Pseudohomogeneous models for

Chapter 10 begins a more detailed treatment of heterogeneous reactors. This chapter continues the use of pseudohomogeneous models for steady-state, packed-bed reactors, but derives expressions for the reaction rate that reflect the underlying kinetics of surface-catalyzed reactions. The kinetic models are site-competition models that apply to a variety of catalytic systems, including the enzymatic reactions treated in Chapter 12. Here in Chapter 10, the example system is a solid-catalyzed gas reaction that is typical of the traditional chemical industry. A few important examples are listed here ... 

The heterogeneous system here is described by a pseudohomogeneous model. For complete heterogeneous systems, see Chapter 6. 

This is a case with negligible mass transfer resistances, as described by the pseudohomogeneous model. For a full heterogeneous system, see Chapter 6. This situation is a bit more complicated compared to the lumped system. We will consider a two-phase system with no mass transfer resistance between the phases and the voidage is equal to s (see Fig. 4.9). The mass balance design equation over the element A/ is ... 

Calculate the heat transfer parameters of the two-dimensional pseudohomogeneous models for the design of the reactor for hydrocarbon oxidation of Section 11.7.3, using the correlations given in Section 11.7.1. Compare the value calculated from the expressions given by (a) Kunii and Smith, (b) Zehner and Schundler. Determine their sensitivity with respect to the solid conductivity. Additional data ... 

The relationships that were derived in an earlier chapter between concentrations and the volume flow, as well as the stoichiometric measures (n t> flp for homogeneous flow reactors (Chapter 3), are also valid for the pseudohomogeneous model for a packed bed discussed here. 

De facto Equation A9.4 is the one-dimensional, pseudohomogeneous model for fixed beds presented in Chapter 5. 

The first eight chapters of this book treat homogeneous reactions. Chapter 9 provides models for packed-bed reactors, but the reaction kinetics are pseudohomogeneous so that the rate expressions are based on fluid-phase concentrations. There is a good reason for this. Fluid-phase concentrations are what can be measured. The fluid-phase concentrations at the outlet are what can be sold. 

In our original system of partial differential equations, to obtain a pseudohomogeneous model the two energy balances can be combined by eliminating the term (Usg/Vb)(Ts — Tg), that describes the heat transfer between the solid and the gas. If the gas and solid temperatures are assumed to be equal (Ts = Tg)19 and the homogeneous gas/solid temperature is defined as T, the combined energy balance for the gas and solid becomes... 

Wei el al. (1984) show that the average temperature difference between the solid and gas phase is yXAA/fi for fixed bed reactors. This then defines one possible criterion for the applicability of the pseudohomogeneous model as given in Table V. Another possible criterion is that given by Vortmeyer and Schaefer (1974), discussed later in this section. 

Although this assumption is often not too restrictive, additional assumptions must be introduced to reduce the mathematical model to the pseudohomogeneous form for reactors that are nonadiabatic or require radial temperature considerations. For the full mathematical model, the following assumptions would be necessary in addition to Eq. (68) first negligible energy accumulation in the gas phase second, no axial diffusion in the gas phase or... 

The differential equations (7.164), (7.165), (7.166), and (7.168) form a pseudohomogene-ous model of the fixed-bed catalytic reactor. More accurately, in this pseudohomogeneous model, the effectiveness factors rji are assumed to be constantly equal to 1 and thus they can be included within the rates of reaction ki. Such a model is not very rigorous. Because it includes the effects of diffusion and conduction empirically in the catalyst pellet, it cannot be used reliably for other units. 

Actual and pseudohomogeneous model results for an industrial reactor... 

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. 

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 heat transfer coefficient h was calculated according to Hand-ley and Heggs (24) with the Reynolds number based upon an equivalent diameter, namely that of a sphere with the same volume as the actual particle. The overall heat transfer coefficient U was calculated from the heat transfer parameters of the two dimensional pseudohomogeneous model (since the interfacial At was found to be negligible), to allow for a consistent comparison with two dimensional predictions and to try to predict as closely as possible radially averaged temperatures in the bed (25). Therefore ... 

For the special case of a simple reaction A — B, the low-dimensional model for a CSTR with premixed feed consists of three differential equations and two algebraic equations. When the mass and thermal micromixing effects are ignored (r) = t]H — 0), cm — (c), (Of) — dfm — (ds), and we get the classical pseudohomogeneous CSTR model... 

The development of mathematical models for the simulation of non-adiabatic fixed-bed catalytic reactors has received considerable attention. In previous work, we have analyzed the two-dimensional and one-dimensional versions of the models (1, 2) which, in turn, were classified as (I) pseudohomogeneous, (II) heterogeneous, but conceptually wrong, and (III) heterogeneous, written in the correct way (Table I). Model equations are in the Appendix. 

A relatively large number of models can be written down for a packed-bed reactor, depending on what is accounted for in the model. These models, however, basically fall into two categories pseudohomogeneous models and heterogeneous models. The various models are described in standard reaction engineering texts — such as those of Carberry ( ), Froment and Bischoff ( ), and Smith ( ), to cite just a few — and in review articles (cf., and so details of their equations will not be reported here. We will, instead, only make some qualitative remarks about the models. 

If the catalyst is dispersed throughout the pellet, then internal diffusion of the species within the pores of the pellet, along with simultaneous reaction(s) must be accounted for if the prevailing Thiele modulus > 1. This aspect gives rise to the effectiveness factor" problem, to which a significant amount of effort, summarized by Aris ( ), has been devoted in the literature. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed-bed reactor model must be a heterogeneous model it cannot be a pseudohomogeneous model. 

Parametric Sensitivity. One last feature of packed-bed reactors that is perhaps worth mentioning is the so-called "parametric sensitivity" problem. For exothermic gas-solid reactions occurring in non-adiabatic packed-bed reactors, the temperature profile in some cases exhibits extreme sensitivity to the operational conditions. For example, a relatively small increase in the feed temperature, reactant concentration in the feed, or the coolant temperature can cause the hot-spot temperature to increase enormously (cf. 54). This sensitivity is a type of instability, which is important to understand for reactor design and operation. The problem was first studied by Bilous and Amundson (55). Various authors (cf. 57) have attempted to provide estimates of the heat of reaction and heat transfer parameters defining the parametrically sensitive region for the plug-flow pseudohomogeneous model, critical values of these parameters can now be obtained for any reaction order rather easily (58). 

The initial and boundary conditions are given in Chapter 9. The present treatment does not change the results of Chapter 9 but instead provides a rational basis for using pseudohomogeneous kinetics for a solid-catalyzed reaction. The axial dispersion model in Chapter 9, again with pseudohomogeneous kinetics, is an alternative to Equation 10.1 that can be used when the radial temperature and concentration gradients are small. 

The experimental values of the effective diffusivities are clearly lower than the values deduced from the theoretical models, even taking into consideration the internal convective flow. Of course, the experimental values depend on the pseudohomogeneous model chosen to represent the alumina particle, but even if the spherical model were used, the values obtained (1,8 times those obtained with the slab model by identification of the variance) would be less than the theoretical values. Thus, the theoretical models based on the porous structure of the particles cannot be used for... 

The process is described by an one-dimensional, pseudohomogeneous, non-steady state dispersion model for an adiabatic fixed bed reactor. The kinetics are modelled by a re-versibll reaction system where each reaction step follows a power law with a reaction order of one in the gas and in the solid component. The temperature dependency of the reaction rate constant follows the Arrhenius law. The equilibrium constant is set to be independent of temperature. 

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