Diffusion equation concentration-dependent diffusivities

For noncoustaut diffusivity, a numerical solution of the conseiwa-tion equations is generally required. In molecular sieve zeohtes, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is ... 

This equation has the same form of that obtained for solid diffusion control with D,j replaced by the equivalent concentration-dependent diffusivity = pDpj/[ pn]Ki l - /i,//i)) ]. Numerical results for the case of adsorption on an initially clean particle are given in Fig. 16-18 for different values of A = = 1 - R. The upt e curves become... 

Salt flux across a membrane is due to effects coupled to water transport, usually negligible, and diffusion across the membrane. Eq. (22-60) describes the basic diffusion equation for solute passage. It is independent of pressure, so as AP — AH 0, rejection 0. This important factor is due to the kinetic nature of the separation. Salt passage through the membrane is concentration dependent. Water passage is dependent on P — H. Therefore, when the membrane is operating near the osmotic pressure of the feed, the salt passage is not diluted by much permeate water. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

When applied to a volume-fixed frame of reference (i.e., laboratory coordinates) with ordinary concentration units (e.g., g/cm3), these equations are applicable only to nonswelling systems. The diffusion coefficient obtained for the swelling system is the polymer-solvent mutual diffusion coefficient in a volume-fixed reference frame, Dv. Also, the single diffusion coefficient extracted from this analysis will be some average of concentration-dependent values if the diffusion coefficient is not constant. 

It can be seen from the table that, in dilute solutions, the diffuse layer may extend some hundreds of angstroms out from the electrode. In contrast, in more concentrated solutions, i.e. 0.1 M, the diffuse layer thickness decreases to < 10 A not much more than the thickness of the Helmholtz layer. As CH has no concentration dependence it remains constant on changing the concentration however, from equations (2.22) and (2,23), CGC decreases as the concentration of the electrolyte increases. Thus, at low concentration ... 

In order to improve the predictive power of the Fickian diffusion theory, a concentration dependent diffusion coefficient is used in Eqs. (15) and (16). Equation (16) is then rewritten and solved with the appropriate boundary conditions ... 

This equation has the same form as that obtained for solid diffusion control with D replaced by the equivalent concentration-dependent diffusivity Da = - nf/nf)2]. Numerical results for the... 

Hydrolysis. NMR results show that TBT carboxylates undergo fast chemical exchange. Even the interfacial reaction between TBT carboxylates and chloride is shown to be extremely fast. The hydrolysis is thus not likely to be a rate determining step. Since the diffusivity of water in the matrix is expected to be much greater than that of TBTO, a hydrolytic equilibrium between the tributyltin carboxylate polymer and TBTO will always exist. As the mobile species produced diffuses out, the hydrolysis proceeds at a concentration-dependent rate. Godbee and Joy have developed a model to describe a similar situation in predicting the leacha-bility of radionuclides from cementitious grouts (15). Based on their equation, the rate of release of tin from the surface is ... 

The development of a scientific understanding of diffusion in liquid-phase polymeric systems has been largely due to Duda et al. (1982), Ju et al. (1981), and Vrentas and Duda (1977a,b, 1979) whose work in this area has been signal. In their most recent work, Duda et al. (1982) have developed a theory which successfiilly predicts the strong dependence of the diffusion coefficient on temperature and concentration in polymeric solutions. The parameters in this theory are relatively easy to obtain, and in view of its predictive capability this theory would seem to be most appropriate for incorporating concentration-dependent diffusion coefficients in the diffusion equation. 

Mn Mgi )2Si04 olivine mixture obtained by Morioka (1981). The concentration profiles are symmetric, indicating that the diffusivities are independent of the concentration of the diffusing ion. On the contrary, possible asymmetry in the diffusion profiles indicate that concentration depends significantly on the diffusing cations. In this case, the interdiffusion coefficient can be obtained by the Boltzmann-Matano equation ... 

In Equation (4.31) the rate constant is either the reaction rate constant or the transport rate constant, depending on which rate controls the dissolution process. If the reaction rate controls the dissolution process, then k. t becomes the rate of the reaction while if the dissolution process is controlled by the diffusion rate, then k j becomes the diffusion coefficient (diffusivity) divided by the thickness of the diffusion layer. It is interesting to note that both dissolution processes result in the same form of expression. From this equation the dependence on the solubility can be seen. The closer the bulk concentration is to the saturation solubility the slower the dissolution rate will become. Therefore, if the compound has a low solubility in the dissolution medium, the rate of dissolution will be measurably slower than if the compound has a high solubility in the same medium. 

