Please design a de$-$ethanizer to perform the separation illustrated in Figure $1.$ The five species in the feed with corresponding flow rates are methane $(C_1,160$kmo$\frac{l}{h}),$ ethane $(C_2,370$kmo$\frac{l}{h}),$ propane $(C_3,240$kmo$\frac{l}{h}),$ n$-$butane $(n{C}_4,25$kmo$\frac{l}{h}),$ and n$-$pentaneane $(($:$n{C}_5,5$kmo$\frac{l}{h}\}.$ Use propane $C_3$ as the reference. The average relative volatilities of $C_1-n{C}_5$ relative to $C_3$ are, respectively, $8\mathrm{.}22,2\mathrm{.}42,1\mathrm{.}00,0\mathrm{.}378,0\mathrm{.}150.$ The desirable flow rate of $C_3$ in the distillate is $2$kmo$\frac{l}{h}.$ The expected flow of $C_2$ in the bottom is $2$kmo$\frac{l}{h}.$ The feed is saturated vapor. Complete the design of this column where design variables include the total number of actual stages, the compositions of distillate and bottoms, the optimum feed plate ${?}^{-1}$ pation, and the reflux ratio.

Figure $1.$ Light Ends De$-$ethanizer ${?}^{[1]}.$

Task $1.$ With the given information in problem statement, $($a$)$ identify the light key $($LK$),$ heavy key $($HK$),$ light non key $($LNK$),$ and heavy non key $($HNK$)$; $($b$)$ compute the fraction recoveries $($FRs$)$ for the LK and HK$.$

Items to be included in your report: $(1)$ Specify all the LK$,$ HK$,$ LNK$,$ and HNK; $(2)$ Show the calculation procedures and results for FRs of LK and HK in both distillate and bottom.

Task $2.$ Use the Fenske equation to determine: $($a$)$ the minimum number equilibrium stages $N_{\text{m}\text{i}\text{n}}$; $($b$)$ the fractional recoveries for all non$-$keys in the distillate and bottom.

Items to be included in your report: $(1)$ Calculation procedures and results for $N_{\text{m}\text{i}\text{n}}$; $(2)$ Calculation procedures and results for FRs of all non$-$keys in both distillate and bottom.

Please design a de$-$ethanizer to perform the separation illustrated in Figure $1.$ The five species in the feed with corresponding flow rates are methane $(C_1,160$kmo$\frac{l}{h}),$ ethane $(C_2,370$kmo$\frac{l}{h}),$ propane $(C_3,240$kmo$\frac{l}{h}),$ n$-$butane $(n{C}_4,25$kmo$\frac{l}{h}),$ and n$-$pentaneane $(($:$n{C}_5,5$kmo$\frac{l}{h}\}.$ Use propane $C_3$ as the reference. The average relative volatilities of $C_1-n{C}_5$ relative to $C_3$ are, respectively, $8\mathrm{.}22,2\mathrm{.}42,1\mathrm{.}00,0\mathrm{.}378,0\mathrm{.}150.$ The desirable flow rate of $C_3$ in the distillate is $2$kmo$\frac{l}{h}.$ The expected flow of $C_2$ in the bottom is $2$kmo$\frac{l}{h}.$ The feed is saturated vapor. Complete the design of this column where design variables include the total number of actual stages, the compositions of distillate and bottoms, the optimum feed plate ${?}^{-1}$ pation, and the reflux ratio.

Figure $1.$ Light Ends De$-$ethanizer ${?}^{[1]}.$