Solution-diffusion model relationships

Figure 1 Summary of solution-diffusion model relationships. Typically the diffusion coefficient, Dj, decreases as the penetrant molecular weight increases. On the other hand, the solubility coefficient, Sj, tends to increase with increasing penetrant molecular weight. As a result, the permeability, Pj, may either decrease...

In order to solve the MDPE in Equation 9.1, it is necessary to understand how y is related to x. There are numerous methods of describing this relationship. Gas separation involves the diffusion of a gaseous mixture through the membrane material. The rate at which a gas, or particular component of a gas mixture, moves through a membrane is known as the flux (total or individual). In order to obtain a general flux model for a gas separation membrane, the fundamental thermodynamic approach incorporating the solution-diffusion model [11] will be used at vacuum permeate conditions (itp 0) ... 

The permeability process through dense polymers is considered to be a solution-diffusion process. The gas dissolves into the polymer surface and diffuses through the film in the direction of lower activity. The solution-diffusion model is embodied in the relationship... 

Four observation were thought to be in disagreement with the diffusion model (1) the lack of a proportional relationship between the electron scavenging product and the decrease of H2 yield (2) the lack of significant acid effect on the molecular yield of H2 (3) the relative independence from pH of the isotope separation factor for H2 yield and (4) the fact that with certain solutes the scavenging curves for H2 are about the same for neutral and acid solutions. Schwarz s reconciliation follows. 

Finely Porous Model. In this model, solute and solvent permeate the membrane via pores which connect the high pressure and low pressure faces of the membrane. The finely porous model, which combines a viscous flow model eind a friction model (7, ), has been developed in detail and applied to RO data by Jonsson (9-12). The most recent work of Jonsson (12) treated several organic solutes including phenol and octanol, both of which exhibit solute preferential sorption. In his paper, Jonsson compared several models including that developed by Spiegler eind Kedem (13) (which is essentially an irreversible thermodynamics treatment), the finely porous model, the solution-diffusion Imperfection model (14), and a model developed by Pusch (15). Jonsson illustrated that the finely porous model is similar in form to the Spiegler-Kedem relationship. Both models fit the data equally well, although not with total accuracy. The Pusch model has a similar form and proves to be less accurate, while the solution-diffusion imperfection model is even less accurate. 

Another model developed [57] for solute diffusion in osmotic dehydration of apple based on solids gain divided by water content M as a function of rate constant K, time (/), and a constant A was given as M = Kt + A. A relationship was established in the form of A" = T C" where rate parameter K is related to temperature T at different sucrose concentrations C. The average activation energy of the process was 28.2 kj/mol. 

Theoretical breakthrough curves for nonlinear systems may be calculated by numerical solution of the model equations using standard finite difference or collocation methods. Such solutions have been obtained by many authors and a brief summary is given in Table 8.4. In all cases plug flow was assumed and the equilibrium relationship was taken to be of cither Langmuir or Freundlich form. As linearity is approached ( ->1.0) the linearized rate models approach the Anzelius model (Table 8.1, model la) while the diffusion models approach the Rosen model (Table 8.1, model la). The conformity of the numerical solution to the exact analytic solution in the linear limit was confirmed by Garg and Ruthven. ... 

Diffusion models can also result in mathematical solutions approximating a first-order relationship. Ham considered diffusion-limited precipitation from a supersaturated solution upon an array of nuclei. This would fit the case where one element in solid solution in a second begins to diffuse to and precipitate oh dislocations. The state of aggregation of the present system probably does not resemble this situation. 

With this relationship for all samples was calculated from ninh This M is used for evaluating the reaction data. The ultracen rifuge (u.c measurements were carried out in a Spinco model E analytical ultracentrifuge, with 0.4% solutions in 90% formic acid containing 2.3 M KCl. By means of the sedimenta- ion diffusion equilibrium method of Scholte (13) we determine M, M and M. The buoyancy factor (1- vd = -0.086) necessary for tSe calculation of these molecular weights from ultracentrifugation data was measured by means of a PEER DMA/50 digital density meter. 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

This solid solution still makes up the bulk of the solid particles after equilibration in an aqueous solution (59), since solid state diffusion is negligible at room temperature in these apatites (60), which have a melting point around 1500°C. These considerations and controversial results justify a thermodynamic analysis of the solubility data obtained by Moreno et al (58 ). We shall consider below whether the data of Moreno et al (58) is consistent with the required thermodynamic relationships for 1) an ideal solid solution, 2) a regular solid solution, 3) a subregular solid solution and 4) a mixed regular, subregular model for solid solutions. 

The interfacial capacitance increases with the DDTC concentration added. The relationship among potential difference t/ of diffusion layer, the electric charge density q on the surface of an electrode and the concentration c of a solution according to Gouy, Chapman and Stem model theory is as follows. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

For glutamic acid (18) and glycine (10) the yield of ammonia varies approximately as the cube root of the concentration. This variation agrees with the diffusion of the spur model which derives from the hypothesis that at higher solute concentrations, water radicals are scavenged which would react with each other in more dilute solution. However, for the effect of cathode rays on the aromatic amino acids phenylalanine, tryptophan, and tyrosine and for cystine, this relationship is inverted, and amino acid destruction decreased with an increase in concentration (29). 

There have been few studies reported in the literature in the area of multi-component adsorption and desorption rate modeling (1, 2,3., 4,5. These have generally employed simplified modeling approaches, and the model predictions have provided qualitative comparisons to the experimental data. The purpose of this study is to develop a comprehensive model for multi-component adsorption kinetics based on the following mechanistic process (1) film diffusion of each species from the fluid phase to the solid surface (2) adsorption on the surface from the solute mixture and (3) diffusion of the individual solute species into the interior of the particle. The model is general in that diffusion rates in both fluid and solid phases are considered, and no restrictions are made regarding adsorption equilibrium relationships. However, diffusional flows due to solute-solute interactions are assumed to be zero in both fluid and solid phases. 

---