From the text, consider the problem outlined in $5\mathrm{.}20($pg $392).$ Consider the following scenarios and identify the

appropriate outputs for each $($note: listing every signal as an output is not a reasonable answer!$).$ Justify your selections.

a$)$ Tank $2$ is a water tower and Tank $1$ manages overflow

b$)$ This is an irrigation system

c$)$ This is an industrial batch reactor system managing two connected, volume sensitive$,$ processes.

$5\mathrm{.}20$ A liquid$-$level system is shown in Figure P$5\mathrm{.}20,$ where two tanks of constant areas

$(A_1,A_2)$ are connected by a pump with pressure difference $p$ and a valve; Tank $1$ has a

volume inflow rate $q_{in}$; Tank $2$ has a volume outflow rate $q_{\text{o}\text{u}\text{t}}$; and $p_a$ is the atmospheric

pressure. The fluid resistances $R$ and $R_1$ are related to the mass flow rate, as described

by Eq$.(5\mathrm{.}5\mathrm{.}14).$ The density of the fluid is $.$

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

and pipe.

$($b$)$ Derive the governing differential equations of the system with liquid heights $h_1$ and

$h_2$ as the dynamic variables.

$($c$)$ Obtain a state$-$space model with $q_{in}$ and $p$ as the inputs, and $h_2$ and $q_{\text{o}\text{u}\text{t}}$ as the

outputs.