Two tanks are connected in the following way:

We have the following assumptions:

The density of the incoming liquid, $,$ is constant.

The cross$-$sectional areas of the two tanks are $A_1$ and $A_2.$

$w_2$ is positive for flow from Tank $1$ to Tank $2.$

The two valves are linear with resistances $R_2$ and $R_3.$ The flow rate through the valves can be determined by: $w=\frac{P}{g_{c}R}$ where $P$ is the pressure difference at the two ends of the valve and $R$ is the resistance and $g_c$ is a constant.

$($a$)$ Develop a model for this system that can be used to find $h_1,h_2,w_2,w_3$ as functions of time for any given variations in inputs.

$($b$)$ Perform a degrees of freedom analysis. Identify all input and output variables.