The following hybrid system shown for separating methanol and water is one way of coupling membrane separators with distillation. The gas permeation membrane separator is designed to produce \(\mathrm{y}_{\mathrm{p}}=60 \mathrm{~mol} \%\) methanol and \(\mathrm{y}_{\mathrm{r}}=30 \mathrm{~mol} \%\) methanol for \(\mathrm{p}_{\mathrm{r}}=1.0 \mathrm{~atm}\) and \(\mathrm{p}_{\mathrm{p}}=0.2 \mathrm{~atm}\). The cut \(\theta=\mathrm{F}_{\mathrm{p}} / \mathrm{F}_{\mathrm{in}}=1 / 3\). Assume ideal gases in the membrane separator. The feeds to the distillation column are both saturated vapors at \(1.0 \mathrm{~atm}\) pressure at mole fractions 0.60 methanol and 0.30 methanol, respectively. Assume CMO is valid. The column has a total condenser and a partial reboiler. VLE data are in Table 2-8. Distillate mole fraction is 0.9 methanol, and bottoms mole fraction is 0.05 methanol. Find:

a. The optimum feed stages and the total number of stages if \(\mathrm{L} / \mathrm{D}=3.25\).

b. \((L / D)_{\min }\).

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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