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Thermal Conductivity of Gases Transient Hot-Wire Method Carla Susana Contreiras Louro Dissertation to obtain the Master Degree in Chemical Engineering Jury President: Dr. Maria Rosinda Costa Ismael (DEQB) Guides: Dr. Maria Gabriela da Silva Bernardo Gil (DEQB) Dr. Ralf Dohrn (Bayer Technology Services) Vogal: Dr. João Manuel Nunes Alvarinhas Fareleira (DEQB) Junho de 2008 Thermal Conductivity of Gases Acknowledgments This master thesis was the final step of my master degree in chemical engineering, was with big efforts and work that I have finally ended this stage of my life. I want to thank for the support and help given by Prof. Dr. Ralf Dohrn from Bayer Technology and Services, Prof. Dr. Gabriela Bernardo Gil and Prof. Dr. João Alvarinhas Fareleira from Instituto Superior Técnico. I also want to say thank you to all the technicians from the laboratory at Bayer Technology Services and also to Eng. José Fonseca, for the help given during my work. To all of them my thanks, because without them I wouldn’t have this great opportunity and I wouldn’t have accomplished it so successfully. I Thermal Conductivity of Gases Resumo Esta tese de mestrado teve como principal objectivo medir a condutividade térmica de compostos no seu estado gasoso, segundo o método transiente do fio quente. Para a calibração do aparelho mediu-se a condutividade térmica do azoto, sendo os resultados obtidos posteriormente comparados com valores da literatura (NIST). Posteriormente foram estudados dois compostos puros e dois sistemas binários gasosos. Para os sistemas binários realizaram-se medições experimentais de condutividade térmica para três diferentes composições molares. Os resultados experimentais obtidos foram comparados com valores de condutividade térmica calculados segundo: a Equação LPUR, o Wassiljeva Model e o Extended Wassiljewa Model. A Equação LPUR prevê o valor da condutividade térmica para compostos gasosos puros e o Wassiljeva Model e Extended Wassiljewa Model prevêem o valor da condutividade térmica de misturas gasosas. Ambos os modelos e a Equação LPUR têm sido continuamente desenvolvidos e aperfeiçoados, e verificou-se que estes se ajustam bem aos resultados experimentais obtidos. Palavras-chave: Condutividade Térmica; Método Transiente do Fio Quente; Equação LPUR; Wassiljewa Model; Extended Wassiljewa Model. II Thermal Conductivity of Gases Abstract The main objective of this master’s thesis was to measure the thermal conductivity of compounds in their gaseous state, according to the transient hot-wire method. For the calibration of the apparatus, the thermal conductivity of nitrogen was measured, and the results were subsequently compared with values from the literature (NIST). Afterwards two pure components were studied and also two gaseous binary systems. For the binary systems, the experimental measurements of thermal conductivity were carried out for three different ratios of molar composition. The obtained experimental results were compared with the thermal conductivity values calculated by: the LPUR Equation, the Wassiljewa Model and the Extended Wassiljewa Model. The LPUR Equation correlates the value of thermal conductivity for pure gaseous components and the Wassiljewa Model and Extended Wassiljewa Model is used for the calculation of the thermal conductivity of gas mixtures. Both models and the LPUR Equation have been continuously developed and improved, and it was checked that they fit well the obtained experimental results. Key-words: Thermal Conductivity; Transient Hot Wire Method; LPUR Equation; Wassiljewa Model; Extended Wassiljewa Model. III Thermal Conductivity of Gases Index ACKNOWLEDGMENTS...........................................................................................I RESUMO............................................................................................................. II ABSTRACT ....................................................................................................... III INDEX............................................................................................................... IV INDEX OF FIGURES........................................................................................... VI INDEX OF TABLES............................................................................................. XI 1. INTRODUCTION .............................................................................................. 1 2. THEORETICAL BACKGROUND .......................................................................... 3 2.1. THERMAL CONDUCTIVITY AND TRANSIENT HOT-WIRE METHOD [25][26] ........................ 3 2.1.1. THERMAL CONDUCTIVITY .................................................................................... 3 2.1.2. THE TRANSIENT HOT-WIRE METHOD ...................................................................... 5 2.1.2.1. The ideal model of the method .................................................................... 5 2.1.2.2 Properties of the Real Model ........................................................................ 7 2.1.2.3 Corrections to the Ideal Model...................................................................... 8 2.2. THE EMPIRICAL MODELS .................................................................................. 16 2.2.1. PURE COMPOUNDS FITTING EQUATION (LPUR EQUATION) ...........................................16 2.2.2. GAS MIXTURES MODELS ....................................................................................17 2.2.2.1 Wassiljeva Equation modified by Mason and Saxena ......................................17 2.2.2.2. Extended Wassiljeva Model ........................................................................18 3. APPARATUS AND EXPERIMENTAL PROCEDURE ............................................. 20 3.1. DESCRIPTION OF THE APPARATUS ...................................................................... 20 3.2. EXPERIMENTAL PROCEDURE .............................................................................. 24 3.2.1. EXPERIMENTAL PROCEDURE FOR THE MEASUREMENT OF PURE COMPONENTS .......................26 IV Thermal Conductivity of Gases 3.2.2 PREPARATION AND EXPERIMENTAL PROCEDURE FOR THE MEASUREMENT OF GAS MIXTURES.......27 4. EXPERIMENTAL RESULTS.............................................................................. 30 4.1. CALIBRATION OF THE APPARATUS ...................................................................... 30 4.2. PURE COMPONENTS ........................................................................................ 