Mass Diffusion and Thermodiffusion in Multicomponent Fluid Mixtures

Abstract

Fick diffusion and thermodiffusion coefficients of multicomponent mixtures are known to depend on the frame of reference adopted for the mass fluxes. This paper further elucidates a recent proposal on how to define mass- and thermodiffusion coefficients that are independent of the frame of reference. The primary purpose of the paper is to emphasize the simplicity and convenience of using such frame-independent mass- and thermodiffusion coefficients.

Appendix: Volume-Average Frame of Reference

The procedure described above for the formulation of frame-independent mass- and thermodiffusion coefficients can be readily extended to diffusion coefficients obtained when the fluxes are relative to the center of volume velocity:

$${{\varvec{J}}}_{i}^{\text{V}}={c}_{i}\left[{{\varvec{u}}}_{i}-\left({\sum }_{i}{\phi }_{i}{{\varvec{u}}}_{i}\right)\right],$$

(33)

where ${\phi }_{i}= {x}_{i}{\widehat{V}}_{i}/\sum_{j}{x}_{j}{\widehat{V}}_{j}$ is the volume fraction of component $i$ with ${\widehat{V}}_{i}$ being the partial molar volume of component $i$. The corresponding elements ${\varPhi }_{ij}$ of the concentration-dependent matrix $\Phi$ for the frame-independent transformation are [29, 46]

$${\varPhi }_{ij}={x}_{i}{\delta }_{ij}-{\phi }_{j}{x}_{i}.$$

(34)

Equation 16 becomes

$$ {\text{D = }}\Phi ^{{ - 1}} \cdot {\text{ D}}^{{\text{V}}} {\text{ }} \cdot \Phi = {\text{W}}^{{ - 1}} \cdot {\text{D}}^{{\text{w}}} \cdot {\text{W = X}}^{{ - 1}} \cdot {\text{D}}^{{\text{X}}} \cdot {\text{X}}, $$

(35)

where ${\text{D}}^{\text{V}}$ is now the Fick diffusion matrix in the volume-average frame of reference [29]. Then Eqs. 31 and 32 are supplemented with [46]

$$ \left( {\begin{array}{*{20}c} {\user2{J}_{1}^{{\text{V}}} } \\ : \\ {\user2{J}_{{n - 1}}^{{\text{V}}} } \\ \end{array} } \right) = - {\text{ }}\left[ {\Phi \cdot {\text{D}} \cdot \Phi ^{{ - 1}} \cdot \left( {\begin{array}{*{20}c} {\nabla c_{1} } \\ : \\ {\nabla c_{{n - 1}} } \\ \end{array} } \right) + \Phi \cdot \left( {\begin{array}{*{20}c} {D_{{T,1}} } \\ : \\ {D_{{T,n - 1}} } \\ \end{array} } \right){\text{ }}\varvec{\nabla }T} \right]. $$

(36)

While the matrices ${\text{W}}$ and ${\text{X}}$ are determined by the composition of the mixture, the matrix $\Phi$ requires knowledge of the partial molar volumes, information that may not be so readily available.