Fully developed flow moving through a 40-cm diameter pipe has the following velocity profile:

Find the volume flow rate Q using the relationship Q = ∫^R₀ 2πrυ dr, where r is the radial axis of the pipe, R is the radius of the pipe, and υ is the velocity. Solve the problem using two different approaches.

(a) Fit a polynomial carve to the velocity data and integrate analytically.

(b) Use multiple-application Simpson’s 1/3 rule to integrate.

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

Radius, r, cm Velocity, v, m/s 7.5 20.0 0.02 10.0 12.5 17.5 15.0 0.427 2.5 5.0 0.914 0.890 0.795 0.543 0.847 0.719 0.204