Matrix Fick diffusivity

In the HSDM model, pore diffusion is neglected and it is assumed that surface diffusion is the dominant mechanism of intraparticle mass transfer. The imsteady-state diffusion process within the particle is taken into accoimt by a diffusion process model assuming a constant matrix of Fick diffusivity. The variation of q,- with the distance along the column and the time is governed by the diffusion equation [49-51]... 

The matrix [D] of Fick diffusion coefficients is a square matrix of dimension n — 1 X n — 1... 

The Fick diffusion coefficients may be termed practical in the sense that the binary coefficient P and the corresponding multicomponent diffusion coefficients can be obtained from composition profiles measured in a diffusion apparatus. The measurement of binary and multicomponent diffusion coefficients, a subject with an extensive literature, is beyond the scope of this book. The interested reader is referred to Dunlop et al. (1972), Cussler (1976) and Tyrrell and Harris (1984) for descriptions of techniques and summaries of experimental results. Most experimental data are reported for [P ]. This matrix must be... 

The matrix of Fick diffusion coefficients in the molar average reference velocity frame for the system acetone(l)-benzene(2)-carbon tetrachloride(3) at a temperature of 25°C and composition = 0.70, X2 = 0.15, X3 = 0.15 has been obtained from the experimental data of Cullinan and Toor (1965) as... 

Strictly speaking, the rules of matrix algebra do not allow us, on the basis of Eqs. 3.2.5 and 2.2.10, to assert that [D] and [B] [r] are equal. The equality of these two matrices is an assumption, albeit the only reasonable way to relate the Fick diffusion coefficients to the Maxwell-Stefan diffusion coefficients >,y. The equality... 

Some assumptions regarding the constancy of certain parameters are usually in order to facilitate the solution of the diffusion equations. For the binary diffusion problems discussed in Chapters 5 (as well as later in Chapters 8-10), we assume the binary Fick diffusion coefficient can be taken to be a constant. In the applications of the linearized theory presented in the same chapters, we assume the matrix of multicomponent Fick diffusion coefficients to be constant. If, on the other hand, we use Eq. 6.2.1 to model the diffusion process then we must usually assume constancy of the effective diffusion coefficient if... 

The first step is to determine the matrix of Fick diffusion coefficients [ >]. The arithmetic average mole fractions will be needed in the evaluation of the [D] these average mole fractions are... 

In their original development of the linearized theory Toor (1964) and Stewart and Prober (1964) proposed that correlations of the type given by Eqs. 8.8.5 and 8.8.7 could be generalized by replacing the Fick diffusivity D by the charactersitic diffusion coefficients of the multicomponent system that is, by the eigenvalues of the Fick matrix [ >]. The mass transfer coefficient calculated from such a substitution would be a characteristic mass transfer coefficient an eigenvalue of [/c]. For example, the Gilliland-Sherwood correlation (Eq. 8.8.5) would be modified as follows ... 

SOLUTION With the Maxwell-Stefan diffusion coefficients and thermodynamic factors as given in Example 8.7.1, the matrix of Fick diffusivities [D] can be evaluated from Eqs. 4.2.12 using the average mole fractions with the following results ... 

The elements of the matrix of low flux mass transfer coefficients may be computed using Eqs. 10.4.25 and 10.4.30. This requires the matrix of Fick diffusion coefficients in the mass average velocity reference frame. This matrix can be computed with the help of Eqs. 4.2.2, from which [D] is obtained in the molar average velocity reference frame, and Eqs. 3.2.11, which allows transformation to the mass average reference velocity frame. Thus, we need the Maxwell-Stefan diffusivities of the three binary pairs in the vapor phase and the molar masses of the three components. 

The arithmetic average mole fractions will be needed in the calculation of the matrix of Fick diffusivities [D]. 

For the system dodecane(l)-hexadecane(2)-hexane(3), Kelt and Anderson (1969) reported the values of the matrix of Fick diffusion coefficients in the volume average reference velocity frame. At a temperature of 25° C and at the composition x = 0.35, X2 = 0.317 the matrix [D ] is... 

