Transient regimes

Leva et al (1949) and Ergun (1952) developed similar useful equations. Later, these were refined and modified by Handley and Heggs (1968 ) and by MacDonald et al (1979). All these equations try to handle the transient regime between laminar and turbulent flow somewhat differently but are based on the same principles. 

It is also of interest to explore the small-time or transient regime. Using the functional form x2A(x) = o, -I x a2, the second entropy, Eq. (227), may be extrapolated to smaller time scales. The thermal conductivities become... 

Figure 10 shows small-time fits to the thermal conductivity using the above functional form. It can be seen that quite good agreement with the simulation data can be obtained with this simple time-dependent transport function. Such a function can be used in the transient regime or to characterize the response to time-varying applied fields. 

The present analysis shows that when a thermodynamic gradient is first applied to a system, there is a transient regime in which dynamic order is induced and in which the dynamic order increases over time. The driving force for this is the dissipation of first entropy (i.e., reduction in the gradient), and what opposes it is the cost of the dynamic order. The second entropy provides a quantitative expression for these processes. In the nonlinear regime, the fluxes couple to the static structure, and structural order can be induced as well. The nature of this combined order is to dissipate first entropy, and in the transient regime the rate of dissipation increases with the evolution of the system over time. 

At high [Cu(II)] and low [H2A] initial concentrations, the Pt electrode potential, used to follow the chemical process, increased monotonously. When both species were present at high initial concentrations, a monotonous decrease was observed. Various non-monotonic transient regimes were found at approximate initial concentrations of [Cu(II)] 10-4 M and [H2A] 10-4 M. Thus, the batch experiments properly illustrate that the system is sensitive to variations of the initial concentrations of ascorbic acid and copper(II) ion, and the observations can be indicative of a transient bi-stability. 

Fig. 5. Two-dimensional parametric diagram of system response at different initial concentrations of reagents in batch A, monotonic growth of Pt potential [Fig. 1(a)] V, monotonic decrease of Pt potential [Fig. 1(b)] O, Pt electrode potential first decreases and then increases in time [Fig. 1(c)] , various nonmonotonic transient regimes [Fig. l(d—f)]. Strizhak, P. E. Basylchuk, A. B. Demjanchyk, I. Fecher, F. Shcneider, F. W. Munster, A. F. Phys. Chem. Chem. Phys. 2000, 2, 4721. Reproduced by permission of The Royal Society of Chemistry on behalf of the PCCP Owner Societies.

On the other hand, if the really relevant phenomena are overlooked, then this could lead to incorrect interpretation of the fitted parameters, and, consequently, invalid predictions, e.g. if they form the basis of a risk assessment. As illustrative examples, consider two cases that highlight the range of convenience of a refinement (1) most transient effects cannot be seen for microorganisms with very small radii, but the influence of the transient regime can be relevant in the description of accumulation data and (2) if there is transport limitation (i.e. the FIAM assumption does not hold), the lability of the com-plexation becomes very relevant for both the flux and the depletion of the medium. A decision about which phenomena to keep and which to neglect for the specific biological system under consideration and the specific measured quantity can only be made on the basis of a close interaction between theoretical and experimental studies. 

Transient Regime. Equation 7 is separable and may be integrated to obtain the time dependence of the oxide thickness. Assuming a finite sputtering rate, the result is... 

Watanabe and Ohnishi [39] have proposed another model for the polymer consumption rate (in place of Eq. 2) and have also integrated their model to obtain the time dependence of the oxide thickness. Time dependent oxide thickness measurement in the transient regime is the clearest way to test the kinetic assumptions in these models however, neither model has been subjected to experimental verification in the transient regime. Equation 9 may be used to obtain time dependent oxide thickness estimates from the time dependence of the total thickness loss, but such results have not been published. Hartney et al. [42] have recently used variable angle XPS spectroscopy to determine the time dependence of the oxide thickness for two organosilicon polymers and several etching conditions. They did not present kinetic model fits to their results, nor did they compare their results to time dependent thickness estimates from the material balance (Eq. 9). More research on the transient regime is needed to determine the validity of Eq. 10 or the comparable result for the kinetic model presented by Watanabe and Ohnishi [39]. 

