Chapter 24, Problem 49P

Fully developed flow of a Bingham plasticfluid moving through a 12-in diameter pipe has the given velocity profile. The flow of a Bingham fluid does not shear the center core, producing plug flow in the region around the centerline.

Radius, r, in	0	1	2	3	4	5	6
Velocity, v, ft/s	5.00	5.00	4.62	4.01	3.42	1.69	0.00

Find the total volume flow rate Q using the relationship $Q={\displaystyle {\int }_{r1}^{r_2}2\pi rvdr+v_c{A}_c}$, where r is the radial axis of the pipe, v is the velocity, $v_c$ is the velocity at the core, and $A_c$ is the cross-sectional area of the plug. Solve the problem using two different approaches.

(a) Fit a polynomial curve to the noncore data and integrate.

(b) Use multiple-application Simpson's rule to integrate.

(c) Find the percent error using the integral of the polynomial fit as the more correct value.