Permeabilities of carbon dioxide and methane in a cellulose acetate membrane were measured as \(\mathrm{P}_{\mathrm{CO} 2}=15.0\) \(\times 10^{-10}\) and \(\mathrm{P}_{\mathrm{CH} 4}=0.48 \times 10^{-10}[\mathrm{cc}(\mathrm{STP}) \mathrm{cm}] /\left[\mathrm{cm}^{2} \mathrm{~s}(\mathrm{~cm} \mathrm{Hg})\right]\) (Chern et al., 1985). The active layer of the membrane is \(0.0001 \mathrm{~cm}\) thick. We want to remove \(\mathrm{CO}_{2}\) from methane. The feed gas is \(32.3 \mathrm{~mol} \% \mathrm{CO}_{2}\), and the remainder is methane. Retentate pressure is \(18.2 \mathrm{~atm}\), and permeate pressure is \(1.08 \mathrm{~atm}\). Feed flow rate is \(12.4 \mathrm{~m}^{3} \mathrm{STP} / \mathrm{s}\). The cut is set as \(\theta=0.310\). Determine the value of \(\mathrm{y}_{\mathrm{CO} 2, \mathrm{out}} \mathrm{y}_{\mathrm{CO} 2, \mathrm{p}, \mathrm{avg}}\), and total area in \(\mathrm{m}^{2}\) for the following configurations:

a. A perfectly mixed membrane module

b. A crossflow membrane module