Pseudohomogeneous model

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 ... 

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... 

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. 

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. 

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. 

Simulation of an Industrial Reactor Using the Pseudohomogeneous Model... 

The above pseudohomogeneous model was used to simulate an industrial reactor with the following data. 

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 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... 

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. 

Assuming thermodynamic equilibrium between phase ( ) and ("), the two-phase model in Eq. (1) can be further simplified. The resulting pseudohomogeneous model is obtained from the sum of the balances of both phases according to... 

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 ... 

Wei [107] in 1982 was the first to come up with a continuous pseudohomogeneous model which allowed to simulate shape-selective effects observed during the alkylation of toluene using methanol to yield xylene isomers on a HZSM-5 catalyst. He treated diffusion and reaction of the xylene isomers inside the pores in a one-dimensional model. The isomer concentration at the pore mouth was set to zero, as a boundary condition. This allowed the model equations to be solved analytically, but it also limited the application of the results to small conversions. 

The most important observation that follows from the above analysis of the multi-mode model is that in almost all practical cases, tubular reactor instabilities arise due to mixing/diffusional limitations at the small scales and spread over the reactor. In contrast, pseudohomogeneous models predict (erroneously) that reactor instabilities (ignition, multiple solutions, etc.) arise due to macromixing limitations at the reactor scale. 

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. 

Pseudohomogeneous Models. The basic assumption that is made in a pseudohomogeneous model is that the reactor can be described as an entity consisting only of a single phase. Since, in reality, two phases are present, the properties used in describing the reactor are so-called "effective" properties which respect the presence of two phases. A comprehensive review of estimating these effective properties has recently been published ( ). ... 

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. 

Jensen and Ray (50) have recently tabulated some 25 experimental studies which have demonstrated steady state multiplicity and instabilities in fixed-bed reactors many of these (cf., 29, 51, 52) have noted the importance of using a heterogeneous model in matching experimental results with theoretical predictions. Using a pseudohomogeneous model, Jensen and Ray (50) also present a detailed classification of steady state and dynamic behavior (including bifurcation to periodic solutions) that is possible in tubular reactors. 

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). 

A related phenomenon is the "wrong-way behavior" of packed-bed reactors, where a sudden reduction in the feed temperature leads to a transient temperature rise. This has been observed (52, 59) and satisfactorily analyzed using a plug-flow pseudohomogeneous model (60). 

Tables 6.30, 6.31 show, respectively, the industrial and pseudohomogeneous model results given by Sheel and Crowe (1969) after introducing the corrections given by Crowe (1992). Table 6.32 gives the results of the pseudo-homogeneous model using the kinetic parameters of Sheel and Crowe (1969) without the corrections of the kinetic parameters provided by Crowe (1989). It is clear from Table...

---