P$6\mathrm{.}34$ Sulfur dioxide $(S{O}_2)$ is manufactured by the oxidation of solid sulfur $S,$ using air $(79$mol$\%N_2,21$mol$\%O_2)$ as the source of oxygen. $S$ and $O_2$ are fed at stoichiometric ratio and the reaction goes to $100\%$ completion. The reaction generates a lot of heat, so cool inert gas would be useful to maintain a cooler reactor temperature. Ideally there would be $5\mathrm{.}0$ moles inert gas per mole $O_2$ fed to the reactor. Eddie Engineer proposes to use nitrogen in the air as the inert gas. In this scheme, air and sulfur are fed to the reactor, and the sulfur dioxide is separated from nitrogen downstream of the reactor. Eddie estimates that the separator will recover $97\%$ of the $S{O}_2$ in a product stream that is $99$mol$\%S{O}_2.$ The other product, which is nitrogen$-$rich, could be recycled to provide sufficient inert gas to the reactor inlet. Draw and label a flow sheet that illustrates Eddie's idea. Complete a DOF analysis, choosing $100$gmo$\frac{l}{m}in$ sulfur feed as a basis. What fraction of the nitrogen$-$rich stream should be recycled? Calculate the flow rates and compositions of all streams. $($Hint: First solve the case with $100\%$ recovery of $S{O}_2,$ to obtain an initial estimate.$)$