If the velocity distribution of a fluid flowing through a pipe is known (Fig. P18.17), the...

If the velocity distribution of a fluid flowing through a pipe is known (Fig. P18.17), the flow rate Q (that is, the volume of water passing through the pipe per unit time) can be computed by Q = ∫ v dA, where v 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 = πr2 and dA = 2πr dr. Therefore,

FIGURE P18.17

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 multiple-application trapezoidal rule. Discuss the results.