Kinetics axial dispersion coefficient

The axial dispersion coefficient Dax was estimated from conventional residence time analysis for nonreacting conditions [14, 47]. The lines shown in Fig. 12.8 demonstrate that essential features of the reactor behavior are well represented. This also indicates a certain reliability of the derived kinetic equation Eq. (30). 

Chromatography is a powerful separation method because it can be carried out easily under experimental conditions such that the two phases of the system are always near equilibrium. This is because the kinetics of the mass transfers between these phases is usually fast. The separation power of a column, under a given set of experimental conditions, is directly a function of the rate of the mass transfer kinetics and of the axial dispersion coefficient. The scientists involved in the development of stationary phases for chromatography have produced excellent packing materials that permit the achievement of a very large number of equilibrium stages (i.e., theoretical plates) in a column. Thus, as we show later in Chapters 10 and 11, the thermodynamics of phase equilibria is often the main... 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

In the ideal model, we assume that the column efficiency is infinite, hence the rate of the mass transfer kinetics is infinite and the axial dispersion coefficient in the mass balance equation (Eq. 2.2) is zero. The differential mass balances for the two components are written ... 

Jaulmes and Vidal-Madjar [51] studied the influence of the mass transfer kinetics on band profiles, using a Langmuir second-order kinetics, and a constant axial dispersion coefficient, D. They derived numerical solutions using a finite difference algorithm. The influence of the rate constant on the band profile at various sample sizes is illustrated in Figure 14.18. As the mass transfer kinetics slows down, the band broadens and the shock layer becomes thicker. When the sample size increases, however, the influence of thermodynamics on the profile becomes more dominant, as shown by the change in shock layer thickness which decreases with increasing sample size. 

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. 

Experimental extraction curves can be represented by this type of model, by fitting the kinetic coefficients (mass transfer coefficient to the fluid, effective transport coefficient in the solid, effective axial dispersion coefficient representing deviations from plug flow) to the experimental curves obtained fi om laboratory experiments. With optimized parameters, it is possible to model the whole extraction curve with reasonable accuracy. These parameters can be used to model the extraction curve for extractions in larger vessels, such as from a pilot plant. Therefore, the model can be used to determine the kinetic parameters from a laboratory experiment and they can be used for scaling up the extraction. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

For theoretical discussions, the Langmuir isotherm q = aCI l+bC) is convenient. For a review of isotherms used in liquid chromatography, see [23]. With Da = 0 this assumption would be equivalent to assuming that the column efficiency is infinite. However, the finite efficiency of an actual column can be taken into account by including in the axial dispersion coefficient the influences on band profiles due to both axial dispersion and the kinetics of mass transfer ... 

Optimal reactor design is critical for the effectiveness and economic viability of AOPs. The WAO process poses significant challenges to chemical reactor engineering and design, due to the (i) multiphase nature of WAO reactions (ii) temperatures and pressures of the reaction and (iii) radical reaction mechanism. In multiphase reactors, complex relationships are present between parameters such as chemical kinetics, thermodynamics, interphase/intraphase intraparticle mass transport, flow patterns, and hydrodynamics influencing reactant mass transfer. Complex models of WAO are necessary to take into account the influence of catalyst wetting, the interface mass-transfer coefficients, the intraparticle effective diffusion coefficient, and the axial dispersion coefficient. " ... 

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. 

The next level of detail in the model hierarchy of Fig. 6.2 is the so-called dumped rate models" (third from the bottom). They are characterized by a second parameter describing rate limitations apart from axial dispersion. This second parameter subdivides the models into those where either mass transport or kinetic terms are rate limiting. No concentration distribution inside the particles is considered and, formally, the diffusion coefficients inside the adsorbent are assumed to be infinite. 

This noninvasive method could allow the differentiation between the various packing materials used in chromatography, a correlation between the chromatographic properties of these materials that are controlled by the mass transfer kinetics e.g., the coliunn efficiency) and the internal tortuosity and pore coimectivity of their particles. It could also provide an original, accurate, and independent method of determination of the mass transfer resistances, especially at high mobile phase velocities, and of the dependence of these properties on the internal and external porosities, on the average pore size and on the parameters of the pore size distributions. It could be possible to determine local fluctuations of the coliunn external porosity, of its external tortuosity, of the mobile phase velocity, of the axial and transverse dispersion coefficients, and of the parameters of the mass transfer kinetics discussed in the present work. Further studies along these lines are certainly warranted. 

In the equilibrium-dispersive model, we assume that the mobile and the stationary phases are constantly in equilibrium. We recognize, however, that band dispersion takes place in the column through axial dispersion and nonequilibrium effects e.g., mass transfer resistances, finite kinetics of adsorption-desorption). We assume that their contributions can be lumped together in an apparent dispersion coefficient. This coefficient is related to the experimental parameters by... 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

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 simplification is a result of the assumption that the coefficients of axial dispersion and of mass transfer kinetics are equal for the two components. The ratio of the two optimum velocities is... 

In the case of a breakthrough curve for a binary mixture (step injection), Liapis and Rippin [32,33] used an orthogonal collocation method to calculate numerical solutions of a kinetic model including axial dispersion, intraparticle diffusion, and surface film diffusion, and assuming constant coefficients of diffusion and... 

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