If the velocity distribution of a fluid flowing through a pipe is known (Fig. P20.17), the flow rate Q (i.e., the volume of water passing through the pipe per unit time) can be computed by , where υ is the velocity, and A is the pipe’s cross-sectional area. (To grasp the meaning of this relationship physically, recall the close connection between summation and integration.) For a circular pipe, A = π r2 and dA = 2πr dr. Therefore,
where r is the radial distance measured outward from the center of the pipe. If the velocity distribution is given by
where r0 is the total radius (in this case, 3 cm), compute Q using the composite trapezoidal rule. Discuss the results.
Fig. P20.17:
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