Transport effective

Figure 12. Comparison between cup average conversion predicted by axisymmetric model with effective transport properties and experimentally measured values for styrene [5] and vinyl acetate [2],...

Many investigators have studied diffusion in systems composed of a stationary porous solid phase and a continuous fluid phase in which the solute diffuses. The effective transport coefficients in porous media have often been estimated using the following expression ... 

FIG. 27 Effective transport coefficient versus porosity from the model of Trinh et al. (Reproduced with permission from Ref. 399.)... 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

Analytical solutions for the closure problem in particular unit cells made of two concentric circles have been developed by Chang [68,69] and extended by Hadden et al. [145], In order to use the solution of the potential equation in the determination of the effective transport parameters for the species continuity equation, the deviations of the potential in the unit cell, defined by... 

Determination of the effective transport coefficients, i.e., dispersion coefficient and electrophoretic mobility, as functions of the geometry of the unit cell requires an analogous averaging of the species continuity equation. Locke [215] showed that for this case the closure problem is given by the following local problems ... 

Effective transport coefficients for unit cell given in Figure 29. (Reprinted with permis-from Ref. 215, Copyright 1998, American Chemical Society.)... 

For the case of a flowing fluid in the a phase, Locke [215] showed that the effective transport equation is given by... 

Akaimi, KA Evans, JW Abramson, IS, Effective Transport Coefficients in Heterogeneous Media, Chemical Engineering Science 42, 1945, 1987. 

Chandrasekhar, S, Liquid Crystals, 2nd ed. Cambridge University Press Cambridge, 1992. Chang, H-C, Multi-Scale Analysis of Effective Transport in Periodic Heterogeneous Media, Chemical Engineering Communications 15, 83, 1982. 

The substrate concentrations may be too low for effective transport into the cells. 

A graphic example of the signihcance of effective transport is provided by an aerobic reductase from Xenophilus azovorans. This was expressed in Escherichia coli, and was able to reduce a range of important sulfonated colorants, even though whole cells were unable to do so (Bliimel et al. 2002). 

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

The EMA method is similar to the volume-averaging technique in the sense that an effective transport coefficient is determined. However, it is less empirical and more general, an assessment that will become clear in a moment. Taking mass diffusion as an example, the fundamental equation to solve is... 

As a second method to determine effective transport coefficients in porous media, the position-space renormalization group method will be briefly discussed. 

It must be pointed out that in a diffusion layer where the ions are transported not only by migration but also by diffusion, the effective transport numbers t of the ions (the ratios between partial currents ij and total current t) will differ from the parameter tj [defined by Eq. (1.13)], which is the transport number of ion j in the bulk electrolyte, where concentration gradients and diffusional transport of substances are absent. In fact, in our case the effective transport number of the reacting ions in the diffusion layer is unity and that of the nonreacting ions is zero. 

Mixtures of aqueous emulsions of oil can be more effectively transported through pipelines if certain antifreeze formulations are added to the system. Stable oil-in-water emulsions for pipeline transmission by using 0.05% to 4% ethoxylated alkylphenol as an emulgator and a freezing-point depressant for water enable pipeline transmission at temperatures below the freezing point of water [736]. 

Equations A12-A14), which represent the ratio of the effective transport time with the time taken for concentration of the isotope to decrease by half. For 1, while... 

Sediment may be added by bulk mixing via imbricate thnisting (Bebout and Barton 2002), dehydration (Class et al. 2000), or melting (Johnson and Plank 1999). The latter two may differ in their P-T conditions and, therefore, residual mineralogy as well as relevant partition coefficients. In general, fluids are less effective transport agents than melts (i.e., trace elements are more soluble in melt than in pure water or even brine), but fluid/solid partitioning can fractionate some elements, notably Ba-Th and U-Th, more than melt/solid. However, as pressure increases, the distinction between fluid and melt decreases as their mutual solubility increases and they approach a critical end-point. 

Unstirred Water Layer Effect (Transport across Barriers in Series and in Parallel)... 

Another group of surveys has focused on the direct modeling of some effective transport phenomena which are essential for predicting parameters that have an important role in underground gas sequestration process such as diffusivity and convection. Azin et al., in 2013, have conducted study regarding correct measurement of diffusivity coefficient [114]. 

In many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

From this illustration we can see that the added detail of the radial temperature profile near the wall that could be provided by CFD simulations does not help in obtaining better estimates for the standard heat transfer parameters. It also implies that experimental efforts to measure temperatures closer to the wall are, in fact, counter-productive. Finally, it is clear that the standard model with plug flow and constant effective transport parameters does not fit satisfactorily to temperature profiles in low-Abeds. These considerations have led us to look for improved approaches to near-wall heat transfer. 

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