Stationary states reaction model

Fig. 12.5. (a) The region of multiple stationary-state behaviour for the Takoudis-Schmidt-Aris model of surface reaction, with = 10 3 and k2 = 2 x 10-3 (b), (c), and (d) show how the stationary-state reaction rate varies with the gas-phase pressure of reactant R for different values of p, giving isola, mushroom, and single hysteresis loop respectively. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A415, 363-87.)... 

FIGURE 3.6. (a) Reaction scheme that corresponds to the decrease and increase in capillary frequency, (b) Model of the phase transfer catalytic reaction in the stationary state. Reaction proceeds between the adsorbed chemical species at the interface after adsorption from the water phase. 

In an extension of analytic techniques used by Leovy (150) and Nicolet (180) for stratospheric calculations, Levy (152) developed a simple stationary-state photochemical model of tropospheric radical chemistry. Assuming R42 to be the sole source of radicals and assuming. radical-radical reactions to be the sole loss paths, with R63 = R66 = i- R65, the following expression was derived ... 

The proposed model for the so-called sodium-potassium pump should be regarded as a first tentative attempt to stimulate the well-informed specialists in that field to investigate the details, i.e., the exact form of the sodium and potassium current-voltage curves at the inner and outer membrane surfaces to demonstrate the excitability (e.g. N, S or Z shaped) connected with changes in the conductance and ion fluxes with this model. To date, the latter is explained by the theory of Hodgkin and Huxley U1) which does not take into account the possibility of solid-state conduction and the fact that a fraction of Na+ in nerves is complexed as indicated by NMR-studies 124). As shown by Iljuschenko and Mirkin 106), the stationary-state approach also considers electron transfer reactions at semiconductors like those of ionselective membranes. It is hoped that this article may facilitate the translation of concepts from the domain of electrodes in corrosion research to membrane research. 

Farkas and Sherwood (FI, S5) have interpreted several sets of experimental data using a theoretical model in which account is taken of mass transfer across the gas-liquid interface, of mass transfer from the liquid to the catalyst particles, and of the catalytic reaction. The rates of these elementary process steps must be identical in the stationary state, and may, for the catalytic hydrogenation of a-methylstyrene, be expressed by ... 

In our approach, we analyze not only the steady-state reaction rates, but also the relaxation dynamics of multiscale systems. We focused mostly on the case when all the elementary processes have significantly different timescales. In this case, we obtain "limit simplification" of the model all stationary states and relaxation processes could be analyzed "to the very end", by straightforward computations, mostly analytically. Chemical kinetics is an inexhaustible source of examples of multiscale systems for analysis. It is not surprising that many ideas and methods for such analysis were first invented for chemical systems. 

I would like to comment on the theoretical analysis of two systems described by Professor Hess, in order to relate the phenomena discussed by Professor Prigogine to the nonequilibrium behavior of biochemical systems. The mechanism of instability in glycolysis is relatively simple, as it involves a limited number of variables. An allosteric model for the phosphofrucktokinase reaction (PFK) has been analyzed, based on the activation of the enzyme by a reaction product. There exists a parameter domain in which the stationary state of the system is unstable in these conditions, sustained oscillations of the limit cycle type arise. Theoretical... 

Let us return to Fig. 1.12(c), where there are multiple intersections of the reaction rate and flow curves R and L. The details are shown on a larger scale in Fig. 1.15. Can we make any comments about the stability of each of the stationary states corresponding to the different intersections What, indeed, do we mean by stability in this case We have already seen one sort of instability in 1.6, where the pseudo-steady-state evolution gave way to oscillatory behaviour. Here we ask a slightly different question (although the possibility of transition to oscillatory states will also arise as we elaborate on the model). If the system is sitting at a particular stationary state, what will be the effect of a very small perturbation Will the perturbation die away, so the system returns to the same stationary state, or will it grow, so the system moves to a different stationary state If the former situation holds, the stationary state is stable in the latter case it would be unstable. 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

The CSTR is, in many ways, the easier to set up and operate, and to analyse theoretically. Figure 6.1 shows a typical CSTR, appropriate for solution-phase reactions. In the next three chapters we will look at the wide range of behaviour which chemical systems can show when operated in this type of reactor. In this chapter we concentrate on stationary-state aspects of isothermal autocatalytic reactions similar to those introduced in chapter 2. In chapter 7, we turn to non-isothermal systems similar to the model of chapter 4. There we also draw on a mathematical technique known as singularity theory to explain the many similarities (and some differences) between chemical autocatalysis and thermal feedback. Non-stationary aspects such as oscillations appear in chapter 8. 

