For a binary distillation column with two feeds,

a. Show that the intersection of the top and bottom operating lines occurs at the feed line for fictitious feed \(F_{T}\) where \(F_{T}=F_{1}+F_{2}, z_{T} F_{T}=z_{1} F_{1}+z_{2} F_{2}\), and \(\mathrm{h}_{\mathrm{F}, \mathrm{T}} \mathrm{F}_{\mathrm{T}}=\mathrm{h}_{\mathrm{F}, 1} \mathrm{~F}_{1}+\mathrm{h}_{\mathrm{F}, 2} \mathrm{~F}_{2}\).

Suggestion: Draw the McCabe-Thiele diagram for the actual column with three operating lines using both actual feed lines. On the same diagram, draw the two operating lines for a column with the single mixed feed \(F_{T}\) (they are unchanged from top and bottom operating lines of a two-feed column) and then determine the feed line for this mixed column.

b. If CMO is valid, show that \(\mathrm{q}_{\mathrm{T}} \approx\left(\mathrm{F}_{1} \mathrm{q}_{1}+\mathrm{F}_{2} \mathrm{q}_{2}\right) / \mathrm{F}_{\mathrm{T}}\) (McCabe and Thiele, 1925).