Axial Dispersion Effects

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) ... 

In practice, the process regime will often be less transparent than suggested by Table 1.4. As an example, a process may neither be diffusion nor reaction-rate limited, rather some intermediate regime may prevail. In addition, solid heat transfer, entrance flow or axial dispersion effects, which were neglected in the present study, may be superposed. In the analysis presented here only the leading-order effects were taken into account. As a result, the dependence of the characteristic quantities listed in Table 1.5 on the channel diameter will be more complex. For a detailed study of such more complex scenarios, computational fluid dynamics, to be discussed in Section 2.3, offers powerful tools and methods. However, the present analysis serves the purpose to differentiate the potential inherent in decreasing the characteristic dimensions of process equipment and to identify some cornerstones to be considered when attempting process intensification via size reduction. 

A similar finite-differenced equivalent for the energy balance equation (including axial dispersion effects) may be derived. The simulation example DISRET involves the axial dispersion of both mass and energy and is based on the work of Ramirez (1976). A related model without reaction is used in the simulation example FILTWASH. 

Our main motivation to develop the specific transient technique of wavefront analysis, presented in detail in (21, 22, 5), was to make feasible the direct separation and direct measurements of individual relaxation steps. As we will show this objective is feasible, because the elements of this technique correspond to integral (therefore amplified) effects of the initial rate, the initial acceleration and the differential accumulative effect. Unfortunately the implication of the space coordinate makes the general mathematical analysis of the transient responses cumbersome, particularly if one has to take into account the axial dispersion effects. But we will show that the mathematical analysis of the fastest wavefront which only will be considered here, is straight forward, because it is limited to ordinary differential equations dispersion effects are important only for large residence times of wavefronts in the system, i.e. for slow waves. We naturally recognize that this technique requires an additional experimental and theoretical effort, but we believe that it is an effective technique for the study of catalysis under technical operating conditions, where the micro- as well as the macrorelaxations above mentioned are equally important. 

This chapter describes the different types of batch and continuous bioreactors. The basic reactor concepts are described as well as the respective basic bioreactors design equations. The comparison of enzyme reactors is performed taking into account the enzyme kinetics. The modelhng and design of real reactors is discussed based on the several factors which influence their performance the immobilized biocatalyst kinetics, the external and internal mass transfer effects, the axial dispersion effects, and the operational stabihty of the immobilized biocatalyst. 

For packed bed reactors, Carberry and Wendel (1963), Hlavacek and Marek (1966), and Carberry and Butt (1975) report that axial dispersion effects are negligible if the reactor length is sufficient. These and other researchers (Young and Finlayson, 1973 Mears, 1976) have developed criteria based on the reactor length for conditions where axial dispersion can safely be neglected. The criterion shown in Table V is a classic criterion for neglecting axial mass dispersion. The works by Young and Finlayson (1973) and Mears (1976) provide more detailed criteria to predict when axial dispersion is unimportant in nonisothermal packed bed reactors. 

Diffusional effects were combined into apparent kinetic rate constants by using commercial-sized catalysts in kinetic experiments. The experiments were designed so that no significant external transport and axial dispersion effects occurred. 

In the preceding sections we concentrated mainly on the coke-conversion selectivity aspects of various reactors. In this section we will briefly discuss the selectivities of major FCC products gasoline, LCO, HCO, and light gases in both transient and steady state reactors. Weekman (1-3) has looked at steady state and transient reactors for gasoline selectivity shifts. However, he did not include axial dispersion effects in his analysis which are important for laboratory FFB reactors, and are accounted for here. 

The data-correlation models described above do not take into account the axial dispersion effect on the performance of the reactor. In small catalyst beds, particularly when they are packed with large catalysts and operated at low liquid flow rates, the axial dispersion may have a significant effect on the reactor performance. Furthermore, Hochman and Effron17 and others have shown that... 

Meats23-2 showed that the catalyst bed-length effect observed during de-nitrogenation of gas oils in pilot-scale reactors can be correlated on the basis of an axial dispersion effect on the reactor performance. Montagna and Shah29 showed that the bed-length effect observed in desulfurization reaction with 22 percent KVTB and 36 percent KATB (see Fig. 4-4) can also be explained on the basis of an axial dispersion effect on the reactor performance. 

It is difficult to ascertain whether the poor performance observed in pilot-scale trickle-bed reactors is due either to ineffective catalyst wetting or to the axial dispersion effects, because both these effects are physically realistic and both occur under similar operating conditions (i.e., low liquid flow, large catalyst size, and shorter beds). It should be noted, however, that the criterion for removing the axial dispersion effect is available. A similar criterion for removing ineffective catalyst wetting is, however, presently not available. 

Shah and Paraskos47 applied their analysis to evaluate the importance of axial dispersion on pilot scale (a) residue hydrodesulfurization, (b) gas-oil hydrocracking, and (c) shale-oil denitrogenation reactor performances. The calculations indicated that the axial dispersion effect is less important in case (c) than in cases (a) and (b). Under certain pilot-scale operations, axial dispersion effects could be significant in cases (a) and (b). 

According to Mears criterion, the minimum bed-length required to eliminate the axial dispersion effect can be expressed as... 

Nonequilibrium designs, in which mass transfer and axial dispersion effects are considered. 

The model of Tayakout et al. [117,118] in addition accounts for the possibility of axial dispersion effects in the tubeside and shellside. The inclusion of axial dispersion effects in regions (1) and (4) necessitates a different set of initial conditions at Z = 0 and a companion set of conditions at Z = L. The effect of pressure drop through the catalytic bed could be included in this type of model using Ergun s equation. 

Evaluate the results obtained in problems 8 and 14, Chapter 4, in terms of the criterion developed by Meats for freedom from axial dispersion effects in tubular reactors [D.E. Meats, Ind. Eng. Chem. Proc. Design Devel., 10, 541 (1971)]. 

Expand upon the radial dispersion model of equations (7-142) and (7-143) to include axial dispersion effects as well. Reduce this model to nondimensional form and express the characteristic parameters in terms of the dimensionless numbers described above, as far as possible. Start with the unsteady-state forms of the model equations. ... 

The basic message in all of the above is essentially that one may avoid axial dispersion effects if it is possible to design with the necessary axial aspect ratio, n, but there is no way in general to avoid radial dispersion effects. 

Table 8.4 Minimum Bed Length to Eliminate Axial Dispersion Effects...

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