The transient regime following electric field inversion aJ.lows more precise determination of the P/Q ratio to be achieved. If the particle does not possess any permanent dipole (P/Q = O) the birefringence remains stationary upon field inversion. If there is a contribution of permanent dipole, then An reaches a minimum An. related at low fields with P/Q according to ... 

The control results for the j0-carotene model (Figure 9.6) agree qualitatively with the simulated control results for the excited population dynamics in Ref. [64] (see Figure 6). Namely, the population dynamics is phase-controllable in both the transient regime while the pulse is acting, and in final regime when the pulse is over, that is, t > 180 ps. 

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

It results in a number of transient regimes to be discussed below. At k — 0 another (quasi-chaotic) kind of solution arises, which has another asymptotic behaviour. 

Solution for a very small parameter k = 0.005 demonstrates a distinctive transient regime with oscillations considerably different at the very beginning from what has been discussed in this Chapter. 

Figure 8.9 serves as a good illustration of different possible transient regimes arising as k is reduced. As stabilization time increases, r oc kT1, the Lotka model reveals a series of quasi-periodic motions, separated by chaotic transient phases. The main trend seen from the analysis of results, is emergence of the periodic motion with a minimal period. To get some important properties of the transient irregular regimes, such as the presence of main frequencies or a white noise, it is useful to analyze the Fourier spec-... 

Without considering transient regimes, we find the asymptotic laws which are satisfied by the wave and the state of gas compressed by it during the motion of the piston (which is long compared to the time of the chemical... 

When used in the time-invariant mode (i.e., in equilibrium), it is a first-order chemical sensor that can yield qualitative and quantitative information based on the LSER paradigm about composition of the vapor mixtures (Fig. 10.13). By acquiring the data in the transient regime, it becomes a second-order sensor and in addition to the composition, information about diffusion coefficients in different polymers is obtained. This is then the added value. It is possible only because the model describing the capacitance change included diffusion. In spite of the complexity of the response function, a good discrimination and quantification has been obtained. 

Kuts et al. [35] studied the relationship between the maximum depth of penetration into the opposing stream, xmM, and the diameter of the particle, dp, in the ranges of dp = 100-1000 pm and xm.M = 0.01-0.11 m the results are illustrated in Fig. 2.3. The results of the theoretical analysis are in the Stokes and the transient regimes jtmax is positively in proportion to dp and dp5, respectively. It can be seen from Fig. 2.3 that the results theoretically predicted fit the experimental data relatively well. 

The mean field approach used here reproduces all the qualitative features but it cannot give all the details of the experimental curves, especially in the transient regime (see Ref. [167]). In fact the coverage dependence in the rate constant of the elementary steps is very crude, in particular it cannot reproduce the local variation of coverage that may be present in the high coverage regime and due to the different facets of the clusters. To address this problem a Monte Carlo simulation approach is necessary [171]. 

J. L. Bassani, D. E. Hawk, and A. Saxena, Evaluation of the C, Parameter for Characterizing Creep Crack Growth Rate in the Transient Regime, in... 

J. L. Bassani, D. E. Hawk, and A. Saxena, Evaluation of the C, Parameter for Characterizing Creep Crack Growth Rate in the Transient Regime, in Nonlinear Fracture Mechanics Volume I—Time Dependent Fracture, eds. A. Saxena, J. D. Landes, and J. L. Bassani, ASTM STP 995, American Society for Testing and Materials, Philadelphia, PA, 1988, pp. 7-26. 

It is the purpose of the present work to try to obtain insight into CsH formation and to estimate the efficiency of the process. For this reason, we performed a spectroscopic investigation of the relative concentrations of the reactants - Cs(7P), and CsH - under various experimental conditions in stationary as well as in transient regimes. A more detailed discussion of these results will be published in J. Chem. Phys. 

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