A distillation column with a partial condenser (PC) and a partial reboiler (PR) is separating a feed that is \(40 \mathrm{~mol} \%\) methanol and \(60 \mathrm{~mol} \%\) water. Feed flow rate is \(200 \mathrm{kmol} / \mathrm{h}\) and is a two-phase feed that is \(70 \%\) liquid. A bottoms that is \(5 \mathrm{~mol} \%\) methanol is produced. Half of the distillate product is a vapor from the partial condenser and the other half of the distillate product is condensed liquid \(\left(\mathrm{D}_{\mathrm{V}}=\right.\) \(\mathrm{D}_{\mathrm{L}}\) ). Reflux is the same composition as the liquid distillate product. The distillate vapor product is \(\mathrm{y}_{\mathrm{D}}=0.85\) mole fraction methanol. The \(\mathrm{PC}\) and the PR are both equilibrium contacting devices (liquid and vapor are in equilibrium). The boilup ratio \((s / B)=0.875\). Operation is at \(1.0 \mathrm{~atm}\) and \(\mathrm{CMO}\) is valid. Use the optimum feed stage. The partial condenser is an equilibrium contact. Step off stages from the top downwards. The top of the column is shown in the figure. VLE data are in Table 2-8.

Table 2-8

a. What is the value of the liquid distillate mole fraction \(\mathrm{x}_{\mathrm{D}}\) ?

b. Find values of \(\mathrm{B}, \mathrm{D}_{\mathrm{V}}\), and \(\mathrm{D}_{\mathrm{L}}\).

c. Determine the number of stages and the optimum feed location (stages below PC).
The \(\mathrm{y}=\mathrm{x}\) intersection of the top operating line is not at \(\mathrm{x}_{\mathrm{D}}\) or \(\mathrm{y}_{\mathrm{D}}\).

TABLE 2-8. Vapor-liquid equilibrium data for methanol water (p = 1 atm) (mol %) Methanol Liquid Methanol Vapor Temp., C 0 0 100 2.0 13.4 96.4 4.0 23.0 93.5 6.0 30.4 91.2 8.0 36.5 89.3 10.0 41.8 87.7 15.0 51.7 84.4 20.0 57.9 81.7 30.0 66.5 78.0 40.0 72.9 75.3 50.0 77.9 73.1 60.0 82.5 71.2 70.0 87.0 69.3 80.0 91.5 67.6 90.0 95.8 66.0 95.0 97.9 65.0 100.0 100.0 64.5