Modeling of Nonideal Reactors

The reader may very well wonder what happened to the chemical reaction, since we have mostly discussed mixing models in this chapter without reference to reaction. In review of the various approaches to modeling nonideal flow effects on reactor performance, however, we find that in fact a number of these have already been treated, although perhaps with different applications in mind. The classes of reactor models we have treated are 

Those making direct use of exit-age distribution information—segregated-flow models. 

Those employing a diffusion/dispersion term in the continuity equation. 

Those employing combinations of ideal reactor models with dead volume, short circuiting, or channeling components. 

Direct application of the residence-time/exit-age data has been treated in application to chemical reactor design in Section 5.2a. In terms of the net effect of the exit-age distribution on conversion, we wrote 

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Other Models of Nonideal Reactors Using CSTRs and PFRs... 

Use of Software Packages to Determine the Model Parameters 988 Other Models of Nonideal Reactors Using CSTRs and PFRs 990... 

The similarity of the behavior of the sequence of a large number of CSTRs to that of a PFR has important applications in the modeling of nonideal reactors, since from the illustration above it is apparent that for 1 < < oo we have obtained a conversion result corresponding to something intermediate between the ideal Umits of the single CSTR and the PFR. However, since the example we have given is a very specific one and since this type of modeling is sufficiently important to be the topic of a subsequent section, we will not pursue the matter further at this point. 

As will be seen, a number of alternative approaches have been developed which provide a considerable degree of flexibility in the modeling of nonideal reactors. 

The reactors with recycle are continuous and may be tanks or tubes. Their main feature is increasing productivity by returning part of unconverted reactants to the entrance of the reactor. For this reason, the reactant conversion increases successively and also the productivity with respect to the desired products. The recycle may also be applied in reactors in series or representing models of nonideal reactors, in which the recycle parameter indicates the deviation from ideal behavior. As limiting cases, we have ideal tank and tubular reactors representing perfect mixture when the recycle is too large, or plug flow reactor(PFR) when there is no recycle. 

At some point in most processes, a detailed model of performance is needed to evaluate the effects of changing feedstocks, added capacity needs, changing costs of materials and operations, etc. For this, we need to solve the complete equations with detailed chemistry and reactor flow patterns. This is a problem of solving the R simultaneous equations for S chemical species, as we have discussed. However, the real process is seldom isothermal, and the flow pattern involves partial mixing. Therefore, in formulating a complete simulation, we need to add many additional complexities to the ideas developed thus far. We will consider each of these complexities in successive chapters temperature variations in Chapters 5 and 6, catalytic processes in Chapter 7, and nonideal flow patterns in Chapter 8. In Chapter 8 we will return to the issue of detailed modeling of chemical reactors, which include all these effects. 

Figure P14-24 C curve for a nonideal reactor Suggest a model using the collection of ideal reactors to model the nonideal reactor.

Modeling of Nonideal Flow or Mixing Effects on Reactor Performance... 

Use of the CSTR sequence as a model for nonideal reactors has been criticized on the basis that it lacks certain aspects of physical reality, such as the absence of backward communication between the individual mixing cell units. Such may be the case nonetheless the mathematical simplicity of the approach makes it very attractive, particularly for systems with complex kinetics, nonisothermal effects, or other complicating factors. 

If the deviations are small then they can be described by the dispersion model (additional dispersive flow is is superimposed on the plug flow) or cell model (cascade of ideal stirred tanks). For larger deviations the calculation of nonideal reactors is generally difficult. A more simply treated special case occurs when the volume elements flowing through the reactor are macroscopically but not microscopically mixed (segregated flow). This case can be solved by the Hofmann-Schoenemann method (see below). 

The mathematical modeling of nonideal catalytic reactors would require reactive CFD simulations. Deutschmann (2001) developed models and tools for the numerical simulation of heterogeneous reactive flows, in which all physical and chemical processes are described in as much detail as possible. However, this approach requires a massive computational effort, because of the broad time and space scales, as well as the presence of reactive species during the gas phase and adsorbed on the active sites. This may make it impossible in practice to simulate those reactors within an acceptable computational time. 

