1. California State University, Chico
CE 150 Fluid Mechanics
G.A. Kallio
Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology
California State University, Chico
CE 150

2. Reading: Munson, et al., Chapter 8
Viscous Flow in Pipes
Reading: Munson, et al., Chapter 8
CE 150

3. Introduction Pipe Flow – important application
Pipe: circular cross section
Duct: noncircular cross section
Piping system may contain
pipes of various diameters
valves & fittings
nozzles (pipe contraction)
diffusers (pipe expansion)
pumps, turbines, compressors, fans, blowers
heat exchangers, mixing chambers
reservoirs
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4. Introduction Typical assumptions
pipe is completely filled with a single fluid (gas or liquid)
phase change possible but course focus is single phase
pipe flow is primarily driven by a pressure difference rather than gravity
steady, incompressible flow
uniform (average) flow at all cross sections
extended Bernoulli equation (EBE) is applicable
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5. Characteristics of Pipe Flow
Laminar vs. turbulent
laminar: Re  2100
transitional:  Re  4000
turbulent: Re  4000
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6. Characteristics of Pipe Flow
Entrance region flow - typically between D ; depends on Re:
Fully developed flow - occurs beyond entrance region; velocity profile is independent of x
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7. Pipe Flow Problems Laminar flow Turbulent flow
Applications: blood flow, bearing lubrication, compact heat exchangers, solar collectors, MEMS fluid devices
Fully-developed flow: exact analysis possible
Entrance region flow: analysis complex; requires numerical methods
Turbulent flow
Applications: nearly all flows
Defies analysis
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8. Pressure and Viscous Forces in Pipe Flow
Entrance region
Flow is accelerating at centerline, or pressure forces > viscous (shear) forces
Flow is decelerating at wall, or viscous forces > pressure forces
Fully-developed region
Non-accelerating flow
Pressure forces equal viscous forces
Work done by pressure forces equals viscous dissipation of energy (into heat)
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9. Fully Developed Laminar Flow
Velocity profile
Volume flow rate
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10. Fully Developed Laminar Flow
Pressure drop
Friction factor
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11. Turbulent Flow
Occurs Re  4000
Velocity at given location:
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12. Characteristics of Turbulent Flow
Laminar flow: microscopic (molecular scale) randomness
Turbulent flow: macroscopic randomness (3-D “eddies”)
Turbulence
enhances mixing
enhances heat & mass transfer
increases pressure drop in pipes
increases drag on airfoils
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13. Characteristics of Turbulent Flow
Velocity fluctuation averages:
Turbulence intensity:
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14. Turbulent Shear Stress
Turbulent eddies enhance momentum transfer and shear stress:
Mixing length model:
Eddy viscosity:
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15. Turbulent Shear Stress
Shear stress distribution:
Mean velocity distribution:
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16. Turbulent Pipe Flow Velocity Profile
For fully-developed flow, the mean velocity profile has been obtained by dimensional analysis and experiments
for accurate analysis, equations are available for each layer
for approximate analysis, the power-law velocity profile is often used:
where n ranges between 6-10 (see Figure 8.17); n = 7 corresponds to many typical turbulent flows
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17. Dimensional Analysis of Pipe Flow
Pressure drop
where  = average roughness height of pipe wall; has no effect in laminar flow; can have significant effect in turbulent flow if it protrudes beyond viscous sublayer (see Table 8.1)
Typical pi terms
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18. Dimensional Analysis of Pipe Flow
Pressure drop is known to be linearly proportional to pipe length, thus:
Recall friction factor:
Pressure drop in terms of f :
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19. Summary of Friction Factors for Pipe Flow
Laminar flow
Turbulent flow in smooth pipes
Turbulent flow in rough pipes
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20. The Moody Chart
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21. Friction Head Loss in Pipe Flow
For a constant-diameter horizontal pipe, the extended Bernoulli equation yields
Head loss due to friction:
If elevations changes are present:
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22. Minor Head Losses in Pipe Flow
Minor losses are those due to pipe bends, fittings, valves, contractions, expansions, etc. (Note: they are not always “minor” when compared to friction losses)
Minor head losses are expressed in terms of a dimensionless loss coefficient, KL:
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23. Minor Head Losses in Pipe Flow
The loss coefficient strongly depends on the component geometry
Entrance: Figures 8.22, 8.24
Exits: Figure 8.25
Sudden contraction: Figure 8.26
Sudden expansion: Figure 8.27
Conical diffuser: Figure 8.29
90º bends: Figures 8.30, 8.31
Pipe fittings: Table 8.2
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24. Noncircular Conduits Friction factors for are usually expressed as
where Reh is the Reynolds number based on the hydraulic diameter (Dh):
Friction factor constants (C) are given in Figure 8.3 for annuli and rectangular cross sections
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25. Common Types of Pipe Flow Problems
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26. Multiple Pipe Systems Analogy to electrical circuits:
Electrical circuits:  e = iR
Pipe flow:  p = Q2 R( f,KL)
Series path: Q = constant,  p’s are additive
Parallel path:  p = constant, Q’s are additive
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27. Pipe Flowrate Measurement
Orifice meter
Venturi meter
Rotameter
Turbine and paddlewheel
Nutating disk meter
Bellows meter
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