Transport equation, dual mode model

One can easily show that the appropriate equation derived from the dual mode sorption and transport models for the steady state permeability of a pure component in a glassy polymer is given by Eq (7) (18) when the downstream receiving pressure is effectively zero and the upstream driving pressure is p ... 

The experimental evidence presented here and in the literature (15) show that the real diffusion coefficient depends on concentration. These results are incompatible with the notion of concentration-independent diffusion coefficients for the dissolved and Langmuir sorbed molecules [D and Djj in equation (15)] as proposed by the dual-mode sorption and transport model ( 13). 

Fredrickson and Helfand incorporated coupling terms in the transport equation and showed that "these terms provide mobility to molecules that are absorbed into microvoids, even if the molecules have an intrinsic diffusion coefficient in the hole phase that vanishes." These investigators proposed experiments that could provide information on the size and topological connectivity of microcavities (51). Related studies have also been made by Chern, Koros, et al. (52) and by Barrer (53). However, Petropoulos (54) has expressed the opinion that the treatment of Fredrickson and Helfand (51 ) as well as that of Barrer (53) "introduces more diffusion parameters than can reasonably be expected to be measurable on the basis of past experience". Petropoulos also showed that these treatments impose certain limitations on the physical meaning of the diffusion parameters. In the same study, Petropoulos has examined the modification of his dual-mode sorption model ( 5) which are necessary if the Langmuir domains are sufficiently extensive to constitute a macroscopically recognizable phase rather than scattered individual sites (or microcavities) (54). ... 

Composition-Dependent Permeability. There are some experimental data 8"19 indicatiug the existence of competition between penetrants in accordance with the postulations of the generalized dual-mode transport model [Eq. (20.6-40)]. It is of interest to examine the magnitude or this competition effect on the overall performance of an actual separator. This is illustrated in Tables 20.6-5 and 20.6-6. Equation... 

Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. Satisfactory agreement is obtained between integral diffusion coefficient from sorption and desorption rate curves and apparent diffusion coefficient from permeation rate curves (time-lag method). These rate curves were also compared to the curves predicted by Fickian-type diffusion equations. 

When the temperature is lower than the critical temperature 7 of gas the solubility of the permeant gas in the membrane may become so high that the diffusion constant no longer remains constant. A modification was done in the above dual transport model with partial immobilization of the Langmuir sorption mode to include the concentration dependency of the diffusion coefficient [153]. Instead of Equation 5.204 the following equation expresses the permeability coefficient ... 

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