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Example 5-2: Kinetic Energy Correction Factor for Laminar Flow of a Newtonian Fluid. We will show later that the velocity profile for the laminar flow of a Newtonian fluid in fully developed flow in a circular tube is parabolic. Because the velocity is zero at the wall of the tube and maximum in the center, the equation for the profile is

This can be used to calculate the kinetic energy correction factor from Eq. (5-38) as follows. First we must calculate the average velocity, V, using which shows that the average velocity is simply one-half of the maximum (centerline) velocity. Thus, replacing V in Eq. (5-38) by Vmax/2 and then integrating the cube of the parabolic velocity profile over the tube cross section gives a = 2. (The details of the manipulation are left as an exercise for the reader.)

Example 5-3: Diffuser. A diffuser is a section in a conduit over which the flow area increases gradually from upstream to downstream, as illustrated in Fig. 5-3. If the inlet and outlet areas (Aj and A2) are known, and the upstream pressure and velocity (Pj and Vj) are given, we would like to find the downstream pressure and velocity (P2 and V2). If the fluid is incompressible, the continuity equation gives V2:

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