Stable steady states

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

Pipeline deflagrations and detonations can be initiated by varions ignition sonrces. The flame proceeds from a slow flame throngh a faster accelerating tnrbnlent flame to a point where a shock wave forms and a detonation transition occnrs, resnlting in an overdriven detonation (see Fignre 4-3). A stable (steady state) detonation follows after the peak overdriven detonation pressnre snbsides. 

As shown on Figs. 8.31 to 8.33 the rate and UWR (or 0) oscillations of CO oxidation can be started or stopped at will by imposition of appropriate currents.33 Thus on Fig. 8.31 the catalyst is initially at a stable steady state. Imposition of a negative current merely decreases the rate but imposition of a positive current of200 pA leads to an oscillatory state with a period of 80s. The effect is completely reversible and the catalyst returns to its initial steady state upon current interruption. 

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 4.1 Transient approach to a stable steady state in a CSTR. 

FIGURE 5.5 Method of false transients applied to a system having two stable steady states. The parameter is the initial temperature Tq. 

Since a stable steady state is sought, the method of false transients could be used for the simultaneous solution of Equations (5.29) and (5.31). However, the ease of solving Equation (5.29) for makes the algebraic approach simpler. Whichever method is used, a value for UAext pQCp is assumed and then a value for Text is found that gives 413 K as the single steady state. Some results are... 

The dynamic behavior of nonisothermal CSTRs is extremely complex and has received considerable academic study. Systems exist that have only a metastable state and no stable steady states. Included in this class are some chemical oscillators that operate in a reproducible limit cycle about their metastable... 

Stability. The first consideration is stability. Is there a stable steady state The answer is usually yes for isothermal systems. 

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 rabbit and l5mx problem does have stable steady states. A stable steady state is insensitive to small perturbations in the system parameters. Specifically, small changes in the initial conditions, inlet concentrations, flow rates, and rate constants lead to small changes in the observed response. It is usually possible to stabilize a reactor by using a control system. Controlhng the input rate of lynx can stabilize the rabbit population. Section 14.1.2 considers the more realistic control problem of stabilizing a nonisothermal CSTR at an unstable steady state. 

Dummy scans are the preparatory scans with the complete time course of the experiment (pulses, evolution, delays, acquisition time). A certain number of these dummy scans are generally acquired before each FID in order to attain a stable steady state. Though time-consuming, they are extremely useful for suppressing artifact peaks. 

Fig. 3.14 shows the heat gain curve, Hq, for one particular set of system parameters, and a set of three possible heat loss, Hl, curves. Possible curve intersection points. A] and C2, represent singular stable steady-state operating curves for the reactor, with cooling conditions as given by cooling curves, 1 and III, respectively. 

The cooling conditions given by curve II, however, indicate three potential steady-state solutions at the curve intersections. A, B and C. By considering the effect of small temperature variations, about the three steady-state conditions, it can be shown that points A and C represent stable, steady-state operating... 

Consider a small positive temperature deviation, moving to the right of point B. The condition of the reactor is now such that the Hq value is greater than that for Hl. This will eause the reactor to heat up and the temperature to increase further, until the stable steady-state solution at point C is attained. For a small temperature decrease to the left of B, the situation is reversed, and the reactor will continue to cool, until the stable steady-state solution at point A is attained. Similar arguments show that points A and C are stable steady states. 

If Xi and A,2are real numbers and both have negative values, the values of the exponential terms and hence the magnitudes of the perturbations away from the steady-state conditions, c, and T, will reduce to zero, with increasing time. The system response will therefore decay back to its original steady-state value, which is therefore a stable steady-state solution or stable node. 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

Show that in the absence of control and feed disturbances (u = v = 0), the system has a singular, stable, steady-state solution of C = 0.1654 and T = 550. This can best be done by carrying out runs with different initial conditions (CO and TEMPO) and plotting the results as a phase-plane, TEMP versus C. 

The solids flux depends on the local concentration of solids, the settling velocity of the solids at this concentration relative to the liquid, and the net velocity of the liquid. Thus the local solids flux will vary within the thickener because the concentration of solids increases with depth and the amount of liquid that is displaced (upward) by the solids decreases as the solids concentration increases, thus affecting the upward drag on the particles. As these two effects act in opposite directions, there will be some point in the thickener at which the actual solids flux is a minimum. This point determines the conditions for stable steady-state operation, as explained below. 

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... 

The experiments were performed at a constant inflow concentration of ascorbic acid ([H2A]) in the CSTR. Oscillations were found by changing the flow rate and the inflow concentration of the copper(II) ion systematically. At constant Cu(II) inflow concentration, the electrode potential measured on the Pt electrode showed hysteresis between two stable steady-states when first the flow-rate was increased, and then decreased to its original starting value. The results of the CSTR experiments were summarized in a phase diagram (Fig. 6). 

In Fig. 6, separate regions of bi-stability, oscillations and single stable steady-states can be noticed. This cross-shaped phase diagram is common for many non-linear chemical systems containing autocatalytic steps, and this was used as an argument to suggest that the Cu(II) ion catalyzed autoxidation of the ascorbic acid is also autocatalytic. The... 

These multiple branches are not equivalent. From the contour lines in Figure 7.8, we can deduce which branch is stable, and which branch is not. The middle branch (C-C1) lies in a range where, for a given value of p, F increases with u. The derivative of F with respect to u is therefore positive which is just the criterion we found for an unstable equilibrium. Any fluctuation of u at constant p will drive the system away from the branch C-C. The opposite holds for the upper and lower branches A-C and A -C that lie in a range where F decreases when u increases. The derivative of F with respect to u is therefore negative and any concentration fluctuation around an equilibrium state along these branches dies out rapidly. The branches A-C and A -C are stable steady-states. 

Let us now And out how the system works. Assume that it starts at a large reduced flow-rate (point A) and reduce the input slowly. Up to the point C, any deviation from the equilibrium curve will die out rapidly. At C, concentration fluctuations become unstable and the system evolves quite rapidly towards D (p is fixed) where it finds a stable steady-state. The system has become unstable because reducing the flow-rate enhances crystallization which through the kinetic factor enhances the rate of precipitation and thereby depletes the residual liquid. The system quenches. Upon reducing the flow-rate further, the stable evolution continues towards point E. 

Figure 38, Chapter 3. A bifurcation diagram for the model of the Calvin cycle with product and substrate saturation as global parameters. Left panel Upon variation of substrate and product saturation (as global parameter, set equalfor all irreversible reactions), the stable steady state is confined to a limited region in parameter space. All other parameters fixed to specific values (chosen randomly). Right panel Same as left panel, but with all other parameters sampled from their respective intervals. Shown is the percentage r of unstable models, with darker colors corresponding to a higher percentage of unstable models (see colorbar for numeric values).

Figure 19. The steady state solutions A0 of the pathway shown in Fig. 18 as a function of the influx vi. For an intermediate influx, two pathways exist in two possible stable steady states (black lines), separated by an unstable state (gray line). The stable and the unstable state annihilate in a saddle node bifurcation. The parameters are k.2 0.2, 3 2.0, K] 1.0, and n 4 (in arbitrary units).

Figure 23. The steady state ATP concentration as a function of maximal ATP utilization Vmi for the minimal model of glycolysis. The letters denoted on the x axis correspond to the different scenarios shown in Fig. 22A D. Bold lines indicate stable steady states. Note that the physiologically feasible region is confined to the interval ATP0 e [0,Ar]. For low ATP usage (Vm3 small), there are three steady states, two of which are stable. However, both stable states are outside the feasible interval.

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