Dispersion coefficients statistical” models

Although the above method can give a simple evaluation of Peclet number for the system, the tailing in the RTD curve can cause significant inaccuracy in the evaluation of the Peclet number. Michell and Furzer67 suggested that a better estimation of the axial dispersion coefficient is obtained if the observed RTD is statistically fitted to the exact solution of the axial dispersion model equation with appropriate boundary conditions. For example, a time-domain solution to the partial differential equation describing the dispersion model, i.e.,... 

It can be seen from Table 2-3 that while the ADE requires an increasing dis-persivity with depth to simulate the data correctly, the FADE simulates the data with a dispersion coefficient and an order of the fractional derivative (a 1.62) that are not statistically different for all depths. Besides, although both the ADE and the FADE capture the shapes of breakthrough curves, inspection of the graphs shows that the FADE simulates the breakthrough curves in unsaturated sand better. Statistics of the model performance shown in Table 2-3 support this observation. The root-mean-square errors are markedly lower for the FADE than for the ADE. 

Note that D, the dispersion coefficient, is not the molecular diffusivity, but a measure of combined dispersive effects inherent in packed bed operations, of which molecular diffusivity is a minor component. As pointed out by Kramers and Alberda (24), eddy diffusivity involves fluctuations of a statistical nature, and should not be applied to macroscopic effects, such as by-passing and mixing. This equation is important because it allows the modeling of chromatographic results using the dispersion coefficient as a free parameter. 

In Section 3.4a we examine a model for the second virial coefficient that is based on the concept of the excluded volume of the solute particles. A solute-solute interaction arising from the spatial extension of particles is the premise of this model. Therefore the potential exists for learning something about this extension (i.e., particle dimension) for systems for which the model is applicable. In Section 3.4b we consider a model that considers the second virial coefficient in terms of solute-solvent interaction. This approach offers a quantitative measure of such interactions through B. In both instances we only outline the pertinent statistical thermodynamics a somewhat fuller development of these ideas is given in Flory (1953). Finally, we should note that some of the ideas of this section are going to reappear in Chapter 13 in our discussions of polymer-induced forces in colloidal dispersions and of coagulation or steric stabilization (Sections 13.6 and 13.7). 

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

Further on, Kunz et al could show that this failure of the simplified dispersion model is not a consequence of the weakness of the Poisson-Boitzmann equation. More elaborate statistical mechanics, using the so-called hypernetted chain equation (HNC), yielded basically the same result. Obviously the problem comes from the neglect of ion-water interactions and their changes near the surface. To introduce such interactions in primitive model calculations, Bostrom et al [see also Refs. 13(b)-13(d)] used Jungwirth s water profile perpendicular to the surface as a basis to model a distance-dependent electrostatic function, instead of a static dielectric constant. Such ideas were used several times over the years, for instance to model activity coefficients of electrolyte solutions. ... 

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