Kinetic Models and Multicomponent Problems

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 

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Kinetic analysis usually employs concentration as the independent variable in equations that express the relationships between the parameter being measured and initial concentrations of the components. Such is the case with simultaneous determinations based on the use of the classical least-squares method but not for nonlinear multicomponent analyses. However, the problem is simplified if the measured parameter is used as the independent variable also, this method resolves for the concentration of the components of interest being measured as a function of a measurable quantity. This model, which can be used to fit data that are far from linear, has been used for the resolution of mixtures of protocatechuic... 

A reactive transport model in a more general sense treats a multicomponent system in which a number of equilibrium and perhaps kinetic reactions occur at the same time. This problem requires more specialized solution techniques, a variety of which have been proposed and implemented (e.g., Yeh and Tripathi, 1989 Steefel and MacQuarrie, 1996). Of the techniques, the operator splitting method is best known and most commonly used. 

Equations 3.15, 3.17 and 3.19 provide the flux relationships in the limiting regimes. There remains the problem of finding the flux relationships in intermediate situations, where the pore size is comparable to the mean free path and the mixture is a multicomponent one. At present, no quantitative kinetic theory exists for flow in the transition region where the dimensions of A and dt are comparable. Therefore different simplified models have been developed. 

The extension of the CNT to homogeneous nucleation in atmospheric, essentially multicomponent, systems have faced significant problems due to difficulties in determining the activity coefficients, surface tension and density of binary and ternary solutions. The BHN and THN theories have been experiences a number of modifications and updates. At the present time, the updated quasi-steady state BHN model [16] and kinetic quasi-imary nucleation theory [24,66], and classical THN theory [25,33] and kinetic THN model constrained by the experimental data... 

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-... 

A qualified question is then whether or not the multicomponent models are really worthwhile in reactor simulations, considering the accuracy reflected by the flow, kinetics and equilibrium model parts involved. For the present multiphase flow simulations, the accuracy reflected by the flow part of the model is still limited so an extended binary approach like the Wilke model sufEce in many practical cases. This is most likely the case for most single phase simulations as well. However, for diffusion dominated problems multicomponent diffusion of concentrated ideal gases, i.e., for the cases where we cannot confidently designate one of the species as a solvent, the accuracy of the diffusive fluxes may be significantly improved using the Maxwell-Stefan approach compared to the approximate binary Fickian fluxes. The Wilke model might still be an option and is frequently used for catalyst pellet analysis. 

To describe the peak shapes of a separation under overload conditions a clear understanding of how the competitive phase equilibria, the finite rate of mass transfer, and dispersion phenomena combine to affect band profiles is required [ 11,66,42,75,76]. The general solution to this problem requires a set of mass conservation equations appropriate initial and boundary conditions that describe the exact process implemented the multicomponent isotherms and a suitable model for mass transfer kinetics. As an example, the most widely used mass conservation equation is the equilibrium-dispersive model... 

Adsorption kinetics of a single particle (activated carbon type) is dealt with in Chapter 9, where we show a number of adsorption / desorption problems for a single particle. Mathematical models are presented, and their parameters are carefully identified and explained. We first start with simple examples such as adsorption of one component in a single particle under isothermal conditions. This simple example will bring out many important features that an adsorption engineer will need to know, such as the dependence of adsorption kinetics behaviour on many important parameters such as particle size, bulk concentration, temperature, pressure, pore size and adsorption affinity. We then discuss the complexity in the dealing with multicomponent systems whereby governing equations are usually coupled nonlinear differential equations. The only tool to solve these equations is... 

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