Stationary-state solutions

In a given experiment p is, in effect, being varied for us (by the exponential decay in (3.19)). The dimensionless uncatalysed reaction rate constant ku would remain constant during a given experiment, but might be varied from one run to another. In our analysis we seek first to determine how ass and pss depend on p for a fixed value of ku. We then investigate how this dependence changes as ku varies. 

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  

Recalling that ku will usually be small, this can be a relatively large value, occurring at low values of the reactant concentration. For the example data in Table 2.1, ku = 10-2 and hence (ass)max= 5 and is achieved when p = 0.1. Somewhat surprisingly, the maximum stationary-state concentration of the 

Turning briefly to the special case of no uncatalysed reaction ku = 0, the various results above take even simpler forms. For the stationary states we then have 

However, we may also see an immediate problem as the reactant concentration n approaches zero, the value of the stationary-state concentration of A becomes infinite, which is physically unrealistic. This problem does not occur if ku is given any non-zero value, no matter how small. 

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The approximate results can be compared with the long time limit of the exact stationary state solution derived in section A3.4.8.3 ... 

If some of the reactions of (A3.4.138) are neglected in (A3.4.139). the system is called open. This generally complicates the solution of (A3.4.141). In particular, the system no longer has a well defined equilibrium. However, as long as the eigenvalues of K remain positive, the kinetics at long times will be dominated by the smallest eigenvalue. This corresponds to a stationary state solution. 

A] = b/a (equation (A3.4.145)) is stationary and not [A ] itself This suggests d[A ]/dt < d[A]/dt as a more appropriate fomuilation of quasi-stationarity. Furthemiore, the general stationary state solution (equation (A3.4.144)) for the Lindemaim mechanism contams cases that are not usually retained in the Bodenstein quasi-steady-state solution. 

The stationary-state solutions to this differential equation are... 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

This reaction and the synthesis of HBr have also received much less attention than the corresponding reactions of the HI system. The problem of the mechanism of the H2+Br2 reaction which Bodenstein and Lind33 found to be complex was later solved independently by Christiansen34, Herzfeld35 and Polanyi36. The well-known mechanism and the kinetic equation resulting from the stationary-state solution are given below... 

This depends on fcls a0, and b0. Points of intersection of these two curves on the flow diagram correspond to conditions where R = L, and hence to stationary-state solutions. If R and L have just one intersection, as shown in Fig. 1.12(a) or (e), there is a unique stationary state. If L cuts R three times, as... 

The purpose of this appendix, in giving detail of the derivation of eqns (1.22)—(1.24), is to demonstrate a method of analysis which will be of particular use in later chapters when we discuss the local stability of a stationary-state solution. We will see here concentrations of different species evolving as the sum of a series of exponential terms which involve first-order rate constants. Later we will see similar sums of exponential terms, where the exponents, although more complicated can also be interpreted as pseudo-first-order rate coefficients. 

The conditions under which the above stationary-state solution loses its stability can be determined following the recipe of 2.6. Again we find that instability may arise, and hence oscillatory behaviour is possible, in this reversible case. The condition for the onset of instability can be expressed in terms of the reactant concentration p < p p, where... 

Equations (3.20) and (3.21) with their stationary-state solutions (3.24) and (3.25) are simple enough to provide a good introduction to some of the mathematical techniques which can serve us so well in analysing these sorts of chemical models. In the next sections we will explain the ideas of local stability analysis ( 3.2) and then apply them to our specific model ( 3.3). After that we introduce the basic aspects of a technique known as the Hopf bifurcation analysis ( 3.4) which enables us to locate the conditions under which oscillatory states are likely to appear. We set out only those aspects that are required within this book, without any pretence at a complete... 

Fig. 3.3. Variation in local stability and character of stationary.state solutions with the values of the trace and determinant of the corresponding Jacobian matrix (see also Table 3.2).

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 ... 

Fig. 3.6. The dependence of local stability and character of the stationary-state solution on the parameters /i amd ku. (a) The locus of Hopf bifurcation points with tr(J) = 0 beneath...

In the previous sections we have implied that the loss of local stability which occurs for a stationary-state solution as the real part of the eigenvalues changes from negative to positive is closely linked to the conditions under which sustained oscillatory responses are born. 

Our approach of regarding a and / as functions of n has served us well. It has allowed us to identify stationary-state solutions and how they vary with the reactant concentration and the uncatalysed reaction rate. By examining the local stability of these solutions we have also been able to obtain simple... 

