MULTIPLE STEADY STATES AND TRANSIENTS

In the previous chapter we showed how nonisothermal reactors can exhibit much more complex behavior than isothermal reactors. This occurs basically because k(T) is strongly temperature dependent Only a single steacfy state is possible in the PFTR, but the CSTR, although (or because) it is described by algebraic equations, can exhibit even more interesting (and potentially even more dangerous) behavior. 

This chapter will also be more mathematical than previous chapters. We need to analyze these equations to find these steady states and examine their stabihty. Some students wiU find this mathematics more than they are comfortable with or than they want to learn. We assure you that this section wiU be short and fairly simple. The intent of this chapter is (1) to show how multiple steady states arise and describe their consequences and (2) to show how mathematical analysis can reveal these behaviors. Attempts at nonmathematical discussions of them makes their occurrence and consequences quite mysterious. 

We wiU close this chapter by summarizing the principles of designing nonisothermal reactors so if you want to skim the more mathematical sections, you should stiU read the last section carefully. 

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However, all rate data for this reaction are not explained simply by this rate expression. At pressures above 10 Torr the rate exhibits multiple steady states, long transients, and rate oscillations ]). Clearly other processes are Involved than those Implied by the simple one state, constant parameter LH model. 

From a dynamical point of view the systems in this class typically have multiple steady states and then the autocatalytic process is associated to transitions between them. A special case with a single steady state, that also belongs to this class, is the excitable dynamics where apart from the basic state an excited meta-stable state also exists, that is effectively stable for short times. In any case, late time approach to the final state after the autocatalytic dynamics is usually a decay-type process already described in the previous Chapter. The final state in most of the problems considered in this Chapter is typically a trivial spatially uniform steady state. Thus, the main focus in this Chapter will be on the transient temporal dynamics, i.e. to understand and quantify the influence of the fluid transport on the progress of the autocatalytic process. As we will show, the situation is somewhat different in open flow systems where the same type of autocatalytic processes can produce a non-trivial long-time asymptotic state with complex spatial structure. 

Although steady-state kinetic methods cannot establish the complete enzyme reaction mechanism, they do provide the basis for designing the more direct experiments to establish the reaction sequence. The magnitude of kcm will establish the time over which a single enzyme turnover must be examined for example, a reaction occurring at 60 sec will complete a single turnover in approximately 70 msec (six half-lives). The term kcJKm allows calculation of the concentration of substrate (or enzyme if in excess over substrate) that is required to saturate the rate of substrate binding relative to the rate of the chemical reaction or product release. In addition, the steady-state kinetic parameters define the properties of the enzyme under multiple turnovers, and one must make sure that the kinetic properties measured in the first turnover mimic the steady-state kinetic parameters. Thus, steady-state and transient-state kinetic methods complement one another and both need to be applied to solve an enzyme reaction pathway. 

This set of first-order ODEs is easier to solve numerically than the algebraic equations that result from setting all the time derivatives to zero. The initial conditions are Uout = do, bout = bo, at t = 0. The long-time solution to the ODEs will satisfy Equations 4.1 provided that a steady-state solution exits and is accessible from the initial conditions. As discussed in Chapter 5, some CSTRs have multiple steady states and the achieved steady state depends on the initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can exhibit oscillations or even a semirandom behavior known as chaos. The method of false transients will then fail to achieve a steady state. Another possibility is a metastable steady state. Operation at a metastable steady state requires a control system and cannot be reached by the method of false transients. Metastable steady states arise mainly in nonisothermal systems and are discussed in Chapter 5. 

Three important (complicating) possibilities were not considered in the treatment of reactors presented in earlier chapters (1) the residence time of the reactant molecules need not always be fully defined in terms of plug flow or fully mixed flow (2) the equations describing certain situations can have more than one solution, leading to multiple steady states and (3) there could be periods of unsteady-state operation with detrimental effects on performance, that is, transients could develop in a reactor. 

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. 

Whenever multiple steady states in a reactor are possible, we must be very concerned that we are operating on the desired steady-state branch. This requires a proper startup procedure to attain the desired steady state and suitable operation limits to make sure that we never exhibit a sufficiently large transient to cause the system to fall off the desired conversion branch. We will consider transients in the CSTR in the next section. 

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. 

