Simplified Reactor Models

The classical batch reactor model is preferrably formulated in terms of mole numbers, Ng (mol), so we may divide the relation by the molecular weight of the species s, M, . The resulting mole balance for the batch reactor yields  

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below  

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.354). 

Similar balance equations with purely laminar diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.353) might suffice. 

It is important to note that the important difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

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As shown in Eq. (23), the heat released (Qr(k )), which cannot be measured, is needed in the GMC algorithm. Here, the EKF algorithm, as used in on-line optimization strategy, coupled with the simplified reactor model, given by [26], is also applied to estimate the heat released (Qr(k)). The reason of using the simplified model, not the exact model of the plant, is because if the exact model were used, too many uncertain/unknown parameters as well as too many unmeasurable states would be involved. That may lead to poor performance of the EKF. Hence, the simplified model with less uncertain/unknown parameters and unmeasurable... 

Simplified reactor models of a deposition process can be classified by... 

Computational fluid dynamics based flow models were then developed to simulate flow and mixing in the loop reactor. Even here, instead of developing a single CFD model to simulate complex flows in the loop reactor (gas dispersed in liquid phase in the heater section and liquid dispersed in gas phase in the vapor space of the vapor-liquid separator), four separate flow models were developed. In the first, the bottom portion of the reactor, in which liquid is a continuous phase, was modeled using a Eulerian-Eulerian approach. Instead of actually simulating reactions in the CFD model, results obtained from the simplified reactor model were used to specify vapor generation rate along the heater. Initially some preliminary simulations were carried out for the whole reactor. However, it was noticed that the presence of the gas-liquid interface within the solution domain and inversion of the continuous phase. 

The first modeling attempt represents a very simplified reactor model as given by Eq. (12)... 

Let us first discuss how we can simplify reactor models with respect to fluiddynamics, i.e. with respect to stagnant parts of the dispersed phase as well as dispersion (axial and radial). Here the experimental results of mock-up studies are very interesting. 

Gorgiin et al. presented the developments of observers for fuel processing reactors such as a partial oxidation reformer, tvater-gas shift and preferential oxidation cleanup reactors [454]. Simplified models tvere used for the observer design. The principle of an observer-based control strategy is to run a simplified reactor model in the controller sofiware. Model parameters are then permanently adopted by measurements during the control process. For each reaction, vhich vas considered in one of the reactors, one measurement vas used to adopt the models. With catalytic partial oxidation and preferential oxidation the reactor temperature and one species... 

The calculated and experimental temperature distribution is shown in Fig. 18. The fit of the axial concentration profile was even better. A perfect fit cannot be expected because of the simplified reactor model and reaction model and the use of constant average parameter values throughout the whole reactor. On the other hand, the agreement between simulation and experiment seems to be sufficient for a study of the behavior of industrial multitube reactors with larger tube diameters. It turned out that for a maximum allowable entrance temperature of 640 K and a conversion larger than 95 %, as demanded by economic considerations, the tube diameter has to be limited to 10 cm, a result that is in excellent agreement with reports on technical units. 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

A simplified mathematical model was developed for the novel OCM reactor. One version of the model, presented here, describes batch operation. A second version addressing continuous flow operation will appear elsewhere [16]. 

In the ASTER reactor deposition experiments were performed in order to compare with the 2D model results. Normalized deposition rates are plotted in Figure 22 as a function of radial position for data taken at 25 and 18 Pa. The deposition takes place on a square glass plate. For each pressure two profile measurements were performed, each profile perpendicular to the other (a and b in Fig. 22). A clear discrepancy is present. The use of the simplified deposition model is an explanation for this. Another recent 2D fluid model also shows discrepancies between the measured and calculated deposition rate [257], which are attributed to the relative simplicity of the deposition model. 

