A distillation column with a partial reboiler and a total condenser is being used to separate a mixture of benzene, toluene, and cumene. The feed is $40$ mole $\%$ benzene, $30$ mole $\%$ toluene, and $30$ mole $\%$ cumene and is input as a saturated vapor. We desire $95\%$ recovery of the toluene in the distillate and $95\%$ recovery of the cumene in the bottoms. The reflux is returned as a saturated liquid, and constant molal overflow $($CMO$)$ can be assumed. Pressure is $1$ atm.

Equilibrium can be represented as constant relative volatilities. Choosing toluene as the reference component, $\backslash$alpha benz$$tol $=2\mathrm{.}25$ and $\backslash$alpha cumene$$tol $=0\mathrm{.}21.$ Find the number of equilibrium stages required at total reflux and the recovery fraction of benzene in the distillate.

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$2.$ Underwood equations

For the distillation problem given in problem $1$ find the minimum reflux ratio. Use a basis of $100$ kg moles$/$hr of feed.

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$3.$ Gilliland correlation

Estimate the total number of equilibrium stages and the optimum feed plate location required for the distillation problem presented in problem $1$ and $2$ if

the actual reflux ratio is set at L $/$D $=2$

please solve it all step by step $.$