3. Thermodynamics of thermal diffusion factor

Here we present the thermodynamic theory based on the modified form of Chapman and Cowling [22] and Kihara [24] as applied to binary hydrocarbon mixtures. This approach involves the calculation of collision integrals of the fluid mixture for a well-defined potential function. The calculation of the transport property collision integrals for gases, whose molecules obey a simple intermolecular potential, enables to explain the transport properties of slightly non-ideal gas mixtures following the isotropic intermolecular interactions. For non-ideal mixtures, in which molecules interact with strongly anisotropic intermolecular interactions, additional contributions are assumed arising from the expansion of non-equilibrium distributions. These anisotropic interactions could affect the thermal diffusion factors significantly [25].

We consider a binary mixture of components i and j. In this mixture the molecules are assumed to interact with an effective pair-wise additive intermolecular potential function (Exp-6), given by

$$u\_{\vec{\eta}}(r) = e\_{\vec{\eta}} \left[ \frac{a\_{\vec{\eta}}}{a\_{\vec{\eta}} - 6} \exp \left( a\_{\vec{\eta}} \{ 1 - r/r\_{m\vec{\eta}} \} \right) \right] \\ \quad - \quad \frac{a\_{\vec{\eta}}}{a\_{\vec{\eta}} - 6} (r\_{m\vec{\eta}}/r)^6 \Bigg| \tag{5}$$

where uij is the potential energy of two molecules of species i and j at a separation distance r, εij is the depth of the potential minimum which is located at rmij, αij determines the softness of the repulsion energy, and kB is the Boltzmann's constant. In this (Exp-6) potential function, the molecules of mixture species are represented by the size parameter (Rmab) and energy parameter (εab/kB).

Using the mth-order Chapman-Cowling approximation, the general form of the thermal diffusion factor (αT) can be given by

$$\mathbf{x}(a\_T)\_m = \frac{5}{2\varkappa\_1 \varkappa\_2 A\_{00}^{(m)}} \left[ \varkappa\_1 A\_{01}^{(m)} \left( \frac{M\_1 + M\_2}{2M\_1} \right)^{0.5} + \varkappa\_2 A\_{0-1}^{(m)} \left( \frac{M\_1 + M\_2}{2M\_2} \right)^{0.5} \right] \tag{6}$$

where, x1 and x2 are the mole fractions. M1 and M2 are the molecular weights of the mixture components 1 and 2. A(m) is a determinant of (2 m + 1) order, whose general term is Aij, where i and j range from -m to +m, including zero. The minor of A(m) obtained by striking out the row and column containing Aij is denoted by the symbol Aij (m). Similarly, the i and jth minor of A00 (m) is denoted by the symbol Aij00 (m). The elements Aij are functions of the mole fractions, molecular weights and collision integrals, which are functions of temperature, molecular size and energy parameters.

From Eqs. (3) and (7) the <sup>m</sup>th-order thermal diffusion ratio ð Þ <sup>k</sup><sup>T</sup> can be defined <sup>m</sup> as

$$(k\_T)\_m = \varkappa\_1 \varkappa\_2 (a\_T)\_m \tag{7}$$

and the collision integrals are given by the following equations:

$$\Omega^{(l)}(n) = \left(\frac{k\_B T}{2\pi\mu}\right)^{0.5} \int\_0^\infty \left(\exp\left(-\gamma^2\right)\gamma^{2n+3}Q^{(l)}(\mathbf{g})d\chi\right.\tag{8}$$

$$Q^{(l)}(\mathbf{g}) = 2\pi \ (\mathbf{1} - \cos^l \chi) b db \tag{9}$$

with

$$\mathbf{y}^2 = \frac{\mu \mathbf{g}^2}{2k\_B T} \tag{10}$$

where μ is the reduced mass of a pair of colliding molecules, and g is the initial relative speed of the colliding pair. The molecules are deflected by the collision through a relative angle χ which is a function of g and the collision parameter b.

