Diffusion, description

Diffusion description of orientational relaxation is only possible when... 

A theoretical analysis of the experimental kinetics for Vk centres in KC1-Tl, as well as for self-trapped holes in a-Al203 and Na-salt of DNA, is presented in [55]. The fitting of theory to the experimental curves is shown in Fig. 4.4. Partial agreement of theory and experiment observed in the particular case of Vk centres was attributed to the violation of the continuous approximation in the diffusion description. This point is discussed in detail below in Section 4.3. Note in conclusion that the fact of the observation of prolonged increase in recombination intensity itself demonstrated slow mobility of defects. In the case of pure irradiated crystals, it is a strong... 

Since typically ro 1 A, and A is restricted in a crystal or liquid by a distance of several A, the condition (4.3.7) can be violated which makes the diffusion description inadequate and an alternative hopping recombination formalism should be used. 

Thus the gas/vapor/liquid-liquid mass transfer is modeled as a combination of the two-film model and the Maxwell-Stefan diffusion description. In this stage model, the equilibrium state exists only at the interface. 

For truly high rejection reverse osmosis membranes, the solution-diffusion description of this process is the most popular and probably the most realistic. In this case, the high osmotic pressure difference between the... 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

W. Nadler and D. L. Stein, Reaction-diffusion description of biological transport processes in general dimension. J. Chem. Phys. 104(5) 1918-1936 (1996). 

The Heaviside step ftinction 0(e+Ae) in Eq.(26) symbolically emphasizes the fact that the FO VP diffusion description is valid when the initial energy level lies above the chaos onset limit s =... 

This model therefore accounts well for the experimental results except at large Q where the theorical shape of the broad component deviates from the experimental one. This deviation mainly comes from the limitation of the diffusive description at large Q (finite jump distance between two successive proton positions). We have also examined the possibility of an anisotropic shape of the restricted volume. We have found that the possible anisotropy is in any case rather small (at most 1.5 to 2)... 

Intuitively, D appears to be well-placed to capture the dynamical signature of the coupling between orientational and translational order. In the energy landscape formalism the time-dependent position rft) of a particle i can be resolved into two components rft) = Rft) + Sft), where Rft) is the spatial position of the particle i in the inherent structure for the basin inhabited at time and S ft) is the intrabasin displacement away from that inherent structure [159], It has been theoretically argued that the replacement of the real positions r (f) by the corresponding inherent structure positions in the Einstein relation yields an equivalent diffusion description [159, 160]. Such a proposition, which has been verified in simulations [159, 160], forms the foundation of the analysis presented here. 

Along these lines, the vapor-Hquid mass transfer is modeled as a combination of the two-film model presentation and the Maxwell-Stefan diffusion description. In this stage model, the equilibrium exists only at the interface. A reasonable simplification for RD is represented by the effective diffusivUy approach, provided that the effective diffusion coefficients are estimated properly. These coefficients can be obtained, for instance, via a relevant averaging of the Maxwell-Stefan diffusivities [42]. 

Figures 10.7 and 10.8 show the liquid-phase compositions for the reboiler and condenser as functions of time. After column start-up, the concentration of methanol decreases continuously whereas the distillate mole fraction of methyl acetate reaches about 90%. A comparison of the rate-based simulation using the Max-well-Stefan diffusion equations (Eq. (10.1)) and experimental results for the liquid-phase composition at the column top and in the reboiler demonstrates their satisfactory agreement. Figure 10.9 shows the simulation results obtained after an operation time of 10000 s with different modeling approaches the model including the Maxwell-Stefan diffusion description, the model with effective diffusion coefficients, and the equilibrium-stage model. Both the Maxwell-Stefan...

The steady-state rate-based model used in this case study was similar to the model from Section 10.4.2, with the pseudohomogeneous mode for the reaction kinetics and the Maxwell-Stefan diffusion description. The model was implemented into ASPEN Custom Modeler together with another, simpler rate-based model, with effective diffusivities. The latter allows starting values for the Maxwell-Stefan-based model to be generated, thus enhancing convergence. 

The diffusion description, elaborated by Christiansen around 1935 (Christiansen 1936) and fully systematised in 1940 by Kramers (Kramers 1940), was an interesting and successful method complementary to transition state theory (TST). It received, however, little or no attention in chemistry circles for a long time (Zambelli 2010). 

Finally, it is appropriate to mention a restriction of above stated approach, which is connected to the fact that the contact layer depth A has been ignored in kinematic calculations (the spots have been considered as flat). This restriction has been overcome recently in the range of diffusive description of particles motion in many-dimensional space, being compounded of translational and rotational coordinates for partners /22/. Boundary condition for such approach permits to take into account the chemical and physical interaction of the active centers on finite distance between them and to vary the spot shape. All above mentioned functional dependences f obtained in the diffusion approach, has been already... 

The simple solution-diffusion description of the steady state transport of a penetrant in a homogeneous polymer matrix indicates that the permeability coefficient, P, is the product of solubility, S, and diffusivity, D, coefficients, that is, a similar form has been assumed to apply for composites comprised of particles. 

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