Finite rate of mass transfer

Considering the end regions of the column as well-mixed stages, with small but finite rates of mass transfer, component balance equations can be derived for end stage 0... 

The apparent dispersion coefficient in Equation 10.8 describes the zone spreading observed in linear chromatography. This phenomenon is mainly governed by axial dispersion in the mobile phase and by nonequilibrium effects (i.e., the consequence of a finite rate of mass transfer kinetics). The band spreading observed in preparative chromatography is far more extensive than it is in linear chromatography. It is predominantly caused by the consequences of the nonlinear thermodynamics, i.e., the concentration dependence of the velocity associated to each concentration. When the mass transfer kinetics is fast, the influence of the apparent axial dispersion is small or moderate and results in a mere correction to the band profile predicted by thermodynamics alone. 

Capillary electrophoresis provides unprecedented resolution. When we conduct chromatography in a packed column, peaks are broadened by three mechanisms in the van Deemter equation (23-33) multiple flow paths, longitudinal diffusion, and finite rate of mass transfer. An open tubular column eliminates multiple paths and thereby reduces plate height and improves resolution. Capillary electrophoresis reduces plate height further by knocking out the mass transfer term that comes from the finite time needed for solute to equilibrate... 

C. Resistance to Mass Transfer In the plate theory, it was assumed that the transfer of solute molecules between the mobile phase and the stationary phase was instantaneous. In the rate theory, it is accepted that there is a finite rate of mass transfer. In addition, molecules of the same species may spend different lengths of time in the stationary and mobile phases (Fig. 1.15). Resistance to mass transfer is represented by the C term of the van Deemter equation. 

In order to assess whether secondary reactions to form CO could be responsible for the experimental CO versus time curve shape, a series-parallel kinetic mechanism was added to the model. Tar and gas are produced in the initial weight loss reaction, but the tar also reacts to form gas. The rate coefficients used are similar to hydrocarbon cracking reactions. Fig. 5 presents the model predictions for a single pellet length. It is observed that the second volatiles maximum is enhanced. For other pellet lengths, the time of the second peak follows the same trends as in the experiments. While the physical model might be improved by the inclusion of finite rates of mass transfer, the porosity is quite large and Lee, et al have verified volatiles outflow is... 

Concentration polarization occurs because of the finite rate of mass transfer from the solution to the electrode surface. Electron transfer between a reactive species in a solution and an electrode can take place only from the interfacial region located immediately adjacent to the surface of the electrode this region is only a fraction of a nanometer in thickness and contains a limited number of reactive ions or molecules. For there to be a steady current in a cell, the interfacial region must be continuously replenished with reactant from the bulk of the solution. That is, as... 

Therefore, Giddings [67] has demonstrated that nonequilibrium effects resulting from the finite rate of mass transfer kinetics can be treated as a contribution to the axial dispersion, itself the result of axial molecular diffusion, the tortuosity of the packing, the anastomosis of the network of interparticle channels where the stream of mobile phase flows, and the nonhomogeneity of the coliunn packing. The axial diffusion and column tortuosity account for the B term of the classical Knox equation ... 

Originally, Houghton [13] derived his equation with the assumption that the mass transfer kinetics is infinitely fast but that axial dispersion caimot be neglected. In view of the previous discussion (Section 10.1), we can extend the validity of the Houghton approach to the case of a finite rate of mass transfer, by lumping axial dispersion and mass transfer contributions into an apparent dispersion coefficient. 

The transport-dispersive model assumes infinitely fast kinetics of adsorption-desorption but a finite rate of mass transfer, following the solid film linear driving force equation... 

For a single-zone equivalent TMB model, an analytical solution is available for a linear isotherm, considering both axial mixing and a finite rate of mass transfer which is accounted for with the linear driving force (LDF) model model 2a) [18]. 

Apparent dispersion coefficient, Dapi The apparent dispersion coefficient lumps all the contributions to axial dispersion arising from axial molecular diffusion, tortuosity, eddy diffusion, and from a finite rate of mass transfer, adsorption-desorption, or other phenomena, such as reactions, in which the eluites may be involved. It is used in the equilibrium-dispersive model of chromatography to ac-coimt for the finite efficiency of the column (Eq. 2.53 and 10.11). See equilibrium-dispersive model. 

Equilibrium-dispersive model Model of chromatography assuming near equilibrium between the stationary and the mobile phases. Specifically, it assumes that the concentrations in these two phases are related by Ae isotherm equation, and that the effect of the finite rate of mass transfer can be lumped together with the axial dispersion coefficient. This model is valid when the column efficiency is larger than a few hundred plates. 

Shock Layer Because the efficiency of actual columns is finite, concentration shocks are not stable. They are eroded by axial dispersion and the finite rate of mass transfer. A steep concentration gradient is formed instead. The steepness of the profile depends on the axial dispersion and the mass transfer resistance. In frontal analysis, a constant pattern, or steady-state profile, forms after an infinite period of time and an infinite migration length. In this case, the shock layer profile... 

First, the system deviates from ideality as there is a finite rate of mass transfer of solute molecules across the chromatographic interface. The contribution to the overall HETP arising from this kinetic control of the sorption-desorption process increases with increasing mobile phase flow-rate. 

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... 

Broadening by finite rate of mass transfer broadening (21-6)... 

Capillary electrophoresis can provide extremely narrow bands. Three mechanisms of band broadening in chromatography are longitudinal diffusion (B in the van Deemter equation 21-7), the finite rate of mass transfer between the stationary and mobile phases (C in the van Deemter equation), and multiple flow paths around particles (A in the van Deemter equation). An open tubular column in chromatography or electrophoresis reduces band broadening (relative to that of a packed column) by eliminating multiple flow paths (the A term). Capillary electrophoresis further... 

The first difficulty arises in that the kinetics of mass transfer between the mobile and stationary phases are not always rapid. If one tries to introduce a finite rate of mass transfer, however, except for the simplest of cases (which conditions are not at all relevant to preparative LC) the set of equations becomes intractable and no solution can be found. Hence we are forced to assume instantaneous mass transfer in order to proceed. 

---