Unique steady-state

We would be remiss in our obligations if we did not point out that the regions of multiple solutions are seldom encountered in industrial practice, because of the large values of / and y required to enter this regime. The conditions under which a unique steady state will occur have been described in a number of publications, and the interested student should consult the literature for additional details. It should also be stressed that it is possible to obtain effectiveness factors greatly exceeding unity at relatively low values of the Thiele modulus. An analysis that presumed isothermal operation would indicate that the effectiveness factor would be close to unity at the low moduli involved. Consequently, failure to allow for temperature gradients within the catalyst pellet could lead to major errors. 

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... 

Figure 18. A simple bistable pathway [96], Left panel The metabolite A is synthesized with a constant rate vi and consumed with a rate vcon V2(A) + V3(A), with the substrate A inhibiting the rate V3 at high concentrations (allosteric regulation). Right panel The rates of vsyn vi const. and vcon V2 (A) + V3(A) as a function of the concentration A. See text for explicit equations. The steady state is defined by the intersection of synthesizing and consuming reactions. For low and high influx v, corresponding to the dashed lines, a unique steady state A0 exists. For intermediate influx (solid line), the pathway gives rise to three possible solutions of A0. The rate equations are specified in Eq. 67, with parameters 0.2, 3 2.0, Kj 1.0, and n 4 (in arbitrary units).

Figure 22. The nullclines corresponding to the minimal model of glycolysis. Depending on the value of the maximal ATP utilization Vm3, the pathway either exhibits a unique steady state or allows for a bistable solution. Note that the nullcline for TP does not depend on VThe corresponding steady states are shown in Fig. 23. Parameters are Vm 3.1, K 0.57, ki 4.0, K i 0.06, and n 4 (the values do not correspond to a specific biological situation).

A and B evolve in time to a unique steady state dictated by C. Steady-state concentrations of A and B show step functions in respect to C. 2 and k-i determine the steepness of the jump. When 2 k-i the curves of the steady-state concentrations of A and B are not symmetric. 

The second assumption employed in this article is that all species designated as intermediates—those that do not enter into a given system as either terminal reactants or products—will be present at constant concentrations. This includes stationary systems that can be described by a unique steady state rather than those which exhibit transient or oscillatory behavior. 

FIGURE 6 Demonstration of the relation between the function and the shape of the phase plane, (a) Phase plane contours of the function d> = (I - 0 - ft) = A. (b) Phase portrait for the special case of negligible desorption and identical absorption rates (< i = a2 = A -y, = y2 = 0). (c) Shape of the non-unique steady-state reaction rates for variations in reactant partial pressures for the case of negligible desorption. 

One of the characteristics of the book is to treat the case with multiple steady states as the general case while the case with a unique steady state is considered a special case. When reading Chapter 2, our readers should notice the systems similarities between chemical and biological systems that make these two areas conjoint while uniquely disjoint from the other engineering disciplines. 

It is important to introduce the reader at an early stage to simple examples of nonlinear models. We will first present cases with bifurcation behavior as the more general case, followed by special cases without bifurcation. Note that this is deliberately the reverse of the opposite and more common approach. We take this path because it sets the important precedent of studying chemical and biological engineering systems first in light of their much more prevalent multiple steady states rather than from the rarer occurrence of a unique steady state. 

The left-hand-side function g(y, K) in (4.87) represents a line in the y-g plane with slope 1 + K and g intercept — find values of yj and yrn with associated multiple (or a unique) steady states of the system, we need to find instances of multiple (or unique) crossing points of this line and the exponential function f(y, K) on the right-hand side of equation (4.87). For this purpose we use the MATLAB m function hetcontbifrange.m. 

Here the coordinates of the eight corners of the cube C serve as the initial values xao, xbo, and yo. These are marked by small circles in each instance. The eight trajectories proceed from the corners of the cube C to the only steady state of the system, marked by a gray. Their final positions at the end of the considered time interval are marked by eight x symbols. After T = 600 time units the trajectories in our example have all converged to the unique steady state of Figure 4.34. 

Finally, we redraw Figure 4.44 for exactly the same parameters, except that we use LeA = 0.11. The trajectory from the same initial value o = (xa(0), xb o), y(o)) = (0.1, 0.1, 0.5) as in Figure 4.44 now goes through one high-temperature loop similar to the infinitely repeated loop in Figure 4.46, but then it spirals around the unique steady state in four and a half loops during 1,200 time units, and it will ultimately settle at the steady state. ... 

Profile of s 12(f) and phase plot of S12 versus sn a unique steady state for large hf values Figure 4.63... 

We note that in the top pair of phaseplots, all our initial values lead to a unique steady state with (xAi(Tend), xsiiTend), y (Tend)) (0.92, 0.05,1.03) as depicted by the red marks. Note further the large phaseplot swing of the black curve that starts at the initial value (xai(0), xbi(0)) = (0.95,0). This curve nearly backs to its start in each of the topmost plots for tank 1. 

One of the differences is our treatment of the multiple steady states case as the general case and considering the unique steady-state case as a special case. 

