The modelling of chemical reactors

The methodology of chemical reactor development requires a thorough interaction between experiments and calculations. In general, we make use of mathematical models for describing the effect of the various phenomena as functions of reaction conditions and dimensions. 

A mathematical model is an equation, or more often a set of equations, describing 

It may be useful to consider the various ways in which we can make use of mathematical models. We can recognize at least five different types of models, according to their objectives (see Shinnar, 1978)  

Modelling has become much more sophisticated since complex computer programmes can be handled easily. However, particularly for learning models, mathematical equations are preferred that are solvable by analytical methods. 

In this book we use models of the first three types, with emphasis on the first one. For the more complicated models of the other two types die reader is referred to more specialized litterature. In modelling chemical reactors we usually proceed stepwise, starting with the smallest scale. On that scale we observe the chemical kinetics of homogeneous or heterogeneous reactions, including adsorption and desorption effects. The well known Langmuir-Hinshelwood equations for reactions at solid surfaces are examples of this sort of model. We call these molecular scale models. 

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Some references for the modeling of chemical reactors include Fogler (1998), Fro-ment and Bischoff (1990), Levenspiel (1998), Missen and colleagues, (1998), and Schmidt (1997). 

Modeling and analysis of chemical reactors is at the core of the chemical engineering discipline. It is one activity that is unique to our discipline and distinguishes it from other branches of engineering. One fundamental difference between the modeling of chemical reactors and non-reacting systems is that the... 

The F curve is another function that has been defined as the normalized response to a particular input. Alternatively, Equation (13-12) has been used as a definition of F(t), and it has been stated that as a result it can be obtained as the response to a positive-step tracer test. Sometimes the F curve is used in the same manner as the RTD in the modeling of chemical reactors. An excellent example is the study of Wolf and White, who investigated the behavior of screw extruders in polymerization processes. 

When the rate equations for coke formation are available, accounting for the effect of deactivation in the modeling of chemical reactors is relatively straightforward an equation for... 

Thoenes.D. "Current problems in the modeling of chemical reactors." Chem.Engng.Sci. 35 (1980) 1840-1853. 

The modeling of chemical reactors, as it is conceived in the following chapters, is not based on the external form of the equipment nor on the reaction taking place in it, nor even on the nature of the medium — homogeneous or not. 

The final goal of the axial dispersion model is its utilization in the modeling of chemical reactors. Below we will consider steady-state models only, which imply that the mass balance of a reacting component i can be written as... 

Much research in this field is in the area of "learning models". This is because of the several phenomena that affect the modelling of chemical reactors such as instabilities, multiple steady states, oscillations, etc. The purpose of controlling is faced with several such as catalyst activity, may not be possible. 

Thoenes,D,"Current Problems in the Modelling of Chemical Reactors", Chem,Eng,Sci, v35 (1980) 1840-1853,... 

Of the 23 studies hsted under Modeling of Chemical Reactors in Sec. 23, a number are optimization oriented. Added to them... 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

In spite of all doubts, mathematical modelling in fine chemicals process development is strongly recommended. The following steps in mathematical modelling of chemical reactors can be distinguished ... 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

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. 

F ure 8-18 Coirqifflison of sterns that cai be described by the models of chemical reaction engineering. Reactions include those of molecailes aid Hidng systems. Possible chemical reactors include conventional reactors as well as the enwronment. 

In the present paper we study common features of the responses of chemical reactor models to periodic forcing, and we consider accurate methods that can be used in this task. In particular, we describe an algorithm for the numerical computation and stability analysis of invariant tori. We shall consider phenomena that appear in a broad class of forced systems and illustrate them through several chemical reactor models, with emphasis on the forcing of spontaneously oscillating systems. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

The principle of thermal recycling is also used in reactors with a boiling layer, in which the heat from the hot region of the reactor is transported to the cold region by circulating solid particles suspended in the gas flow.15 Methods of the theory of chemical reactor regulation have been successfully used in other sciences as well. We note the model of Belousov-Zhabotinskii, proposed for the description of heart disease, of spasmatic contractions of the cardiac muscle. 

If the behaviour of complex chemical (in our case catalytic) reactions is known, it will be clear in what way these reactions can be carried out under optimal conditions. The results of studying kinetic models must be used as a basis for the mathematical modelling of chemical reactors to perform processes with probable non trivial kinetic behaviour. It is real systems that can appear to show such behaviour first far from equilibrium, second nonlinear, and third multi dimensional. One can hardly believe that their associated difficulties will be overcome completely, but it is necessary to approach an effective theory accounting for several important problems and first of all provide fundamentals to interpret the dependence between the type of observed kinetic relationships and the mechanism structure. 

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. 

G Eigenberger, Practical Problems in the Modelling of Chemical Reactions in Fixed Bed Reactors , Chem Eng Process 1984,18, 55-65... 

Several length-scales have to be considered in a number of applications. For example, in a typical monolith reactor used as automobile exhaust catalytic converter the reactor length and diameter are on the order of decimeters, the monolith channel dimension is on the order of 1 mm, the thickness of the catalytic washcoat layer is on the order of tens of micrometers, the dimension of the pores in the washcoat is on the order of 1 pm, the diameter of active noble metal catalyst particles can be on the order of nanometers, and the reacting molecules are on the order of angstroms cf. Fig. 1. The modeling of such reactors is a typical multiscale problem (Hoebink and Marin, 1998). Electron microscopy accompanied by other techniques can provide information on particle size, shape, and chemical composition. Local composition and particle size of dispersed nanoparticles in the porous structure of the catalyst affect catalytic activity and selectivity (Bell, 2003). 

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