Steady Incompressible Flow In Pressure Conduits Chapter - 8 4n5i3a

Chapter 8 Steady Incompressible Flow in Pressure Conduits

Energy Equation for Real Incompressible Fluid under steady state •

𝑧1 +

𝑝1 𝛾

+

𝑉12 2𝑔

= 𝑧2 +

𝑝2 𝛾

+

𝑉22 2𝑔

+ ℎ𝐿 (4.9)

• Isothermal conditions are assumed to eliminate thermodynamic effects • Energy Equation for ideal Incompressible Fluid under steady state •

𝑧1 +

𝑝1 𝛾

+

𝑉12 2𝑔

= 𝑧2 +

• Bernoulli’s Theorem

𝑝2 𝛾

+

𝑉22 2𝑔

(4.10)

8.1 Laminar and Turbulent Flow • For laminar flow n = 1, hL varies as V • For turbulent flow, n = 1.75 for very smooth pipes and n = 2.00 for rough pipes • If the velocity is reduced from a high value, the line BC will not be retraced. Instead, the points lie along curve CA • Point B is known as the higher critical point, and A as lower critical point. • However, velocity is not only the factor which discriminate laminar and turbulent flow. The criterion is Reynolds number, • 𝑅𝑁 =

𝐷𝑉𝜌 𝜇

=

𝐷𝑉 ν

• RN is dimensionless

(8.1)

8.2 Critical Reynolds Number • Critical Reynolds Number has no definite value • However, for normal cases of flow in straight pipes of uniform diameter and usual roughness, the critical value may be taken as RN = 2,000 • For water at 20oC the kinematic viscosity is 1.00 x 10-6 m2/s, and for this case the critical Reynolds number is obtained when • 𝐷𝑉𝑐𝑟𝑖𝑡 = 𝑅𝑁 ν = 2,000 𝑋 10−6 = 0.002 m2/s • Thus, for a pipe 25 mm in diameter, 0.002

• 𝑉𝑐𝑟𝑖𝑡 = = 0.08 m/s 0.025 • Or if the velocity were 0.8 m/s the diameter would be only 2.5 mm. Velocities or pipe diameters as small as these are not often encountered with water flowing in practical engineering. Thus often flow is turbulent in nature except for very viscous oils

Example 8.1 • An oil (s = 0.85, ν = 1.8 x 10-5 m2/s) flows in a 10-cm-diameter pipe at 0.50 L/s. Is the flow laminar or turbulent? • Solution: • •

𝑄 0.0005 𝑉= = = 0.0637 m/s 𝐴 𝜋×0.052 𝐷𝑉 0.10 𝑋 0.067 𝑅𝑁 = = = 354 −5 ν 1.8 𝑋10

• Since 𝑅𝑁 < 2,000, the flow is laminar.

8.3 Hydraulic Radius • For conduits having noncircular cross sections, some value other than the diameter must be used for the linear dimension in the Reynolds number. Such a characteristics is the hydraulic radius, defined as 𝐴

𝑅ℎ = (8.2) 𝑃 • Where A is the cross-sectional area of the flowing fluid, and P is the wetted perimeter, that portion of the perimeter of the cross section where there is between fluid and solid boundary. For circular pipe flowing full, 𝜋𝑟2 2𝜋𝑟

𝑟

𝐷

• 𝑅ℎ = = , 𝑜𝑟 𝑜𝑟 𝐷 = 4𝑅ℎ 2 4 • Thus Rh is not the radius of the pipe, and hence the term “radius” is misleading. • 𝑅ℎ is a convenient means for expressing the shape as well as the size of a conduit, since for the same cross-sectional area the value of 𝑅ℎ will vary with the shape.

• For noncircular conduits substitute 4Rh for D in Eq. (8.1) to evaluate Reynolds number

Friction Coefficient, f • For Laminar flow 𝑓 = 64/𝑅𝑁 (8.23) • For turbulent flow f is not fully understood yet. All is available from experimentation.

8.11 Pipe Roughness (e) • The degree of unevenness on the internal pipe surface is called pipe roughness/absolute roughness. • e/D is called relative roughness, where D is the pipe diameter.

Nominal thickness of the viscous sublayer (δl) and Absolute roughness (e)

f-values for different types of flows

8.12 Chart for Friction Factor

The chart shows four zones: 1-Laminar flow zone, 2-Critical range zone, not certain about flow type, 3-Transition zone, where value depends upon RN and e/D both, 4-Complete turbulence zone, f only depends upon e/D. For new smooth pipes, accuracy is ±5% and that new rough pipes ±10%. f is not known before the time. It usually ranges from 0.01 to 0.07.

