Fickian diffusion model, sorption

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

The kinetic reactions occurring in the sorption of Ni, Cd, and Zn on goethite during a period of 2 hours to 42 days at pH 6 were hypothesized to occur via a three-step mechanism using a Fickian diffusion model (1) sorption of trace elements on external surfaces (2) solid-state diffusion of trace elements from external to internal sites and (3) trace element binding and fixation at positions inside the goethite particle (Bruemmer et al., 1988). 

THE BERENS-HQPFENBERG MODEL. The Berens and Hopfenberg model considers the sorption process in glassy polymers as a linear superposition of independent contributions of a rapid Fickian diffusion into pre-existing holes or vacancies (adsorption) and a slower relaxation of the polymeric network (swelling).(lS) The total amount of sorption per unit weight of polymer may be expressed as... 

Farnworth [14] reported a numerical model describing the combined heat and water-vapor transport through clothing. The assumptions in the model did not allow for the complexity of the moisture-sorption isotherm and the sorption kinetics of fibers. Wehner et al [30] presented two mechanical models to simulate the interaction between moisture sorption by fibers and moisture flux through the void spaces of a fabric. In the first model, diffusion within the fiber was considered to be so rapid that the fiber moisture content was always in equilibrium with the adjacent air. In the second model, the sorption kinetics of the fiber were assumed to follow Fickian diffusion. In these models, the effect of heat of sorption and the complicated sorption behavior of the fibers were neglected. 

Of particular importance is the timescale over which diffusion occurs under various conditions of relative humidity (RH) and temperature. The RH determines the equilibrium moisture concentration, whereas higher temperatures will accelerate the moisture sorption process. In order to predict the moisture profile in a particular structure, it is assumed that Fickian diffusion kinetics operate. It will be seen later that many matrix resins exhibit non-Fickian effects, and other diffusion models have been examined. However, most resin systems in current use in the aerospace industry appear to exhibit Fickian behaviour over much of their service temperatures and times. Since the rate of moisture diffusion is low, it is usually necessary to use elevated temperatures to accelerate test programmes and studies intended to characterize the phenomenon. Elevated temperatures must be used with care though, because many resins only exhibit Fickian diffusion within certain temperature limits. If these temperatures are exceeded, the steady state equilibrium position may not be achieved and the Fickian predictions can then be inaccurate. This can lead to an overestimate of the moisture absorbed under real service conditions. 

Several diffusion models have been used to propose transport mechanism of liquid, vapour and gas molecules through the polymer. A model described by Pick s laws is frequently used and known as Case I or Fickian diffusion. The diffusion behaviour in the rubbery polymers, represented by permeation, migration and sorption processes, can be described by the equation of Pick s first law ... 

These methods are interested in studying the distinction between the pure dualmode sorption/transport model curve and the actual sorption and permeation experiment curve that seems to contain the various unsolved appearances for diffusion and sorption of gases in glassy polymer. It becomes obvious that the deviations from the Fickian model in an experimental transport or sorption/desorp-tion curve for a gas in a glassy polymer are not necessarily consistent with the onset of concomitant diffusion and relaxation [11], but are just owing to the dual-mode model. That is to say, only the combinations of parameters of the dual mode model make the curve either fit with or deviation from the Fickian model curve. 

Another consideration in modeling the uptake of moisture by adhesives is that the uptake behavior can be different in absorption and desorption and can change with number of sorption cycles. This is illustrated in Fig. 31.13b. It can be seen that the rate of moisture absorption and the equilibrium moisture content increase from the first to second absorption cycle, although there is little difference between the second and third absorption cycles. Desorption is faster than absorption and more closely resembles Fickian diffusion. Mubashar et al. (2009b) proposed a method of incorporating the moisture history effects illustrated in Fig. 31.13b in a finite element-based predictive methodology. 

The above case of dual-sorption model can be considered a special case of what has been termed anomalous diffusion in polymers. Because organic vapors or liquids can interact strongly with a polymer and cause it to swell, an extreme case of anomalous diffusion occurs when the mass uptake (time-integrated flux) into the polymer is totally controlled by the stress gradient between the swollen and unswollen regions rather than by the concentration gradient. This was first characterized by Alfrey and coworkers [31] and referred to as case II diffusion. Fickian diffusion leads to an initial mass uptake of a polymer film or sheet exposed to a swelling solvent that follows the expression [32] ... 

Sousa et al [5.76, 5.77] modeled a CMR utilizing a dense catalytic polymeric membrane for an equilibrium limited elementary gas phase reaction of the type ttaA +abB acC +adD. The model considers well-stirred retentate and permeate sides, isothermal operation, Fickian transport across the membrane with constant diffusivities, and a linear sorption equilibrium between the bulk and membrane phases. The conversion enhancement over the thermodynamic equilibrium value corresponding to equimolar feed conditions is studied for three different cases An > 0, An = 0, and An < 0, where An = (ac + ad) -(aa + ab). Souza et al [5.76, 5.77] conclude that the conversion can be significantly enhanced, when the diffusion coefficients of the products are higher than those of the reactants and/or the sorption coefficients are lower, the degree of enhancement affected strongly by An and the Thiele modulus. They report that performance of a dense polymeric membrane CMR depends on both the sorption and diffusion coefficients but in a different way, so the study of such a reactor should not be based on overall component permeabilities. 

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. 

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