Continuous stirred tank reactor mathematics

It is useful to examine the consequences of a closed ion source on kinetics measurements. We approach this with a simple mathematical model from which it is possible to make quantitative estimates of the distortion of concentration-time curves due to the ion source residence time. The ion source pressure is normally low enough that flow through it is in the Knudsen regime where all collisions are with the walls, backmixing is complete, and the source can be treated as a continuous stirred tank reactor (CSTR). The isothermal mole balance with a first-order reaction occurring in the source can be written as... 

Yet who would have thought the old man to have had so much hlood in him This title, given by Prof. Rutherford Aris and his collaborator W.W. Farr to their recent paper [Chem. Eng. Sci., 41 (1986) 1385], is a phrase used by Lady Macbeth (Macbeth, V, 1, 42-44). Fierce, isn t it Apparently, they mean it to imply that traditional theoretical problems in the dynamics of chemical reactions, in particular the known problem of the dynamics of the continuous stirred tank reactor (CSTR), are far from being exhausted. Novel mathematical approaches provide new results oriented to physico-chemical comprehension. This current trend is confirmed by the present volume. 

The semibatch reactor is a cross between an ordinary batch reactor and a continuous-stirred tank reactor. The reactor has continuous input of reactant through the course of the batch run with no output stream. Another possibility for semibatch operation is continuous withdrawal of product with no addition of reactant. Due to the crossover between the other ideal reactor types, the semibatch uses all of the terms in the general energy and material balances. This results in more complex mathematical expressions. Since the single continuous stream may be either an input or an output, the form of the equations depends upon the particular mode of operation. 

The rational design of a reaction system to produce a desired polymer is more feasible today by virtue of mathematical tools which permit one to predict product distribution as affected by reactor type and conditions. New analytical tools such as gel permeation chromatography are beginning to be used to check technical predictions and to aid in defining molecular parameters as they affect product properties. The vast majority of work concerns bulk or solution polymerization in isothermal batch or continuous stirred tank reactors. There is a clear need to develop techniques to permit fuller application of reaction engineering to realistic nonisothermal systems, emulsion systems, and systems at high conversion found industrially. A mathematical framework is also needed which will start with carefully planned experimental data and efficiently indicate a polymerization mechanism and statistical estimates of kinetic constants rather than vice-versa. 

Classical chemical reaction engineering provides mathematical concepts to describe the ideal (and real) mass balances and reaction kinetics of commonly used reactor types that include discontinuous batch, mixed flow, plug flow, batch recirculation systems and staged or cascade reactor configurations (Levenspiel, 1996). Mixed flow reactors are sometimes referred to as continuously stirred tank reactors (CSTRs). The different reactor types are shown schematically in Fig. 8-1. All these reactor types and configurations are amenable to photochemical reaction engineering. 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

The continuous stirred-tank reactor (CSTR) is a mathematical model that describes an important class of continuous reactors—continuous, steady, well-agitated tank reactors. The CSTR model is based on two assumptions ... 

Example 4.10 Mathematical Model of a Continuous Stirred Tank Reactor (CSTR)... 

For each of the chemical species participating in the mechanism, a mass balance equation must be written. The appropriate form of the mass balance for the specific type of reactor at hand must be used. Two of the most common types of reactors used in industry are the CSTR (continuous stirred tank reactor) and the tubular reactor. The corresponding mathematical models for their idealized forms, based on transport phenomena equations and available in any standard chemical reactor text [17, 18], are the ideal CSTR and the ideal model for the plug flow tubular reactor (PFR). The ideal CSTR model is given by Equation 12.1 ... 

In the design of ideal batch and continuous stirred tank reactors, we assume uniform current distribution at the electrodes to simplify the mathematical... 

