Chemical reactors model formulation

Chemical reaction models play a major role in chemical engineering as tools for process analysis, design, and discovery. This chapter provides an introduction to structures of chemical reaction models and to ways of formulating and investigating them. Each notation in this chapter is defined when introduced. Chapter 3 gives a complementary introduction to chemical reactor models, including their physical and chemical aspects. 

It is common practice when applying CFD to chemical reactor modeling, to solve a set of independent transport equations consisting of upto N — 1 species transport equations the mixture continuity equation and the momentum equation and the internal (or thermal) energy equation formulated in terms of temperature. 

Leonard [97] was apparently the first to use the term Large Eddy Simulation. He also introduced the idea of filtering as a formal convolution operation on the velocity field and gave the first general formulation of the method. Since Leonard s approach form the basis for application of LES to chemical reactor modeling, we discuss this approach in further details. 

The area averaging theory described in this section is based on the papers by [229, 243, 44, 46, 47, 22, 10, 24, 115, 135, 16[. The main object in this section is to provide the necessary theorems to derive the cross-sectional averaged equations that coincide with the conventional ID multiphase chemical reactor model from first principles. To formulate these theorems we chose to adopt the same concepts as were used deriving the corresponding single phase equations... 

The first question the reader might ask is why do I bring this up If we can develop a proper mathematical description of a chemical reactor we should be able to use it for any purpose we want. This brings us to the basic problem of chemical reactor modeling Most chemical reactors of interest are too complex to be given an exact mathematical formulation. We assume that there exists a true mathematical relation. 

Many elements of a mathematical model of the catalytic converter are available in the classical chemical reactor engineering literature. There are also many novel features in the automotive catalytic converter that need further analysis or even new formulations the transient analysis of catalytic beds, the shallow pellet bed, the monolith and the stacked and rolled screens, the negative order kinetics of CO oxidation over platinum,... 

Keeping in mind the discussion leading to Eq. (3), the formulation of a CFD model for a chemical reactor can be broken down into the following broad steps ... 

Minimal bounds on the production quantity are most often process dependent. Typically, a minimal campaign length is required if for example a critical mass is necessary to initiate a chemical reaction. The same is valid for maximal bounds on the production quantity. The rationale here is that a cleaning operation may be required every time a certain amount has been produced. Finally, batch size restrictions often arise in the chemical industry, if for example the batch size is determined by a reactor load or, as discussed above, the processing time for a certain production step is independent of the amount of material processed. In these scenarios, when working with model formulations using a discrete time scale, it is important that the model formulation takes into account that lot sizes may comprise of production in several adjacent periods. 

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. 

Mathematical interpretations of the homogeneous chemical reactor have been presented along with the major assumptions attendant in formulating the models. The general assumptions and limitations of the equations are discussed here. 

Many of the equations employed in unstructured and non-segregated models derive from those of enzymatic kinetics (Sinclair and Kristiansen, 1987 Nielsen and Nikolajsen, 1988). Cells are considered as chemical reactors that support thousands of complex reactions catalyzed by enzymes that allow the conversion of substrates into secreted products. The equation formulated by Michaelis and Menten represents the enzymatic conversion rate of a unique substrate into one product (Equation 14). 

By use of the above mentioned framework and mathematical tools, and by reasonable assumptions based on analysis of experimental data we are hopefully able to formulate proper models providing reasonable predictions of the chemical reactor performance. In the following sects, the basic mathematical and conceptual tools determining the general reactor technology fundamentals are given and discussed. 

Based on the considerations presented above, it was found necessary to carry out a critical evaluation of the various model formulations available aiming to achieve a proper description of the multiphase flow regimes occurring in chemical reactors. 

To analyze and design chemical reactors more effectively and to obtain insight into the operation, we adopt reaction-based design formulation. In this section, we derive the reaction-based design equations for the three ideal reactor models. Reaction-based design equations of other reactor configurations are derived in Chapter 9. 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

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