 ;

(Pe), =-vd, Dea

since the turbulent Peclet number approaches 2 at high Reynolds number. An empirical correlation developed by Wen and Fan (1975) is: - 1

(Pe), 0.008

sc=- P

when Dm is the molecular diffusivity for the gas species of interest. The majority of correlation efforts have been centered around the Peclet number for eddy dispersion. It has been recognized that the overall Peclet number is dominated by the turbulent Peclet number when the particle Reynolds number is greater than 100, its value approaching approximately 11 for radial dispersion and 2 for axial dispersion. For the radial effective diffusivity, Froment (1967) found that at high Reynolds numbers the empirical correlation factor of Fahien and Smith (1955) given by (Pe),/[l 19.4(dp/dt)2] approaches 11. Using this as the turbulent contribution, Eq. 14.39 may be written as: (Pe),

. -

where K has been set to 1.5. The stochastic model of Gunn (1969) for radial dispersion in a packed bed relates the turbulent Peclet number to the probability of a molecule at the entrance traveling a certain distance. This model also yields a value of 11 for the Peclet number at high Reynolds number. As for axial diffusion, the axial effective diffusivity can be obtained from:

dp=6Vp/Sp

Eflective thermal conductivity (radial)

1 -=-

--

800 and 0.03 c d p< 0.2

where

(Pe), = (ReSc)

-

.>

(kal/m hr°C) 100-250 6-20 (Typically 11 for Re > 100) 0.01-10 (Typically 2 for Re > 10) --- 15-73

0.3 (ReSc)

0.5 + 1 + 3.8/(ReSc)

=-

< Re < 400 and 0.28 < Sc < 2.2

This correlation, which is similar to that proposed by Bischoff and Levenspiel (1962), s for a maximum that exists in the indicated range of Reynalds number. Typical ranges of transport properties and coefficients are summarized in Table 14.4.

1 4 4 DETERMINATION OF TRANSPORT PROPERTIES A N D COEFFICIENTS

Fixed-beds packed with inert pellets are usually used for the determination of transport properties and transfer coefficients. The solution of a steady-state mass or heat balance equation is compared with concentration or temperature measurements for the determination. For thermal properties, for instance, radial temperature

I

TRANSPORT PROPERTIES AND

L

EXPERIMENTAL METHODS501 Figure 14.5 Dependence of effective wall heat transfer coefficient on reactor lengrh for various conditions. (Dewasch and Froment 1972; reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, Inc.)

profile (Coberly and Marshall 1951; Michelsen 1979), axial temperatures (Yagi and Wakao 1959) or centerline temperatures (Qlbrich et al. 1966) are measured.

These properties can also be determined from a moment analysis of the transient response at the bed outlet obtained by introducing a pulse at the inlet. While this procedure is more convenient experimentally, possible dependence of the properties on transient conditions has not yet been fully resolved. For this reason, a steady-state experiment may be preferred for the determination. The correlations given earlier are on4y for approximate calculations of the properties. For any systematic study of a fixed-bed, these properties should be determined for the reactor under consideration using the experimental methods described in this section. Perhaps the simplest to determine is the effective wall heat transfer coefficient (based on tube wall temperature) used in conjunction with the one-dimensional model. A heat balance for a bed packed with inert pellets is given by:

with

T = Ti,,, z =0

where Tw is the constant wall temperature. Integrating this equation over the reactor length (2)gives:

where the mean outlet temperature Te is obtained fram the radial profile at the outlet by:

whose solution (Coberly and Marshall 1951) is given by:

where Effective wall heat transfer coefficients determined in this way by DeWasch and Frornent (1972) are shown in Figure 1 4 5 It is seen that the effective transfer coefficients are higher at the inlet and approach a limiting value toward the outlet. In general, this limiting value is reached at a smaller length for a smaller ratio of dt(dp. Note that h, is a lumped quantity averaged over the reactor length, and theikfore, is a constant for a given reactor. For the determination of effectivethermal conductivity and wall heat transfer coefficient, the following heat balance equation is solved:

and JO and J 1 are Bessel functions of the first kind of order zero and one, respectively. Here, the eigenvalues bn are defined by the relationship:

TRANSPORT PROPERTIES AND EXPERIMENTAL METHODS 503

When the value of /3z is greater than 0.2, the series in Eq. 14.49 converges so rapidly that only the first term is significant. Therefore, for pz > 0.2, the equation can be reduced to:

If the temperature along the tube centerline is measured, for instance, and used in Eq. 14.52, a plot of the left hand side versus z should give a straight line, yielding flbf as its slope from which A,, can be calculated, and also yielding the logarithmic term in the right hand side as its intercept from which h, can be calculated. Various methods of calculating A, and h, result depending on how Eq. 14.49 or Eq. 14.52 is used, given certain temperature measurements. DeWasch and Froment, for instance, suggest the selection of A,, and hw based on the rninimization of the function F = Z(Ticalc- Tiexp)2 using a radial profile. One way of writing a heat balance for the two-dimensional model is to replace the boundary condition of Eq* 14.48 by:

Effective axial transport properties can be determined using an adiabatic reactor. Steady state mass and heat balances result in second-order ordinary differential equations when the axial dispersion is taken into consideration, solutions of which can readily be obtained. Based on these solutions and temperature or concentration measurements, the effective transport properties can be calculated in a manner similar to the procedures used for the radial transport properties. As indicated earlier, a transient experiment can also be used for the determination. Here, experimental and analytical procedures are illustrated for the determination of the effective axial transport property for mass. An unsteady state mass balance for an adiabatic reactor can be written as:

ac

ac -+----= at'

a2

where 2 = z/Z, f=" t / ~T, = Z / V and from steady state is given by:

P ,

1

a 2 c

a,, a22 = vZ/D,,. The concentration deviation

In such a case, the solution becomes:

