What pressure drop per foot of tube is caused by the shear stress in Problem 7.17?Problem...

What pressure drop per foot of tube is caused by the shear stress in Problem 7.17?

Problem 7.17

For water flowing in a 0.1-in.-diameter tube, the velocity distribution is parabolic (see Example 4.2). If the average velocity is 2 fps, determine the magnitude of the shear stress at the tube wall.

Example 4.2

Let us consider the case of an incompressible flow, for which the flow area is circular and the velocity profile is parabolic (see

Figure 4.3), varying according to the expression

where υmax is the maximum velocity, which exists at the center of the circular passage (i.e., at r = 0), and R is the radial distance to the inside surface of the circular area considered.

Figure 4.3 A parabolic velocity profile in a circular flow passage.

The above velocity-profile expression may be obtained experimentally. It will also be derived theoretically in Chapter 8 for the case of laminar flow in a circular conduit. This expression represents the velocity at a radial distance, r, from the center of the flow section. As the average velocity is of particular interest in engineering problems, we will now consider the means of obtaining the average velocity from this expression.

At the station where this velocity profile exists, the mass rate of flow is

For the present case of incompressible flow, the density is constant. Solving for the average velocity, we have