Fixed Bed Dispersion Models

The pseudo-homogeneous fixed bed dispersion models are divided into three categories The axial dispersion model, the conventional two-dimensional dispersion model, and the full two-dimensional axi-symmetrical model formulation. The heterogeneous fixed bed dispersion models can be grouped in a similar way, but one dimensional formulations are employed in most cases. 

Assuming that the bed packing is uniform, the local void fraction equals the overall holdup thus the void parameter can be canceled out in several model equations. 

For reactive flows in packed beds a set of cross sectional average balance equations is written for the gas-solid multiphase mixture [3, 5]. 

Within a porous body the flow of a fluid is resisted by viscous and geometric (tortuosity) effects. A porous media friction term is therefore added to the right hand side of the momentum equation. The physical meaning of different terms in the equation is explained in sect 3.4.6. 

For flows through a porous packed bed the pressure drop is generally dominated by the bed friction and for fixed bed processes the velocity is normally not very large, hence the momentum balance for the bulk gas phase can be reduced to (6.13)  

Several parameterizations for the friction factor, /, are given in the literature. A parameterization valid for spheres over a relatively broad range of particle Reynolds numbers is frequently used [3]  

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Observations on fixed-bed dispersion models The role of the interstitial fluid (with S. Sundar-esan and N.R. Amundson). AJ.Ch.EJ. 26,529-536 (1980). 

Here x is the conversion of SiH4. combines the effect of the molar expansion in the deposition process as well as the change in the volumetric flow and the dispersion coefficient, D, with temperature. At low pressures and small Re in LPCVD reactors the dispersion occurs mainly by molecular diffusion, therefore, we have used (D/D0) = (T/T0)l 65. e is the expansion coefficient and the stoichiometry implies that e = (xi)q, the entrance mole fraction of SiH4. The expansion coefficient, e is introduced as originally described by Levenspiel (33) The two reaction terms refer to the deposition on the reactor wall and wafer carrier and that on the wafers, respectively. The remaining quantities in these equations and the following ones are defined at the end of the paper. The boundary conditions are equivalent to the well known Danckwerts1 boundary conditions for fixed bed reactor models. 

Examination of the criteria for significant dispersion in fixed bed reactors shows that in practical cases of fixed bed reactor modeling, axial dispersion of mass and heat as well as radial dispersion of mass are negligible, which should be proven by the criteria summarized in Table 4.10.8. Then the mass and heat balance equations (4.10.125) and (4.10.126) simplify to ... 

Two-Dimensional Fixed Bed Reactor Model If we neglect axial dispersion of heat, the heat balance according to the two-dimensional reactor model is as follows [Eq. (4.10.126), Section 4.10.7.1] ... 

When a number of competing reactions are involved in a process, and/or when the desired product is obtained at an intermediate stage of a reaction, it is important to keep the residence-time distribution in a reactor as narrow as possible. Usually, a broadening of the residence-time distribution results in a decrease in selectivity for the desired product. Hence, in addition to the pressure drop, the width of the residence-time distribution is an important figure characterizing the performance of a reactor. In order to estimate the axial dispersion in the fixed-bed reactor, the model of Doraiswamy and Sharma was used [117]. This model proposes a relationship between the dispersive Peclet number ... 

The units of rv are moles converted/(volume-time), and rv is identical with the rates employed in homogeneous reactor design. Consequently, the design equations developed earlier for homogeneous reactors can be employed in these terms to obtain estimates of fixed bed reactor performance. Two-dimensional, pseudo homogeneous models can also be developed to allow for radial dispersion of mass and energy. 

Equations 12.7.28 and 12.7.29 provide a two-dimensional pseudo homogeneous model of a fixed bed reactor. The one-dimensional model is obtained by omitting the radial dispersion terms in the mass balance equation and replacing the radial heat transfer term by one that accounts for thermal losses through the tube wall. Thus the material balance becomes... 

Equations 12.7.48 and 12.7.39 provide the simplest one-dimensional mathematical model of tubular fixed bed reactor behavior. They neglect longitudinal dispersion of both matter and energy and, in essence, are completely equivalent to the plug flow model for homogeneous reactors that was examined in some detail in Chapters 8 to 10. Various simplifications in these equations will occur for different constraints on the energy transfer to or from the reactor. Normally, equations 12.7.48 and 12.7.39... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Kjaer (K9) gives a very comprehensive study of concentration and temperature profiles in fixed-bed catalytic reactors. Both theoretical and experimental work is reported for a phthallic anhydride reactor and various types of ammonia converters. Fair agreement was obtained, but due to the lack of sufficiently accurate thermodynamic and kinetic data, definite conclusions as to the suitability of the dispersed plug flow model could not be reached. However, the results seemed to indicate that the... 

Ideal flow is studied and represented using the classic dispersion or dispersed plug-flow model of Levenspiel (1962). Recall the material balance of a fixed-bed reactor with perfect radial mixing (eq. (3.285)) ... 

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). 

LPCVD Reactor Models. First-Order Surface Reaction. The traditional horizontal-wafer-in-tube LPCVD reactor resembles a fixed-bed reactor, and recent models are very similar to heterogeneous-dispersion models for fixed-bed reactors (21,167,213). To illustrate CVD reactor modeling, this correspondence can be exploited by first considering a simple first-order surface reaction in the LPCVD reactor and then discussing complications such as complex reaction schemes, multicomponent diffusion effects, and entrance phenomena. 

Fixed-bed reactors may exhibit axial dispersion. If axial dispersion is important for reactor simulation, analysis, or design, a variant of the one-dimensional homogeneous model that contains an axial dispersion term may be used. Approximate criteria to determine if mass and heat axial dispersion have to be considered are available (see, e.g., Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990). 

The third and fourth condition are fulfilled by Tarhan [25]. Axial dispersion is fundamentally local backmixing of reactants and products in the axial, or longitudinal direction in the small interstices of the packed bed, which is due to molecular diffusion, convection, and turbulence. Axial dispersion has been shown to be negligible in fixed-bed gas reactors. The fourth condition (no radial dispersion) can be met if the flow pattern through the bed already meets the second condition. If the flow velocity in the axial direction is constant through the entire cross section and if the reactor is well insulated (first condition), there can be no radial dispersion to speak of in gas reactors. Thus, the one-dimensional adiabatic reactor model may be actualized without great difficulties. ... 

Because of the analogy between simulated and true counter-current flow, TMB models are also used to design SMB processes. As an example, the transport dispersive model for batch columns can be extended to a TM B model by adding an adsorbent volume flow Vad (Fig. 6.38), which results in a convection term in the mass balance with the velocity uads. Dispersion in the adsorbent phase is neglected because the goal here is to describe a fictitious process and transfer the results to SMB operation. For the same reason, the mass transfer coefficient feeff as well as the fluid dispersion Dax are set equal to values that are valid for fixed beds. 

Morbidelli et al. [41] discussed a numerical procedure for the calculation of numerical solutions of the GRM model in the case of an isothermal, fixed-bed chromatographic column with a multicomponent isotherm. These authors considered two different models for the inter- and intra-particle mass transfers. These models can either take into account the internal porosity of the particles or neglect it. They include the effects of axial dispersion, the inter- and intra-particle mass transfer resistances, and a variable linear mobile phase velocity. A generalized multicomponent isotherm, initially proposed by Fritz and Schliider [34] was also used ... 

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