Chapter 24, Problem 42P

If the velocity distribution of a fluid flowing through a pipe is known (Fig. P24.42), the flow rate Q (that is, the volume of waterpassing through the pipe per unit time) can be computed by $Q=\int vdA$, where v is the velocity and A is the pipe's crosssectional area. (To grasp the meaning of this relationship physically, recall the close connection between summation and integration.) For a circular pipe, $A=\pi r^2\text{and }dA=2\pi rdr$. Therefore,

$Q={\displaystyle {\int }_0^{r}v\left(2\pi r\right)dr}$

FIGURE P24.42

wherer is the radial distance measured outward from the center of the pipe. If the velocity distribution is given by

$v=2{\left(1-\frac{r}{r_0}\right)}^{1/6}$

where $r^0$ is the total radius (in this case, 3 cm), compute Q using the multiple-application trapezoidal rule. Discuss the results.