Diffusion coefficient molecular origins

Here the vector rj represents the centre of mass position, and D is usually averaged over several time origins to to improve statistics. Values for D can be resolved parallel and perpendicular to the director to give two components (D//, Dj ), and actual values are summarised for a range of studies in Table 3 of [45]. Most studies have found diffusion coefficients in the 10 m s range with the ratio D///Dj between 1.59 and 3.73 for calamitic liquid crystals. Yakovenko and co-workers have carried out a detailed study of the reorientational motion in the molecule PCH5 [101]. Their results show that conformational molecular flexibility plays an important role in the dynamics of the molecule. They also show that cage models can be used to fit the reorientational correlation functions of the molecule. 

The sub-grid-scale turbulent Schmidt number has a value of Scsgs 0.4 (Pitsch and Steiner 2000), and controls the magnitude of the SGS turbulent diffusion. Note that due to the filtering process, the filtered scalar field will be considerably smoother than the original field. For high-Schmidt-number scalars, the molecular diffusion coefficient (T) will be much smaller than the SGS diffusivity, and can thus usually be neglected. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

As already said, Taylor s effective model contains a contribution in the effective diffusion coefficient, which is proportional to the square of the transversal Peclet number. Frequently this term is more important than the original molecular diffusion. After his work, it is called Taylor s dispersion coefficient and it is generally accepted and used in chemical engineering numerical simulations. For the practical applications we refer to the classical paper (Rubin, 1983) by Rubin. The mathematical study of the models from Rubin (1983) was undertaken in Friedman and Knabner (1992). 

The Molecular Origins of Mass Diffusivity. In a manner directly analogous to the derivations of Eq. (4.6) for viscosity and Eq. (4.34) for thermal conductivity, the diffusion coefficient, or mass diffusivity, D, in units of m /s, can be derived from the kinetic theory of gases for rigid-sphere molecules. By means of summary, we present all three expressions for transport coefficients here to further illustrate their similarities. 

In complete accord with a simple numerical evaluation, / — Pmaxr/D0 and the magnitude of DQ decreases the turbulent diffusion is independent of the molecular diffusion coefficient. Let us carefully consider the structure of the quantity r = vX. It is obvious that in a turbulent flow we cannot directly determine a quantity which is linear in the fluctuation velocity. It is no accident that the square of the velocity figured in the original equation. It is precisely the mean square of the velocity and its spectral representation that may be determined in a turbulent flow. Therefore, consistently performing all the calculations, we obtain... 

This equation allows one to consider the cumulative distribution of small-intestinal transit time data with respect to the fraction of dose entering the colon as a function of time. In this context, this equation characterizes well the small-intestinal transit data [173, 174], while the optimum value for the dispersion coefficient D was found to be equal to 0.78 cm2 s 1. This value is much greater than the classical order of magnitude 10 5 cm2 s 1 for molecular diffusion coefficients since it originates from Taylor dispersion due to the difference of the axial velocity at the center of the tube compared with the tube walls, as depicted in Figure 6.5. 

The diffusion coefficient is a physical property that represents the speed of molecular diffusion. Recently it was shown that D changes during chemical reactions. Here, we describe the origin of the change in D. Intuitively, it may be easily understood that D decreases when molecular size increases because of the association reaction. In some cases, the relationship between D and the molecular size is well described by the Stokes-Einstein equation. The Stokes-Einstein equation is expressed by [9-13]... 

The results imply that the diffusion coefficient represents the thermally activated transport of electrons through the particle network. Indeed, these and subsequent studies have been interpreted with models that involve trapping of conduction band electrons or electron hopping between trap sites [158, 159]. An unexpected feature of the diffusion constants reported by Cao et al. is that they are dependent on the incident irradiance. The photocurrent rise times display a power law dependence on light intensity with a slope of -0.7. The data could be simulated if the diffusion constant was assumed to be second order in the electron concentration, D oc n. The molecular origin of this behavior is not well understood and continues to be an active area of study [157, 159]. 

Dissolving 0 to 2x 10-4 mole/liter of KMnC>4 in water decreases the diffusion coefficient by 25%. A very large change in the diffusion coefficient is observed in solutions of methylen-blue (the molecular weight m = 317) the presence of 6 x 10-4 mole/liter of this substance decreases the diffusion coefficient to half the original value at room temperature. 

In this regime the typical distance from the origin of motion increases as the square root of time. Thus, the dispersion in turbulent flows at long times is analogous to molecular diffusion or random walks with independent increments and comparison of Eq. (2.24) with (2.16) relates the turbulent diffusion coefficient, Dt, to the integral of the Lagrangian correlation function, Tl, as... 

Bilous and Counas [B17] have used an equation derived originally for wetted-wall gas-absorption towers to evaluate Ejif as a function of the molar velocities C and H. The basic assumption is that the actual gas flow pattern behaves as if there were a stagnant film of thickness t adjacent to the tube wall, through udiich light component is transported by molecular diffusion, with diffusion coefficient D. As will be shown later, E j in this model is given by... 

The first part of this equation originates from intermolecular collisions (molecular diffusion) and the second from collisions between the molecules and the pore walls (e.g. Knudsen diffusion). Equation (9.338) is valid, strictly speaking, only for equimolar diffusion in a binary solution (Na =-Nb). In porous media, the diffusion coefficient calculated from (9.338) should be corrected for porosity and tortuosity according to eq. (9.103). 

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