A four-zone SMB with one column per zone is separating pyrene (P) and anthracene (A). The columns are \(100.0 \mathrm{~cm}\) long, and the switch time \(\mathrm{t}_{\mathrm{sw}}=312.5 \mathrm{~s}\). The solute velocities have been measured as \(\mathrm{u}_{\mathrm{P}}=0.0664 \mathrm{v}\) and \(\mathrm{u}_{\mathrm{A}}=0.1066 \mathrm{v}\) ( \(\mathrm{v}\) is interstitial velocity). The isotherms are linear, and \(\varepsilon_{\mathrm{e}}=0.37\) and \(\varepsilon_{\mathrm{p}}=0\). The following superficial velocities are used:

\(\mathrm{v}_{3 . \text { super }}=1.332 \mathrm{~cm} / \mathrm{s}, \mathrm{v}_{2 . \text { super }}=1.628 \mathrm{~cm} / \mathrm{s}, \mathrm{v}_{1 . \text { super }}=0.888 \mathrm{~cm} / \mathrm{s}\), and \(\mathrm{v}_{4 . \text { super }}=2.0 \mathrm{~cm} / \mathrm{s}\) (the geometry of the SMB is shown in Figure 20-14B).

a. What are the linear isotherm values if \(\mathrm{q}\) and \(\mathrm{c}\) are both in \(\mathrm{g} / \mathrm{L}\) ?

b. Are the conditions for separation in Eqs. (20-30) satisfied?

c. Determine the product superficial velocities.

d. Determine the times required and the distances traveled for the fastest (enters as feed at time 0 ) and the slowest (enters at time \(312.4+\mathrm{s}\) ) anthracene to exit the SMB. The time required can be important for unstable compounds.

e. Optional. Determine the time and distance traveled for pyrene.