Modeling of stirred reactors

In the following sub-section, state of the art CFD modeling of stirred reactors is reviewed with reference to these requirements. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Y.Q. Cui, R.G.J.M. Van der Lans, HJ. Noorman, and K.C.A.M. Luyben. Compartment mixing model for stirred reactors with multiple impellers. Transactions of IChemE, 74 261-271,... 

FIGURE 1.7 Network of zones model for stirred reactors 2 x 10 x 10 network. Flow through different zones was specified using experimental measurements (from Mann and Mavros, 1982). 

Deans and Lapidus (1960) made use of the analogy between a tubular reactor and a sequence of continuous flow stirred tank reactors for the modelling of heterogeneous reactors. In cell models the reactor is subdivided into small finite elements, the height of each being usually equal to one pellet diameter and each of these elements is considered to be a small CSTR in which the fluid phase is perfectly mixed. In the simple cell model the temperature and concentration conditions of any given cell are dependent only on those of the previous row of cells. 

Eoint, then the ordinary differential equations become two-point oundary value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Determination of Conversion Levels Based on the Cascade Model of Stirred-Tank Reactors... 

As opposed to the model of a reactor with internal heat removal due to boiling, the temperature in each zone of the tubular turbulent reactors, with external heat removal, is determined not only by the heat balance inside the reaction zone, but also by the amount of reaction heat removed through the wall. For an estimation of the temperature change in the cooling zone of the reaction mixture at vigorous cross-sectional stirring (turbulent flow), the following equation is true [2] ... 

Chapter 5 gives an overview of novel green reactors and the application of the CFD technique in modelling of green reactors. This chapter presents detailed discussions on a number of novel reactors, namely, the microreactor, microwave reactor and spinning disc reactor. A brief introduction on CFD and the application of CFD in modelling laminar mixing in a stirred tank reactor is presented. 

Hlavacek, V., Kubicek, M., and Visnak, K. Modeling of chemical reactors XXVI. Multiplicity and stability analysis of a continuous stirred tank reactor with exothermic consecutive reactions A B C. Chem. Eng. Sci. 27, 719-742, 1972. 

For the modeling of a reactor we need solutions of the equations of the balances of mass, energy, and impulse. For isothermal operation the energy balance is not needed. The impulse balance mostly only serves to calculate the pressure drop of a reactor. The definition of a suitable control space for balancing is important. In the simplest case, the variables - such as temperature and concentrations - are constant within the control space (stirred tank reactor). However, in many cases the system variables depend on the location, for example, in the axial direction in a tubular reactor. Then infinitesimal balances (differential equations) have to be solved to obtain integral data. 

The only relatively simple way of modelling a stirred reactor with imperfect macro-mixing but rapid micro-mixing, is to make use of the tanks-in-series model, described in section 7.2.7.i, or of a more complex "black box model such as will be discussed in the next section. 

Industrial-scale SSP is carried out in moving packed bed, fluidized bed, and stirred bed reactors Mallon and Ray have published a brief discussion of idealized models of these reactors [75]. Fluidized beds have a serious drawback because of the high consumption of gas needed to keep the bed in a fluidized state, and residence time distribution is unfavorable to high conversions. Stirred beds in series are a possible solution, depending on economic details. 

It is useful to relate the space-velocity to other simple models of practical reactor performance. Expressions describing conversion in model reactors are provided later (section 2.5). It is convenient to summarize expressions for the (well-stirred) simple batch reactor and the single-pass, plug flow reactor here, considering a reaction under complete mass transport control. For the batch reactor, the limiting current is given by ... 

In the case of lumped processes, hybrid model design focuses on reducing a complex set of static relationships with mat r fit parameters by overall fiizzy sub-models. These sub-models can be derived from measurements. A polymer reactor has been used as an example. In the liquid phase reactor, which is ideally stirred, monomer reacts to polymer by means of a single site metallocene catalyst. The inputs of the reactor are listed in Table 30.2 the load Fin, the hydrogen fraction of the feed xh2, the activated catalyst fraction Xc,a, the non-activated catalyst fraction Xc,na, the monomer fraction and the jacket temperature 7 adfe,. The most important outputs are the molecular weight distribution MWD (or q, the chain termination probability which is inversely proportional to MWD) and the conversion or (feed mass fraction conversion). The model is described in detail by Roffel and Betlem (2003). The requirements for the model of this reactor are ... 

K. K. Boon, "A Flexible Mathematical Model for Analy2ing Industrial P. F. Furnaces," M.S. thesis. University of Newcasde, AustraUa, Sept. 1978. R. H. Essenhigh, "A New AppHcation of Perfectly Stirred Reactor (P.S.R.) Theory to Design of Combustion Chambers," TechnicalEeport FS67-1 (u), Peimsylvania State University, Dept, of Euel Science, University Park, Pa., Mar. 1967. 

Specific reactor characteristics depend on the particular use of the reactor as a laboratory, pilot plant, or industrial unit. AH reactors have in common selected characteristics of four basic reactor types the weH-stirred batch reactor, the semibatch reactor, the continuous-flow stirred-tank reactor, and the tubular reactor (Fig. 1). A reactor may be represented by or modeled after one or a combination of these. SuitabHity of a model depends on the extent to which the impacts of the reactions, and thermal and transport processes, are predicted for conditions outside of the database used in developing the model (1-4). 

Fig. 2. Schematic models of a plug flow electrochemical reactor (PFER) and a stirred tank electrochemical reactor (STER).

A stirred tank sometimes can be modeled as having a fraction Ot in bypass and a fraction of the reactor volume stagnant. The material balance then is made up of... 

Guichardon etal. (1994) studied the energy dissipation in liquid-solid suspensions and did not observe any effect of the particles on micromixing for solids concentrations up to 5 per cent. Precipitation experiments in research are often carried out at solids concentrations in the range from 0.1 to 5 per cent. Therefore, the stirred tank can then be modelled as a single-phase isothermal system, i.e. only the hydrodynamics of the reactor are simulated. At higher slurry densities, however, the interaction of the solids with the flow must be taken into account. 

Non-ideal reactors are described by RTD functions between these two extremes and can be approximated by a network of ideal plug flow and continuously stirred reactors. In order to determine the RTD of a non-ideal reactor experimentally, a tracer is introduced into the feed stream. The tracer signal at the output then gives information about the RTD of the reactor. It is thus possible to develop a mathematical model of the system that gives information about flow patterns and mixing. 

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