Fickian laws, equations

Before writing down the mass balance equations, we need to define the constitutive flux equations for the macropore and the micropore. These flux equations are assumed to follow the following Fickian law equations ... 

The development of the theory of solute diffusion in soils was largely due to the work of Nye and his coworkers in the late sixties and early seventies, culminating in their essential reference work (5). They adapted the Fickian diffusion equations to describe diffusion in a heterogeneous porous medium. Pick s law describes the relationship between the flux of a solute (mass per unit surface area per unit time, Ji) and the concentration gradient driving the flux. In vector terms. 

By analogy to the description of dispersion in rivers, the dispersive flux relative to the mean flow, Fdis, can be described by an equation of the First Fickian Law type (see Eq. 24-42) ... 

The nice fittings of the previous functions to the release data generated from (4.16) and (4.17), respectively, verify the argument that the power law can describe the entire set of release data following combined release mechanisms. In this context, the experimental data reported in Figures 4.8 to 4.10 and the nice fittings of the power-law equation to the entire set of these data can be reinterpreted as a combined release mechanism, i.e., Fickian diffusion and a Case II transport. 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

Equation 1 applies to stationary states, and Equation 2 to nonsta-tionary states of diffusion. Equation 2 can be derived from Equation 1 by considering the rate of accumulation of diffusate at a given point in the medium it reduces to Equation 3 when the diffusion coeflBcient is a constant (9). If the latter condition is satisfied, the diffusion process is said to be ideal, or Fickian, and Equations 1 and 3 represent Ficks first and second laws of diffusion respectively. 

Since Fick s first and second laws of diffusion are valid independent of whether I) is a function of c, only or not and also of the form of initial and boundary conditions of a particular experiment, it is quite inadequate to specify this particular type of sorption as Fickian. The term Fickian" should be applied more generally to all mass transport phenomena which are governed by Eq. (1), i. e., the Pick diffusion equation. 

Note the similarity of this equation to the form of Fick s first law as shown in Eq. [1-4] the difference is that Darcy s law is for water flow while Fick s first law is for mass transport by Fickian processes.)... 

A multicomponent Fickian diffusion flux on this form was first suggested in irreversible thermodynamics and has no origin in kinetic theory of dilute gases. Hence, basically, these multicomponent flux equations represent a purely empirical generalization of Pick s first law and define a set of empirical multi-component diffusion coefficients. 

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. 

Here m is the ratio of uptake due to relaxation relative to the total uptake, M( ). The quantities fD and fDR are, respectively, the series solution for Fick s second law and a related series which includes the coupling constant. For 1, the behavior reduces to classical Fickian diffusion. For 1, that is relaxation very slow compared to diffusion, the equation reduces to the sum of an independent Fickian and a first order relaxation process. This simple model was originally proposed and widely used by Berens and Hopfenberg (10). 

Compared with heat transfer, the process of moisture transport is slower by a factor of approximately 10. For example, moisture equilibration of a 12 mm thick composite, at 350 K, can take 13 years whereas thermal equilibration only takes 15 s. Fick adapted the heat conduction equation of Fourier, and his (Pick s) second law is generally considered to be applicable to the moisture diffusion problem. The one-dimensional Fickian diffusion law, which describes transport through the thickness, and assumes that the moisture flux is proportional to the concentration gradient, is ... 

Deviations from Pick s law become more pronounced at high temperatures and are often found for materials immersed in certain liquids, notably solvents, hydrocarbons and sea water. However, in many laminates, even where moisture transport is by a non-Fickian process, the results obtained using Pick s equations together with appropriate apparent D and apparent values are not very different from the observed behaviour. 

Combining equations [12.1] and [12.2] yields the famihar governing equation for one-dimensional isotropic Fickian diffusion, also known as Pick s Law, and given by ... 

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 ... 

In the case of Fickian diffusion, the diffusion obeys Fick s second law and the order n in (1) is equal to 0.5. In the case of bead without rotation, the drug release was explained by the above empirical equation. The order n was estimated to be... 

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