Q$2.$

The irreversible exothermic reaction takes place in a continuous stirred tank reactor.

The heat released by the reaction is removed using cooling water. A proportional

controller is used to adjust coolant flow rate so as to keep the reactor temperature

at desired value. The controller has been designed so that the controlled reactor

exhibits typical underdamped second$-$order temperature response

characteristics when it is disturbed by feed flow rate.

a$.$ The mass feed flow rate to CSTR varies suddenly from $0\mathrm{.}3$ to $0\mathrm{.}4k\frac{g}{s},$ and the

temperature of the reaction mixture, initially at $90C,$ increases to $92C.$ What is

the gain of the transfer function that relates changes in reactor temperature to

changes in feed flow rate? $($Remember transfer function of second order process and

use final value theorem$)$

b$.$ The operator notes that the resulting response is slightly oscillatory with a value

of $0\mathrm{.}25($A$/$B in Figure$)$ occurring at times $1000$ and $3060s$ after the change is

initiated. Find damping factor and natural period of oscillation.

c$.$ Draw the underdamped response $($in Excel$),$ find decay ratio and rise time.

Overshoot $=$exp$(\frac{-}{\sqrt[\phantom{2}]{1-{}^2}})$

$y^{\text{'}}(t)=K_p[1-\frac{1}{\sqrt[\phantom{2}]{1-{}^2}}{e}^{-\frac{t}{}}sin(t+)]$

$=ta{n}^{-1}[\frac{\sqrt[\phantom{2}]{1-{}^2}}{}]$

$=\frac{\sqrt[\phantom{2}]{1-{}^2}}{}$

$T=\frac{2}{\sqrt[\phantom{2}]{1-{}^2}}$