37 4.2.1. BLOWING AGENT 1 (BA1)..................................................................................37 4.2.2. BLOWING AGENT 2 (BA2)..................................................................................40 4.3. GAS MIXTURES .............................................................................................. 44 4.3.1. BLOWING AGENT 3 (BA3) AND BLOWING AGENT 1 (BA1) ............................................44 4.3.1.1. 26% BA3 – 74% BA1 .................................................................................44 4.3.1.2. 52% BA3 – 48% BA1 .................................................................................45 4.3.1.3. 75% BA3 – 25% BA1 .................................................................................47 4.3.2. BLOWING AGENT 2 (BA2) AND BLOWING AGENT 1 (BA1) ............................................48 4.3.1.1. 32% BA2 – 68% BA1 .................................................................................49 4.3.1.1. 55% BA2 – 45% BA1 .................................................................................50 4.3.1.1. 73% BA2 – 27% BA1 .................................................................................51 5. ANALYSIS OF THE EXPERIMENTAL RESULTS................................................. 53 5.1. ANALYSIS OF PURE COMPONENTS ...................................................................... 53 5.2. ANALYSIS OF GAS MIXTURES ............................................................................ 55 5.2.1. THE BLOWING AGENT 1 AND BLOWING AGENT 3 MIXTURE............................................55 5.2.2. THE BLOWING AGENT 1 AND BLOWING AGENT 2 MIXTURE............................................71 5.3. TESTING OF THE EXTENDED WASSILJEWA MODEL FOR OTHER MIXTURE ...................... 88 5.4. ALL MIXTURES ............................................................................................... 91 6. CONCLUSIONS .............................................................................................. 94 7. BIBLIOGRAPHY ............................................................................................ 96 8. APPENDIX .................................................................................................... 99 8.1. EXPERIMENTAL RESULTS .................................................................................. 99 V Thermal Conductivity of Gases Index of Figures Figure 1. The operating range of the transient hot-wire instrument.............................15 Figure 2. Scheme of the thermal conductivity apparatus used. ...................................20 Figure 3. Picture of the apparatus itself. ..................................................................21 Figure 4. Picture with some of the electronic part. ....................................................21 Figure 5. A scheme and a photo with the interior system of the measuring cell. ...........23 Figure 6. Schematic representation of the Teflon® sealing for the connections in the head of the cell. ...........................................................................................................23 Figure 7. Schematic representation of the Wheatstone bridge. ...................................24 Figure 8. Graphs of ∆T = f (ln t). ............................................................................26 Figure 9. Representation of the valves and tubing system of the apparatus. ................28 Figure 10. Thermal conductivity function of pressure for nitrogen; first calibration. .......30 Figure 11. Thermal conductivity function of pressure for nitrogen; final calibration. ......31 Figure 12. Thermal conductivity function of temperature for nitrogen at 1 bar..............33 Figure 13. Relative error of the correlation obtained from the experimental points with nitrogen for P = 1 bar, relatively to correlation of the NIST reference data. .................34 Figure 14. Relative error of the correlation obtained from the experimental points with nitrogen for P = 1 bar, relatively to correlation of José Fonseca. .................................34 Figure 15. Relative error of the correlation obtained from the experimental points with nitrogen for P = 2 bar, relatively to correlation of the NIST reference data. .................35 Figure 16. Relative error of the correlation obtained from the experimental points with nitrogen for P = 5 bar, relatively to correlation of the NIST reference data. .................36 Figure 17. Relative error of the correlation obtained from the experimental points with nitrogen for P = 10 bar, relatively to correlation of the NIST reference data.................36 Figure 18. Thermal conductivity function of pressure for Blowing Agent 1. ...................38 Figure 19. Thermal conductivity function of temperature for Blowing Agent 1 at 1 bar...38 Figure 20. Relative error of the correlation obtained from the experimental points with Blowing Agent 1 for P = 1 bar, relatively to correlation of Nelson Oliveira. ...................40 Figure 21. Thermal conductivity function of pressure for Blowing Agent 2. ...................41 Figure 22. Thermal conductivity function of temperature for Blowing Agent 2 at 1 bar...41 Figure 23. Relative error of the correlation obtained from the experimental points with Blowing Agent 2 for P = 1 bar, relatively to correlation of the NIST reference data. ......43 Figure 24. Relative error of the correlation obtained from the experimental points with Blowing Agent 2 for P = 1 bar, relatively to correlation of the DIPPR data....................43 VI Thermal Conductivity of Gases Figure 25. Thermal conductivity function of pressure for the mixture 26% BA3 - 74% BA1. ..........................................................................................................................45 Figure 26. Thermal conductivity function of pressure for the mixture 52% BA3 – 48% BA1. ....................................................................................................................46 Figure 27. Expectable behaviour of the thermal conductivity function of pressure when there is a leak in the apparatus. .............................................................................47 Figure 28. Thermal conductivity function of pressure for the mixture 75% BA3 – 25% BA1. ....................................................................................................................48 Figure 29. Thermal conductivity function of pressure for the mixture 32% BA2 – 68% BA1. ....................................................................................................................49 Figure 30. Thermal conductivity function of pressure for the mixture 55% BA2 – 45% BA1. ....................................................................................................................50 