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Fox, Pritchard, & McDonald

Introduction to Fluid Mechanics

Chapter 8

Internal Incompressible

Viscous Flow
http://he-cda.wiley.com/WileyCDA/HigherEdTitle.rdr?productCd=0471742996


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Main Topics

Entrance Region

Fully Developed Laminar FlowBetween Infinite Parallel Plates

Fully Developed Laminar Flow in a PipeTurbulent Velocity Profiles in

Fully Developed Pipe Flow

Energy Considerations in Pipe Flow

Calculation of Head Loss

Solution of Pipe Flow Problems

Flow Measurement


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Entrance Region


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Fully Developed Laminar Flow

Between Infinite Parallel PlatesBoth Plates Stationary


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Transformation of Coordinates


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Shear Stress Distribution

Volume Flow Rate


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Flow Rate as a Function of Pressure Drop

Average and Maximum Velocities




Between Infinite Parallel PlatesUpper Plate Moving with Constant Speed, U




in a PipeVelocity Distribution

Shear Stress Distribution




in a PipeVolume Flow Rate

Flow Rate as a Function of Pressure Drop




in a PipeAverage Velocity

Maximum Velocity



Turbulent Velocity Profiles in



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Energy Considerations in

Pipe FlowEnergy Equation


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Energy Considerations in

Pipe FlowHead Loss


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Major Losses: Friction Factor


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Laminar Friction Factor

Turbulent Friction Factor


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Minor Losses Examples: Inlets and Exits; Enlargements and

Contractions; Pipe Bends; Valves and Fittings


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Minor Loss: Loss Coefficient, K

Minor Loss: Equivalent Length, Le


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Pumps, Fans, and Blowers


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Noncircular Ducts

Example: Rectangular Duct


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Energy Equation


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Major Losses


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Minor Losses


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Single Path Find Dp for a given L, D, and Q

Use energy equation directly

Find L for a given Dp, D, and QUse energy equation directly


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Single Path (Continued) Find Q for a given Dp, L, and D

1. Manually iterate energy equation and friction factorformula to find V (orQ), or

2. Directly solve, simultaneously, energy equation andfriction factor formula using (for example) Excel

Find D for a given Dp, L, and Q1. Manually iterate energy equation and friction factorformula to find D, or

2. Directly solve, simultaneously, energy equation andfriction factor formula using (for example) Excel


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Multiple-Path Systems

Example:


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Multiple-Path Systems Solve each branch as for single path

Two additional rules1. The net flow out of any node (junction) is zero

2. Each node has a unique pressure head (HGL)

To complete solution of problem1. Manually iterate energy equation and friction factor

for each branch to satisfy all constraints, or

2. Directly solve, simultaneously, complete set ofequations using (for example) Excel


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Flow Measurement

Direct Methods Examples: Accumulation in a Container;

Positive Displacement Flowmeter

Restriction Flow Meters for Internal Flows Examples: Orifice Plate; Flow Nozzle; Venturi;

Laminar Flow Element


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Flow Measurement

Linear Flow Meters Examples: Float Meter (Rotameter); Turbine;

Vortex; Electromagnetic; Magnetic; Ultrasonic


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Flow Measurement

Traversing Methods Examples: Pitot (or Pitot Static) Tube; Laser

Doppler Anemometer

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