$5\mathrm{.}22$ The liquid$-$level system in Figure P$5\mathrm{.}22$ consists a single tank of constant area A$,$ which

has an orifice at height h$\_($o$),$ a pump with pressure difference $\backslash$Delta p$,$ and a valve $($R$)$;q$\_($in$)$ is a

volume inflow rate, R is related to the mass flow rate, and p$\_($a$)$ is the atmospheric

pressure. Let the area and discharge coefficient of the orifice be A$\_($o$)$ and C$\_($d$),$ respectively.

The density of the fluid is $\backslash$rho $.$ Assume that the liquid height h is always larger than h$\_($o$).$

$($a$)$ Draw a free$-$body diagram for the tank, and auxiliary plots for the pump, valve,

and orifice.

$($b$)$ Derive the governing differential equation of the system in terms of h$.$

Figure P$5.22$