ARTICLE IN PRESS. Fluid Phase Equilibria xxx (2007) xxx xxx. Received 4 May 2007; received in revised form 2 July 2007; accepted 2 July 2007

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1 Fluid Phase Equilibria xxx (2007) xxx xxx Thermal conductivity of polyurethane foam cell gases: Improved transient hot wire cell data of isopentane + n-pentane mixtures Extended Wassiljewa-model Ralf Dohrn a,, José M. Fonseca a,c, Reinhard Albers b, Jacqueline Kušan-Bindels b, Isabel M. Marrucho c a Bayer Technology Services GmbH, Fluid Properties & Thermodynamics, Building B310, D Leverkusen, Germany b Bayer Material Science, PUR-EMEA-IS Technical Insulation, Building B108, D Leverkusen, Germany c CICECO, Departamento de Quimica, Universidade de Aveiro, P Aveiro, Portugal Received 4 May 2007; received in revised form 2 July 2007; accepted 2 July 2007 Abstract The thermal conductivity of pure components and binary mixtures of n-pentane, isopentane and nitrogen was measured at temperatures between 309 and 414 K and at pressures up to 1.0 MPa using an apparatus based on the transient hot wire method. These mixtures are important in the assessment of the progress of the thermal insulation capacity of closed cell polyurethane foams during the total life time of the respective applications. The experimental thermal conductivity data were correlated with the Wassiljewa mixing rules as modified by Mason and Saxena, and predicted using two new empirical models, with average deviations under 0.4% and 0.6%, respectively. Significant changes were performed in the apparatus, with the design of a new measuring cell and the simplification of the tubing system. A new experimental procedure was also developed leading to a considerable improvement in the precision of the results Elsevier B.V. All rights reserved. Keywords: Extended Wassiljewa model; n-pentane; Isopentane; Vapor-phase thermal conductivity; Transient hot-wire method 1. Introduction Due to their high thermal insulating capacity, rigid polyurethane (PUR) and polyisocyanurate (PIR) foams are used in a large number of applications, from thermal insulation boards to pipe insulation, technical refrigeration processes and industry applications. In the construction of domestic refrigerators for example, PUR foam is the preferred insulating material, presenting several advantages. Self-adhesive rigid foam systems enable a weight-saving sandwich construction to be produced in a single operation and the excellent thermal insulating properties permit a relatively small wall thickness. The insulation capacity of the PUR foam is mainly due to the gases, blowing agents, trapped inside the closed foam cells. These gases account for 60 65% of the heat transfer through the foam. The effectiveness of a blowing agent as an insulator is Corresponding author. Fax: address: ralf.dohrn@bayertechnology.com (R. Dohrn). characterized by its thermal conductivity in the gas phase relative to the thermal conductivity of air (typically 27 mw m 1 K 1 at 300 K) that it displaces [1]. Until around 1995, CFC-11 was the most widely used blowing agent, presenting a value for the gas phase thermal conductivity as low as 8.3 mw m 1 K 1 at 300 K [2]. However this compound has a high Ozone Depletion Potential, and within the Montreal Protocol, is was decided to abolish its use as a blowing agent. Then a search for the best substitute began. Research in North America started focusing on the HCFCs but later, and due to regulations, their attention was turned to the HFCs, characterized by zero Ozone Depletion Potential. These compounds present some advantages when compared to hydrocarbons, the solution found in Europe after the phase-out of CFC-11, with lower gas phase thermal conductivity and the fact of being non-flammable. Nevertheless, Europe s choice for hydrocarbons (mostly cyclopentane and cyclopentane mixtures) was made taking into high consideration direct emissions of global warming substances [3]. Cyclopentane and cyclopentane mixtures present a zero Ozone Depletion Potential /$ see front matter 2007 Elsevier B.V. All rights reserved. doi: /j.fluid

2 2 R. Dohrn et al. / Fluid Phase Equilibria xxx (2007) xxx xxx and a much lower Global Warming Potential than HFCs while retaining excellent thermophysical properties for this particular application. But the constitution of the gas phase inside the foam cells is not constant within the lifetime of the foam. With time, it loses a considerable part of its thermal efficiency due to diffusion processes, as the air slowly diffuses into the cells and the blowing agent simultaneously diffuses out. According to a research carried out at the Oak Ridge National Laboratory, USA, the aging of PUR foams is characterised by an initial rapid increase in the values of thermal conductivity, fact which is attributed to diffusion of air into the cells of the foam, followed by a more gradual increase which is attributed to diffusion of the blowing agent out of the cells [4]. The kinetics of aging due to the mentioned diffusion processes is also dependent on several other properties of the foam, like the type of isocyanate used, type of polyol, cell size, cell openness, blowing