Tw - T T w - Tin

, Jo@n

4) e-ghgz

.hn~l(hn)

where An are the roots of Jo(x)= 0. Similar analysis as for the determination of A,, and h, given above applies. The effectivediffusivity in the radial direction can be determined &om concentration measurements and the solution of the following mass balance:

T=-

r

R

z

,z'=-

z

where C, is the steady state concentration. Taking the Laplace transform of Eq. 14.56 and solving the resulting second-order ordinary differential equation gives a solution in the following form:

When a concentration pulse is introduced to the bed and the response at the outlet is measured, this outlet concentration, when multiplied by t n and integrated over t E (O,m), yields the nth-order moment:

vR

P,,, = -

arz

The moment can also be obtained from:

This cquatiorl is essentially equivalent to Eq. 14.47, However, the boundary conditiuri at the wall is a C / a ~= 0. The solutiorl (Olbrich et al. 1966) is given by:

M,,= lim (-1)" dsn

where Cm is the concentration approached asymptotically in the bed of infinite length, and qi are the roots of the equation Jl(7i) = 0. For a bed of sufficient length, the exit concentration may be taken as C,.

Since = c(s,l;P,.), the moment obtained from Eq. 14.59 will contain P,. as the only unknown, When the moment is determined by graphical integration of the outlet response, P,, can be calculated by equating Eq. 14S8 to Eq. 14.59. In this case, only the first moment (n = 1) is required for the determination, which will yield fairly accurate results since the higher order moments are not involved. No satisfactory comparisons have been made between the properties

d "Cl(s)

S-.O

cl(s)

i


determined from steady state and transient experiments. For design purposes, therefore, the properties determined from steady-state experiments may be preferred.

MI/M2; quantity defined in Eq. 14.50 molecular weight n t h moment molar flux total pressure Peclet number axial, mass Peclet number given in Eq. 14.42 Peclet number for molecular diffusion given in Eq. 14.40 radial, mass Peclet number given in Eq. 14.41 Peclet number for turbulent diffusion in Eq. 14.39 vZ/Dea VR /Dp,Z Prandtl number for fluid, C p p / h radial coordinate r/R; average pore radius rate of reaction radius Reynolds number d p G / p gas constant global rate Laplace transform variable surface area of a cylinder Schmidt number, p/pD, time

Summary

Transport properties and coefficients have been presented in this chapter, including theories, correlations, and experimental methods. While correlations are useful for estimating the transport properties and coefficients, it is desirable for any detailed study of a reactor to determine these properties experimentally using the methods presented in this chapter. NOTATION

6, (Bi)h C Cb

C, dP dl

D

0,. 12 De,K

a, Der

Dk

D m ,12 DIM D P

fe G h hb hc

he hw

(-AH) JDPJH

Jot

k

k, L

J1

solutions of Eq. 14.51 Biot number for heat given in Eq. 14.34 concentration bulk-fluid concentration specific heat content concentration at the exit of a sufficiently long bed particle diameter tube diameter effective diffusivity effective molecular diffusivity effective Knudsen diffusivity effective diffusivity in the axial direction in a fixed-bed effective diffusivity in the radial direction in a fixed-bed Knudsen diffusivity molecular diffusivity macropore diffusivity micropore diffusivity correction factor for pore geometry given in Eqs. 14.3 and 14.4 mass rate per unit area film heat transfer coefficient radiant heat transfer coefficient coolant-side heat transfer coefficient tube-side heat transfer coefficient in a one-dimensional reactor model wall heat transfer coefficient in a two-dimensional reactor model heat liberated by reaction per mole j factor for mass and heat, respectively Bessel functions of the first kind of order zero and one, respectively Boltzmann constant film mass transfer coefficient ratio of volume to external surface area of pellet; effective distance between particles

t/r

temperature bulk-fluid temperature reactor outlet temperature pellet surface temperature tube wall temFerature reactor inlet temperature overall heat transfer coefficient based on he superficial fluid velocity average molecular velocity volume of a cylinder mole fraction pellet coordinate, axial coordinate z/z reactor length Greek Letters a

fl 6

constant factor in Eq, 14.32 quantity defined in Eq. 14.50; quantity defined in Eq. 14.28 stagnant fluid thickness

TRANSPORT PROPERTIES AND EXPERIMENTAL METt.iOI3S

1

porosity bed porosity macropore porosity micropore porosity tortuosity factor effectiveness factor thermal conductivity; mean free path of a molecule effective thermal conductivity gas thermal conductivity solid thermal conductivity effective thermal conductivity in axial direction effective thermal conductivity in radial direction dynamic contribution to A,, in Eq. 14.26 static contribution to A,, in Eq. 14.26 conduction component of h.: in Ey. 14.27 radiation component of A.: in Eq. 14.27 viscosity kinematic viscosity density parameters given in Table 14.1 parameters given in Table 14,2 z/v generalized Thiele modulus quantity defined by Eq. 14.20

5' h A,

A, As

hea

Aer A8 A," A,C

A,' P V

PROBLEMS

14.1. The following results have been obtained using a Wicke and Kallenbach diffusion cell:

The same catalyst pellets are to be used for combustion of C6H8in excess . Assume for simplifications air. Find the effective diffusivity of CbHg, (DCBHg). that temperature, pressure, and the constants in the Lennard-Jones potential function remain the same. (Ans. 0.017 cm2/s) 14.2. One set of data obtained by Coberly and Marshall (1951) for a bed (dt = 5in) packed with inert pellets is given below for the centerline temperatures. z (in)

5.75

11.5

17.25

23.0

34.5

46.0

T(OF)

115

126

156

160

187

197

507

For this run, G = 475 ib/ft2hr, Tw = 220°F, and Tin = 10O0F. Determine from the data the effective thermal conductivity in the radial direction her and the wall heat transfer coefficient h,. Use the physical properties of air. 14.3. Consider an adiabatic fixed-bed. Suppose the bed is packed with inert pellets.

a. Write the unsteady-state, dimensionless heat balance for the bed such that the Peclet number appears as the only constant in the equation. b. Rewrite the equation in of temperature deviation from the steady state value. c. Suppose that at time zero, a temperature impulse of unit magnitude is introduced at the inlet and that the outlet response is recorded. What is the expression for the first moment? Is this equal to the area under the outlet response curve?

