Develop a simple backmixed model to evaluate the performance of a gas-separation system. Here a feed gas enters a high-pressure chamber with a mole fraction of A of \(x_{\mathrm{Af}}\) and leaves enriched with a mole fraction of \(x_{\mathrm{Ae}}\). The exit stream leaving the high-pressure chamber is often referred to as the retentate.

A membrane separates the high-pressure chamber from a permeate side maintained at a low pressure. A permeate stream leaves this section. Exit mole fractions and the permeate flow rates are to be calculated from the model. Generate equations to calculate these quantities.