$(solve$ each part $on$ a paper and explain and make proper block diagram $on$ a paper sheet,you solved $it$ previously assuming constant volume that $is$ wrong$)$

QUESTION I

Consider the cylindrical tank in the Fig$-$

ure mixing a pure component A stream

and an inert solvent $S$ stream. Assume

perfect flow control. A PI level controller

with a gain $K_c=0\mathrm{.}2mi{n}^{-1}($with an aggressive $$manipulates

the outflow with the two inflow streams

being independent. Initially the system

is at steady state with the salient de$-$

sign and operating conditions noted in

the Figure. The initial hold up is $50\%$

of full tank capacity. Answer the follow$-$

ing questions,

reaction ${}_A=k{C}_A($isother$mal$ reaction$)$

volume $is$ not constant.

$($a$)$ Using linearization, derive a linear dynamic ODE model relating the outlet A concentration

$c_A$ to the two independent flow rate setpoints, $F_A^{SP}$ and $F_S^{SP}.$

$($b$)$ Present the ODE model above as a block diagram with appropriate transfer functions.

$($c$)$ Obtain an analytic expression for $c_{A(t)}$ for the linear ODE derived in part $($a$)$ for a step change

in the solvent flow rate of size $F_{S_0}.$

$($d$)$ It is proposed to use a PI controller that manipulates $F_A^{SP}$ to hold $c_A.$ The integral time ${}_{\text{I}}$

is set to twice the mixing tank residence time at the initial steady state and the controller

gain $K_c$ is chosen for a dominant closed loop pole pair damping coefficient of $0\mathrm{.}5.$ Calculate

$K_c$ and ${}_1.$

$($e$)$ Obtain an analytical expression for the response of $c_{A(t)}$ to a step change in the solvent rate

of size $F_{s0}$ with the composition controller in part $($d$)$ on antomatic.

$($f$)$ What is the maximum deviation from setpoint in the elosed loop $c_A$ response.

Solve each part commpletely and explain explain with proper reasoning