Transport coefficients, effect

V.P. Karpov and E.S. Severin 1980, Effects of molecular transport coefficient on the rate of turbulent combustion, Fizika Goreniya I Vzryva 16(1) 45-51, translated by Plenum Publishing Corporation. 

The Chemkin package deals with problems that can be stated in terms of equation of state, thermodynamic properties, and chemical kinetics, but it does not consider the effects of fluid transport. Once fluid transport is introduced it is usually necessary to model diffusive fluxes of mass, momentum, and energy, which requires knowledge of transport coefficients such as viscosity, thermal conductivity, species diffusion coefficients, and thermal diffusion coefficients. Therefore, in a software package analogous to Chemkin, we provide the capabilities for evaluating these coefficients. ... 

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]. 

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 ... 

It can be noted that in general this result predicts that the ratio of the dispersion coefficient to the free-solution diffusion coefficient is different from the ratio of the effective mobility to the free-solution mobility. In the case of gel electrophoresis, where it is expected that the (3 phase is impermeable (i.e., the gel fibers), the medium is isotropic, and the a phase is the space between fibers, the transport coefficients reduce to... 

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

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

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... 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

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

Dimensionless numbers (Reynolds number = udip/jj., Nusselt number = hd/K, Schmidt number = c, oA, etc.) are the measures of similarity. Many correlations between them (known also as scale-up correlations) have been established. The correlations are used for calculations of effective (mass- and heat-) transport coefficients, interfacial areas, power consumption, etc. 

Requirements regarding laboratory liquid-liquid reactors are very similar to those for gas-liquid reactors. To interpret laboratory data properly, knowledge of the interfacial area, mass-transfer coefficients, effect of contaminants on mass-transport processes, ionic characteristics of the system, etc. is needed. Commonly used liquid-liquid reactors have been discussed by Doraiswamy and Sharma (1984). 

Methods that compensate for nonequilibrium effects in the situation of E-parametrized coefficients are very complicated, and are sometimes not firmly grounded. Because the electron temperature also gives reasonable results without correction methods, the rate and transport coefficients were implemented as a function of the electron energy, as obtained from the PIC calculations presented in Figure 25. 

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

The presence of a transporter can be assessed by comparing basolateral-to-apical with apical-to-basolateral transport of substrates in polarized cell monolayers. If P-gp is present, then basolateral-to-apical transport is enhanced and apical-to baso-lateral transport is reduced. Transport experiments are in general performed with radioactively labeled compounds. Several studies have been performed with Caco-2 cell lines (e.g. Ref. [85]). Since Caco-2 cells express a number of different transporters, the effects measured are most probably specific for the ensemble of transporters rather than for P-gp alone. P-gp-specific transport has been assayed across confluent cell layers formed by polarized kidney epithelial cells transfected with the MDR1 gene [86], Figure 20.11 shows experimental data obtained with these cell lines. A rank order for transport called substrate quality was determined for a number of compounds [86]. The substrate quality is a qualitative estimate, but nevertheless allows an investigation of the role of the air/water (or lipid/water) partition coefficient, log Kaw, for transport as seen in Fig. 20.11(A). For most of the compounds, a linear correlation is observed between substrate quality and log Kaw- However, four compounds are not transported at all despite their distinct lipophilicity. A plot of the substrate quality as a function of the potential of a... 

The transport of heat between latitude bands is assumed to be diffusive and is proportional to the temperature difference divided by the distance between the midpoints of each latitude band. This is the temperature gradient. In this simulation all these distances are equal, so the distance need not appear explicitly. The temperature gradient is multiplied by a transport coefficient here called diffc, the effective diffusion coefficient. The product of the diffusion coefficient and the temperature gradient gives the energy flux between latitude zones. To find the total energy transport, we must multiply by the length of the boundary between the latitude zones. In... 

Other parameters of the simulation are specified in subroutine SPECS. The quantity solcon is the solar constant, available here for tuning within observational limits of uncertainty. The quantity diffc is the heat transport coefficient, a freely tunable parameter. The quantity odhc is the depth in the ocean to which the seasonal temperature variation penetrates. In this annual average simulation, it simply controls how rapidly the temperature relaxes into a steady-state value. In the seasonal calculations carried out later in this chapter it controls the amplitude of the seasonal oscillation of temperature. The quantity hcrat is the amount by which ocean heat capacity is divided to get the much smaller effective heat capacity of the land. The quantity hcconst converts the heat exchange depth of the ocean into the appropriate units for calculations in terms of watts per square meter. The quantity secpy is the number of seconds in a year. 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

The diffusion model and the hydraulic permeation model differ decisively in their predictions of water content profiles and critical current densities. The origin of this discrepancy is the difference in the functions D (T) and /Cp (T). This point was illustrated in Eikerling et al., where both flux terms occurring in Equation (6.46) were converted into flux terms with gradients in water content (i.e., VA) as the driving force and effective transport coefficients for diffusion, A), and hydraulic permeation,... 

As expected, the overall reaction rate increases with increasing catalyst volume rate. The effect can be explained in terms of the dependency of the mass transport coefficient ll on the Re munber (Eq. 12). Due to the increase of the volume hold-up of the aqueous phase, the residence time of the organic phase decreases, so that the observed conversion degrees do not change within the limits of the investigated regime. 

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