Lecture 30

Review of Fluid Flow and Heat Transfer

 

Fully Developed Flow

  • In the above figure the fluid velocity profile is shown at different sections in a pipe.
  • It has a flat profile at the entrance and develops into a fully developed profile after some distance, called entrance length.
  • Fully developed flow implies that the velocity profile does not change along length.
  • Detailed analysis indicates that the non-dimensional entrance length (Lh /D) is ~ 0.06 Re.
  • This is small in turbulent flow (Lh /D ~ 6-10). The subscript h is used to denote hydrodynamic entrance length.
  • Since velocity profile is same, it implies that wall shear stress is same.
 
Laminar Flow in Pipes
  • For  axi-symmetric, incompressible, steady flow V = V(r,z).
  • If the flow is also fully developed, we can write the following:

    • Continuity Equation


          

  • Vr and Vz represent the velocities in r and z directiions respectively.
  • Since flow is fully developed, = 0

  • We can calculate the following:
    • rVr is independent of r.
    • Since Vr at r = R  is 0, Vr is 0 everywhere.
    • Hence there is only Vz , and Vz is only a function of r .

    r -  Momentum Equation

       

  • Since Vr is zero everywhere, we can write .
  • This implies that p is only a function of z