Analytical and Numerical Solutions of the Kinetic Models

Note that in Eq. 14.48, for the sake of simplicity, the absolute error Arjg — At]Q i) is referred here to the approximate value (Atjg i), not to the true value (Arje). In practice, the result is the same as long as the relative error is small. The relative error given by Eq. 14.48 is obviously small if the second term of the RHS of Eq. 14.45 is small compared to the first one, i.e., if the mass transfer resistances are small, and the column is highly efficient. If this is not true, however, and the second term in the RHS of Eq. 14.45 is dominant, Eq. 14.48 becomes 

The relative error tends toward 0 with decreasing height of the concentration step, and tends toward for very large heights, which might be quite large xmder some 

If the second term of the RHS of Eq. 14.45 accoxmts for a fourth of the plate height, then the limit error made with a high value of k and bCo = 1 is only 25% under the most unfavorable set of experimental conditions. This illustrates the fact that deviations between experimental results and the prediction of the equilibrium-dispersive model will occur preferably at high mobile phase flow velocities. We see from Eq. 14.48 that the error introduced by the use of the equilibrium-dispersive model instead of a kinetic model decreases with increasing values of both Dakf/u = St/Pe and k /(1 -I- The value of D is of the 

we can conclude that, as long as the mass transfer kinetics is reasonably fast, the equilibrium-dispersive model can be used as a first approximation to predict shock layer profiles. As a consequence, the results of calculations of band profiles, breakthrough curves, or displacement chromatograms made with this model can be expected to agree well vsdth experimental results. Conclusions based on the s) stematic use of such calculations have good predictive value in preparative chromatography. 

The kinetics of adsorption-desorption is rarely slow in preparative chromatography, and most examples of a slow kinetics are foimd in bioaffinity chromatography, or in the separation of proteins. Thus, there are few cases in which a reaction-kinetic model is appropriate. This model is important, however, because there are many cases where it is convenient to model the finite rate of the mass transfer 

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A test problem comparing the analytical and numerical solutions of the same problem using a finite element model 12) is illustrated in Figure 2. The solution corresponds to the case of a continuously stirred tank reactor (CSTR) in which first-order kinetics are assumed, and the rates of reaction are comparable to those we have observed in the laboratory (5,10). 

If the effect of dispersion is not taken into accoimt in the apparent number of reaction units (N ), there will be a large difference between the solutions of the Thomas model and the transport-dispersive or the reaction-dispersive models. This is illustrated in Figure 14.14, which compares the analytical solution of the Thomas model and the numerical solution of the reaction-dispersive model. The front of the latter solution is less steep than that of the former because the Thomas model does not take into accoimt axial dispersion, but only the kinetics of adsorption-desorption. 

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... 

The differential equations are stiff that is, several processes are going on at the same time, but at widely differing rates. This is a common feature of chemical kinetic equations and makes the numerical solution of the differential equations difficult. A steady state is never reached, so the equations cannot be solved analytically. Traditional methods, such as the Euler method and the Runge-Kutta method, use a time step, which must be scaled to fit the fastest process that is occurring. This can lead to large number of iterations even for small time scales. Hence, the use of Stella to model this oscillatory reaction would lead to an impossible situation. 

In this expository article, the basic mathematical model of some simple electrochemical processes was discussed. The model is based on the concept of conservation of charge within the electrolyte. The boundary conditions, on the other hand, are problem-specific. The subject of electrode kinetics is central to the proper specification of the boundary conditions. In their most general form, the conditions are nonlinear, leading to a nonlinear boundary value problem. This is closely tied to the nonlinear polarization curves. The analytical solution of the mathematical model is formidable and for moderately simple two-dimensional regions is impossible to obtain. The only feasible approach is numerical simulation. The use of high-speed digital computers is an essential tool in solving such problems. 

Solutions were obtained, either analytically or numerically, on a computer. The quenched-reaction, kinetic model considered that the nucleation sequence of reactions evolves to some time (the quenching time) and then promptly halts. Both kinetic models yield a result having the same general form as the statistical model, namely,... 

A number of kinetic models of various degree of complexity have been used in chromatography. In linear chromatography, all these models have an analytical solution in the Laplace domain. The Laplace-domain solution makes rather simple the calculation of the moments of chromatographic peaks thus, the retention time, the peak width, its number of theoretical plates, the peak asymmetry, and other chromatographic parameters of interest can be calculated using algebraic expressions. The direct, analytical inverse Laplace transform of the solution of these models usually can only be calculated after substantial simplifications. Numerically, however, the peak profile can simply be calculated from the analytical solution in the Laplace domain. 

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. 

Theoretical models (analytical and numerical), developed for simulation of the BOHLM and BAHLM transport kinetics, are based on independent experimental measurements of (a) individual mass-transfer coefficients of the solutes in boundary layers and (b) facilitating parameters of the liquid membrane (LMF potential) and lEM potential in the case of ion-exchange membrane (lEM) application. Satisfactory correlation between experimental and simulated data is achieved. 

At the intermediate wavelengths no useful analytic forms of solution are known. On the other hand, (138) yields readily to numerical solutions. Such results show a smooth interpolation between characteristic hydrodynamic and free-particle behavior. Notice, however, that in this region it is essential to treat the collisions and molecular flow on equal footing for this reason it would be inappropriate to apply either (154) or (155). Since the intermediate k range is particularly relevant to neutron inelastic scattering studies of liquids and related computer molecular dynamics simulations, the validity of our kinetic model solutions is of interest. 

Given this failure of the continuum model, it is evidently necessary to treat the solvent as an assembly of molecules. A hard-sphere model is the first approximation. Kinetic theory of diffusion in dilute gases, where the mean free path is much greater than the collision diameter, is well established it can be extended with some success to dense gases, where the two quantities are more nearly equal, and (more speculatively) to hard-sphere models of liquids, where they are comparable. For these highly mathematical theories the reader may consult more specialised works [14]. Analytical solutions are not always to be expected numerical solutions may be required. Computer-simulation calculations have had considerable success, and with the advent of fast computers have become a major source of understanding of real systems (cf., e.g.. Section 7.3.4.5). 

We shall present here an equilibrium simulation of the transport of a solute across a liquid-liquid interface, which permits to measure the rate constant. This work has been done with the same rationale than other recent molecular dynamics studies of chemical kinetics /5,6/. The idea is to obtain by simulation, at the same time, a computation of the mean potential as a function of the reaction coordinate and a direct measure of the rate constant. The mean potential can then be used as an input for a theoretical expression of the rate constant, using transition state /7/, Kramers /8/ or Grote-Hynes /9/ theories for instance. The comparaison can then be done in order to give a correct description of the kinetics process. A distinct feature of molecular dynamics, with respect to an experimental testing of theoretical results, is that the numerical simulations have both aspects, theoretical and experimental. Indeed, the computation of mean potentials, as functions of the microscopic models used, is simple to obtain here whereas an analytical derivation would be a heavy task. On the other hand, the computation of the kinetics constant is more comparable to an experimental output. 

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