Transport coefficients axial dispersion coefficient

In packed beds, the main parameters of transport of adsorbates are the axial dispersion coefficient and the fluid-to-particle mass transfer coefficient. The other important parameter, the intraparticle diffusion coefficient, is not dependent on type of adsorption contactor and the treatment described in Chapter 4 can be applied. 

In (12.7.2-1), u is taken to be the mean (plug flow) velocity through the vessel, and Da is an axial dispersion coefficient to be obtained by means of experiments. One important application is the modeling of fixed beds, as discussed in detail in Chapter 11, and then it is usually termed an "effective" transport model, with Da = Dea- The axial dispersion model has also been used to approximately describe a variety of other reactors. 

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 moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

If the radial diffusion or radial eddy transport mechanisms considered above are insufficient to smear out any radial concentration differences, then the simple dispersed plug-flow model becomes inadequate to describe the system. It is then necessary to develop a mathematical model for simultaneous radial and axial dispersion incorporating both radial and axial dispersion coefficients. This is especially important for fixed bed catalytic reactors and packed beds generally (see Volume 2, Chapter 4). 

Equation 5.29 shows that the variance of the response curve is separable into contributions from the axial dispersion, and from the external and internal mass transfer. Measurements at different velocities lead to an estimate of all transport coefficients. 

In the bubble column the velocity profile of recirculating liquid is shown in Fig. 27, where the momentum of the mixed gas and liquid phases diffuses radially, controlled by the turbulent kinematic viscosity Pf When I/l = 0 (essentially no liquid feed), there is still an intense recirculation flow inside the column. If a tracer solution is introduced at a given cross section of the column, the solution diffuses radially with the radial diffusion coefficient Er and axially with the axial diffusion coefficient E. At the same time the tracer solution is transported axially Iby the recirculating liquid flow. Thus, the tracer material disperses axially by virtue of both the axial diffusivity and the combined effect of radial diffusion and the radial velocity profile. 

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. 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

A similar comparison is shown in Figure 14.15b, using the transport-dispersive model with a number of transfer units, Nm, equal to 2000, and a finite coefficient of axial dispersion, Di, such that... 

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. 

Thermal axial dispersion must be treated with care. Even if axial dispersion of mass is negligible, the same may not be true for heat transport. The dispersion coefficient that appears in the thermal Peclet number is very different from the dispersion coefficient of the mass Peclet number. The combination of a plug-flow model for the mass balance and a dispersion... 

The practical importance of this Taylor diffusion analysis lies in the justification of the effective transport models to take into account complicated velocity and concentration profiles in a simple manner, as well as providing a theoretical framework for the dispersion coefficient, D . Similar results have been worked out for turbulent flow, packed columns, and other situations. For correlations of the axial dispersion coefficients, see Himmelblau and Bischoff [4] and Wen and Fan [2]. 

In the equilibrium dispersive model it is assumed - like in the ideal model - that the mobile and the stationary phases are permanently in the equilibrium state. In addition to adsorption and convection all band-broadening transport effects are considered. Thereby, all kinetic effects like axial dispersion, film and pore diffusion are lumped together in the apparent dispersion coefficient Djp. As the basis of the model the following partial differential equation can be written ... 

Heat and Mass Transfer Using the film theory, both phenomena mainly depend on the film and gas stream thickness and the type of reaction. Other parameters are the interfacial area, the residence time and the axial dispersion. Good mass and heat transport presume a good fiow equipartition in the channels. In mesh reactors the mesh open area determines the interfacial area. Mass transfer coefficients ki a from 3 to 8 L s and higher values in catalytic systems can be achieved [25]. 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

As a rule of thumb, axial dispersion of heat and mass (factors 2 and 3) only influence the reactor behavior for strong variations in temperature and concentration over a length of a few particles. Thus, axial dispersion is negligible if the bed depth exceeds about ten particle diameters. Such a situation is unlikely to be encountered in industrial fixed bed reactors and mostly also in laboratory-scale systems. Radial mass transport effects (factor 1) are also usually negligible as the reactor behavior is rather insensitive to the value of the radial dispersion coefficient. Conversely, radial heat transport (factor 4) is really important for wall-cooled or heated reactors, as such reactors are sensitive to the radial heat transfer parameters. 

The mixing effect is a combination of turbulence and molecular diffusion. Both are first order transport processes therefore we define, in analogy to the diffusion coefficient, the axial dispersion or mixing coefficient 0 as follows... 

Many models and correlations have been developed for the radial effective thermal conductivity (see the review by Kulkarani and Doraiswamy 1980). The thermal dispersion coefficient K used in Chapter 9 is related to the effective conductivity by X = pCpK. For nonadiabatic fixed-beds, the main heat conduction is in the radial direction, and thus, the radial conductivity is much more important than the axial conductivity. The axial conductivity represents the conduction superimposed on the bulk flow, which is quite small relative to the heat transport by the bulk flow. Therefore, most work has been directed to the radial conductivity. 

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