Figure 31. Thermal conductivity function of pressure for the mixture 73% BA2 – 27% BA1. ....................................................................................................................51 Figure 32. Relative error between the experimental values of thermal conductivity for nitrogen with the predicted values with the LPUR equation. .......................................54 Figure 33. Relative error between the experimental values of thermal conductivity for Blowing Agent 1 with the predicted values with the LPUR equation. ............................54 Figure 34. Relative error between the experimental values of thermal conductivity for Blowing Agent 2 with the predicted values with the LPUR equation. ............................55 Figure 35. Thermal conductivity function of the temperature for 1 bar for the mixture BA3 – BA1 and pure components...................................................................................56 Figure 36. Thermal conductivity function of the temperature for 2 bar for the mixture BA3 – BA1 and pure components...................................................................................56 Figure 37. Thermal conductivity function of the temperature for 5 bar for the mixture BA3 – BA1 and pure components...................................................................................57 Figure 38. λ function of the molar composition at 1 bar for the mixture BA3 – BA1. ......58 Figure 39. Relative deviations between the experimental data and the Wassiljewa Model at 1 bar, for the mixture BA3 – BA1. ........................................................................58 Figure 40. λ function of the molar composition at 2 bar for the mixture BA3 – BA1. ......59 Figure 41. Relative deviations between the experimental data and the Wassiljewa Model at 2 bar, for the mixture BA3 – BA1. ........................................................................59 Figure 42. λ function of the molar composition at 5 bar for the mixture BA3 – BA1. ......60 Figure 43. Relative deviations between the experimental data and the Wassiljewa Model at 5 bar, for the mixture BA3 – BA1. ........................................................................60 Figure 44. ε function of the pressure at some temperatures for the system BA3 – BA1. The lines are the values obtained by eq. 43. ..................................................................62 VII Thermal Conductivity of Gases Figure 45. Deviations of ε represented as a function of the pressure, for the system BA3 – BA1. ....................................................................................................................62 Figure 46. ε function of the temperature at some pressures for the system BA3 – BA1. The lines are the values obtained by eq. 43. ..................................................................63 Figure 47. Deviations of ε represented as a function of the temperature, for the system BA3 – BA1. ...........................................................................................................63 Figure 48. λ function of the molar composition at 1 bar for the system BA3 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. .............64 Figure 49. Relative deviations between the experimental data and the Extended Wassiljewa Model at 1 bar, for the mixture BA3 – BA1................................................65 Figure 50. λ function of the molar composition at 2 bar for the system BA3 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. .............65 Figure 51. Relative deviations between the experimental data and the Extended Wassiljewa Model at 2 bar, for the mixture BA3 – BA1................................................66 Figure 52. λ function of the molar composition at 5 bar for the system BA3 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. .............66 Figure 53. Relative deviations between the experimental data and the Extended Wassiljewa Model at 5 bar, for the mixture BA3 – BA1................................................67 Figure 54. Comparison of the individual data points measured with the values obtained for the same conditions of pressure and temperature using the LPUR Equation and the Extended Wassiljewa Model, for the system BA3 – BA1. .............................................68 Figure 55. Relative errors between the experimental thermal conductivity and the data obtained for the same conditions of pressure and temperature using the LPUR equation and the Extended Wassiljewa Model, for the system BA3 – BA1...................................69 Figure 56. Experimental results for the mixture 26% of Blowing Agent 3 and 74% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. ..................................................................................70 Figure 57. Experimental results for the mixture 52% of Blowing Agent 3 and 48% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. ..................................................................................70 Figure 58. Experimental results for the mixture 75% of Blowing Agent 3 and 25% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. ..................................................................................71 Figure 59. Thermal conductivity function of the temperature for 1 bar for the mixture BA2 – BA1 and pure components...................................................................................72 Figure 60. Thermal conductivity function of the temperature for 2 bar for the mixture BA2 – BA1 and pure components...................................................................................72 VIII Thermal Conductivity of Gases Figure 61. Thermal conductivity function of the temperature for 3 bar for the mixture BA2 – BA1 and pure components...................................................................................73 Figure 62. λ function of the molar composition at 1 bar for the mixture BA2 – BA1........74 Figure 63. Relative deviations between the experimental data and the Wassiljewa Model at 1 bar, for the mixture BA2 – BA1. ........................................................................74 Figure 64. λ function of the molar composition at 2 bar for the mixture BA2 – BA1. ......75 Figure 65. Relative deviations between the experimental data and the Wassiljewa Model at 2 bar, for the mixture BA2 – BA1. ........................................................................75 Figure 66. λ function of the molar composition at 3 bar for the mixture BA2 – BA1. ......76 Figure 67. Relative deviations between the experimental data and the Wassiljewa Model at 3 bar, for the mixture BA2 – BA1. ........................................................................76 