Separation of Water and Methanol: $(60$ points$)$

A liquid mixture of water and methanol is to be distilled in a fractionating column at $101\mathrm{.}3$kPa

pressure. Consider a total condenser and partial reboiler. The feed of $1000$kmol$\frac{e}{h}$ is liquid and it

contains $44$mole$\%$ methanol and enters at $327\mathrm{.}6K.$ The boiling point of the saturated liquid feed is

$367\mathrm{.}6K.$ The amount of liquid fed back to the column at the top is $1\mathrm{.}3$ times the distillate product.

The average heat capacity of the feed is $160k\frac{J}{k}g$ mole. $K$ and the average latent heat $32000k\frac{J}{k}g$

moles. The overhead product contains $99\%$ of methanol by moles and the bottoms contains $1\%$ of

methanol by moles.

Vapor Liquid Equilibrium values for methanol$-$water can be calculated from the following equation

$y=\frac{7\mathrm{.}15x}{1+6\mathrm{.}15x}-0\mathrm{.}33x(1-x)$

$($a$)$ What are the distillate and bottoms flow rates $($D and B$)?$

$($b$)$ What is $N_{\text{m}\text{i}\text{n}},$ the minimum number of stages required? $($corresponding to $R$ and $V_B=$ infinity$)$

$($c$)$ What is q$?$

$($d$)$ What are the number of stages $($N$)$ required? Write the equations for the Rectifying Section and

Stripping Section Operating lines. Find the liquid and vapor flowrates in the rectifying and stripping

sections. Find the vapor boilup ratio. Solve either graphically or analytically.

$($e$)$ Where is the optimal feed stage?

$**$Hint $-$ It is useful to find the point of intersection of the q$-$line, ROL and SOL to identify the x$-$

coordinates at which the stepping of stages switches from the ROL to the SOL. This will give you the

value of the feed stage.