CE 150 Fluid Mechanics

1. CE 150Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico CE 150

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

3. 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

4. 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

5. Characteristics of Pipe Flow • Laminar vs. turbulent • laminar: Re  2100 • transitional: 2100  Re  4000 • turbulent: Re  4000 CE 150

6. 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

7. 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

8. 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

9. Fully Developed Laminar Flow • Velocity profile • Volume flow rate CE 150

10. Fully Developed Laminar Flow • Pressure drop • Friction factor CE 150

11. Turbulent Flow • Occurs Re  4000 • Velocity at given location: CE 150

12. 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

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

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

15. Turbulent Shear Stress • Shear stress distribution: • Mean velocity distribution: CE 150

16. 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

17. 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

18. 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

19. Summary of Friction Factors for Pipe Flow • Laminar flow • Turbulent flow in smooth pipes • Turbulent flow in rough pipes CE 150

20. The Moody Chart CE 150

21. 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

22. 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

23. 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

24. 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

25. Common Types of Pipe Flow Problems CE 150

26. 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

27. Pipe Flowrate Measurement • Orifice meter • Venturi meter • Rotameter • Turbine and paddlewheel • Nutating disk meter • Bellows meter CE 150