Band broadening transfer, resistance

Resistance to sass transfer in both the stationary and mobile phase prevents the existence of an instantaneous eguilibriuB. Under sost practical conditions this is the dosinant cause of band broadening. 

The plate theory assumes that an instantaneous equilibrium is set up for the solute between the stationary and mobile phases, and it does not consider the effects of diffusional effects on column performance. The rate theory avoids the assumption of an instantaneous equilibrium and addresses the diffusional factors that contribute to band broadening in the column, namely, eddy diffusion, longitudinal diffusion, and resistance to mass transfer in the stationary phase and the mobile phase. The experimental conditions required to obtain the most efficient system can be determined by constructing a van Deemter plot. 

System efficiency good packing quality and low band-broadening due to axial dispersion and mass transfer resistance are necessary... 

Band broadening effects such as dispersion and mass transfer resistance are represented by the number of tanks (or stages) N. This can be explained by evaluating the moments of the analytical solution of Eq. 6.96. For linear isotherms and the injection... 

Chapters 10 to 13 review the solutions of the equilibrium-dispersive model for a single component (Chapter 10), and multicomponent mixtures in elution (Chapter 11) and in displacement (Chapter 12) chromatography and discuss the problems of system peaks (Chapter 13). These solutions are of great practical importance because they provide realistic models of band profiles in practically all the applications of preparative chromatography. Mass transfer across the packing materials currently available (which are made of very fine particles) is fast. The contribution of mass transfer resistance to band broadening and smoothing is small compared to the effect of thermodynamics and can be properly accounted for by the use of an apparent dispersion coefficient independent of concentration (Chapter 10). 

The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of ad-sorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. 

In this model we assume that the contributions of the mass transfer resistances to band broadening are negligible, but that the kinetics of adsorption-desorption is slow. So the behavior of the chromatographic system is described by the mass balance Eq. 6.40. If we assume now that the kinetics of adsorption-desorption is of first order, we have the kinetic equation ... 

In nonlinear chromatography, most of band broadening is due to the thermodynamic contribution. As soon as is significant, the kinetic contribution becomes small compared to the thermod5mamic contribution and the difference between these two equations has little practical consequence. The contribution Hq is significant only when mass transfer kinetics is slow and the contribution of the mass transfer resistances to the band profile is greater than the contribution originating from the nonlinear behavior of the isotherm. 

The behavior of chromatographic columns operated in gradient elution, under linear conditions i.e., assuming linear isotherms for all the solutes) has been studied theoretically by numerous authors [2,4-10]. The most comprehensive treatment is that based on the linear solvent strength (LSS) theory of Snyder et al. [5,6]. This theory has formd widespread acceptance [7,8] and has been extended to include the contributions of the various mass transfer resistances to band broadening [9-11]. It assumes the injection of infinitesimal pulses of a feed and a linear gradient of the volume fraction of a mobile phase modifier, cf). 

Axial dispersion, D When a band migrates along a column packed with non-porous particles, it spreads axially because of the combination effects of axial diffusion and the inhomogeneity of the pattern of flow velocity in a packed bed. This combination of effects is accounted for by a single term, proportional to the axial dispersion coefficient. It is independent of the mass transfer resistance and of the other contributions of kinetic origin to band broadening. 

Efficiency, N The column efficiency characterizes the combined effects of the sources of band broadening due to axial dispersion and mass transfer resistance. It is derived from the width of the elution peak observed as the response to the injection of a small, narrow pulse of a dilute solution of a compoimd. It is difficult to correct for the contribution of the extracolumn sources of band broadening which have to be kept small. In preparative and nonlinear chromatography, there is a correlation between the colmnn efficiency and both the steepness of the shock layer and the duration of the band beyond the retention time However, the column efficiency is essentially a concept of linear chromatography, and it is difficult to extend to and use in nonlinear chromatography, except through the shock layer thickness concept. 

Linear addition rule Rule stating that the contributions of axial dispersion and all the sources of mass transfer resistance to the band broadening are additive. This rule is valid in linear chromatography, but has limited applicability in nonlinear chromatography. 

A mathematical analysis of a crossflow magnetically stabilized fluidized bed chromatograph has been presented (14). The geometry of this system is similar to the rotating annular chromatograph and therefore the modeling approach is quite similar to that reported here. A parametric sensitivity study was conducted and the results indicated that the extent of band broadening was most sensitive to two factors. These factors were the external resistance to mass transfer and the width of the feed band. 

Equation 16 shows that the peak variance or band broadening is comprised of individual contributions from different aspects of the separation process. The first term in equation 16 represents the contribution of the width of the feed band to the peak variance. The second term represents the contribution to band broadening from dispersion due to eddy diffusion. The third term represents the contribution of mass transfer effects external to the particles while the fourth term represents the contribution of diffusional resistances within the stationary phase. The significance of each term relative to the total variance depends upon the operating parameters, the column and packing dimensions and the size of the solute. 

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