A fractional distillation column will be designed to separate $30000$kgmol$/$h of a mixture of $40$ mole $\%$ benzene and $60$ mole $\%$ toluene into an overhead product containing $97$ mole $\%$ benzene and a bottom product containing $98$ moel $\%$ toluene. A reflux ratio of $3\mathrm{.}5$mol to $1$mol of product is to be used. The molal latent heats of benzene and toluene are $7360$ and $7960ca\frac{l}{g}$mol, respectively. Benzene and toluene form an ideal system with a relative volatility of about $2\mathrm{.}5$; use the equilibrium data given in Table $2\mathrm{.}1.$ The feed has a boiling point of $95C$ at a pressure of $1$atm. Calculate the following

a$.(10)$ The flow rate of distillate $($D$),$ overhead product $($V$)$ and bottom product $($W$)$ per hour.

b$.(20)$ The number of ideal trays and the position of the feeding tray, minimum reflux ratio and the minimum number of ideal trays for three different cases:

$($i$)(10)$ When the feed is saturated liquid $($liquid at its boiling point$)$;

$($ii$)(10)$ When the feed is a mixture of vapor $(50\%)$ and liquid $(50\%).$

Table $2\mathrm{.}1$ Liquid$-$Vapor Equilibrium data for benzene$($A$)-$toluene$($B$)$ system

$\backslash$table$[[x_A,1\mathrm{.}00,0\mathrm{.}78,0\mathrm{.}581,0\mathrm{.}411,0\mathrm{.}258,0\mathrm{.}13,0],[y_A,1\mathrm{.}00,0\mathrm{.}9,0\mathrm{.}777,0\mathrm{.}632,0\mathrm{.}456,0\mathrm{.}261,0]]$