Heterogeneous models

FIG. 19 Scheme of a simple fluid confined by a chemically heterogeneous model pore. Fluid modecules (grey spheres) are spherically symmetric. Each substrate consists of a sequence of crystallographic planes separated by a distance 8 along the z axis. The surface planes of the two opposite substrates are separated by a distance s,. Periodic boundary conditions are imposed in the x and y directions (see text) (from Ref. 77). 

Fig. 2.3 According to the homogeneous accretion model (a), iron-containing material (black) and silicate-containing material (colorless) condensed out at the same time, i.e., the proto-Earth consisted of a mixture of the two. The concentration of iron in the Earth s core took place later. According to the heterogeneous model (b), the iron condensed out of the primeval solar nebula first, while the silicates later formed a crust around the heavy core. From Jeanloz (1983)...

In the heterogeneous model, the metals condensed first and formed the core, while the silicates, which condensed later, formed an outer layer—the mantle. 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

The catalytic site is chiral and analogous to that proposed for the isospedfic heterogeneous catalysts (Section 3.2), but, unlike the heterogeneous model site, interconversion between enantiomeric complexes is assumed to be possible, after each insertion step, when the metal atom is pentacoordinated.147,149 The analysis of the nonbonded interactions at the catalytic site showed that a si chain favors the formation of the A complex, which in turn favors the re coordination and insertion of the successive monomer unit, thus assuring the syndiospedfic propagation of the chain. 

Induced heterogeneity model, 30 241, 251 Industrial catalytic cracking, 6 271 Industrial reactions see also specific reactions... 

Keywords. Oxygen absorption. Three-phase mass transfer. Dispersed phase. Heterogeneous model. Fermentation, Bulk oxygen concentration... 

Mathematical models of packed bed reactors can be classified into two broad categories (1) one-phase, or pseudohomogeneous, models in which the reactor bed is approximated as a quasi-homogeneous medium and (2) two-phase, or heterogeneous, models in which the catalyst and fluid phases and the heat and mass transfer between phases are treated explicitly. Although the... 

De Wasch and Froment (1971) and Hoiberg et. al. (1971) published the first two-dimensional packed bed reactor models that distinguished between conditions in the fluid and on the solid. The basic emphasis of the work by De Wasch and Froment (1971) was the comparison of simple homogeneous and heterogeneous models and the relationships between lumped heat transfer parameters (wall heat transfer coefficient and thermal conductivity) and the effective parameters in the gas and solid phases. Hoiberg et al. (1971)... 

The analysis has shown that PAI may only be negative, and PAB ( both positive and negative. Therefore, the thermal effect accompanying a reversible stretching of the model depends on the ratio between p and PA,n and may be a function of strain even at small strains. Besides, Poisson s ratio for such a heterogeneous model may exeed 0.5, Direct measurements of Poisson s ratio for a number of various oriented crystalline polymers are consistent with this suggestion (see Table 5). 

Figure 5. Correlation of C02 isotherms according to the heterogeneous model (Equation 3)...

Considering heterogeneous models for the film structure, we realize that if PVC with its low permeability were the continuous phase, there should only be small increases in permeability with the addition of EVA polymer. Such effects have been observed for a system of butadiene-based polymer modifier added to PVC to increase the impact strength (1). Addition of 15% modifier increased the permeability less than 10%. Electron micrographs of this film showed that the butadiene-based modifier was dispersed in the PVC phase. 

Here we develop a pseudohomogeneous model first and investigate how to solve this model numerically. This is later followed by a more rigorous heterogeneous model. 

Note that the results of our simulation via the pseudohomogeneous model tracks the actual plant very closely. However, since the effectiveness factors r]i were included in a lumped empirical fashion in the kinetic parameters, this model is not suitable for other reactors. A heterogeneous model, using intrinsic kinetics and a rigorous description of the diffusion and conduction, as well as the reactions in the catalyst pellet will be more reliable in general and can be used to extract intrinsic kinetic parameters from the industrial data. 

Simulation of Industrial Units Using the more Rigorous Heterogeneous Model... 

Here we develop a heterogeneous model that is based on the more rigorous dusty gas model for diffusion and reaction in porous catalyst pellets. 

The industrial rates obtained earlier from the pseudohomogeneous model actually include diffusional limits and are suitable for the specific reactor with the specific catalyst particle size for which the data was extracted. Such pseudohomogeneous models do not account explicitly for the catalyst packing of the reactor. On the other hand, heterogeneous models account for the catalyst explicitly by considering the diffusion of reactants and of products through the pores of the catalyst pellet. 

The set of constants in columns 3 and 4 above are not intrinsic rate constants. They can, however, be used as starting values or as the initial guess in an iterative scheme to obtain kinetics that are suitable for the heterogeneous model. 

The heterogeneous model of this section is used to extract the intrinsic kinetic constants from the industrial data. The iteration scheme to find these constants is as follows. 

The rate constants of the pseudohomogeneous model are used as starting values or the initial guess in the heterogeneous model. The results obtained from the heterogeneous model with these settings will show lower conversion (as is to be expected) when compared with the results of the actual industrial plant and of the... 

For components resulting from a single reaction, such as styrene, benzene, toluene, and carbon dioxide, the molar flow rates from the industrial reactor are divided by the corresponding rates obtained from the heterogeneous model and multiplying factors are established for the respective four reactions. 

The remaining two rate constants are multiplied by suitable factors to give the best match between the heterogeneous model and the industrial data, or equivalently, the data obtained from the pseudohomogeneous model. 

The above five steps are repeated until the results for the heterogeneous model are very closely agreeing with the industrial data. 

The intrinsic frequency factors At from this iterative procedure are listed in the second column of the previous table as Ai intrinsic . The results from the heterogeneous model with these intrinsic rates and the industrial data is as follows where MFR denotes the Molar Flow Rate, Pe is the exit pressure in bar and Tg the exit temperature in K. 

Results of the heterogeneous model with the intrinsic rate constants (found iteratively) and the output data of the industrial reactor... 

Industrial reactor MFR Conversion % MFR Heterogeneous model Conversion %... 

Numerically the heterogeneous model involves I VPs for the reactor and BVPs for the catalyst pellets. These problems can be solved as before via MATLAB. 

Develop the heterogeneous model of this section and a MATLAB algorithm for its numerical solution. Verify your work for the supplied industrial data. 

S. Elnashaie, G. Ibrahim, Heterogeneous Modeling for the Alcoholic Fermentation Process, Applied Biochemistry and Biotechnology, 19(1), 71-101, 1988... 

F.M. Alhabdan, M.A. Abashar, S. Elnashaie, A flexible software package for industrial steam reformers and methanators based on rigorous heterogeneous models, Mathematical and Computer Modeling, 16, 77-86, 1992... 

Also here the equilibria can be formally given by formulas equivalent to (17) and (20). In the first case A/i0,o —27 kcal./mole and a 20 kcal./ mole. Contrary to the case with nickel, the isosteric heats fit closely to the results of both plots at higher coverages but deviate markedly at lower ones, the heterogeneous model giving the best fit here. 

Fischer CH, Quarini GL. Three-dimensional heterogeneous modeling of distillation tray hydraulics. Proceedings of AIChE Annual Meeting, Miami Beach, 1998. 

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