agents, initial cell gas composition, etc., as well as in the interactions between the blowing agents and the gases in the external environment, defined by solubility values and diffusion coefficients. This work is part of a project focusing on the evolution in time of the thermal conductivity of the gas phase inside the foam cells, centering the search for a blowing agent that can offer good properties during the whole lifetime of the foam. Taking into account that nitrogen is the more abundant component in air, the study of the thermal conductivity of mixtures consisting of nitrogen and blowing agents can give an insight of the efficiency of the aged foam. To understand the behaviour of the thermal conductivity of gas mixtures, reliable experimental data are essential. Therefore, we improve our experimental techniques and measure thermal conductivities of important pure cell gas components and of their relevant binary mixtures. Based on these experimental data, we plan to develop a predictive semi-empirical method for the thermal conductivity of gas mixtures. This work consists of three parts 1. Experimental method: A new measuring cell, based on the transient hot-wire method, was designed, along with other modifications in the existing apparatus and a new experimental procedure was developed which lead to a significant improvement of the precision of the results. 2. Experimental data: The thermal conductivity of three pure components, n-pentane, isopentane and nitrogen, and of the binary system n-pentane + isopentane was measured at temperatures between 309 and 414 K and pressures up to 1.0 MPa. 3. Modeling: A correlative model is presented, that allows the calculation of thermal conductivity values for any given pressure and temperature, for pure compounds. A new extended Wassiljewa mixing rule model was developed, based on the Wassiljewa mixing rule [5] modified by Mason and Saxena [6,7]. A study of how the parameter ε changes with temperature and pressure was performed, culminating in the development of a mathematical model which allows the calculation of values for this parameter for any given conditions of temperature, pressure and composition. 2. Experimental method Fig. 1. New measuring cell. The experimental thermal conductivity measurements were carried out in an apparatus based on the transient hot-wire method. A detailed description of the apparatus and the technical details are given elsewhere [8 12], so only a description of the changes performed in the measuring cell are given here. The new measuring cell, schematically represented in Fig. 1, has a diameter of 48 mm and it consists of two parallel chambers (diameter 16 mm each) with platinum wires of different lengths (ratio of lengths = 0.37). The second wire was used to compensate the end effects. The new cell is now slightly shorter than before (260 mm instead of 270 mm) with the major difference being the supports of the very thin (10 m) platinum wires which constitute the basis of the hot-wire method. The new support system for the wires eliminates the flexible parts used until now, increasing substantially the accuracy in the positioning of the wires in the center of the cavities in the cell. This leads to a significant improvement of the quality of the results and it has also other experimental advantages: the highly sensitive wires are now more stable, breaking much less frequently and the welding of new wires when necessary is now a simpler process. The fundamental part of this new system is located in the lower part of cell that now has a stainless steel disk, inside which the support of the wire is connected with a screw. Fig. 2 shows the support part of one of the wires, on the right, which possesses a six-sided fitting that connects to the disk keeping it from rotating. On the left of the photo, it is possible to see a screw that connects on its outside with the disk, and on its inside

3 R. Dohrn et al. / Fluid Phase Equilibria xxx (2007) xxx xxx 3 Fig. 2. Detail of the parts that constitute the new system for support of the platinum wires. with the support of the wires. In the new cell, the tightening of the screws causes an axial displacement of the parts supporting the wires, allowing the stretching of the platinum wires to their ideal position. The basic theory behind the transient hot-wire method is given by Healy et al. [13]. The essential feature of the method is the precise determination of the transient temperature with a thin metallic wire. This is determined from measurements of the resistance of the wire over a period of a few seconds followed by the initiation of the heating cycle, with a T = (2.000 ± 0.025) K. For cylindrical wires, with radius r 0, the ideal temperature rise T on the surface of the wire can be calculated using Eq. (1). q 4at T id = ln 4πλ(T ref,ρ ref ) r0 2C (1) where λ(t ref,ρ ref ) is the thermal conductivity at the temperature and density reference conditions, a is the thermal diffusivity, a = λ/(ρc P ), q is the heat flow through the wire, t is the time, and C = exp(γ) = is the exponential of Euler s constant γ. We use several corrections to the ideal transient hot-wire method: corrections due to the