Figure 68. ε function of the pressure at some temperatures for the system BA2 – BA1. The lines are the values obtained by eq. 43. ..................................................................78 Figure 69. Deviations of ε represented as a function of the pressure, for the system BA2 – BA1. ....................................................................................................................78 Figure 70. ε function of the temperature at some pressures for the system BA2 – BA1. The lines are the values obtained by eq. 43. ..................................................................79 Figure 71. Deviations of ε represented as a function of the temperature, for the system BA2 – BA1. ...........................................................................................................79 Figure 72. λ function of the molar composition at 1 bar for the system BA2 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. .............80 Figure 73. Relative deviations between the experimental data and the Extended Wassiljewa Model at 1 bar, for the mixture BA2 – BA1................................................81 Figure 74. λ function of the molar composition at 2 bar for the system BA2 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. .............81 Figure 75. Relative deviations between the experimental data and the Extended Wassiljewa Model at 2 bar, for the mixture BA2 – BA1................................................82 Figure 76. λ function of the molar composition at 3 bar for the system BA2 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. .............82 Figure 77. Relative deviations between the experimental data and the Extended Wassiljewa Model at 3 bar, for the mixture BA2 – BA1................................................83 Figure 78. Comparison of the individual data points measured with the values obtained for the same conditions of pressure and temperature using the LPUR Equation and the Extended Wassiljewa Model, for the system BA2 – BA1. .............................................84 Figure 79. Relative errors between the experimental thermal conductivity and the data obtained for the same conditions of pressure and temperature using the LPUR equation and the Extended Wassiljewa Model, for the system BA2 – BA1...................................85 IX Thermal Conductivity of Gases Figure 80. Experimental results for the mixture 32% of Blowing Agent 2 and 68% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. ..................................................................................86 Figure 81. Experimental results for the mixture 55% of Blowing Agent 2 and 45% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. ..................................................................................86 Figure 82. Experimental results for the mixture 73% of Blowing Agent 2 and 27% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. ..................................................................................87 Figure 83. ε function of the pressure at some temperatures for the system of nitrogen and methane. The lines are the values obtained by eq. 43.........................................88 Figure 84. Deviations of ε represented as a function of the pressure, for the system of nitrogen and methane...........................................................................................89 Figure 85. ε function of the temperature at some pressures for the system of nitrogen and methane. The lines are the values obtained by eq. 43.........................................89 Figure 86. Deviations of ε represented as a function of the temperature, for the system of nitrogen and methane...........................................................................................90 Figure 87. ε as a function of the temperature at 1 bar. ..............................................91 Figure 88. ε as a function of the temperature at 5 bar. ..............................................92 Figure 89. ε as a function of the pressure at 353K. ...................................................92 Figure 90. ε as a function of the pressure at 413K. ...................................................93 X Thermal Conductivity of Gases Index of Tables Table I. Parameters values, a and b, for each isotherm for nitrogen............................32 Table II. Thermal conductivity values at the pressure of 1 bar for the present work. .....32 Table III. Values of parameters, c and d, for nitrogen at 1 bar. ..................................33 Table IV. Values of parameters, c and d, for nitrogen at 2 bar....................................35 Table V. Values of parameters, c and d, for nitrogen at 5 bar. ....................................35 Table VI. Values of parameters, c and d, for nitrogen at 10 bar. .................................36 Table VII. Parameters values, a and b, for each isotherm for Blowing Agent 1..............39 Table VIII. Thermal conductivity values at the pressure of 1 bar for the present work. ..39 Table IX. Parameters values, c and d, for Blowing Agent 1 at 1 bar.............................39 Table X. Parameters values, a and b, for each isotherm for Blowing Agent 2. ...............42 Table XI. Thermal conductivity values at the pressure of 1 bar for the present work. ....42 Table XII. Parameters values, c and d, for Blowing Agent 2 at 1 bar. ..........................42 Table XIII. Parameters values, a and b, for each isotherm for the mixture 26% BA3 74% BA1. ............................................................................................................45 Table XIV. Parameters values, c and d, for mixture 26% BA3 - 74% BA1 at 1 bar. ........45 Table XV. Parameters values, a and b, for each isotherm for the mixture 52% BA3 – 48% BA1. ....................................................................................................................46 Table XVI. Parameters values, c and d, for mixture 52% BA3 – 48% BA1 at 1 bar. ........46 Table XVII. Parameters values, a and b, for each isotherm for the mixture 75% BA3 – 25% BA1. ............................................................................................................48 Table XVIII. Parameters values, c and d, for mixture 75% BA3 – 25% BA1 at 1 bar.......48 Table XIX. Parameters values, a and b, for each isotherm for the mixture 32% BA2 – 68% BA1. ............................................................................................................49 Table XX. Parameters values, c and d, for mixture 32% BA2 – 68% BA1 at 1 bar. .........50 Table XXI. Parameters values, a and b, for each isotherm for the mixture 55% BA2 – 