Radial hydrogen diffusion

Fig. 7.33 Radial hydrogen diffusion in yttrium observed visually. Microscope photographs in transmitted light of (a) a 300 nm thick yttrium film on sapphire, covered by a 30nm thick palladium disk of 1.1 mm diameter at various times t after introduction of H2 (10 Pa). Within a few seconds transparent p-YHs-j is formed beneath the Pd, which adsorbs and dissociates hydrogen. The remaining Y surface...

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

The MD simulations show that second shell water molecules exist and are distinct from freely diffusing bulk water. Freed s analytical force-free model can only be applied to water molecules without interacting force relative to the Gd-complex, it should therefore be restricted to water molecules without hydrogen bonds formed. Freed s general model [91,92] allows the calculation of NMRD profiles if the radial distribution function g(r) is known and if the fluctuation of the water-proton - Gd vector can be described by a translational motion. The potential of mean force in Eq. 24 is obtained from U(r) = -kBT In [g(r)] and the spectral density functions have to be calculated numerically [91,97]. 

And again the residual stresses change the symmetry of the task the diffusion process in the hollow cylinder obeys the law of flat symmetry. The hydrogen absorption rate increases from the side of the outer boundary. Acceleration of the process kinetics is caused by reducing the compression stresses in the radial direction, which are changed with the tensile stresses. The solution of tasks (5) and (6) is well known. Thus, it is not difficult to obtain the solution of task (4). This solution describes the hydrogen absorption kinetics from the two surfaces of the hollow cylinder at given character of the residual stresses distribution... 

The first three types (pellets, extrudates and granules) are primarily used in packed bed operations. Usually two factors (the diffusion resistance within the porous structure and the pressure drop over the bed) determine the size and shape of the particles. In packed bed reactors, cooled or heated through the tube wall, radial heat transfer and heat transfer from the wall to the bed becomes important too. For rapid, highly exothermic and endothermic reactions (oxidation and hydrogenation reactions, such as the ox-... 

Very recently, Lavenda devised an interesting method of solution of the Kramers problem in the extreme low-friction limit. He was able to show that it could be reduced to a formal Schrddinger equation for the radial part of the hydrogen atom and thus be solved exactly. One particular form of the long-time behavior of the rigorous rate equation coincides with that obtained by Kramers with the quasi-stationary hypothesis and may thus clarify the implications of this hypothesis. The method of Lavenda is reminiscent of that used by van Kampen but applied to a Smoluchowski equation for the diffusion of the energy. 

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