finite radius of the wire (5 m) and due to the existence of an outer isothermal boundary layer. The calculations of the thermal conductivity are performed from a linear plot of the temperature rise versus logarithm of time. For the evaluation of the experimental thermal conductivity data, the second virial coefficient and the vapor-phase heat capacity of the investigated substances (pure component or mixture) at the temperature and pressure conditions of the experiments had to be known. The virial coefficients of the pure components and of the mixtures have been determined from vapor phase density data, which have been calculated with the Lee Kesler Plöcker model [14,15] within the process simulator ASPEN PLUS. The same model has been used to calculate the heat capacity of the pure components and of the mixtures for the pressure and temperature conditions of the thermal conductivity measurements. The density of the mixtures was calculated using the Lee Kesler Plöcker model [14,15] from the partial pressure of the first component and from the final pressure of the mixture, as well as from the registered temperature. The mixtures were prepared directly in the measuring cell, one of the main changes in the experimental procedure, with a much more precise and accurate monitoring of the pressure and temperature conditions updated every minute. Care was taken in order to prevent any possible condensation during the preparation of the mixtures. In an equally important improvement in the experimental procedure, the experiments were started at the highest pressure at several temperatures, and then continued to the lower pressures at the same temperatures, ensuring that all the experiments for a certain composition were performed with the same gas mixture. In the old procedure, the measuring cell was refilled from a mixing cell for each isotherm. All these changes in the experimental procedure allowed further improvements of the apparatus, with significant simplifications of the tubing system. Nitrogen was used straight from a high pressure bottle supplied by Linde (purity > %) with no further purification; n-pentane (CAS No ) and isopentane (2-methyl butane, CAS No ) were supplied by Kraemer and Martin (>99%) and used after degassing processes by successive freezing/vacuum/melting cycles. Before starting the thermal conductivity measurements, the precision and accuracy of the apparatus was checked by measuring the thermal conductivity of the vapor phase of nitrogen at four different temperatures between 309 and 381 K and at pressures up to 0.8 MPa. An excellent agreement was found between our results and the data calculated with the NIST Chemistry Webbook [16], with an overall average absolute deviation of 0.13% for the extrapolation to 0.1 MPa and of 0.21% for a pressure of 0.8 MPa. 3. Experimental results The results for the pure compounds, n-pentane and isopentane, are depicted in Figs. 3 and 4. The pressure dependence of the thermal conductivity at each temperature followed the usual behaviour of the gas phase thermal conductivity, i.e., a slight increase of the thermal conductivity with pressure was observed. The thermal conductivity performance of PUR cell gases is usually compared at a pressure of 0.1 MPa. Therefore, it is a standard procedure to fit the isothermal data to a linear correlation to find the thermal conductivity at 0.1 MPa. These Fig. 3. Pressure and temperature dependence of the thermal conductivity of n-pentane.

4 4 R. Dohrn et al. / Fluid Phase Equilibria xxx (2007) xxx xxx Fig. 4. Pressure and temperature dependence of the thermal conductivity of isopentane. data were correlated with a simple linear equation λ = a (1) P + a (0) (2) where P is the pressure in MPa, λ in mw m 1 K 1 and a (1) and a (0) are the coefficients of the linear equation. For n-pentane, our results, extrapolated to a pressure of 0.1 MPa, agree well with data from Vargaftik [17] (AAD 1.07%) and Smith et al. [18] (AAD 0.95%), while other data sources show a lower temperature dependence of the thermal conductivity, leading to average absolute deviations of 3.82% [16] and 2.83% [19]. For isopentane, our data, extrapolated to a pressure of 0.1 MPa, agree best with the values calculated using the NIST Chemistry Webbook [16] (AAD 1.52%) and are in fair agreement with DIPPR 801 data [19] (AAD 2.57%). Two other literature sources [20,21] show systematically lower values (AAD 4.25% and AAD 6.14%) than all other sources for isopentane. Besides the measurements for the pure compounds, three different mixtures of n-pentane with isopentane were prepared. The compositions of the studied mixtures were the following: Mixture A, mol% of isopentane and mol% of n- pentane, Mixture B, mol% of isopentane and mol% of n-pentane and Mixture C, mol% of isopentane and Fig. 6. Pressure and temperature dependence of the thermal conductivity of Mixture B, mol% of isopentane and mol% of n-pentane mol% of n-pentane. No mixture data for this system were found in the literature. Figs. 5 7 shows the results obtained for the three different mixtures. Table 1 presents the extrapolated values of thermal conductivity for the pure compounds and the studied mixtures, for