The Calculated Coefficient of Thermal Conductivity of the Binary Mixture

Introduction. Much attention is paid to the study of the influence of temperature conditions on the reliability and safety of the operation of engineering structures in the cryolithozone, both in our country ^[1,2,3] and abroad ^[4,5,6]. Earlier, in particular, it was noted that in order to increase the reliability and safety of linear structures in the cryolithozone (which also include highways), it is necessary to take into account the influence of the thermal factor on the operational characteristics of road surfaces and foundations during their design and construction ^[7,8,9]. To reduce the negative impact of cryogenic processes caused by the temperature factor, it is proposed to use both special heat-protective structures ^[10,11,12] and mixtures of materials with the necessary thermal insulation and heat storage properties ^[13,14,15]. When choosing design solutions, the accuracy of the forecast of the thermal regime of the cryolithozone highways largely depends on the accuracy of determining the thermophysical properties of the materials of road clothing and road foundation soils ^{[16,17,18,19]}. Artificial dispersed rocks (sand-based mixtures) are widely used in the construction of road clothes ^[20,21,22]. Such mixtures, as a rule, are binary, that is, they consist of two components: a binder and a filler. In principle, even simple wet sand can be considered as a binary mixture^[13,23].

The purpose of this work was to compare the calculated values of the thermal conductivity coefficient of binary mixtures determined by the formulas of K.Lichtenecker ^[24,25] and P.Schwerdtfeger ^[26].

Method. P. Shverdtfeger's formula has the form ^[26]

Here and further it is accepted:  ; m = Vvc/Vo. - thermal conductivity coefficients of the filler and binder, W/mK; Vvc,Vo – the volume of the filler and the total volume occupied by the binder and filler, m3.

If the filler completely occupies only the pore space of the binder, then the parameter "m" will be equal to the porosity of the binder. Otherwise, for a binary mixture, the parameter "m" means the concentration of one of the components in the mixture.

K.Lichtenecker's formula has a simpler form ^[24,25]

Taking into account that it is possible to write

We will introduce the "inconsistency parameter" of calculations using two formulas, which we will write in the following form

Then, the difference in the deviation of calculations in percentages will be determined by the formula

Taking into account formulas (1) and (3), the parameter can be found by the following formula

Or in the form of Results and discussion.

 According to the obtained formulas, variant calculations were carried out, which are presented in the form of graphs in the figures. At the same time, two types of binary mixtures were considered: thermal insulation (the coefficient of thermal conductivity of the binder is greater than the coefficient of thermal conductivity of the filler) and heat storage (the coefficient of thermal conductivity of the binder is less than the coefficient of thermal conductivity of the filler). For the first mixtures, the range of variation in the ratio of the thermal conductivity coefficient of the filler to the binder varied in the range of 0.05 to 0.3. And for the second type of binary mixtures from 1.0 to 2.0.

 Fig.1 and Fig.2 show graphs characterizing the change in the values of the dimensionless parameter "?" from the parmeter "g", which characterizes the ratio of the thermal conductivity coefficients of the filler to the binder, at different concentrations of the filler in the binary mixture "m".

Fig.1. Change of the parameter "? = a ·c" for the heat-accumulating binary mixture depending on the concentration of the filler "m".

Fig.2. Change of the parameter "? = a ·c" for the thermal insulation binary mixture depending on the filler concentration "m".

A comparison of the graphs in the figures shows a multiple change in the parameter "?" during the transition from one type of binary mixtures to another, in almost the entire range of changes in the concentration of filler in the mixture.

Fig.3 and Fig.4 show the percentage values of the "coefficient of disagreement" of the calculated values calculated according to different formulas for heat-accumulating and heat-insulating mixtures with different filler concentrations in the binary mixture.

Fig.3. The value of the "coefficient of disagreement" as a percentage in the calculations

coefficient of thermal conductivity of a heat-accumulating binary mixture

according to various formulas

Fig.4. The value of the "coefficient of disagreement" as a percentage in the calculations

coefficient of thermal conductivity of a heat-insulating binary mixture

according to various formulas

As can be seen from the graphs in the figures, for heat-accumulating binary mixtures, both formulas give relatively similar results in a small range of initial parameters. For thermal insulation mixtures, the results differ significantly in almost the entire range considered. Moreover, the discrepancy for some ranges of changes in filler concentrations is hundreds and thousands of percent, which indicates a complete disagreement of the results obtained.  The validity of the application of one or another formula in different ranges of changes in the initial parameters for thermal insulation binary mixtures needs separate special studies.

It should be noted that the results obtained and the conclusions drawn can also be used to compare the formulas of K.Lichteneker and V.I.Odelevsky^[27]. This follows from the results of work ^[16], where the calculation formulas of V.I.Odelevsky and P. Shverdtfeger were compared, which showed that both formulas give approximately the same results in a wide range of initial data and any of them can be used in practical calculations: the difference in determining the thermal conductivity coefficient of a binary mixture does not exceed ten percent, which is permissible error in engineering practice. It follows from the above results that the calculated values obtained by the formula of K. Lichtenecker differ significantly from the calculated values obtained by the formula of P. Shverdtfeger in a very wide range of initial data. Therefore, additional research should be carried out to determine the rational scope of using the formulas of K. Lichtenecker, P. Shverdtfeger and V.I.Odelevsky in determining the thermal conductivity coefficient of binary mixtures. And, in particular, when calculating thermal insulation binary mixtures. It is necessary to determine the range of changes in the initial data, in which all formulas give similar results, when the deviations of the calculated values do not exceed ten percent permissible in engineering practice.

Conclusion. The thermal conductivity coefficients obtained by calculation according to the formulas of K. Lichteneker and P. Shverdtfeger are compared.

The studies were carried out in the range of changes in the ratio of the coefficients of thermal conductivity of the filler to the binder from 0.05 to 0.3 (thermal insulation mixtures) and from 1.0 to 2.0 (heat-accumulating mixtures). The range of filler concentration in both the first and second cases varied from 0.1 to 0.4.

It is shown that for heat-accumulating binary mixtures, the calculated values are quite close in a wide range of initial data. For thermal insulation binary mixtures, the results differ significantly. The discrepancy is in the hundreds of percent.  At this stage of research, it is impossible to conclude that it is advisable to use one or another calculated dependence to determine the thermal conductivity coefficient of a heat-insulating binary mixture. The obtained quantitative results and conclusions are valid not only for comparing the formulas of K. Lichtenecker and P. Shverdtfeger, but also for calculations performed according to the formula of V.I. Odelevsky.

The article has both applied and methodological significance and can be useful both to design engineers of various objects in the cryolithozone, and to graduate students and students studying in the directions 1.6. and 2.1.

In the future, additional research should be carried out to determine the rational use of the formulas of K. Lichtenecker, P. Shverdtfeger and V.I. Odelevsky in determining the thermal conductivity coefficient of binary mixtures. And, in particular, when calculating thermal insulation binary mixtures. It is necessary to determine the range of changes in the initial data, in which all formulas give similar results, when the deviations of the calculated values do not exceed ten percent permissible in engineering practice.

In addition, it is of interest to assess the impact of the accuracy of determining or calculating the thermal conductivity coefficient of binary mixtures on specific design parameters. For example, when determining the depth of the active layer of the road base or the thermal resistance of the pavement. It is also of interest to compare the calculated and experimental methods for determining the thermal conductivity coefficients of both binary and multicomponent mixtures of building materials.