I

REFERENCES Baiker, A., M. New and W. Richarz, Chem. Eng. Sci., 37, 643 (1982). Bird, R.B.,W.E. Stewart and E.N. Ligh tfoot, Transport Phenomena. Wiley, New York (1960). Bischoff, K.B. and 0. Levenspiel, Chem. Eng. Sci., 17, 245 (1962). Butt, J.B., AIChE J., 11, 106 (1965). Butt, J.B.,D.H. Downing and J.W. Lee, Ind. Eng. Chem. Fund., 16, 270 (1977). Carberry, J.J., AIChE J., 6, 460 (1960). Chu, J.C., J. Kaiil and W.A. Wetterath, Chern. Eng. Prog., 49, 141 (1953). Caberly, C.A. and W.R. Marshall, Jr., Chem. Eng. Prog. 47, 141 (1951). Crider, J.E. and A.S. Foss, AIChE J., 11, 102 (1965). DeWasch, A.P. and G.F. Froment, Chem. Eng. Sci., 27, 567 (1972). Dogu, G . and J.M. Smith, AIChE J., 21. 58 (1975). Dwivedi, P.N. and S.N.Upadhay, Ind. Eng. Chem. Proc. Des. Dev., 16, 157 (1977). Fahien, R.W.and J.M. Smith, AIChE J., 1, 25 (1955). Feng, C.F.and W.E. Stewart, Ind. Eng. Chem. Fund., 12, 143 (1973). Frornent, G.F.,Ind. Eng. Cheln, 59, 27 (1967). Frost, A.C.,AIChE J., 27, 813 (1981). Gunn, D.J., Trans. Inst. Chem. Eng., 47, T351 (1969). Kulkarani, B.D.and L.K.Doraiswamy, Cat. Rev., 22, 43 1 (1980). Kunii, D.and 0. Levenspiel, Fluidization Engineering, Wiley, New York (1969). Kunii, D.and J.M. Smith, AIChE J., 6, 71 (1960). Li, C.H. and B.A. Finlayson, Chem. Eng. Sci., 32. 1055 (1977). Masamune, S. and J.M. Smith, J. Chern. Eng, Data, 8, 54 (1963). McGreavy, C.and M.A. Siddiqui, Chern. Eng. Sci., 35, 3 (1980). Michelsen, M.L., Chem. Eng. J., 18, 67 (1979). Olbrich, W.E., J.B.Agnew and O.E. Potter, Trans. Inst, Chern. Eng., 46, T207 (1966)+ Reid, R.C.. J.M. Prausnitz and T.K. Sherwood, The Properties cfGasc.s and Liquids. Chap, 11, McGraw-Hill, New York (1977). Satterfield, C.N., Mass Transfer in Heterogeneous Catalysis, MIT Press, Cambridge, Mass. (1970).

Sehr, R.A,, Chem. Eng. Sci., 2, 145 (1958). Wakao, N. and J.M. Smith, Chem. Eng. Sci., 17, 825 (1962). Wakao, N. and T. Funazkri, Chem. Eng. Sci., 33, 1375 (1978). Wakao, N., S. Kaguel and T. Funazkri, Chem. Eng. Sci., 34, 325 (1979). Wen, C.Y. and L.T. Fan, Modelsfor Flow Systems and Chemical Reactors, Marcel-Dekker, New York (1975). Whitaker, S., AIChE J., 18, 361 (1972). Wicke, E. and R. Kallenbach, Kolloid-Z, 17, 135 (1941). Wolff, H.J., K.H. Radeke and D. Gelbin, Chem. Eng. Sci., 34, 101 (1979). Woodside, W. and J.H. Messmer, J. Appl. Phys., 32, 1688 (1961). Yagi, S. and N. Wakao, AIChE J., 5, 79 (1959). Yagi, S., D. Kunii and N. Wakao, AIChE J., 6. 543 (1960).

SUBJECT INDEX

Acidity of catalyst, 6 Brønsted and Lewis acid, 6 correlations, in of, 31, 32 determination of, 27 Activation energy homogeneous vs. heterogeneous, 5 diffusional effect on, 104 Active site, 55 of acidic catalysts, 6, 27 Activity factor, 165 Activity, on-line estimation of, 465 Adsorption apparatus for, 25 chemical, 20, 21, 24 energetics of, 22 heat of, 43 hysteresis of physical, 43 inhibition, 62 in surface reaction, 55 isotherms, 23, 35 multilayer, 35 physical, 20, 2 1, 40, 4 1 types of physical, 40 Arrhenius number, 108 Aspect ratios definitions of, 289 role of, in design, 291 Atom migration mechanism of, 202 kinetics of sintering for, 21 1 Autotherrnic reactor, 309 Average pore radius, 44 Axial dispersion boundary conditions in the model of, 288 coefficients, 499 CSTR analogy, 326 in fixed-bed, 287 in trickle-bed. 443

Bernoulli equation, 294 Biot numbers, 107 Blowout, 312 Bubble phase, 405 Bubble properties, 419 BET (Brunauer-Emmett-Teller) theory, 35 surface area, 39 Calculus of variations, 459 Carbonium ion, 6 Catalyst characterization of, 35 classification of, 5 definition of, 3 for, 10 uses of, 8 Catalyst deactivation. See Chemical deactivation; Physical deactivation Catalyst preparation impregnation, 10 single pore model of, 14 equilibration time for, 17 parameter determination of, 18 precipitation, 10 calcination, 1 I drying, 11 Catalyst regeneration, 180, 201, 470 Catalytic activity correlations of, 27 distribution of, 18, 126, 141 Catalytic converter, 312, 314 Cell model for fixed-bed, 297 for fluidized-bed, 415 Chapman-Enskog formulas, 484, 49 1 Chemical adsorption (chemisorption). See Adsorption

