Transient and steady-state behaviors

Patankar, S. V., and D. B. Spalding. 1974. A calculation procedure for the transient and steady-state behavior of shell-and-tube heat exchangers. In N. H. Afgan and E. V. Schliinder (eds.). Heat Exchangers Design and Theory Sourcebook. New York McGraw-Hill, pp. 155-176. 

In the present communication we report on the effects of voltage, temperature and electrode surface area on the transient and steady-state behavior of the system. The change in the rate of C2Hi 0 production can exceed the rate of 0 pumping by a factor of 400 and is proportional to the anodic overvoltage both at steady state and during transients. 

During the period in which Horgan and Lange were carrying on their studies, Clegg (37) presented a careful study of respiratory control which followed a more basic control approach than that used previously. The schematic representation of the system is shown in Figure 4. This work, which is seldom referenced, represented one of the first attempts at specifying the relative importance of the peripheral and central chemoreceptors to respiratory dynamics and stability. The model exhibited realistic transient and steady-state behavior under various conditions. 

Since short and long times are relative terms, it is useful to determine the times over which transient and steady state behaviors will predominate and how this time regime is affected by the electrode radius. 

Beretta, A., Groppi, G., Lualdi, M., et al. (2009). Experimental and Modeling Analysis of Methane Partial Oxidation Transient and Steady-State Behavior of Rh-Coated Honeycomb Monoliths, Ind. Eng. Chem. Res., 48, pp. 3825-3836. 

Ruhstaller, B., Carter, S.A., Barth, S., Riel, H Riess, W, and Scott, J.C. (2001) Transient and steady-state behavior of space charges in multilayer organic light-emitting diodes. J. Appl. Phys., 89,4575. 

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. 

There is similar transient and steady-state electrochemical promotion behavior of Pt on YSZ and Pt on Ti02 [Figure 39(a) and (b)] and similar backspillover mechanism of Pt on YSZ and of Pt on Ti02 as manifested by XPS (Figure 21). 

The above references mainly describe rheological behavior of graphite fiber (length 0.5-16 cm) in poly-ether-ketone-ketone (PEKK) at 370°C. Anthors conclnde that the transient and steady-state rheological properties of these materials are different from the unfilled melt. 

A strategy employing both steady-state and transient information to optimize the parameters simultaneously failed to achieve an acceptable compromise between the transient and steady-state uniform model predictions. This inability of optimized parameters obtained from transient studies to predict observed steady-state behavior is one potential kinetic fingerprint of nonuniform surfaces. 

Letwimolnun et al. [2007] used two models to explain the transient and steady-state shear behavior of PP nanocomposites. The first model was a simplified version of the stmcture network model proposed by Yziquel et al. [1999] describing the nonlinear behavior of concentrated suspensions composed of interactive particles. The flow properties were assumed to be controlled by the simultaneous breakdown and buildup of suspension microstructure. In this approach, the stress was described by a modified upper-convected Jeffery s model with a modulus and viscosity that are functions of the suspension structure. The Yziquel et al. model might be written ... 

Tye325 explained that separator tortuosity is a key property determining transient response of a separator and steady-state electrical measurements do not reflect the influence of tortuosity. Fie recommended that the distribution of tortuosity in separators be considered some pores may have less tortuous paths than others. He showed mathematically that separators with identical average tortuosities and porosities could be distinguished by their unsteady-state behavior if they have different distributions of tortuosity. 

If the reaction mechanism contains more than one or at most two steps, the full solution becomes very complicated and we will have to solve for the rates and coverages by numerical methods. Although the full solution contains the steady state behavior as a special case, it is not generally suitable for studies of the steady state as the transients may make the simulation of the steady state a numerical nightmare. 

For the very low density varieties of the cases shown in Figure 2 and, more particularly, Figure 4 (curve 7), for which initiation is slow compared to both termination (release from end of template) and polymerization, a simpler treatment, in which the interference of one ribosome with another is totally neglected, should suffice. In this case an equation of the form of Eq. (1), herein only applied to the problem of DNA synthesis, should be valid, but Eqs. (2) and (3) should be modified to account for repetitive initiation at site 1 and continuing release from site K, respectively Eqs. (4) and (6) will not apply. In the even more restricted (but perhaps biochemically relevant) case in which, in addition to neglecting ribosome interference, one may also neglect the back reaction (kb x 0), one may solve this system of equations (Eq. (1), plus Eqs. (2) and (3) modified as described) very easily by taking Laplace transforms.13 This is the only case with repetitive initiation for which we have been able to find solutions for the transient, as well as steady state, behavior. 

Whether such bulk gettering effects are transient or steady state is unclear. The mechanisms of such effects and whether they are due to bulk or surface phenomena is also unknown. Thus it is clear that bulk thermodynamic information used in a cursory way does not help in rationalizing observed sulfur-poisoning behavior. Furthermore, since most metals of catalytic interest do not form stable bulk sulfides under typical reaction conditions, the observed severe poisoning by sulfur suggests that surface rather than bulk thermodynamics may be required. 

Even though OSC is an inherently transient phenomenon, it appears that there is a relationship between steady-state reaction rates and OSC [3,17J. For the CO-oxidation, WGS, and steam-reforming reactions, it has been shown that rates can be enhanced by contact between the precious metals and ceria. Furthermore, high-lemperature treatments, which are known to deactivate the OSC properties of pure ceria, also remove the promotional effects associated with ceria [3,20,221, Given that SO2 affects OSC, one should expect SOt to influence the steady-state behavior of ceria-supported catalysts, if OSC is related to these reactions. 

Both qualitative and quantitative insight can be garnered from transient X -i, i-t and r -t measurements in quiescent or stirred solutions, while measurements of steady-state behavior are best performed under well-defined hydrodynamic conditions. Typically, a rotating disc electrode (RDE), or a related method, is used to specify and/or modulate the hydrodynamic boundary layer thickness, 8. With an RDE the boundary layer is specified by... 

The time constants characterizing metabolic transients are typically very rapid compared to those of cell growth and process dynamics therefore, the transient mass balances can be simplified to consider only the steady-state behavior. " Eliminating the time derivative in Eq. (4) and rearranging the equation yields ... 

This set of first-order ODEs is easier to solve numerically than the algebraic equations that result from setting all the time derivatives to zero. The initial conditions are Uout = do, bout = bo, at t = 0. The long-time solution to the ODEs will satisfy Equations 4.1 provided that a steady-state solution exits and is accessible from the initial conditions. As discussed in Chapter 5, some CSTRs have multiple steady states and the achieved steady state depends on the initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can exhibit oscillations or even a semirandom behavior known as chaos. The method of false transients will then fail to achieve a steady state. Another possibility is a metastable steady state. Operation at a metastable steady state requires a control system and cannot be reached by the method of false transients. Metastable steady states arise mainly in nonisothermal systems and are discussed in Chapter 5. 

Changes in the production rate come from the plant s management and are rather infrequent (e.g., once every two weeks, or a month, or longer period). The time required by the plant to reach the new operating level is much shorter than the periods noted above. Consequently, the transient dynamic behavior is very short-lived and not very important. Therefore, we can assume that the plant always operates at steady state. Figure 25.6 demonstrates this pseudo-steady-state behavior. It is clear, then, that steady-state balances are sufficient for material balance control. 

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). 

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