45% BA1. ............................................................................................................50 Table XXII. Parameters values, c and d, for mixture 55% BA2 – 45% BA1 at 1 bar. .......51 Table XXIII. Parameters values, a and b, for each isotherm for the mixture 73% BA2 – 27% BA1. ............................................................................................................52 Table XXIV. Parameters values, c and d, for mixture 73% BA2 – 27% BA1 at 1 bar. ......52 Table XXV. Parameters values of the eq. 37 after a fitting process with the experimental data obtained for pure components in this work. ......................................................53 Table XXVI. ε values obtained after a fitting process for the different pressures and temperatures for the system BA3 – BA1. ..................................................................61 XI Thermal Conductivity of Gases Table XXVII. Parameters values for the Extended Wassiljewa Model (eq. 43) after a fitting process with the experimental ε. ............................................................................61 Table XXVIII. ε values obtained after a fitting process for the different pressures and temperatures for the system BA2 – BA1. ..................................................................77 Table XXIX. Parameters values for the Extended Wassiljewa Model (eq. 43) after a fitting process with the experimental ε. ............................................................................77 Table XXX. ε values obtained after a fitting process for the different pressures and temperatures for the system nitrogen and methane..................................................90 Table XXXI. Parameters values for the Extended Wassiljewa Model (eq. 43) after a fitting process with the experimental ε. ............................................................................91 Table A1.1. Selected results of the calibration for Nitrogen. .......................................99 Table A1.2. Selected results of the measurements for Blowing Agent 1. ....................100 Table A1.3. Selected results of the measurements for the mixture 26% BA3 – 74% BA1. ........................................................................................................................103 Table A1.4. Selected results of the measurements for the mixture 52% BA3 – 48% BA1.. ........................................................................................................................105 Table A1.5. Selected results of the measurements for the mixture 75% BA3 – 25% BA1. ........................................................................................................................108 Table A1.6. Selected results of the measurements for Blowing Agent 2.. ...................111 Table A1.7. Selected results of the measurements for the mixture 55% BA2– 45% BA1. ........................................................................................................................114 Table A1.8. Selected results of the measurements for the mixture 32% BA2– 68% BA1.. ........................................................................................................................116 Table A1.9. Selected results of the measurements for the mixture 73% BA2– 27% BA1.. ........................................................................................................................118 XII Thermal Conductivity of Gases 1. Introduction In the present work, measurements of the thermal conductivity of gases using the transient hot wire method were performed, which is the method recommended by IUPAC for this thermophysical property Due to their high thermal [5] . insulating capacity, rigid polyurethane (PUR) and polyisocyanurate (PIR) foams are used in a large number of applications, e. g. for thermal insulation boards, pipe insulation, technical refrigerant processes or in the appliance industry. The insulation efficiency of the PUR foam is mainly due to the gases trapped inside the closed cells (mainly blowing agents and carbon dioxide), which are responsible for 60 to 65% of the heat transfer through the foam. As the foam gets older it loses a considerable part of its thermal efficiency, due to diffusion into and out the foam (air diffuses slowly into the cells mixing with the blowing agent that at the same time diffuses out), then the composition of the gas in the closed cells is changing with time and the thermal conductivity of the gas mixture rises. This investigation is focused in the thermal conductivity of the gases used as blowing agents. The target is to explore blowing agents that have a very high thermal efficiency in the new foam and that are also superior to existing blowing agents during the lifetime of the foam. Until a few years ago, the chlorofluorocarbons (CFC‘s) were the most commonly use blowing agents. This family of substances had a great success in the market due to his excellent properties in refrigeration: not flammable, low toxicity, high stability, high inertness, high thermal efficiency, good compatibility with lubricants and low costs. However these substances have harmful effects due to ozone layer depletion and they contribute to the greenhouse effect. Due to their high stability they remain in the atmosphere until they go up into the stratosphere were they finally are broken down by ultraviolet radiation releasing a chlorine atom. To protect the environment the international community made some efforts, like establishing agreements limiting the use and production of some CFC’s and the development of alternative fluids. For this purpose, in 1987 the Montreal Protocol, an international treaty designed to protect the ozone layer by phasing out the production or eliminating a number of substances that are responsible for ozone depletion, was negotiated and signed. Since the Montreal Protocol came into effect, the atmospheric concentrations of the most important chlorofluorocarbons and related chlorinated hydrocarbons have either leveled off or decreased [22] . 