Chemical deactivation coking. See Coking dependent, 151, 177 diffusion-deactivation, 164 fouling, 15 1 independent, 151, 156 kinetics of, 155 nature of, 152 poisoning, 151 shell-progressive, 164, 167 uniform, 164 Coarsening, 201 Coking combustion kinetics of, 181 effect on diffusivity, 154, 189 multilayer kinetics of, 183 nature of, 154 pore plugging, induced by, 189 regeneration of coked catalyst, 180, 402, 470 Compensation effect, 66 Complex reactions reaction network, synthesis of, 81 Wei's treatment, 225 Conservation equations fixed-bed, 286 simplification of, 290, 292 slurry reactor, 433 trickle-bed, 437, 441 Constant conversion policy, 458, 460 Continuous reaction/regeneration, 402, 422 Control of reactor affected by deactivation, 467 Crank-Nicholson method, 301 Crystallite migration kinetics of, 218 mechanism of, 202 Damkoehler number, 107, 112 Darcey permeability, 98 Davidson-Harrison model, 408 Deactivation. See Chemical deactivation; Physical deactivatian Demanding reaction, 20 Design of reactor. See Fixed-bed; Fluidizedbed design; Multiphase reactors Differential reactor, 72 Diffusion in catalyst pellet, 98

Diffusion (Continued) effect of, on activation energy, 104 effect of, on reaction order, 105 effect of, on reaction rate, 95 importance of, 115 Knudsen, 484 molecular, 484 on surface, 98, 208 Diffusivity. See Effective diffusivity Dispersion (of catalyst), 19 Dispersion coefficients (axial and radial), 287, 493, 497 experimental determination of, 502, 503 Dusty gas model, 98 Effective diffusivity comparison between molecular and Knudsen, 487 correction factor for, 484 definition of, 98 experimental determination of, 488 intermolecular parameters for, 485 Knudsen, 98, 484 macro- and micropores, 487 molecular, 99, 484 random pore model for, 487 Wicke-Kallenbach cell for, 489 Effectivenessfactor definition of, 95 external, 109 for macro- and micropores, 148 for multiphase reactions, 242, 246, 439 for nonuniform activity, 126 generalized, 117 internal, 109 nonisothermal, 108 overall, 108 shape normalization for, 102 Effective thermal conductivity, 490 Elutriation of fines, 406, 427 Emulsion phase, 405 Equivalence of fixed-bed models for axial dispersion, 334, 346 for radial dispersion, 335 Ergun relationship, 404 Experimental reactors, 71 Explicit method. See Numerical methods Extinction point, 312

Facile reaction, 19 Fanning equation, 293 Fast fluidization, 414 Film heat transfer coefficient (gas-solid), 492 Film mass transfer coefficient gas-liquid, 237 gas-solid, 49 1 in fluidized-bed, 420, 42 1 liquid-solid, 239 Finite-difference method. See Numerical methods Fixed-bed autothermic, 309 catalytic converter, 312 conservation equations for, 286 multiple reactions in, 388 numerical methods for, 296 parametric sensitivity in, 295, 317 pressure drop in, 293 radial, 312 reactor/heat exchanger coupling, 309 velocity change in, 293 Fixed-bed design approach of reactor point effectiveness for, 331 axial dispersion, 346 dependent deactivation, 363, 366 diffusion-affected reactions, 337 general case of chemical deactivation, 361, 367 multiple reactions, 388 shell-progressive chemical deactivation, 355, 367 sintering, 374, 379 uniform chemical deactivation, 348, 367 variable velocity, 345 classical approach for, 285 for parametrically sensitive reactions, 317 Flow regimes. See Fluidization; Trickle-bed Fluidization bubbling, 404 classification of particles for, 414 effect of particle size on, 413 fast, 414 flow regimes in, 404 minimum (incipient) velocity for, 403 regime of bubble-free, 414 slugging, 404

Fluidization (Continued) terminal velocity for, 405 turbulent, 414 Fluidized-bed design bubble properties for, 419 coke removal in, 402 comparisons of two phase models. 412 Davidson-Harrison model, 408 flow reversal in, 415, 417 Kunii-Levenspiel model, 410 non-bubbling beds at high velocity, 422 two phase model, 407 variable bubble property model, 415 Fluxes, 98 pressure effect on, 100 Fouling. See Chemical deactivation Gas-solid reactions. See Noncatalytic gassolid reactions Generalized effectiveness factor, 1 18 Generalized Thiele modulus, 118 Global rate as affected by nonuniform activity. 126, 36 1 definition of, 71 for multiphase reactions, 240, 243 for multiple reactions, 130, 383 for reactions affected by chemical deactivation, 168, 345 for reactions affected by sintering, 222. 374 for reactions affected by transport resistances, 120, 337 Graham's relationship, 100 Heat of adsorption, 22, 30 Heat transfer coefficients film. See Film heat transfer coefficient (gas-solid) overall and wall, 496 Heterogeneousmodel, 286 Hollow pellet, 141 Homogeneous reactor, 285 Hat spot, 305 Hydrogenation/dehydrogenation, 63 Hydrogenolysis, 5 Ignition line, 316 Implicit method. See Numerical methods Impregnation. See Catalyst preparation

Incipient fluidization. See Fluidization Interfacial heat and mass transport, 106 Integral reactor, 72 Internal age distribution, 423 Internal (intraphase) transport resistance, 106 Interphase (external) transport resistance, 106 Intrinsic kinetics determination of, 76 experimental reactors for, 7 1 transport criteria for, 74, 138 Intrinsic rate, 71, 95

Monolithic catalyst bed, 312, 314 Multibed reactor/heat exchanger, 452 Multilayer model for coking, 183 for physisorption, 35 Multiphase reactors, 431. See also Slurry reactor; Trickle-bed Multiple steady state for effectiveness factor, 109 in interfacial transport (reactor point effectiveness), 123 in reactor/heat exchanger coupling, 3 11 in stability consideration, 269 number of stable, 262

