A solution containing $30$ wt$.\%$ benzene $($B$),45$ wt$.\%$ toluene $($T$)$ and $25$ wt$.\%$ xylene $($X$)$ is fed

into a distillation process as shown in the attached diagram. The overhead product from the first

column contains $($by wt$.)91\mathrm{.}40\%$ B$,8\mathrm{.}30\%$ T$,$ and $0\mathrm{.}30\%$ X$.$ The overhead product from the second

column contains $4\mathrm{.}25\%$ B$,91\mathrm{.}60\%$ T$,$ and $4\mathrm{.}15\%$ X$.$ One$-$third of the bottom product of the second

column is recycled into the first column. It is also required that the benzene in the first column

overhead product be $20$ times that of the second column overhead product, and $2$ times that of the

second column bottom product. Assuming a basis of $100$ kg fresh feed, carry out the following

operations:

$($a$)$ Carry out DOF analysis for all possible units and $$mixing points$$ that mass balance can be carried

out.

$($b$)$ Derive and show all the independent simultaneous linear mass balance equations for the system

using all the information provided.

$($c$)$ Solve the mass balance equations using the Excel Solver only $($as demonstrated in class$/$recitation$).$

The formula used in each cell must be shown in the submitted work as well as the computed cell result.