One-Dimensional Flow in a Tube

We will apply the steady state momentum balance to a fluid in plug flow in a tube, as illustrated in Fig. 5-6. (The stream tube may be bounded by either solid or imaginary boundaries the only condition is that no fluid crosses the boundaries other that through the inlet and outlet planes.) The shape of the cross section does not have to be circular it can be any shape. The fluid element in the slice of thickness dx is our system, and the momentum balance equation on this system is 

The forces acting on the fluid result from pressure (dFP), gravity (dFg), wall drag (dFv), and external shaft work (SW = —Fext dx, not shown in Fig. 5-6)  

rw is the stress exerted by the fluid on the wall (the reaction to the stress exerted on the fluid by the wall), and Wp is the perimeter of the wall in the cross section that is wetted by the fluid (the wetted perimeter ). After substituting the expressions for the forces from Eq. (5-43) into the momentum balance equation, Eq. (5-42), and dividing the result by — pA, where A = Ax, the result is 

Comparing this with the Bernoulli equation [Eq. (5-33)] shows that they are identical, provided 

We see that there are several ways of interpreting the term ef. From the Bernoulli equation, it represents the lost (i.e., dissipated) energy 

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Similarly, the flow in a runner can be modeled as an axisymmetric one-dimensional flow in a tube of radius R. For a stationary wall, we have the pressure equation as... 

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