Stability steady state operation

We can, however, consider the stability of each of the three operating points in Example 14-7 with respect to the inevitable small random fluctuations in operating conditions, including cA, in steady-state operation. Before doing this, we note some features of the rate law as revealed in Figure 14.4. There is a maximum value of (- rA) at cA = 1.166 mol m-3. For cA < 1.166, the rate law represents normal kinetics ( rA) increases as cA increases for cA > 1.166, we have abnormal kinetics (—rA) decreases as cA increases. We also note that (-rA) in equation (C), the rate law, represents the (positive) rate of disappearance of A by reaction within the CSTR, and that (—rA) in equation (D), the material balance, represents the (positive) net rate of appearance of A by flow into and out of the reactor. As noted above, in steady-state operation, these two rates balance. 

In summary, the stationary-states at C3 and C3 are stable, and the state at Q is unstable, and cannot be achieved in steady-state operation. It can, however, be stabilized by a control system involving a feedback loop. For a simple system, as considered here, there... 

The stellarators are similar to Tokamaks, both employing a closed toroidal magnetic field for basic plasma confinement. The stabilization in stellarators is, however, provided by special helical windings16. They possess some potential advantages as compared to the Tokamaks with respect to confinement properties and the possibility of steady state operation. Technical and technological limitations such as the small size of the presently available stellarators and the lack of appropriate, sufficiently intense heating systems have inhibited progress comparable to that which Tokamaks have so far achieved. 

To achieve a steady-state environment, some actions must be taken before starting each impedance measurement for a PEMFC system. For example, Wagner [23] prepolarized the cell for at least 15 min at the measuring potential. The current densities before and after measurement were taken to prove the stability of the cell during measurement times. Guo et al. [38] operated a fuel cell at 0.6 V for 20 h to reach its steady-state operating current. Pickup et al. [34, 39] ran a H2/02 fuel cell for 30 min at 0.5 V before the impedance measurements were performed. In Gode et al. s work [40], the cell was mn galvanostatically for 1 h prior to the impedance measurement. 

In many chemical plants the main reaction is a highly exothermic one. Consequently, the hot reactor effluent is often used to preheat the feed to the reactor (Figure 25.2a). Such feed-effluent heat exchange may lead to serious stability problems in the operation of the reactor. In particular, if 7/ is the temperature of the feed to the reactor, the temperature of the reactor effluent Tr follows the sigmoidal curve shown in Figure 25.2b. Furthermore, a heat balance around the heat exchanger shows that 7 depends linearly on T, (Figure 25.2b). Therefore, the steady-state operation will correspond to one of the three possible points Pi, Pi, or Pi, where the two lines intersect. 

Figure 6.16 Steady-state operating diagram for a PFR. [After D.D. Perlmutter, Stability of Chemical Reactors, reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ, (1972).]...

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). 

Most of the commonly used membranes used in DMFC exhibit acceptable performance stability after thousands of hours under steady-state operation, including perfluorinated sulfonic acid, PTFE-co-HFP radiation-grafted poly- styrenes, and sPAE [85]. However, membrane durability results under unsteady-state operation are stUl scarce. Siroma et al. [114] have shown that a significant fraction of Nation dissolved after 1 week of DMFC operation with methanol solution. 

Non-lsothermal CSTR Operation Multiple Steady States and Stability... 

For the purpose of stability analysis, we need to linearize the DAE model around a steady state operating point [x z ], which is in the following form in the case of the general model (1-2) ... 

Another class of time-dependent problems of concern to the reactor physicist are questions on fuel burnup, poison production and burnup, breeding ratio, and the like. These problems differ from those on reactor stability in that they involve time scales measured in hours (or years) in contrast to stability problems which are concerned with fractions of a second. Reactor-analysis problems, such as the determination of critical mass and neutron-density distributions, are based on the steady-state operating condition of the reactor. The day-to-day operation of the reactor at steady state involves, however, long-time changes in the fuel concentration. Except in the case of circulating-fuel reactors, the fuel is introduced into the reactor according to some predetermined cycle. As the fuel is consumed, some gradual adjustments can be made by means... 

The experiments have only given an insight into what is a complicated problem. What is required are comprehensive theoretical steady-state and stability studies of all of the parameters involved in order to optimize a design for given set of operating conditions. 

We have identified dehydrogenation catalysts with high activity, Stability for prolonged steady-state operation, excellent selectivity to product and operability at practical steam/ethyl-toluene feed ratios. 

Steady state operation is invariably concerned with stability since one must question whether or not a system will eventually converge to a steady state when it is subjected to perturbations, i.e., changes in operating conditions. Furthermore, steady state solutions do permit certain limited statements to be made about stability. For example, Gavalas (1968) has shown that, given multiple solutions of a steady state equation, there are (2m + 1) steady states except under exceptional circumstances, and that at least m of them are unstable.. It will also be seen that... 

The type of stability analysis carried out above is not mathematically rigorous. However, it is a very useful way to understand the concept of stable and unstable operating points. In mathematical terms, the above analysis has shown that a steady-state operating point will be unstable if... 

The number of (ission neutrons which are delayed usually is considered as an important factor in reactor safety. However, if reactivity is added as a linear rate function at low initial reactor power, the delayed neutrons have little influence upon the rcactiiity addition ahove prompt critical. Tliey will influence the stability and steady-state operational behavior of th( reactor, though, and are neces.sary to damp the power oscillation following a reactivity addition. 

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