The dimensionless collision integrals of the above equations can be expressed as follows:

$$\boldsymbol{\Omega}^{(l,n)\*} = \left(\frac{\mathbf{4}}{\sigma^2(1+n)!}\right)^{-1} \mathbf{1} - \frac{\mathbf{1} + (-1)^l}{2(1+l)} \bigg( \begin{array}{c} \mu \\ \end{array} \frac{\mu}{2\pi k\_B T} \bigg)^{0.5} \boldsymbol{\Omega}^{(l)}(n) \tag{11}$$

ð Þ where Ω <sup>l</sup>;<sup>n</sup> <sup>∗</sup> is the dimensionless collision integral reduced with respect to that of the diameter σ of a rigid elastic sphere.

Using the first-order approximation, Eq. (7) is written as

$$(a\_T)\_1 = \left(6\mathbf{C}\_{12}^\* - 5\right) \left[\frac{\boldsymbol{\omega}\_1 \mathbf{S}\_1 - \boldsymbol{\omega}\_2 \mathbf{S}\_2}{\mathbf{x}\_1^2 Q\_1 + \mathbf{x}\_2^2 Q\_2 + \boldsymbol{\omega}\_1 \mathbf{x}^{12} Q\_{12}}\right] \tag{12}$$

Thermodynamics of Thermal Diffusion Factors in Hydrocarbon Mixtures DOI: http://dx.doi.org/10.5772/intechopen.75639

where the parameters S1, S2, Q1 and Q2 are given by

$$\Delta S\_1 = \frac{M\_1}{M\_2} \left(\frac{2M\_2}{M\_1 + M\_2}\right)^{0.5} \frac{\Omega\_{11}^{(2,2)} \ell}{\Omega\_{12}^{(1,1)} \ell} \left(\frac{\sigma\_{11}}{\sigma\_{12}}\right)^2 \left(-\frac{4M\_1M\_2A\_{12}^\*}{\left(M\_1 + M\_2\right)^2} - \frac{15M\_2(M\_2 - M\_1)}{2\left(M\_1 + M\_2\right)^2} \tag{13}$$

$$\mathbf{S}\_{2} = \frac{M\_{2}}{M\_{1}} \left(\frac{2M\_{1}}{M\_{1} + M\_{2}}\right)^{0.5} \frac{\Omega\_{22}^{(2.2)} \zeta}{\Omega\_{12}^{(1.1)} \zeta} \left(\frac{\sigma\_{22}}{\sigma\_{12}}\right)^{2} - \frac{4M\_{1}M\_{2}A\_{12}^{\*}}{\left(M\_{1} + M\_{2}\right)^{2}} - \frac{15M\_{1}\left(M\_{1} - M\_{2}\right)}{2\left(M\_{1} + M\_{2}\right)^{2}} \tag{14}$$

$$\begin{aligned} Q\_1 &= \left(\frac{2}{M\_2(M\_1+M\_2)}\right) \left(\frac{^{2M\_2}}{^{(M\_1+M\_2)}}\right) \left(\begin{array}{c} \Omega\_{11}^{(2,2)}\\ \Omega\_{12}^{(1,1)\*} \end{array}\right) \left(\begin{array}{c} \Omega\_{11}^{(2,2)}\\ \Omega\_{12}^{(1,1)\*} \end{array}\right) \left(\begin{array}{c} \sigma\_{11} \\ \sigma\_{12} \end{array}\right) \\ &\left[\left(\begin{array}{c} 5 \\ \end{array}\right) - \frac{6}{5}B\_{12}^\* \end{aligned} \tag{15}$$

$$\begin{split} Q\_{2} &= \left(\frac{2}{M\_{1}(M\_{1}+M\_{2})}\right) \binom{2M\_{1}}{(M\_{1}+M\_{1})}^{0} \binom{\Omega\_{22}^{(2,2)}}{\Omega\_{12}^{(1,1)} \mathrm{k}} \binom{\sigma\_{22}}{\sigma\_{21}}^{2} \\ & \left[ \left(\frac{2}{5} - \frac{6}{5}B\_{12}^{\*}\right) \mathrm{M}\_{2}^{2} + 3\mathrm{M}\_{1}^{2} + \frac{8}{5}M\_{1}\mathcal{M}\_{2}\mathcal{A}\_{12}^{\*} \right] \left( \\ Q\_{12} &= 15 \left(\frac{M\_{1}-M\_{2}}{M\_{1}+M\_{2}}\right)^{2} \left( \left(\frac{2}{5} - \frac{6}{5}B\_{12}^{\*}\right) + \frac{4M\_{1}M\_{2}\mathcal{A}\_{12}^{\*}}{(M\_{1}+M\_{2})^{2}} \right) \left( \left(1 - \frac{12}{5}B\_{12}^{\*}\right) \right) \\ & \quad + \frac{8(M\_{1}+M\_{2})}{5\left(M\_{1}\mathcal{M}\_{2}^{\*}\right)^{0.5}} \frac{\Omega\_{11}^{(2,2)^{\sharp}}}{\Omega\_{12}^{(1,1)^{\sharp}}} \left( \frac{\mathcal{Q}\_{22}^{(2,2)^{\sharp}}}{\Omega\_{12}^{(1,1)^{\sharp}}} \right) \left( \frac{\mathcal{q}\_{11}}{\Omega\_{12}} \right)^{2} \left( \frac{\mathcal{q}\_{22}}{\Omega\_{12}} \right)^{2} \end{split} \tag{17}$$

