Steady-state solution, stability

To pursue this question we shall examine the stability of certain steady state solutions of Che above equaclons by the well known technique of linearized stability analysis, which gives a necessary (but noc sufficient) condition for the stability of Che steady state. 

The next step should clarify why the unstable growth of the variable x occurs through a stable state at the bifurcation point. To determine the stability of the bifurcation point, it is necessary to examine the linear stability of the steady-state solution. For Eq. (1), the steady-state solution at the bifurcation point is given as jc0 = 0. So, let us examine whether the solution is stable for a small fluctuation c(/). Substituting Jt = b + Ax(f) into Eq. (1), and neglecting the higher order of smallness, it follows that... 

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. 

Figure 3.17. Phase-plane representations of reactor stability. In the above diagrams the point -I- represents a possible steady-state solution, which (a) may be stable, (b) may be unstable or (c) about which the reactor produces sustained oscillations in temperature and concentration.

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. 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

Consider a steady-state solution (XsT s). This solution is stable if the system will return to it following a small perturbation away from it. To decide this, we lineaiize the equations about the steady state and examine the stability of the linear equations. First we subtract the steady-state version of these equations from the transient equations to obtain... 

Fig. 10. Linear stability diagram illustrating the branching of a new inhomogeneous steady-state solution. The regions (a), (6), (c) are defined as in Fig. 9. A = 2 D, = 1.6 10-3, D2= 8 10 3. The critical mode /a is the integer that gives to B(n) its minimal value Bc.

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

Fig. 5. Steady-state solution of deterministic rate equations to which a stochastic term has been added. Low noise level (mean absolute magnitude of fluctuations). Note increase in noise level near lower marginal stability point.

With A = 0.06 M and the rate constants of Ref. 14b, these equations admit a unique homogeneous steady-state solution (HSS). It is well known that the irreversible Oregonator 14 and its reversible counterpartl4b exhibit homogeneous limit cycle oscillations for realistic values of rate constants and buffered concentrations. My purpose here is to explore several other features of the reversible model (F) which explain a variety of observed behaviors in closed and open stirred reactors. To that end I begin with the stability properties of the unique HSS, as displayed in the partial phase diagram of Fig. 1. 

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). 

Exchange of Stability. The TDGL method provides a procedure for determining the stability of the patterned states near the bifurcation point. Let us sketch the main ideas for the = 1 steady state patterns under the simplification that the developmental time scale is very long compared to the time to generate patterns, i.e. we neglect the time dependence of a and b in (36). With this (36) yields steady state solutions Vi obeying... 

For T = 16 s, the single nephron model undergoes a supercritical Hopf bifurcation at a = 11 (outside the figure), fn this bifurcation, the equilibrium point loses its stability, and stable periodic oscillations emerge as the steady-state solution. For a = 19.5, at the point denoted PDla 2 in Fig. 12.5, this solution undergoes a period-... 

When studying the stability of the steady-state, time-dependent calculations are needed (see [7]). It can also be used as a simple method to compute the steady-state solution. A time-dependent approach using the Lesaint-Ravian technique for the normal stress components and the Baba-Tabata scheme for the shear stress component is developed by Saramito and Piau ([34]). This method allows one to obtain rapidly stationary solutions of the PTT models. Convergence with mesh refinement is obtained as well as oscillation-free solutions. 

We obtained three different steady states. The stability of these states can be verified by assigning these values as the initial conditions. If we start with a stable steady state solution as the initial condition, the process remains at the stable steady state solution. If we start with an unstable steady state solution, the process moves to one of the steady state solutions. 

Still be very sensitive to a particular variable. On the other hand, an unstable condition is such that the least perturbation will lead to a finite change and such a condition may be regarded as infinitely sensitive to any operating variable. Sensitivity can be fully explored in terms of steady state solutions. A complete discussion of stability really requires the study of the transient equations. For the stirred tank this was possible since we had only to deal with ordinary differential equations for the tubular reactor the full treatment of the partial differential equations is beyond our scope here. Nevertheless, just as much could be learned about the stability of a stirred tank from the heat generation and removal diagram, so here something may be learned about stability from features of the steady state solution. 

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. 

Linear stability analysis is carried out on dynamical equations linearized about the steady-state solution (Mj,Vj). The steady-state solution is stable if the eigenvalues of the system of linearized equations are negative, unstable if those are positive, and indeterminate otherwise. 

Fig. 3.6. Effect of the recycling reaction on birhythmicity. A series of bifurcation diagrams are represented for increasing values of the maximum rate of product recycling, o- (in s ) (a) 0 (b) 0.5 (c) 0.6 (d) 1.2 (e) 1.3 (f) 1.4 (g) 1.5 (h) 2. Each diagram shows the steady-state concentration of substrate, o, and the mtiximum concentration of substrate in the course of oscillations, < m> s a function of parameter (qvlk ) equal to the steady-state concentration of product. The curve yielding the steady-state level of substrate is therefore identical with the product nullcline. The sohd and dashed lines denote, respectively, stable and unstable branches of periodic or steady-state solutions. Parameter values are <7=10s, L = 5xl0, /iC=10, m = 4, q = l, k = 0.06s. Periodic re mes were obtained by numerical integration of eqns (3.1). The stability properties of the steady state were determined by Unear stabUity analysis. Birhythmicity is apparent in (d)-(f), while in (h) two distinct instabiUty domains appear as a function of parameter v (Moran Goldbeter, 1984).

---