Mass transfer resistances, combining

An alternative strategy for fast liquid chromatography uses short columns packed with small particles operated at high flow rates and often elevated temperatures to separate simple mixtures under conditions were resolution is compromised but still adequate for identification purposes [252-258]. Small diameter particles provide larger plate numbers by virtue of their relatively small interparticle mass transfer resistance combined with a shallow increase in the reduced plate height as the reduced mobile... 

In contrast, physical adsorption is a very rapid process, so the rate is always controlled by mass transfer resistance rather than by the intrinsic adsorption kinetics. However, under certain conditions the combination of a diffiision-controUed process with an adsorption equiUbrium constant that varies according to equation 1 can give the appearance of activated adsorption. 

Methods for analysis of fixed-bed transitions are shown in Table 16-2. Local equilibrium theoiy is based solely of stoichiometric concerns and system nonlinearities. A transition becomes a simple wave (a gradual transition), a shock (an abrupt transition), or a combination of the two. In other methods, mass-transfer resistances are incorporated. 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

The expression for the effectiveness factor q in the case of zero-order kinetics, described by the Michaelis-Menten equation (Eq. 8) at high substrate concentration, can also be analytically solved. Two solutions were combined by Kobayashi et al. to give an approximate empirical expression for the effectiveness factor q [9]. A more detailed discussion on the effects of internal and external mass transfer resistance on the enzyme kinetics of a Michaelis-Menten type can be found elsewhere [10,11]. 

Acrivos, A., On the combined effect of longitudinal diffusion and external mass transfer resistance in fixed bed operations. Chem. Eng. Sci. 13, 1 (1960). 

Chlorination processes in bubble column reactors<9> are unusual in showing a significant gas-phase resistance to mass transfer. It will be seen from the low value of the Henry law constant 3 in the list of data for the example below, that the solubility of chlorine in toluene is much greater than the solubility of either the carbon dioxide or oxygen considered in the previous examples. This means that when the gas-phase mass transfer resistance is taken in combination with the liquid-phase resistance according to equation 4.19 which is derived in Volume 2, Chapter 12, then the gas side contribution to the resistance is much greater if 3 is small. 

HETP Height equivalent to a theoretical plate. A measure of the combined effects of axial mixing and finite mass transfer resistance in causing deviations from ideal (equilibrium) behavior in a chromatographic column or in a countercurrent contact system. The definitions of HETP in these two cases are somewhat different,... 

Both mass transfer resistances - as can be seen from the analytical solutions -act in a different way and cannot be combined to a so-called overall transfer resistance . The introduction of such a lumped parameter will hide essential physical effects, evoked by these two resistances separately. 

In order to avoid the unfavorable process conditions, different flue-gas treatment processes for combustion plants based on catalytic filters were developed, which combine fly-ash removal with SCR of ISKh with NH3 [4—8], The advantages of these processes are space and treatment-cost savings, reduced internal and external mass transfer resistances compared to honeycomb SCR catalysts, heat recovery from offgases with good efficiency, and low corrosion problems due to the removal of both dust and NOx at high temperatures. 

As mentioned, from the reaction kinetics viewpoint the behavior of zeolite catalysts shows large variability. In addition, the apparent kinetics can be affected by pore diffusion. The compilation of literature revealed some kinetic equations, but their applicability in a realistic design was questionable. In this section we illustrate an approach that combines purely chemical reaction data with the evaluation of mass-transfer resistances. The source of kinetic data is a paper published by Corma et al. [7] dealing with MCM-22 and beta-zeolites. The alkylation takes place in a down-flow liquid-phase microreactor charged with catalyst diluted with carborundum. The particles are small (0.25-0.40 mm) and as a result there are no diffusion and mass-transfer limitations. 

The orthogonal collocation polynomial approximation using a single parameter trial function was employed to solve equations (l)-(3), In addition to the solution for time concentration and activity profiles, effectiveness factors representing the combined effect of mass transfer resistance and poisoning in terms of pellet surface conditions were computed according to... 

A very detailed study of the combined effects of axial dispersion and mass-transfer resistance under a constant pattern behavior has been conducted by Rhee and Amundson [10]. They used the shock-layer theory. The shock layer is defined as a zone of a breakthrough curve where a specific concentration change occurs (i.e., a concentration change from 10% to 90%). The study of the shock-layer thickness is a new approach to the study of column performance in nonlinear chromatography. The optimum velocity for minimum shock-layer thickness (SLT) can be quite different from the optimum velocity for the height equivalent to a theoretical plate (HETP) [9]. 

Several models use the mass balance in Eq. 2.2 (ideal and equUibrimn-disper-sive models. Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accoimts also for the effect of the mass transfer resistances. This is legitimate imder certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models. Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or FOR model, see later Sections 2.1.7 and 2.2.4). 

This coefficient combines the broadening effects of axial dispersion and the mass transfer resistances. The former effects decrease with increasing mobile phase velocity while the latter increase, hence there is an optimum velocity for which H is minimum. The solution of the general rate model shows that H is related to the column parameters through the equation... 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

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