Question $2$

A steam heated stirred tank $($similar to the one shown in Figure Q$3)$ has the following operating

conditions and constant parameters:

Mass flow rate, $w={10}^{4}k\frac{g}{h}r$

Density, $=1000k\frac{g}{m^3}$

Volume of the tank, $V=100m^3$

Heat capacity, $C_{}=1$kca$\frac{l}{k}g.C$

Latent heat of vaporization, $H_v=600$kca$\frac{l}{k}g$

Figure Q$2$: Steam$-$heated stirred$-$tank system.

The controlled variable, $CV,$ is the tank temperature $(T),$ the manipulated variable, $MV,$ is the

steam rate $(w_s),$ and the disturbance variable DV$,$ is the inlet temperature $(\{$:$T_i).$ The set point of

the $CV$ is $T_{sp}=150C$ and the steady state inlet temperature $\frac{?}{\text{b}\text{a}\text{r}}(T{)}_i$ is $40C.$

$($i$)$ Process Modelling and transfer function

$($a$)$ Develop a dynamic model for this system that will describe transient operation and that

can be used to find the tank temperature $T.$

$($b$)$ What is the steady state steam flow rate at the set point temperature?

$($c$)$ Write the dynamic model in terms of deviation variables and derive the process transfer

function, $G_P=T^{\text{'}}\frac{s}{W^{\text{'}}}(s)$ and the disturbance transfer function

$G_d=T^{\text{'}}\frac{s}{T_i^{\text{'}}}(s)$ for this process. What are the orders of both transfer function?

Define and calculate the values of process gains and time constants for both transfer

functions $($include units$)?$

$($ii$)$ Process control design

$($a$)$ It is desired to control the tank outlet temperature, $T$ :

Propose a feedback control method and sketch the schematic diagram

Suggest a feedforward control method and sketch the schematic diagram.

$($b$)$ Construct the block diagram of the feedback control system.

$($c$)$ Determine the closed$-$loop transfer function $T\frac{s}{T_i}(s)$ of the system and using the

final value theorem calculate the offset for a unit step changes in disturbance.

$($d$)$ Determine the closed$-$loop transfer function $[T\frac{s}{T_{sp}}(s)]$ of the system and using the

final value theorem calculate the offset for a unit step changes in set$-$point.