CE 150 Fluid Mechanics

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CE 150 Fluid Mechanics. G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico. Viscous Flow in Pipes. Reading: Munson, et al., Chapter 8. Introduction. Pipe Flow – important application

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### CE 150Fluid Mechanics

G.A. Kallio

Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology

California State University, Chico

CE 150

### Viscous Flow in Pipes

Reading: Munson, et al., Chapter 8

CE 150

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

CE 150

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
• uniform (average) flow at all cross sections
• extended Bernoulli equation (EBE) is applicable

CE 150

Characteristics of Pipe Flow
• Laminar vs. turbulent
• laminar: Re  2100
• transitional: 2100  Re  4000
• turbulent: Re  4000

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Characteristics of Pipe Flow
• Entrance region flow - typically between 20-120D ; depends on Re:
• Fully developed flow - occurs beyond entrance region; velocity profile is independent of x

CE 150

Pipe Flow Problems
• Laminar 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

CE 150

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)

CE 150

Fully Developed Laminar Flow
• Velocity profile
• Volume flow rate

CE 150

Fully Developed Laminar Flow
• Pressure drop
• Friction factor

CE 150

Turbulent Flow
• Occurs Re  4000
• Velocity at given location:

CE 150

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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Characteristics of Turbulent Flow
• Velocity fluctuation averages:
• Turbulence intensity:

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Turbulent Shear Stress
• Turbulent eddies enhance momentum transfer and shear stress:
• Mixing length model:
• Eddy viscosity:

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Turbulent Shear Stress
• Shear stress distribution:
• Mean velocity distribution:

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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

CE 150

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

CE 150

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 :

CE 150

Summary of Friction Factors for Pipe Flow
• Laminar flow
• Turbulent flow in smooth pipes
• Turbulent flow in rough pipes

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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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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:

CE 150

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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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

CE 150

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

CE 150

Pipe Flowrate Measurement
• Orifice meter
• Venturi meter
• Rotameter