$(15$ pt$)$ The variable volume, variable temperature tank shown in Figure $P7\mathrm{.}1$ is used to heat a

process stream from temperature $T_{in}$ to temperature $T.$ Assuming constant thermophysical properties

of the liquid and square$-$root dependence of the outlet flowrate on the height, mass and energy

balances on the tank will give the following process model:

$A\frac{dh}{dt}=F_{in}-c\sqrt[\phantom{2}]{h}$

$Ah\frac{dT}{dt}=F_{in}(T_{in}-T)+\frac{Q}{c_p}$

where $h$ is the height of the liquid in the tank, $F_{in}$ is the feed flowrate, $T_{in}$ is the feed temperature, $A$ is

the cross$-$sectional area of the tank, $T$ is the temperature in the tank, $$ is the density, $c_p$ is the heat

capacity and $Q$ is the heat input to the tank.

L

Figure P$7\mathrm{.}1$

$($a$)$ Considering $F_{in},T_{in}$ and $Q$ to be the inputs of the system, linearize the system equations around a

given steady state and check for local asymptotic stability.

$($b$)$ If the temperature $T$ is the output, derive the transfer functions of the system.