Diffusion equations binary

The type of treatment described here was originally introduced by Scott and Dullien [4], who confined attention to isothermal isobaric diffusion in binary mixtures. Similar equations were independently published shortly after by Rothfeld [5], and the method was later extended to multi-component mixtures by Silveston [6], Perhaps the most complete exposition is given by Mason and Evans [7],... 

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

This formulation assumes that the continuum diffusion equation is valid up to a distance a > a, which accounts for the presence of a boundary layer in the vicinity of the catalytic particle where the continuum description no longer applies. The rate constant ky characterizes the reactive process in the boundary layer. If it approximated by binary reactive collisions of A with the catalytic sphere, it is given by kqf = pRGc(8nkBT/m)1 2, where pR is the probability of reaction on collision. 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

When one refers to the diffusion equation, it is usually the binary diffusion equation. Although theories for multicomponent diffusion have been extensively developed, experimental studies of multicomponent diffusion are limited because of instrumental analytical error and theoretical complexity, and there are yet no reliable diffusivity matrix data for practical applications in geology. Multicomponent diffusion is hence often treated as effective binary diffusion by treating the component under consideration as one component and combining all the other components as the second component. 

The one-dimensional binary diffusion equation with constant diffusion coefficient is (Equation 3-10)... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

Therefore, in the transformed components, the diffusion is decoupled, meaning that the diffusion of one component is independent of the diffusion of other components. The equation for each w, can be obtained given initial and boundary conditions using the solutions for binary diffusion. The final solution for C is C = Tw. When the diffusivity matrix is not constant, the diffusion equation for a multicomponent system can only be solved numerically. 

Simple component exchange between solid phases is accomplished by diffusion. If only two components (such as Fe " and Mg) are exchanging, the diffusion is binary. The boundary condition is often such that the exchange coefficient between the surfaces of two phases is constant at constant temperature and pressure. The concentrations of the components on the adjacent surfaces may be constant assuming interface equilibrium. The solution to the diffusion equation... 

The second law of thermodynamics dictates that ct is positive when Vp2 0 therefore, I>0. Hence, based on Equation Al-15, diffusion in binary solutions is always down the chemical potential (or activity) gradient. Comparing J2 in Equation Al-14 with Pick s law and assuming constant molar density p, we have... 

In this section it is shown how the multicomponent equations just discussed can be applied to the discussion of binary systems. First a summary is given of the various notations used in discussing two-component systems. Then some important special results are given for the diffusion and thermal diffusion in binary systems. These equations are used as the starting point for the discussions in Sec. IV. 

In this chapter we shall treat some particular instances of the system (3.1.15) and the related phenomena. Thus in 3.2, we shall concentrate upon binary ion-exchange and discuss the relevant single nonlinear diffusion equation. It will be seen that in a certain range of parameters this equation reduces to the porous medium equation with diffusivity proportional to concentration. Furthermore, it turns out that in another parameter range the binary ion-exchange is described by the fast diffusion equation with diffusivity inversely proportional to concentration. It will be shown that in the latter case some monotonic travelling concentration waves may arise. 

If the diffusivities are functions of concentration, the Boltzmann-Matano method, described in Section 4.3 for the binary case, can be employed if the initial and boundary conditions are appropriate. The diffusion equations are... 

P 21] The mixing of gaseous methanol and oxygen was simulated. The equations applied for the calculation were based on the Navier-Stokes (pressure and velocity) and the species convection-diffusion equation [57]. As the diffusivity value for the binary gas mixture 2.8 x 10 m2 s 1 was taken. The flow was laminar in all cases adiabatic conditions were applied at the domain boundaries. Compressibility and slip effects were taken into account The inlet temperature was set to 400 K. The total number of cells was —17 000 in all cases. 

Mass Transport in Binary Mixtures and the Diffusion Equation, 60... 

MASS TRANSPORT IN BINARY MIXTURES AND THE DIFFUSION EQUATION... 

For the solution of a salt composed of two ionizable species (binary electrolyte), the four basic equations can be combined to yield the convective diffusion equation for steady-state systems ... 

We have found that in most pratical cases, only three terms (n = 2) are required. This usually reduces the error to less than 5%, which makes the numerical results as accurate as the physical quantities (D ) used in computing the diffusion fluxes, and in the original derivation of this form of the diffusion equation (5,7). In addition, for vector lengths of ten (10) or larger and for five diffusion flux term (n v ) is as fast as doing the usual differencing for the binary approximation (V-D Vp ). 

The Stokes-Einstein equation can also be used to estimate the diffusivity of binary liquid mixtures... 

The ordinary diffusion equations have been presented for the case of a gas in absence of porous medium. However, in a porous medium, whose pores are all wide compared to the mean free path and provided the total pressure gradient is negligible, it is assumed that the fluxes will still satisfy the relationships of Stefan-Maxwell, since intermolecular collisions still dominate over molecule-wall collisions [19]. In the case of diffusion in porous media, the binary diffusivities are usually replaced by effective diffusion coefficients, to yield... 

Simplifications in the energy and diffusion equations for unimolecular reactions in binary mixtures... 

Diffusion equations often are written differently from those given in Section E.2.1 [6]. In particular, multicomponent diffusion coefficients differing from D j often are introduced so that diffusion velocities may be expressed directly as linear combinations of gradients. The multicomponent diffusion coefficients are defined so that they reduce to D for binary mixtures [6]. Use of diffusion equations involving multicomponent diffusion coefficients is being made increasingly frequently. 

We will still only deal with binary mixtures in the following, as the diffusion coefficients DAK of mixtures of more than two components are often unknown, and therefore the diffusion in these mixtures cannot be quantitatively calculated. The diffusion equations (2.328) and (2.338) for binary mixtures can often be simplified considerably. 

In order to analyze multicomponent diffusion processes we must be able to solve the continuity equations (Eq. 1.3.9) together with constitutive equations for the diffusion process and the appropriate boundary conditions. A great many problems involving diffusion in binary mixtures have been solved. These solutions may be found in standard textbooks, as well as in specialized books, such as those by Crank (1975) and Carslaw and Jaeger (1959). 

Some assumptions regarding the constancy of certain parameters are usually in order to facilitate the solution of the diffusion equations. For the binary diffusion problems discussed in Chapters 5 (as well as later in Chapters 8-10), we assume the binary Fick diffusion coefficient can be taken to be a constant. In the applications of the linearized theory presented in the same chapters, we assume the matrix of multicomponent Fick diffusion coefficients to be constant. If, on the other hand, we use Eq. 6.2.1 to model the diffusion process then we must usually assume constancy of the effective diffusion coefficient if... 

For a binary system, under conditions of small mass transfer fluxes, the unsteady-state diffusion equations may be solved to give the fractional approach to equilibrium F defined by (see Clift et al., 1978)... 

This relation is referred to as the Maxwell-Stefan model equations, since Maxwell [65] [67] was the first to derive diffusion equations in a form analogous to (2.302) for dilute binary gas mixtures using kinetic theory arguments (i.e., Maxwell s seminal idea was that concentration gradients result from the friction between the molecules of different species, hence the proportionality coefficients, Csk, were interpreted as inverse friction or drag coefficients), and Stefan [92] [93] extended the approach to ternary dilute gas systems. It is emphasized that the original model equations were valid for ordinary diffusion only and did not include thermal, pressure, and forced diffusion. 

Wilke [103] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Pick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

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