Q$2.$ A cylindrical vessel with a cross$-$sectional area of $4m^2$ is used to mix two liquid streams $(4\frac{m^3}{hr}$ and $2\frac{m^3}{hr})$ having a concentration of active material at $5\frac{kg}{m^3}$ and $2\frac{kg}{m^3},$ respectively. At steady state, the liquid level is $3m$ in the vessel. The exit point is $1\mathrm{.}0m$ below the bottom of thevessel. Only the first stream is indicating frequent changes$(+-5\%)$ in the flow rate while the flow rate of the second stream remains constant.

$($i$)$ Examine the effect of these changes in Stream $1$ on the

$($a$)$ level in the vessel,

$($b$)$concentration of the active material in the exit stream and

$($c$)$ total flow rate out of the system.

Derive necessary transfer functions.

$($ii$)$ If the second stream flow rate can be manipulated to keep the concentration of the active material in the exit stream under control, design an appropriate controller for the system and

verify its efficacy for the job. The concentration sensor is having a time constant of $5$min.