We know that a PDE is stable as a linear approximation (see Sect. 2). Whence from eqns. (137) and (138) we establish that, at sufficiently low um and vout and t - oo, a solution of the kinetic equations for homogeneous systems tends to a unique steady-state point localized inside the reaction polyhedron with balance relationships (138) in a small vicinity of a positive PDE. If b(c(0)) = 6(c,n) vinjvout, then at low v,n and eout the function c(t) is close to the time dependence of concentrations for a corresponding closed system. To be more precise, if vm -> 0, uout -> 0, vmjvOM, c(0), cin are constant and c(O) is not a boundary PDE, then we obtain max c(t) — cc](t) -> 0, where ccl(t) is the solution of the kinetic equations for closed systems, ccl(0) = c(0),and is the Euclidian norm in the concentration space. 

Thus if the flow velocity in a completely flowing (homogeneous) system is higher than a certain value, the balance polyhedron contains a unique steady-state point that is globally stable, i.e. every solution for the kinetic equations (139) lying in Da tends to it at t - oo. Note that a critical value for the flow velocity at which this effect is obtained can depend on the choice of balance polyhedron (gas pressure). 

A root (cA )i corresponding to a negative sign, always lies on the segment [0, 1] whereas that of (cA)2 is always above unity. Hence on the segment [0, 1] there exists a unique steady state for the fast subsystem. Since the segment... 

Equations of "fast motions are linear and have a unique steady-state solution... 

Let us examine the properties of eqn. (152) under the assumption of oriented connectivity. Let us fix some co-invariant simplex D0 zt 0, , 2,- = C > 0. Da has a unique steady state z°. Vector z° is positive since, due to the connectivity of the reaction digraph, no steady-state points exist on the boundary Da. Indeed, if we assume the opposite (some components z° are zero), we obtain kJt for such i and j as 2° 0 and z° = 0. But from this it follows that, moving along the direction of arrows in the graph of the reaction mechanism, we cannot get from the substances for which 2° 0 to those for which z° = 0, and this is contrary to oriented connectivity (the arrows in the reaction graph correspond, naturally, to the elementary reactions with non-zero rate constants). 

On the basis of the structure for a bipartite graph of the reaction mechanism, it is possible to formulate a sufficient condition (174) for the uniqueness of a steady state. Applying it to concrete reactions, it is possible to establish the parametric areas for which either a unique steady state exists or there is a multiplicity of such states. 

Similar alterations in the phase portrait also take place with increasing PA,. The only difference is that at low PA, the unique steady state is charac-... 

Dynamic studies can be performed as previously. We will only note that, like eqns. (5), the system (31) has no limit cycles. In addition, the unique steady state is always stable. If there are three steady states (xx < x2 < x l), two are stable (Xj and x3) and one (the middle steady state x2) is unstable. 

The stability of steady states is analyzed [139] like the investigation performed for the three-step mechanism. In stable steady state, the inequality dg(0o)/d0o > df(0o)/ddo is fulfilled. In the unstable steady state, the sign of this inequality reverses. It can easily been shown that the unique steady state is always stable. If there are three steady states, the outer are stable and the middle is unstable. It can be suggested that the addition to the three-step adsorption mechanism of the impact step that is linear with respect to the intermediate does not produce any essential changes in the phase pattern of the system. The only difference is that at k. x = k 2 = 0 the dynamic model corresponding to the two-route mechanism can have only one boundary steady state (60 = 0, 9C0 = 1). 

Fig. 19. Relaxation of reaction rate towards the unique steady state from different initial conditions (60, 0CO) for the case from Fig. 18(d). (0O, 8C0) = (0, 0), (0.5, 0.5), (1, 0), and (0, 1) for curves 1-4, respectively.

There are also two factors that have already been noted in the numerical analysis of the kinetic model of CO oxidation (1) fluctuations in the surface composition of the gas phase and temperature can lead to the fact that the "actual multiplicity of steady states will degenerate into an unique steady state with high parametric sensitivity [170] and (2) due to the limitations on the observation time (which in real experiments always exists) we can observe a "false hysteresis in the case when the steady state is unique. Apparently, "false hysteresis will take place in the region in which the relaxation processes are slow. 

Linearized or asymptotic stability analysis examines the stability of a steady state to small perturbations from that state. For example, when heat generation is greater than heat removal (as at points A— and B+ in Fig. 19-4), the temperature will rise until the next stable steady-state temperature is reached (for A— it is A, for B+ it is C). In contrast, when heat generation is less than heat removal (as at points A+ and B— in Fig. 19-4), the temperature will fall to the next-lower stable steady-state temperature (for A+ and B— it is A). A similar analysis can be done around steady-state C, and the result indicates that A and C are stable steady states since small perturbations from the vicinity of these return the system to the corresponding stable points. Point B is an unstable steady state, since a small perturbation moves the system away to either A or C, depending on the direction of the perturbation. Similarly, at conditions where a unique steady state exists, this steady state is always stable for the adiabatic CSTR. Hence, for the adiabatic CSTR considered in Fig. 19-4, the slope condition dQH/dT > dQG/dT is a necessary and sufficient condition for asymptotic stability of a steady state. In general (e.g., for an externally cooled CSTR), however, the slope condition is a necessary but not a sufficient condition for stability i.e., violation of this condition leads to asymptotic instability, but its satisfaction does not ensure asymptotic stability. For example, in select reactor systems even... 

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