Problem #$2(35$ points$)$: The liquid coming out of a steady state chemical reactor has $=50lb\frac{b_m}{f}{t}^3$ and $=5cP,$ both of which can be assumed constant, and a gauge pressure of $10.$ The diameter of the pipe that leaves the reactor is $8$ inches $($actual$).$ This fluid then enters a pump. It leaves the pump at a gauge pressure of $100,$ and the diameter of the pipe that leaves the pump is $5$ inches $($actual$)$ with a surface roughness of $0\mathrm{.}002$ inches.

The pump itself can be assumed frictionless, but friction cannot be ignored in the pipe that is downstream of the pump. This pipe is has a constant diameter of $5$ inches, is $500$ feet long, and contains six standard $90$ degree elbows. The velocity through the pipe is $7$ feet per second, and the outlet of the pipe is at the same height as the inlet.

A$)(5$ points$)$ What is the velocity of the fluid in the $8$ inch pipe that enters the pump?

B$)(10$ points$)$ How much pump work is required for each pound $($mass$)$ of fluid?

C$)(15$ points$)$ What is the gauge pressure at the end of the $500$ foot section of $5$ inch pipe?

D$)(5$ points$)5$ inches was the "economic diameter" of the pipe at the time it was built. Since then, however, the material used to build the pipe has gotten considerably more expensive, while the cost of the power to run the pump is essentially unchanged. If they were going to replace the pipe today, would the economic diameter be larger or smaller than $5$ inches?