Cascade of Ideal CSTR

The cascade consists of a series of ideal continuously operated stirred tank reactors, CSTR, connected one after the other. The outlet function of one CSTR is 

For a cascade of N tanks of equal space time T we obtain for the RTD  

The cumulative RTD curve can be calculated from this by integration. 

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Each of the individual CSTR s that make up the cascade can be analyzed using the techniques and concepts developed in Section 8.3.1. The present section indicates how one may manipulate the key relations developed earlier to obtain equations that simplify the analysis of a cascade of ideal CSTR s. 

We begin by indicating a few generalizations that are relevant to the treatment of batteries of stirrled tank reactors. Consider the cascade of ideal CSTR s shown in Figure 8.10. For any individual reactor denoted by the subscript i the basic design equatidn developed earlier as equation 8.3.4 is appropriate ... 

Plots used in the graphical analysis of cascades of ideal CSTR s. 

Size Comparisons Between Cascades of Ideal Continuous Stirred Tank Reactors and Plug Flow Reactors. In this section the size requirements for CSTR cascades containing different numbers of identical reactors are compared with that for a plug flow reactor used to effect the same change in composition. 

Remark 1 The reactor network consists of ideal CSTRs and PFRs interconnected in all possible ways (see superstructure of reactor network). The PFRs are approximated as a cascade of equal volume CSTRs. The reactors operate under isothermal conditions. 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

A comparison of the various types of reactor concepts, in a general sense, is actually only possible between the batch, the CSTR and the PFR. The cascade of CSTRs, depending on the number of vessels n in the series, more or less behaves as an ideal mixer for n->l or an ideal plug flow for n- - . The fed-batch reactor is more difficult to situate. Although the concentration of compounds important for the rate of reaction can be controlled optimally during the whole fed period, the reactor volume is only partially utilized, especially in the beginning. Nevertheless, this reactor concept certainly has decisive advantages in many cases, as shown by the fact that it is one of the most widely used. 

The representation of different types of reactor units in the approach proposed by Kokossis and Floudas (1990) is based on the ideal CSTR model, which is an algebraic model, and on the approximation of plug flow reactor, PFR units by a series of equal volume CSTRs. The main advantage of such a representation is that the resulting mathematical model consists of only algebraic constraints. At the same time, however, we need to introduce binary variables to denote the existence or not of the CSTR units either as single units or as a cascade approximating PFR units. As a result, the mathematical model will consist of both continuous and binary variables. 

Let us turn now to the second of the problems mentioned earlier—determination of the combination of CSTRs that is best suited to achieving a specified conversion level. We begin by considering the case of a cascade of two arbitrarily sized ideal CSTRs operating under isothermal conditions and then briefly treat the problem of using multiple identical CSTRs in series. Consider the two cascade configurations shown in Figure 8.12. For the first reactor, equation... 

Consider a reactor network that consists of a cascade of two weU-stirred reactors that differ in size but behave as ideal CSTRs. Prior to initiation of a trial designed to determine the average residence time for the cascade, the reactors are operating at steady state at a volumetric flow rate V. The volumes of the first and second CSTRs are and respectively. 

In the development of the -CSTR model, the concentration of a nonreactive tracer leaving the second reactor in the cascade of identical ideal stirred tanks varies with time in the following manner when the concentration of tracer in the feed to the first CSTR undergoes a step change from zero... 

For reactions with positive order, the performance of such a cascade reactor has a specific function between an ideal plug flow reactor and a single CSTR. This can easily be understood comparing the reactant concentration as function of the reactor volume. In a PFR the concentration and, therefore, the transformation rate diminishes with increasing volume from the reactor entrance to the outlet. The low specific performance of a CSTR can be explained by the overall low concentration corresponding to the outlet concentration. In the cascade, the concentration diminishes stepwise from one vessel to the next. This is shown schematically for a series with N=5 vessel in Figure 3.22. With increasing number of equal sized vessels the concentration profile approaches that of a PFR. 

A cascade of stirred tanks (Figure 4.10.18) is the simplest combination of ideal reactors. This configuration is also used in the chemical industry since the principal disadvantage of a single stirred tank can be bypassed, namely, that a CSTR operates with a low concentration requiring a reactor with a large volume (Section 4.10.2.7). Thus it is helpful to use a sequence of reactors where only the last tank is operated at the final concentrations ofthe reactants. Furthermore, a cascade of CSTRs is useful to model real reactors as we will learn in Section 4.10.5. 

Equation (4.10.134) can be also derived based on the following consideration A real tubular reactor can be regarded as a cascade consisting of N CSTRs, and for a high value of N we approach plug flow behavior. Thus the condition of a negligible influence of backmixing is that the Da number needed to reach a certain conversion in the cascade should not deviate from the value of Da needed in an ideal PFR by more than 5%. For a first-order reaction, this condition and Eqs. (4.10.32) and (4.10.25) lead to ... 

Ideal reactors have idealized flow patterns. Four cases are important, the uniformly mixed batch reactor, the plug flow reactor (PFR), the continuous stirred tank reactor (CSTR), and a cascade of CSTRs. Real reactors are arbitrarily complicated, but can be regarded as composed of elements of ideal reactors. Modeling is possible, if we know how to account for non-ideal flow. 

For ideal isothermal reactors, the conversion of a reactant A can be calculated by one parameter, the Damkoehler number. (For a cascade of CSTRs we also need the number of CSTRs.) For a reaction order n and a rate constant k, Da equals for a batch reactor (t = reaction time) and r (r = resi-... 

It may be described by comparison to a stirred tank cascade of as many tanks as necessary to obtain the behavior of the real reactor the result of this comparison is called "number of equivalent tanks". In this model the CSTR equals 1 ideal tank and the PFR is equivalent to a cascade with an infinite number of tanks. This description corresponds to the mathematical cell model. 

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. 

A "cascade" comprises a number of CSTR s in series in the idealized model perfect mixing is assum for each reactor. 

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