What follows is an attempt to describe the performance of nonideal reactors using some simple statistical models that can be represented by analytical solutions. This... 

OTHER MODELS FOR NONIDEAL REACTORS 10.5.1 Moments of Residence Time Distributions... 

In the next section, some models of nonideal flow through reactors will be introduced. Each model will contain a parameter (or parameters) that may have to be determined from tracer response experiments. For a given model, the unknown parameters usually can be determined from the moments of the tracer curve. 

Different reactor networks can give rise to the same residence time distribution function. For example, a CSTR characterized by a space time Tj followed by a PFR characterized by a space time t2 has an F(t) curve that is identical to that of these two reactors operated in the reverse order. Consequently, the F(t) curve alone is not sufficient, in general, to permit one to determine the conversion in a nonideal reactor. As a result, several mathematical models of reactor performance have been developed to provide estimates of the conversion levels in nonideal reactors. These models vary in their degree of complexity and range of applicability. In this textbook we will confine the discussion to models in which a single parameter is used to characterize the nonideal flow pattern. Multiparameter models have been developed for handling more complex situations (e.g., that which prevails in a fluidized bed reactor), but these are beyond the scope of this textbook. [See Levenspiel (2) and Himmelblau and Bischoff (4).]... 

These two types of deviations occur simultaneously in actual reactors, but the mathematical models we will develop assume that the residence time distribution function may be attributed to one or the other of these flow situations. The first class of nonideal flow conditions leads to the segregated flow model of reactor performance. This model may be used... 

Ideal flow is introduced in Chapter 2 in connection with the investigation of kinetics in certain types of ideal reactor models, and in Chapter 11 in connection with chemical reactors as a contrast to nonideal flow. As its name implies, ideal flow is a model of flow which, in one of its various forms, may be closely approached, but is not actually achieved. In Chapter 2, three forms are described backmix flow (BMF), plug flow (PF), and laminar flow (LF). 

Equation 13.5-2 is the segregated-flow model (SFM) with a continuous RTD, E(t). To what extent does it give valid results for the performance of a reactor To answer this question, we apply it first to ideal-reactor models (Chapters 14 to 16), for which we have derived the exact form of E(t), and for which exact performance results can be compared with those obtained independently by material balances. The utility of the SFM lies eventually in its potential use in situations involving nonideal flow, wheic results cannot be predicted a priori, in conjunction with an experimentally measured RTD (Chapters 19 and 20) in this case, confirmation must be done by comparison with experimental results. 

A mathematical model for nonideal flow in a vessel provides a characterization of the mixing and flow behavior. Although it may appear to be an independent alternative to the experimental measurement of RTD, the latter may be required to determine the parameters) of the model. The ultimate importance of such a model for our purpose is that it may be used to assess the performance of the vessel as a reactor (Chapter 20). 

The tanks-in-series (TIS) model for a reactor with nonideal flow uses the ITS flow model described in Section 19.4.1 and illustrated in Figure 19.11. The substance A is now a reacting species (e.g., A - products) instead of a tracer. 

In this section, we apply the axial dispersion flow model (or DPF model) of Section 19.4.2 to design or assess the performance of a reactor with nonideal flow. We consider, for example, the effect of axial dispersion on the concentration profile of a species, or its fractional conversion at the reactor outlet. For simplicity, we assume steady-state, isothermal operation for a simple system of constant density reacting according to A - products. 

In the quantitative development in Section 24.4 below, we assume the flow to be ideal, but more elaborate models are available for nonideal flow (Chapter 19 see also Kastanek et al., 1993, Chapter 5). Examples of types of tower reactors are illustrated schematically in Figure 24.1, and are discussed more fully below. An important consideration for the efficiency of gas-liquid contact is whether one phase (gas or liquid) is dispersed in the other as a continuous phase, or whether both phases are continuous. This is related to, and may be determined by, features of the overall reaction kinetics, such as rate-determining characteristics of mass transfer and intrinsic reaction. 