Carry out a review of the literature on diffusion in the vicinity of critical points and spinodal lines. Your review should concentrate on summarizing the data that has been measured as well as on contrasting the various theories that have been proposed to explain diffusion phenomena near critical conditions. Consider the question Can the fact that the matrix of Fick diffusion coefficients is close to being singular near the spinodal be exploited in a commercially useful separation process ... 

Explore the structure of the matrix of Fick diffusion coefficients for a gas-vapor mixture of acetone (l)-benzene(2)-helium(3). The Maxwell-Stefan diffusion coefficients in the vapor phase are D = 2.93 X 10 m s, D13 = 31.8 X 10 mVs,... 

Change the order in which the components in Exercise 4.7 are numbered and recalculate the matrix of Fick diffusion coefficients at the same compositions you used in that exercise. 

Calculate the matrix of vapor-phase Fick diffusion coefficients in the four component mixture acetone(l)-methanol(2)-2-propanol(3)-water(4). The composition of the vapor is... 

Compute the matrix of Fick diffusion coefficients in the mass average reference velocity frame for a mixture of MEK(l)-2-butanol(2)-hydrogen(3). The vapor composition (mole fraction) is = 0.4737 and 2 0.0579. The Maxwell-Stefan diffu-... 

The development of the Toor-Stewart-Prober method in Section 8.4 is based on the assumptions that the molar density and the matrix of Fick diffusion coefficients in the molar average reference velocity frame can be assumed constant along the diffusion path. Develop the theory anew in the mass average reference velocity frame that is, assuming and [D° ] can be considered constant. You will need to work with mass fluxes and mass diffusion fluxes 7). 

Correlations of numbers of transfer units developed for binary systems may be used to compute numbers of transfer units for multicomponent systems as described in Section 12.1.5. An alternative method that follows the ideas put forward by Toor in his development of the linearized theory of mass transfer is to generalize binary correlations by replacing the binary diffusivity with the matrix of Fick diffusion coefficients (in much the same way that we generalized correlations of binary mass transfer coefficients in Section 8.8.2). Let the number of transfer units in a binary system be expressed as... 

Matrix of Fick diffusion coefficients in mass average velocity reference frame [mVs]... 

Matrix of Fick diffusion coefficients relative to a reference diffusivity [ — ] Matrix of turbulent diffusion coefficients [m /s] zth eigenvalue of [D] [m7s]... 

A form of Fick s law describes the diffusion of gases through the amorphous polymer matrix. The diffusion coefficient has been observed to follow an Arrhenius relationship, characteristic of an activated process. 

As the Fick diffusion coefficient is directly related to the adsorption isotherm and the Maxwell-Stefan diffusion coefficient using the thermodynamic matrix F, we can describe the diffusion of alkane mixture using the Maxwell-Stefan theory. For this, we have assumed that the Maxwell-Stefan diffusivity is independent of the loading of the zeolite. This suggests a possible industrial application for the separation of linear and branched alkanes using a zeolite membrane. 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Figure 2a represents the concentration profile of the tin species during the service life of the coating. The diffusion in the polymer matrix is represented by Fick s second law for nonsteady state flow ... 

This refers to the depletion of the tin species at a given point in the matrix as a function of time. The diffusion across the boundary layer is given by Fick s first law for stationary state flow ... 

FIGURE 22.2 Schematic illustration of a diffusion-controlled matrix system for which the diffusion process is typically governed by Fick s Law (Equation 22.9). 

The methods of determination of the reaction matrix [AT] are considered in Refs. 167, 181, 183, 184 and 186. Another important matrix parameter entering into the linearized film mass transport equation is the multicomponent diffusion matrix /). The latter results from the transformation of the Maxwell-Stefan Eqs. (1) to the form of the generalized Fick s law (83). Matrix [D] is generally a function of... 

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