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 $1\mathrm{.}2.$

$(11)$

$[32$ Marks$]$