Residence time stationary state

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. 

Fic. 6.3. The dependence of the reactant concentration and the reaction rate kIa5S at the stationary state on (a) flow rate and (b). residence time for a first-order reaction in a CSTR. Rate... 

When P0 = i, the two roots of eqn (6.22) are exactly equal. The ignition and extinction points are coincident at ires = 64/27 multistability is lost. For larger inflow concentrations of B the stationary-state extent of reaction increases smoothly with the residence time and the distinction between the flow and thermodynamic branch is lost. 

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. 

Figure 6.14 shows these stationary-state solutions as a function of residence time for various small values of k2. The non-zero states exist over a limited range of ires they lie on the upper and lower shores of a closed curve, known as an isola . The size of the isola decreases as k2 increases. At each end of the isola there is a turning point in the locus, corresponding to extinction or washout. There are no ignition points in these curves. 

Put another way, for a given k2 there is a range of residence times over which the stationary-state eqn (6.58) has real solutions given by... 

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. 

If k2 is larger than L and R do not have any intersections, except that at the origin, for any residence time, because the flow line can never have a low-enough slope. The only stationary-state solution is that corresponding to zero extent of reaction, so the (1 — ass)-rres bifurcation diagram is now almost completely featureless, although there is, in fact, a unique solution for all residence times. 

If the rate constant k2 = 0.1, then the minimum gradient which L can have is 4k2 = 0.4. This is steeper than both the tangents for this system and might correspond to the flow line Ll in Fig. 6.16(a). There are no ignition or extinction points in the stationary-state locus (Fig. 16.6(b)) which has a unique solution for all residence times. 

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. 6.22. 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.)...

These equations describe the time dependence of the two variables—the concentration a and the temperature rise 9—in terms of five parameters. Of the latter, the residence time is again the one most easily varied during a given experiment, so we look to plot the stationary-state values of a and 9 against tres. The remaining four parameters are 0ad, tn, 6c, and y, and we might look to vary these independently between experiments. 

Fig. 7.2. Thermal or flow diagram for the first-order non-isothermal reaction (FON1) in a non-adiabatic CSTR the rate curve R and the flow line L both depend on the dimensionless residence time, but their intersections still correspond to stationary-state solutions—and tangen-cies to points of ignition or extinction. Note that R has a non-zero value at zero conversion. Exact numerical values correspond to 0ai = 10, t, = tn = A ...

If the reaction is not sufficiently exothermic, so this inequality is not satisfied, then the reaction curve R is much flatter, as shown in Fig. 7.3(c). There are now no points of tangency as the residence time, and hence the gradient of L, varies. There is only ever one intersection and hence only ever one stationary state for any given rres (Fig. 7.3(d)). 

For finite, rather than infinite, values of the dimensionless Newtonian cooling time, the stationary-state condition is given by eqn (7.21). Thus, even with the exponential approximation, both R and L involve the residence time. The correspondence between tangency and ignition or extinction still holds,... 

Balakotaiah, V. and Luss, D. (1984). Multiplicity features of reacting systems. Dependence of the stationary-states of a CSTR on residence time. Chem. Eng. Sci., 39, 1709-22. 

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.)... 

For the no reaction state ass = a0, the relaxation time given by eqn (8.10) is simply equal to the residence time. In terms of the eigenvalue, we have A = - l/tres, which is negative. The stationary state is always stable, irrespective of a0 and kl. Chemistry makes no contribution (formally we have l/tch,ss = 0, so the chemical time goes to infinity) the perturbation of a does not introduce any B to the system, so no reaction is initiated. The recovery of the stationary state is achieved only by the inflow and outflow. 

These arguments thus confirm the alternation of stability with the three branches of stationary-state solutions, as shown in Fig. 8.1. We can also make quantitative comments. As the residence time becomes very long, so the relaxation time for the unstable branch tends to - oo for the stable non-zero state trelax decreases as tres increases, tending to the value l/fc o as tres tends to oo. 

When the residence time is varied so that we approach an ignition or extinction point in the stationary-state locus, then the flow and reaction curves L and R become tangential. The condition for tangency is R = L and 8R/da = SL/da. Thus the difference between the slopes of R and L decreases to zero. From eqn (8.17) we see that the tangency condition also causes the value of the eigenvalue A to tend to zero. An alternative interpretation, in... 

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... 

Both are always real and both are always negative. This stationary state is thus always stable to small perturbations which decay monotonically to zero in an exponential manner. The relaxation times (there are now two and the decay occurs as the sum of two exponentials) are related to the residence time and to the inverse of the rate constant for catalyst decay. For instance, the decay of the perturbation in A can be written in the form... 

If tr(J) is negative, which will be favoured by short residence times, this upper stationary state is stable. Under such conditions the system as a whole has three stationary states, two of which are stable and one unstable. The concentrations of A and B will adjust to one of the stable states and will remain at the chosen one (which will depend on the previous history of our experiment) provided it receives only small perturbations. The middle state,... 

If k2 is greater than ys, we know there will be no isola and no Hopf bifurcation point. For k2 < /g, but greater than 9/256, P2 is positive. This means that the emerging limit cycle will be unstable. The limit cycle grows as the residence time is reduced below the bifurcation point t s surrounding the upper stationary state which is stable. 

We should also consider the behaviour along the top of the isola, on the part of the branch lying at longer residence times than the Hopf point. For Tres > t s, and with k2 still in the above range, the uppermost stationary state is unstable and is not surrounded by a stable limit cycle. The system cannot sit on this part of the branch, so it must eventually move to the only stable state, that of no conversion. Thus we fall off the top of the isola not at the long residence time turning point, but earlier as we pass the Hopf bifurcation point. 

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... 

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