1 Thermal Conductivity of Gases In 1997, the Kyoto Protocol was signed. It is the protocol of the International Convention on Climate Change with the main objective of reducing the greenhouse effect that causes climate change. The Kyoto Protocol is an agreement under which industrialized countries considerably reduce their collective emissions of greenhouse gases (carbon dioxide, methane, nitrous oxide, hydrofluorocarbons (HFC‘s)...) by 5.2 % compared to the year 1990 [22] . So the replacement of such fluids is pointed out as an urgent need. The most promising fluids to replace the completely halogenated blowing agents must not contain chlorine atoms. Hydrocarbons have been chosen in Europe [3] . In this thesis, two binary mixtures with different compositions of a Blowing Agent 1 (BA1) plus a Blowing Agent 3 (BA3) and a Blowing Agent 1 (BA1) plus a Blowing Agent 2 (BA2) were studied; this study was based on the measuring of their thermal conductivity; however the results obtained are independent of the properties of the foam matrix, the solid part of the foam. Blowing Agent 1 (BA1) and Blowing Agent 1 (BA1) mixtures have zero Ozone Depletion Potential and a significantly lower Global Warming Potential than HFC’s and, also have excellent thermophysical properties (high vapour pressure and low thermal conductivity, particularly at lower temperatures). All the measurements in this thesis were carried out at pressures up to 1.0 MPa, and at temperatures between 304K and 394K. In conclusion, the purpose of this investigation is to achieve a better understanding of the thermal conductivity of blowing agents; always trying to study mixtures that combine the best of two worlds, low thermal conductivity (large molecules) and at the same time high vapour pressure (small molecules) for cell stability (typical for appliances, like refrigerators). 2 Thermal Conductivity of Gases 2. Theoretical Background 2.1. Thermal Conductivity and Transient Hot-Wire Method [25][26] 2.1.1. Thermal Conductivity The thermal conductivity, λ, physically is the property of a material that indicates its ability to conduct heat, in other words the thermal conductivity of a fluid measures its propensity to dissipate energy (produce entropy) when disturbed from equilibrium by the imposition of a temperature gradient, ∇ T. For the isotropic fluids the thermal conductivity coefficient, λ, is defined by the linear, phenomenological relationship know as Fourier’s law, QC = − λ × ∇ T eq. 1 Where QC represents the instantaneous flux of heat, relative to the average motion of the fluid, in response to the imposed, instantaneous temperature gradient. This conductive heat flux is the macroscopic manifestation of the energy transported down the temperature gradient by the molecules themselves tending to equalize the temperature. However the impossibility of measuring local heat fluxes and of realizing the accompanying thermodynamic state in practice means that eq. 1 cannot be employed directly as a working equation. All measurements must be based on some integral effect and the accompanying thermodynamic state inferred by averaging. The main difficulty in performing accurate measurements of the thermal conductivity of fluid lies in the isolation of the conduction process from other mechanism of heat transfer. In turn, this arises from the contradictory requirement of imposing a temperature gradient on the fluid while preventing its motion. The imposition of a temperature gradient in a compressible fluid in the gravitational field of the earth inevitably creates a state motion (natural convection) so that pure conduction in a fluid is very difficult to achieve. The success of transient techniques for measurement of thermal conductivity of fluids is based on the fact that the characteristic time for the acceleration of the fluid by buoyancy forces is much longer than the propagation time of a temperature wave originated by a strong and localized temperature gradient. 3 Thermal Conductivity of Gases The advantages of the transient hot-wire technique are that it permits the user to obtain the thermal conductivity by use of an exact working equation resulting from a careful mathematical model of the instrument and to eliminate convective contributions to the heat transfer from the measurement. The working equation corresponds to an ideal relation between ideally measured variable and the thermal conductivity. The departure between this idealized mathematical model and the real experimental situation is represented by a consistent set of small, additive corrections. The transient hot-wire technique is an absolute technique and the instruments based on its principle are considered primary instruments and are capable of providing the highest accuracy possible at present. Fundamental equations The starting point for the formulation of the working equations for transient technique to measure the thermal conductivity of a fluid is the equation of energy conservation that, for a viscous, isotropic and incompressible fluid, with temperature dependent properties, can be written: ρ ρ ρ ρ DU = −∇.Q − P (∇.v ) − S : ∇v Dt eq. 2 Where U is the internal energy, t the time, P the hydrostatic pressure, hydrodynamic velocity of the fluid, S the tensor stress, r v the r Q the heat flux vector and ρ the density. The notation D/Dt represents the substantive derivative. On the assumption that the perturbation of the temperature is small and that a localequilibrium thermodynamic state exists, eq. 2 can be transformed to ρ .cv ρ α DT DP DT = −∇.Q + φ − T . P .− α P + k T Dt DT Dt kT Where cv is the heat capacity at constant volume, αP eq. 3 the isobaric expansion coefficient, r kT the isothermal compressibility and φ = S : ∇v is the rate of internal energy increase DP << α P , and owing to viscous dissipation. Transient techniques are operated so that kT DT the last equation can be written as: ρ .c P ρ DT = −∇.Q + φ Dt eq. 4 4 Thermal Conductivity of Gases where cp is the heat capacity at constant pressure. A general solution of eq. 4 is not possible; thus it is necessary to apply a number of further restrictions before it can be employed as the basis of determinations of thermal conductivity. In the first place we must assure that the temperature gradients to be produced are small, so that a nearequilibrium state is maintained. Secondly, any fluid movement must be avoided so that r v =0 and consequently φ =0. As already mentioned this is a difficult condition to achieve because any temperature gradient imposed on a fluid inevitably creates a state of motion owing to density differences: natural convection. It is therefore necessary to make measurements of the thermal conductivity in such a way that the effect of convection is negligible even if it is unavoidable. Under these conditions, the substantive derivative can be replaced by the partial derivative. The heat flux vector can be written in general r r Q = −λ .∇T + QR In which λ eq. 5 r is the thermal conductivity and QR is the heat flux arising radiation. Although there is always some contributions from radioactive transport there are some r circumstances under which it is negligible so that to formulate an ideal theory QR is assumed negligible in the present discussion. Thus, for an isotropic fluid with a temperature-independent thermal conductivity, density and heat capacity, eq. 5 can be written as: ρ .C P ∂T = λ.