Problem Statement

A pipe network consists of a vertical pipe which draws water from a well that is open to the atmosphere, a pump, and a

$500-ft$ long horizontal pipe which discharges to an open$-$topped tank through a submerged exit. Both pipe sections are

smooth and have the same diameter, which can be specified in $\frac{1}{4}"$ increments from $1$ to $3.$ The pipe costs $$0\mathrm{.}15(L{D}^2),$

where L is the pipe length in ft and D is the diameter in inches.

The pump is located at ground level between the vertical and horizontal sections of pipe. The pump curve is

$h_p=(5-0\mathrm{.}001Q^2)$

where $h_p$ is the pump head in feet, $Q$ is the flowrate in gpm$,$ and $$ is a dimensionless parameter that specifies the size of

pump$/$motor combination chosen $($for simplicity, $$ can have any value from $1$ to $10).$ The cost of the pump$/$motor is $$100.$

If the flowrate is $35$ gpm and the well depth is $15$ ft $,$ determine the pipe diameter D and pump$/$motor size $$ that minimizes

the capital cost of the system. $($The depth of the well is measured from the surface of the storage tank to the surface of the

well.$)$ You will probably have to solve this problem numerically by calculating the total capital cost over the range of

allowable diameters.

Validation

Perform a parameter study by independently varying $($a$)$ the pipe cost and $($b$)$ the pump$/$motor cost each by $+-10\%$

and recalculating the optimum diameter.

For the optimum diameter and corresponding value of $$ calculated in the original solution, plot the system curve

and pump curve on the same graph and show that they intersect at the desired operating point flowrate.