CE 150 Fluid Mechanics

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CE 150Fluid Mechanics

• G.A. Kallio
• Dept. of Mechanical Engineering, Mechatronic
  Engineering Manufacturing Technology
• California State University, Chico

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Viscous Flow in Pipes

• Reading Munson, et al., Chapter 8

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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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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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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

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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

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Pressure and Viscous Forces in Pipe Flow

• Entrance region
• Flow is accelerating at centerline, or pressure
  forces gt viscous (shear) forces
• Flow is decelerating at wall, or viscous forces gt
  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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Fully Developed Laminar Flow

• Velocity profile
• Volume flow rate

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Fully Developed Laminar Flow

• Pressure drop
• Friction factor

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Turbulent Flow

• Occurs Re ? 4000
• Velocity at given location

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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

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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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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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Summary of Friction Factors for Pipe Flow

• Laminar flow
• Turbulent flow in smooth pipes
• Turbulent flow in rough pipes

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The Moody Chart
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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

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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

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Common Types of Pipe Flow Problems
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Multiple Pipe Systems

• Analogy to electrical circuits
• Electrical circuits ? e iR
• Pipe flow ? p Q2 R( f,KL)
• Series path Q constant, ? ps are additive
• Parallel path ? p constant, Qs are additive

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Pipe Flowrate Measurement

• Orifice meter
• Venturi meter
• Rotameter
• Turbine and paddlewheel
• Nutating disk meter
• Bellows meter