In addition to the similarity between the heat conduction equation and the diffusion equation, erosion is often described by an equation similar to the diffusion equation (Culling, 1960 Roering et al., 1999 Zhang, 2005a). Flow in a porous medium (Darcy s law) often leads to an equation (Turcotte and Schubert, 1982) similar to the diffusion equation with a concentration-dependent diffu-sivity. Hence, these problems can be treated similarly as mass transfer problems. 

The square root of time dependence also holds for concentration-dependent diffusivity and the D value in the above equation would be a kind of average D across the profile. 

If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

For three-dimensional diffusion, if there is spherical symmetry (i.e., concentration depends only on radius), the diffusion equation can be transformed to a one-dimensional type by redefining the concentration variable w = rC. This transformation would work for a solid finite sphere, a spherical shell, an infinite sphere with a spherical hole in the center, or an infinite sphere. 

Many diffusion problems cannot be solved anal3dically, such as concentration-dependent D, complicated initial and boundary conditions, and irregular boundary shape. In these cases, numerical methods can be used to solve the diffusion equation (Press et al., 1992). There are many different numerical algorithms to solve a diffusion equation. This section gives a very brief introduction to the finite difference method. In this method, the differentials are replaced by the finite differences ... 

Derive the explicit numerical algorithm for solving a diffusion equation for concentration-dependent D. 

The theoretical lines in Figure 2 are calculated assuming constant values of D0 with the derivative d In p/d In c calculated from the best fitting theoretical equilibrium isotherm (Equation 8). The theoretical lines give an adequate representation of the experimental data suggesting that the concentration dependence of the diffusivity is caused by the nonlinearity of the relationship between sorbate activity and concentration as defined by the equilibrium isotherm. The diffusivity data for other hydrocarbons showed similar trends, and in no case was there evidence of a concentration-dependent mobility. Similar observations have been reported by Barrer and Davies for diffusion in H-chabazite (7). 

Alternatively, coupled diffusion equations with concentration-dependent diffusivities and comparison with experimental results can be solved with numerical methods (see Kirkaldy and Young [1] and Glicksman [4]). 

If the diffusion process is coupled with other influences (chemical reactions, adsorption at an interface, convection in solution, etc.), additional concentration dependences will be added to the right side of Equation 2.11, often making it analytically insoluble. In such cases it is profitable to retreat to the finite difference representation and model the experiment on a digital computer. Modeling of this type, when done properly, is not unlike carrying out the experiment itself (provided that the discretization error is equal to or smaller than the accessible experimental error). The method is known as digital simulation, and the result obtained is the finite difference solution. This approach is described in more detail in Chapter 20. 

The purpose of this paper is to (1) document published studies of vaporization in vacuum of the Group 4 and 5 transition metal carbides and uranium carbide, and determine the temperature dependence of their equilibrium CVCs (C/MeCVC) and vaporization rates (Vecvc, g/cmzs) (2) document published studies on chemical diffusion of these carbides, and develop and compare data to a model describing the concentration dependence of the chemical diffusion coefficients and (3) develop diffusion-coupled vaporization equations which predict changes of surface composition (Cs, units of C/M and g/cm3 of C), average composition (Cavg units of C/M and g/cm3 of C), and C mass loss (M, units of g/cm2 of C). 

Modeling diffusion-coupled vaporization processes associated with non-stoichiometric carbides requires the use of the chemical diffusion coefficient, D, for the calculation of temporal C concentrations. Clearly D will have a strong concentration dependence. In principle, the concentration dependence should be calculable from measured D (NC,T) and ac(Nc,T) values. However, our attempts and those of Wakelkamp31 to verify the correlation have been unsuccessful for TiC, ZrC, VC, and TaC. The disparity is probably based in the approximations used to derive these equations. For example, Howard and Lidiard32 have shown that the right hand sides of equations 3.10 and 3.11 are approximate and proposed that additional terms are needed. 

Our concentration dependent diffusion coefficient model equation 3.21 gives an average diffusion coefficient, Davg, which is constant over the concentration range of the diffusion process. In a similar manner we derive the following equation for aavg ... 

---