Chapter 23, Problem 28P

The velocity profile of a fluid in a circular pipe can be represented as

$v=10{\left(1-\frac{r}{r_0}\right)}^{1/n}$

where $v=$ velocity, $r=$ radical distance measured out from the pipes centerline, $r_0=$ the pipe's radius, and $n=a$ parameter. Determine the flow in the pipe if $r_0=0.75$ and $r_0=0.75$ and $n=7$ using

(a) Romberg integration to a tolerance of 0.1%,

(b) Two-point Gauss-Legendre formula, and

(c) The MATLAB quad function. Note that flow is equal to velocity times area.