Parabolic systems diffusion modeling

The results strongly depend on the crystallographic orientation of the substrate and on the crystal imperfection density. The time-dependence of 3D Me-S bulk alloy formation obeys a parabolic rate law (Fig. 3.65) as found for many other systems. The results were discussed in terms of a semiinfinite-linear diffusion model assuming mutual diffusion of Me and S and reversible 2D Meads overlayer formation. The following time-dependence of q(,AE,i) was derived... 

This stochastic model of the flow with multiple velocity states cannot be solved with a parabolic model where the diffusion of species cannot depend on the species concentration as has been frequently reported in experimental studies. Indeed, for these more complicated situations, we need a much more complete model for which the evolution of flow inside of system accepts a dependency not only on the actual process state. So we must have a stochastic process with more complex relationships between the elementary states of the investigated process. This is the stochastic model of motion with complete connections. This stochastic model can be explained through the following example we need to design some flowing liquid trajectories inside a regular porous structure as is shown in Fig. 4.33. The porous structure is initially filled with a fluid, which is non-miscible with a second fluid, itself in contact with one surface of the porous body. At the... 

Finally, Kvaalen et al have shown that the system of equations of the ideal model for a multicomponent system (see later, Eqs. 8.1a and 8.1b) is strictly h5q3er-bolic [13]. As a consequence, the solution includes two individual band profiles which are both eluted in a finite time, beyond the column dead time, to = L/u. The finite time that is required for complete elution of the sample in the ideal model is a consequence of the assumption that there is no axial dispersion. It contrasts with the infinitely long time required for complete elution in the linear model. This difference illustrates the disparity between the hyperbolic properties of the system of equations of the ideal model of chromatography and the parabolic properties of the diffusion equation. 

The solution flow is normally maintained under laminar conditions and the velocity profile across the channel is therefore parabolic with a maximum velocity occurring at the channel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately determinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffusion equation for mass transport within the rectangular duct may be described by... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

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