Reactor steady state multiplicity

There is a voluminous literature on steady-state multiplicity, oscillations (and chaos), and derivation of bifurcation points that define the conditions that lead to onset of these phenomena. For example, see Morbidelli et al. [ Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, Carberry and Varrria (eds), Marcel Dekker, 1987], Luss [ Steady State Multiplicity and Uniqueness... 

Morbidelli, M., Varma, A., and Arts, R., Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, 973-1055, New York Marcel Dekker, 1986. 

Fig. 13. The effect of the reactor volume V on the stability of steady states. Multiple steady states 45). (Copyright by Pergamon Press. Reprinted with permission.)...

Some specific aspects in the modeling of gas-liquid continuous-stirred tank reactors are considered. The influence of volatility of the liquid reactant on the enhancement of gas absorption is analyzed for irreversible second-order reactions. The impact of liquid evaporation on the behavior of a nonadiabatic gas-liquid CSTR where steady-state multiplicity occurs is also examined. 

Packed-bed reactors are discussed qualitatively, particularly with respect to their models. Features of the two basic types of models, the pseudohomogene-ous and the heterogeneous models, are outlined. Additional issues — such as catalyst deactivation steady state multiplicity, stability, and complex transients and parametric sensitivity — which assume importance in specific reaction systems are also briefly discussed. 

Steady State Multiplicity, Stability, and Complex Transients. This subject is too large to do any real justice here. Ever since the pioneering works of Liljenroth (41), van Heerden (42), and Amundson (43) with continuous-flow stirred tank reactors, showing that multiple steady states — among them, some stable to perturbations, while others unstable — can arise, this topic has... 

Jensen and Ray (50) have recently tabulated some 25 experimental studies which have demonstrated steady state multiplicity and instabilities in fixed-bed reactors many of these (cf., 29, 51, 52) have noted the importance of using a heterogeneous model in matching experimental results with theoretical predictions. Using a pseudohomogeneous model, Jensen and Ray (50) also present a detailed classification of steady state and dynamic behavior (including bifurcation to periodic solutions) that is possible in tubular reactors. 

A feature related to steady state multiplicity and stability is that of "pattern formation", which has its origins in the biological literature. Considering an assemblage of cells containing one catalyst pellet each, Schmitz (47, 53) has shown how non-uniform steady states - giving rise to a pattern - can arise, if communication between the pellets is sufficiently small. This possibility has obvious implications to packed-bed reactors. 

The classical problem of steady-state multiplicity in a continuous stirred tank reactor (CSTR) was brought to popular attention in 1953 in the theoretical article by Van Heerden. " Large amounts of experimental work which measured these steady states were performed by the group of Schmitz beginning in 1970. Schmitz also wrote two excellent reviews on multiplicity, stability, and sensitivity of steady states in chemical reactors and the application of bifurcation theory to determine the presence of steady-state multiplicity in chemical reactors.Even these reviews are not inclusive and it is our intention in this subsection to only provide a background to the novice in reactor design. 

Vejtasa, S.A. Schmitz, R.A. An experimental study of steady state multiplicity and stability in an adiabatic stirred reactor. AIChE J. 1970,16, 410 19. Schmitz, R.A. Multiplicity, stability, and sensitivity of states in chemically reacting systems - a review. Adv. Chem. Ser. 1975, 148, 156-211. Razon, L.F. Schmitz, R.-A. Multiplicities and instabilities in chemically reacting systems - a review. Chem. Eng. Sci. 1987, 42, 1005-1047. Uppal, A. Ray, W.H. Poore, A.B. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci. 1974, 29, 967-985. 

The analysis of steady-state multiplicity of a nonisothermal chemical reactor is complicated due to the number of parameters involved and the exponential nonlinearity in the temperature dependence of the kinetic function. In this and the next subsection, the results of the analysis of steady-state multiplicity are presented. The derivations of these results are detailed in a review chapter by Morbidelli et al. (1986). 

While constructed to encompass all types of reactors, the theory is mainly oriented toward continuous stirred-tank reactors and steady-state multiplicity. [For batch reactions, the TOOLBOX returns the not very enlightening information that no steady state can be sustained for the hydrogen-oxygen reaction network 9-50 (deficiency zero in batch) it gives no warning that the system is explosive.]... 

Piston flow reactors lack any internal mechanisms for memory. There is no axial dispersion of heat or mass. What has happened previously has no effect on what is happening now. Given a set of inlet conditions (flin, 7i , Text), only one output (flout, 7 out)is possible. A PFR cannot exhibit steady-state multiplicity unless there is some form of external feedback. External recycle of mass or heat can provide this feedback and may destabilize the system. Figure 14.7 shows an example of external feedback of heat that can lead to the same multiple steady states possible with a CSTR. Another example is when the vessel walls or packing has significant thermal capacity. In such cases, a second heat balance must be added to supplement Equation 14.16. See Section 10.6 for a comparable result. 

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