Quasi-stationary variables

The second assumption (z = 0) of the Bodenstein hypothesis as a consequence of the first (e = 0) is reasonable. Just arguing from a mathematical point of view assumption 1 is neither a necessary nor sufficient condition for the validity of assumption 2, For this reason in the case of the method of quasi-stationary variables one avoids assumption 1 and takes for the changing concentration... 

The example proves the validity of the method of quasi-stationary variables under certain conditions. But this fact does not imply the answer to the question on the relative values of rate constants A , and concentrations... 

In typical situations, V /x (Tables 5.2 and 5.4). This means that air flow temperature varies much faster than the stack temperature. Physically, air enthalpy is 4 orders of magnitude smaller than the enthalpy of BP and the largest time scale in the system is determined by BP heating (Achenbach, 1995). Therefore, flow temperature can be considered as a quasi-stationary variable i.e. the time derivative in Eq. (5.77) can be omitted. For Tgir we thus get... 

Classifying variables into fast or slow is a typical approach in chemical kinetics to apply the method of (quasi)stationary concentrations, which allows the initial set of differential equations to be largely reduced. In the chemically reactive systems near thermodynamic equihbrium, this means that the subsystem of the intermediates reaches (owing to quickly changing variables) the stationary state with the minimal rate of entropy production (the Rayleigh Onsager functional). In other words, the subsys tern of the intermediates becomes here a subsystem of internal variables. 

The dynamics of mass-exchange between a mono-dispersive ensemble of hydrate inhibitor drops and a hydrocarbon gas has been considered in Section 21.1 within the framework of the above-made assumptions in a quasi-stationary approximation. In dimensionless variables ... 

The purpose of tidal corrections is a reduction from the time variable actual state to a quasi—stationary, time—invariant model of the physical world. Here we will concentrate on tidal reductions for diurnal and semi-diurnal periodical effects on absolute heights, induced by astronomical and earth tides. 

With respect to the previous methods, it can be remarked that an eigenvalue analysis allows the fast and the slow variables to be defined, without having to identify the quasi-stationary species or the reactions at quasi-equilibrium. 

When a chosen adsorber input x is modulated periodically in the quasi-stationary state, all the output variables of the adsorber become periodic functions of time as well. In the general non-isothermal case, four adsorber outputs can be defined pressure P and temperature Tg of the gas... 

Abbr Fo, Fm, Fs and FvJ constant, Kiximum, quasi-stationary and variable fluorescence. Tvs quenching of Fv (e.g. Fm-Fs). T time for quenching from Fm to Fs. Tv/T quenching rate of Fv. T1/2J half time for fluorescence rise. Ch cou>lemantary area of rise. 

The following sections are devoted to the review of the most relevant models of ground motion acceleration processes aimed to satisfy the code requirements. Namely, the stationary, quasi-stationary, and fully nonstationary as well as spatially variable stochastic spectrum-compatible ground motion will be discussed. 

In the previous sections the most common procedures to determine stationary/quasi-stationary power-spectral density functions compatible with a given response spectrum have been presented. It should be noted at the outset that the ground motion time histories generated from quasi-stationary stochastic processes have energy content only variable with time. Although this approach is convenient and accurate for the seismic analysis of traditional linear behaving struc-mres, it does not lead to a comprehensive description of the seismic phenomenon. 

An alternative approach has been proposed by Cacciola (2010) the author s contribution allows a straightforward evaluation of a non-separable power-spectral density function compatible with a target response spectrum. In the model proposed by Cacciola (2010), it is assumed that the nonstationary spectrum-compatible evolutionary ground motion process is given by the superposition of two independent contributions the first one is a fully nonstationary known counterpart which accounts for the time variability of both intensity and frequency content the second one is a corrective term represented by a quasi-stationary zero-mean Gaussian process that adjusts the nonstationary signal in order to make it spectrum compatible. Therefore the grotmd motion can be split in two contributions ... 

To describe the behavior of the system near the quasi-stationary state, the variables are introduced... 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

This transition differs from the one discussed in Section Ill.l.(i) insofar as one of the steady states is a focus (which is, of course, only possible in a two-variable system), and the current does not monotonically increase but overshoots its stationary value, toward which it slowly relaxes. Also, here the increase in current density is accompanied by an accelerated front, whereas the delayed relaxation of the steady state occurs on a spatially quasi-homogeneous electrode. 

As seen above, laser assisted and controlled photofragmentation dynamics can conceptually be viewed in two different ways. The time-dependent viewpoint offers a realistic time-resolved dynamical picture of the basic processes that are driven by an intense, short laser pulse. For pulses characterized by a long duration (as compared to the timescales of the dynamics), the laser field can be considered periodic, allowing the (quasi-) complete elimination of the time variable through the Floquet formalism, giving rise to a time-independent viewpoint. This formalism not only offers a useful and important interpretative tool in terms of the stationary field... 

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