Thermal Conductivity of Liquid Mixtures: Model of the Dependence on Concentration

Abstract

A model of the dependence of thermal conductivity of liquid mixtures on their concentration is constructed on the basis of a series connection of the thermal resistances of their components. The equivalent resistance model of the “Filaments” type is found to serve as a valid basis for describing the phenomenon of additional thermal resistance obtained in experiments carried out on mixtures. From the model, it follows that the thermal conductivity coefficient of a mixture is not an additive value. The dependence of the additional thermal resistance on the concentration becomes more pronounced with a greater ratio of thermal conductivities of pure components. Taking into account the effect of increasing thermal resistance due to the scattering of phonons on the inhomogeneities of the system, it was possible to obtain a sufficient agreement with experimental data on thermal conductivity for selected mixtures having various natures, including those with a lower critical solution temperature.

Appendices

Appendix A: An Alternative Way of Calculating the Thermal Conductivity of the Components

Let us derive the thermal conductivity of a mixture from the elementary kinetic theory of phonons. Consider a surface fragment with area S perpendicular to the temperature gradient in the binary mixture. Then, the heat fluxes through this surface from left to right and from right to left can be obtained. The phonons of both components of the mixture contribute to each of the fluxes. In addition, the phonons associated with each component can be reflected from the surface depending on which molecule is on the other side of it.

The heat flux along the x-axis can be written for the one-component case as the difference between fluxes from left to right and from right to left:

$${J_{\text{q}}} = {1 \over 6}n{v_{\text{s}}}S{i \over 2}k\left[ {T\left( {x - \lambda } \right) - T\left( {x + \lambda } \right)} \right] = - {1 \over 6}nv_{\text{{s}}}S{i \over 2}k{{{\text{d}}T} \over {{\text{d}}z}}2\lambda .$$

where i—is the number of degrees of freedom of a particle. It is assumed here that phonons pass through the surface from a distance equal to their free path length. The coefficient 1/6 reflects the fact that phonons can move in 6 independent directions; i is the number of degrees of freedom of the molecule. Bearing in mind the Fourier law for heat flux, the coefficient of thermal conductivity of a homogeneous substance can be written as follows:

$$\kappa = {1 \over 3}n{v_{\text{s}}}{i \over 2}k\lambda = {1 \over 3}\lambda {v_{\text{s}}}\rho c.$$

The same approach can be applied for a two-component environment. For the heat flux, it can be concluded that:

$${J_{\text{q}}} = {1 \over 6}{n_1}{v_{{\text{s1}}}}S{i \over 2}k\left[ {T\left( {x - {\lambda _1}} \right) - T\left( {x + {\lambda _1}} \right)} \right] + {1 \over 6}{n_2}{v_{{\text{s}}2}}S{i \over 2}k\left[ {T\left( {x - {\lambda _2}} \right) - T\left( {x + {\lambda _2}} \right)} \right].$$

Here it is assumed that the heat capacities of the components are the same.

Let us expand these terms in a series, keeping in mind that the mean free path of molecules of each type is different:

$${J_{\text{q}}} = {1 \over 6}{n_1}{v_{{\text{s}}1}}S{i \over 2}k{{{\text{d}}T} \over {{\text{d}}x}}2{\lambda _1} + {1 \over 6}{n_2}{v_{{\text{s}}2}}S{i \over 2}k{{{\text{d}}T} \over {{\text{d}}x}}2{\lambda _2}.$$

As a result, we get:

$${J_{\text{q}}} = - \left( {{1 \over 3}{n_1}{v_{{\text{s}}1}}{i \over 2}k{\text{ }}{\lambda _1} + {1 \over 3}{n_2}{v_{{\text{s}}2}}{i \over 2}k{\text{ }}{\lambda _2}} \right){{{\text{d}}T} \over {{\text{d}}x}}S.$$

The free path length of each type of phonon can be calculated using the formula for reflecting phonons obtained above:

$${\lambda _1} = x{\lambda _{10}} + \left( {1 - x} \right){\lambda _{10}}\left( {1 - \gamma } \right),$$

$${\lambda _2} = \left( {1 - x} \right){\lambda _{20}} + x{\lambda _{20}}\left( {1 - \gamma } \right).$$

here the free path with subscript “0” refers to the pure component. On the other hand,

$${n_1} = nx,\;{n_2} = n\left( {1 - x} \right).$$

So,

$${\kappa _1} = {\kappa _{10}}\left[ {x + \left( {1 - x} \right)\gamma } \right],$$

$${\kappa _2} = {\kappa _{20}}\left[ {\left( {1 - x} \right) + x\gamma } \right].$$

Appendix B: Case of Different Intermolecular Distances in Components

Let us construct a thermal conductivity model for a mixture in the case of having different average intermolecular distances (a_i). For the coefficient of thermal conductivity, then it can be written as follows:

$$\kappa = \frac{1}{3}\lambda v_{\text{s}} \rho c\left[ {1 - \frac{{\left( {\frac{{a_{1} }}{{a_{3} }} - 1} \right)^{2} }}{{\left( {\frac{{a_{1} }}{{a_{3} }} + 1} \right)^{2} }}} \right].$$

Then, the thermal conductivity of the first component will depend on the presence of the second component due to scattering on its molecules:

$$\kappa_{1} = \frac{1}{3}v_{\text{s}1} \rho_{1} c_{1} \left[ {x\lambda_{1} + \left( {1 - x} \right)\lambda_{1} \left\{ {1 - \frac{{\left( {\frac{{a_{1} }}{{a_{3} }} - 1} \right)^{2} }}{{\left( {\frac{{a_{1} }}{{a_{3} }} + 1} \right)^{2} }}} \right\}} \right].$$

A similar expression can be written for the second component:

$$\kappa_{2} = \frac{1}{3}v_{\text{s}2} \rho_{2} c_{2} \left[ {\left( {1 - x} \right)\lambda_{2} + x\lambda_{2} \left\{ {1 - \frac{{\left( {\frac{{a_{2} }}{{a_{3} }} - 1} \right)^{2} }}{{\left( {\frac{{a_{2} }}{{a_{3} }} + 1} \right)^{2} }}} \right\}} \right].$$

Let us denote:

$1 - \frac{{\left( {\frac{{a_{1} }}{{a_{3} }} - 1} \right)^{2} }}{{\left( {\frac{{a_{1} }}{{a_{3} }} + 1} \right)^{2} }} = \gamma_{1}$, $1 - \frac{{\left( {\frac{{a_{2} }}{{a_{3} }} - 1} \right)^{2} }}{{\left( {\frac{{a_{2} }}{{a_{3} }} + 1} \right)^{2} }} = \gamma_{2} .$

It is easy to show that this formula has similar properties to one constructed for a mixture having components of various masses. If the average intermolecular distances of pure substances are close, then $\gamma_{1} = \gamma_{2}$ and the formulas will coincide with those obtained above.