Transverse diffusion coefficient

Extrapolation, to liquid density, of thermalization time in gaseous water (also approximate) by Christophorou et al. (1975), based on drift velocity and transverse diffusion coefficient measurement, which gives tth 2.0 x 10-1+ s... 

The permeability of solutes across lipid bilayers is a product of the partition coefficient and the transverse diffusion coefficient [30]. Bilayer polymerization can alter solute diffusion by modifying either or both of these processes. In order to examine the effect of polymerization on bilayer permeability a nonionic solute of moderate permeability, [3H-glucose], was encapsulated in the vesicles prior to polymerization, removed from the exterior after polymerization, and its permeation across the bilayer was measured periodically [31]. Quantitative measurements of the 3H-glucose leakage revealed that the formation of linear polymer chains from methacryloyl lipids reduced the permeability coefficient to 0.3 to 0.5 of that of the unpolymerized lipid vesicles. A larger reduction (two orders of magnitude) was only found when crosslinked polymer networks were formed [31]. 

The transverse diffusion coefficient I) can be expressed by the porosity of wood V, the transverse bound water diffusion coefficient Dbt of wood and the vapor diffusion coefficient... 

Here u(f) is the inhibitor and a x, t) is the activator variable. In the semiconductor context u t) denotes the voltage drop across the device and a(x, t) is the electron density in the quantum well. The nonlinear, nonmonotonic function /(a, u) describes the balance of the incoming and outgoing current densities of the quantum well, and D(a) is an effective, electron density dependent transverse diffusion coefficient. The local current density in the device is j a, u) = (/(a, u) + 2a), and J = j jdx is associated with the global current. Eq. (5.22) represents Kirchhoff s law of the circuit (5.3) in which the device is operated. The external bias voltage Uq, the dimensionless load resistance r R, and the time-scale ratio e = RhC/ra (where C is the capacitance of the circuit and Ta is the tunneling time) act as control parameters. The one-dimensional spatial coordinate x corresponds to the direction transverse to the current flow. We consider a system of... 

Other transport coefficients are the transverse diffusion coefficient... 

FIGURE 1.9 Anisotropic ion diffusion in high electric field, E. The diffusion along E (controlled by the longitudinal diffusion coefficient, Dy) is faster than that in the two perpendicular directions (controlled by the transverse diffusion coefficient, Z)j ). The diffusion along any vector r is determined by its projections on the E axis and the orthogonal plane (dashed segments). 

The ion diffusion along the gap will be controlled by the transverse diffusion coefficient (2.2.4) that is normally smaller than Dn but higher than the zero-field isotropic diffusion coefficient, D. Therefore, the resolving power will be somewhat lower for dispersive IMS-ADD than for DT IMS at equal drift voltage and gas... 

In principle, two kinds of diffusion coefficients must be introduced Dj, the longitudinal diffusion coefficient in the direction of the field, and Dj, the transversal diffusion coefficient, perpendicular to the direction of the field. As long as the carriers are in thermal equilibrium with the liquid, Dj = Dj. Differences occur when hot carriers are involved. 

The energy distribution functions in liquid argon and liquid xenon are shown in Figure 16. These results should serve only as illustrative examples. The precision and accuracy of the calculations depend critically on the input data for Aq and A (see Gushchin et al, 1982). Aq can be taken as constant, while Aj is extracted from mobility measurements. The mobility data available in the literature show considerable scatter (see Chapter 3). A direct estimation of the electron mean energy comes from the measurement of the transversal diffusion coefficient. The ratio of diffusion coefficient and mobility is a measure of the mean energy. 

In anisotropic systems, the transverse diffusion coefficient contains a contribution associated to the term which raises the orientational degeneracy. Dj thus remains positive and finite even near threshold contrary to the isotropic case. For instance in the example defined in Eq. (II.12) one finds... 

If the diffusion coefficient of species A is less tlian tliat of B (D < D ) tlie propagating front will be planar. However, if is sufficiently greater than tire planar front will become unstable to transverse perturbations and chaotic front motion will ensue. To understand tire origin of tire mechanism of tire planar front destabilization consider tire following suppose tire interface is slightly non-planar. We would like to know if tire dynamics will tend to eliminate this non-planarity or accentuate it. LetZ)g The situation is depicted schematically in figure... 

Several NMR pulse sequences have been developed for measuring the diffusion coefficient. The most successful and currently used experiment is the modified spin-echo experiment (PGSE) proposed by Stejskal in 1967.62 As the echo signal is due to the transversal components of magnetic moments, the echo attenuation normalized for the attenuation due to relaxation can be written as63 ... 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

Here keg- describes the dynamic process (e.g. the transverse relaxation rate R2, or the diffusion coefficient D) and r describes a time constant typical for the experimental setup. By use of Eq.(l) the kegf can be written as follows ... 

The second equality in equation (6.28) is a definition of longitudinal dispersion coefficient, Dl- Taylor (1953) assumed that some of the terms in equation (6.28) would cancel and that longitudinal convective transport would achieve a balance with transverse diffusive transport. He then solved the second equality in equation (6.28), for a fully developed tubular fiow, resulting in the relation... 

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). 

Here D, D , and Dr are, respectively, the longitudinal, transverse, and rotational diffusion coefficients of the chain averaged over the internal degree of freedom, h an external field, and v and angular velocity of the chain induced by a flow field in the solution. Furthermore, I is the unit tensor and 91 is the rotational operator defined by... 

When the solution is dilute, the three diffusion coefficients in Eq. (40a, b) may be calculated only by taking the intramolecular hydrodynamic interaction into account. In what follows, the diffusion coefficients at infinite dilution are signified by the subscript 0 (i.e, D, 0, D10> and Dr0). As the polymer concentration increases, the intermolecular interaction starts to become important to polymer dynamics. The chain incrossability or topological interaction hinders the translational and rotational motions of chains, and slows down the three diffusion processes. These are usually called the entanglement effect on the rotational and transverse diffusions and the jamming effect on the longitudinal diffusion. In solving Eq. (39), these effects are taken into account by use of effective diffusion coefficients as will be discussed in Sect. 6.3. 

D, collective diffusion coefficient along the transverse direction... 

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