a pressure of 0.1 MPa, obtained by application of Eq. (2). This equation was also used to interpolate the values of thermal conductivity for 0.2 and 0.5 MPa, also presented in the same table. All the experimental data points are provided as Supplementary material. The temperature dependence of the thermal conductivity is also represented by a linear equation λ = b (1) T + b (0) (3) where T is the temperature in K, b (1) and b (0) are the coefficients. In Tables 2 6, the experimental coefficients of the pressure and temperature dependence, from Eqs. (2) and (3), for the pure compounds and for each of the three investigated mixtures for the system of n-pentane + isopentane are presented. The linear temperature dependence of the experimental thermal conductivity at 0.1 MPa of the three mixtures is compared with the thermal conductivity of the pure compounds in Fig. 8. In all cases, the thermal conductivity appears to change linearly with the temperature. In some cases, the values of thermal Fig. 5. Pressure and temperature dependence of the thermal conductivity of Mixture A, mol% of isopentane and mol% of n-pentane. Fig. 7. Pressure and temperature dependence of the thermal conductivity of Mixture C, mol% of isopentane and mol% of n-pentane.

5 R. Dohrn et al. / Fluid Phase Equilibria xxx (2007) xxx xxx 5 Table 1 Values of thermal conductivity for the pure compounds and the studied mixtures, for different pressures and temperatures, calculated by means of Eq. (2) T (K) λ (mw m 1 K 1 ) 0.1 MPa 0.2 MPa 0.5 MPa n-pentane Mixture A: mol% i-c mol% n-c Mixture B: mol% i-c mol% n-c Mixture C: mol% i-c mol% n-c Isopentane Table 2 Pressure and temperature dependence of the thermal conductivity of n-pentane Table 4 Pressure and temperature dependence of the thermal conductivity of Mixture A, mol% of isopentane and mol% of n-pentane Table 5 Pressure and temperature dependence of the thermal conductivity of Mixture B, mol% of isopentane and mol% of n-pentane Table 6 Pressure and temperature dependence of the thermal conductivity of Mixture C, mol% of isopentane and mol% of n-pentane Table 3 Pressure and temperature dependence of the thermal conductivity isopentane Fig. 8. Temperature dependence of the thermal conductivity at 0.1 MPa of n- pentane, isopentane and the respective mixtures.

6 6 R. Dohrn et al. / Fluid Phase Equilibria xxx (2007) xxx xxx Table 7 Values obtained for the parameters in Eq. (4), for the fitting of the experimental values of thermal conductivity for the pure compounds studied in this work Data set A B C D Average deviation (%) No. of data points N 2 (NIST) N 2 (experiment) n-pentane Isopentane conductivity obtained for the mixtures are smaller than those obtained for pure n-pentane. This may be due to scattering of the experimental data. However, the possibility of the existence of a minimum should not be disregarded. 4. Modeling 4.1. Correlation of the pure-component thermal conductivity We propose a simple equation to correlate the thermal conductivity of pure compounds at different temperatures and pressures, assuming linear dependencies of the thermal conductivity of the temperature and of the pressure. λ = A + BT + CP + DTP (4) This equation was tested for nitrogen, with both the reference data calculated using the NIST Chemistry Webbook and the experimental data obtained in this work, and for the two hydrocarbons studied here. The pure component coefficients of Eq. (4) and the average deviations are presented in Table 7. The lowest absolute average deviation (0.08%) was found for the data calculated from NIST Chemistry Webbook for nitrogen, the highest for n-pentane (0.67%). It is worth mentioning that the higher values for the average deviation observed for the experimental data series are due to the scattering inherent in the experimental data Extended Wassiljewa model The thermal conductivity of the mixtures studied in this work was initially correlated using the Wassiljewa mixing rules [5] modified by Mason and Saxena [6,7]. The gas mixture thermal conductivity, λ m, using the Wassiljewa mixing rules can be calculated using Eq. (5), n y i λ i λ m = nj=1 (5) y j A ij i=1 where n is the number of components of the mixture, λ i is the thermal conductivity of pure component i, y i and y j are mole fractions of i and j, respectively. The Wassiljewa function, A ij, can be calculated as proposed by Mason and Saxena in Eq. (6), A ij = ε[1 + (λ tri/λ trj ) 1/2 (M i /M j ) 1/4 ] 2 [8(1 + M i /M j )] 1/2 (6) where M is the molecular weight, ε an adjustable parameter near unity and λ tr is the monatomic value of the thermal conductiv- Table 8 Values obtained for the fitting of ε for different pressures and temperatures, for the binary system n-pentane + isopentane P (MPa) ε K K K K K ity. The ratio of translational thermal conductivities, λ tri /λ trj,is calculated as proposed by Roy and Thodos [22,23], λ tri = Γ j[exp(0.0464t ri ) exp( T ri )] (7) λ trj Γ i [exp(0.0464t rj ) exp( T rj )] where T r is the reduced temperature and Γ is given by Eq. (8). ( Tc M 3 ) 1/6 Γ = 210 (8) P 4 c The parameter ε is usually considered to be constant for every mixture and independent of pressure and temperature. In