Optimization (Continued) maximum principle method for, 459, 461 subject to catalyst deactivation, 458 Orthogonal collocation, 303 Oswald ripening, 2 11 Oxidation of SO2, 304

j-factors and j-factor analogy, 492 Kelvin equation, 41 Kinetics determination of intrinsic, 76 experimental reactors for intrinsic, 71 for complex reactions, 81 intrinsic, 71, 95 Langrnuir-Hinshelwood, 60 model discrimination for, 77 of surface reactions, 55 Kunii-Levenspiel model, 410 Langmuir-Hinshelwood kinetics adsorption inhibition, 62 consistency of, 66 rate-controlling step for, 60 two-step model for, 67 Langrnuir-Hinshelwood mechanism, 60 Lewis acid, 6 Lewis number definition of, 107 stability as affected by, 266 Local activity factor, 465 Lumping, 81 Mass transfer coefficient. See Film mass transfer coefficient Maximum principle, 272, 459 Micro- and macropores diffusivities in, 487 distribution of, 44 effectiveness factors for, 148 Mixing-cup averages, 307, 336 Molecular effusion, 55

Naphthalene oxidation, 305, 327 Nature of chemical deactivation, 152 Non-bubbling beds at high velocities, 422 Noncatalytic gas-solid reactions as applied to deactivation, 167 shell progressive, 160 shrinking core model for, 162 uniform, 159 Nonisothermal effectiveness factor for gas phase reactions, 243 for multiphase reactions, 440 Nonuniform activity distribution, 126 Numerical methods averaging method of Ahmed and Fahien, 304 Crank-Nicholson method, 301 collocation method, 303 explicit method, 297 finite-difference formulas, 298 for multiple reactions, 132 for solving fixed-bed conservation equations, 296 implicit method, 297 line method, 302 Liu's method, 301 stability and error estimation, 301 On-line estimation of catalyst deactivation, 465 Optimization calculus of variations for, 459 dynamic programming for, 452 equilibrium constrained, 450 for sequencing, 471

I

I

Parametric sensitivity, 295, 314, 320 Partial wetting, 241, 441, 442 Peclet numbers, 289, 291, 334, 498, 499, 502 Pellet center concentration, 122, 333 characteristic length, 102 design, 138, 194 hollow or partially impregnated, 141 shape normalization, 102 Phase diagram for exothermic reactions, 317 Physical adsorption (physisorption), 20, 2 1, 35, 40, 41, 43 Physical deactivation definition of, 201 effects of, 202 sintering, 201 Poisoning. See Chemical deactivation Pore blocking, 154, 189 Pore size distribution bidisperse (bimodal), 44 mercury penetration method for, 44 physisorption method for, 41 Prandtl number, 492 Precipitation method. See Catalyst preparation Pressure drop in fixed-bed, 293 in fluidized-bed, 404 Principle of optimality, 452 Promoters, 12 Pseudo-homogeneous model, 285 Pseudo-steady state assumption, 164

Radial dispersion, 335 Radial flow reactor, 312 Random pore model, 487 Raney nickel, 1 1 Rate-controlling step, 60 Recycle reactor, 75 Reactors differential, integral, CSTR, transport, 72 fixed-bed, 285, 331

Reactors (Continued) fluidized-bed, 40 1 slurry, 434 trickle-bed, 438 Reactor design. See Fixed-bed; Fluidizedbed; Slurry reactor; Trickle-bed Reactor point effectiveness assumptions involved in, 332 definition of, 120 for reactions affected by chemical deactivation, 168, 345 for reactions affected by physical deactivation (sintering), 222, 372 for reactions affected by transport resistances, 122, 337 multiple reactions, 130, 383 multiple species, 125, 344 nonuniform activity, 126 pellet center concentration for, 122 Regeneration of coked catalyst, 181, 402 of deactivated catalyst, 180, 201 Region of bubble-free operation, 414 of safe-operation, 317 Reynolds number, 498 Rideal (Eley-Rideal) mechanism, 60 Riser cracker, 422 Runaway criteria first-order reaction, 294 general, 314 n th-order reaction, 295 Sabatier's principle, 28 Schmidt number, 498 Selectivity as affected by catalyst deactivation, 258 as affected by diffusion, 251 as affected by transport resistances, 251 in complex reactions, 255 Shape normalization, 102 Shell-progressive model(SPM), 160 Shrinking core model, 162 Single pore model, 14 Sinrering behavior of, 221 mechanisms and kinetics, 21 1 surface diffusion, 208 Slurry reactor, 236 Stability criterion due to Gavalas, 262

Stability (Continued) multiple solutions, 269 single pellet problem in, 26 1 slope conditions, 263 unit Lewis number and extension, 266 Structure-insensi tive, structure-sensitive reactions, 19, 20 s, 10 Surface diffusion, 98, 208 Surface reactions, 55 Synthesis of kinetic structure, 8 1 Temperature-programmed desorption (TPD), 31 Terminal velocity, 405 Thiele modulus for first-order reaction, 97 general, 108 generalized, 118 Thomas method, 302 Titration method, 27 Transformation ratio, 84 Transport criteria for intrinsic kinetics, 74, 138

Transport properties and transfer coefficients, 483 Transport reactor, 72 Trickle-bed design of, 437, 441 flow regimes in, 432 mass transfer in, 236 Turbulent fluidization, 4 14 Two phase models, 407 Two-step kinetic model, 67 Ultra-high vacuum (UHV) techniques, 34 Variable bubble property model, 415 Velocity as affected by reaction, 293 Volcano curve. 28 Voorhics relationship, 155 Weisz-Prater criterion, 139 Wetting efficiency. See Partial wetting Wicke-Kallenbach chamber, 489