∇ 2 .T ∂t eq. 6 Equation 6 is the basis for all transient experimental methods for the measurement of the thermal conductivity. 2.1.2. The Transient Hot-wire Method 2.1.2.1. The ideal model of the method A transient thermal-conductivity measurement is one in which a time-dependent perturbation, in a form of a heat flux, is applied to a fluid initially in equilibrium. The thermal conductivity is obtained from an appropriate working equation relating the 5 Thermal Conductivity of Gases observe response of the temperature of the fluid to the perturbation. In principle, one can devise a wide variety of techniques of this kind differing in the geometry of the fluid sample employed and the nature of the time-dependent perturbation applied to it. However, the only geometrical arrangement which has gained general acceptance for application over a wide range of conditions is one in which the perturbing heat flux is applied by means of electrical dissipation in a thin, cylindrical wire. The perturbing heat flux itself has been applied in a manner of forms, including pulse, ramp and sinusoidal functions. However, most often the perturbation has been applied in a form of a step-up function, which is the case that will be explained in detail. In the ideal model of this instrument an infinitely-long, vertical, line source of heat possessing zero heat capacity and infinite thermal conductivity is immersed in a infinite isotropic fluid, with physical properties independent of temperature and in thermodynamic equilibrium with the line source at t=0. The transfer of energy from the line source, when a stepwise heat flux, q per unit length is applied, is assumed to be entirely conductive. We define the temperature rise in the fluid at a distance r from the wire, at a time t as, ∆T (r , t ) = T (r , t ) − T0 eq. 7 Where T0 is the equilibrium temperature of the fluid. Then to obtain ∆T(r,t) eq. 6 is to be solved subject to the boundary conditions, ∆T (r , t ) = 0 for t ≤ 0, any r lim r →0 ∆T (r , t ) = 0 lim r →0 r for t > 0, r = ∞ ∂T q =− = constant 2πλ ∂r for t ≥ 0, r = 0 eq. 8 eq. 9 eq. 10 With the additional condition that the thermal diffusivity, k =λ/ρ.CP, is constant. The solution of eq. 6 is, ∆Tid (r , t ) = − r2 E1 4πλ 4kt q eq. 11 Where E1(x) is the exponential integral with the expansion, 0 e− y E1 ( x ) = ∫ dy = −γ − ln + x + O x 2 y x ( ) eq. 12 6 Thermal Conductivity of Gases With γ =0.5772157...being Euler’s constant. If the line source is replaced by a cylindrical wire of radius r0, which assumes a uniform temperature, equal to that in the fluid of the ideal model at r=r0, then, for small values of r2/4kt, it is obtained the following equation. ∆Tid (r0 , t ) = T (r0 , t ) − T0 = q 4kt r02 + ... ln 2 + 4πλ r0 C 4kt eq. 13 Where C=exp( γ ). If the wire radius is chosen such that the second term on the righthand side of the eq. 13 is less than 0,01% of ∆Tid, it becomes clear that, in this ideal arrangement, the temperature rise of the wire is given by: ∆Tid (r0 , t ) = 4kt ln 2 4πλ r0 C q eq. 14 Eq. 14 is the fundamental working equation of the transient hot-wire technique. It suggests the possibility of obtaining the thermal conductivity of the fluid from the slope of the line ∆Tid vs ln t, while the thermal diffusivity may be obtained from its intercept or, more correctly, from the absolute value of ∆Tid at a fixed time. Any practical implementation of this method of measurement inevitably deviates from the ideal model. However, the success of the experimental method rests upon the fact that, by proper design, it is possible to construct an instrument that matches very closely the ideal description of it, making some of the deviations of negligible significance and others very small. 2.1.2.2 Properties of the Real Model In practice the hot wire used at any industrial installation has a length, diameter, heat capacity and thermal conductivity at the ends. The conduction phenomena will not be non axial, because the finite length of the wire causes an axial flux of energy. The heat dissipation is time dependent, because the wire temperature is changing during the measurement. The fluid around the wire, inside the measuring cell, is limited by the cell walls and has physical properties that dependent on the time and temperature. The radiation and convection phenomenons exist. The mains error sources are the heat transference by radiation and convection and also by conduction along the metal connections. 7 Thermal Conductivity of Gases 2.1.2.3 Corrections to the Ideal Model Corrections due to conditions at the Inner Boundary The practical version of a transient hot-wire instrument employs a thin metallic wire as both the heat source and the monitor temperature rise. The non-zero radius of such a wire, and the differences between its physical properties and those of the fluid, require modification of the ideal model to the inner boundary of the fluid. The effect of the nonzero radius alone is readily found by solving eq. 6 subject to the new condition, which replaces eq. 10, that ∂T −q = ∂r 2πλa at r0 = a for any t ≥ 0 eq. 15 At large values of 4kt/r2, the solution for the temperature rise of the fluid is 2 q 4kt a ln + ∆T (r , t ) = 2 4πλ r0 C kt eq. 16 This equation reveals that the temperature history of the fluid is independent of the radius of the hot-wire. It is, therefore, unnecessary in the construction of an instrument to secure accurate cylindricity of the hot wire. Owing to the non-zero heat capacity of the wire, (ρ.cp)w per unit volume, some of the heat flux generated within it is required to raise the temperature of the wire itself; it is, therefore, not conducted to the fluid. Moreover, because of the finite thermal conductivity of the wire material, λw, a radial temperature gradient exists in the wire. By solving the two coupled heat conduction equations for the wire, 0<r<a, and the fluid, a<r<∞, it is possible to deduce the temperature profile in both materials as a function of the time. Because the metallic heat source itself is also employed as a resistance thermometer in the measurements, the quantity required from the analysis is the average temperature rise of a cross-section of the infinitely long wire, ∆Tw. For sufficiently large values of kt/a2, this temperature rise is related to that of the ideal model by the following equation. ∆Tid (a, t ) = ∆TW + δT1 eq. 17 8 Thermal Conductivity of Gases Where the correction δT1 is δT1 = 2 2 2 4kt a (ρ .cP )W − (ρ .c p ) − q a − a + λ ln 2 4πλ a C 2λt 4πλ 2kt 4kW t 2λW q [ ] eq. 18 and kw that is the thermal diffusivity of the wire material, is equal to kW = λW (ρ .cP )W eq. 19 The last term in this correction is time-independent and, therefore, has no influence on the determination of the thermal conductivity