Autocatalyst stationary states

The stationary-state concentration / ss of the autocatalyst shows a linear dependence on the reactant concentration p, as shown in Fig. 3.1.The locus for ass(p) shows a maximum ... 

Influence of autocatalyst inflow on multiple stationary states... 

Fig. 6.8. (a) Schematic three-dimensional representation of the stationary-state surface (1 — ass)-tres- 0 showing the folding at low autocatalyst inflow concentrations which gives rise to ignition, extinction, and multiplicity, (b) The projection onto the P0-t , parameter plane of the two lines of fold points in the stationary-state surface, forming a typical cusp at P0 = gT,e> = 7 inside this cusp region the system has multiple stationary states outside, there is... 

Thus, reversibility decreases the range of inflow concentrations over which multiple stationary states can exist. If the reactor has no autocatalyst in the inflow, multistability exists over some range of residence times, no matter how small the equilibrium constant becomes. Otherwise, increasing the inflow concentration decreases the extent of reversibility (i.e. raises the minimum value for Kc) over which multistability can be found. 

Fig. 6.12. The influence of autocatalyst inflow on the reaction rate curve R for a system with reversibility, = 9 (a) po = showing multiple stationary-state intersections ... 

A number of points need to be made about this result. First, unlike the similar relationship (6.11), eqn (6.50) only applies at the stationary state. Secondly, the numerator really contains two contributionsrthe inflow of B, as P0, and the amount of A that has been converted to B, as 1 — ass. The denominator then shows that the stationary-state concentration of the autocatalyst is always less than this. Of course this shortfall between the amount of B present in the reactor and that which has flowed in or been produced merely.reflects the number of such molecules which have then reacted further to produce C. Thus, the denominator increases as the rate... 

We are again concerned with intersections of R and L on the flow diagram. The larger the value of k2, the steeper the minimum gradient of the flow line and hence L will not cut as far into R as tres varies. In particular we may lose the possibility of one or even both tangencies between the curves, and hence lose points of ignition and extinction. To illustrate the effect of the autocatalyst decay through if2 on the stationary-state response we can consider a CSTR which is fed only by the reactant A, so po = 0. 

Fig. 6.14. Stationary-state loci for reaction with no autocatalyst inflow, but autocatalyst decay and k2 < -j. The zero-reaction state 1 — a = 0 exists as a solution for all conditions the non-zero solutions form a closed curve (isola) which grows as k2 is decreased. The isola patterns shown are for k2 =, 8, 2o, and jj in order of increasing size.

We can now consider how the relationship between isolas and unique stationary states, and indeed any other new patterns of behaviour, is affected by the inflow of some autocatalyst. In such a case / 0 will be non-zero. 

Fig. 8.1. Indication of local stability or instability for the simple cubic autocatalytic step without decay solid curves indicate branches of stable stationary-state solutions, broken curves correspond to unstable states, (a) Stationary-state locus with no autocatalyst inflow, fl0 = 0, with one stable solution, 1 - = 0, corresponding to zero reaction (b) stationary-state locus...

For systems with inflow of both reactant and autocatalyst, the stationary-state solutions cannot be obtained explicitly. The alternation in stability when there are multiple solutions described above is, however, quite general and is strictly followed even when b0 0. 

Equation (8.13) gives the appropriate form for l/treiax( = — X) for the cubic autocatalysis model with no inflow of autocatalyst. The condition for the turning point in the stationary-state locus (there is only one) is = 4. 

Fig. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

Even for the present simple case, for which the inflow does not contain the autocatalyst, we have seen a variety of combinations of stable and unstable stationary states with or without stable and unstable limit cycles. Stable limit cycles offer the possibility of sustained oscillatory behaviour (and because we are in an open system, these can be sustained indefinitely). A useful way of cataloguing the different possible combinations is to represent the different possible qualitative forms for the phase plane . The phase plane for this model is a two-dimensional surface of a plotted against j8. As these concentrations vary in time, they also vary with respect to each other. The projection of this motion onto the a-/ plane then draws out a trajectory . Stationary states are represented as points, to which or from which the trajectories tend. If the system has only one stationary state for a given combination of k2 and Tres, there is only one such stationary point. (For the present model the only unique state is the no conversion solution this would have the coordinates a,s = 1, Pss = 0.) If the values of k2 and tres are such that the system is lying at some point along an isola, there will be three stationary states on the phase... 

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... 

Fig. 9.4. (a) The dependence of the stationary-state concentration of reactant A at the centre of the reaction zone, a (0), on the dimensionless diffusion coefficient D for systems with various reservoir concentrations of the autocatalyst B curve a, / = 0, so one solution is the no reaction states a0i>8 = 0, whilst two other branches exist for low D curves b and c show the effect of increasing / , unfolding the hysteresis loop curve d corresponds to / = 0.1185 for which multiplicity has been lost, (b) The region of multiple stationary-state profiles forms a cusp in the / -D parameter plane the boundary a corresponds to the infinite slab geometry, with b and c appropriate to the infinite cylinder and sphere respectively. 

FIG. 9.5. A typical stationary-state solution for the dimensionless concentration profiles a (p) and P (p) for cubic autocatalysis with decay. The reactant concentration shows simply a central minimum, but the autocatalyst profile has three extrema, including two non-central maxima. 

A typical stationary-state solution is shown in Fig. 9.5 for the specific choice of the three parameters D = 5.2 x 10-3, / = 0.08, and k2 = 0.05. There is considerable consumption of the reactant near to the centre of the reaction zone, but the ass has a similar form to that found in the absence of catalyst decay—a hanging chain with a central minimum and no inflection point. The concentration profile for the autocatalyst has / ss increasing beyond its reservoir concentration near to the edge, but falling again as it approaches the centre. Thus the autocatalyst profile can have non-central extrema, but is always symmetric about p = 0. 

So far almost all aspects of the stationary-state and even the time-dependent behaviour of this reaction-diffusion system differ only qualitatively from that found in the corresponding CSTR. In this section, however, we can consider a variation for which there can be no parallel in the well-stirred system—that of a reaction-diffusion cell set up with asymmetric boundary conditions. Thus we might consider our infinite slab with separate reservoirs on each side, with different concentrations of the autocatalyst in each reservoir. (For simplicity we will take the reactant concentration to be equal on each side.) Thus if we identify the reservoir concentration for p < — 1 as / L and on the other side (p + 1) as / R, the simple boundary conditions in eqn (9.11) are replaced by... 

At low p0, the system has a high stationary-state concentration of A relative to that of the autocatalyst. Typically, both profiles have a maximum at the centre of the reaction zone, p = 0, as shown in Fig. 9.11 (a). High reactant concentrations favour larger concentrations of the autocatalyst B and lower... 

When the concentration of the autocatalyst in the inflow is not zero, the stationary-state relationship becomes... 

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. 

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