Particle flux diffusive

In the fluid model the momentum balance is replaced by the drift-diffusion approximation, where the particle flux F consists of a diffusion term (caused by density gradients) and a drift term (caused by the electric field ) ... 

For the radical neutrals, boundary conditions are derived from diffusion theory [237, 238]. One-dimensional particle diffusion is considered in gas close to the surface at which radicals react (Figure 14). The particle fluxes in the two z-directions can be written as... 

In order to be able to explain the observed results plasma modeling was applied. A one-dimensional fluid model was used, which solves the particle balances for both the charged and neutral species, using the drift-diffusion approximation for the particle fluxes, the Poisson equation for the electric field, and the energy balance for the electrons [191] (see also Section 1.4.1). 

Finally, the diffusion of a chemical may be influenced by another diffusing compound or by the solvent. The latter effect is known as solute-solvent interaction it may become important when solute and solvent form an association that diffuses intact (e.g., by hydration). This may be less relevant for neutral organic compounds, but it plays a central role for diffusing ions. But even for noncharged particles the diffusivities of different chemicals may be coupled. The above example of the glycerol diffusing in water makes this evident in order to keep the volume constant, the diffusive fluxes of water and glycerol must be coupled. 

There are two arenas for describing diffusion in materials, macroscopic and microscopic. Theories of macroscopic diffusion provide a framework to understand particle fluxes and concentration profiles in terms of phenomenological coefficients and driving forces. Microscopic diffusion theories provide a framework to understand the physical basis of the phenomenological coefficients in terms of atomic mechanisms and particle jump frequencies. 

To describe quantitatively the diffusion-controlled tunnelling process, let us start from equation (4.1.23). Restricting ourselves to the tunnelling mechanism of defect recombination only (without annihilation), the boundary condition should be imposed on Y(r,t) in equation (4.1.23) at r = 0 meaning no particle flux through the coordinate origin. Another kind of boundary conditions widely used in radiation physics is the so-called radiation boundary condition (which however is not well justified theoretically) [33, 38]. The idea is to solve equation (4.1.23) in the interval r > R with the partial reflection of the particle flux from the sphere of radius R ... 

Nernst-Planck equation — This equation describes the flux of charged particles by diffusion and electrostatic forces. When the ion with charge ze is distributed at concentration c in the potential, cp, it has a one-dimensional flux of the ion, / = -Ddc/dx - (zF/RT) Dcdcp/dx [i]. This can be derived from the concept that the force caused by the gradient of the electrochemical potential is balanced with frictional force by viscosity, t], of the medium. When a spherical ion with radius ro is in the inner potential, cp, the gradient of the electrochemical potential per ion is given by... 

For turbulence it is convenient to describe particle flux in terms of an eddy diffusion coefficient, similar to a molecular diffusion coefficient. Unlike a molecular diffusion coefficient, however, the eddy diffusion coefficient is not constant for a given temperature and particle mobility, but decreases as the eddy approaches a surface. As particles are moved closer and closer to a surface by turbulence, the magnitude of their fluctuations to and from that surface diminishes, finally reaching a point where molecular diffusion predominates. As a result, in turbulent deposition, turbulence establishes a uniform aerosol concentration that extends to somewhere within the viscous sublayer. Then molecular diffusion or particle inertia transports the particles the rest of the way to the surface. 

In equilibrium the net particle flux is zero when drift and diffusion are the two active transport processes, tfift + f[Pg.61]

To get a sense of the physics behind diffusion, diffusion fluxes and diffusion potentials need to be defined in terms of driving forces. Pick s first law is used to correlate the diffusion flux density with the concentration gradient of the diffiisants, which yields the diffusion coefficient as the factor of proportionality. For a system with two components and one-dimensional diffusion in the z-direction, the particle flux ji is related to the gradient of concentration, 5c,/5z, of these particles as... 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

Obviously, this quantity has no relation to the kinetic processes observed in the corresponding nonequilibrium system. For example, if we disturb the homogeneous distribution of particles, the rate ofthe resulting diffusion process is associated with the net particle flux (difference between fluxes in opposing directions) which is zero at equilibrium. 

The particle flux resulting from simultaneous diffusion and migration in an external force held can be obtained by summing the two effecl.s to give... 

Because the dispersion forces are attractive, they tend to increase the rate of particle transport to the surface. When the diffusion path is long compared with the range of operation of the dispersion forces, the attractive effects on diffusion can be neglected (Fig. 2,11). The sink boundary condition is retained, however, and the particle flux can be calculated by solving the diffusion equation in the absence of an external force field with the condition = 0 at a distance dp/2 from the surface. The particle flux is... 

The situation is quite different for particle diffusion. In this case, v/D )s> I and even weak fluctuations in the viscous sublayer contribute significantly to transport. Consider a turbulent pipe flow. In the regions near the wall, the curvature can be neglected and the instantaneous particle flux can be written as follows ... 

The concentration in the bulk of the fluid is approximately equal to [n], because the volume of fluid bounded by S and the wall is small. Outside region S, the particle flux toward the bottom of the chamber Is given by [n]cj, because the concentration gradients and. therefore, diffusion are negligible. 

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