this work, a new model was developed; an Extended Wassiljewa mixing rule, taking into consideration how the parameter ε changes with temperature and pressure. Table 8 presents the obtained values for the parameter ε, for different pressures and temperatures, in the study of the binary system n-pentane + isopentane. For this hydrocarbon mixture, ε lies close to one, varying between and 1.034, having a larger dependence on the temperature for higher pressures. Table 9 contains the values for ε calculated for the binary system nitrogen and cyclopentane [12]. For this system, ε varies between and for the considered temperature and pressure ranges. There is no reasonable explanation for the variation of ε with temperature and pressure. In fact, it was verified that ε varies in a completely different manner for the three binary systems considered in this work. For the system nitrogen + n-pentane Table 9 Values obtained for the fitting of ε for different pressures and temperatures, for the binary system nitrogen + cyclopentane P (MPa) ε K K K K K

7 R. Dohrn et al. / Fluid Phase Equilibria xxx (2007) xxx xxx 7 Table 10 Obtained values for the parameters in Eq. (9), for two of the binary systems to which the Extended Wassiljewa model was applied Binary system A 1 A 2 A 3 A 4 Average deviation (%) n-c 5 + i-c N 2 + cyclo-c for example (Bayer Technology Services GmbH, unpublished binary system data), the variation in the values of ε is much stronger than for the system n-pentane + isopentane, and a maximum occurs for lower values of temperature and pressure, precisely the conditions for which ε presents a minimum for the last system. For a third binary system, nitrogen and cyclopentane [12], the variation of ε with pressure and temperature does not follow any of the trends verified for the previously mentioned systems and its variation is also significantly stronger than what was verified for the system n-pentane + isopentane. This matter should be object of further studies in a future work. The ε values for different temperatures and pressures have been correlated with the newly developed four-parameter Eq. (9). ε = A 1 e A 2P T A 3P+A 4 (9) This equation was verified for values of ε obtained for three binary systems, with very good agreement. For the system n-pentane and isopentane, the application of the equation for temperatures between 333 and 413 K and pressures up to 0.5 MPa resulted in an average deviation of 0.10% of ε, with all the data points having an error smaller than 0.22%. For the binary system of nitrogen and n-pentane (Bayer Technology Services GmbH, unpublished binary system data), the results showed an average deviation of 0.14%, with all the points having an error inferior to 0.30%. For this system Eq. (9) was verified for values of pressure up to 1 MPa. Finally, the model was also tested for the system nitrogen and cyclopentane [12] up to a pressure of 1 MPa, with an average deviation of 0.19%. Table 10 presents the calculated values for the parameters of Eq. (9) for two of the binary system to which the model was applied. Fig. 9 shows the influence of the composition of the n- pentane + isopentane mixture on the thermal conductivity, at several temperatures between and K, for a pressure of 0.1 MPa. The lines in the graph refer to thermal conductivity values estimated by the application of the new Extended Wassiljewa model. The deviations from the experimental points to this model are presented in Fig. 10. The plot denotes a good agreement between the experimental values and the new Extended Wassiljewa model, with an average error of 0.38%. The same calculations were performed for other values of pressure up to 0.5 MPa. In these two particular cases, the use a four-parameter equation for the prediction of ε may not be entirely justifiable, due to the small change in the values of this parameter. For a pressure of 0.1 MPa for example, the variation is only for the range of temperatures between and K. For a pressure of 0.5 MPa, the variation in the same range of temperatures is close to It is arguable the utility of the method developed here as the use of a single value of ε, independent of the pressure and the temperature, can lead to results of comparable quality. It is important to consider however the limited range of temperature and pressure considered. Nevertheless, when other systems are considered, such as the one with nitrogen and cyclopentane [12] (in which ε has a variation of 0.10 in the range of temperatures between and K for a pressure of 0.1 Mpa), the need of an equation to discriminate ε for different values of pressure and temperature is undeniable. Considering a pressure of 0.1 MPa and the temperature range mentioned above, the prediction of values of thermal conductivity using a single value of ε, independent of the pres- Fig. 9. Composition dependence of the vapor-phase thermal conductivity of mixtures of n-pentane and isopentane at 0.1 MPa. The lines in the graph refer to the application of the new Extended Wassiljewa model. Fig. 10. Deviations from the experimental points to the Extended Wassiljewa model for a pressure of 0.1 MPa, for the binary system of n-pentane and isopentane.

