Uniqueness of steady states

M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks. Arch, Rational Mech. Anal. 132(4), 311 370 (1995). 

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. 

There is no doubt that studies for the establishment of new classes of mechanisms possessing an unique and stable steady state are essential and promising. On the other hand, it is of interest to construct a criterion for uniqueness and multiplicity that would permit us to analyze any reaction mechanism. An important contribution here has been made by Ivanova [5]. Using the Clark approach [59], she has formulated sufficiently general conditions for the uniqueness of steady states in a balance polyhedron in terms of the graph theory. In accordance with ref. 5 we will present a brief summary of these results. As before, we proceed from the validity of the law of mass action and its analog, the law of acting surfaces. Let us also assume that a linear law of conservation is unique (the law of conservation of the amount of catalyst). 

Here all second-order cycles are even. As shown in ref. 5, one of the reasons for the non-uniqueness of steady state [violation of condition (174)] can be the presence of a cycle composed by the positive paths for which nj.-ifiijpj > n r=x aijp.. In our case this cycle (branching cycle) will be the cycle C3 for which / 21 p13 = 2 x 2 = 4, an a23 = 2 x 1 = 2, and the necessary condition for the uniqueness of a steady state is not fulfilled. A comprehensive numerical analysis of several steady states for a given system will be performed in Chap. 5. 

The basic results in the analysis of non-linear mechanisms using graphs, were obtained by Clark [29], who developed a detailed formalism, and Ivanova [30, 31]. On the basis of Clark s approach, Ivanova formulated sufficiently general conditions for the uniqueness of steady states in terms of the graph theory. She suggested an algorithm that can be used to obtain (see Chap. 3, Sect. 5.4)... 

Chapter 8 will reveal that this is one version of a well-known fluidized-bed reactor model due to Davidson and Harrison). Using this, make an analysis of the uniqueness of steady-state operation. 

The analysis of nonlinear dynamics of chemical process systems has a long tradition. Most emphasis has been on chemical reactors initiated by the seminal work of Bilous and Amundson [10], van Heerden [109], and Aris and Amundson [3]. Comprehensive reviews have been given by Razon and Schmitz [86] or Elnashaie and Elshishini [19]. Multiplicity analysis of non-RD can be traced back to the paper of Rosenbrock [91] where stability and hence uniqueness of steady-states of a binary distillation column is demonstrated under quite general assumptions. 

Ivanova, 1979. Conditions of uniqueness of steady state related to the structure of the reaction mechanism. Kinet. Katal. 20, 1019-1023 (in Russian). 

In the treatment of steady-state pipeline network problems so far we have tacitly assumed that there is a unique solution for each problem. For certain types of networks the existence of a unique solution can indeed be rigorously established. The existence and uniqueness theorems for formulation C were proved by Duffin (DIO) and later extended by Warga (Wl). In Warga s derivation the governing relation for each network element assumes the form,... 

Phillips (1980) andPhillips and Rainbow (1993) have stated that each species of aquatic BMO exhibits unique uptake and elimination kinetics for a particular HOC. The ramification of this statement is revealed in the following simple one-compartment model, which is often used for the determination of steady-state BCFs and Ks s. 

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. 

The Weissenbeig Rheogoniometer (49) is a complex dynamic viscometer that can measure elastic behavior as well as viscosity. It was the first rheometer designed to measure both shear and normal stresses and can be used for complete characterization of viscoelastic materials. Its capabilities include measurement of steady-state rotational shear within a viscosity range of 10-1 —13 mPa-s at shear rates of 10-4 — 104 s-1, of normal forces (elastic effect) exhibited by the material being sheared, and of an oscillatory shear range of 5 x 10-6 to 50 Hz, from which the elastic modulus and dynamic viscosity can be determined. A unique feature is its ability to superimpose oscillation on steady shear to provide dynamic measurements under flow conditions all measurements can be made over a wide range of temperatures (—50 to 400°C). 

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. 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

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. 

With A = 0.06 M and the rate constants of Ref. 14b, these equations admit a unique homogeneous steady-state solution (HSS). It is well known that the irreversible Oregonator 14 and its reversible counterpartl4b exhibit homogeneous limit cycle oscillations for realistic values of rate constants and buffered concentrations. My purpose here is to explore several other features of the reversible model (F) which explain a variety of observed behaviors in closed and open stirred reactors. To that end I begin with the stability properties of the unique HSS, as displayed in the partial phase diagram of Fig. 1. 