These have only one true stationary-state solution, ass = pss = 0 (as t - go), corresponding to complete conversion of the initial reactant to the final product C. This stationary or chemical equilibrium state is a stable node as required by thermodynamics, but of course that tells us nothing about how the system evolves in time. If e is sufficiently small, we may hope that the concentrations of the intermediates will follow pseudo-stationary-state histories which we can identify with the results of the previous sections. In particular we may obtain a guide to the kinetics simply by replacing p by p0e et wherever it occurs in the stationary-state and Hopf formulae. Thus at any time t the dimensionless concentrations a and p would be related to the initial precursor concentration by... 

In the next few sections we will concentrate on the form of the governing equations (4.24) and (4.25) with the exponential approximation to f(0) as given by (4.27). We will determine the stationary-state solution and its dependence on the parameters fi and k, the changes which occur in the local stability, and the conditions for Hopf bifurcation. Then we shall go on and use the full power of the Hopf analysis, to which we alluded in the previous chapter, to obtain expressions for the growth in amplitude and period of the emerging oscillatory solutions. 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

When the Hopf bifurcation at p is supercritical (/ 2 < 0) the system has just a single stable limit cycle. This emerges at p and exists across the range p < p < p, within which it surrounds the unstable stationary-state solution. The limit cycle shrinks back to zero amplitude at the lower bifurcation point p%. This behaviour is qualitatively the same as that shown with the simplifying exponential approximation and is illustrated in Fig. 5.4(a). 

Fig. 5.5. A typical phase portrait for a system with ft < ft < ft, showing a stable stationary-state solution (singular point) surrounded first by an unstable limit cycle (broken curve) and then by a stable limit cycle (solid curve). The unstable limit cycle separates those initial conditions, corresponding to points in the parameter plane lying within the ulc, which are attracted to the stationary state from those outside the ulc, which are attracted on to the stable limit cycle and hence which lead to oscillations.

Fig. 5.8. Variation in stationary-state intersection relative to the maximum and minimum in the g(ot, 6) = 0 nullcline with the quotient M/K for a system with y = 0.2. (a) Intersection below maximum, M/K = 1.6 a given trajectory moves quickly to the g(a, 0) = 0 nullcline which it then moves along to the stationary-state solution, (b) Intersection above the minimum, M/K = 20 again a given trajectory will approach the stationary state along the g(a, 0) = 0 nullcline. (c) Intersection lying between the extrema, M/K = 5 now the stationary state is not approached and the time-dependent solutions cycle around the phase plane on the g(x, 6) = 0 nullcline (slow motion) with rapid jumps from one branch to the other (fast motion) at the turning points, (d), (e) Schematic representation of the relaxation oscillations for the conditions in (c).

If this value for a is substituted into the mass-balance eqn (6.3), it gives da/dt = 0. This is thus the stationary-state solution. Once it has been achieved the concentration remains constant. Note that at the stationary state the chemical reaction rate is not zero rather it is given by /c, ass, and this rate of conversion of A to B just balances the net rate of mechanical inflow of A to the reactor. 

Fig. 6.9. Domains of attraction for the two stable branches of stationary-state solution systems which have initial extents of conversion lying within the shaded region evolve to the lower branch those in the unshaded region approach the highest extent of reaction. In the region of multiple stationary states, the middle branch acts as a separatrix.

Other stationary-state solutions must then satisfy the reduced equation... 

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. 

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. 

Equations (7.16a) and (7.16b) correspond to our single first-order exothermic reaction occurring in 9 CSTR fed by reactants at the oven temperature, with the exponential approximation made to the Arrhenius temperature dependence of the reaction rate constant. Stationary-state solutions cor-repond to values of the dimensionless concentration a and temperature rise 9 for which da/dr and dO/dt are simultaneously equal to zero, i.e. 

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 ...

Here F represents the functional form of the left-hand sides of the various stationary-state equations, x is the stationary-state solution such as the extent of reaction, the temperature excess, etc., and rres is the parameter we have singled out as the one which can be varied during a given experiment (the distinguished or bifurcation parameter). All the remaining parameters are represented by p, q, r, s,. For example, in eqn (7.21) the role of x could be played by the extent of reaction 1 — ass, with p = 0ad and q = tN for isothermal autocatalysis, x can again be the extent of reaction, with p = P0, q = k2, and r = jcu. 

This expression only gives the gradient of the locus provided the x and Tres values actually correspond to stationary-state solutions, i.e. provided they satisfy F = 0. Then the gradient varies as x and tres vary along the locus. 

Thus the upper root of eqn (8.10), which gives the middle branch of stationary-state solutions and requires the minus sign above, has a negative value for trelax. It then follows that the eigenvalue A for this branch is positive, so perturbations grow. This is an unstable state. 

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