A12\* , B12\* and C12\* are functions of the collision integrals as given by

$$A\_{12}^{\*} = \begin{pmatrix} \Omega\_{12}^{(2,2)} \\ \Omega\_{12}^{(1,1)} \end{pmatrix} \tag{18}$$

$$B\_{12}^\* = \left. \frac{\mathbf{5} \Omega\_{12}^{(1,2)\*} - \mathbf{4} \Omega\_{12}^{(1,3)}}{\Omega\_{12}^{(1,1)\*}} \right\rangle \Bigg( \tag{19}$$

$$\mathbf{C}\_{12}^\* = \begin{pmatrix} \frac{\Omega\_{12}^{(1,2)} \ell}{\Omega\_{12}^{(1,1)} \ell} \end{pmatrix} \begin{pmatrix} & & & & \tag{20} \\ & & & \tag{20} \end{pmatrix}$$

The details on the various parameters are given elsewhere [21].

To consider the effects of pressure and unlike interaction parameters, Eq. (21) for C12\* was modified empirically as follows:

$$\mathbf{C}\_{12}^\* = \begin{array}{c} \Omega\_{12}^{(1,2)} \\ \overline{\Omega\_{12}^{(1,1)\*}} \end{array} \Big| \left( \left[ \mathsf{sfp} \left( p\_1^\* f\_1 + p\_2^\* f\_2 + p\_3^\* f\_3 \right) \right] \right) \tag{21}$$

where pi are the mixture parameters and fi terms are given by

$$f\_1 = \varkappa\_1 P\_x \, ^\ast / T\_x \, ^\ast$$

$$f\_2 = f\_1 \, ^\ast \tag{22}$$

$$f\_3 = f\_1 \, ^\ast$$

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where the reduced pressure and temperature are given in terms of energy and size parameters of the molecules

$$\begin{aligned} P\_{\mathbf{x}}{}^{\*} &= P R\_{\mathbf{m} \mathbf{x}}^{3} / \varepsilon\_{\mathbf{x}} \\ T\_{\mathbf{x}}{}^{\*} &= kT / \varepsilon\_{\mathbf{x}} \end{aligned} \tag{23}$$

The following van der Waals mixing rules were applied to determine the mixture properties:

$$R\_{\text{max}}^3 = \sum\_{i\_2, j=1}^2 \mathbf{x}\_i \mathbf{x}\_j R\_{\text{mij}}^3 \tag{24}$$

$$
\varepsilon\_{\mathbf{x}} R\_{m\mathbf{x}}^3 = \sum\_{i\_2, j=1}^2 \varkappa\_i \mathbf{x}\_j e\_{ij} R\_{mij}^3 \tag{25}
$$

$$R\_{mij} = \left(R\_{mij} + R\_{mji}\right) / 2 \tag{26}$$

$$
\varepsilon\_{\vec{\text{ij}}} = \left(\varepsilon\_{\vec{\text{ii}}} \varepsilon\_{\vec{\text{jj}}}\right)^{0.5} \tag{27}
$$

Since the properties of the reservoir fluids depend on the fluid compositions, temperature and pressure, and since the collision integrals do not account for the pressure effects of the liquid mixtures, the empirical Eq. (22) was applied to explain the properties of the reservoir fluid mixtures under both temperature and pressure as needed.