In a bubble-column reactor for a gas-liquid reaction, Figure 24.1(e), gas enters the bottom of the vessel, is dispersed as bubbles, and flows upward, countercurrent to the flow of liquid. We assume the gas bubbles are in PF and the liquid is in BMF, although nonideal flow models (Chapter 19) may be used as required. The fluids are not mechanically agitated. The design of the reactor for a specified performance requires, among other things, determination of the height and diameter. 

Our treatment of Chemical Reaction Engineering begins in Chapters 1 and 2 and continues in Chapters 11-24. After an introduction (Chapter 11) surveying the field, the next five Chapters (12-16) are devoted to performance and design characteristics of four ideal reactor models (batch, CSTR, plug-flow, and laminar-flow), and to the characteristics of various types of ideal flow involved in continuous-flow reactors. Chapter 17 deals with comparisons and combinations of ideal reactors. Chapter 18 deals with ideal reactors for complex (multireaction) systems. Chapters 19 and 20 treat nonideal flow and reactor considerations taking this into account. Chapters 21-24 provide an introduction to reactors for multiphase systems, including fixed-bed catalytic reactors, fluidized-bed reactors, and reactors for gas-solid and gas-liquid reactions. 

We will not attempt to solve the preceding equations except in a few simple cases. Instead, we consider nonideal reactors using several simple models that have analytical solutions. For this it is convenient to consider the residence time distribution (RTD), or the probability of a molecule residing in the reactor for a time f. 

The next level of sophistication is to use the nonideal reactor models developed in this chapter. These are fairly simple to calculate, and the results tell us how serious these nonideaUties might be. 

For any more complex flow pattern we must solve the fluid mechanics to describe the fluid flow in each phase, along with the mass balances. The cases where we can still attempt to find descriptions are the nonideal reactor models considered previously in Chapter 8, where laminar flow, a series of CSTRs, a recycle TR, and dispersion in a TR allow us to modify the ideal mass-balance equations. 

The modeling of chemical batch reactors has been chosen as the starting point for the roadmap developed in this book. The simplified mathematical models presented in the first sections of the chapter allow us to focus the attention on different aspects of chemical kinetics, whereas the causes of nonideal behavior of chemical batch reactors are faced in the last chapter. 

In Chaps. 5 and 6 model-based control and early diagnosis of faults for ideal batch reactors have been considered. A detailed kinetic network and a correspondingly complex rate of heat production have been included in the mathematical model, in order to simulate a realistic application however, the reactor was described by simple ideal mathematical models, as developed in Chap. 2. In fact, real chemical reactors differ from ideal ones because of two main causes of nonideal behavior, namely the nonideal mixing of the reactor contents and the presence of multiphase systems. 

The second type of nonideal models takes into account the possible formation of donor-acceptor complexes between monomers. Essentially, along with individual entry of these latter into a polymer chain, the possibility arises for their addition to this chain as a binary complex. A theoretical analysis of copolymerization in the framework of this model revealed (Korolev and Kuchanov, 1982) that the statistics of the succession of units in macromolecules is not Markovian even at fixed monomer mixture composition in a reactor. Nevertheless, an approach based on the "labeling-erasing" procedure has been developed (Kuchanov et al., 1984), enabling the calculation of any statistical characteristics of such non-Markovian copolymers. 

The reactors treated in the book thus far—the perfectly mixed batch, the plug-flow tubular, and the perfectly mixed continuous tank reactors—have been modeled as ideal reactors. Unfortunately, in the real world we often observe behavior very different from that expected from the exemplar this behavior is tme of students, engineers, college professors, and chemical reactors. Just as we must learn to work with people who are not perfect, so the reactor analyst must learn to diagnose and handle chemical reactors whose performance deviates from the ideal. Nonideal reactors and the principles behind their analysis form the subject of this chapter and the next. 

The basic ideas or concepts used to characterize and model nonideal reactors are really few in munber. Before proceeding further, a few selected examples of nonideal mixing and modeling from the author s experiences will be presented. 

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