Steady-state multiplicity and stability

Nonlinear equations may admit no real solntions or mnltiple real solutions. For example, the quadratic equation can have no real solutions or two real solutions. Thus, it is important to know whether a given equation governing the behavior of an engineering system can admit more than one solution, since it is related to the issue of operation and performance of the system. In this subsection, criteria for the existence of multiple steady-state solutions to the governing equations of a CSTR and tubular reactors and, subsequently, the stability of these multiple steady states are presented. 

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So far there is a number of experimentally oriented papers dealing with steady-state multiplicity and stability, however, the majority of them are devoted to homogeneous systems. The important papers dealing with heterogeneous systems are discussed below. 

Steady-State Multiplicity and Stability A simple example of steady-state multiplicity is due to the interaction between kinetics and heat transport in an adiabatic CSTR. For a first-order reaction at steady state, Eq. (19-13) gives... 

There is a voluminous literature on steady-state multiplicity, oscillations (and chaos), and derivation of bifurcation points that define the conditions that lead to onset of these phenomena. For example, see Morbidelli et al. [ Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, Carberry and Varrria (eds), Marcel Dekker, 1987], Luss [ Steady State Multiplicity and Uniqueness... 

A feature related to steady state multiplicity and stability is that of "pattern formation", which has its origins in the biological literature. Considering an assemblage of cells containing one catalyst pellet each, Schmitz (47, 53) has shown how non-uniform steady states - giving rise to a pattern - can arise, if communication between the pellets is sufficiently small. This possibility has obvious implications to packed-bed reactors. 

Vejtasa, S.A. Schmitz, R.A. An experimental study of steady state multiplicity and stability in an adiabatic stirred reactor. AIChE J. 1970,16, 410 19. Schmitz, R.A. Multiplicity, stability, and sensitivity of states in chemically reacting systems - a review. Adv. Chem. Ser. 1975, 148, 156-211. Razon, L.F. Schmitz, R.-A. Multiplicities and instabilities in chemically reacting systems - a review. Chem. Eng. Sci. 1987, 42, 1005-1047. Uppal, A. Ray, W.H. Poore, A.B. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci. 1974, 29, 967-985. 

Morbidelli, M., Varma, A., and Arts, R., Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, 973-1055, New York Marcel Dekker, 1986. 

Luss.O. "Steady-state multiplicity and stability". (Proceedings of MATO ASI on "Multiphase Reactors". Portugal, 1980). 

Luss, D., and Medellin, P. Steady state multiplicity and stability in a counter-currently cooled tubular reactor. Chem. React. Eng., Proc. Eur. Symp., B4, 47-56, 1972. 

Vejtasa, S.A. and Schmitz, R.A., An experimental study of steady state multiplicity and stability in an adiabatic stirred reactor, AIChE /., 16,410-419,1970. 

Thus, as the interaction intensifies, steady-state multiplicity and stability come into play. In the case of uniformly coupled bimolecular reactions kij = kjkj), the interaction is not strong enough to induce steady-state multiplicity the system behaves similarly to Case II above. 

R.F. Heinemann, K.A. Overholser, and G.W. Reddien. Multiplicity and Stability of the Hydrogen-Oxygen-Nitrogen Flame The Influence of Chemical Pathways and Kinetics on Transitions between Steady States. AIChE J., 4 725-734,1980. 

Packed-bed reactors are discussed qualitatively, particularly with respect to their models. Features of the two basic types of models, the pseudohomogene-ous and the heterogeneous models, are outlined. Additional issues — such as catalyst deactivation steady state multiplicity, stability, and complex transients and parametric sensitivity — which assume importance in specific reaction systems are also briefly discussed. 

Steady State Multiplicity, Stability, and Complex Transients. This subject is too large to do any real justice here. Ever since the pioneering works of Liljenroth (41), van Heerden (42), and Amundson (43) with continuous-flow stirred tank reactors, showing that multiple steady states — among them, some stable to perturbations, while others unstable — can arise, this topic has... 

The classical problem of steady-state multiplicity in a continuous stirred tank reactor (CSTR) was brought to popular attention in 1953 in the theoretical article by Van Heerden. " Large amounts of experimental work which measured these steady states were performed by the group of Schmitz beginning in 1970. Schmitz also wrote two excellent reviews on multiplicity, stability, and sensitivity of steady states in chemical reactors and the application of bifurcation theory to determine the presence of steady-state multiplicity in chemical reactors.Even these reviews are not inclusive and it is our intention in this subsection to only provide a background to the novice in reactor design. 

When multiple steady states exist, the stability of these solutions and the startup condition must be contemplated to ensure that the desired steady state is attained. 

The above discussion may involve both control and dynamics of steady state (single and multiple), unsteady state (transition between two states) and unstable operations of the bioreactor resulting from typical non-linear response of biological systems. In spite of the fact that the stability analysis of non-linear systems is quite advanced, experimental confirmation of multiple steady states and instabilities lags behind (for a review of theory and experiments see (45)). An excellentexample of experimental demonstration of unstable operation of a continuous reactor is in (46). 

In modds of tubular reactors, material and energy balance are expressed by partial differential equations in time and space variaUes. Althou detailed numerical studies have been made in order to duddate the transient behaviour of tubular reactors, analytical studies have largely been confined to the question of existence, multiplicity, and stability of the reactm steady-state profiles, since the dimination of transirait behaviour often reduces tbe balance equations to a system of ordinary differential equations. 

Rheological studies can provide useful information on the stability and internal microstructure of the multiple emulsions. Some attention was given to this subject in recent years, and the results help clarify certain aspects of stability and release properties of the multiple emulsions (Benichou et al., 2002b). Oscillatory dynamic tests and steady-state analyses, and parameters such as shear or complex modulus (G ), the lag phase between stress and strain (5), the storage modulus (G ), and the loss modulus (G"), provide quantitative characterization of the balance between the viscous and elastic properties of the multiple emulsions. Oscillatory measurements generally indicate that multiple emulsions are predominantly viscous in that the loss modulus... 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

The problem of ignition and extinction of reactions is basic to that of controlling the process. It is interesting to consider this problem in terms of the variables used in the earlier discussion of stability. When multiple steady-state solutions exist, the transitions between the various stable operating points are essentially discontinuous, and hysteresis effects can be observed in these situations. 

E. E. Selkov, Stabilization of energy charge, generation of oscillation and multiple steady states in energy metabolism as a result of purely stoichiometric regulation. Eur. J. Biochem. 59, 151 157 (1975). 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

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