Transient behavior

The time-dependent mass and energy balances are given by Eqs. 10.2.a-4 and 5 dx. 

Analytical solution of this system ofdifferential equations is not possible. Therefore Aris and Amundson [32] linearized it by a Taylor expansion, about the steady- 

Expanding and Q,(T) in Taylor series and neglecting second-order terms leads to  

These equations (10.4.b-5, 6) are linear differential equations, whose solutions are combinations of exponentials of the form exp[mt/T], where the values of m are solutions of the characteristic equation  

If Oq 0, at least one of the roots will be positive, and the solution will diverge for t - 00. If 0 = 0 and Oo 0, the roots will be purely imaginary numbers, with oscillatory solutions for x and y. Thus, the necessary and sufficient conditions for stability (i.e., x and T return to the steady state after removal of the perturbation or x and y - 0 as r - oo) are Eqs. 10.4.b-8 and 9. In terms of the physical variables those equations can be written as follows  

The solutions will only go to zero as t oo when the real parts of the roots are negative [e.g., see Himmelblau and Bischoff, 1968]. The solution of (10.4.2-7) is 

---

Selzer Y and Manler D 2000 Scanning electrochemical microscopy. Theory of the feedback mode for hemispherical ultramicroelectrodes steady-state and transient behavior Anal. Chem. 72 2383... 

Process Systems. Because of the large number of variables required to characterize the state, a process is often conceptually broken down into a number of subsystems which may or may not be based on the physical boundaries of equipment. Generally, the definition of a system requires both definition of the system s boundaries, ie, what is part of the system and what is part of the system s surroundings and knowledge of the interactions between the system and its environment, including other systems and subsystems. The system s state is governed by a set of appHcable laws supplemented by empirical relationships. These laws and relationships characterize how the system s state is affected by external and internal conditions. Because conditions vary with time, the control of a process system involves the consideration of the system s transient behavior. 

Hlavacek, V, Analysis of a Complex Plant Steady-State and Transient Behavior, Computers and Chemical Engineeiing, 1 1977, 75-100. (Review article)... 

It appears that the observed breakdown must be explained in terms of the transient behavior of stress-induced defects even though the stresses are well within the nominal elastic range. In lithium niobate [77G06] and aluminum oxide [68G05] the extent of the breakdown appears to be strongly influenced by residual strains. In the vicinity of the threshold stress, dielectric relaxation associated with defects may have a significant effect on current observed in the short interval preceding breakdown. 

The analysis of the transient behavior of the packed bed reactor is fairly recent in the literature 142-145)- There is no published reactor dynamic model for the monolith or the screen bed, which compares well with experimental data. 

Vermeulen, D. P. and Fortuin, J. M. H., Experimental verification of a model describing the transient behavior of a reaction system approaching a limit cycle or a runaway in a CSTR, Chem. Eng. Sci., 41, 1089-1095 (1986). 

Steady state flow occurs transient behavior is not analyzed. 

All the supported catalysts used gave TCE conversions less than 20% for the wet oxidation at 310 K, except for the 5 wt.% CoOx/Ti02, which had a steady-state conversion of 45% via a transient behavior in activity up to 1 h on stream (Fig. 1). Subsequently, there was negligible TCE conversion for the bare Ti02 during continuous operating hours near 6. 

In industry, as well as in a test reactor in the laboratory, we are most often interested in the situation where a constant flow of reactants enters the reactor, leading to a constant output of products. In this case all transient behavior due to start up phenomena have died out and coverages and rates have reached a constant value. Hence, we can apply the steady state approximation, and set all differentials in Eqs. (142)-(145) equal to zero ... 

The steady state is disturbed and the system exhibits transient behavior when at least one of its parameters is altered under an external stimulus (perturbation). Transitory processes that adjust the other parameters set in (response) and at the end produce a new steady state. The time of adjustment (transition time, relaxation time) is an important characteristic of the system. 

Most SECM experiments at liquid-liquid interfaces have principally involved the determination of the steady-state tip current response as a function of the separation between the tip and the interface (approach curve measurements). However, in some situations complementary information can be gleaned from the transient behavior (as illustrated below for SECMIT). We therefore describe models for the time-dependent problem from which the steady-state characteristics can be developed from the longtime limit. 

