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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 150 fluid mechanics

CE 150Fluid Mechanics

G.A. Kallio

Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology

California State University, Chico

CE 150

viscous flow in pipes

Viscous Flow in Pipes

Reading: Munson, et al., Chapter 8

CE 150

introduction
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

introduction1
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

CE 150

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

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characteristics of pipe flow1
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

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pipe flow problems
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
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
Fully Developed Laminar Flow
  • Velocity profile
  • Volume flow rate

CE 150

fully developed laminar flow1
Fully Developed Laminar Flow
  • Pressure drop
  • Friction factor

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

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characteristics of turbulent flow
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 flow1
Characteristics of Turbulent Flow
  • Velocity fluctuation averages:
  • Turbulence intensity:

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

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turbulent shear stress1
Turbulent Shear Stress
  • Shear stress distribution:
  • Mean velocity distribution:

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turbulent pipe flow velocity profile
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
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 flow1
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
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
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 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 flow1
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

CE 150

noncircular conduits
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
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
Pipe Flowrate Measurement
  • Orifice meter
  • Venturi meter
  • Rotameter
  • Turbine and paddlewheel
  • Nutating disk meter
  • Bellows meter

CE 150