X-ray line broadening, 27 Zcolites, 7

AUTHOR INDEX

Agnew, J.H., 500, 502, 507 Akella, L.M., 84, 86, 90, 130, 136, 316, 321, 329 Amenomiya, A., 31, 34, 52 Ames, W.F., 303, 329 Amhed, M., 303, 326, 329, 400 Arnundson, N.K., 266, 280, 323, 329 Anderson, J.B., 74, 90 Anderson, J.R., 1 1, 13, 24, 26, 52 Anderson, R.B., 14, 52, 53 Angelino, H., 420, 429 Aris, R., 81, 90, 102, 103, 117, 149, 150, 251, 263, 269, 270, 280, 452, 457, 458, 479 Austin, L.G., 147, 150 Baddour, R.F., 309, 329 Baker, R.T.K., 154, 199 Baiker, A., 489, 507 Baldi, G., 441, 443, 448 Barber, M.A., 154, 199 Barkelew, C.R., 294, 329 Barrer, R. M., 7, 52 Bartholomew, C.H., 196, 199 Becker, E.R., 194, 199 Beekman, J.W., 154, 199 Bellman, R., 452, 479 Bennett, C.O., 117, 150 Bensei, H.A., 27, 52 Beveridge, G.S.G., 449, 479 Berty, J.M., 73, 90 Bird, R.B., 98, 150, 286, 329, 484, 490, 507 Bischoff, K.B., 62, 70, 74, 90, 101, 117, 119, 139, 140, 150, 158, 162, 199,294,319, 329, 401, 499, 507 Bliss, G.A., 459, 479 Bliss, G.S., 239, 250 Bond, G.C., 5, 7, 24, 28, 52 Bosdekker, E.R., 153, 199

Botty, M.C., 188, 199 Boudart, M., 23, 53, 64, 65, 67, 70, 80 Box, G.E.P., 77, 90 Brian, P.L.T., 238, 249, 309, 329 Brown, C.E., 117, 150 Brunauer, S., 35, 52 Buffham, B.A., 443. 448 Bulvidas, L.J., 293, 329 Burghardt, A., 335, 400 Butt, J.B., 60, 61, 66, 70, 112, 114, 150, 154, 173, 174, 175, 187, 190, 195, 199, 290, 329, 341, 349, 354, 359, 400, 474, 479, 490, 493, 507 Buzzi-Ferraris, G., 77, 90 Caldebank, P. H., 237, 238, 249, 250 Cappelli, A., 309, 329, 477, 479 Carberry, J.J., 20, 21, 52, 64, 70, 73, 90, 110, 111, 112, 150, 167, 199, 253, 277, 280, 290, 291, 304, 305, 326, 329, 492, 507 Carbonell, R.G., 241, 250 Carnahan, B., 301, 329 Carver, M.B., 303, 329 Caughanowr, D. R., 83, 90 Cervello, J., 50, 53 Chen, H.C., 14, 52 Chen, M., 230, 233 Chen, N.Y., 7, 52 Chludzinski, J.J., Jr., 154, 199 Chou, A., 458, 479 Chu, J.C., 492, 507 Chu, Y. F., 222, 233 Churchill, R.V., 17, 52 Coberly, C.A., 500, 501, 506, 507 Cohan, L. H., 43, 52 Cole, J.D., 334, 400 Collina, A., 294, 309, 329, 477, 479 Constable, F.H., 66, 70 Copelowitz, J., 270, 280

Corbett, W.E., 148, 150 Corbetta, D., 477, 479 Crider, J.E., 468, 479, 497, 507 Crowe, C.M., 459, 460, 479 Crump, J.G., 230, 233 Cunningham, R.E., 100, 150 Cvetanovic, R.J., 31, 34, 52 Daddyburjor, D.B., 194, 199, 212, 233 Dalla Betta, R.A., 202, 233 Darton, R.C., 404, 429 Davidson, J.F., 404, 407, 414, 420, 429 Dawson, P.T., 53 Degtyarev, G.L., 461, 479 DeJong, J.A.H., 413, 429 Delmon, D., 190, 199 Denbigh, K.G., 449, 479 Dente, M., 105, 150, 294, 309, 329 DePauw, R., 258, 280 DeWasch, A.P., 496, 500, 507 Dharwadkar, A., 239, 250 DiNapoli, N.M., 100, 150 Dogu, G., 489, 507 Daraiswamy, L. K., 493, 494, 507 Downing, D.M., 112, 150, 173, 199, 493 Dudukovic, M. P., 242, 250, 441, 442, 448 Dumez, F.J., 77, 90, 186, 199, 309, 329 Dwivedi, P.N., 491, 507 Eberly, P.E., Jr., 26, 52 Eischens, R.P., 63, 70 Emmett, P.A., 35, 39, 52 Evans, R.B., 98, 150 Everett, D.H., 66, 70

Eymery, J. P., 309, 329 Fahien, R.W., 303, 326, 329, 400, 498, 507 Fan, L.S., 415, 418, 429 Fan, L.T., 499, 508 Farrauto, R.J., 25, 52 Feates, F.S., 154, 199 Feng, C.F., 98, 150, 189, 487, 507 Finlayson, B.A., 304, 329, 497, 507 Finneran, J.A., 293, 329 Flynn, P. C., 202, 233 Formi, L., 27, 52 Forsythe, G. E., 15, 53 Forzatti, P., 77, 90 Foss, A. S., 497, 507

Fournier, C.D., 452, 479 Freel, J., 25, 53 Froment, G.F., 62, 70, 77, 90, 139, 150, 154, 158, 186, 190, 199, 258, 280, 294, 309, 319, 329, 401, 496, 498, 500, 507 Frost, A.C., 489, 507 Frost, A.S., 468, 479 Fryer, C., 415, 418, 429 Funazkri, T., 492, 508 Furusaki, S., 70

Hiraki, I., 415, 429 Hite, R. H., 100, 150 Hlavacek, V., 294, 297, 312, 329 Hofmann, H. P., 437, 448 Hong, J.C., 467, 468, 479 Horio, M., 413, 415, 429 Hosten, J . , 77, 90 Hougen, O.A., 60, 67, 70, 77, 91 Hurwitz, H., 64, 67, 70, 77, 91 Hutchinson, P., 81, 90