S.5.2 Transient two-phase-flow pressure drop. Calculation of transient behavior in a complex flow network containing a compressible fluid in two-phase states was... 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

In simulating transient flows in pipeline networks, the importance of accuracy cannot be over-emphasized. Because the transient behaviors are less well-understood, they are often rich in surprises. Physical intuition affords less guidance in these situations than in steady-state phenomena. Rachford and Dupont (R2) provided two instructive and deceptively simple examples to illustrate the interaction between regulators and compressors and the oscillatory response which can produce pressures higher than the supply pressure through reinforcement. 

Ehlig-Economides, C. "Use of the Pressure Derivative for Diagnosing Pressure-Transient Behavior," J. Petr. Technol.. October 1988, 1280-1282. 

The kinetic equations describing these four steps have been summarized and discussed in the earlier paper and elsewhere (1,5). They can be combined with conservation laws to yield the following non-linear equations that describe the transient behavior of the reactor. In these equations the units of the state variables T, M, and I are mols/liter, while W is in grams/liter. The quantity A (also mols/liter) represents that portion of the total polymer that is unassociated — i.e. reactive. 

Villegier, A.S., Blanc, G., Glowinski, J., Tassin, J.P. Transient behavioral sensitization to nicotine becomes long-lasting with monoamine oxidases inhibitors. Pharmacol. Biochem. Behav. 76 267, 2003. 

Analysis of CSTR Cascades under Nonsteady-State Conditions. In Section 8.3.1.4 the equations relevant to the analysis of the transient behavior of an individual CSTR were developed and discussed. It is relatively simple to extend the most general of these relations to the case of multiple CSTR s in series. For example, equations 8.3.15 to 8.3.21 may all be applied to any individual reactor in the cascade of stirred tank reactors, and these relations may be used to analyze the cascade in stepwise fashion. The difference in the analysis for the cascade, however, arises from the fact that more of the terms in the basic relations are likely to be time variant when applied to reactors beyond the first. For example, even though the feed to the first reactor may be time invariant during a period of nonsteady-state behavior in the cascade, the feed to the second reactor will vary with time as the first reactor strives to reach its steady-state condition. Similar considerations apply further downstream. However, since there is no effect of variations downstream on the performance of upstream CSTR s, one may start at the reactor where the disturbance is introduced and work downstream from that point. In our generalized notation, equation 8.3.20 becomes... 

Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column. 

Assuming that after their previous swap the two walks were sufficiently long to be in the asymptotic regime, this means that transient behavior has elapsed and the system has relaxed to equilibrium for the respective parameters. Then, the joint configurational probability density just before the current swap is simply... 

The squares and full lines of Fig. 11 summarize their results. The scatter of the experimental points seems mainly due to the analysis of the transient behavior the diffusion coefficient D and hence the solubility s = P/D fluctuate much more than the steady-state permeation coefficient P. Their Arrhenius lines are described by ... 

Indeed, it is worth noting that by itself, a permeation rate proportional to p°50 could be consistent with any value whatever for the ratio of monatomic to diatomic species in the solid, if the diatomic species is very immobile. For in such case, the permeation flux would be carried entirely by the monatomic species, whose concentration always goes as p0 50. However, a sizable diatomic fraction would significantly modify the transient behavior of the permeation after a change in gas pressure. Although neither Van Wieringen and Warmholtz nor Frank and Thomas published details of the fit of their observed transients to the predictions of diffusion theory, it is unlikely that any large discrepancies would have escaped their attention. 

Transient behavior of the catalyst at high oxidation state. The significance of the redox type reactions between reaction medium and catalyst is depicted in Figure 7, where the conversion of CO to CO2 has been stimulated under shift conditions. After a long range pretreatment of the catalyst in a mixture of t O/ (0,30 bar 1 0) a periodic operation has been established consisting of a testing phase of 1 min with a mixture of CO, H 0, N ... 

Transient behavior of the catalyst at low oxidation state. After a reduction of the catalyst in a CO/ mixture, the transient shift operation, depicted in Figure 8 (see also (7)), re-... 

Pr q/Pro = 5,0 are relevant sorption effects of CO- but not of H2 thus only the 1 wavefronts represent rather tne shift conversion). Therefore it seems conceivable that there are two different mechanisms which participate in the CO shift conversion which is also in agreement with the established two different sorption mechanisms for 1 0 and with the transient behavior, depicted on Figure 6. 

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... 

---