Explicit equations for the height and position of the first component shock for binary mixtures with competitive Langmuir isotherms under ideal conditions

Explicit equations for the height c1S and retention time tR,1 of the pure first component shock in the case of a narrow rectangular injection pulse of a binary mixture with competitive Langmuir isotherms were derived within the frame of the equilibrium theory. The height of the first shock is obtained as an only positive root of a quartic equation. Hence, it was shown that, for binary Langmuir systems, the individual concentration profiles at the column outlet can be expressed entirely in closed-form. In addition, a novel, simple parametric representation that gives the trajectory of the first shock in the distance–time diagram as a function of c1S was derived. The practical relevance of the new equations was demonstrated by utilizing them for optimization of batch chromatography. It was shown that c1S increases and tR,1 decreases with increasing duration of injection for given feed concentrations when the pure first component plateau is eroded during elution. The derivative of the cycle time with respect to the duration of injection is always more than unity. For this reason, the maximum productivity of more retained component is obtained when the duration of injection is selected so that the purity constraint can be fulfilled by having 100% yield. For the less retained component, an implicit expression for the maximum productivity was derived. When the injected loadings are constant, tR,1 decreases with increasing feed concentrations while c1S and the cycle time are independent of them. In addition, the productivities of both components always increase with increasing feed concentrations.