A distillation column with a total condenser uses a flash distillation system as the partial reboiler. The \(200 \mathrm{kmol} / \mathrm{h}\) of feed is \(55 \mathrm{~mol} \%\) methanol and \(45 \mathrm{~mol} \%\) water and is a saturated liquid at a pressure of \(1.0 \mathrm{~atm}\). VLE data are in Table 2-8. Operation is at \(1.0 \mathrm{~atm}\). CMO can be assumed to be valid. The reflux is returned as a saturated liquid. The distillate is \(95 \mathrm{~mol} \%\) methanol, and the bottoms is \(10 \mathrm{~mol} \%\) methanol. The flash distillation system is also at \(1.0 \mathrm{~atm}\). A pump pressurizes \(\mathrm{r}\), and a heat exchanger heats liquid from the column before it is fed to the flash chamber. The vapor product from the flash chamber is fed to the column as \(\mathrm{v}\). The liquid product from the flash system is the bottoms product B of the distillation system. The fraction of feed to the flash system (r) that becomes vapor (v) in the flash system is \(\mathrm{f}=\sigma / \bar{\tau}=0.6923\). Hint: Draw and label the streams for the flash system used as the partial reboiler.

a. Find distillate flow rate \(\mathrm{D}\) and bottoms flow rate \(\mathrm{B}\).

b. Determine the flow rates \({ }_{\tau}\) and \({ }_{v}\).

c. Find the mole fraction of the feed to the flash system, and the mole fraction of the vapor from the flash system.

d. Derive and plot the operating lines and feed line for the distillation system. Stepping off stages from the bottom upwards, find the number of stages needed and the optimum feed stage.