A distillation column is connected to another separation device. The fresh feed to the column is \(\mathrm{F}=1000.0 \mathrm{kmol} / \mathrm{h}\) of a saturated vapor that is \(20.0 \mathrm{~mol} \%\) ethanol and \(80.0 \mathrm{~mol} \%\) water. The feed is input at its optimum feed stage. The distillation column has a partial condenser and no reboiler. The column is heated with direct injection of pure steam that is a saturated vapor at \(1.0 \mathrm{~atm}\). The ratio bottoms flow rate/steam flow rate \(=B / S=5.0\). The reflux ratio \(\mathrm{L}_{0} / \mathrm{D}=1.0\). The distillate from the column is \(70.0 \mathrm{~mol} \%\) ethanol and the bottoms is \(1.0 \mathrm{~mol} \%\) ethanol. The column operates at a pressure of \(1.0 \mathrm{~atm}\), the reflux is returned as a saturated liquid, and CMO is assumed to be valid. VLE data are in Table 2-1.

The distillate is sent to a separation device that, because it operates on a different principle than distillation, is not limited by the presence of an azeotrope. The device produces product \(\mathrm{P}\) and recycle stream \(\mathrm{R}\) that is a saturated liquid returned to the distillation column at its optimum feed stage. The product \(\mathrm{P}\) concentration from the device is \(98.7 \mathrm{~mol} \%\) ethanol.

a. Find steam flow rate \(S\), bottoms flow rate \(B\), distillate flow rate \(D\), product flow rate \(\mathrm{P}\), and recycle rate \(\mathrm{R}\)-all in \(\mathrm{kmol} / \mathrm{h}\)-and find mole fraction of ethanol in recycle stream \(x_{R}\).

b. For the distillation column, find the optimum feed plate for the recycle stream, the optimum feed plate for the fresh feed, and the total number of equilibrium stages required.

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TABLE 2-1. Vapor-liquid equilibrium data for ethanol and water at 1 atm y and x in mole fractions XEtoh Xw YEtoh Yw T, C 0 1.0 0 1.0 100 0.019 0.981 0.170 0.830 95.5 0.0721 0.9279 0.3891 0.6109 89.0 0.0966 0.9034 0.4375 0.5625 86.7 0.1238 0.8762 0.4704 0.5296 85.3 0.1661 0.8339 0.5089 0.4911 84.1 0.2337 0.7663 0.5445 0.4555 82.7 0.2608 0.7392 0.5580 0.4420 82.3 0.3273 0.6727 0.5826 0.4174 81.5 0.3965 0.6035 0.6122 0.3878 80.7 0.5079 0.4921 0.6564 0.3436 79.8 0.5198 0.4802 0.6599 0.3401 79.7 0.5732 0.4268 0.6841 0.3159 79.3 0.6763 0.3237 0.7385 0.2615 78.74 0.5732 0.4268 0.6841 0.3159 79.3 0.6763 0.3237 0.7385 0.2615 78.74 0.7472 0.2528 0.7815 0.2185 78.41 0.8943 0.1057 0.8943 0.1057 78.15 1.00 0 1.00 0 78.30