Non-Fickian transport

The previous sections discussed the advection-dispersion equation and variants such as the mobile-immobile conceptualization, which are based on the key assumption that mechanical dispersion is Fickian. In other words, the advection-dispersion equation (Eq. 10.5) is strictly valid only under perfectly homogeneous 

To account for the effect of a sufficiently broad, statistical distribution of heterogeneities on the overall transport, we can consider a probabilistic approach that will generate a probability density function in space (5) and time (t), /(i, t), describing key features of the transport. The effects of multiscale heterogeneities on contaminant transport patterns are significant, and consideration only of the mean transport behavior, such as the spatial moments of the concentration distribution, is not sufficient. The continuous time random walk (CTRW) approach is a physically based method that has been advanced recently as an effective means to quantify contaminant transport. The interested reader is referred to a detailed review of this approach (Berkowitz et al. 2006). 

A variety of specific mathematical formulations of the CTRW approach have been considered to date, and network models have also been applied (Bijeljic and Blunt 2006). A key result in development of the CTRW approach is a transport equation that represents a strong generalization of the advection-dispersion equation. As shown by Berkowitz et al. (2006), an extremely broad range of transport patterns can be described with the (ensemble-averaged) equation 

The CTRW approach accounts naturally for transport in preferential pathways, with mass transfer to stagnant and slow flow regions the CTRW can account for these physical transport mechanisms, as well as other factors that influence transport of reactive contaminants, such as sorption. 

Of specific interest here are the analyses by Cortis and Berkowitz (2004) of transport in partially saturated, laboratory columns. Three typical breakthrough curves from a series of miscible displacement experiments in partially saturated 

FIGURE 4.16 Relative mass uptake, of a slab with con- 

FIGURE 4.17 Schematic of the concentration-distance curve used in the calculation of the diffusivity. 

A convenient method of predicting whether the transport of a solvent in an amorphous polymer is FicMan or non-Fickian is to examine the diffusional Deborah number. Deg. This number is defined as the ratio of a characteristic relaxation time for the polymer-solvent system to the characteristic time of the diffusion process (Vrentas et al., 1975). Fickian transport is observed when either Dee 0.1 or Dee 10, whereas non-Fickian transport is observed when Dee = 1. 

Alfrey et al. (1966) proposed the following classification of the diffusional processes  

Case I, or Fickian transport the diffusion time scale is much longer than the relaxation time scale. (Rubbery polymers usually exhibit such behavior.) 

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Therefore, Eq. 10.5 is limited in its applicability, as are variations of this equation such as the mobile-immobile one (see Sect. 10.2). We discuss non-Fickian transport in detail in Sect. 10.3. 

The analysis was limited in part by the scarcity of measurements, and clear discrepancies between measured and calculated values may be observed. As discussed in Chapter 10, tailing effects often are due to non-Fickian transport behavior, which was not accounted for in this model. Interestingly, the field-scale retardation coefficient values of the reactive contaminants were smaller by an order of magnitude than their laboratory values, obtained in an accompanying experiment. 

Berkowitz B, Emmanuel S, Scher H (2008) Non-Fickian transport and multiple rate mass transfer in porous media Water Resour Res 44, D01 10.1029/2007WR005906 Bijeljic B, Blunt MJ (2006) Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 42, W01202, D01 10.1029/2005WR004578 Blunt MJ (2000) An empirical model for three-phase relative permeability. SPE Journal 5 435-445... 

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

By plotting the fractional amount of absorbed or desorbed protein, 00, versus time, the difflisivity can be evaluated from the slope of the initial 60% of the total release, with an accuracy of 1 %. This analysis is valid for systems that behave in a Fickian manner, such as an equilibrium swollen hydrogel with evenly dispersed protein. However, non-Fickian transport phenomena are t q)ically observed for swelling hydrophilic polymers initially in the glassy state, as described by Peppas and Lustig (1985). 

Indeed, entirely different responses can be observed as the time required for relaxation of stresses varies relative to the time required to change the local concentration at a point in the material. The dependence of mass absorbed Aft on time at early time in a sorption run is often used to categorize the type of transport process occurring. A simple dependence on the square root of time is an indication of Fickian transport (recall eq. 3a), whereas dependence a > 0.5 on is indicative of non-Fickian transport and is termed anomalous. 

Pick s first and second laws were developed to describe the diffusion process in polymers. Fickian or case I transport is obtained when the local rate of change in the concentration of a diffusing species is controlled by the rate of diffusion of the penetrant. For most purposes, diffusion in rubbery polymers typically follows Fickian law. This is because these rubbery polymers adjust very rapidly to the presence of a penetrant. Polymer segments in their glassy states are relatively immobile, and do not respond rapidly to changes in their conditions. These glassy polymers often exhibit anomalous or non-Fickian transport. When the anomalies are due to an extremely slow diffusion rate as compared to the rate of polymer relaxation, the non-Fickian behaviour is called case II transport. Case II sorption is characterized by a discontinuous boundary between the outer layers of the polymer that are at sorption equilibrium with the penetrant, and the inner layers which are unrelaxed and unswollen. 

Berens A (1977) Diffusion and relaxation in glassy polymer powders 1. Fiekian diffusion of vinyl chloride in poly(vinyl choride). Polymer 18(7) 697-704 Berens A, Hopfenberg H (1978) Diffusion and relaxation in glassy polymer powders 2. Separation of diffusion and relaxation parameters. Polymer 19(5) 489-496 Bond DA (2005) Moisture diffusion in a fiber-reinforced composite part 1 - non-Fickian transport and the effect of fiber spatial distribution. J Compos Mater 39(23) 2113-2141 Cai LW, Weitsman Y (1994) Non-Fickian moisture diffusion in polymeric composites. J Compos Mater 28(2) 130-154... 

In this section the analogy between heat and mass transfer is introduced and used to solve problems. The specific estimation relationships for permeants in polymers are discussed in Section 4.2 with the emphasis placed on gas-polymer systems. This section provides the necessary formulas for a first approximation of the diffusivity, solubility, and permeability, and their dependence on temperature. Non-Fickian transport, which is frequently present in high activity permeants in glassy polymers, is introduced in Section 4.3. Convective mass transfer coefficients are discussed in Section 4.4, and the analogies between mass and heat transfer are used to solve problems involving convective mass transfer. Finally, in Section 4.5 the solution to Design Problem III is presented. 

In conclusion. Case 11 and non-Fickian transport behaviors are frequently present in glassy polymer systems. Case II transport particularly was found to be associated with sharp penetrant fronts and linear mass uptake with time, whereas in the non-Fickian transport the mass uptake is proportional to f", where < n < 1. Methanol absorbed in PMMA exhibits Case n diffusion characteristics at relatively low temperatures, whereas at higher temperatures a more peculiar behavior is noticed. 

Sarti, G. C. and F. Doghieri. 1994. Non-Fickian Transport of Alkyl Alcohols Through Prestretched PMMA. Chem. Eng. Sci., 49(5), 733-748. 

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