$(30$ points$)$ A Peerless Type $4$AE$11$ centrifugal pump $($Fig$.1$ below$)$ is tested at $1750$ rpm using a flow system with the layout shown in Fig. $2.$ The water level in the inlet reservoir is $3\mathrm{.}5$ ft above the pump centerline; the inlet line consists of $6$ ft of $5$ in $.$ diameter straight cast$-$iron pipe $(=0\mathrm{.}00085ft),$ a standard elbow, and a fully open gate valve. Kinematic viscosity of water at $80F$ is $v=0\mathrm{.}927{10}^{-5}f\frac{t^2}{s}.$ Kinematic viscosity of water at $180F$ is $v=0\mathrm{.}380{10}^{-5}f\frac{t^2}{s}.$ The total minor losses coefficient is $K=2\mathrm{.}5.$ The vapor pressure for water at $80F$ is $p_v=0\mathrm{.}507.$ The vapor pressure for water at $180F$ is $p_v=7\mathrm{.}51.$ Calculate:

a$.$ The net positive suction head available $($NPSHa$)$ at the pump inlet at a volume flow rate of $1000$ gpm of water at $80F.$

b$.$ Compare with the net positive suction head required $($NPSHr$)$ by the pump at this flow rate $($Fig$.1).$ SHOW YOUR WORK ON THE PLOT. SHOW YOUR WORK ON THE PLOT. Will cavitation occur at $80F$ and $1000$ gpm$?$

c$.$ Plot NPSHa and NPSHr for water at $80F$ and $180F$ versus volume flow rate $($this is the suction head vs flowrate plot outlining the NPSHa and NPSHr curves$).$ Will cavitation occur at $180F$ and $1000$ gpm$?$ TIP: Take NPSHr curve $($longer one$)$ from the Fig. $1.$ The NPSHr is the same in the two plots with different NPSHa. Think about how to construct curves based on the energy equation and suction head at flowrates ranging from $0$ to $1500$ gpm$.$