Gandhi, H., 26, 53 Gates, B.C., 4, 29, 53, 60, 70 Gavalas, G.R., 81, 90 Geldart, D., 413, 429 Gianetto, A., 441, 443, 448 Gjostein, N.A., 202, 212, 215, 233 Glastonbury, J.R., 238, 250 Golikeri, S.V., 77, 91 Goodwin, R.D., 181, 198, 199, 403, 429 Gorring, R.L., 167, 199 Goto, S., 238, 250, 437, 448 Grace, J.R., 420, 429 Gross, S., 81, 9 1 Groves, F.R., 452, 479 Gruyaert, F., 459, 460, 479 Gunn, D.J., 499, 507 Gunn, R. D., 100, 150

Il'in, A.M., 271, 280 Iordache, C., 302, 329

Hagstrum, H.D., 35, 53 Haldemann, R. G., 188, 199 Hales, H.B., 238, 249 Halsey, G.D., 42, 53 Hannay, N.B., 53 Hansford, R.C., 30, 53 Harnsberger, H.F., 17, 53 Harriott, P., 238, 250 Harris, P.S., 154, 199 Harrison, D., 404, 407, 420, 429 Hartman, J., 120 Hayward, D.A., 26, 53 Hawley, M.C.. 293, 329 Hegedus, L.L., 190, 199, 314, 329 Hermana, E., 50, 53 Herskowitz, M., 241, 250, 437, 442, 448 Hicks, J.S., 108, 150 Hicks, R. E., 293, 329 Hill, T.L., 36, 53 Hill, W.J., 77, 90

Jackson, R., 100, 150, 266, 268, 280 Jacob, S.M., 8 1, 9 1 Jarvi, G.A., 196, 199 Johanson, L.N., 67, 70 John, T.M., 294, 329 Johnson, O., 31, 53 Johnston, H. F., 414, 429 Jones, H., 217, 233 Juvekar, V.A., 237, 250 Kabel, R.L., 67, 70 Kaguel, S., 492, 508 Kallenbach, R., 488, 508 Kaiil, J., 492, 507 Kane, P.F., 35, 53 Kashchiev, D., 210, 221, 233 Kato, K., 415, 419, 420, 429 Katz, S., 449, 479 Kehoe, J.P.G., 112, 150, 195, 199 Kehoe, P.W.K., 414, 429 Kernball, C., 63, 70 Khang, S. J., 155, 199 King, C.J., 100, 150 Kittrell, J.R., 67, 70, 77, 91 Klingman, K.J., 186, 199 Knözinger, H., 26, 53 Komiyama M., 17, 53 Koppel. L.B., 83, 90, 461, 479 Kovarik, F.S., 474, 479 Kubicek, M., 312, 329 Kulkarani, B.D., 493, 494, 507 Kulkarini, A.A., 110, 150 Kunii, D., 404, 406, 407, 410, 415, 427, 429, 492, 494, 495, 507

LaNauze, R.D., 404, 429 Lapidus, L., 77, 91, 263, 280, 302, 329 Larrabee, G.B., 35, 53 Lee, J.C.M., 266. 270. 272, 280 Lee, J.W., 112, 150, 173, 174, 199,493 Lee, K.Y., 457, 479 Leva, M., 406, 429, 448 Levec, J., 437, 442, 448 Levenspiel, O., 74, 76, 91, 155, 199, 286, 329, 404, 406, 407, 410, 423, 427, 429, 458, 479, 499, 507 Levins D. M., 238, 250 Levinter, M. E., 154, 199 Li, C. H., 497, 507 Lifshitz, I.M., 211, 233 Lightfoot, E.N., 98, 150, 286, 329, 484, 490, 507 I Linsen, B. G., 13, 53 Liu, S. L., 146, 150, 298, 329 Liu, Y.A., 77, 91 Logeais, B.A., 309, 329 Luss, D., 77, 90, 91, 109, 148, 150, 263, 266, 270, 272, 274, 280, 323, 329, 468, 479 Luther, H.A., 301, 329 Marek, M., 294, 329 Marshall, W.R., 500, 501, 506, 507 Masamune, S., 167, 199, 490, 507 Mason, E. A., 98, 150 Massimila, L., 414, 429 Maxted, E. B., 152, 199 McCune, R.C., 202, 233 McGreavy, C., 489, 507 McRae, E.G., 35, 53 Mears, D.E., 65, 70, 74, 91, 139, 150 Mehta, D., 293, 329 Mehta, P.S., 468, 479 Melo, F., 50, 53 Merrill, R. P., 13, 15, 17, 53 Messmer, J. H., 490, 508 Mezaki, R. A., 67, 70 Michelsen, M.A., 500, 507 Mihail, R., 302, 329 Mikus, O., 314, 329 Miller, D.J., 26, 53 Millman, M.C., 449, 479 Mills, G.A., 153, 199 Mills, P.L., 441, 442, 448 Mingle, J.O., 148, 150

Minhas, S., 304, 329 Misic, D., 237, 250 Mkhlin, S.G., 303, 329 Moo-Young, M. B., 237, 249, 250 Moore, W.J., 56, 70 Morbidelli, M., 142, 150, 295, 329 Morgan, C.Z., 253, 281 Mori, S., 419, 429 Moss, R.L., 13, 53 Murray, J.D., 419, 420, 429 Nace, D.M., 81, 91, 197, 258, 281 New, M., 489, 507 Nguen, K., 314, 329 Niiyana, H., 433, 448 Nomden, J.F., 413, 429

