Dual permeability models

The dual continuum model (DCM) formulation is comparable to the dual permeability model (DKM) formulation (TRW Environmental Safety Systems, Inc., 2000). The DCM and DKM conceptualizations provide separate continua for the matrix and the fractures. The dual continua are coupled throughout the model domain by transfer functions for heat and mass transfer between the fractures and matrix. Use of a DCM increases the complexity of the numerical model used in the simulations, but offers the potential to realistically partition flow between matrix and fractures. Mass flow across the matrix/fracture interface is directionally dependent. When liquid pressure in the matrix exceeds the pressures in the fractures (i.e., P > P//), liquid flow, Qi, from the matrix to the fracture continuum is defined by... 

There are clearly opportunities for further research in this area, particularly in relating the homogenisation methods to the upscaling and multiscale methods. The results of [23] that show the need for dual permeability models as effective media for 2-phase flow are particularly interesting. Dual permeability behaviour has been observed in the field, [35], in apparently unfractured reservoirs. 

The ideas involved in dual-porosity and dual-permeability models could be more widely applicable to upscaling than is generally realised. By using models that are designed to characterise behaviour involving two time scales, the need for fine scale models can be reduced. 

The permeability of a polymer to a penetrant depends on the multiplicative contribution of a solubility and a mobility term. These two factors may be functions of local penetrant concentration in the general case as indicated by the dual mode model. Robeson (31) has presented data for CO2 permeation in... 

Nonlinear, pressure-dependent sorption and transport of gases and vapors in glassy polymers have been observed frequently. The effect of pressure on the observable variables, solubility coefficient, permeability coefficient and diffusion timelag, is well documented (1, 2). Previous attempts to explain the pressure-dependent sorption and transport properties in glassy polymers can be classified as concentration-dependent and "dual-mode models. While the former deal mainly with vapor-polymer systems (1) the latter are unique for gas-glassy polymer systems (2). 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

Comparing the curves in Fig. 2 shows that representing the permeability versus pressure data by either model provides a satisfactory fit to the data over the pressure range of 1 to 20 atm. However, at pressures less than 1 atm. the two models differ in their prediction regarding the behavior of the permeability-pressure curve [Fig. 2]. While the matrix model predicts a strong apparent pressure dependence of the permeability in this range (solid line), the dual-mode model predicts only a weak dependence (broken line). 

Figure 2. Permeability of CO2 in conditioned polycarbonate at 35 °C. The experimental data are from Ref. 15. The solid curve is the calculated permeability based on the matrix model and the broken curve is calculated from the dual-mode model. Parameters used in the curves are given in the text.

The dual-mode model proved rather successful to describe the isotherms and permeabilities at higher pressures. 

Based on the recent study of the permeabilities of Kapton polyi-mlde to CO2/CH mixtures (2]L, ), it is expected that for systems which can be described by the generalized dual mode model (17), the permeability ratio of CO2 over CH in a mixture can be approximated to within about 20% by using the respective pure component permeabilities. Consequently, for the present general discussion, pure-component values will be used in Equation 7 in the following sections for discussing this Important system. 

Useful informetion regarding the characteristics of a particular separator aimngement often can be obtained through parametric studies. Separator specifications used in the following calculation are shown in Table 20.6-1 unless stated otherwise. Experimental permeability parameters, according to the dual-mode model, obtained from dense film-pure ges measurements34 36 are summerized in Table 20.6-2. For simplicity, only the results for a binaiy-component feed will he discussed. 

The fit of these expressions to experimental results is very good. At low pressure regimes, the fit was shown to be even better than that of dual sorption expressions. Except for these regimes, the two models seem to do equally well in describing sorption and permeability data. Concentration dependent diffusivity and permeability have been considered before mainly for vapors. The new aspect of the matrix model is that it broadens these effects to fixed gases. The important difference between the matrix and dual sorption models is in the physical picture they convey of gas transport and interaction with the polymer. Additional experimental evidence will be needed to determine the preference of these different physical representations. 

Vogel, T. Gerke, H. H. Zhang, R. van Genuchten, M. T. (2000) Modelling flow and transport in a two-dimensional dual-permeability system with spatially variable hydraulic properties,... 

The following three multicomponent transport models have been used to explain the depression of the permeability of a component in a mixture relative to its pure component value (Fig. 21) the Petropoulos model and the competitive sorption model, both of which assume that direct competition for diflfiisive pathways within the glass is negligible, and a more general permeability model in which direct competition can occur between penetrant molecules for both sorption sites and diffusion pathways. All three of the models presented here are based upon the framework of the dual-mode model. It is worth mentioning that the site-distribution model has recently been extended to accoimt for diffusion (98) and that free volume models exist for transport in glassy polymers (99). 

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 ... 

For better understanding the diverse relaxation behavior of confined polymers, researchers have utilized models or simulation tools to capture the kinetic features of the material at the molecular level, aiming to represent the results observed in experiments. The FVHD model, which has been widely employed in characterizing physical aging in bulk polymers, is reformulated to describe the relaxation behavior of polymers under nanoconfinement. A dual mechanism combines the effect of vacancy diffusion and lattice contraction, and was recently applied with time-dependent internal length scales to characterize the free volume reduction in the aging process [169]. The dual mechanism model (DMM) fits the data of thin film permeability fairly well. The potential predictive capability of the DMM model depends on the accuracy of the relationship between the internal length and time scale on the description of complex material dynamics [161]. 

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