Fluid Flow 3

Fluid Flow 3

1. Flow in Pipes

   • Darcy-Weisbach
     • h = f(L/D)(V^2/2g)
       • h = head loss (m) f = friction factor L = pipe length (m) D = diameter of pipe (m) V = flow veloctiy (ms-1) g = gravitational field strength (kgms-2)
     • Gives the loss of pressure head due to friction in pipe
     • Friction factor (f) is a function of Reynold's number and relative roughness
       • How it is calculated depends on the Reynold's number of the flow
   • Velocity Profiles
     • Laminar Flow
       • Hagen-Poiseuille Law
         • Justifies assumptions: - Newtonian Fluid Behaviour - No slip condition
         • U(r) = dp/L * 1/4mu * (R^2 - r^2)
           • For laminar flow Where U(r) = velocity at radius r  dp = pressure difference 
         • Q = dp/L * pi*D^2/128*mu
           • For discharge in the pipe: Q = volumetric flowrate dp = pressure dropL = pipe lengthd = diameter mu = dynamic viscosity 
         • h(f) = 32*mu*u*L / rho*g*D^2
           • h(f) = frictional head loss  mu = dynamic viscosity  u = velocity  L = pipe length  rho = density  g = gravitational field strength D = diameter
           • f = 64/Re
             • We can rewrite this to Darcy-Weisbach form and show that f = 64(mu/rho*u*D) 
       • Re < 2300
     • Turbulent Flow
       • Re > 4000
       • Blasius Formula
         • f = 0.316*Re^-1/4
         • Only valid for SMOOTH pipes
         • Empirical formula
       • Hydraulically Smooth
         • Have no laminar sub-layer
         • 1/f^0.5 = 2log10(Re*f^0.5 / 2.51)
           • Must be solved iteratively
       • Hydraulically Rough
         • 1/f^0.5 = 2log10(3.7D/Ks)
       • Turbulent flows have a region of laminar flow next to boundary layer
         • It has thickness delta and height k
       • Colebrook-White Equation
         • Calculates f for use in Darcy-Weisbach equation
           • 1/f^0.5 = -2log10(Ks/3.7D + 2.51/Re*f^0.5)