Oblad, A.G., 153, 199 Ogunye, A.F., 461, 479 Olbrich, W.E., 500, 502, 507 Othmer, D.F., 402, 429 Penabhan, L., 266, 280 Panchenkov, G.M., 154, 199 Pansing, W.F., 402, 429 Partridge, B.A., 407, 412, 429 Pasquou, I., 105, 150 Pelossof, A.A., 242, 250 Pereira, C.J., 243, 250 Peters, M.H., 415, 418, 419, 429 Petersen, E.E., 74, 91, 117, 140, 147, 150, 166, 190, 199 Plank, C.J., 197, 199 Pliskin, W.A., 63, 70 Potter, O.E., 415, 418, 429, 500, 502, 507 Prater, C.D., 74, 91, 105, 139, 150 Prausnitz, J.M., 490, 507 Price, T.H., 290, 329, 354, 400 Pulverrnacher, B., 202, 218, 219, 233 Pyle, D.L., 412, 429 Rathor, M.N., 443, 448 Ray, W.H., 449, 458, 461, 479 Raymond, L.R., 266, 288 Reh, L., 414, 429 Reid, R.C., 490, 507 Reilly, M.H., 312, 329 Rester, S., 103, 150 Richardson, J.F., 404, 406, 429

Richardson, J.T., 230, 233 Richarz, W., 489, 507 Roberts, G.W., 120, 150, 277, 281 Roberts, S.M., 452, 479 Rowe, P.N., 405, 407, 412, 420, 429 Ruckenstein, E., 122, 150, 202, 212, 218, 219, 221, 222, 233, 374, 376, 378, 400 Sada, E., 260, 281 Sams, W.N., 468, 479 Satterfield, C.N., 4, 6, 7, 8, 13, 25, 27, 40, 53, 120, 150, 239, 241, 242, 250, 437, 448, 488, 507 Schechter, R.S., 449, 479 Schmidt, L.D., 230, 233 Schmitz, R.A., 312, 329 Schuit, G.C.A., 4, 29, 53, 60, 70 Sehr, R.A., 490, 508 Servida, A., 142, 150 Shah, Y.T., 437, 443, 448 Sharma, M.M., 237, 250 Shelef, M., 26, 53 Sherwood, T.K., 238,242,249,250,490,507 Shulman, R.A., 64, 67, 70, 77, 91 Sicardi, S., 441, 443, 448 Siddiqui, M.A., 489, 507 Sinfelt, J.H,, 21, 53, 64, 67, 70, 77, 91 Sirazetdinov, T.K., 461, 479 Sit, S.P., 420, 429 Slyozov, V.V., 211, 233 Smith, J.M., 44, 45, 46, 53, 148, 150, 167, 189, 199, 236, 237, 238, 239, 241, 242, 250, 312, 329, 432, 433, 434, 437, 442, 448, 487, 489, 490, 494, 498, 507 Smolitskiy, K.C., 303, 329 Somarjai, G.A., 208, 210, 211, 233 Specchia, V., 441, 443, 448 Spenadel, L., 23, 53 Sprys, J.W., 202, 233 Squires, A. M., 413, 429 Stewart, W.E., 98, 150, 189, 286, 303, 329, 484, 487, 490, 507 Sweeney, T.L., 415, 418, 419, 429 Sylvester, N.D., 239, 250 Szekely, J., 449, 479 Szepe, S., 155, 199, 458, 479 Tamaru, J., 29, 53 Tan, C.S., 242, 250

Tanaka, K., 29, 53 Tanatarov, M.A., 154, 199 Teller, E., 35, 53 Therien, N., 459, 479 Thiele, E.W., 97, 150 Thomas, J.M., 23, 53 Thomas, W.J., 23, 53 Toarney, R.D., 414, 429 Trapnell, B.M.W., 26, 53 Tsotsis, T.T., 148, 150 Turkevich, J.F., 63, 70 Turner, D.H., 413, 429 Turner, J.C.R., 449, 479 Upadhay, S.N., 491, 507 Van Eeek, M.W., 239, 250 Van Heerden, C., 309, 329 Vannice, M.A., 65, 70 Van Welsenaere, R.J., 294, 329 Varghese, P., 194, 199 Varma, A., 128, 150, 243, 250, 295, 329 Villadsen, J.V., 303, 329 Vincent, R.P., 13, 15, 53 Voge, H.H., 7, 53, 253, 281 Voltz, S.E., 81, 91 Voorhies, A., 15 1, 155, 199, 472, 479 Votruba, J., 297, 314, 329 Waite, R.J., 154, 199 Wakao, N., 189, 487, 492, 495, 500, 508 Walen, N., 293, 329 Walker, P.L., 147, 150 Wang, J.B., 128, 150, 243, 250 Wanke, S. E., 202, 233 Ward, J.W., 32, 53

Wason, W.R., 15, 53 Watson, K.M., 60, 70 Weatherbee, C.D., 196, 199 Weekman, V.W., Jr., 73, 81, 91, 180, 199, 258, 281, 422, 429, 470, 479 Wehner, J. F., 288, 329 Wei, J,, 194, 199, 255, 256, 261, 278, 281 Weisz, P.B., 7, 52, 74, 91, 105, 108, 139, 150, 181, 198, 199, 403, 429 Weller, S.W., 64, 70 Wen, C.Y., 168, 199, 260,281,413,415,419, 420, 429, 499, 507 Wetterath, W.A., 492, 507 Wheeler, A., 42, 53, 164, 165, 199 Whitaker, S., 49 1, 508 White, D., 305, 329 Wicke, E., 488, 508 Wilhelm, R.H., 288, 329 Wilkes, J.O., 301, 329 Williams, R.J.J., 100, 150 Winquist, B.H.C., 27, 52 Wolf, E.E., 194, 199 Woodside, W., 490, 508 Wynblatt, P., 202, 212, 215, 217, 218, 222, 233 Yacono, C., 420, 429 Yagi, S., 495, 500, 508 Yang, K.H., 67, 70, 77, 91 Yates, D.J.C., 21, 53 Yerushalmi, J., 413, 429 Yortsos, Y.C., 148, 150 Yoshida, K., 415, 429 Zaleski, T., 335, 400 Zenz, F.A., 402, 429

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