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

Reading: Munson, et al., Chapter 8 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 steady, incompressible flow 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 CE 150

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 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

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 CE 150

Characteristics of Turbulent Flow Velocity fluctuation averages: Turbulence intensity: CE 150

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

Turbulent Shear Stress Shear stress distribution: Mean velocity distribution: CE 150

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 CE 150

The Moody Chart CE 150

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

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 CE 150

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

Common Types of Pipe Flow Problems 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 Turbine and paddlewheel Nutating disk meter Bellows meter CE 150