Q$4.$

In ethylene glycol production plant, ethylene oxide is removed from water$-$glycols

mixture in a flash column operating constant temperature of $105C$ and a constant

pressure of $1$atm. At steady state condition, the temperature and flow rate of feed

stream are $55C$ and $113$kmo$\frac{l}{m}in,$ respectively, the flowrates of vapor and liquid

products are $72\mathrm{.}74$kmo$\frac{l}{m}in$ and $40\mathrm{.}26$kmo$\frac{l}{m}in.$ The amount of liquid mixture in

flash column is $120$kmol. The flowrates of feed and product streams and pressure

don't change with time. Assume that only controlled variable is operating

temperature of flash column. The temperature of feed stream acts as a disturbance

and temperature of flash column is controlled by adjusting the steam flow rate.

The specific heat of capacities of feed stream, vapor product and liquid product are

$81\mathrm{.}55,35\mathrm{.}53$ and $101\mathrm{.}9k\frac{J}{k}$molK. $($The enthalpies of streams can be obtained directly

by multiplying specific heat of capacity and temperature of corresponding stream$).$ The

latent heat of condensation of steam is $2257k\frac{J}{k}g.$ Transfer function is to be

developed by neglecting change in the flowrates of vapor and liquid products with

temperature.

a$.$ Obtain the transfer functions relating controlled variable and manipulated

variable, $G_p(s)$ and the controlled variable and disturbance, $G_d(s).$

b$.$ Draw the block diagram of closed loop response $(G_m(s)=G_r(s)=1).$

c$.$ Obtain the expression of closed loop response as a function of $G_c.$

d$.$ Determine the offset when the disturbance is changed with unit step function if

PID controller is used with a gain of $5,$ reset time of $0\mathrm{.}5$ and derivative time of $0\mathrm{.}1$ min.

Please write your answer by satep step and clearly. I'll remember to evaluate your answer.