Fully developed flow of a Bingham plastic fluid 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 = r1∫r2 2prv dr + vc Ac, where r is the radial axis of the pipe, v is the velocity, vc is the velocity at the core, and Ac 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.