The schematic below represents a single equilibrium stage in a distillation column.

The following component balance may be derived for a binary system under assump$-$

tions of constant molar holdup, equi$-$molar overflow, negligible vapor holdup, and

constant relative volatility:

$M\frac{d{x}_1}{dt}=L(x_0-x_1)+V(y_2-y_1)$

where

$y_1=\frac{x_1}{1+(-1)x_1}.$

In the above, $M$ is the molar holdup, $$ is the relative volatility, $L$ and $V$ are the

total liquid and vapor molar flow rates, $x_0$ and $x_1$ are the mole fractions of the

streams entering and leaving the tray, and $y_2$ and $y_1$ are the mole fractions of the

entering and exiting vapor streams.

$($a$)$ Derive a transfer function model relating $x_1^{\text{'}}$ to changes in the liquid flow rate

$L^{\text{'}}$ and entering liquid composition $x_0^{\text{'}},$ linearizing if and where necessary.

$($b$)$ Draw a block diagram representation of your transfer function model.

$($c$)$ Determine the response $x_1^{\text{'}}(t)$ to the unit impulse input, $x_0^{\text{'}}(t)=(t).$