To the theory of thermal conductivity of binary liquids

An analytical description of the dependence of the ratio of thermal conductivity of heterogeneous liquids in the form of function on concentration and temperature is given. It has been proven that the main factor that plays a major role in the process of thermal conductivity is the interaction of volume molecules with surface molecules.


Introduction
This paper will provide some analytical and experimental results on the dependence of the thermal conductivity ratio N of liquid mixed solutions in the form of function from temperature T and relative concentration cr с с [ , where c -concentration of foreign additive mixed with homogeneous composition, cr c percolation threshold [1], which is also often referred to as the flow effect. In the first part of the article, we will elaborate on the analytical description of dependency ,T

N [
for the simplest case of binary solutions, and then give its generalization in case of an arbitrary number of additives, the concentration of which will be denoted by a dimensionless parameter i [ , where index 1, 2,... , i p and p full amount of additional liquids. As to the physical properties of such heterogeneous mixtures, they are quite understandable and easily explained from the point of view of the basic laws of the theory of heat transfer in heterogeneous structures, detailed, for example, in the monograph [2]. Although this monograph reflects the basic principles of the theory of heat transfer not for liquids, but for heterogeneous crystals, the general physical principle described in it can easily be transferred to liquid mixture solutions.
Since there is always a boundary setting when describing the phenomena of heat transfer, the key point of the theory below will be to take into account the connection between volume molecules and surface molecules, which we will consider to be equilibrium when solving the Boltzmann equation. This means that they must have their own temperature equal to the temperature of the thermostat 0 T and be described as an equilibrium distribution function, as opposed to volumes, which are considered quasi-equilibrium but with temperature 0 T T z (details below). It should be noted that we have not found a solution to the problem, taking into account the phenomenologically introduced additional integral of collisions 2 associated with the interaction of volume molecules with surface molecules, in the literature known to us on the classical kinetic equation of Boltzmann. The second part of the work will provide the results of experimental measurements of thermal conductivity of liquid mixtures for a fairly wide range of solutions.

Description of the thermal conductivity of the mixture solutions
In this section, we will elaborate on the analytical description of the thermal conductivity ratio of liquid solutions, using the basic principles of the theory of nonequilibrium processes [3], [4], and the general positions of the physical properties of surfaces [5]. To this end, it is convenient to take advantage of the principle of additive heat flows for the case of binary liquids, which means equality where 00 N thermal conductivity of the main liquid, 11 N the thermal conductivity of the liquid added, and the coefficients 01 N and 10 N in terms of dimensions correspond to thermal conductivity, but in their physical sense are responsible for heat transfer at the boundary of contact of different phases. In the event that there is no sharp boundary between the contact area of the mixed liquids, the formula (2) is greatly simplified and can be presented in the next very compact form where abbreviated designations are introduced 0 0 0 N N and 1 1 1 N N , which correspond to the usual thermal conductivity ratios in the main liquid and in the additive.
Each of these coefficients can be easily assessed on the basis of the classical theory of liquids, using the isotropic gas-kinetic approximation of the thermal conductivity factor, similar to how it was done, for example, in the work of the [2] (compare with with monograph [6]) for diffusion ratios, i.e.
where energy of molecule is n p p f p dp m T p f p dp where mass m should be understood by mass 0,1 m . If we now use the medium time and carry out a simple integration on momentums, we will come to the formula where the average relaxation time W we have to calculate with the help of a simple algorithm, proposed in the work [2]. However, unlike the approach presented in this work, when it comes not to diffusion, but to thermal conductivity, the physical staging here is quite different. Indeed, in the absence of a connection with the thermostat, Boltzmann's kinetic equation for distribution function p f , where p is classical momentum of molecule, we can write in the form (see refs. [7] - [10]) where the collision integral has a classic look 1 1  (10) is assigned to the average potential energy of interaction between the closest z to each other surface molecules, s n concentration of surface molecules. With that said, the equation (7) can then be presented in the next additive form where the second integral collisions, describes the interaction of volume molecules with surface thermostat molecules, and is defined as  (11) can be searched using the method of successive approximations. Indeed, in the first approximation of the hierarchy of the reverse times (13) we have the right to put the voluminous integral of collisions to zero, that is, Its solution, as it is known (see [12] - [14]) defines a quasi-equilibrium distribution function that can be presented as a where P medium chemical potential of 3D molecules, and T T t their quasi-equilibrium temperature. Substituting a solution (15) in the equation (11), we will have in linear approximation on temperature differences  T n e 0 T 0 s P n e 0 ¦ ¦ (16) So, as it should be (see, for example, [14]), we come to the equation where the average time of surface relaxation is introduced due to the ratio From what can be seen that another additional condition must be fulfilled, namely the pulse of the volume molecule is subordinated to inequality 2 q p t . (24) Before we start calculating the average from the expression (23) according to the definition (17), we need to estimate for the average values of chemical potentials P and s P . This is the easiest way to do this if you take advantage of the general definition [15]. Indeed, for the average value of the chemical potential of the liquid we have