This argument shows that for the first-order reaction model the stationary state always has some sort of stability to perturbations. In fact, this is only a first step and will not reveal Hopf bifurcations or oscillatory solutions, should they occur-. A full stability analysis of typical flow-reaction schemes will appear in the next chapter. 

Figure 6.6(b) is better approximated by a cubic form, rate ocy2(l — y). Cubic autocatalysis has already provided us with behaviour of interest in chapter 2. In the remainder of this chapter we consider the stationary-state responses of schemes with this feedback mechanism in flow reactors. We will consider three models, with increasingly varied possible behaviour first an autocatalytic step on its own next we allow the autocatalytic species to undergo a subsequent reaction finally we add an uncatalysed reaction in competition with the autocatalysis. The local stability of such systems is... 

Figures 6.19(a-d) show the four different types of bifurcation diagram where the stationary-state extent of reaction is plotted as a function of residence time for the model with the uncatalysed reaction included for the special case of no catalyst inflow, 0Q = 0. Three patterns have been seen before (a) unique, (b) isola, and (c) mushroom, in the absence of the un-...

We now turn to the non-isothermal reaction system in a non-adiabatic CSTR, as studied in 7.2.4—6. We begin with the simplified model with exponential approximation to the Arrhenius law, and to systems for which the inflow and ambient temperatures are the same (y = 0 and gc = 0), This system has two unfolding parameters gad and rN. The stationary-state equation and its various derivatives are... 

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

Fig. 9.7. Non-stationary behaviour in the diffusive autocatalysis model showing sustained temporal and spatial oscillations with D = 5.2 x 10 3, / = 0.08, and k2 = 0.05 (a) z = 0 or 465 (the oscillatory period) (b) z = 115 (c) z = 140 (d) z = 160 (e) z = 235. The limit cycle obtained by plotting the concentrations at the centre of the reaction zone, a,s(0) and /J (0), versus each other is shown in (f). The broken curve in (a) is the unstable stationary-state profile about...

We may also briefly consider the behaviour of the simple autocatalytic model of chapters 2 and 3 under reaction-diffusion conditions. In a thermodynamically closed system this model has no multiplicity of (pseudo-) stationary states. We now consider a reaction zone surrounded by a reservoir of pure precursor P. Inside the zone, the following reactions occur ... 

Non-linearities arising from non-reactive interactions between adsorbed species will not be our main concern. In this section we return to variations of the Langmuir-Hinshelwood model, so the adsorption and desorption processes are not dependent on the surface coverage. We are now interested in establishing which properties of the chemical reaction step (12.2) may lead to multiplicity of stationary states. In particular we will investigate situations where the reaction step requires the involvement of additional vacant sites. Thus the reaction step can be represented in the general form... 

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

With this identification, the stable stationary-state behaviour (found for the cubic model with 1 < A < 4) corresponds to oscillations for which each amplitude is exactly the same as the previous one, i.e. to period-1 oscillatory behaviour. The first bifurcation (A = 4 above) would then give an oscillation with one large and one smaller peak, i.e. a period-2 waveform. The period doubling then continues in the same general way as described above. The B-Z reaction (chapter 14) shows a very convincing sequence, reproducing the Feigenbaum number within experimental error. 

The specific models we will analyse in this section are an isothermal autocatalytic scheme due to Hudson and Rossler (1984), a non-isothermal CSTR in which two exothermic reactions are taking place, and, briefly, an extension of the model of chapter 2, in which autocatalysis and temperature effects contribute together. In the first of these, chaotic behaviour has been designed in much the same way that oscillations were obtained from multiplicity with the heterogeneous catalysis model of 12.5.2. In the second, the analysis is firmly based on the critical Floquet multiplier as described above, and complex periodic and aperiodic responses are observed about a unique (and unstable) stationary state. The third scheme has coexisting multiple stationary states and higher-order periodicities. 

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

Previously, we considered the case where heat and mass flows are coupled in a reaction diffusion system with heat effects, in which the cross coefficients Zrq. Zqr. and LlS, LSl have vanished (Demirel, 2006). Here, we consider the other three cases. The first involves the stationary state balance equations. In the second case, there is no coupling between the heat flow and chemical reaction with vanishing coefficients Zrq and Zqr. Finally, in the third, there is no coupling between the mass flow and chemical reaction because of vanishing cross-coefficients of ZrS and LSl. The thermodynamically coupled modeling equations for these cases are derived and discussed briefly in the following examples. 

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