Fig. 1.13. Cubic autocatalysis with relatively high inflow concentration of the catalyst, b0 = |

Singularity theory for cubic autocatalysis with uncatalysed reaction [Pg.203]

Returning to the simple cubic autocatalysis model above, we shall be more interested later in the relationship between the rate and the extent of conversion. This is shown for various values of b0 in Fig. 1.7. If b0 = 0 (Fig. 1.7(a)), the rate curve is both a minimum and zero at no conversion (i.e. there is a double root at the origin) and has a further zero at complete conversion (a = 0, a0 — a = a0). The rate has a maximum value of (4/27)/ccao occurring two-thirds of the way across the diagram (a = a0). There is also a point of inflection at 50 per cent conversion, a0 — a — a0. [Pg.11]

Fig. 1.15. Flow diagram for cubic autocatalysis showing different stabilities of multiple stationary-state intersections. |

As an example we can again take our cubic autocatalysis with inflow of B, so Fx and Fz are as given above, leading to [Pg.199]

Fig. 1.9. Consecutive first-order reactions and cubic autocatalysis, showing pseudosteady-state predictions for the intermediate concentrations. Initial concentrations and rate constants are given in Table 1.1. |

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 [Pg.147]

Comparisons between properties of the reaction rate curve R for cubic autocatalysis without decay and the first-order non-isothermal system in an [Pg.191]

Figure C3.6.14 Space-time (y,t) plot of the minima (black) in the cubic autocatalysis front ( )(y,t) in equation C3.6.16 showing the nature of the spatio-temporal chaos. |

This reduction to a single variable is similar to that possible in the case of cubic autocatalysis when the species B is infinitely stable (i.e. k2 - 0) in chapter 6. In fact there are many qualitative parallels between the adiabatic non-isothermal reaction and autocatalysis without decay, as we shall see later. [Pg.189]

Fig. 6.22. The 14 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. (Reprinted with permission from Balakotaiah, V. (1987). Proc. R. Soc., A41J, 193.) |

Fig. 6.7. The onset of stationary-state multiplicity represented on a flow diagram for cubic autocatalysis with P0 = (a) the reaction rate curve R (b) three typical flow lines L, with [Pg.151]

The previous chapter has provided some indication of the behaviour which can be exhibited by the simple cubic autocatalysis model. In order to make a full analysis, it is convenient both for algebraic manipulation and as an aid to clarity to recast the rate equations in dimensionless terms. This is meant to be a painless procedure (and beloved of chemical engineers even though traditionally mistrusted by chemists). We aim wherever possible to make use of symbols which can be quickly identified with their most important constituents thus for the dimensionless concentration of A we have a, with / for the dimensionless concentration of B. Once this transformation has been achieved, we can embark on a quite detailed and comprehensive analysis of the behaviour of this prototype chemical oscillator. [Pg.57]

Fig. 1,10. Actual time-dependent concentration of intermediate A for consecutive first-order reactions with cubic autocatalysis showing pseudo-steady-state behaviour, pre-oscillatory evolution, an oscillatory period, and then the return to pseudo-steady-state behaviour. |

Fig. 7.6. The unfolding of a hysteresis loop as condition (7.46) is satisfied. For the FONI model this occurs as 0ad decreases through 4 for cubic autocatalysis the unfolding corresponds to fl0 |

Fig. 6.6. Dependences of reaction rate R on extent of reaction y typical of self-accelerating (autocatalytic) systems (a) prototype quadratic autocatalysis (b) prototype cubic autocatalysis. |

In between these tangencies, the curves R and L have three intersections, so the system has multiple stationary states (Fig. 7.3(b)). We see the characteristic S-shaped curve, with a hysteresis loop, similar to that observed with cubic autocatalysis in the absence of catalyst decay ( 4.2). [Pg.189]

The species HOI is then rapidly converted to iodide, by reaction first with I" to produce I2 which is then reduced by the arsenite of hydrogen sulphite. Thus, identifying A with the reactant iodate and B with iodide, the system shows a cubic autocatalysis with rate proportional to [IOJ][I-]2 at constant pH. [Pg.13]

Autocatalysis will play a central role in driving the oscillations and other non-linear phenomena of interest in this book. Usually, an autocatalytic process will be combined in a larger mechanism with other steps. Before considering such systems, however, we investigate the properties and behaviour of autocatalysis on its own—in particular how the concentrations and rate vary with time and with respect to each other. We start with quadratic autocatalysis, and then look at the cubic form. [Pg.7]

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. [Pg.211]

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). [Pg.196]

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See also in sourсe #XX -- [ Pg.2 , Pg.14 , Pg.18 ]

See also in sourсe #XX -- [ Pg.487 ]

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