Thermal Conductivity Calculation Model for Refrigerant Mixtures in the Vapor Phase

Thermal conductivity measurements of ten refrigerant mixtures (R-404A, R-406A, R-407C, R-409A, R-410A, R-415A, R-507A, R-227ea/R-134a 61.5/38.5, R-227ea/R-134a 88.8/11.2 and R-227ea/R-134a 45/55) in the gas phase are analyzed. The thermal conductivity was studied with the same experimental setup, which improved the reliability of the results, excluding thereby systematic errors caused by using different methods to measure thermal conductivity. The reported experimental data have 1.5 % to 2.5 % uncertainty at 0.95 confidence level with a coverage factor of k = 2. Equations for calculating thermal conductivity depending on temperature and pressure are given for each mixture. The equations for thermal conductivity on the dew line and in the ideal gas state are obtained. A model for predicting thermal conductivity on a wide range of state parameters is proposed based on obtained experimental data and the theory of thermodynamic similarity. A comparison of the experimental data with the calculation model gives the standard deviation at 0.4 % to 2.1 %, which does not exceed the measurement error.


Introduction
The terms of the international agreements on ozone layer protection and prevention of global climate change impose certain restrictions on refrigerants widely used in various domestic and industrial applications. The exit strategy is to create a large variety of new mixed refrigerants that meet the modern environmental safety and energy efficiency requirements. Mainstreaming such refrigerant mixtures requires exact knowledge of their thermophysical properties.
Obviously, it is impossible to investigate the thermophysical properties of all refrigerant compositions experimentally, not only because of a large number of refrigerants, but also due to the labor-and time-consuming nature of such measurements. This circumstance has led to development of particular methods for calculating and predicting the refrigerant properties for their industrial use. A property calculation model should meet several criteria, including a minimum set of source data, high accuracy of results, as well as fast and simple computational operations. One of the ways to develop effective models for predicting thermophysical properties that would meet the noted requirements is to apply the similarity theory in order to generalize the data on a wide range of refrigerants.

