Fickian-type models

The appendices present the parameters and empirical correlations necessary for the models discussed in the book. They also give basic information on the use of the orthogonal collocation technique for the solution of non-linear two-point boundary value differential equations which arise in the modelling of porous catalyst pellets and the estimation of clFectiveness factors. The application of orthogonal collocation techniques to equations resulting from the Fickian type model as well as models based on the more rigorous Stefan-Maxwell equations are presented. 

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

B.l SIMPLE APPLICATION OF THE ORTHOGONAL COLLOCATION TECHNIQUES FOR A FICKIAN-TYPE DIFFUSION-REACTION MODEL FOR POROUS CATALYST PELLETS... 

Techniques for a Fickian-type Diffusion-Reaction Model for Porous Catalyst Pellets 436... 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

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. 

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. 

To use the above model to predict the durability of the epoxy/mild steel joints (see Fig. 8.6) first requires a knowledge of the rate of diffusion of water into the adhesive layer in the joint. The diffusion and solubility coefficient of water into the bulk adhesive was measured and the diffusion process was also found to be of the concentration independent Fickian type [147]. Further, work [147,149,150] has also shown that the diffusion of water into the adhesive layer may often be modelled by assuming Fickian diffusion. Support for this comes, for example, from studies which have employed tritiated water and so enabled the water concentration in the joint to be directly measured good agreement with the calculated water concentration profile was obtained. 

Rodriguez (2006) was able to locate numerous sets of data on both the soil turnover rates and depth of soil activity/disturbance for several types of macrofauna. These were converted to particle biodiffusion coefficients and summaries of the data appear graphically in Figure 13.2. Shown in this figure are cumulative probability distributions of this coefficient for earthworms, ants and termites and vertebrates. Included within the earthworm particle data set are four sorbed-phase chemical data points. These Dbs data are for PBCs, which were extracted from concentration profiles using a Fickian diffusion model. The PCB profiles were obtained in soils with abundant earthworm populations. The reader should note that the range of I>bs values for soils is within those for sediments see Figure 13.1 in comparison. 

For non-ideal systems, on the other hand, one may use either D12 or D12 and the corresponding equations above (i.e., using the first or second term in the second line on the RHS of (2.498)). In one interpretation the Pick s first law diffusivity, D12, incorporates several aspects, the significance of an inverse drag D12), and the thermodynamic non-ideality. In this view the physical interpretation of the Fickian diffusivity is less transparent than the Maxwell-Stefan diffusivity. Hence, provided that the Maxwell-Stefan diffusivities are still predicable for non-ideal systems, a passable procedure is to calculate the non-ideality corrections from a suitable thermodynamic model. This type of simulations were performed extensively by Taylor and Krishna [96]. In a later paper, Krishna and Wesselingh [49] stated that in this procedure the Maxwell-Stefan diffusivities are calculated indirectly from the measured Fick diffusivities and thermodynamic data (i.e., fitted thermodynamic models), showing a weak composition dependence. In this engineering approach it is not clear whether the total composition dependency is artificially accounted for by the thermodynamic part of the model solely, or if both parts of the model are actually validated independently. 

Cylindrical samples of a water-swellable polymer were immersed in water for different times. The profile of the water front moving from the surface to the inner core of the sample is visualized (Ghi et al. 1997) and compared with model calculations of the water transport. The Fickian nature of the diffusion process is proved. As a result of stress formation during the diffusion processes, cracks were formed. From analysis of T2 the presence of two types of water within the polymer follows a less mobile phase of water that is interacting strongly with the polymer matrix and a more mobile phase within the cracks. 

For porous membranes the mass transport mechanisms that prevail depend mainly on the membrane s mean pore size [1.1, 1.3], and the size and type of the diffusing molecules. For mesoporous and macroporous membranes molecular and Knudsen diffusion, and convective flow are the prevailing means of transport [1.15, 1.16]. The description of transport in such membranes has either utilized a Fickian description of diffusion [1.16] or more elaborate Dusty Gas Model (DGM) approaches [1.17]. For microporous membranes the interaction between the diffusing molecules and the membrane pore surface is of great importance to determine the transport characteristics. The description of transport through such membranes has either utilized the Stefan-Maxwell formulation [1.18, 1.19, 1.20] or more involved molecular dynamics simulation techniques [1.21]. 

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. 

The models mentioned so far are limited in their application as they represent only first order reaction kinetics with Fickian diffusion, therefore do not allow for multicomponent diffusion, surface diffusion or convection. Wood et al. [16] applied the algorithms developed by Rieckmann and Keil [12,44] to simulate diffusion using the dusty gas model, reaction with any general types of reaction rate expression such as Langmuir-Hinshelwood kinetics and simultaneous capillary condensation. The model describes the pore structure as a cubic network of cylindrical pores with a random distribution of pore radii. Transport in the single pores of the network was expressed according to the dusty gas model as... 

Whereas the value of n equals 0.5, the mechanism of diffusion described by Case I of Fickian model, when n values range between 0.5 and 1 that exhibits anomalous transport model and when n value equals 1, Case II of non-Fickian model is used to determine. Fickian model describing the rate of diffusion of penetrant molecule is much less than the relaxation rate of polymer chains while Case II of non-Fickian diffusion representing rate of diffusion is rapid than relaxation process. For the anomalous transport model, both solvent diffusion rate and polymer relaxation rate are comparable. Figure 27.2 shows various types of non-Fickian model anomalous transport and Case II of non-Fickian are included in this group. ... 

In some cases the Fickian model does not accurately represent moisture uptake in adhesives. This is illustrated in O Fig. 31.12a, which shows the uptake plot for an epoxide immersed in water at SO C. The experimental data appears to indicate Fickian diffusion however, the best fit of the Fickian diffusion equation to the data indicates equilibrium is reached more slowly than predicted by Fickian diffusion. This type of behavior is sometimes termed pseudo-Fickian behavior. In general, anomalous behavior is seen at high temperatures and humidity. 

In some cases, a pseudo-equilibrium stage is reached in the moisture uptake, which is then followed by a secondary uptake process. This is illustrated in O Fig. 31.13a. This type of behavior indicates that the absorbed moisture has triggered an additional moisture uptake mechanism. For example, differential hygroscopic expansion may lead to cracking and moisture uptake by capillary action. In order to model the moisture uptake shown in Fig. 31.13a, Mubashar et al. (2009b) proposed a dual Fickian model with a Heaviside step function. This model was termed the delayed dual Fickian model, where the secondary uptake is modeled by a power law. The mass uptake by a delayed dual Fickian model is, hence, given by ... 

Charles Darwin in 1881 reported that by ingesting soil at depth and depositing it upon the surface earthworms caused surface objects to migrate downward. Soil turnover rates, v in cm/year, and depths of activity h, in cm, are the measurements typically taken on cast production by earthworms. A random particle displacement concept applied over time as the conventional definition of a diffusion coefficient allows using the two measurements to yield a biodiffusion coefficient, Dbs = v h (cm /year). This approach was first used by Mclachlan et al. (2002) the results that appear here are those of Rodriguez (2006), who extended the earlier work. Unlike sediment bioturbation where chemical tracers and Fickian diffusion-type mathematical models, such as Equation 13.1, are used with chemical profile data to yield Dbs and h values directly, no such approaches have been used for estimating surface soil bioturbation parameters. 

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