Stationary-state patterns

If the CSTR is fed with both A and B, so p0 0, then a fifth pattern of response can also be found over a narrow range of experimental conditions. This is shown in Fig. 6.19(e) and has both a breaking wave and an isola. In total such a bifurcation diagram shows three extinction points and only one ignition. 

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Fig. 6.20. Typical forms for the boundaries in the /)0-ic2 parameter plane separating regions of diflerent stationary-state patterns for the system with uncatalysed reaction (a) the hysteresis bocndai ies (b) the isola boundaries (c) relative positions of boundaries in (a) and (b), with four regions visible (d) enlargement of area close to cusp in isola boundaries, showing existence of...

Chapter 6 considered isothermal autocatalysis in an open system here we study a classic case of thermal feedback. A rich variety of stationary-state patterns (bifurcation diagrams) are generated and considered here alongside those of the previous isothermal example. Flow diagrams are again illuminating and singularity theory provides a systematic approach. After study a reader should be able to ... 

Figure 7.5 is quantitatively correct only for the special case of the exponential approximation to the Arrhenius rate law. However, the figure is also qualitatively correct for the exact Arrhenius form with non-zero y, provided y < 4. No new stationary-state patterns are introduced. 

If the temperature difference 0C between the heat bath and the inflow is greater than zero, we can have the opposite effect to Newtonian cooling, with a net flow of heat into the reactor through the walls. With his possibility, two more stationary-state patterns can be observed, giving a total of seven different forms—the same seven seen before in cubic autocatalysis with the additional uncatalysed step (the two new patterns then required negative values for the rate constant) or with reverse reactions included and c0 > ja0 ( 6.6). 

The important feature of these equations is that the winged cusp point exists for physically acceptable values of the various quantities (x, tres, 0ad, and tn > 0, 9C > — y 1) provided y stays within the above range. The system can be unfolded from the singular point by varying 0ad, tN, and 9C, and in this way all seven of the stationary-state patterns shown in Fig. 7.8. 

The pitchfork singularity can be unfolded to give five stationary-state patterns unique, hysteresis loop, isola, mushroom, and hysteresis loop plus isola, as shown in Fig. 7.9. Note that, for this system, the hysteresis loop is reversed from the typical S-shape seen previously. 

Fig. 8.10. Stationary-state patterns showing multiplicity and Hopf bifurcation points distinguished by the curve A in Fig. 8.9 stable states are indicated by solid curves, unstable states by broken curves and Hopf bifurcation points by solid circles.

Oscillatory behaviour and exotic stationary-state patterns of extent of reaction versus flow rate in the simplest of open systems survive. 

FIGURE 2 The birth and growth of limit cycle oscillations in the I - a, jS, Tr space for a system with non-zero e and k displaying a mushroom stationary-state pattern. Oscillatory behaviour originates from a supercritical Hopf bifurcation along the upper branch and terminates via homoclinic orbit formation. 

A graphical route to evaluating the stationary-state patterns for the general case of non-zero catalyst inflow (3q > 0) has been given elsewhere [2,3]. As well as isolas, mushrooms and unique dependences of 1 - on may be found (Figs. 1b and c). Figure 2 divides the 3o t2 parameter plane into regions in which each of these different responses occur. 

ISOTHERMAL AUTOCATALYSIS IN THE CSTR EXOTIC STATIONARY-STATE PATTERNS (ISOLAS AND MUSHROOMS) AND SUSTAINED OSCILLATIONS... 

The stationary-state response curves, or bifurcation diagrams shown in Figs 1.13(b) and 1.12(f), represent two of the simplest possible patterns monotonic variation and a single hysteresis loop respectively. These are the only qualitatively different responses possible for the cubic autocatalytic step on its own. They are also found for a first-order exothermic reaction in an adiabatic flow reactor (see chapter 6). With only slightly more complex chemical mechanisms a whole array of extra exotic patterns can be found, such as those displayed in Fig. 1.14. The origins of these shapes will be determined in chapter 4. 

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

The character and local stability of the stationary-state solution, eqns (3.24) and (3.25), now depend only on /z, the dimensionless initial concentration of the reactant. There are four different patterns in this system ... 

In this section, therefore, we briefly investigate the stationary-state and Hopf bifurcation patterns that are found with the exact Arrhenius temperature dependence. 

Using the qualitative stability analysis from the flow diagram as outlined in 1.7.3 and 6.1.5, we find the following pattern. For conditions where there is a single stationary state, it is always stable to perturbations in regions of multiple solutions, the uppermost and lowest are stable whilst the middle... 

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. 

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 see quite persuasively how an isola pattern arises, by considering the flow diagram. The reaction rate curve R is shown in Fig. 6.15. For short residence times, the flow line L is steep and so only intersects with R at the origin there is thus only one stationary state, corresponding to zero extent of reaction. 

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. 6.17. Three-dimensional representation of the folding of the stationary-state surface (1 - ,s)-rres-K2 slices through this surface with constant k2 give mushroom, isola, or unique patterns. A similar surface arises in the (1 — ass)-rres-/ 0 space.

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-...

Fig. 6.19. The five stationary-state loci patterns found when the uncatalysed conversion of A to B is included in the model (a) unique, (b) isola (c) mushroom (d) single hysteresis loop (breaking wave) (e) hysteresis loop + isola.

Fig. 6.23. The seven different qualitative forms for the stationary-state locus for cubic autocatalysis with reversible reactions and inflow of all species, with c0 > )a0 the broken line represents the equilibrium composition which is approached at long residence times. These patterns are the same as those found for the irreversible system with an uncatalysed step—see Fig. 6.19. (Reprinted with permission from Balakotaiah, V. (1987). Proc. R. Soc., A411, 193.)...

With the exponential approximation (y 0) and the assumption that the inflow and ambient temperatures are equal, we have a stationary-state equation which links ass to tres and which involves two other unfolding parameters, 0ad and tn. Depending on the particular values of the last two parameters the (1 — ass) versus rres locus has one of five possible qualitative forms. These different patterns are shown in Fig. 7.4 as unique, single hysteresis loop, isola, mushroom, and hysteresis loop plus isola. The five corresponding regions in the 0ad-rN parameter plane are shown in Fig. 7.5. This parameter plane is divided into these regions by a straight line and a cusp, which cut each other at two points. 

With the full Arrhenius rate law, an extra unfolding parameter y is introduced. Even then, however, the appropriate stationary-state condition and its derivatives for the winged cusp cannot be satisfied simultaneously (at least not for positive values of the various parameters). Thus we do not expect to find all seven patterns. 

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