Steady-state behavior existence

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. 

One limit of behavior considered in the models cited above is an entirely bulk path consisting of steps a—c—e in Figure 4. This asymptote corresponds to a situation where bulk oxygen absorption and solid-state diffusion is so facile that the bulk path dominates the overall electrode performance even when the surface path (b—d—f) is available due to existence of a TPB. Most of these models focus on steady-state behavior at moderate to high driving forces however, one exception is a model by Adler et al. which examines the consequences of the bulk-path assumption for the impedance and chemical capacitance of mixed-conducting electrodes. Because capacitance is such a strong measure of bulk involvement (see above), the results of this model are of particular interest to the present discussion. 

More evidence that thermal equilibrium is not attained is the existence of a memory effect. It has been observed that the kinetics of doping depends on the wait time spent in the insulating state [15]. After 105 s in the undoped state, steady-state behavior is still not obtained. This means that a slow relaxation process is taking place in the film maintained in the insulating state. This effect has been quite well characterized, but no microscopic explanation has yet been given [16]. 

Transient vs. Steady-State Behavior in Permeability Determinations. The foregoing derivations raise some intriguing speculations about the measurement and determination of permeabilities for the respective components in a mixture. Thus, if a true or complete steady-state condition exists during the experiment, whereby all of the feed stream passes through the membrane, then the ratio V/F = and the ratio L/F = 0. That is, it can be said that no reject phase whatsoever is produced. 

No other reactor configuration (batch, semibatch, plug flow) exhibits the range of dynamic behavior of the CSTR. However, over a finite time interval, other configurations can sometimes exhibit a narrower range of dynamic behavior. If a semibatch reactor is operated such that the rate of reaction is just balanced by the rate of dilution from the feed, a pseudo-steady state may exist. In this case, the concentration of reactants and products in the reactor will remain constant over the time interval necessary to fill the reactor [27]. This may be exploited to provide constant polymer properties during the filling and start-up of a CSTR or CSTR train. 

Here Aft and Ab are the forward and backward reaction fluxes, respectively. Chemical detailed balance (or equilibrium steady state) occurs when Afj = Ab in every reaction, while mathematical detailed balance assumes that total forward rate is equal to total backward rate. As the Schldgl reaction system is cubic (from the trimolecular reaction) there may be three states depending on the set of parameters, such as a bistable state with two stable steady states separated by an unstable steady state. There exist many biological examples of bistability and switching behavior when A < 0 ... 

Three methods exist for studying non-steady-state behavior ... 

These functions represent the solutions for the case 7 > 0, which, as previously noted, implies a negative temperature coefficient for the system. This case describes a stable system that oscillates about the steady-state condition existing prior to t = 0. The case 7 < 0 yields a divergent behavior with time, the functions w t) and 0(0 increasing monotonically. 

Condition [9] is sufficient but not necessary to insure uniqueness of the steady state. If it is violated, multiple steady states will exist in cells for which Fj(yj+) > 1 > Fj(yj-). Typical behavior of the Fj(yj) curves with cell ntomber is illustrated in Figure 2, which shows that for M/H > 1, yj-i increases with bed depth, and ym,j increases, but less rapidly, to ym- Simultaneously, the Fj(yj) curve rises, yj- approaches yj+, and at some point in the bed, yj+ = yj- = yj,t fj(yj+) = Fj(yj ) = Fj(yj,t) — the trifurcation point. If Fj(yj t) multiple steady states exist for some catalyst particles in the bed in the region where Fj (yj+) > 1 > Fj (yj-). However, if Fj(yj. . ) < 1, then all catalyst particles have a unique steady state, and the reactor profile is unique. Fj(yj ) 1 is both a necessary and sufficient condition for uniqueness of the steady state. 

The behavior of liquid flow in micro-tubes and channels depends not only on the absolute value of the viscosity but also on its dependence on temperature. The nonlinear character of this dependence is a source of an important phenomenon - hydrodynamic thermal explosion, which is a sharp change of flow parameters at small temperature disturbances due to viscous dissipation. This is accompanied by radical changes of flow characteristics. Bastanjian et al. (1965) showed that under certain conditions the steady-state flow cannot exist, and an oscillatory regime begins. 

The fact that the concentration of G-actin at steady-state in the presence of ATP varies with the number of filaments may have some biological significance indeed, in cells, large pools of G-ADP-actin may accumulate in regions where a large number of short filaments exist. This behavior is the direct consequence of two combined features of actin polymerization namely, the hydrolysis of ATP, and the relatively slow rate of ATP exchange for ADP on G-actin. 

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... 

The capacitance determined from the initial slopes of the charging curve is about 10/a F/cm2. Taking the dielectric permittivity as 9.0, one could calculate that initially (at the OCP) an oxide layer of the barrier type existed, which was about 0.6 nm thick. A Tafelian dependence of the extrapolated initial potential on current density, with slopes of the order of 700-1000 mV/decade, indicates transport control in the oxide film. The subsequent rise of potential resembles that of barrier-layer formation. Indeed, the inverse field, calculated as the ratio between the change of oxide film thickness (calculated from Faraday s law) and the change of potential, was found to be about 1.3 nm/V, which is in the usual range. The maximum and the subsequent decay to a steady state resemble the behavior associated with pore nucleation and growth. Hence, one could conclude that the same inhomogeneity which leads to pore formation results in the localized attack in halide solutions. 

In order for an equilibrium to exist between E -E S and ES, the rate constant kp would have to be much smaller than k i However, for the majority of enzyme activities, this assumption is unlikely to hold true. Nevertheless, the rapid equilibrium approach remains a most useful tool since equations thereby derived often have the same form as those derived by more correct steady-state approaches (see later), and although steady-state analyses of very complex systems (such as those displaying cooperative behavior) are almost impossibly complicated, rapid equilibrium assumptions facilitate relatively straightforward derivations of equations in such cases. 

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