Your plant has a catalytic system for burning methane in the stoichiometric quantity of air. All of the methane is converted to carbon dioxide and water vapor. The flue gas is cooled, and the water vapor is condensed out of the gas stream. Assume that air is \(21 \%\) oxygen and \(79 \%\) nitrogen. The gas that will be fed to the permeator consists of carbon dioxide and nitrogen. This gas is sent to a perfectly mixed membrane separator equipped with a silicone rubber membrane with an effective membrane thickness of \(1.0 \mu \mathrm{m}\). The pressure on the permeate side is \(1.02 \mathrm{bar}\), and the membrane unit is operating at \(25^{\circ} \mathrm{C}\). Choose a basis of \(1.0 \mathrm{kmol}\) methane burned/minute. Data are in Table 19-2. Assume ideal gas.

a. What is the flow rate and the concentration of the gas that is fed to the permeators?

b. If the cut \(\theta=0.20\), what value of retentate pressure is needed to obtain a permeate product that is 30 \(\mathrm{mol} \%\) carbon dioxide? What is corresponding retentate concentration?

c. With \(\theta=0.20\), what is the highest value of \(\mathrm{y}_{\mathrm{p}}\) that can be obtained by increasing \(\mathrm{p}_{\mathrm{r}}\) ?

d. What membrane area in \(\mathrm{m}^{2}\) is required for a basis of burning \(1.0 \mathrm{kmol} / \mathrm{min}\) methane for part \(\mathrm{b}\) ?

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TABLE 19-2. Permeabilities of gases in Barrers in polymer membranes TC He H2 CO2 O2 CH4 N2 Ref Membrane Poly [1-(trimethylsilyl)-1-propyne] Poly (dimethylsiloxane) Poly (dimethylsiloxane) Poly (4-methyl pentene) Natural rubber Natural rubber 25 100 Natural rubber Silicone rubber Ethyl cellulose Ethyl cellulose Ethyl cellulose Poly (2.6-dimethyl phenylene oxide) Polystyrene Polystyrene Polystyrene Polycarbonate Butyl rubber Butyl rubber Acetyl cellulose Cellulose acetate Nylon 6 Nylon 66 PIM-1 (polymers of intrinsic porosity-ladder type) PIM-7 28,000 7730 - 4970 DR 25 230 - 3240 605 300 N 35 - - 4550 781 351 DR 93 32 - 8.0 N 25 23.7 90.8 99.6 17.7 6.12 N 25 - - 153.0 9.43 DR 25 31 49 131 24 30 8.1 G 25 300 550 2700 25 53.4- 113 35 500 800 250 G 15 - 3.0 N 5.0 12.4 3.4 DR 30 35.7 49.2 47.5 11.2 7.47 3.29 G 25 - - 75 15 4.43 N 20 16.7 - 10.0 2.01 - 0.32 N 35 - - 12.4 2.9 - 0.52 DR 30 40.8 56.0 23.3 7.47 2.72 2.55 G 25 19 - 8.0 1.4 - 0.30 N 25 8.42 5.2 1.30 0.33 N 35 - 5.18 - 0.324 DR 22 13.6- - 0.43 0.14 N 35 - - 4.75 0.82 0.15 DR 30 0.53- 0.16 0.038- N 25 1.0 - 0.17 0.034 0.008 G 1300 2300 370 12.5a 92.3b R ] 860 1100 190 62.1a 42.1b R