D$7.$ We are separating a mixture of ethane, propane, n$-$butane and n$-$pentane in a distillation column operating at $5\mathrm{.}0$ atm. The column has a total condenser and a partial reboiler. The feed flow rate is $1000\mathrm{.}0$kmole$\frac{s}{h}r.$ The feed is a saturated liquid. Feed is $8\mathrm{.}0$ mole $\%$ ethane, $33\mathrm{.}0$ mole $\%$ propane, $49\mathrm{.}0$ mole $\%$ n$-$butane and $10\mathrm{.}0$ mole $\%$ n$-$pentane. A $98\mathrm{.}0\%$ recovery of propane is desired in the distillate. A $99\mathrm{.}2\%$ recovery of $n-$butane is desired in the bottoms. Use the optimum feed stage. Use DePriester chart. Assume relative volatility is constant at value calculated at the bubble point temperature of the feed.

a$.$ Find $N_{\text{m}\text{i}\text{n}}$ from the Fenske equation

b$.$ Find $(\frac{L}{D}{)}_{\text{m}\text{i}\text{n}}$ from the Underwood equation

c$.$ Use $\frac{L}{D}=1\mathrm{.}2(\frac{L}{D}{)}_{\text{m}\text{i}\text{n}}$ and estimate $N$ and $N_{\text{F}\text{e}\text{e}\text{d}}$ from Gilliland correlation.