Multiple stationary states

Volumetric heat generation increases with temperature as a single or multiple S-shaped curves, whereas surface heat removal increases linearly. The shapes of these heat-generation curves and the slopes of the heat-removal lines depend on reaction kinetics, activation energies, reactant concentrations, flow rates, and the initial temperatures of reactants and coolants (70). The intersections of the heat-generation curves and heat-removal lines represent possible steady-state operations called stationary states (Fig. 15). Multiple stationary states are possible. Control is introduced to estabHsh the desired steady-state operation, produce products at targeted rates, and provide safe start-up and shutdown. Control methods can affect overall performance by their way of adjusting temperature and concentration variations and upsets, and by the closeness to which critical variables are operated near their limits. 

Fig. 15. Temperature vs heat generation or removal in estabHshing stationary states. The heavy line (—) shows the effect of reaction temperature on heat-generation rates for an exothermic first-order reaction. Curve A represents a high rate of heat removal resulting in the reactor operating at a low temperature with low conversion, ie, stationary state at a B represents a low rate of heat removal and consequently both a high temperature and high conversion at its stationary state, b and at intermediate heat removal rates, ie, C, multiple stationary states are attainable, c and The stationary state at c ...

An unusual feature of a CSTR is the possibility of multiple stationary states for a reaction with certain nonlinear kinetics (rate law) in operation at a specified T, or for an exothermic reaction which produces a difference in temperature between the inlet and outlet of the reactor, including adiabatic operation. We treat these in turn in the next two sections. 

If feed at a specified rate and T0 enters a CSTR, the steady-state values of the operating temperature T and the fractional conversion fA (for A —> products) are not known a priori. In such a case, the material and energy balances must be solved simultaneously for T and fA. This can give rise to multiple stationary states for an exothermic reaction, but not for an endothermic reaction. 

The existence of possible multiple stationary-states, as illustrated in Example 14-8, raises further questions (for adiabatic operation) ... 

What is the range of Rvalues for which multiple stationary-states exist ... 

Figure 14.6 Illustration of range of feed temperatures (T 0 to T ) for multiple stationary-states in CSTR for adiabatic operation (autothermal behavior occurs for T0 > T )...

Figure 14.7 Representation of multiple stationary-states on an enthalpy-temperature diagram corresponding to (b) in Figure 14.5...

For an endothermic reaction, whether the operation is adiabatic or nonadiabatic, there is no possibility of multiple stationary-states because of the negative slope of the /A versus T relation in the energy balance (see equation 14.3-11). This is illustrated schematically in Figure 14.8. 

Figure 14.8 Illustration of solution of material and energy balances for an endothermic reaction in a CSTR (no multiple stationary-states possible)...

Procaccia, I. Ross, J. (1978). Stability and relative stability in reactive systems far from equilibrium. II. Kinetic analysis of relative stability of multiple stationary states. J. Chem. Phys., 67, 5565-71. 

A response showing multiple stationary states requires that the inflow concentration of B be significantly less than that of A. Multiple intersections and tangencies are only possible if... 

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

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. 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

This chapter and chapter 7 introduce open systems. Though of the simplest kind, they permit multiple stationary states. These are truly stationary and may be quite different from the state of final equilibrium. After a study of this chapter, which deals with stationary states in isothermal systems, the reader should be able to ... 

Influence of autocatalyst inflow on multiple stationary states... 

Fig. 6.8. (a) Schematic three-dimensional representation of the stationary-state surface (1 — ass)-tres- 0 showing the folding at low autocatalyst inflow concentrations which gives rise to ignition, extinction, and multiplicity, (b) The projection onto the P0-t , parameter plane of the two lines of fold points in the stationary-state surface, forming a typical cusp at P0 = gT,e> = 7 inside this cusp region the system has multiple stationary states outside, there is... 

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.

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. 

Fig. 6.12. The influence of autocatalyst inflow on the reaction rate curve R for a system with reversibility, = 9 (a) po = showing multiple stationary-state intersections ... 

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). 

The condition on the size of the dimensionless adiabatic temperature rise is similar to the condition for multiple stationary states in terms of the dimen-... 

It cuts. the axis at 0ad = 4 as 1/tn tends to zero (adiabatic limit). We have already seen that this is the condition for transition from multiple stationary states (hysteresis loop) to unique solutions for adiabatic reactors, so the line is the continuation of this condition to non-adiabatic systems. Above this line the stationary-state locus has a hysteresis loop this loop opens out as the line is crossed and does not exist below it. Thus, as heat loss becomes more significant (l/iN increases), the requirement on the exothermicity of the reaction for the hysteresis loop to exist increases. 

It is, however, possible to have multiple stationary states even for parameter values lying below the hysteresis line. These multiplicities are associated with the cusp in the parameter plane. The equation describing the full cusp is most easily presented in the form... 

We also have the hint of a new type of degeneracy associated with systems possessing multiple stationary states. It is possible for both the trace and the determinant of the Jacobian matrix to become zero simultaneously this gives the system two eigenvalues which are both equal to zero. These double-zero eigenvalue situations are important because they represent conditions at which a Hopf bifurcation point with an associated homoclinic orbit first appears. In the present case, tr(J) = det(J) = 0 only when k2 = Vg, but then the isola has shrunk to a point. 

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

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. 

The unfolding of the hysteresis loop gives rise to a cusp in the D-f ex parameter plane, as shown in Fig. 9.4(b). Also shown there are the cusps for infinite cylinder and spherical geometries. For the latter, multiple stationary states cease for / ei = 0.1129 and 0.1078 respectively, values still smaller than the 5 for the CSTR. 

Kay, S. R. and Scott, S. K. (1988). Multiple stationary states, sustained oscillations and transient behaviour in autocatalytic reaction diffusion equations. Proc. R. Soc., A418, 345-64. 

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