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**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

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**Reading: Munson, et al., Chapter 8**

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

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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 CE 150

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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 CE 150

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**Characteristics of Pipe Flow**

Laminar vs. turbulent laminar: Re 2100 transitional: Re 4000 turbulent: Re 4000 CE 150

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**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 CE 150

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**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 CE 150

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**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

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**Fully Developed Laminar Flow**

Velocity profile Volume flow rate CE 150

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**Fully Developed Laminar Flow**

Pressure drop Friction factor CE 150

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Turbulent Flow Occurs Re 4000 Velocity at given location: CE 150

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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 CE 150

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**Characteristics of Turbulent Flow**

Velocity fluctuation averages: Turbulence intensity: CE 150

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**Turbulent Shear Stress**

Turbulent eddies enhance momentum transfer and shear stress: Mixing length model: Eddy viscosity: CE 150

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**Turbulent Shear Stress**

Shear stress distribution: Mean velocity distribution: CE 150

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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

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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 CE 150

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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 : CE 150

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**Summary of Friction Factors for Pipe Flow**

Laminar flow Turbulent flow in smooth pipes Turbulent flow in rough pipes CE 150

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The Moody Chart CE 150

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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: CE 150

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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

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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 CE 150

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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

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**Common Types of Pipe Flow Problems**

CE 150

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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, p’s are additive Parallel path: p = constant, Q’s are additive CE 150

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**Pipe Flowrate Measurement**

Orifice meter Venturi meter Rotameter Turbine and paddlewheel Nutating disk meter Bellows meter CE 150

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