2. Thermal diffusion coefficients

We consider a binary fluid mixture. The total diffusive mass flux of component 1 of the mixture is given by [21, 23]

$$\overrightarrow{J}\_{1}^{(t)} = - (\rho\_{m}^{2}/\rho) \mathbf{M}\_{1} \mathbf{M}\_{2} \mathbf{D}\_{12} \left[ \nabla \mathbf{x}\_{1} + \frac{\mathbf{M}\_{1} \mathbf{x}\_{1}}{RT} \left( \frac{\overline{V\_{1}}}{M\_{1}} - \frac{1}{\rho} \right) \nabla P / F\_{1} - \kappa\_{T} \nabla \ln T \right] \tag{1}$$

where, ρ<sup>m</sup> is total molar density, M is molecular weight, T is temperature, x<sup>1</sup> is mole fraction of component 1, ρ is mass density, P is pressure, R is gas constant, V<sup>1</sup> is the partial molar volume of component 1, D<sup>12</sup> is molecular diffusion coefficient, κ<sup>T</sup> is thermal diffusion ratio of component 1, ∇ is the gradient operator, and

$$F\_1 = \left(\partial \ln f\_1 / \partial \ln \mathcal{x}\_1\right)\_{T,P} \tag{2}$$

where f1 is the fugacity of component 1.

The first, second and third parts of Eq. (1) arise due to the molecular diffusion, pressure diffusion and thermal diffusion. The thermal diffusion factor α<sup>T</sup> of component 1 is defined as

$$
\kappa\_T = \kappa\_T / \mathfrak{x}\_1 \mathfrak{x}\_2 \tag{3}
$$

For a binary mixture, the thermal diffusion factor of component 2 has the opposite sign.

Here we consider a one dimension case in steady state, and assume that there are no convection and gravity segregation. Therefore, the mass flux can be assumed to be zero. Under these conditions, the composition and the temperature gradients are related through the following equation [2]:

$$d\infty\_1/dz = (a\_T \ \propto\_1 \ \propto\_2 d\ln T/dz) \tag{4}$$