Experimental Data and Their Generalization
Thermal conductivity was measured by the stationary method of coaxial cylinders in the temperature range from 300 K to 430 K and pressures from 0.1 MPa to 2.1 MPa. A detailed description of the measurement procedure and the experiment can be found in [1,2]. The error of the experimental data on thermal conductivity was equal to 1.5 % to 2.5 %, while the temperature error was 0.05 K, and the pressure error did not exceed 4 kPa. The research objects (refrigerant solutions) and thermal conductivity measurement intervals are given in Table 1. Ten mixed refrigerants were covered by the study. Their thermal conductivity was studied using the same experimental installation. It improved the reliability of the results, excluding thereby systematic errors caused by applying different methods for measuring thermal conductivity.
Experimental data were approximated by the empirical dependence on temperature and pressure [1]: where T is temperature, K, p is the pressure, MPa, and λ is the thermal conductivity, mW·m −1 K −1 . The equation coefficients are represented in Table 1. As an example, Fig. 1 shows experimental data and calculations of isotherms for one of the mixtures by the dependence (1).
As is known, thermal conductivity of gaseous refrigerants outside the critical region is represented in the following form [12,13] where λ 0 is the thermal conductivity in the ideal gas state (p 0 = 0.101325 MPa) and Δλ (p) is the excess thermal conductivity depending on the pressure or density (ρ). The choice of normal atmospheric pressure, rather than zero pressure, as an ideal gas state is due to practical considerations, since measurements of thermal conductivity (1) (p, T) = a 0 + a 10 T 100 + a 20 100 T + p a 11 T 100 + a 21 100 T + p 2 a 12 T 100 + a 22 100 T , (2) (T, p) = 0 (T) + Δ (p), at p 0 are convenient and easy to perform. In addition, the dependence λ(p) in a dilute gas differs from that for a continuous medium.
Thermal conductivity in the ideal gas state for the studied refrigerants was determined on the basis of the measurement results. λ 0 values were obtained dually: extrapolating the thermal conductivity isotherms of the vapor mixture to atmospheric pressure (p 0 = 0.101325 MPa) and doing calculations in accord to the generalized Eq. 1. Both methods showed similar results within the measurement errors. To preserve the uniformity when describing the properties across the entire range of the state parameters, a second method was chosen for calculation; λ 0 was obtained from Eq. 1 The coefficients of Eq. 3 are presented in Table 2. The thermal conductivity on the dew line was determined in a similar way. The data obtained were approximated by a quadratic temperature function The polynomial coefficients are given in Table 2. Following the general approaches of the thermodynamic similarity theory, it is possible to obtain an equation for λ 0 as a temperature function in reduced units. Temperature T m = 0.9 T c was chosen as the normalizing temperature, where T c is the critical temperature. This choice of T m , as well as p 0 , is related to the convenience of measurements (the exclusion of the influence of the critical point). T c for the refrigerant mixtures was calculated from the data for pure components using the Lee-Kesler mixing rules [14]. The following set of mixing rules was used: where V c is the critical volume, Z c is the compressibility factor, R is the gas constant, P c is the critical pressure, ω is the acentric factor, x is the molar composition, i, j, k is the component identification. As seen from Fig. 2, λ 0 data for all ten refrigerants are described by a quadratic dependence (Eq. 11) with a standard deviation of 0.85 %, which is less than the estimated error in thermal conductivity measurements: where T r = T/T m , λ 0r = λ 0 (T)/λ 0 (T m ).
Similarly, approximation dependence for the thermal conductivity on the dew line (Fig. 3) can be obtained with a standard deviation of 1.45 %  where λ dr = λ d (T)/λ d (T m ). Equations 11 and 12 enable calculating λ 0 and λ d for a wide class of refrigerant mixtures in a technically important temperature interval by a single measurement of vapor thermal conductivity at 0.9 T c , and the critical parameters and the acentric factor of pure components.
To determine the excess thermal conductivity, let us take advantage of the fact that the simplest correlations for Δλ are obtained when density is used as an independent variable. If experimental data were not available, the density was calculated from the Lee-Kesler equation of state in the approximation of the three-parameter Pitzer correlation [14]. Practice has demonstrated that the Lee-Kesler equation of state allows estimating the density of mixed refrigerants in the vapor phase with an error of 0.5 % to 1 %.
The excess thermal conductivity was obtained as the difference between the experimental values of thermal conductivity and the smoothed values of λ 0 at the measured points. General processing of all the data depending on the reduced density (ρ r ) showed that with a standard deviation of 0.75 % they are well described by a linear equation: where ρ r (T,p) = ρ (T,p)/ρ c , ρ c is the critical density. The excess thermal conductivity is shown in Fig. 4. Comparing the results calculated by Eqs. 2, 11, 12 and 13 with the experimental data, we have found that the standard deviation for all refrigerants lies in the range of 0.4 % to 2.1 %, which does not exceed the estimated measurement errors. The suggested approach enables calculating the thermal conductivity of mixed refrigerants in a wide temperature range based on two experimentally determined values of λ 0 (T m ) and λ d (T m ). The possibility to determine these quantities using only data for the pure components is also of particular interest. For this reason, we considered the dependence of the λ 0 (T m )/C p 0 (T m ) and λ d (T m )/C p 0 (T m ) complexes on the molecular weight M (Fig. 5). Here C p 0 (T m ) is the ideal-gas heat capacity [15][16][17]. The heat capacity C p 0 is a strictly additive quantity; therefore, it is easily calculated through the ideal-gas heat capacity of the components.
It is seen from Fig. 5 that these complexes in the range of molecular weights of the mixture of 70 g·mol −1 to 115 g·mol −1 are well approximated by the following equations: The standard deviation amounted to 1.2 % and 2.5 %, respectively. In Eqs. 14 and 15, the dimension of λ 0 (T m ) and λ d (T m ) is mW·m −1 ·K −1 , C p 0 (T m )-J·mol −1 ·K −1 , and M-g·mol −1 . The R409A mixed refrigerant was excluded from data generalization, since deviations from the approximation dependencies for this refrigerant amounted to 8.3 % for λ 0 and 11.8 % for λ d . Apparently, the reason for such discrepancies is that this is the only mixture with chlorine-based components. Calculations showed that the replacement of fluorine by chlorine leads to an increase in the heat capacity by 1.4 times, and the molecular weight by 1.9 times and, as a result, to falling out of the general pattern.  The proposed approaches were tested using experimental data [18] for four compositions of the R-32 + R-125 freon mixture (Fig. 6). The first method gave a standard deviation of 0.9 % to 1.5 %, and the second one-1.8 % to 5.8 %, which indicates the reliability of the proposed methods for assessing thermal conductivity of mixed refrigerant vapors.
In addition, the first prediction method has been applied to Novec 7000 coolant [19]. It is a hydrofluoroether with zero ozone depletion potential and low global warming potential. We have tested only the first method, since its molecular weight is 200 g·mol −1 . This prediction method describes experimental thermal conductivity data for Novec 7000 coolant with a standard deviation of 1.25 %.

Conclusions
Thus, two methods for predicting thermal conductivity of refrigerant vapors in a wide range of state parameters have been developed. In the first method, the density and critical parameters of the mixtures are calculated based on the equation of state and Lee-Kesler mixing rules using data for pure components. Further, the thermal conductivity of the refrigerant mixture in the whole interval from the ideal gas state to the dew line is estimated using just a single experimental value of the ideal-gas thermal conductivity measured at 0.9 T c .
In contrast to the first method, the second one relies on the values obtained by Eqs. (14) and (15) instead of the experimental values of λ 0 and λ d . Thus, the calculation does not involve the experimental data on a mixture. A comparison of the experimental data with the calculations has shown that the standard deviation was 0.4 % to 2.1 %, which does not exceed the measurement error. It can be concluded that the proposed methods for predicting thermal conductivity are also applicable to other mixtures consisting of the same components as the studied 10 mixed refrigerants. These components are R-22, R-32, R-125, R-134a, R-152a and R-227ea. However, care should be taken when calculating the second method: analyze the composition and take into account the limitation on the molecular weight of the mixture. The first method is more universal. Calculations for the well-studied refrigerants R-22, R-134a and R227ea showed that the difference between the calculated thermal conductivity values and the experimental ones practically does not change up to temperature and pressure of 450 K and 4 MPa, respectively. It should also be noted that the first prediction method can be extended to other type refrigerants, such as hydrofluoroethers.
Author Contributions E.P. Raschektaeva contributed to the experimental study, calculations and the main text of the article. S.V. Stankus participated in the generalization and analysis of the data and in the editing of the article. All authors read and approved the final manuscript.
Funding This work was supported by the state contract with IT SB RAS (121031800219-2).

Data Availability
The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.