Two-Dimensional Pseudo-Homogeneous Models

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... 

Curve 5 corresponds to no-heat transfer through the solid and this predicts a hot spot that is far too important. Such a model is no improvement at all with respect to the two-dimensional pseudo-homogeneous model of Sec. 11.7. It is interesting also to note that, for the conditions used in these calculations, the solid temperature only exceeds the gas temperature by 1 or 2°C This is generally so in industrial reactors. Finally, the radial mean temperatures of the two-dimensional models are significantly different from the temperature predicted by the one-dimensional models. Provided the physical data are available the two-dimensional models would definitely have to be preferred for the simulation of this reactor. 

Calculate the heat transfer parameters of the two-dimensional pseudo homogeneous models for the design of the reactor for hydrocarbonoxidation of Ex. 11.7.c, using the correlations given in Sec. 11.7.a. Compare the value of calculated from the expressions... 

In order to simulate the behavior of an industrial membrane reactor, the model described in Sect. 4.2.2, i.e., a two-dimensional pseudo-homogeneous model, is fitted to natural gas steam reforming. Chemical-physical properties of reactions and components involved are taken from literature [17, 18] as functions of temperature, pressure, and mixture composition. The kinetic equations for the reactions scheme composed by 5.1, 5.2 and the overall reaction ... 

For each region a mean value of the void fraction was calculated and a hydraulic radius was defined which was used in a pressure drop correlation. Martin [20] divided the bed into two regions a wall and a bulk region. He calculated for both different flow rates and a different rate of heat transfer. Carbonell [2] also used a two zone model for his analysis of the dispersion phenomena. In more recent work Vortmeyer et al. [5>6] tried to use the complete radial void fraction profile, and so did Chang [3]. They followed the same itinerary outlined by Lerou and Froment [l] and Marivoet et al. [2l]. Starting from the void fraction profile the radial velocity profile is calculated. With both profiles the effective thermal conductivity is established and the temperature and concentration profiles can be calculated by means of a two dimensional pseudo homogeneous model for the reactor. 

Illustration 12.7 indicates how to estimate an effective thermal conductivity for use with two-dimensional, pseudo homogeneous packed bed models. 

The conventional two-dimensional pseudo-homogeneous reactor model consists of the continuity equation (11.1) and the simplified momentum equation (11.3) defined in connection with the pseudo-homogeneous dispersion model. The species mass and temperature equations are extended to 2D by adding postulated diffusion terms in the radial space dimension [3]. 

As an example, the mass and energy balances for a two-dimensional pseudo-homogeneous packed bed reactor model are represented by the following equations ... 

In order to minimize the experimental error, special probes were used 25 miniature thermocouples located at 9 different radial positions introduced in an asbesteous honeycomp in a single cross-section of a 50 mm diameter column or 6 capillaries for concentration sampling at 6 radial positions. The large amount of simultaneously obtained data was recorded automatically on a data storage device and subsequently processed with the help of the computer program FIBSAS /19/ for a two-dimensional pseudo-homogeneous dispersion model of a fixed-bed reactor. 

This equation may be used as an appropriate form of the law of energy conservation in various pseudo homogeneous models of fixed bed reactors. Radial transport by effective thermal conduction is an essential element of two-dimensional reactor models but, for one-dimensional models, the last term must be replaced by one involving heat losses to the walls. 

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. 

Full Pseudo-Homogeneous Two-Dimensional Axi-symmetric Model... 

A more rigorous pseudo-homogeneous two-dimensional axi-symmetric model can be obtained reducing the governing averaged equations, that can be derived using any of the local averaging procedures described in sect 3.4, for the particular axi-symmetric tube flow problem. 

Three different versions of the basic one dimensional model were developed, two heterogeneous models and a pseudo-homogeneous model. The difference in the two heterogeneous models is the way the sorbent was installed in the reactor bed. One model version considers that Li2Zr03 and the reforming catalyst are coated on two different particles, while in the other case there is one particle with both catalytic and capture properties. The steam methane reforming and the water-gas shift reaction kinetics are taken from Xu and Proment [14], but corrected for different properties of the catalyst. 

The effectiveness factor is defined as the ratio of the rate of reaction with diffusional resistances over the rate of reaction at bulk conditions. It follows that the effectiveness factor is not constant along the reactor length and therefore has to be calculated at each axial point along the reactor length. In the case of a two dimensional model, the effectiveness factor radial variations should also be calculated. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed bed reactor model must be described by a heterogeneous model, and pseudo-homogeneous models cannot be used except in a few very special cases as discussed earlier. 

Partial differential equations are involved, and a simple analytical solution is usually impossible. One has to use advanced numerical techniques and computing aids to solve such models. The two-dimensional models use the effective transport concept to formulate the flux of heat and mass in the radial direction. This flux is superimposed upon the transport by the overall continuity equation for the key reacting component A and the energy equation. For a single reaction and at steady state, the equations can be written for the pseudo-homogeneous model as follows ... 

In the following section a two-dimensional model will be described that is used for the computation of temperature and concentration profiles inside a packed bed membrane reactor for hydrogen production. For simplicity, only a pseudo-homogeneous model will be described. The extension of the heterogeneous model is analogous to the ID model. 

With respect to reaction-zone models, all types listed in Table 12.1 can be implemented. In the following, for sake of simplicity, only pseudo-homogeneous models are presented. IMR models can be one- or two-dimensional. Usually, axial mixing phenomena are neglected and therefore two model types can be implemented to describe IMR behaviour ... 

A dynamic pseudo-homogeneous two-dimensional model is employed for the description of the reactor behaviour. 

The choice of a model to describe heat transfer in packed beds is one which has often been dictated by the requirement that the resulting model equations should be relatively easy to solve for the bed temperature profile. This consideration has led to the widespread use of the pseudo-homogeneous two-dimensional model, in which the tubular bed is modelled as though it consisted of one phase only. This phase is assumed to move in plug-flow, with superimposed axial and radial effective thermal conductivities, which are usually taken to be independent of the axial and radial spatial coordinates. In non-adiabatic beds, heat transfer from the wall is governed by an apparent wall heat transfer coefficient. ... 

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