Isotherms for dilute amounts of toluene and xylene adsorbed on silica gel from n-heptane are linear at low concentrations (Matz and Knaebel, 1991). For toluene, \(\mathrm{q}_{\mathrm{Tol}}=\mathrm{K}_{\mathrm{Tol}}^{\prime}(\mathrm{T}) \mathrm{x}_{\mathrm{Tol}}\) where \(\mathrm{q}_{\mathrm{Tol}}\) is \(\mathrm{g}\) toluene/ \(\mathrm{g}\) adsorbent and \(\mathrm{x}_{\mathrm{Tol}}\) is in \(\mathrm{g}\) toluene/ \(\mathrm{g}\) solution (mass fraction) (Matz and Knaebel, 1991). The linear constants are functions of temperature. For toluene, \(\mathrm{q}_{\mathrm{Tol}}=\) \(17.46 \mathrm{x}_{\mathrm{Tol}}\) at \(0^{\circ} \mathrm{C}, \mathrm{q}_{\mathrm{Tol}}=7.77 \mathrm{x}_{\mathrm{Tol}}\) at \(30^{\circ} \mathrm{C}, \mathrm{q}_{\mathrm{Tol}}=5.16 \mathrm{x}_{\mathrm{Tol}}\) at \(35^{\circ} \mathrm{C}\), and \(\mathrm{q}_{\mathrm{Tol}}=1.23 \mathrm{x}_{\mathrm{Tol}}\) at \(80^{\circ} \mathrm{C}\). Isotherms for xylene at low concentrations have similar form to those for toluene, \(\mathrm{q}_{\mathrm{xy}}=\mathrm{K}_{\mathrm{xy}}^{\prime}(\mathrm{T}) \mathrm{x}_{\mathrm{Txyl}}\) where \(\mathrm{q}_{\mathrm{xy}}\) is \(\mathrm{g}\) xylene/g adsorbent and \(\mathrm{x}_{\mathrm{xy}}\) is in \(\mathrm{g}\) xylene/g solution (mass fraction). Isotherms for xylene: \(\mathrm{q}_{\mathrm{xy}}=22.36 \mathrm{x}_{\mathrm{xy}}\) at \(0^{\circ} \mathrm{C}, \mathrm{q}_{\mathrm{xy}}=12.03 \mathrm{x}_{\mathrm{xy}}\) at \(30^{\circ} \mathrm{C}, \mathrm{q}_{\mathrm{xy}}=6.28\) \(\mathrm{x}_{\mathrm{xy}}\) at \(35^{\circ} \mathrm{C}\), and \(\mathrm{q}_{\mathrm{xy}}=2.01 \mathrm{x}_{\mathrm{xy}}\) at \(80^{\circ} \mathrm{C}\). Wall heat capacities can be ignored, heat of adsorption is negligible, and no adsorption of \(n\)-heptane. Find appropriate zone, product, and desorbent interstitial velocities (based on column diameter) for an SMB separating toluene and xylene if interstitial feed velocity \(v_{F}=1.0 \mathrm{~cm} / \mathrm{min}\) and temperature is \(30^{\circ} \mathrm{C}\). Switching time is \(100.0 \mathrm{~min}\) and \(\mathrm{L}=15.0 \mathrm{~cm}\). Physical properties: \(\varepsilon_{\mathrm{e}}=0.43, \varepsilon_{\mathrm{p}}=0.48, ho_{\mathrm{s}}=\) \(2100 \mathrm{~kg} / \mathrm{m}^{3}, ho_{\mathrm{f}}=684 \mathrm{~kg} / \mathrm{m}^{3}, \mathrm{~K}_{\mathrm{d}}=1.0, \mathrm{C}_{\mathrm{ps}}=920 \mathrm{~J} / \mathrm{kg}^{\circ} \mathrm{C}\), and \(\mathrm{C}_{\mathrm{pf}}=1841 \mathrm{~J} /\left(\mathrm{kg}{ }^{\circ} \mathrm{C}\right)\).