In a $30-$tray distillation column, temperature T $($C$)$ on the $8-$th tray is measured and controlled to minimize energy consumption and to get the right distillate concentration. Feed$-$flowrate F $($kg$/$min$)$ entering the column is a disturbance. Steam flowrate S $($kg$/$min$)$ in the reboiler can be manipulated. The models are given below.

Process:

T$($s$)=(-0\mathrm{.}2)/(2$s$+1)$ F$($s$)+20/(6$s$+1)$ S$($s$)$

The steam valve$$s dynamics is negligible compared to the process. The following linear relationship holds between the valve output and controller signal that activates the valve:

S$($t$)-$S$\_$ss$=4($p$($t$)-$p$\_$ss$)$

where $$ss$$ denotes steady$-$state. The sensor is first order with time constant equal to $0\mathrm{.}05$ min. When T increases from $110$ to $140$ C$,$ the sensor output increases from $6$ to $18$ mA$.$ Nominally the system operates at steady$-$state: T$\_$ss$=120$ C; F$\_$ss$=2$kg$/$min; S$\_$ss$=4$kg$/$min$.$

Draw the closed loop diagram showing all the important variables in right units and labeling the blocks with the right transfer functions.

Let the controller be a Proportional controller with gain K$\_$c$=10.$ Calculate and sketch the closed$-$loop response T$($t$)$ when the set$-$point increases from $120$ C to $130.$

Let the controller be a Proportional controller with gain K$\_$c$=10.$ Calculate and sketch the closed$-$loop responsed S$($t$)$ and T$($t$)$ when the feedflowrate increases from $2$ to $2\mathrm{.}5$ kg$/$min