A methanol$/$n$-$propanol solution containing $35$ mole $\%$ methanol will be rectified in a plate

column equipped with a partial reboiler and a total condenser at $101\mathrm{.}3$kPa to produce a top

product with $95$ mole$\%$ purity in methanol and a bottom product with $95$ mole$\%$ purity in n$-$

propanol. The external reflux ratio is $\frac{5}{3}.$ It is suggested that the feed at $1000$kmol$\frac{e}{h}$ can be

introduced to the column from the $5^{th}$ stage at $5$ different thermal conditions, as shown in

the table below.

a$)$ Calculate the q$-$parameter for each feed condition, write it in the Table, and mark all

feed conditions on the T$-$xy diagram,

b$)$ Find the number of theoretical stages for each case using the Lewis method and

discuss the influence of feed thermal condition on the number of stages. Which feed

condition do you suggest for the column design?

c$)$ It is claimed that the energy cost for running the distillation column can be controlled

by adjusting the feed thermal conditions. Do you agree with this proposal? Please justify

your answer.

Note: Please show the derivation of all equations. Write your assumptions and justify your

assumptions. For the stage$-$by$-$stage calculations, draw each stage, show the composition

of more volatile components in vapor and liquid phases, and show your calculations clearly.