8 8 R. Dohrn et al. / Fluid Phase Equilibria xxx (2007) xxx xxx 5. Conclusions Fig. 11. Graphic representation of the thermal conductivity values predicted using Eqs. (4) and (9), vs. the experimental data points at similar conditions of pressure and temperature, for the binary system n-pentane and isopentane. sure and temperature, leads to an average deviation of 1.5%, with errors that can go up to 3.7% for particular data points, while the use of the four parameter equation developed in this work results in an average deviation of 0.8% with a maximum error of 2.0% for individual data points. Once again, the consideration of broader ranges of pressure and temperature will only contribute to higher errors in the use of a single value of ε. In conclusion, it is safe to say that the use of the four parameter equation developed in this work brings considerable advantages in the prediction of thermal conductivity values when the variation of ε is higher than 0.1. However, and given the difficulty in predicting the behaviour of ε for different systems, the systematic use of this equation can constitute a good approach to ensure a more accurate prediction of thermal conductivity values for mixtures. The Extended Wassiljewa model was also applied to the system of nitrogen with n-pentane (Bayer Technology Services GmbH, unpublished binary system data), with an average error of 0.14% for a pressure of 0.1 MPa, increasing to 0.35% for a pressure of 1 MPa. It is possible to combine the two empirical models developed, Eqs. (4) and (9), to predict values of thermal conductivity for any temperature, pressure and composition. This was made with very satisfactory results for the studied system as it is observable in Fig. 11. The average deviation was 0.56% for 564 data points. Regarding the plot, it is necessary to take into consideration once again that the scattering of the points is caused by the scattering of the experimental data. The simultaneous application of these two models for the prediction of thermal conductivity data for any conditions of pressure, temperature and composition has also shown very good results for the binary system of nitrogen and n-pentane (Bayer Technology Services GmbH, unpublished binary system data), with an average deviation of 0.50% for 438 data points, with all the deviations for the individual data points under 1.5%. The thermal conductivity of three pure components and of the binary system of n-pentane and isopentane was measured at temperatures between 309 and 414 K and at pressures up to 1.0 MPa using an apparatus based on the transient hot wire method. These measurements were made after profound changes in the apparatus, including the re-design of the measuring cell, with an improvement in the support system of the thin platinum wires that constitute the basis of the transient hot-wire method. Further simplifications were made possible in the experimental set up, on account of a new experimental procedure developed in order to improve the precision of the results. The experimental thermal conductivity data was correlated with the Wassiljewa mixing rules as modified by Mason and Saxena, and predicted successfully using two new empirical models, a correlative model for the calculation of thermal conductivity values for any given pressure and temperature, for pure compounds, and the Extended Wassiljewa model through which the calculation of thermal conductivity of mixtures is possible for any pressure, temperature and composition, using a four parameter equation. For future work, further measurements should be conducted for the gathering of precise, accurate and reliable experimental data that will allow the further testing of the applicability of the two models developed here for other systems, now without the need of measuring so many experimental points. List of symbols a thermal diffusivity b cell radius C P heat capacity C constant, C = exp(γ) P total pressure q heat flow through the wire r radius of the wire t time T temperature rise of the wire T temperature Greek letters γ Euler s constant λ thermal conductivity ρ density Subscripts 0 designates the surface of the wire c critical conditions id ideal P at constant pressure r reduced value ref at reference conditions Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi: /j.fluid

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