Nonlinear, Ideal Model of SMB

The transient behavior of a continuous countercurrent multicomponent system was considered in detail by Rhee, Aris and Amimdson [22,23] from the perspective of the equilibrium theory, i.e., assuming that axial dispersion and the mass transfer resistances are negligible and that equilibrium is established everywhere, at every time along the colinnn. The final steady-state predicted by the equilibrium theory is simply a uniform concentration throughout the colimm, with a transition at one end or the other. Therefore, the equilibriinn theory analysis is of lesser practical value for a coimtercurrent system, which normally operates rmder steady-state conditions, than for a fixed-bed (i.e., an SMB) system, which normally operates under transient conditions. The equilibrium theory analysis, however, reveals that, under different experimental conditions, several different steady-states are possible in a coimtercurrent system. It shows how the evolution of the concentration profiles may be predicted in order to determine which state is obtained in a particular case. 

In general, the feed injection in a countercurrent system takes place at one end of the column. In a long enough column, equilibrium is reached at a certain distance from this zone. The size of this zone depends only on the fluid and solid phase concentrations of the feed stream components and on the flow rate ratio, my. It is independent of the initial state of the column [22,23]. 

1 Optimization of the Operating Conditions for a Nonlinear Isotherm Using the Triangle Method 

One approach for selecting the operation conditions consists in applying a McCabe-Thiele-like analysis to an ideal stage-by-stage model of the unit [13]. This approach can be applied to systems described by any kinds of isotherm but it is limited to binary separations. 

A second approach is the triangle method which was developed based on the equilibrium theory model which assumes that the adsorption equilibrium is established everywhere at any time in the column. The equivalent TMB configuration with a four-section emit will be considered here. The model equations consist in four sets of mass balance equations, one for each section j j = 1,- , 4), with the relevant boundary conditions and the integral material balances at the column ends and at the nodes of the unit [16,28]. These equations were given earlier, in Section 17.2 (Eqs. 17.4 to 17.6). 

---

In ideal chromatography, we assume that the column efficiency is infinite, or in other words, that the axial dispersion is negligibly small and the rate of the mass transfer kinetics is infinite. In ideal chromatography, the surface inside the particles is constantly at equilibrium with the solution that percolates through the particle bed. Under such conditions, the band profiles are controlled only by the thermodynamics of phase equilibria. In linear, ideal chromatography, all the elution band profiles are identical to the injection profiles, with a time or volume delay that depends only on the retention factor, or slope of the linear isotherm, and on the mobile phase velocity. This situation is unrealistic, and is usually of little importance or practical interest (except in SMB, see Chapter 17). By contrast, nonlinear, ideal chromatography is an important model, because the profiles of high-concentration bands is essentially controlled by equilibrium thermodynamics and this model permits the detailed study of the influence of thermodynamics on these profiles, independently of the influence of the kinetics of mass transfer... 

---