Expansion timescale

Fig. 24. The likelihood of a DYR r-process for given combinations of the electron fraction Ye and the entropy per baryon s. A SoS-like r-process is expected for a suitable superposition of conditions between the black lines. The results inferred from an initial NSE phase at low s are smoothly connected to those of various nuclear network calculations for high s values. In the latter cases, the assumed expansion timescales imply that the freeze-out of the charged-particle induced reactions is reached after dynamical timescales Tdyn in excess of about 50 - 100 ms. The two dotted lines represent the contours of successful r-processing for Tdyn = 50 ms (left line) and 100 ms (right line) (see [59] for details)...

It turns out that the CSP approximation dominates the full wavefunction, and is therefore almost exact till t 80 fs. This timescale is already very useful The first Rs 20 fs are sufficient to determine the photoadsorption lineshape and, as turns out, the first 80 fs are sufficient to determine the Resonance Raman spectrum of the system. Simple CSP is almost exact for these properties. As Fig. 3 shows, for later times the accuracy of the CSP decays quickly for t 500 fs in this system, the contribution of the CSP approximation to the full Cl wavefunction is almost negligible. In addition, this wavefunction is dominated not by a few specific terms of the Cl expansion, but by a whole host of configurations. The decay of the CSP approximation was found to be due to hard collisions between the iodine atoms and the surrounding wall of argons. Already the first hard collision brings a major deterioration of the CSP approximation, but also the role of the second collision can be clearly identified. As was mentioned, for t < 80 fs, the CSP... 

Hysteresis was generally observed in the compression-expansion cycles of the force-area isotherms, indicating that the timescale for relaxation of the fully compressed film back to its expanded state was slower than the movement of the barrier of the Langmuir trough. Our studies, like many others, imply that monolayers are metastable and that reversible thermodynamics can only be applied to their analysis with caution. 

Equilibrium (i.e. local steady-state) ionization leads in this regime to solar-corona-like conditions where col-lisional ionization is balanced by recombination and the degree of ionization is fixed by the temperature alone, the electron density cancelling out. However, here departures from equilibrium occur because the time taken to establish ionization equilibrium is not negligible with respect to the timescale of expansion. 

If the flow rate is increased so that Peclet number Pe l, then there is a timescale at which transversal molecular diffusion smears the contact discontinuity into a plug. In Taylor (1993), Taylor found an effective long-time axial diffusivity proportional to the square of the transversal Peclet number and occurring in addition to the molecular diffusivity. After this pioneering work of Taylor, a vast literature on the subject developed, with over 2000 citations to date. The most notable references are the article (Aris, 1956) by Aris, where Taylor s intuitive approach was explained through moments expansion and the lecture notes (Caflisch and Rubinstein, 1984), where a probabilistic justification of Taylor s dispersion is given. In addition to these results, addressing the tube flow with a dominant Peclet number and in the absence of chemical reactions, there is... 

In the paragraph from Section 3.1, we will give a detailed derivation of the effective equations. Our technique is motivated by the paper by Rubinstein and Mauii, 1986, where the analysis is based on the hierarchy of timescales and a corresponding two-scale expansion. For k —0, our approach gives the effective... 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

We assume stationarity and radiative equilibrium for the energy balance because the radiative timescales are short in respect to the hydrodynamic timescales soon after the initial increase in luminosity. Spherical symmetry is assumed. According to detailed numerical models (Falk and Arnett, 1977 Muller, personal communication, 1987 Nompto, 1987 Nomoto et aL, 1987) and also analytical solutions for strong shock waves in spherical expanding enveloped (Sedov, 1959) density profiles are taken which are given by the self-similar expansion of an initial structure i.e. 

According to the previous section, we shall start by considering X and P as fast degrees of freedom, relaxing on a much more rapid timescale than the orientational coordinates and momenta of the solute and the solvent cage. Many different projection schemes are available to handle stochastic partial differential operators. Here we choose to adopt a slightly modified total time ordered cumulant (TTOC) expansion procedure, directly related to the well known resolvent approach. In order to make this chapter self-contained, we summarize the method in the Appendices and its application to the cases considered here and in the next section. 

Electricity generated from wind power can be highly variable at several different timescales on a second by second basis, as well as hourly, daily, and even seasonally. In order to correctly assess the relevance of intermittent power sources in the expansion of the power system, the concept of capacity credit is commonly used. It expresses the amount of installed conventional power that can be avoided or replaced by intermittent power sources. This capacity credit is the fraction of the installed renewable power for which no double investment is needed. For example, 1,000 MW of installed wind power with a capacity credit of 30 percent, can avoid a 300 MW investment in conventional dispatchable power (Voorspools and D haeseleer, 2006). In good locations with a high mean wind velocity and with an efficient wind turbine, capacity credits may reach 33-38 percent (Sherif et al., 2005). 

A simulation for 1 fs after perturbation can cover only an infinitesimal area of the potential surfece, unlike simulations for a longer timescale, which can better explore the rugged potential surface. In this sense, our simulation of energy transfer during 1 fs after perturbation does not intend to analyze the kinetic process of energy transfer, but intends to study the shape of the local potential surface around an instantaneous structure formed at a finite temperature. Therefore, it is reasonable to analyze the results of such simulations from the viewpoint of series expansion of potential energy. 

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