Ternary Mixture Separation

a$)$ A distillation column with a partial reboiler and a total condenser is being used to separate a mixture of benzene $($B$),$ toluene $($T$),$ and cumene $($C$).$ The feed is $40$mol$\%$ benzene, $30$mol$\%$ toluene and $30$ mol$\%$ cumene. The feed is input as a saturated vapor. We desire $99\%$ recovery of the toluene in the bottoms and $98\%$ recovery of the benzene in the distillate. The reflux is returned as a saturated liquid, and CMO can be assumed. Equilibrium can be represented as constant relative volatilities. Choosing toluene as the reference component, ${}_{BT}=2\mathrm{.}25$ and ${}_{CT}=0\mathrm{.}210.$

Use the Fenske equation to find the number of equilibrium stages required at total reflux and the recovery fraction of cumene in the bottoms.

b$)$ For the distillation problem given in part $($a$),$ find the minimum reflux ratio by use of the Underwood equations. Use a basis of $100$ moles of feed $\frac{?}{h}.$ Clearly state your assumptions.

c$)$ For $R=1\mathrm{.}25R_{\text{m}\text{i}\text{n}},$ find the total number of equilibrium stages required for the distillation problem presented in parts $($a$)$ and $($b$).$ Use the Gilliland correlation. Estimate the optimum feed plate location.