Diffusion coefficients ordinary molecular

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

The species diffusivity, varies in different subregions of a PEFC depending on the specific physical phase of component k. In flow channels and porous electrodes, species k exists in the gaseous phase and thus the diffusion coefficient corresponds with that in gas, whereas species k is dissolved in the membrane phase within the catalyst layers and the membrane and thus assumes the value corresponding to dissolved species, usually a few orders of magnitude lower than that in gas. The diffusive transport in gas can be described by molecular diffusion and Knudsen diffusion. The latter mechanism occurs when the pore size becomes comparable to the mean free path of gas, so that molecule-to-wall collision takes place instead of molecule-to-molecule collision in ordinary diffusion. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows... 

This equation is by no means easy to solve in a straightforward manner. By use of suitable approximations, Vandershce et al. [42] derived two expressions relating two of the most important parameters of the FIA curve (travel time and peak width) to the essential characteristics of an ordinary FIA system (namely the reactor length, /, and radius, t, and the flow-rate, cf), together with the molecular diffusion coefficient ... 

Ordinary diffusion can cause significant isotope fractionations. In general, light isotopes are more mobile and hence diffusion can lead to a separation of light from heavy isotopes. For gases, the ratio of diffusion coefficients is equivalent to the inverse square root of their masses. Consider the isotopic molecules of carbon in CO2 with masses and C 0 0 having molecular weights of 44 and 45. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

Here the are the gas-phase mole fractions, the K are the gas-phase mass fractions, W is the mean molecular weight of the gaseous mixture, Dkj is the ordinary multicomponent diffusion coefficient matrix, and the Dj are the thermal diffusion coefficients. 

Fick s law is derived only for a binary mixture and then accounts for the interaction only between two species (the solvent and the solute). When the concentration of one species is much higher than the others (dilute mixture), Fick s law can still describe the molecular diffusion if the binary diffusion coefficient is replaced with an appropriate diffusion coefficient describing the diffusion of species i in the gas mixture (ordinary and, eventually, Knudsen, see below). However, the concentration of the different species may be such that all the species in the solution interact each other. When the Maxwell-Stefan expression is used, the diffusion of... 

For a multicomponent flow the ordinary or molecular diffusion coefficient for each species can be calculated on the basis of its binaiy coefficient with each of the others [60] ... 

Generally, this vector contains three components, which correspond to the mechanisms characterizing the behavior of the property carriers during their movement. The molecular, convective and turbulent moving mechanisms can together contribute to the vector flux formation [3.6]. In the relation below (3.12), Dj- is the ordinary diffusion coefficient of the property. Dj-, represents the diffusion coefficient of the turbulences and w is the velocity flow vector, then the general relation of the transport flux of the property is ... 

SRT/ nMif = Vi is the mean molecular speed Dij is the ordinary diffusion coefficient M, is the molecular weight of component i... 

First, retention does not yield Dj directly, but rather the Soret coefficient, which is the ratio of to the ordinary diffusion coefficient (T>). Because compositional information is contained in alone, an independent measure of D must be available. Second, a general model for the dependence of on composition has not been established therefore, the dependence must be determined empirically for each polymer-solvent system. Fortunately, Dj is independent of molecular weight, and for certain copolymers, the dependence of Dj on chemical composition has been established. With random copolymers, for example, Dj is a weighted average of the Dj values for the corresponding homopolymers, where the weighting factors are the mole fractions of each component in the copolymers [9]. As a result, the composition of random copolymers can be determined by combining thermal FFF with any technique that measures D. 

When treating diffusion of solutes in porous materials where diffusion is considered to occur only in the fluid inside the pores, it is common to refer to an effective diffusivity, DABeg, which is based on (1) the total cross-sectional area of the porous solid rather than the cross-sectional area of the pore and (2) on a straight path, rather than the actual pore path, which is usually quite tortuous. In a binary system, if pore diffusion occurs only by ordinary molecular diffusion, Fick s law can be used with an effective diffusivity that can be expressed in terms of the ordinary diffusion coefficient, DAB, as... 

