Multispecies diffusion

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

If a diffusion component is present as two or more different species, the diffusion of the component is often referred to as multispecies diffusion (Zhang et al., 1991a,b). Multispecies diffusion is distinguished from multicomponent diffusion in that in the former case, the multiple species are from one component. 

Self-diffusion and tracer diffusion are described by Equation 3-10 in one dimension, and Equation 3-8 in three dimensions. For interdiffusion, because D may vary along a diffusion profile, the applicable diffusion equation is Equation 3-9 in one dimension, or Equation 3-7 in three dimensions. The descriptions of multispecies diffusion, multicomponent diffusion, and diffusion in anisotropic systems are briefly outlined below and are discussed in more detail later. 

In a silicate melt or aqueous solution, a component may be present in several species. The species may interconvert and diffuse simultaneously. For example, the H2O component in silicate melt can be present as at least two species, molecular H2O (referred to as H20m) and hydroxyl groups (referred to as OH) (Stolper, 1982a). The diffusion of such a multispecies component is referred to as multispecies diffusion (Zhang et al., 1991a,b). Starting from Equation 3-5d, the one-dimensional diffusion equation for this multispecies component can be written as... 

There are two methods to write the diffusion equation for a multispecies component. One is to write the diffusion equation for the conserved component, and then relate the species concentrations by the reaction(s). Using one-dimensional H2O diffusion as an example, the diffusion equation is Equation 3-22a ... 

Equation 3-80b would not be applicable and the third equation would involve the reaction and diffusion of a species, which is similar to the second way to mathematically describe the diffusion of a multispecies component. 

The second way to write the diffusion equations for a multispecies component is to write the diffusion-reaction equation for each species. Starting from Equation 3-5b, the diffusion-reaction equation for each species is... 

The total concentration (w) of a multispecies component is independent of species interconversions of the t)q)e of Reaction 3-81, but is affected by the diffusion flux of individual species. Because each species may have a distinct dif-fusivity, the diffusion equation for w may be written as... 

In summary, the diffusion behavior of both H2O and CO2 demonstrates the importance of understanding the role of speciation in diffusion, and the very different consequences due to that role. Diffusion of a single-species component (such as Ar) usually does not depend on its own concentration (when the concentration is low), but depends on the melt composition. For a multispecies component, speciation affects the diffusion behavior. For H2O, speciation makes the diffusion behavior very complicated even at low H2O concentrations, total H2O diffusivity still depends on H2O content (because the species concentrations are not proportional), in addition to the dependence on melt composition. If species concentrations are proportional to each other and hence to the total concentration of the component, then the diffusivity is independent of the concentration of the component, as in the case of CO2 diffusion. Many multispecies components probably satisfy this condition that the concentrations of... 

Figure 4. Total time required to compute the diffusion fluxes for grids of 10, 20, 40, and 80 points. These times are obtained using a vectorized multispecies...

W.W. Jones and J.P. Boris, "A MultiSpecies Diffusion Algorithm," submitted to Comp and Chem. (1980). 

Molecular Derivation of the Model. The molecular theory of surface complexes is a special case of a multispecies lattice model. The surface complexes thus are interpreted as molecular species immobilized (relative to the 10-ps time scale for diffusive motion of an ion in aqueous solution) on an array of M sites that represent surface functional groups. If there are two different surface species on the sites (e.g., a protonated and an unprotonated surface hydroxyl group) and if each site has z nearest neighbors, then any distribution of the two species (call them A and B) over the sites must satisfy the general conditions (23)... 

Liu, C., Shang, J. Zachara, J. M. (2011). Multispecies diffusion models A study of uranyl species diffusion. Water Resources Research, 47. 

If amperometric sensors work with constant U, the response time can be short and they can be used in real-time measurements. On the other hand, for a multispecies analysis system, it is necessary to draw the entire 1(U) curve (or to use a data acquisition system) to locate the different diffusion plateaus. Such sensors, like coulometric sensors, are more adapted for... 

Stefan-Maxwell Approach A more rigorous approach to multispecies diffusion effects is known as the Stefan-Maxwell diffusion model and should be used for higher order models seeking the greatest accuracy. The Stefan-Maxwell equation for multicomponent diffusion flux in the x direction is shown as... 

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