Characteristic distance

How large a simulation do we need Once more this depends on the system and properties of interest. From a spatial correlation fiinction (x(O)x(r) relating values computed at different points r apart, we may define a characteristic distance over which the correlation decays. The sumdation box size L should be significantly larger than in order not to influence the results. 

Grady and Asay [49] estimate the actual local heating that may occur in shocked 6061-T6 Al. In the work of Hayes and Grady [50], slip planes are assumed to be separated by the characteristic distance d. Plastic deformation in the shock front is assumed to dissipate heat (per unit area) at a constant rate S.QdJt, where AQ is the dissipative component of internal energy change and is the shock risetime. The local slip-band temperature behind the shock front, 7), is obtained as a solution to the heat conduction equation with y as the thermal diffusivity... 

An alternative procedure rescales the coordinates of each atom at each time step [24]. The atomic coordinate x and the characteristic distance for repeating boundary conditions, d, are rescaled to values uc and fid, respectively, where... 

The basic correlation for packed tubes is derived from those of empty tubes by properly reinterpreting some critical variables. The most important change is in the characteristic distance that is changed from the tube diameter to the particle diameter. Other corrections are also used. The transfer correlations are based on dimensional analysis, expressed as either the Nusselt-type (1930) or the Colbum-type (1933) equations. For empty tubes at high Re numbers ... 

These simple models are based on the assumption of constant drift velocity i.e., particles are assumed to achieve their final charge instantaneously. This is a reasonable assumption in the case of large particles, the charging of which is governed by field-driven ion motion. The characteristic distance, x% corresponding to the time constant in Eq. (13,53) is given by... 

It is easy to see that, under normal circumstances, the characteristic distance is much shorter than the effective length of the precipitation system. 

A large body of experimental evidence confirms that covalent bonds have characteristic distances depending on bond type. Carbon-carbon single and double bond lengths are around 1.54A and 1.32A, respectively, while partial double bond distances, e.g., in benzene, are about 1.40A. 

Transition states, because they represent a molecule in which bonds are being made (or broken), necessarily contain partial bonds. There are no experimental data, however, that can tell us how long these bonds are, or whether partial bonds even have characteristic distances. 

Characteristic length [Eq. (121)] L Impeller diameter also characteristic distance from the interface where the concentration remains constant at cL Li Impeller blade length N Impeller rotational speed also number of bubbles [Eq, (246)]. N Ratio of absorption rate in presence of chemical reaction to rate of physical absorption when tank contains no dissolved gas Na Instantaneous mass-transfer rate per unit bubble-surface area Na Local rate of mass-transfer per unit bubble-surface area Na..Average mass-transfer rate per unit bubble-surface area Nb Number of bubbles in the vessel at any instant at constant operating conditions N Number of bubbles per unit volume of dispersion [Eq. (24)] Nb Defined in Eq. (134)... 

The boundary conditions for the system are (1) that at the surface of the hygroscopic material the partial pressure of water is determined by that of the saturated salt solution (Ps) and (2) that at a characteristic distance from the surface (8) the partial pressure of water vapor is given by the chamber pressure (Pc). 

The experiments are based on the fact that the probability of energy transfer between the two chromophores depends strongly on their separation distance [314]. For the Np-Py pair the characteristic distance of the process is 28.9 A [315]. Thus, the occurrence of NRET between Np and Py in dilute mixed solutions of polymers carrying either Np or Py signals the existence of interpolymeric association. By assessing the extent of energy transfer under various circumstances it becomes possible to monitor closely the interactions between polymer chains. 

In an interval of time At, advection carries a non-reacting solute across a distance L = vx At, and the solute is dispersed over a characteristic distance L VDLAt (e.g., Crank, 1975). For any given At, then, advection and dispersion observed at a length scale L are equally important wherever... 

Casimir and Polder also showed that, at very long range (i.e., separations greater than a characteristic distance R of a few hundred angstrom units), the dispersion interaction takes the modified asymptotic form... 

The thickness of the layer of the adsorbed molecules is the characteristic distance scale for fractal surface. (3) Van der Waals attraction forces between solid/gas interactions and the liquid/gas surface tension forces are contributed to the grand potential of the system. 

The characteristic distances for FRET can be reliably calculated from the spectral properties of the donor (D) and acceptor (A). Importantly, FRET can be reliably predicted to occur for any D-Apair, so that the system can be wavelength-adjusted to match the wavelengths of laser diode sources. The extent of FRET depends on the proximity of the donor and acceptor. 

Figure 2.11. Evolution of the experimental characteristic distance X (deduced from the best Ht to the data using Eq. (2.19)) as a function of the polymer coil hydrod5mamic radius Rg (deduced from viscosimetric measurements) for the PVA-Vac polymer. ( ) Air-water interface ( ) Oil-water interface. (Adapted from [31].)...

We note that all the films designed by e,f and g in Fig. 12 are transparent and exhibit no phase structure when examined by TEM, but differ somewhat in the degree of mixing when studied by the fluorescence technique. In fact, it was found that Ic/Ia for film e was sHghtly but definitely larger than those of the others. For the donor-acceptor pair used in this study, the characteristic distance is 2.7-2.9 nm [106], so that composition fluctuation on this scale in the films could be detected by routine TEM observations. The failure to see the phase structure in e, therefore, may be due to a great decrease in composition fluctuation by a strong intercomponent interaction. 

Remember that the solutions of linear box models usually comprise exponential functions in time of the form e ft, where the specific rate constant, k, has the dimension of T 1. As Eq. 24-21 demonstrates, in the case of concentration profiles along a river, the relevant functions are of the form e " , where x is length along the river (see Eq. 24-14). The parameter e, which describes one or several processes in the moving water volume, has dimension L-1. Then xp = s 1 is the characteristic distance of the particular process p or of the sum of all processes. The relative size of different ep values describes the relative importance of different processes occurring along the river. 

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