For relatively large Lewis numbers and with appropriate feedback, we observe that convergence to the system s unique stable steady state is quite swift and straightforward from any initial value. To illustrate, we now plot the 3D trajectories that emanate from the eight corners of the (xa, xi>, y) unit cube... 

For even larger values of hf, the system eventually reaches a unique fixed steady state that is stationary and involves no limit cycle at all, just as we have seen to be the case for small values of hf in Figure 4.52. For example, for hf = 0.0065, the phase plot starts at sn = 1.27 and S12 = 0.2 in the bottom plot of Figure 4.63 and moves in two spiral loops toward the asymptotic steady state with sn 1.285 and si2 0.17, as depicted in Figure 4.63. 

Region 3 of Figure 7.27 is characterized by having a unique stable steady state with conversion, yield and productivity characteristics very close to those of the unstable steady state of region 2. 

In the determination of steady state reaction kinetic constants of enzyme-substrate reactions, FABMS also provides some very unique capabilities. Since these studies are best performed in the absence of glycerol in the reaction mixture, the preferred method is that which analyzes aliquots which are removed from a batch reaction at timed intervals. Quantitation of the reactants and products of interest is essential. When using internal standards, generally, the closer in mass the ion of interest is to that of the internal standard, the better is the quantitative accuracy. Using these techniques in the determination of kinetic constants of trypsin with several peptide substrates, it was found that these constants could be easily measured (8). FABMS was used to follow the decrease in the reactant substrate and/or the increase in the products with time and with varying concentrations of substrate. Rates of reactions were calculated from these data for each of the several substrate concentrations used and from the Lineweaver-Burk plot, the values of Km and Vmax are obtained. 

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. 

Let us consider a structure for the multitude of steady states for eqns. (158) or (160) in the positive orthant. For linear systems z = Kz it forms either a ray (in the case of the unique linear law of conservation) emerging from zero, or a cone formed at the linear subspace ker K intersection with the orthant. The structure for the multitude of steady states for the systems involving no intermediate interactions is also rather simple. Let us consider the case of only one linear law of conservation ZmjZ, = c = const, and examine the dependence of steady-state values zf on c. Using eqn. (160), we obtain... 

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. 

The Wicke and Eigenberger models are models for an ideal adsorption layer. They have been examined at the Institute of Catalysis, Siberian Branch of the U.S.S.R. Academy of Sciences [93-104,108,109] independently of Wicke and Eigenberger (the first publications refer to 1974). It was shown [93-96] that, for the detailed mechanisms of catalytic reactions either with the steps that are linear with respect to intermediates or with non-linear steps (but containing no interactions between various intermediates), the steady state of the reaction is unique and stable (autocatalytic steps are assumed to be absent). Thus the necessary condition for the multiplicity of steady states is the presence of steps for the interaction between various intermediates in the detailed reaction mechanism [93-100]. Special attention in these studies was paid to the adsorption mechanism of the general type permitting the multiplicity of steady states [102-104]... 

These properties characterize both the type and the stability of steady-state points in system (5). If the steady-state point is unique, it is stable. If there are several steady states, then at least one of them is unstable. Stable and unstable steady states alternate. 

The stability of steady states is analyzed [139] like the investigation performed for the three-step mechanism. In stable steady state, the inequality dg(0o)/d0o > df(0o)/ddo is fulfilled. In the unstable steady state, the sign of this inequality reverses. It can easily been shown that the unique steady state is always stable. If there are three steady states, the outer are stable and the middle is unstable. It can be suggested that the addition to the three-step adsorption mechanism of the impact step that is linear with respect to the intermediate does not produce any essential changes in the phase pattern of the system. The only difference is that at k. x = k 2 = 0 the dynamic model corresponding to the two-route mechanism can have only one boundary steady state (60 = 0, 9C0 = 1). 

There are also two factors that have already been noted in the numerical analysis of the kinetic model of CO oxidation (1) fluctuations in the surface composition of the gas phase and temperature can lead to the fact that the "actual multiplicity of steady states will degenerate into an unique steady state with high parametric sensitivity [170] and (2) due to the limitations on the observation time (which in real experiments always exists) we can observe a "false hysteresis in the case when the steady state is unique. Apparently, "false hysteresis will take place in the region in which the relaxation processes are slow. 

The load schedule specifies for normal at-power operation a unique plant steady state at each power level over the normal operating range, typically from 25 to 100% of full power. This includes the values of all plant forcing functions such as turbo-machine power inputs and reactor and cooler heat rates. For normal operating transients the load schedule gives conditions at which a transient begins and ends. If the transient is an upset event, then while it begins from a point on the load schedule it may terminate at some stable off-normal condition not found on the load schedule. 

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