One unique but normally undesirable feature of continuous emulsion polymerization carried out in a stirred tank reactor is reactor dynamics. For example, sustained oscillations (limit cycles) in the number of latex particles per unit volume of water, monomer conversion, and concentration of free surfactant have been observed in continuous emulsion polymerization systems operated at isothermal conditions [52-55], as illustrated in Figure 7.4a. Particle nucleation phenomena and gel effect are primarily responsible for the observed reactor instabilities. Several mathematical models that quantitatively predict the reaction kinetics (including the reactor dynamics) involved in continuous emulsion polymerization can be found in references 56-58. Tauer and Muller [59] developed a kinetic model for the emulsion polymerization of vinyl chloride in a continuous stirred tank reactor. The results show that the sustained oscillations depend on the rates of particle growth and coalescence. Furthermore, multiple steady states have been experienced in continuous emulsion polymerization carried out in a stirred tank reactor, and this phenomenon is attributed to the gel effect [60,61]. All these factors inevitably result in severe problems of process control and product quality. 

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. 

These equations remain valid for bioreactors provided that one employs a suitable mathematical representation of the rate of disappearance of the substrate that is the limiting reagent. In Illustration 13.3 we employ an alternative form of the design equation to determine the holding time necessary to achieve a specified degree of conversion in a strictly batch bioreactor. This illustrative example also indicates how overall yield coefficients are employed as a vehicle for taking the stoichiometry of the reaction into account. Illustration 13.4 describes how one type of semibatch operation (the fed-batch mode) can be exploited to combine the potential advantages of batch and continuous flow operation of a stirred-tank reactor. 

Reactors systems that have plug flow characteristics (that is, iV > 5 or Bo > 7, see Sect. 3.3) differ from CSTR types in that in the former case there is a narrow residence time distribution. The whole mathematical theory of continuous cultivation discussed so far has concerned the well-mixed stirred tank reactor. 

In order to describe adequately the hydrodynamics of the experimental fixed bed reactor, it is necessary to take into account the axial dispersion in the mathematical model. The time dependent continuity equation including axial dispersion for a fixed bed reactor is given by a partial differential equation (pde) of the parabolic/hyperbolic class. These types of pde s are difficult to solve numerically, resulting in long cpu times. A way to overcome these difficulties is by describing the fixed bed reactor as a cascade of perfectly stirred tank reactors. The axial dispersion is then accounted for by the number of tanks in series. For a low degree of dispersion (Bo < 50) the number of stirred tanks, N, and the Bodenstein number. Bo, are related as N Bo/2 [8].The fixed bed reactor is now described by a system of ordinary differential equations (ode s). No radial gradients are taken into account and a onedimensional model is applied. Mass balances are developed for both the gas phase and the adsorbed phase. The reactor is considered to be isothermal. 

A semi-batch reactor is more difficult to analyze mathematically because at least one of the reactant or product species enters or leaves the system boundaries, thus specific applications should be modeled [1,5]. However, the most typical application for a semi-batch reactor is the presence of one reactant initially contained in a stirred tank reactor and a second reactant continuously added to the reactor, with no flow out of the reactor. The addition of a gas to participate in a liquid-phase reaction is one of the more common situations involving a semi-batch reactor, especially because the rate of addition of the gas can be controlled to keep its partial pressure essentially constant as well as providing quantitative information about the rate of reaction. In addition, there is frequently little or no change in the volume of the liquid phase. Well-mixed autoclave reactors coupled with gas pressure controllers, mass flow meters and computers can nicely provide continuous, real-time rate data related to heterogeneous catalysts used in such gas/liquid systems [6-8]. Again, it must be emphasized that experiments must be performed and/or calculations made to verify that no heat or mass transfer limitations exist. 

The commercially used emulsion polymerization reactors (stirred-tank and continuous-loop) are designed to achieve perfect mixing. As will be discussed in Section 6.4.5, perfect mixing is not always achieved. Nevertheless, this flow model allows a good prediction of the emulsion polymerization reactor performance with a moderate mathematical effort, and it will be used here. Macroscopic balances (i.e., considering the reactor as a whole) are used. For the sake of generality, inlet and outlet streams are included in the balances. Both terms should be removed for batch operation, the outlet term should be eliminated in semibatch and both maintained in continuous processes. 

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