The kinetic theory of dilute gases accounts for collisions between spherical molecules in the presence of an intermolecular potential. Ordinary molecular diffusion coefficients depend linearly on the average kinetic speed of the molecules and the mean free path of the gas. The mean free path is a measure of the average distance traveled by gas molecules between collisions. When the pore diameter is much larger than the mean free path, collisions with other gas molecules are most probable and ordinary molecular diffusion provides the dominant resistance to mass transfer. Within this context, ordinary molecular diffusion coefficients for binary gas mixtures are predicted, with units of cm /s, via the Chapman-Enskog equation (see Bird et al., 2002, p. 526) ... 

The following strategy should be used to calculate the interpellet axial dispersion coefficient and the mass transfer Peclet number in packed catalytic tubular reactors (see Dullien, 1992, Chap. 6). Initially, one should calculate a simplified mass transfer Peclet number (i.e., Pesimpie) based on the equivalent diameter of the catalytic pellets, equivalent, the average interstitial fluid velocity through the packed bed, (Uj>intetstitiai, and the ordinary molecular diffusion coefficient of reactant A, a, ordinary-... 

The Stokes-Einstein equation for liquid-phase ordinary molecular diffusion coefficients in binary mixtures suggests that the product of Hab and the solvent viscosity /u-b should scale linearly with temperature T. Cite references (i.e., equations) from the literature and evaluate the product of Hab and /xb in terms of its scaling-law dependence on temperature for low-density gases. In other words ... 

The three-halves power of dimensionless temperature in the expression for eA( ) is based on the temperature dependence of gas-phase ordinary molecular diffusion coefficients when the catalytic pores are larger than 1 p.m. In this pore-size regime, Knudsen diffusional resistance is negligible. The temperature dependence of the collision integral for ordinary molecular diffusion, illustrated in Bird et al. (2002, pp. 526, 866), has not been included in ea) ). The thermal energy balance given by equation (27-28), which includes conduction and interdiffu-sional fluxes, is written in dimensionless form with the aid of one additional parameter,... 

The temperature dependence of the effective intrapellet diffusion coefficient conforms to the assumption that ordinary molecular diffusion provides the dominant resistance to mass transfer in the pores, relative to Knudsen diffusion. This is valid when the pore diameter is larger than 1 tim. Gas-phase diffusivities are approximately proportional to the three-halves power of absolute temperature. Hence,... 

The interphase mass transfer coefficient of reactant A (i.e., a,mtc), in the gas-phase boundary layer external to porous solid pellets, scales as Sc for flow adjacent to high-shear no-slip interfaces, where the Schmidt number (i.e., Sc) is based on ordinary molecular diffusion. In the creeping flow regime, / a,mtc is calculated from the following Sherwood number correlation for interphase mass transfer around solid spheres (see equation 11-121 and Table 12-1) ... 

We now turn to the influence of the ordinary diffusion coefficient D in polymer separation. We noted earlier that D is related to the hydrodynamic radius of a polymer molecule by the Stokes-Einstein equation. Consequently, parameter D reflects the dimensions of the polymer chain or, for a given polymer class, the molecular mass. Thus we have a situation in which the two parameters controlling retention in thermal FFF, D and Dj, depend upon two entirely distinct polymer characteristics D depends upon the physical dimensions of the polymer chain and thus on its molecular mass while Dj depends only upon the chemical composition of the polymeric material. 

Example The diffusion coefficient in a liquid benzene/cyclohexane mixture (Dmoi) is 1.4 X 10 m s (300 K), whereas the thermal coefficient Dthermai is about 5 X 10 m s (Zhang, 2006). For steady state, the fluxes (in the z-direction) by thermal and ordinary molecular diffusion are equal ... 

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