Bubble instability is one of the complications of this process. Only recently did this matter receive theoretical attention. As pointed out by Jung and Hyun (28), there are three characteristic bubble instabilities axisymmetric draw resonance, helical instability, and metastability where the bubble alternates between steady states, and the freeze line moves from one position to another. Using linear stability analysis, Cain and Denn (62) showed that multiple steady state solutions are possible for the same set of conditions, as pointed out earlier. However, in order to study the dynamic or time-dependent changes of the process, transient solutions are needed. This was recently achieved by Hyun et al. (65), who succeeded in quite accurately simulating the experimentally observed draw resonance (28). 

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... 

Changing the exit gas pressure also gave three multiple steady state responses in which the same set of operating parameters produced different reactor profiles. Finally, a rough estimate for the location of the bifurcation points was given for the coal moisture, steam feed rate, and exit gas pressure transient response runs. 

In section 2.2.7, multiple steady states in a heterogeneous chemical reaction (one dependent variable) and a jacketed stirred tank reactor (two dependent variables) were analyzed. Both stable and unstable steady states were obtained. The transient behavior of the system was found to depend on the initial conditions. The methodology and Maple programs presented in this chapter should be valid for any system of IVPs with multiple steady states. In section 2.2.8, phase plane behavior of a jacketed stirred tank reactor was analyzed. The program provided should be of use for analyzing phase plane behavior of different chemical systems. A total of ten different examples were presented in this chapter. 

This problem exhibits multiple steady states. Obtain all the steady states by equating the transient term to zero in all the equations. For mathematical convenience, express steady state P, T, and Pp in terms of steady state Tp using the first three equations. Use the steady state equation for Tp (after eliminating all other dependent variables) to obtain the multiple steady states. Solve the dynamic problem using the initial conditions P(0) = 0.1, T(0) = 600, Pp(0) = 0 and Tp(0) = 761 and plot the dynamic profiles for t = 0..15. Can you change the initial conditions to obtain a different steady state (see examples 2.2.6 and 2.2.7)... 

In chapter 3.2 we obtained multiple steady states (three states) for this problem for the values of the parameters = 0.2, p = 0.8 and y = 20. Solve this transient problem using numerical method of lines for two different initial conditions u(x,0) = 1 and u(x,0) = 0 What do you observe Can you obtain all the three steady states discussed in example 3.2.2 Consider the shrinking core problem discussed in example 5.2.6. Redo this problem if the particle is rectangular instead of spherical. The governing equations are ... 

The lumped model considered in section 5.2.1.8 provides a useful first step towards an understanding of the general behaviour of the porous catalyst pellet. However, it is limited in its validity since the true nature of the problem is distributed and internal concentration and temperature gradients have very important effects on steady state as well as transient behaviour of the system. For example, the lumped model predicts multiple steady states for cases for which the distributed system has a unique solution. 

Chemical relaxation techniques have been employed to study the rates of elementary reaction steps. The two most useful variables for the system control are the concentrations of the reactants and the reactor temperature. The dynamic responses from the system after the changes of these variables are related to the elementary steps of the catalytic processes. Chemical relaxation techniques can be divided into two general groups, which are single cycle transient analysis (SCTA) and multiple cycle transient analysis (MCTA). In SCTA, the reaction system relaxes to a new steady-state and analysis of this transition furnishes information about intermediate species. In MCTA, the system is periodically switched between two steady-states, e.g. by periodically changing the reactant concentration. 

Stability analysis could prove to be useful for the identification of stable and unstable steady-state solutions. Obviously, the system will gravitate toward a stable steady-state operating point if there is a choice between stable and unstable steady states. If both steady-state solutions are stable, the actual path followed by the double-pipe reactor depends on the transient response prior to the achievement of steady state. Hill (1977, p. 509) and Churchill (1979a, p. 479 1979b, p. 915 1984 1985) describe multiple steady-state behavior in nonisothermal plug-flow tubular reactors. Hence, the classic phenomenon of multiple stationary (steady) states in perfect backmix CSTRs should be extended to differential reactors (i.e., PFRs). 

This whole field of oniquotess and stability has been reviewed recently by Aris [100]. As previously mentioned the possibility of multiple steady states seriously complicates the design of the reactor. Indeed, transient computations have to be performed in order to make sure that the correct steady-state profile throughout the reactor is predicted. Another way would be to check the possibility of multiple steady states on the effectiveness factor chart for every point in the reactor. This... 

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