06,Viscous Flow in Ducts,06 - 2,Objectives,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.,06 - 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.,06 - 4,Laminar 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 Flow,Turbulent Flow Flow,06 - 5,Laminar and Turbulent Flows: Reynolds Number,Definition,Physical Significance,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,06 - 6,The Entrance Region,Entrance 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.,06 - 7,Entrance Length of Circular Pipes,06 - 8,Head Loss,Fx_visc = twAsurf = tw(2pRL),SFx = DppR2 + prgR2Dz 2ptwRL,Fx_other = 0,= prgR2Dz,= rg(pR2L) sinf, Divided by prgR2,(1),(2),06 - 9,Head Loss Friction Factor,Head Loss,To predict head loss hf, we need to be able to calculate w.,Darcy-Weisbach Equation,(e = wall roughness height),(3),(4),Friction Factor f,e is significant for turbulent flows, but not for laminar flows.,Pressure Drop,06 - 10,Friction Factor: Fully Developed Lamina Flow,Velocity Profile,Wall Shear Stress,Friction Factor f,Head Loss hf,06 - 11,Friction Factor : Turbulent Flow,Background,Need EES, Excel, Matlab or other software to solve for f.,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.,e has no effect on laminar flow,(e = wall roughness height),Since analytical solutions for turbulent flows do not exist, all available formulas are all from interpolation of experimental data.,Colebrook Formula,06 - 12,Turbulent Flow: Moody Chart,Moody Chart plot of the Colebrook formula and the lamina flow equation.,Probably the most famous and useful figure in fluid mechanics with 15% accuracy.,06 - 13,Recommended Roughness, e, for Commercial Ducts*,* From White 6th ed. Table 6.1,06 - 14,Example 6.1 (White 6th ed. Ex. 6.7),Oil, with = 900 kg/m3 and = 0.00001 m2/s, flows at 0.2 m3/s through 500 m of 200 mm-diameter cast iron pipe. Determine (a) the head loss and (b) the pressure drop if the pipe slopes down at 10 in the flow direction.,Problem,V,Red,e/d,Solution,Use the Colebrook formula (or Moody Chart).,Turbulent flow. Use Colebrook formula.,06 - 15,Example 6.1: f, hf and Dp,hf,hf = 117 m,Dp = 266 kPa,Dp,f,f = 0.0227 (use Excel),06 - 16,Four Types of Pipe Flow Problems,Given d, L, and V or Q, , , and g, computer the head loss or Dp (head loss problem, Red is known).,Given d, L, hf, , , and g, compute the velocity V or flow rate Q (flow rate problem, Red is unknown).,Given Q, L, hf, , , and g, compute the diameter d of the pipe (sizing problem, Red is unknown).,Given Q, d, L, hf, , , and g, computer the pipe length L (Red is known).,In design and analysis of piping systems, 4 problem types are encountered.,Types 1 and 4 are well suited to the Moody chart.,Types 2 and 3 are common engineering design problems, i.e., selection of pipe diameters to minimize construction and pumping costs. However, iterative approach required since either V or D is unknown, and both are in the Reynolds number.,With the help of suitable computer software (EES, Matlab, Mathcad, Mathematica, etc.), the iteration may not be needed.,06 - 17,Type 2: Given d, L, hf, r, m & g, Compute V or Q,Iteration,Explicit Formula,Guess an initial f, based on e/d,where,06 - 18,Example 6.2 (White 6th ed. Ex. 6.9): Direct Solution,Oil, with = 950 kg/m3 and = 0.00002 m2/s, flows through a 30-cm-diameter pipe 100 m long with a head loss of 8 m. The roughness ratio e/d = 0.0002. Find the average velocity and flow rate.,