Mass balance models, chromatography

A much simpler model in concept is the Craig model of chromatography. This is more a phenomenological model in that it tries to mimic the physical processes occurring in the column. As we will see, it uses most of the same assumptions as are necessary for the solution of the mass balance model and it also gives closely similar results. 

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

The equilibrium models of chromatography are given by the mass balance equation given in Equation 10.8 and a proper isotherm equation, q = f(C), should be used to relate the mobile phase and stationary phase concentrations. 

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

All mathematical models of chromatography consist of a differential mass balance equation for each component involved and the equation expresses mass conservation in the process [13, 109], In the ED model the mass balance equation for a single component is expressed as follows ... 

All terms of the mass balance for liquid chromatography have now been specified in detail, which represents the most extended model discussed. Table 6.1 summarizes these elements. 

In the following the most relevant models for liquid chromatography are derived in a bottom-up procedure related to Fig. 6.2. To illustrate the difference between these models their specific assumptions are discussed and the level of accuracy and their field of application are pointed out. The mass balances are completed by their boundary conditions (Section 6.2.7). For the favored transport dispersive model a dimensionless representation will also be presented. 

The simplest model takes into account convective transport and thermodynamics only. It assumes local equilibrium between mobile and stationary phase. This model, also called the ideal or basic model of chromatography, was described first by Wicke (1939) for the elution of a single component. Subsequently, De Vault (1943) derived the correct form of the mass balance. 

Guiochon, G., Czolc, M. The physical sense of simulation models of liquid chromatography propagation through a grid or solution of the mass balance equation, Anal. Chem., 1990, 62, 189-200. 

Since Eq. 2.2 contains two functions, C, and another equation or relationship between them is necessary for its solution. Depending on the model of chromatography used, Eq. 2.2 will be accompanied by a mass balance in the stationary phase and a kinetic equation, by a lumped mass transfer kinetic equation, or by an adsorption isotherm (Section 2.1.3). 

The detailed study of the mass transfer kinetics is necessary in certain problems of chromatography in which the column efficiency is low or moderate. Complex models are then useful. The most important ones are the General Rate Model [52,62] and the FOR model (see next Section) [63]. To study the mass transfer kinetics, these models need to consider separately the mass balance of the feed components in the two different fractions of the mobile phase the one that percolates through the bed of the solid phase (column packed with fine particles or monolithic column) and the one that is stagnant inside the pores of the packing material. 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-... 

Figures 4.26A and 4.26B compare the results of the experimental determination of isotherms using the traditional mass balance method (MMB) and those obtained with MMC. The adsorption isotherm predicted by MMC deviates significantly from the isotherm data obtained by MMB. This may be due to the limited applicability of the Langmuir competitive model for the modeling of the adsorption behavior even of such simple systems as p-cresol and phenol in reversed-phase chromatography. Figures 4.26C and 4.26D compare the results obtained by MMB and HMMB for the same system. Over most of the concentration range, the agreement between the experimental data and the results of these two methods is...

The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. 

The ideal model of chromatography, which has great importance in nonlinear chromatography, has little interest in linear chromatography. Along an infinitely efficient column, with a linear isotherm, the injection profile travels unaltered and the elution profile is the same as the injection profile. We also note here that, because of the profound difference in the formulation of the two models, the solutions of the mass balance equation of chromatography for the ideal, nonlinear model and the nonideal, linear model rely on entirely different mathematical techniques. 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

The equilibrium-dispersive model had been discussed and studied in the literature long before the formulation of the ideal model. Bohart and Adams [2] derived the equation of the model as early as 1920, but it does not seem that they attempted any calculations based on this model. Wicke [3,4] derived the mass balance equation of the model in 1939 and discussed its application to gas chromatography on activated charcoal. In this chapter, we describe the equilibrium-dispersive model, its historical development, the inherent assumptions, the input parameters required, the methods used for the calculation of solutions, and their characteristic features. In addition, some approximate analytical solutions of the equilibrium-dispersive model are presented. 

The model of Lapidus and Amundson [5] is the focal point of study of linear and nonlinear chromatography. Since in chromatography we have two independent variables, z and t, and two dependent variables, the concentrations of the solute in the mobile and the stationary phases, C and Cg, respectively, two equations are required for the model to permit the calculation of C(x, t) and Cs x, t). The model of Lapidus and Amundson considers a set of two partial differential equations for a single component. The first equation is the mass balance equation (Eq. 2.2)... 

Accordingly, the equilibrium-dispersive model of chromatography for a single component is represented by one single partial differential equation, the mass balance equation... 

In the equilibrium-dispersive model of chromatography, however, we assume that Eq. 10.4 remains valid. Thus, we use Eq. 10.10 as the mass balance equation of the component, and we assume that the apparent dispersion coefficient Da in Eq. 10.10 is given by Eq. 10.11. We further assume that the HETP is independent of the solute concentration and that it remains the same in overloaded elution as the one meastued in linear chromatography. As shown by the previous discussion this assxunption is an approximation. However, as we have shown recently [6], Eq. 10.4 is an excellent approximation as long as the column efficiency is greater than a few hundred theoretical plates. Thus, the equilibriiun-dispersive model should and does account well for band profiles under almost all the experimental conditions used in preparative chromatography. In the cases in which the model breaks down because the mass transfer kinetics is too slow, and the column efficiency is too low, a kinetic model or, better, the general rate model (Chapter 14) should be used. 

We have discussed the theory of system peaks in linear chromatography [20]. The discussion is based on the use of the equilibrium-dispersive model. The mass balance equations are written for the n components of the sample and for the p additives ... 

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. 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

This chapter deals essentially with the apphcations of the theory of chromatography to the calculation of solutions of the SMB model in different cases of general interest. The theoretical tools required are a general model of the SMB process and a model for its columns. The former is an integral mass balance that is easy to write. The possible column models were described in the previous chapters. Finally, an accurate model of the competitive isotherms of the feed components is necessary. 

General rate model Model of chromatography taking into account separately the contributions of each of the various sources of mass transfer resistances. It includes a bulk and a pore mass balance equations, and the relevant kinetic equations. 

For a number of nonlinear and competitive isotherm models analytical solutions of the mass balance equations can be provided for only one strongly simplified column model. This is the ideal model of chromatography, which considers just convection and neglects all mass transfer processes (Section 6.2.3). Using the method of characteristics within the elegant equilibrium theory, analytical expressions were derived capable to calculate single elution profiles for single components and mixtures (Helfferich and Klein, 1970 Helfferich and Carr 1993 Helfferich and Whitley 1996 Helfferich 1997 Rhee, Aris, and Amundson, 1970 ... 

Band profiles in chromatography are obtained as solutions to the differential mass-balance equations for the mixture components (Eq. 11). The models differ in the way such balance equations are simplified and solved. 

Guiochon et al. [3] also arrived at equations for retention and efficiency, based upon the solution of differential mass balance equations for chromatography using the Ideal Model of chromatography. This makes the major assumption that the column efficiency is infinite, under which conditions it is possible to reach an analytical solution of the equations. Their equation for capacity factor converges with that of Snyder et al. at high values of efficiency and has the virtue of simplicity ... 

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