Consider the stirred tank heating system with constant holdup shown below. The liquid inlet stream consists of a single component with mass flowrate $w_i$ and inlet temperature $T_l.$ The tank contents ure agitated and heated using an electrical heater that provides a heating rate, $Q.$ It is desired to maintain the temperature of the liquid leaving the tank at a temperature $T$ with the help of the heater before it continues to a downstream process.

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a$)$ On the diagram, indicate which quantities are manipulated $($MV$),$ controlled $($CV$),$ and potential disturbance variables $($DV$).$

$15$ b$)$ Find the dynamic model of the temperature of the liquid leaving the tank. State all assumptions.

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c$)$ Find an expression for the steady state value of the temperature of the liquid leaving the tank.

d$)$ Using the deviation $($perturbation$)$ variables:

$x^{\text{'}}=x-\stackrel{}{x}$

Linearize if needed, and express the dynamic model found in part $b$ in the following deviation form:

$\frac{d{T}^{\text{'}}}{dt}={}_1{T}_i^{\text{'}}+{}_2{T}^{\text{'}}+{}_3{Q}^{\text{'}}$

What are the expressions for ${}_1,{}_2$ and ${}_3?$

$6/6$e$)$ Find the transfer functions that relates $T^{\text{'}}$ to $T_i^{\text{'}}$ and $T^{\text{'}}$ to $Q^{\text{'}}.$

$\frac{2}{6}$ f$)$ If the heating rate, $Q^{\text{'}},$ is increased by a constant magnitude ofM. Find an expression of the new steady state value of the temperature of the liquid leaving the tank.