CHAPTERSchool of Engineering Mechanical EngineeringENGR 320FLUID MECHANICS Tulong Zhu, All rights reserved. 06Viscous Flow in DuctsEsc06 - 2Objectives Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow. Calculate the major and minor losses associated with pipe flow in piping networks and determine the pumping power requirements.Esc06 - 3 Recall - because of the no-slip condition, the velocity is not uniform, and at the walls of a duct flow is zero.Incompressible Flow: Velocity and Mass Flow Rate We are often interested only in Vavg, which we usually call just V (drop the subscript for convenience). Keep in mind that the no-slip condition causes shear stress and friction along the pipe walls. For pipes of constant diameter and incompressible flow, V (average) stays the same since Q1 = Q2.VV21 For pipes with variable diameter, m is still the same due to conservation of mass, but V1 V2.d221V1d1m.m.V2Esc06 - 4Laminar and Turbulent Flows Can be steady or unsteady. Always unsteady. There are always random, swirling motions in a turbulent flow. Can be one-, two, three- dimensional. Always three-dimensional. However, it can be 1-D or 2-D in the mean. Has regular, predictable behavior. Has irregular or chaotic behavior. Cannot predict exactly there is some randomness associated with it. Analytical solution possible. No analytical solution exits so far! Its too complicated. Occurs at low Reynolds numbers. Occurs at high Reynolds numbers.Laminar FlowTurbulent Flow FlowEsc06 - 5Laminar and Turbulent Flows: Reynolds NumberDefinition For circular pipes/ducts,Physical Significance For circular pipes/ducts, These values are approximate, different books may give slightly different values. For a given application, these values depends upon Pipe roughness Vibrations Upstream fluctuations, disturbances (valves, elbows, etc. that may disturb the flow), .Critical Reynolds Number For circular pipes/ducts, the critical Reynolds number is the Reynolds number at which transition from lamina flow starts,Esc06 - 6The Entrance RegionEntrance Length A boundary layer is a layer of fluid in the immediate vicinity of a bounding surface, over which the velocity gradient is not equal to zero. Boundary Layer The profile develops downstream over several diameters called the entry length Le. Le/d is a function of Re. Consider a round pipe of diameter d. Esc06 - 7Entrance Length of Circular Pipes Laminar flow Turbulent flowEsc06 - 8Head Loss p1 = p2 + Dpp2fu(r) (r) wrr = RLz1z2x Incompressible mass conservation: Steady flow energy equation (hp = ht =0): Steady flow momentum equationFx_press = p1A p2A = Dp(pR2)Fx_visc = wAsurf = w(2pRL)SFx = DppR2 + pgR2Dz 2pwRLFx_other = 0= pgR2Dz= g(pR2L) sinf Fx_grav =gV sinf Divided by pgR2(1)(2)Esc06 - 9Head Loss Friction Factor From Eqs. (1) and (2),Head Loss To predict head loss hf, we need to be able to calculate w.Darcy- Weisbach Equation Through dimensional analysis, we can get a more general equation, the Darcy- Weisbach Equationf is the Darcy friction factor,( = wall roughness height) Alternative equation for f . From Eqs. (3) and (4), (3)(4)Friction Factor f is significant for turbulent flows, but not for laminar flows.Pressure Drop From Eqs. (1)Esc06 - 10Friction Factor: Fully Developed Lamina Flow For fully developed Poiseuille flow (slow viscous incompressible flow through a constant circular cross-section) in a round pipe of diameter d, radius R,Velocity ProfileWall Shear Stress From = du/drFriction Factor f Head Loss hf Esc06 - 11Friction Factor : Turbulent Flow Colebrook (1938) proposed the famous Colebrook Formula ( Probably the most famous and widely used among engineers): Background Need EES, Excel, Matlab or other software to solve for f. Approximations of the above formula also exist, for example, Haaland Formula The Colebrook formula is valid for both smooth and rough walls. The Haaland formula varies less than 2% from Colebrook. The Colebrook formula is accurate to 15% due to roughness size, experimental error, curve fitting of data, etc. has no effect on laminar flow( = wall roughness height) Since analytical solutions for turbulent flows do not exist, all available formulas are all from interpolation of experimental data.Colebrook FormulaEsc06 - 12Turbulent Flow: Moody ChartMoody Chart plot of the Colebrook formula and the lamina flow equation. Probably the most famous and useful figure in fluid mechanics with 15% accuracy.Esc06 - 13Recommended Roughness, , for Commercial Ducts*SteelSteel metal, new0.000160.0560 Stainless, new0.0000070.00250 Commercial, new0.000150.04630 Riveted0.013.070 Rusted0.0072.050 Iron Cast, new0.000850.2650 Wrought, new0.000150.04620 Galvanized, new0.00050.1540 Asphalted cast0.00040.1250 BrassDrawn, new0.0000070.00250 PlasticDrawn, tubing0.000005