Stationary-state behaviour

The stationary states are given by the solutions of a cubic equation  

As seen in chapter 6, this may have multiple solutions over some range of residence times if b0 a0. The range of residence times is given by 

If b0 a0 or if tres lies outside this range, the system has just one stationary state. (Note that if b0 = ia0 exactly, then t s = t , = fftMo)-1 and the value of ass at this residence time is ass = la0.) 

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Fig. 2.4. Computed concentration.histories for autocatalytic model with rate constants given exactly as in Table 2.1 (a) exponential decay of precursor (b) intermediate concentrations a(t) and 6(r), showing initial pseudo-stationary-state behaviour but subsequent development of an oscillatory period of finite duration, 1752 s < t < 3940 s.

Stationary-state behaviour for systems with catalyst decay... 

This decoupling of the concentrations of A and B may appear initially to be only a small modification, but it really has far-reaching effects. We will see below that even the stationary-state behaviour can be much more complex, and there is a much greater flexibility in the pattern of local stabilities the uppermost and lowest states are no longer always stable, nor are unique states, and oscillatory responses are now possible. 

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. 

We should first recall the stationary-state behaviour for this case. If the reaction rate constant for the catalyst decay step is large compared with that for the autocatalytic step, so that k2 > iV, the system can only ever have one stationary state. This state corresponds to no net conversion of A to B, so ass = 1. For slower decay rates, k2 < Vs non-zero stationary states exist over a range of residence times t 9 < ires < t+s in the form of an isola. The extents of conversion along the branches of the isola are given by... 

In the course of any given experiment we may vary the residence time. In between experiments there are now two parameters which we can alter the decay rate constant k2 and the inflow concentration of autocatalyst fi0. We thus wish to divide up the parameter plane into different regions, within each of which our experiments will reveal qualitatively different responses. We have already achieved this for the stationary-state behaviour, yielding regions of unique, isola, and mushroom patterns (see Fig. 6.18). We will now add the... 

A solution of the reaction-diffusion equation (9.14) subject to the boundary condition on the reactant A will have the form a = a(p,r), i.e. it will specify how the spatial dependence of the concentration (the concentration profile) will evolve in time. This differs in spirit from the solution of the same reaction behaviour in a CSTR only in the sense that we must consider position as well as time. In the analysis of the behaviour for a CSTR, the natural starting point was the identification of stationary states. For the reaction-diffusion cell, we can also examine the stationary-state behaviour by setting doi/dz equal to zero in (9.14). Thus we seek to find a concentration profile cuss = ass(p) which satisfies... 

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.)... 

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. 

Despite its simplicity, our model displays an unexpected variety of stationary-state behaviour, with isolas and mushrooms for the dependence of the extent of conversion on residence time or flow-rate. There is also a rich variety of stability and character, with stable and unstable nodes and foci. It is also the simplest isothermal scheme to show sustained oscillatory responses. 

They are of great value, illustrating all aspects of the stationary-state behaviour unique and multiple solution, hysteresis and jumps between different branches (ignition and extinction or washout), and the effects of reversibility and of non-zero inlet concentration of the autocatalyst. The algebraic analyses are, by comparison, far less transparent, although their forms can also be expressive. 

If we apply the term stationary state to those systems in which the concentration of growing chains does not vary throughout the reaction, then it is evident that none of the systems comprised under Case 1 can involve a stationary state. The population of growing chains is declining throughout most of the reaction and falls to zero before all the monomer has been consumed. The method of dealing with this type of kinetic behaviour, devised by Pepper et al., [26] has been used effectively by others [40b, 109, 110]. 

In open, or flow, reactors chemical equilibrium need never be approached. The reaction is kept away from that state by the continuous inflow of fresh reactants and a matching outflow of product/reactant mixture. The reaction achieves a stationary state , where the rates at which all the participating species are being produced are exactly matched by their net inflow or outflow. This stationary-state composition will depend on the reaction rate constants, the inflow concentrations of all the species, and the average time a molecule spends in the reactor—the mean residence time or its inverse, the flow rate. Any oscillatory behaviour may now, under appropriate operating conditions, be sustained indefinitely, becoming a stable response even in the strictest mathematical sense. 

