Effective molecular diffusion coefficient

In an aquifer, the total Fickian transport coefficient of a chemical is the sum of the dispersion coefficient and the effective molecular diffusion coefficient. For use in the groundwater regime, the molecular diffusion coefficient of a chemical in free water must be corrected to account for tortuosity and porosity. Commonly, the free-water molecular diffusion coefficient is divided by an estimate of tortuosity (sometimes taken as the square root of two) and multiplied by porosity to estimate an effective molecular diffusion coefficient in groundwater. Millington (1959) and Millington and Quirk (1961) provide a review of several approaches to the estimation of effective molecular diffusion coefficients in porous media. Note that mixing by molecular diffusion of chemicals dissolved in pore waters always occurs, even if mechanical dispersion becomes zero as a consequence of no seepage velocity. 

Equation [3-17] does not hold at very low seepage velocities because mechanical dispersion no longer dominates Fickian mass transport. When the mechanical dispersion coefficient becomes less than the effective molecular diffusion coefficient, the longer travel times associated with lower velocities do not result in further decreases in Fickian mass transport. 

In a sand having a median grain size of 1 mm and porosity of 0.25, how high must specific discharge be to make the mechanical dispersion coefficient equal to the effective molecular diffusion coefficient ... 

Dispersion The movement of aggregates of molecules under the influence of a gradient of concentration, temperature, etc. The effect is represented by Pick s law with a dispersion coefficient substituted for molecular diffusion coefficient. The rate of transfer = -Dg(dC/3z). 

Previously (e.g.. Ref. 344), it has been noted that Eq. (26) will still be valid if the point concentration variable is replaced by the average concentration however, the diffusion coefficient was fonnd to differ from the molecular diffusion coefficient obtained in the pnre flnid. This diffusion coefficient was termed the effective diffusion coefficient. The... 

Tybsre m is the total mass of analyte collected, D the molecular diffusion coefficient, A the area of the diffusion channel, L the diffusion path length, C the analyte concentration in the air, and Tt the sampling time. In deriving equation (8.7) it was assumed t. that the sorbent is effective sink for the analyte and,... 

NMR signals are highly sensitive to the unusual behavior of pore fluids because of the characteristic effect of pore confinement on surface adsorption and molecular motion. Increased surface adsorption leads to modifications of the spin-lattice (T,) and spin-spin (T2) relaxation times, enhances NMR signal intensities and produces distinct chemical shifts for gaseous versus adsorbed phases [17-22]. Changes in molecular motions due to molecular collision frequencies and altered adsorbate residence times again modify the relaxation times [26], and also result in a time-dependence of the NMR measured molecular diffusion coefficient [26-27]. 

Fig. 8. Graph illustrating the effect of particle size (left scale) and column length (right scale) on the pressure drop (vertical scale) required for obtaining SOOQ plates in S min with soliiies h. iving different molecular diffusion coefficients. (1) f> 3 x I0 cm /sec (2)...

Use of the proper diffusion coefficient Replace the molecular diffusion coefficient by the effective diffusion coefficient of fluid in the porous structure. Representative values for gases and liquids in porous solids are given by Weisz (1959). 

Turbulent flow means that, superimposed on the large-scale flow field (e.g., the Gulf Stream), we find random velocity components along the flow (longitudinal turbulence) as well as perpendicular to the flow (transversal turbulence). The effect of the turbulent velocity component on the transport of a dissolved substance can be described by an expression which has the same form as Fick s first law (Eq. 18-6), where the molecular diffusion coefficient is replaced by the so-called turbulent or eddy diffusion coefficient, E. For instance, for transport along the x-axis ... 

Diw is the molecular diffusion coefficient of the chemical in water, x is tortuosity, and aL is the (longitudinal) dispersivity (dimension L). The first term describes molecular diffusion in a porous medium (Eq. 18-57), the second the effect of dispersion (Eq. 22-52). Typical values of the dispersivity aL for field systems with flow distances of up to about 100 m lie between 1 and 100 m. Since aL depends strongly on the scale... 

It is not unreasonable to use the left-hand side of this equation as the definition of the effective diffusion constant K, the more so as it will be shown that any distribution tends to normality. With this definition K is the sum of the molecular diffusion coefficient, D, and the apparent diffusion coefficient k = oP-U2I 48D, which was discovered by Taylor in his first paper (Taylor 1953, equation (25)). Equation (26), however, is true without any restriction on the value of p, or on the distribution of solute. The constant 1/48 is a function of the profile of flow, and for so-called piston flow with x — 0 this constant is zero and K = D as it should. 

We write the exact solution of the diffusion in a velocity field with a single Fourier-component. The expression for the effective diffusion contains the molecular diffusion coefficient as a factor this ensures correct behavior of the result with respect to time reversal. 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

Implicit in this model is the assumption that molecular diffusivity and Henry s Law constant are directly and inversely proportional, respectively, to the gas flux across the atmosphere-water interface. Molecular diffusion coefficients typically range from 1 x 10-5 to 4 x 10-5 cm2 s-1 and typically increase with temperature and decreasing molecular weight (table 5.3). Other factors such as thickness of the thin layer and wind also have important effects on gas flux. For example, wind creates shear that results in a decrease in the thickness of the thin layer. The sea surface microlayer has been shown to consist of films 50-100 pm in thickness (Libes, 1992). Other work has referred to this layer as the mass boundary layer (MBL) where a similar range of film thicknesses has been... 

The overall process can be affected by pore diffusion and external mass transfer. Molecular diffusion coefficients DPB may be calculated by Aspen Plus. Effective pore diffusion may be estimated by the relation DP = Dpb( j,/tp) = 0.1 DPE, in which ep is the particle porosity and rp the tortuosity. Furthermore, the Thiele modulus and internal effectiveness can be calculated as ... 

For ideal gases the effective binary diffusion coefficient can be calculated from molecular properties (see Appendix A). The film thickness, 5, is determined by hydrodynamics. Correlations are given in the literature which allow the calculation of the transfer coefficient in the case of equimolar counterdiffusion, kf, rather than the film thickness, 5 ... 

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