Transport coefficient, calculation

Bacri J., Gomes A. M., Fieni J. M., Thouzeau F. and Birolleau J. C. (1989) Transport coefficient calculation in air-argon mixtures explana-... 

Since the static memory function affects the sound speed the two kinetic equations do not give the same hydrodynamic limit. On the other hand, it is known that transport coefficients calculated in the two descriptions are the same. ... 

One alternative approach to the calculation of the diffusion and other transport coefficier is via an appropriate autocorrelation function. For example, the diffusion coefficie... 

The rate of extraction depends on the mass transport coefficient (f), the phase contact area (F) and the difference between the equilibrium concentration and the initial concentration of the dissolved component, which is usually expressed as the driving force of the process (a). The rate of extraction (V) can be calculated as shown in Equation (135) ... 

Gianturco F. A., Serna S., Sanna N. Dynamical decoupling in the quantum calculations of transport coefficients. I. Coupled state results for He-N2 gaseous mixture, Mol. Phys. 74, 1071-87 (1991). 

Specific heat of each species is assumed to be the function of temperature by using JANAF [7]. Transport coefficients for the mixture gas such as viscosity, thermal conductivity, and diffusion coefficient are calculated by using the approximation formula based on the kinetic theory of gas [8]. As for the initial condition, a mixture is quiescent and its temperature and pressure are 300 K and 0.1 MPa, respectively. 

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. 

In order to calculate the rates for electron impact collisions and the electron transport coefficients (mobility He and diffusion coefficient De), the EEDF has to be known. This EEDF, f(r, v, t), specifies the number of electrons at position r with velocity v at time t. The evolution in space and time of the EEDF in the presence of an electric field is given by the Boltzmann equation [231] ... 

II. 51 Pa for H2, and 0.55 Pa for Si2H6. With these partial pressures, and the data on electron collisions as given in Table II, first the EEDF can be calculated, followed by the calculation of the electron transport coefficients and electron impact rates. From a comparison of a Maxwellian EEDF and the two-term Boltzmann... 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

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. 

Parameters representing the catalyst bed were taken from Briggs et al. (1977,1978) transport coefficients for heat and mass were calculated from well-established correlations, whereas the kinetic parameters came from measurements on Russian catalysts (Ivanov and Balzhinimaev, 1987) of about the same composition as the commercial catalyst used by Briggs. These parameters are collected in Table VI. The model used by Silveston et al. contained, therefore, no adjustable parameters. 

T. Ihle, E. Tiizel, and D. M. Kroll, Equilibrium calculation of transport coefficients for a fluid-particle model, Phys. Rev. E 72, 046707 (2005). 

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. 

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-... 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

Finally, we attack the problem of the transport coefficients, which, by definition, are calculated in the stationary or quasi-stationary state. The variation of the distribution functions during the time rc is consequently rigorously nil, which allows us to calculate these coefficients from more simple quantities than the generalized Boltzmann operators which we call asymptotic cross-sections or transport operators. 

In general, experiments using transient methods utilize solutions to Eq. (92) (Sect. 4.2.2.3) to obtain so-called experimentally derived diffusion coefficients. The following sections will show briefly the common transient methods of experimentation used to obtain test data for calculations of the transport coefficient. 

Reactions carried in aqueous multiphase catalysis are accompanied by mass transport steps at the L/L- as well as at the G/L-interface followed by chemical reaction, presumably within the bulk of the catalyst phase. Therefore an evaluation of mass transport rates in relation to the reaction rate is an essential task in order to gain a realistic mathematic expression for the overall reaction rate. Since the volume hold-ups of the liquid phases are the same and water exhibits a higher surface tension, it is obvious that the organic and gas phases are dispersed in the aqueous phase. In terms of the film model there are laminar boundary layers on both sides of an interphase where transport of the substrates takes place due to concentration gradients by diffusion. The overall transport coefficient /cl can then be calculated based on the resistances on both sides of the interphase (Eq. 1) ... 

The overall mass transport coefficient for mass transport from the dispersed organic phase into the continuous aqueous phase was then calculated according to the Calderbank equation (Eq. 4) ... 

In an equilibrium system, all times t are equivalent and can thus be averaged over. Transport coefficients can be calculated as integrals over these functions, e.g.. 

F eOH FH20, and Fmgoh) for different solvated acidic polymers are presented in a way that allows some interesting comparisons and the calculation or estimation of the elements of the transport matrix Ljj. In many publications, these transport parameters are reported as a function of the solvent content and are expressed as the number of solvent molecules (i.e., water) per sulfonic acid group. Because of the importance of percolation effects in all considered transport coefficients, we have converted these solvent contents to solvent volume fractions, except for proton conductivities, as shown in Figures 17 and 18. 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

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