Please, solve g$.$ A jacketed vessel is used to cool a process stream as shown in Figure $1.$ The following information is available:

Volume of the vessel, V$,$ is constant; Volume of the jacket, $V_j,$ is constant.

Heat losses are negligible.

Tank and jacket contents are well$-$mixed.

Wall thermal capacities are negligible.

Heat removed from vessel by the coolant is $Q_c=UA(T-T_j)kca\frac{l}{h}r$ where $U=K{q}_i^{0\mathrm{.}8}$kca$\frac{l}{h}{r}^{-}{m}^2-{?}^{0}C$

Inlet flowrate $q_F(\frac{m^3}{h})$ and inlet temperature of the coolant, $T_i,$ are constant but inlet temperature $T_F$ and the coolant flowrate, $q_j$ can vary.

a$.$ List the variables which are functions of time as inputs and outputs.

b$.$ Derive a dynamic model for this system by stating any additional assumptions.

c$.$ Apply degrees of freedom do find which variables has to be defined.

d$.$ What are the probable disturbances and manipulated variables?

e$.$ Choose one disturbance and suggest three different control configurations $($feedback$,$ feedforward and feedback$/$feedforward$.$ Draw them schematically.

f$.$ Convert the model you developed in terms of deviation variables and linearize any non$-$linear terms.

g$.$ Find transfer functions as

$T^{\text{'}}(s)=G_1(s)Q^{\text{'}}(s)+G_2(s)T_F^{\text{'}}(s)$

$T^{\text{'}}(s)=G_3(s)Q^{\text{'}}(s)+G_4(s)T_F^{\text{'}}(s)$