Existence of Multiple Stationary States

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. 

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states 

In the catalyzed gas-phase decomposition A - B + C, suppose A also acts as an inhibitor of its own decomposition. The resulting rate law (a type of Langmuir-Hinshelwood kinetics, Chapter 8) is  

Since this is a gas-phase reaction, and the total number of moles changes, the density varies, and q must be linked to q0. Using a stoichiometric table, we have  

Species Initial moles Change Final moles 

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In addition to openness and feedback, a third condition is often found in conjunction with oscillation, though it has not yet been proven to be necessary the existence of multiple stationary states. Most chemists are aware of the steady state condition, wherein the rate of change of the concentration of an intermediate can be equated to zero. Thus, the concentration of this intermediate can be obtained by solving an algebraic, rather than a differential equation. The familiar solution to this equation is a single concentration for a given set of initial conditions. [The reader may be familiar with a mechanism k k... 

Further work is necessary to extend the range of flows to the blowoflF and flashback limits for a series of droplet diameters, tube diameters, tube lengths, and fuel-to-air ratios. The possible existence of multiple stationary states for the same external conditions and the eflFects of nitrogen in the fuel, air preheating, ignition prior to complete evaporation, other fuels, and other atomization patterns should also be investigated. 

These results represent the first indication of the existence of multiple stationary states for an organic photochemical reaction. Two different responses (high or low concentration of violet radicals) can be observed for the same external constraints in the reactor (identical photon flux, flow rate, and initial concentration). 

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. 

A system in which the dependent variables are constant in time is said to be in a steady or stationary state. In a chemical system, the dependent variables are typically densities or concentrations of the component species. Two fundamentally different types of stationary states occur, depending on whether the system is open or closed. There is only one stationary state in a closed system, the state of thermodynamic equilibrium. Open systems often exhibit only one stationary state as well however, multistability may occur in systems with appropriate elements of feedback if they are sufficiently far from equilibrium. This phenomenon of multistability, that is, the existence of multiple steady states in which more than one such state may be simultaneously stable, is our first example of the universal phenomena that arise in dissipative nonlinear systems. 

In Chap. 5 we discussed reaction diffusion systems, obtained necessary and sufficient conditions for the existence and stability of stationary states, derived criteria of relative stability of multiple stationary states, all on the basis of deterministic kinetic equations. We began this analysis in Chap. 2 for homogeneous one-variable systems, and followed it in Chap. 3 for homogeneous multi-variable systems, but now on the basis of consideration of fluctuations. In a parallel way, we now follow the discussion of the thermod3mamics of reaction diffusion equations with deterministic kinetic equations, Chap. 5, but now based on the master equation for consideration of fluctuations. 

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 ... 

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. 

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. 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Bernstein and Churchill (3) constructed an essentially identical combustion chamber, closely reproduced the experimental values of Chen and Churchill (I), and confirmed the existence of the predicted multiple stationary states. The exact number of stable stationary states remains somewhat uncertain, since the separation of the states within the two groupings was within the range which might be attributed to irrepro-ducibility in setting the flow rates and inlet temperature. They also discovered that these flames produced only 5-32 ppm NO., with the particular value depending primarily on the location of the flame front inside the tube. 

This is a known condition for the existence of multiple roots of equation (6.37). When A < 0, equation (6.37) has three stationary states of which yu y3 are stable on the other hand, when A > 0, there is only on stable stationary state. The system crossing the sensitive state changes its properties from a bistable state it becomes a monostable state. 

In Chap. 2 we analyzed single variable linear and non-linear systems with single and multiple stable stationary states by use of the deterministic equations of chemical kinetics. We introduced species-specific affinities and the concept of an excess work with these we showed the existence of a thermodynamic state function 4> and compiled its many interesting properties, see (2.15 2.19), including its relation to fluctuations as given by the stationary solution of the master equation, (2.34). We continue this approach here by turning to systems with more than one intermediate, [1]. 

Fig. 7.3. The existence and loss of multiplicity as the dimensionless adiabatic temperature 9ai decreases through 4 (a) flow diagram for 6ai = 7.2 (b) corresponding stationary-state locus (1 — ot ,)-r 9 (c) flow diagram for 0a<1 = 3.6 (d) corresponding stationary-state locus.

Fig. 12.3. Stationary-state fractional coverage for adsorption and reaction involving two vacant sites (a) k2/K = 36 showing multiplicity, (b) multiplicity in absence of desorption now one solution corresponding to a fully covered surface exists for all reactant pressures.

Figure 4a shows Xq the stationary state value of X for the nonlinear system as a function of P. With this set of parameters no multiple steady states exist but the steady states marked by the broken line are unstable and evolve to stable limit cycles. The amplitude of the limit cycles is shown in Figure 4B. 

The model also predicted up to six stationary states in addition to the observed one. These seven states were in two groups three in the upstream and four in the downstream region of the tube. Because of irreversible modification of the reactor by cementing in a liner with 4.76-mm i.d. to obtain laminar fiow, the multiplicity predicted for turbulent fiow could not be tested. They speculated that one of the additional predicted stationary states might exist physically but that the other five were probably mathematical artifacts of the idealizations in the model or in the method of solution. 

In reactor dynamics it is particularly important to find out if multiple stationary points exist or if sustained oscilladons can arise. Bifurcation analysis is aimed at locating the set of parameter values for which multiple steady states will occur. We apply biftircation analysis to learn whether or not multiple steady states are possible. An outline of what is on the CD-ROM follows. 

The broad principles underlying the existence of stationary states (both nodal and focal stabilities), simple oscillatory states and multiple stabilities, involving competitive chain branching and non-branching modes and the interaction with heat release and heat loss, are clear. The more complex oscillatory waveforms do not emerge naturally from this simple recipe, and greater chemical complexity, not yet taken into account, is probably their cause. 

This set of equations gives the possibility to calculate the dynamic behavior of a CSTR and, what is more important, the characteristics of the steady state. In contrast to free radical polymerization where multiple steady states and even oscillating regimes can exist owing to the second-order chain termination reaction and the gel effect [54], Eqs. (3.22) and (3.23) have only one stationary solution. Hereinafter, steady-state parameters of the MWD are discussed. 

We can conclude from Eq. (6.194) that the state of vanishing X concentration is a stationary state Tl = 0 the formation of any X implies the pre-existence of X). Yet, the slightest variation in [X] immediately leads to a spontaneous multiplication. In this sense the state is unstable as displayed by Eq. (6.201). When the back reaction is not taken into account (the back reaction becomes ever more important at large values of [X]) it is not possible to reach a stable state. Only taking the back reaction into account leads to a further stationary state (here the equilibrium state), but this is now stable ... 

The roots of spontaneous combustion studies are deeply seated in physico-chemical interactions due to non-linear kinetics, heat release and heat dissipation. Discontinuous responses to slowly varying conditions are features that are most familiar as ignition and extinction, but the transitions can also be between stationary or oscillatory states. This presentation is concerned with oscillations and multiple stabilities that owe their existence to complex thermoKinetic interactions. 

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