A "run$-$of$-$the$-$river" micro$-$hydropower plant is operated for small$-$scale power generation in which

$200$ gallons per minute of water is taken from a stream and delivered through $800$ feet of $3-$inch

diameter PVC pipe to a turbine $100$ feet lower than the source. The turbine$/$generator at the end of

the pipe has an efficiency of $40\%.$

a$.$ What fraction of the $100$ feet of available elevation head will be lost to friction in the pipe?

$($Hint: see Figure $8\mathrm{.}35$ of Masters, $2^("$nd $")$ edition.$)$

b$.$ What is the linear velocity $($ft$//$s$)$ of the water diverted into the pipe?

c$.$ How much electricity $($kWh$)$ is delivered by this micro$-$hydro plant in one $30-$day month?

d$.$ By what percentage can the monthly electricity production be increased if $4-$inch diameter PVC

pipe is substituted for the $3-$inch PVC pipe, assuming the same volumetric flow rate?