Diffusion mass fraction

In the presence of diffusion (mass fraction Y of reactants) and for a fast chemical reaction at the wall ... 

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

Here solute concentration C and Cp (in permeate) are expressed as mass fractions, D is the diffusion coefficient of the solute and y is the distance from the membrane. Rearranging and integrating from C - Cf when y = / the thickness of the film, to C = Cw, the concentration of solute at the membrane wall, when y=0, gives ... 

In these equations x and y denote independent spatial coordinates T, the temperature Tib, the mass fraction of the species p, the pressure u and v the tangential and the transverse components of the velocity, respectively p, the mass density Wk, the molecular weight of the species W, the mean molecular weight of the mixture R, the universal gas constant A, the thermal conductivity of the mixture Cp, the constant pressure heat capacity of the mixture Cp, the constant pressure heat capacity of the species Wk, the molar rate of production of the k species per unit volume hk, the speciflc enthalpy of the species p the viscosity of the mixture and the diffusion velocity of the A species in the y direction. The free stream tangential and transverse velocities at the edge of the boundaiy layer are given by = ax and Vg = —ay, respectively, where a is the strain rate. The strain rate is a measure of the stretch in the flame due to the imposed flow. The form of the chemical production rates and the diffusion velocities can be found in (7-8). 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

Example 3.2 Consider a combustor using air and methane under steady conditions. A stoichiometric reaction is assumed and diffusion effects are to be neglected. The reactants and products are uniform in properties across their flow streams. Find the mass fraction of carbon dioxide in the product stream leaving the combustor. 

Solid PET feedstock for the SSP process is semicrystalline, and the crystalline fraction increases during the course of the SSP reaction. The crystallinity of the polymer influences the reaction rates, as well as the diffusivity of the low-molecular-weight compounds. The crystallization rate is often described by the Avrami equation for auto-accelerating reactions (1 — Xc) = cxp(—kc/"), with xc being the mass fraction crystallinity, kc the crystallization rate constant and n a function of nucleation growth and type. 

FIGURE 6.11 Characteristic parametric variations of dimensionless temperature T and mass fraction m of fuel, oxygen, and products along a radius of a droplet diffusion flame in a quiescent atmosphere. j is the adiabatic, stoichiometric flame temperature, pA is the partial density of species A, and p is the total mass density. The estimated values derived for benzene are given in Section 2b. 

When one gas diffuses into another, as A into B, even without the quasi-steady-flow component imposed by the burning, the mass transport of a species, say A, is made up of two components—the normal diffusion component and the component related to the bulk movement established by the diffusion process. This mass transport flow has a velocity Aa and the mass of A transported per unit area is pAAa. The bulk velocity established by the diffusive flow is given by Eq. (6.58). The fraction of that flow is Eq. (6.58) multiplied by the mass fraction of A, pA/p. Thus,... 

Defining mA as the mass fraction of A, one obtains the following proper form for the diffusion of species A in terms of mass fraction ... 

Cygan RT, Wright K, Fisler DK, Gale JD, Slater B (2002) Atomistic models of carbonate minerals bulk and surface structures, defects, and diffusion. Mol Simul 28 475-495 Davis AM, Hashimoto A, Clayton RN, Mayeda TK (1990) Isotope mass fractionation during evaporation of Mg2Si04. Nature 347 655-658... 

From the above equations, it could be easily understood that the key properties of the analytes are the hydration radius, the p A, the diffusion coefficient, and the mass fraction. They will play a role in the separation. Another important factor is the /tjff of the ion and the fidi of the EOF in accordance with Equation (8). For /tapp.AT Oj they need to have the same sign or the /teff of the ion must be smaller than the effective mobility of the EOF. 

Methane-air diffusion flames are selected for the example to be studied here. The temperature T and species mass fractions Yi (for species i) in such flames are functions of the mixture fraction Z, which varies from zero in air to unity in fuel and measures the fraction of the material present that came from the fuel. Figure 25.2 is a schematic illustration of major profiles in the methane-air diffusion flame as functions of Z, obtained by the rate-ratio as3miptotics described above. The work to be reported here adds to this picture the chemistry relevant to the production of oxides of nitrogen. 

Figure 25.5 Comparison between steady-state radical levels predicted at Z° and peak radical mass fractions from full numerical calculations. Hollow symbols show results with the detailed mechanism, while the solid symbols provide results using only the skeletal mechanism. Calculations for methane-air diffusion flames at p = 1 bar and fuel and oxidizer stream temperatures of Tf = To = 300 K. A = Yo,B = Yh...

The full treatment of multicomponent diffusion requires a diffusion matrix because the diffusive flux of one component is affected by the concentration gradient of all other components. For an N-component system, there are N-1 independent components (because the concentrations of all components add up to 100% if mass fraction or molar fraction is used). Choose the Nth component as the dependent component and let n = N 1. The diffusive flux of the components can hence be written as (De Groot and Mazur, 1962)... 

The diffusion couple discussed above consists of two halves of the same phase. If the two halves are two minerals, such as Mn-Mg exchange between spinel and garnet (Figure 3-5), there would be both partitioning and diffusion. Define the diffusivity in one half (x < 0) to be D, and in the other half (x > 0) to be D. Both and are constant. Let w be the concentration (mass fraction) of a minor element (such as Mn). The initial condition is... 

Crystal growth Consider the case for crystal growth along one direction (hence a one-dimensional problem). Define the initial interface to be at x = 0 and the crystal is on the side with negative x (left-hand side) and the melt is on the positive side (Section 3.4.6). Due to crystal growth, the interface advances to the positive side. Define the interface position at time t to be at x = Xq, where Xq > 0 is a function of time. Let w be the mass fraction of the main equilibriumdetermining component then the diffusion equation in the melt is... 

One-dimensional diffusive dissolution With the above general discussion, we now turn to the special case of one-dimensional crystal dissolution. Use the interface-fixed reference frame. Let melt be on the right-hand side (x > 0) in the interface-fixed reference frame. Crystal is on the left-hand side (x < 0) in the interface-fixed reference frame. Properties in the crystal will be indicated by superscript "c". For simplicity, the superscript "m" for melt properties will be ignored. Diffusivity in the melt is D. Diffusivity in the crystal is D. The concentration in the melt is C (kg/m ) or w (mass fraction). The initial concentration in the crystal is or simplified as or if there would be no confusion from the context. It is assumed that the interface composition rapidly reaches equilibrium. In the following, diffusion in the melt is first considered, and then diffusion in the crystal. 

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