Characteristic-time model

Three models were used to simulate the observed extraction rates for the oily components. Each of employed the effective diffusivity De as an adjustable parameter. They were the Characteristic Time Model (3) two other published models here designated Single Sphere Models I (4,5) and II (6). In these models it is assumed that the solute is extracted from a particulate bed composed of porous... 

Characteristic Time Model This model was developed by Reverchon and Osseo in 1994. As in model I it is assumed that the extraction is uniform along the bed (and in the form used) that the external film resistance can be neglected (see Equations 8 and 9 in reference 3). 

The Characteristic Time Model was always found to give the worst fit and the Single Sphere Model II (which is not shown and which allows for film resistance) was not found to be appreciably better than the computationally simpler Model I for these systems. The values of De obtained were about two orders of magnitude lower than the estimated binary diffusivities. 

It is important to note that this model contains no characteristic time. It thus implies that the power-law parameters are independent of shear rate. Of course such a model cannot describe the low-shear-rate portion of the curve, where the viscosity approaches a constant value. Several empirical equations have been proposed to allow for the transition to Newtonian behavior over a range of shear rates. It was noted in the discussion of the Weissenberg number earlier in this chapter that the variation of 77 with y implies the existence of at least one material property with units of time. The reciprocal of the shear rate at which the extrapolation of the power-law line reaches the value of tiq is such a characteristic time. Models that can describe the approach to tIq thus involve a characteristic time. Examples include the Cross equation [64] and the Carreau equation [65], shown below as Eqs. 10.55 and 10.56 respectively. 

However, although it allowed a correct description of the current-voltage characteristics, this model presents several inconsistencies. The main one concerns the mechanism of trap-free transport. As noted by Wu and Conwell [1191, the MTR model assumes a transport in delocalized levels, which is at variance with the low trap-free mobility found in 6T and DH6T (0.04 cm2 V-1 s l). Next, the estimated concentrations of traps are rather high as compared to the total density of molecules in the materials (see Table 14-4). Finally, recent measurements on single ciystals [15, 80, 81] show that the trap-free mobility of 6T could be at least ten times higher than that given in Table 14-4. 

An even more serious problem concerns the corresponding time scales on the most microscopic level, vibrations of bond lengths and bond angles have characteristic times of approx. rvib 10-13 s somewhat slower are the jumps over the barriers of the torsional potential (Fig. 1.3), which take place with a time constant of typically cj-1 10-11 s. On the semi-microscopic level, the time that a polymer coil needs to equilibrate its configuration is at least a factor of the order larger, where Np is the degree of polymerization, t = cj 1Np. This formula applies for the Rouse model [21,22], i. e., for non-... 

In choosing a model, the user can optimize fate assessment efforts by delineating first, the source release patterns and second, the dominant dynamical processes. Taking the intramedia processes first, one can address model criteria by considering the ratio of characteristic times. The advection time is the principal length scale of the domain L divided by the average flow speed u i.e. 

A number of other spectroscopies provide information that is related to molecular structure, such as coordination symmetry, electronic splitting, and/or the nature and number of chemical functional groups in the species. This information can be used to develop models for the molecular structure of the system under study, and ultimately to determine the forces acting on the atoms in a molecule for any arbitrary displacement of the nuclei. According to the energy of the particles used for excitation (photons, electrons, neutrons, etc.), different parts of a molecule will interact, and different structural information will be obtained. Depending on the relaxation process, each method has a characteristic time scale over which the structural information is averaged. Especially for NMR, the relaxation rate may often be slower than the rate constant of a reaction under study. 

Although the usefulness and performance of continuous and discrete time models strongly depends on the particular problem and solution characteristics, our... 

To more fully appreciate the equilibrium models, like SCRF theories, and their usefulness and limitations for dynamics calculations we must consider three relevant times, the solvent relaxation time, the characteristic time for solute nuclear motion in the absence of coupling to the solvent, and the characteristic time scale of electronic motion. We treat each of these in turn. 

Figure 5.1. Closures for the chemical source term can be understood in terms of their relationship to the joint composition PDF. The simplest methods attempt to represent the joint PDF by its (lower-order) moments. At the next level, the joint PDF is expressed in terms of the product of the conditional joint PDF and the mixture-fraction PDF. The conditional joint PDF can then be approximated by invoking the fast-chemistry or flamelet limits, by modeling the conditional means of the compositions, or by assuming a functional form for the PDF. Similarly, it is also possible to assume a functional form for the joint composition PDF. The best method to employ depends strongly on the functional form of the chemical source term and its characteristic time scales.

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

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