Diffusional flux density

The diffusional flux density (Eq. 3.35) is the difference between the mean velocities of solute and water. In mass flow (such as that described by Poiseuille s law Eq. 9.11), vs equals vw, so JD is then zero such flow is independent of All and depends only on AP. On the other hand, let us consider All across a membrane that greatly restricts the passage of some solute relative to the movement of water, i.e., a barrier that acts as a differential filter vs is then considerably less than vw, so JD has a nonzero value in response to its conjugate force, All. Thus Jo helps express the tendency of the solute relative to water to diffuse in response to a difference in osmotic pressure. 

LP is the hydraulic conductivity coefficient and can have units of m s-1 Pa-1. It describes the mechanical filtration capacity of a membrane or other barrier namely, when An is zero, LP relates the total volume flux density, Jv, to the hydrostatic pressure difference, AP. When AP is zero, Equation 3.37 indicates that a difference in osmotic pressure leads to a diffusional flow characterized by the coefficient Lo Membranes also generally exhibit a property called ultrafiltration, whereby they offer different resistances to the passage of the solute and water.14 For instance, in the absence of an osmotic pressure difference (An = 0), Equation 3.37 indicates a diffusional flux density equal to LopkP. Based on Equation 3.35, vs is then... 

On the other hand, due to one of the basic principles of nonequilibrium thermodynamics, the diffusional flux density of components 1 and 2 in a binary mixture (solution) is written as (Rehage et al., 1970 Ba2arov, 1976 Gurov, 1978)... 

As the diffusional field strength depends on the coordinate jc in the diffusion layer, the diffusion flux density (in contrast to the total flux density) is no longer constant and the concentration gradients dCjIdx will also change with the coordinate x. 

Gas Phase Diffusion. The gas phase diffusion limitation arises when the diffusional flux of molecules to the surface of the droplet is less than the maximum possible flux of gas across the surface as given by Equation 2. Under these circumstances the gas density near the surface of the droplet (n ) is smaller than average volume density (n). The situation can be simply analyzed by writing the rate equation for the total number of molecules N in the neighborhood of the droplet ... 

For the total mass conservation of a single-phase fluid, / represents the fluid density p. jr represents the diffusional flux of total mass, which is zero. For flow systems without chemical reactions, d> = 0. Therefore, from Eq. (5.12), we have the continuity equation as... 

The relationship between the -> current density (j) and the diffusional flux (/ ) is... 

The fundamental driving force for diffusive transport is the gradient of the chemical potential. At low molecular densities this is proportional to the concentration gradient, C. The equation for the local steady-state diffusional flux, J, is... 

One more algebraic eqnation is required to solve for all unknown molar densities at Zk+i It is not advantageons to write the mass balance at the catalytic surface (i.e., at xatx-i-i) because the no-slip boundary condition at the wall stipnlates that convective transport is identically zero. Hence, one relies on the radiation boundary condition to generate eqnation (23-46). Diffusional flux of reactants toward the catalytic snrface, evalnated at the surface, is written in terms of a backward difference expression for a first-derivative that is second-order correct, via equation (23-40). This is illnstrated below at Xwaii = x x+i for equispaced data ... 

Step 11. Write all the boundary conditions that are required to solve this boundary layer problem. It is important to remember that the rate of reactant transport by concentration difhision toward the catalytic surface is balanced by the rate of disappearance of A via first-order irreversible chemical kinetics (i.e., ksCpJ, where is the reaction velocity constant for the heterogeneous surface-catalyzed reaction. At very small distances from the inlet, the concentration of A is not very different from Cao at z = 0. If the mass transfer equation were written in terms of Ca, then the solution is trivial if the boundary conditions state that the molar density of reactant A is Cao at the inlet, the wall, and far from the wall if z is not too large. However, when the mass transfer equation is written in terms of Jas, the boundary condition at the catalytic surface can be characterized by constant flux at = 0 instead of, simply, constant composition. Furthermore, the constant flux boundary condition at the catalytic surface for small z is different from the values of Jas at the reactor inlet, and far from the wall. Hence, it is advantageous to rewrite the mass transfer equation in terms of diffusional flux away from the catalytic surface, Jas. 

The equations which govern the electrolyte are similar to those which govern the semiconductor with the exceptions that homogeneous reactions can frequently be neglected and that convective transport of ionic species may be important. For dilute electrolytic solutions (less than 3 Af) the flux density of an ionic species can be expressed in terms of migrational, diffusional, and convective components, i.e.,... 

For reaction (1.3.2), the following relationship exists between the current density and the diffusional flux of O ... 

Here pj is the loeal mass density of species i, j is the local diffusional flux of i relative to the mass average veloeity v, and is the loeal rate of production of i per unit volume as the result of one or more chemical reactions. As before, the eorresponding equations for the extrapolated bulk phases must be subtraeted from this equation, various manipulations performed to convert the resulting equation to a statement that an integral over the reference surface S must vanish, and the arbitrary extent of S invoked to argue that the integrand itself must be zero (compare Equations 5.28 through 5.32). 

In the course of a diffiisional experiment, the concentrations of the diffusing components vary in a restricted pliysical space (a diffusional cell), which leads to changes in the density and volume in parts of the diffusional cell with respect to a immovable coordinate system. That is why choosing a location of the reference plane of the section, through which passes a unit of the substance amount, is not a simple matter, and the coordinate system for flux density is fixed in different ways (Erdey-Gruz, 1974 Read et al., 1977 Malkin and Chalykh, 1979), which leads to different diffusion coefficients, according to Equation 31. 

It has already been noted that the flux of material to the rotating disc electrode is uniform over the whole surface, and it is therefore possible to discuss the mass transport processes in a single direction, that perpendicular to the surface (i.e. the z direction). Furthermore, it has been noted that the velocity of movement of the solution towards the surface, is zero at the surface and, close to the surface, proportional to Hence, even in the real situation it is apparent that the importance of convection drops rapidly as the surface is approached. In the Nernst diffusion layer model this trend is exaggerated, and one assumes a boundary layer, thickness 6, wherein the solution is totally stagnant and transport is only by diffusion. On the other hand, outside this layer convection is strong enough for the concentration of all species to be held at their bulk value. This effective concentration profile must, however, lead to the same diffusional flux to the surface (and hence current density) as it found in the real system. 

Diffusion is the process at the molecular level, and it is determined by the random character of the motion of individual molecules. The rate of diffusion is proportional to the average velocity of molecules. It is not obviously possible to track the diffusion process completely in this temporary framework (about several nanoseconds). Therefore, to increase the diffusion rate in accordance with the model of the diffusion-growth of nanowhiskers and Pick s first law (the flux density of matter is proportional to the diffusion coefficient and concentration gradient), an additional diffusional flux to the base of a whisker is used, that is, the force O, whose direction is perpendicular to the z axis, is applied to the deposited silicon atoms. Correspondingly, this force has the nature of intermolecular interaction. 

The cxirrently used description of homogeneous diffusion of volatile by-products in polymer media during reversible polycondensations is due to Secor [44]. It considers polymer molecules immobile. The flux of small molecules has a negligible convective contribution only the diffusional flux with respect to the polymer needs be considered, and the microscopic mass balance of a generic volatile component (usually, but not always, a by-product) Y and a group A belonging to the polymer may be written, neglecting density variations, as in Eq. (27). 

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

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