Question $1$

A fractionation tower is being supplied with a new input stream of $100mo\frac{l}{m}in,$ consisting of $50$mol$\%$ of component A and $50$mol$\%$ of component $B.$ The gas leaving the upper section of the tower contains $97$mol$\%$ of component A and is completely condensed. This condensed gas stream is then divided into two equal parts: The top product stream, which contains $89\%$ of the component A initially fed to the tower. The reflux stream, which is recirculated back to the upper section of the tower. The liquid exiting the lower section of the tower is directed to a partial reboiler where $45\%$ of the liquid is vaporized. This vaporized stream, referred to as the vaporization stream, is recycled back to the tower. The remaining liquid from the reboiler is the bottom product stream. The relationship that describes the compositions of the streams leaving the reboiler is as follows:

$y_{A,6}=2\mathrm{.}25y_{A,7}$

where $y_{A,6}$ represents the mole fraction of component $A$ in the vapor stream $(6),$ and $y_{A,7}$ represents the mole fraction of Component $A$ in the liquid stream $(7).$

a$)$ Label the flowchart of the separation process.

b$)$ Do degrees of freedom analysis for all systems.

c$)$ Calculate the molar amounts of the overhead and bottom products.

d$)$ Calculate the mole fraction of component $A$ in the bottom product.