e/D-Values

e-Values

Example 8.4

8.14 Empirical Equation For Pipe Flow • The presentation of friction loss in pipes given in sections 8.1 to 8.12 incorporates the best knowledge available on this subject, as far as application to Newtonian fluids (Sec. 1.11). • Some empirical design formulas have been developed, applicable only to specific fluids and conditions but very convenient in a certain range. • For example Hazen Williams formula, applicable only to the flow of water in pipes larger than 5 cm and at velocities less than 3 m/s, but widely used in the waterworks industry. This formula is given in the form .63 0.54 0 • English units: 𝑉 = 0.85𝐶𝐻𝑊 𝑅ℎ 𝑆 • Where Rh (m) is the hydraulic radius, and S = hL/L, the energy gradient. The advantage of this formula over the standard pipe-friction formula is that the roughness coefficient CHW is not a function of the RN and trial solutions are therefore eliminated. • Another empirical formula is the Manning formula, which is 1 𝑛

2/3

• 𝑉 = 𝑅ℎ . 𝑆 1Τ2

8.15 Minor Losses In Turbulent Flow • Losses due to the local disturbances of the flow in conduits such as changes in cross section, projecting gaskets, elbows, valves, and similar items are called minor losses. In the case of a very long pipe or channel, these are usually insignificant in comparison with the fluid friction in the length considered. But if the length is short, these so called minor losses may be major losses. • Minor losses are expressed as kV2/2g, where k must be determined for each case.

𝑝1 𝑉12 𝑝2 𝑉22 𝑧1 + + = 𝑧2 + + + ℎ𝑒′ + ℎ𝐿 𝛾 2𝑔 𝛾 2𝑔

𝑝1 𝑉12 𝑝3 𝑉32 𝑧1 + + = 𝑧3 + + + ℎ𝐿 + ℎ𝑛′ 𝛾 2𝑔 𝛾 2𝑔

•

𝐴3 𝑉3 = 𝐴2 𝑉2

•

𝑉3 =

•

𝑉3 = 𝐴2 𝑉2

•

𝑉32 = (𝐴2 )2 𝑉22

•

𝑉32 2𝑔

•

𝑉32 2𝑔

•

𝑉32 2𝑔

•

𝑉32 2𝑔

𝐴2 𝑉 𝐴3 2 𝐴

3

𝐴

3

=

𝐴2 2 𝑉22 (𝐴 ) 2𝑔 3

=

𝜋Τ4.𝐷22 2 𝑉22 (𝜋Τ4.𝐷2) 2𝑔 3

=

𝐷22 2 𝑉22 (𝐷2) 2𝑔 3

=

𝐷2 4 𝑉22 (𝐷 ) 2𝑔 3

𝑝1 𝑉12 𝑝2 𝑉22 𝑧1 + + = 𝑧2 + + + ℎ𝐿 𝛾 2𝑔 𝛾 2𝑔

𝑝0 𝑉02 𝑝3 𝑉32 ′ 𝑧0 + + − ℎ𝑒 − ℎ𝐿1 + ℎ𝑝 − ℎ𝐿2 = 𝑧3 + + 𝛾 2𝑔 𝛾 2𝑔

2 𝑝0 𝑉02 𝑝 𝑉 1 1 𝑧0 + + − ℎ𝑒′ − ℎ𝐿1 = 𝑧1 + + 𝛾 2𝑔 𝛾 2𝑔

8.25 Pipes In Series • ℎ = ℎ𝐿1 + ℎ𝐿2 + ℎ𝐿3 + ⋯ + ℎ𝐿𝑛 • 𝑄1 = 𝑄2 = 𝑄3 = … = 𝑄𝑛

8.26 Pipes In Parallel • 𝑄 = 𝑄1 + 𝑄2 + 𝑄3 • ℎ = ℎ𝐿1 = ℎ𝐿2 = ℎ𝐿3

(8.60) (8.61)

Head loss equation in Q form • ℎ𝐿 =

𝐿 𝑉2 𝑓 𝐷 2𝑔

=

𝐿 𝑓 𝐷

𝑄2 1 𝐴2 2𝑔

=

𝐿 𝑓 𝐷

𝑄2

1 π𝐷2 2 2𝑔

1 12.1

𝐿 2 𝑓 5𝑄 𝐷

=

4

• In Book 2: 𝑓 = 4𝑓

• ℎ𝐿 =

1 3

𝐿 2 𝑓 5𝑄 𝐷

• • • •

H = hL1 + hL2 H = hL1 + hL3 hL2 = hL3 Q1 = Q2 + Q3

• • • •

H = hL1 + hL3 H = hL2 + hL3 hL1 = hL2 Q1 + Q2 = Q3

For long pipe (𝐿 > 1000𝐷) minor losses may be neglected

𝑍1 +

𝑝1 𝛾

+

𝑉12 2𝑔

− ℎ𝐿𝐴 − ℎ𝐿𝐶 = 𝑍2 +

𝑝2 𝑉22 + 2𝑔 𝛾

𝑍1 +

𝑝1 𝛾

+

𝑉12 2𝑔

− ℎ𝐿𝐴 = 𝑍𝑝 +

𝑝𝑝 𝛾

𝑉𝑝2

+ 2𝑔

Assignment # 6 • Book 1: Examples 8.1, 8.4 to 8.9 • Book 2: Examples 14.1, 14.3 to 14.10