Example $5.$X$7$: A plate column equipped with a total condenser and a kettle$-$type reboiler is used to separate $100$kmo$\frac{l}{h}$ of a benzene$-$toluene solution containing $50$mol$\%$ benzene into a distillate product containing $95$mol$\%$ benzene and bottom product containing $5$mol$\%$ benzene. The column is operated at $101\mathrm{.}3$kPa. The feed is partially vaporised and is one$-$third vapour and two$-$thirds liquid. Determine the following:

$($a$)$ The minimum number of plates graphically and analytically using an average $$ of $2\mathrm{.}39$

$($b$)$ The minimum reflux ratio

$($c$)$ The number of plates if the operating reflux ratio is $30\%$ in excess of the minimum.

$($d$)$ The feed plate location

$($e$)$ The number of plates required if the overall stage efficiency is $75\%$

The equilibrium data for benzene$-$toluene system at $101\mathrm{.}3$kPa :

$\backslash$table$[[x,0,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9,0\mathrm{.}95,1],[y,0,0\mathrm{.}205,0\mathrm{.}369,0\mathrm{.}502,0\mathrm{.}618,0\mathrm{.}706,0\mathrm{.}789,0\mathrm{.}849,0\mathrm{.}907,0\mathrm{.}955,0\mathrm{.}978,1]]$