QUESTION $1$

A cooling coil is integrated inside a stirred tank reactor to release heat generated during a reaction process. To maintain a constant reactor temperature, a proportional controller is utilized to control the flow rate of the coolant. The controller has been built in a manner that the regulated reactor displays common underdamped second$-$order temperature responses when it is disrupted by the flow rate of the feed or by the temperature changes of the coolant.

$1\mathrm{.}1$ What is the gain of the transfer function $($under feedback control$)$ that relates the changes in the reactor temperature to that of the flow rate of the feed if the flow rate of the feed to the reactor changes from $0\mathrm{.}4k\frac{g}{s}$ to $0\mathrm{.}5k\frac{g}{s}$ and the reactor temperature changes from $100C$ to $102C?$

$1\mathrm{.}2$ What is the complete transfer function, if it is observed that the response is oscillating with a maximum of $102\mathrm{.}5C$ and $102C$ taking place at $1000$ seconds and $3060$ seconds after the change has started? $($NB$.$ Use the analytical approach to get your answers$).$

$1\mathrm{.}3$ If the operator failed to notice the rise time, estimate $t_r$ based on the results in question $1\mathrm{.}1$ and $1\mathrm{.}2.$

$[32$ Marks$]$

QUESTION $2$

Given the transfer function below:

$G(s)=\frac{Y(s)}{U(s)}=\frac{10}{(5s+1)(3s+1)}$

What is $y(t)$ for the subsequent inputs:

$2\mathrm{.}1$ Step input of height $M$

$2\mathrm{.}2$ Unit impulse input $(t)$

$2\mathrm{.}3sin(t)$

$2\mathrm{.}4$ Unit rectangular pulse, where $h=1)$

fin $-\{\begin{array}{cc}0,& t<mn\; id="EditorElement\_273235">0</mn>\\ h,& 0t=t\end{array}$

$[33$ Marks$]$

QUESTION $3$

$3$ Use the Taylor series to linearize the following $($show all your steps$)$:

$3\mathrm{.}1$

$f(x)=5x^2-2x,$ where $x=3$

$3\mathrm{.}2$

$f(x)=-3x^3+30x,$ where $x=-3$

$[20$ Marks$]$