Concentration of the autocatalyst

This can be integrated in closed form, and the simplest expressions again emerge if the initial concentration of the autocatalyst is zero. The reaction shows a typical autocatalytic induction period, followed by an acceleration through a period of rapid consumption. Figure 1.6 shows the variation of concentration and rate with time for a system with a0 = 0.1 mol dm-3 and b0 — 0.001 mol dm-3. 

For larger inflow concentrations of the autocatalyst B, i.e. with b0 > a0/8, the curves R and L can only intersect once, whatever the flow rate. As shown in Fig. 1.13, we then have a monotonic dependence of ass on kf. 

The reaction rate curve R is zero at complete conversion and also has low (but non-zero) values close to 1 — a = 0, with a maximum close to two-thirds conversion (actually at 1 — a = — / 0). Importantly, R does not depend on the residence time rres, although it does vary if / 0 is changed. The flow line L is zero when 1 — a = 0 since the inflow and outflow have the same composition (no conversion of A to B). The gradient of the flow line (Fig. 6.7(b)) is given by 1 /Tres, so it is steep for short residence times (fast flow rates) and relatively flat for long rres. (Note how tres actually compares fres and lch, so short residence times are those that are much less than the chemical timescale etc.) The flow line is, however, unaffected by the inflow concentration of the autocatalyst f 0. 

The final term in eqn (6.22) involves the discriminant (1 — 8/30)1/2. For real roots, we therefore require that the dimensionless inflow concentration of the autocatalyst should be less than , so b0 < ja0 in real terms. 

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... 

The full expressions for R and L have been given in eqns (6.52) and (6.53). The only difference between these forms and those of the previous subsection is that the reaction rate curve R now involves the inflow concentration of the autocatalyst. The flow line L, its dependence on the residence time, and its minimum gradient are all independent of po. 

There is another type of time dependence possible in this system. If the inflow concentration of the autocatalyst is adjusted so that b0 - a0, then the ignition and extinction points merge at trcs = (k1ao) 1, with ass = Iu0 Under these special conditions, the coefficient of the term in (Aa)2 in the rate equation, and hence in the denominator of eqn (8.21), becomes zero as well as those of the lower powers in A a. Thus the inverse time dependence disappears, and the only non-zero term governing the decay of perturbation is that in (Aa)3 ... 

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. 

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... 

In stage b, the chlorite produced in stage a reacts with the remaining iodide to produce more iodine in a reaction that is accelerated both by the growing concentration of the autocatalyst, I2, and by the declining concentration of the inhibitor, I. The stoichiometry during this part of the reaction is... 

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

This richness arises primarily from the non-linear dependence of the reaction rate on the concentration of the autocatalyst B. 

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. 

At P only A is present, the concentrations of the autocatalyst and its degradation product vanish (x = b = 0). P thus represents a kind of "frozen" state at which no dyneunical compensation exists. 

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