Mathematical modelling reactor application

The several industrial applications reported in the hterature prove that the energy of supersonic flow can be successfully used as a tool to enhance the interfacial contacting and intensify mass transfer processes in multiphase reactor systems. However, more interest from academia and more generic research activities are needed in this fleld, in order to gain a deeper understanding of the interface creation under the supersonic wave conditions, to create rehable mathematical models of this phenomenon and to develop scale-up methodology for industrial devices. 

The derivation and development of a mathematical model which is as general as possible and incorporates detailed knowledge from phenomena operative in emulsion polymerization reactors, its testing phase and its application to latex reactor design, simulation, optimization and control are the objectives of this paper and will be described in what follows. 

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).]... 

Many wastewater flows in industry can not be treated by standard aerobic or anaerobic treatment methods due to the presence of relatively low concentration of toxic pollutants. Ozone can be used as a pretreatment step for the selective oxidation of these toxic pollutants. Due to the high costs of ozone it is important to minimise the loss of ozone due to reaction of ozone with non-toxic easily biodegradable compounds, ozone decay and discharge of ozone with the effluent from the ozone reactor. By means of a mathematical model, set up for a plug flow reactor and a continuos flow stirred tank reactor, it is possible to calculate more quantitatively the efficiency of the ozone use, independent of reaction kinetics, mass transfer rates of ozone and reactor type. The model predicts that the oxidation process is most efficiently realised by application of a plug flow reactor instead of a continuous flow stirred tank reactor. 

We will attempt to keep the mathematical details as brief as possible so that we will not lose sight of the principles of the design and operation of chemical reactors. The student will certainly see more applied mathematics here than in any other undergraduate course except Process Control. However, we wiU try to indicate clearly where we are going so students can see that the mathematical models developed here are essential for describing the application at hand. 

The analytical predictor, as well as the other dead-time compensation techniques, requires a mathematical model of the process for implementation. The block diagram of the analytical predictor control strategy, applied to the problem of conversion control in an emulsion polymerization, is illustrated in Figure 2(a). In this application, the current measured values of monomer conversion and initiator feed rate are input into the mathematical model which then calculates the value of conversion T units of time in the future assuming no changes in initiator flow or reactor conditions occur during this time. 

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. 

Barbieri, G., Scura, F. and Brunetti, A. (2008) Mathematical modelling ofPd-alloy membrane reactors, in Inorganic Membranes Synthesis, Characterization and Applications, vol. 13 (eds R. Mallada and M. Menendez), Membrane Science and Technology, Elsevier BV, Chapter 9 (ISSN... 

The present chapter is not meant to be exhaustive. Rather, an attempt has been made to introduce the reader to the major concepts and tools used by catalytic reaction engineers. Section 2 gives a review of the most important reactor types. This is deliberately not done in a narrative way, i.e. by describing the physical appearance of chemical reactors. Emphasis is placed on the way mathematical model equations are constructed for each category of reactor. Basically, this boils down to the application of the conservation laws of mass, energy and possibly momentum. Section 7.3 presents an analysis of the effect of the finite rate at which reaction components and/or heat are supplied to or removed from the locus of reaction, i.e. the catalytic site. Finally, the material developed in Sections 7.2 and 7.3 is applied to the design of laboratory reactors and to the analysis of rate data in Section 7.4. 

To enable assessment of the feasibility of the BSR concept for a particular application, it is necessary to have an adequate mathematical model that describes the reactor performance in terms of pressure drop and reactant conversion. Ingredients of such mathematical models are relations that describe transport of momentum, mass, and heat in the reactor and relations that describe the kinetics of the reaction that is carried out. Furthermore, an estimate of the costs of structuring the catalyst is needed. 

To assess the feasibility of the BSR as a competitor of the monolithic reactor, the parallel-passage reactor, and the lateral-flow reactor, it is necessary to do case studies in which the performance and price of these reactors are compared, for certain applications. To allow such case studies, two tools are needed (1) mathematical models of the reactors that predict the reactor performance, and (2) an optimization routine that, given a mathematical reactor model and a set of process specifications, finds the optimum reactor configuration. Furthermore, data are needed on costs, safety, availability, etc. In this section, five mathematical models of different complexity for the bead-string reactor (BSR) are presented that can be numerically solved on a personal computer within a few hours down to a few minutes. The implementation of the reactor models in an optimization routine, as well as detailed cost analyses of the reactor, are beyond the scope of this text. 

I have made an attempt to provide sufficient information to understand and to define the specific role of computational flow modeling in reactor engineering applications. Discussions on the main features of reactor engineering, computational flow modeling and their interrelationship will help to select appropriate models, and to apply these computational models to link reactor hardware to reactor performance. Mathematical modeling of flow processes (including turbulent flows, multiphase flows and reactive flows) and corresponding numerical methods to solve these model... 

Penlidis, A. Macgregor, J.F. Hamielec, A.E. Mathematical-modeling of emulsion polymerization reactors—a population balance approach and its applications. ACS Symp. Ser. 1986, 313, 219-240. 

Saldivar, E. Dafniotis, P. Ray, W.H. Mathematical modeling of emulsion copolymerization reactors. I. Model formulation and application to reactors operating with micellar nucleation. J. Macromol. Sci. Rev. Macromol. Chem. Phys. 1998, C38 (2), 207-325. 

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