Species Fluxes

This is an explicit solution of the Stefan-Maxwell equations for the diffusion fluxes. The species flux vectors are then given by... 

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

Figure 13-5 is the box model of the remote marine sulfur cycle that results from these assumptions. Many different data sets are displayed (and compared) as follows. Each box shows a measured concentration and an estimated residence time for a particular species. Fluxes adjoining a box are calculated from these two pieces of information using the simple formula, S-M/x. The flux of DMS out of the ocean surface and of nss-SOl back to the ocean surface are also quantities estimated from measurements. These are converted from surface to volume fluxes (i.e., from /ig S/(m h) to ng S/(m h)) by assuming the effective scale height of the atmosphere is 2.5 km (which corresponds to a reasonable thickness of the marine planetary boundary layer, within which most precipitation and sulfur cycling should take place). Finally, other data are used to estimate the factors for partitioning oxidized DMS between the MSA and SO2 boxes, for SO2 between dry deposition and oxidation to sulfate, and for nss-SO4 between wet and dry deposition. 

The gradient of species concentration is in a direction opposite to the thermal gradient, see the green arrows. At places where the front is concave toward the unburnt gas, the species flux is locally divergent. The flux of reactive species into the reactive zone decreases, leading in turn to... 

A number of definitions are needed to set up the discussion of species fluxes. In particular, several different types of velocities are encountered. 

Species fluxes calculated by either the multicomponent (Section 12.7.2) or the mixture-averaged (discussed subsequently in Section 12.7.4) formulations are obtained from the diffusion velocities V, which in turn depend explicitly on the concentration gradients of the species (as well as temperature and pressure gradients). Solving for the fluxes requires calculating either all j-k pairs of multicomponent diffusion coefficients Dy, or for the mixture-averaged diffusion coefficient D m for every species k. 

Unfortunately, resolution of the conservation problem requires knowledge of species flux, and hence details of the specific problem and discretization method. Therefore it is not possible in the general setting of the present discussion to give a universal solution. Nevertheless, a software author and users of a simulation code must be aware of the difficulty, and consider its resolution when setting up the difference approximations to the particular system of conservation equations. 

The problem under consideration is shown schematically in Figure 9.9. The governing equation for this problem was given previously in Equation (9.8), where the species fluxes, are determined from Equation (9.7). Equation (9.7), for i = 1 to J — 1, provides a linear set of independent ordinary differential equations. To solve for all J components, an additional equation is needed. For this we use the conservation of species ... 

A mathematical description of an electrochemical system should take into account species fluxes, material conservation, current flow, electroneutrality, hydrodynamic conditions, and electrode kinetics. While rigorous equations governing the system can frequently be identified, the simultaneous solution of all the equations is not generally feasible. To obtain a solution to the governing equations, we must make a number of approximations. In the previous section we considered the mathematical description of electrode kinetics. In this section we shall assume that the system is mass-transport limited and that electrode kinetics can be ignored. 

Species flux can be described by the Nernst-Planck equation,... 

Here we illustrate the utility of the function defined in the previous section. By "correcting off the counter ion contribution to the mass flux (see the form of equation [3]), neutral species fluxes are highlighted they are the departure of from zero. 

We have the following unknown boundary values the two species nearsurface concentrations Cyo and Cb,o, the two species fluxes, respectively G and G n, the additional capacitive flux Gc, and the potential p, differing (for p > 0) from the nominal, desired potential pnom that was set, for example, in an LSV sweep or a potential step experiment. Five of the six required equations are common to all types of experiments, but the sixth (here, the first one given below) depends on the reaction. That might be a reversible reaction, in which case a form of the Nernst equation must he invoked, or a quasi-reversible reaction, in which case the Butler-Volmer equation is used (see Chap. 6 for these). Let us now assume an LSV sweep, the case of most interest in this context. The unknowns are all written as future values with apostrophes, because they must, in what follows below, be distinguished from their present counterparts, all known. 

The column vector of species fluxes along a pore segment is then... 

We mentioned earlier that the mass diffusion equation is analogous to the heat diffusion (conduction) equation, and thus wc need comparable boundary conditions to determine the species concentration distribution in a medium. Two common types of boundary conditions are the (1) specified species concentration, which corresponds to specified temperature, and (2) specified species flux, which corresponds to specified heat flux. 

Using Pick s law, Ihe constant species flux boundary condition for a diffusing species A at a boundary at. v = 0 is expressed, in the absence of any blowing or suction, as... 

Second, it is assumed that the recycling mass flux and its ancillary trace species fluxes are at any time dependent on the amount of crust in existence at that time. This dependence can take the form of various functions, and a variable geometry parameter E is introduced, which describes the crust to mantle recycling flux (t)recycimg function of crust mass in existence. 

Cosmic rays are particles that arrive in Earth coming from the Sun, our Galaxy or other galaxies. Primary species fluxes give a constraint in source type and composition and primary acceleration mechanisms. AMS-02 will precisely determine individual element fluxes with 1 < Z < 26 in the energy... 

Here, c is the concentration and j is the flux of the species in question due to the diffusion process. The species flux vector j plays the same role in the diffusion problem as the heat flux vector q plays in the transfer of heat by conduction. In the classical diffusion problem,... 

Figure 13-5 is the box model of the remote marine sulfur cycle that results from these assumptions. Many different data sets are displayed (and compared) as follows. Each box shows a measured concentration and an estimated residence time for a particular species. Fluxes adjoining a box are calculated from these two pieces of information using the simple formula, S = Mix. The flux of DMS out of the ocean surface and of nss-SOj back to the ocean surface are also quantities estimated from measurements. These are converted from surface to... 

The analysis of mass transfer in electrochemical cells requires the use of equations that describe the condition of electroneutrality (which applies for the entire elecnolyte outside the double layer at an electrode), species fluxes, mass conservation, current density, and fluid hydrodynamics. Often, mass transport events are rate limiting, as compared to kinetics processes at the electrode surface, in which case the overall electrode reaction rate is solely dependent on species mass transfer (e.g., during high-rate electroplating of some metals and for those elecnochemical reactions where the concentration of reactant in solution is low). 

Sea River discharge, kmVyr N species Fluxes, tons/yr... 

Fig. 29. The effect of reactor aspect ratio on species flux uniformity along the wafer radius.

To infer a dry deposition rate from an eddy correlation measurement, a nondivergent vertical species flux should exist. Nondivergence essentially stipulates that quasi-one-dimensional transport exists. The nondivergence assumption is, in fact, equivalent to the constant-flux-layer assumption of the surface layer in practical terms, nondivergence is best satisfied in relatively flat topography for which a substantial fetch over the terrain exists. 

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