Multiplicity of steady states

The obtained result gives a desired answer regarding the validity of the Horiuti-Boreskov form. So, the presentation of the overall reaction rate of the complex reaction as a difference between two terms, overall rates of forward and backward reactions respectively, is valid, if we are able to present this rate in the form of Equation (77). We can propose a reasonable hypothesis (it has to be proven separately) that it is always possible even for the nonlinear mechanism, if the "physical" branch of reaction rate is unique, i.e. multiplicity of steady states is not observed. As it has been proven for the MAE systems, the steady state is unique, if the detailed mechanism of surface catalytic reaction does not include the step of interaction between the different surface intermediates (Yablonskii et ah, 1991). This hypothesis will be analyzed in further studies. 

Besides, as was pointed out in the Preface, there exists a number of largely unexplained, practically relevant phenomena, occurring in purely electro-diffusional systems and potentially related to the multiplicity of steady states [19]—[21]. Finally, uniqueness results could be valuable for the numerical analysis of semiconductor models. 

J. Yet Who Would Have Thought the Old Man to Have had so Much Blood in Him —Reflections on the Multiplicity of Steady States of the Stirred Tank Reactor... 

OLD MAN TO HAVE HAD SO MUCH BLOOD IN HIM —REFLECTIONS ON THE MULTIPLICITY OF STEADY STATES OF THE STIRRED TANK REACTOR... 

The notions of uniqueness and multiplicity of steady states in the development of chemical reactor analysis. In W.F. Furler (ed.), A Century of Chemical Engineering, (pp. 389-404). New York Plenum Press, 1982. 

Yet who would have thought the old man to have had so much blood in him —Reflections on the multiplicity of steady states of the stirred rank reactor (with W.W. Farr). Chem. Eng. Sci. 41, 1385-1402 (1986). (Reprint J)... 

Continuous reactions in a non-isothermal CSTR-I. Multiplicity of steady states (with P. Cicarelli). Chem. Eng. Sci. 49,621-631 (1994). 

The essential topics of this review article are the experimental aspects of multiplicity of steady states as well as periodic activity of open chemical reacting systems catalyst-gas. In the last two decades a great number of theoretical papers were published on this subject which indicated a number of pathological phenomena to be expected in chemically reacting systems. The next step toward a deeper understanding of... 

The paper is not equation oriented since after the period of theoretical investigation, only a small percentage of experimental papers published is completely supported with theory and very often only a qualitative explanation is presented. Hence in this paper we shall review the experimental information published in the literature concerning multiplicity of steady states and periodic activity in the systems catalyst-gas, making an attempt to explain qualitatively these phenomena on the basis of the theory developed.1 The number of experimental observations surveyed here which are not supported by a theory will surely indicate that there are many roads open for fundamental research in this area. 

Condition 2 The necessary condition for multiplicity of steady states is given by B > B. ... 

The multiplicity of steady states and associated bifurcation phenomena are associated with open systems which allow more than one stationary nonequilibrium state for the same set of parameters. This multiplicity of a steady state may result from many causes. The most important sources of multiplicity in chemical/biological engineering processes are ... 

The reaction is exothermic. For (3 = 1.2 the exothermicity factor, and 7 the dimensionless activities factor, find the range of a, the dimensionless preexponential factor, that gives multiplicity of steady states using a suitable MATLAB program with Kc = 0, the dimensionless heat transfer coefficient of the cooling jacket. Choose a value of a in the multiplicity region and obtain the multiple steady-state dimensionless temperatures and concentrations. 

This appendix gives an introduction to multiplicity of steady states and to bifurcation and chaos in Chemical and Biological Engineering. 

Sets of quasi-steady-state equations can have several solutions, which correspond to several steady-state rate values of complex reaction in open systems (multiplicity of steady states). It has been shown that the necessary condition here is the presence of an interaction step between various intermediates in the complex reaction mechanism. Let us discuss this result in more detail. 

For several cases, e.g. for linear pseudo-steady-state equations (linear mechanisms), the steady state is certain to be unique. But for non-linear mechanisms and kinetic models (which are quite common in catalysis, e.g. in the case of dissociative adsorption), there may be several solutions. Multiplicity of steady-states is associated with types of reaction mechanisms. 

In conclusion of the discussion of reaction dynamics in closed systems, it can be suggested that the principal problems here have been solved closed systems "have been closed . The case is different for open systems. Progress in their study has been extensive. A large number of publications are devoted to the analysis of various dynamic peculiarities (multiplicity of steady states, self-oscillations, stochastic self-oscillations) in various open systems. It can hardly be said that most problems here are completely clear. 

A similar claim for heterogeneous systems is, generally speaking, wrong. Indeed, gas concentrations rapidly become close to some values controlled by the balance equations and concentration ratios for the input gas flow. But in close proximity to this value any dynamic behaviour is possible, i.e. a multiplicity of steady states, self-oscillations, etc. The surface state can, however, vary in a rather complicated manner. Figuratively speaking, nontrivial dynamic behaviour of heterogeneous systems cannot be "inhibited (by a heavy flow). 

Let us emphasize the most essential conclusion that can be drawn in this section a sufficient condition for the uniqueness of steady states in catalytic reactions is the absence of interaction steps for various intermediates in the detailed reaction mechanisms. Their presence is a necessary condition for the multiplicity of steady-state values for the catalytic reaction rates. This principal statement possesses an evident discrimination property. If some experiment is characterized by the multiplicity of steady states and its interpretation suggests a law of acting surfaces, the description of this experiment implies a detailed mechanism that must contain interaction steps of various intermediates. 

To study the problem concerning the uniqueness and multiplicity of steady states it is necessary to consider one more type of cycle that is more general compared with oriented cycles. We will call them Clark (or Clark-Ivanova) cycles. 

Investigations with the graphs of non-linear mechanisms had been stimulated by an actual problem of chemical kinetics to examine a complex dynamic behaviour. This problem was formulated as follows for what mechanisms or, for a given mechanism, in what region of the parameters can a multiplicity of steady-states and self-oscillations of the reaction rates be observed Neither of the above formalisms (of both enzyme kinetics and the steady-state reaction theory) could answer this question. Hence it was necessary to construct a mainly new formalism using bipartite graphs. It was this formalism that was elaborated in the 1970s. 

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