Solution of the Breakthrough Curve under Constant Pattern Condition

The same approaches that were successful in linear chromatography—the use of either one of several possible liunped kinetic models or of the general rate model — have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotiienn provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. 

Finally, we note that the solution of the general rate model for a multicomponent mixture can be applied simply to the single component case by choosing 1 as the number of components. Thus, the general rate model and its solution are discussed in the next chapter to avoid repetitive discussions. 

1 Solution of the Breakthrough Curve under Constant Pattern Condition 

1 Anal3 ical Solution for the Constant Pattern Profile 

Bohart and Adams [4] were the first to recognize the existence of this constant pattern behavior. They assumed irreversible adsorption and the control of the mass transfer kinetics by the rate of adsorption. Later Wicke [5] gave an analytical solution in the case of irreversible adsorption, assuming that the mass transfer kinetics are controlled by the diffusion rate inside the particles. The asymptotic nature of constant pattern behavior has been discussed by Cooney and Lightfoot [6], who demonstrated its existence for aU convex-upward isotherms. 

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