Models CSTR

Reliable kinetie data are of paramount importanee for sueeessful modelling and seale-up of preeipitation proeesses. Many data found in the literature have been determined assuming MSMPR eonditions, analogous to the CSTR model in reaetion engineering. Here, a method developed by Zauner and Jones (2000a) is outlined. 

In the second model (Figure 5.1b), the mixed-flow or continuous well-mixed or continuous-stirred-tank (CSTR) model, feed and product takeoff are both continuous, and the reactor contents are assumed to be perfectly mixed. This leads to uniform composition and temperature throughout the reactor. Because of the perfect mixing, a fluid element can leave the instant it enters the reactor or stay for an extended period. The residence time of individual fluid elements in the reactor varies. 

Residence time distribution curves for the n-CSTR model. 

The CSTR model, on the other hand, is based on a stirred vessel with continuous inflow and outflow (see Fig. 1.2). The principal assumption made when deriving the model is that the vessel is stirred vigorously enough to eliminate all concentration gradients inside the reactor (i.e., the assumption of well stirred). The outlet concentrations will then be identical to the reactor concentrations, and a simple mole balance yields the CSTR model equation ... 

The CSTR model can be derived from the fundamental scalar transport equation (1.28) by integrating the spatial variable over the entire reactor volume. This process results in an integral for the volume-average chemical source term of the form ... 

The first combination - well macromixed and well micromixed - is just the CSTR model for a stirred reactor wherein the scalar is assumed to be constant at every point in the reactor. The second combination - well macromixed and poorly micromixed - corresponds to a statistically homogeneous flow and is often assumed when deriving CRE micromixing models. The third combination - poorly macromixed and well micromixed - is often... 

Example 9Ji. The nonisothermal CSTR modeled in Sec. 3.6 can be linearized (sec Prob. 6.12) to give two linear ODEs in teinis of perturbation variables. 

Example 2. Let us now test the robust linear regulator under model mismatch. For this application let us consider again the CSTR of Example 1. Here, we assumed that the CSTR model dynamics is given by... 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

Figure 7-4 Slurry reactor (left) for well-mixed gas-solid reactions and fluidized bed reactor (center) for liquid-solid reactions. At the right is shown a riser reactor in which the catalyst is carried with the reactants and separated and returned to the reactor. The slurry reactor is generally mixed and is described by the CSTR model, while the fluidized bed is described by the PFTR or CSTR models.

Figure 8-15 Sketch of Los Angeles and simplified CSTR model of air in the LA basin.

While the CSTR model is extremely primitive, the ideas of size dependence and differences in r for air flow before and after ignition explain many of the features of these troublesome chemical reactors. 

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. 

The analogy between the above equations and the CSTR model can be easily realized. From this analogy we can define the modified flow rate with the following physical significance ... 

If, in the system examined, we can neglect spatial differences in the reactant concentrations, a continuous stirred tank reactor (CSTR) model for a reactor can be used. A set of equations is constructed accounting for the process of the totality of reactions under examination at a constant volume. It is then supplemented by a new factor which accounts for the substance exchange with the ambient medium. As usual, concentration equations are used that are analogues to those for substance quantities since the reaction system volume is assumed to be unchanged... 

We have used CO oxidation on Pt to illustrate the evolution of models applied to interpret critical effects in catalytic oxidation reactions. All the above models use concepts concerning the complex detailed mechanism. But, as has been shown previously, critical. effects in oxidation reactions were studied as early as the 1930s. For their interpretation primary attention is paid to the interaction of kinetic dependences with the heat-and-mass transfer law [146], It is likely that in these cases there is still more variety in dynamic behaviour than when we deal with purely kinetic factors. A theory for the non-isothermal continuous stirred tank reactor for first-order reactions was suggested in refs. 152-155. The dynamics of CO oxidation in non-isothermal, in particular adiabatic, reactors has been studied [77-80, 155]. A sufficiently complex dynamic behaviour is also observed in isothermal reactors for CO oxidation by taking into account the diffusion both in pores [71, 147-149] and on the surfaces of catalyst [201, 202]. The simplest model accounting for the combination of kinetic and transport processes is an isothermal continuously stirred tank reactor (CSTR). It was Matsuura and Kato [157] who first showed that if the kinetic curve has a maximum peak (this curve is also obtained for CO oxidation [158]), then the isothermal CSTR can have several steady states (see also ref. 203). Recently several authors [3, 76, 118, 156, 159, 160] have applied CSTR models corresponding to the detailed mechanism of catalytic reactions. 

Solution Polymerization in a CSTR. Although many polymerization reactors in use by industry have the residence time distribution of a CSTR, they may not, at first glance, have the appearance of a CSTR (cf. Figure 1). Nevertheless, CSTR models, perhaps with some allowance for imperfect micromixing, are successfully employed to describe these reactors. Thus the behavior of the CSTR is of great practical interest. 

Tubular Reactor with Dispersion As discussed earlier, a multistage CSTR model can be used to simulate the RTD in pilot and commercial reactors. The dispersion model, similar to Fick s molecular diffusion law with an empirical dispersion coefficient De replacing the diffusion coefficient, may also be used. 

The above equation is often converted to dimensionless variables and solved. The solution of this partial differential equation is recorded in the literature [Otake and Kunigata, Kagaku Kogaku, 22 144 (1958)]. The plots of E(tr) versus /. are bell-shaped, similar to the response for a series of n CSTRs model (Fig. 19-7). A relation between a2(tr), n, and Pe (for the closed-ends condition) is... 

Understanding Reactor Flow Patterns As discussed above, a RTD obtained using a nonreactive tracer may not uniquely represent the flow behavior within a reactor. For diagnostic and simulation purposes, however, tracer results may be explained by combining the expected tracer responses of ideal reactors combined in series, in parallel, or both, to provide an RTD that matches the observed reactor response. The most commonly used ideal models for matching an actual RTD are PRF and CSTR models. Figure 19-9 illustrates the responses of CSTRs and PFRs to impulse or step inputs of tracers. 

There have been many hybrid multiscale simulations published recently in other diverse areas. It appears that the first onion-type hybrid multiscale simulation that dynamically coupled a spatially distributed 2D KMC for a surface reaction with a deterministic, continuum ODE CSTR model for the fluid phase was presented in Vlachos et al. (1990). Extension to 2D KMC coupled with ID PDE flow model was described in Vlachos (1997) and for complex reaction networks studied using 2D KMC coupled with a CSTR ODEs model in Raimondeau and Vlachos (2002a, b, 2003). Other examples from catalytic applications include Tammaro et al. (1995), Kissel-Osterrieder et al. (1998), Qin et al. (1998), and Monine et al. (2004). For reviews, see Raimondeau and Vlachos (2002a) on surface-fluid interactions and chemical reactions, and Li et al. (2004) for chemical reactors. 

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