Another approach to scale-up is the use of simplified models with key parameters or lumped coefficients found by experiments in large beds. For example, May (1959) used a large scale cold reactor model during the scale-up of the fluid hydroforming process. When using the large cold models, one must be sure that the cold model properly simulates the hydrodynamics of the real process which operates at elevated pressure and temperature. 

Ford WPJ, Walsh FC, Whyte I (1992) Simplified batch reactor models for the removal of metal ions from solution Inst Chem Eng Symp Ser, 1992, 127(Electrochem Eng Environ 92)111 Chem Abstr 117 (1992) 197712x... 

A much more interesting case of chaotic dynamics of the reactor can be obtained from the study of the self-oscillating behavior. Consider the simplified mathematical model (8) and suppose that the reactor is in steady state with a reactant concentration of Prom Eq.(8) the equilibrium point [x, y ] can be deduced as follows ... 

Eq.(34) are a set of differential equations, which lead a flow in a four dimensional phase space R. This flow can be simplified to three dimensional phase space R when the dynamics of the jacket can be considered negligible respect to the reactor s dynamics. Putting dxA/dr = 0 the dimensionless jacket s temperature X4 can be eliminated from Eq.(34), and the simplified mathematical model of the reactor can be written as... 

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

If the liquid phase is reacting and batch, the system becomes dynamic as the liquid phase concentrations change with time. To simplify the reactor model, we consider the common case of constant gas-phase concentration. Furthermore, the liquid phase is considered to be under complete mixing condition. 

We have conducted experiments to improve the selectivity of the reaction for hydrogen peroxide formation, with a special attention to the influence of the reactor will. These experimental results are interpreted by a simplified kinetic model, which is fairly well explained by knowledge of the rate of the free radical reaction. 

D.S. Dandy and M.E. Coltrin. A Simplified Analytical Model of Diamond Growth in Direct Current Arcjet Reactors. J. Mater. Res., 10(8) 1993-2010,1995. 

We neglected in our model the back reactions, in order to simplify the already quite complicated process. In some of the steps this does not correspond to the reality with the exception of the last step because NH3 is removed from the surface and the reactor. We do not believe that the inclusion of the back reactions will alter significantly the conclusions of the present very simplified reaction model except via a reduced reaction rate. The surface coverages should remain essentially unchanged and they are the prominent information available from our model. Further on the removal of NH3 introduces a drag on the reaction process in the direction of smaller importance of the back reactions. As a result of our model we found in the... 

Vertical CVD Reactors. Models of vertical reactors fall into two broad groups. In the first group, the flow field is assumed to be described by the one-dimensional similarity solution to one of the classical axisymmetric flows rotating-disk flow, impinging-jet flow, or stagnation point flow (222). A detailed chemical mechanism is included in the model. In the second category, the finite dimension of the susceptor and the presence of the reactor walls are included in a detailed treatment of axisymmetric flow phenomena, including inertia- and buoyancy-driven recirculations, whereas the chemical mechanism is simplified to a few surface and gas-phase reactions. 

Chemical kinetics plays a major role in modeling the ideal chemical batch reactor hence, a basic introduction to chemical kinetics is given in the chapter. Simplified kinetic models are often adopted to obtain analytical solutions for the time evolution of concentrations of reactants and products, while more complex kinetics can be considered if numerical solutions are allowed for. 

The model reduction procedure must be adapted to the use of the simplified models and to the availability of experimental data needed to evaluate the unknown parameters, as discussed in Chap. 3. In general, more complex models are used for the design of the reactor and for the simulation of the entire process, whereas more simplified models are best fit for feedback control. In the following chapters it is shown that fairly accurate results are obtained when a strongly simplified kinetic model is used for control and fault diagnosis purposes. 

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 Fig. 4.5, temperature profiles are reported in subcritical, critical, and supercritical conditions. Supercritical solutions of the simplified mathematical model proposed by Semenov are, however, purely theoretical since the assumption of negligible reactant conversion becomes very unrealistic. As an example, in the worst case where

theory predicts an infinitely increasing temperature in the reactor. 

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