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 are left with the behaviour of the intermediates A and B. A common approach to kinetic models involving relatively reactive species is to apply the pseudo-stationary-state (PSS) hypothesis. 

Stability relates to the behaviour of a system when it is subjected to a small perturbation away from a given stationary state (or if fluctuations occur naturally). If the perturbation decays to zero, the system has some in-built tendency to return back to the same state. In this case it is described as locally stable. (The qualification local means that very large perturbations may have different consequences.) We will introduce the relatively simple mathematical techniques required to determine this local stability of a given state in Chapter 3. It will also be useful before then to reduce the reaction rate equations (2.1)—(2.3) to their simplest possible form by introducing dimensionless variables and quantities. 

At this stage, however, we may proceed to the important features of the model s behaviour qualitatively from Fig. 2.4. For some range of time or of reactant concentration the pseudo-stationary state described by eqns (2.15) and (2.16) become locally unstable. Let this range be denoted... 

The conditions under which the above stationary-state solution loses its stability can be determined following the recipe of 2.6. Again we find that instability may arise, and hence oscillatory behaviour is possible, in this reversible case. The condition for the onset of instability can be expressed in terms of the reactant concentration p < p p, where... 

In addition to the general aims set out at the beginning of this chapter we have discovered a wealth of specific detail about the behaviour of the simple kinetic model introduced here. Most results have been obtained analytically, despite the non-linear equations involved, with numerical computation reserved for confirmation, rather than extension, of our predictions. Much of this information has been obtained using the idea of a pseudo-stationary state, and regarding this as not just a function of time but also as a function of the reactant concentration. Stationary states can be stable or unstable. 

The concentrations a and P vary in time as they approach or move away from any particular stationary state. Often it is convenient to visualize the time-dependent behaviour another way, by plotting the variation of one concentration against that of the other, in what is known as the a-/ phase plane. 

These changes in stability and character are marked on the stationary-state loci in Fig. 3.5. This also shows clearly that the loss of stability occurs as the two loci cross, when /z = 1. At this point eqn (2.84) is also satisfied with tr(J) = 0, corresponding to the special case (f) of 3.2.1 and holding the promise of the onset of oscillatory behaviour. We return to this point later. 

The question of what happens to the system in the range of instability, and how the concentrations of A and B vary as they move away from the unstable stationary state, leads us to the study of sustained oscillatory behaviour. Before a full appreciation of the latter can be obtained, however, we must rehearse the relevant theoretical background. Fortunately the autocatalytic model is again an exemplary system with which to introduce at least the basic aspects of the Hopf bifurcation, and we will do this in the next section. 

This quick test does not, however, tell us that there will be only one stable limit cycle, or give any information about how the oscillatory solutions are born and grow, nor whether there can be oscillations under conditions where the stationary state is stable. We must also be careful in applying this theorem. If we consider the simplified version of our model, with no uncatalysed step, then we know that there is a unique unstable stationary state for all reactant concentrations such that /i < 1. However, if we integrate the mass-balance equations with /i = 0.9, say, we do not find limit cycle behaviour. Instead the concentration of B tends to zero and that for A become infinitely large (growing linearly with time). In fact for all values of fi less than 0.90032, the concentration of A becomes unbounded and so the Poincare-Bendixson theorem does not apply. 

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 behaviour exhibited by this model is relatively simple. There is only ever one limit cycle. This is born at one bifurcation point, grows as the system traverses the range of unstable stationary states, and then disappears at the second bifurcation point. Thus there is a qualitative similarity between the present model and the isothermal autocatalysis of the previous chapter. The limit cycle is always stable and no oscillatory solutions are found outside the region of instability. 

First, can we expect any oscillatory behaviour Instability is possible only if k < e 2. This requirement is satisfied here. From the data in Table 4.4, the Hopf bifurcation points for this system occur for n = 0.207 and n = 0 058. For our example, the initial value /r0 = 0.5 exceeds the upper bifurcation point, so the system at first has a stable pseudo-stationary state to approach, with dss x 10 and ass x 4.54 x 10 4. From Fig. 4.3 we may also estimate that the approach to this state will be monotonic since the initial conditions lie outside the region of damped oscillations. 

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

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