Extract meta-, ortho-, and para-xylenes from \(n\)-hexane using \(\beta, \beta^{\prime}-\) thiodipropionitrile as solvent. Solvent and diluent ( \(n\)-hexane) are immiscible. Feed flow rate is \(1000.0 \mathrm{~kg} / \mathrm{h}\). Feed is \(\mathrm{x}_{\mathrm{m}-\mathrm{xy}}=0.5 \mathrm{wt} \% \mathrm{~m}-\) xylene, \(\mathrm{x}_{0-\mathrm{xy}}=0.6 \mathrm{wt} \% \mathrm{o}-\mathrm{xylene}\), and \(\mathrm{x}_{\mathrm{p}-\mathrm{xy}}=0.4 \mathrm{wt} \% \mathrm{p}\)-xylene in \(\mathrm{n}-\) hexane. Solvent flow rate is \(20,000 \mathrm{~kg} / \mathrm{h}\). Entering solvent is pure. We desire \(96 \%\) recovery of p-xylene in solvent. Operation is at \(25^{\circ} \mathrm{C}\) and \(1.0 \mathrm{~atm}\). Equilibrium data are: \(\mathrm{K}_{\mathrm{d}, \mathrm{m}-\mathrm{xy}}=0.050, \mathrm{~K}_{\mathrm{d}, \mathrm{o}-\mathrm{xy}}=0.150\) and \(\mathrm{K}_{\mathrm{d}, \mathrm{pxy}}=0.080\) where \(\mathrm{K}_{\mathrm{d}, \mathrm{A}}=\mathrm{y}_{\mathrm{A}}\) (in solvent) \(\mathrm{x}_{\mathrm{A}}\) (in diluent) (Perry and Green, 1984). Use a simple countercurrent cascade.

a. Find outlet \(\mathrm{p}\)-xylene weight fraction, \(\mathrm{x}_{\mathrm{N}, \mathrm{p}-\mathrm{xy}}\).

b. Find \(\mathrm{N}\).

c. Find \(x_{N, 0-x y}\).

d. Find \(x_{N, m-x y}\).

e. What is minimum solvent flowrate ( \(\mathrm{N}\) approaches infinity)?