from the slope of the line ∆tid vs ln t. Of the remaining time-dependent terms, only the first is significant in most applications. It arises solely from the finite heat capacity of the wire and causes the measured temperature rise to fall below the ideal value at short times. By the choice of suitably small radius and long measurement times, the magnitude of the correction may readily be limited to at most 0.5% [25] of the temperature rise, and it falls rapidly with increasing time so that eq. 18 is entirely adequate for its calculation. The choice of wires with radius of only a few microns implies that for gases at low density the dimensions of the wire are comparable with the mean free path of the gas molecules. Under these conditions, the temperature of the fluid at the wire, T (a,t), will differ from that of the wire, Tw(a,t), owing to the temperature-jump effect. The temperature jump effect is expressed by Smoluchowski equation ∂T TW (a, t ) − T (a, t ) = − g S ∂r r =a eq. 20 Where gS is an empirical factor proportional to the mean free path. A first order analysis of the consequences of the temperature jump leads to the conclusion that the principal effect is merely to shift all the measured ∆Tw vs ln t points along the temperature axis by a constant amount q 2gS 4πλ a δTK = eq. 21 compared to their positions in the absence of the temperature jump. Thus, the determination of the thermal conductivity from the slope of the line is unaffected. A second order analysis reveals that this shift is compounded with a small change in the 9 Thermal Conductivity of Gases slope of the line. However, the change in the thermal conductivity deduced from this slope is almost exactly compensated by the change in the temperature to which the measured thermal conductivity is referred, which results directly the shift δTK. Nevertheless, because a significant correction to the thermal conductivity may be necessary at low densities and because no reliable values for the factor gs are available, it is prudent in order to preserve the highest accuracy to exclude from measurements a low-density fluid region. For practical purposes, the lower density limit is approximately ρ lim ≈ 10 2 q(2.N A .σ 2 .λ .a.t ) −1 eq. 22 where σ is a rigid sphere diameter for the molecule and NA is the Avogadro’s number. For example for helium this limit corresponds to a pressure 0.5 MPa at 25°C. This limitation is not a severe one and above the density ρlim, the temperature-jump effect may be safely ignored. Corrections due to conditions at the Outer Boundary A practical instrument of the transient hot-wire type must incorporate an outer boundary for the fluid. Simplicity dictates that this boundary should be cylindrical, and it is located at r0 = b. During the initial phase of the transient temperature rise, the thermal wave spreading out from the wire will be unaffected by the presence of the boundary. However, as time goes on the heat flux at r0 = b rise to a non-negligible value, and this causes the temperature rise of the wire to fall below that of the ideal model. The introduction of the outer boundary requires the modification of the eq. 9 of the basic problem to read ∆T (r0 , t ) = 0 for r0 = b and any t ≥ 0 eq. 23 A solution to the modified problem for the practical situation when b/a>>1 and 4kt/a2>>1 has been given by Fisher. The temperature rise of the wire in the finite enclosure is related to that of the ideal model by the equation, ∆Tid (a, t ) = ∆TW (a, t ) + δT2 eq. 24 Here, the ‘outer-boundary correction’, δT2, is given by the expression 10 Thermal Conductivity of Gases δT2 = − g v2 .k .t q 4kt ∞ πY0 ( g v )2 ln + exp ∑ 2 2 4πλ b C v =1 b [ ] eq. 25 in which gV are the consecutive roots of J0(gV)=0 and Y0 is the zeroth-order Bessel function of the second kind. As would be expected intuitively, the correction increases with the time and the thermal diffusivity of the fluid, and decreases as the radius at the outer boundary increases. By a suitable selection of the radius of the outer boundary and the measurement time, the correction δT2 can be limited to 0.5% of the fluid temperature rise, even in gases at elevated densities or liquids; the correction is never significant in practice, owing to their low thermal diffusivity. Corrections due the Variable Physical Properties of the Fluid In the ideal model, it is supposed that the physical properties of the fluid: ρ, λ, cp and its viscosity, µ, are temperature independent. In reality these quantities are usually mild functions of temperature for both gases and liquids. Considering first the effect of introducing a variable fluid density, for the case of an infinitely long wire in an infinite fluid. The transient heating of the fluid now induces density variations which provide the buoyancy forces necessary to generate a velocity field. The convective motion has, in general, radial and longitudinal components; however, in the case of an infinitely long wire, only the radial component contributes to the heat transfer. Associated with the relative motion of the fluid there must, of course, be an irreversible generation of heat through viscous dissipation. In addition, some energy is expended reversibly in the expansion of the fluid. In both gases and liquids, an iterative solution of the fluid dynamic processes shows that all these effects contribute only a small amount to the temperature rise of the wire. In a practical thermal conductivity cell, where the heat source must be of finite length and must be attached to both ends to relatively massive supports and where the fluid is bounded by a finite wall, the foregoing analysis does not describe all of the effects. First, owing to the finite length of the wire, the one-dimensional regime of velocity and heat transfer characteristic of the infinite wire will not prevail over the entire length of the wire. In particular, as soon as the transient heating is begun, a three-dimensional temperature field develops in the fluid near the ends of the wire. The buoyancy forces which are generated cause upward acceleration of the fluid near the wire and cooler fluid from the bottom is brought upwards, cooling the wire faster then if there were conduction alone. It takes some time for this effect to become important by extending over a significant fraction of the wire length. However the flow patterns will extent over enough length of the wire making its average temperature rise become significantly 11 Thermal Conductivity of Gases different from that characteristic of the pure conduction regime. As this instant the observed temperature rise of the wire, suitably corrected by other effects, will depart from that of the ideal model. The problem of a transient, natural convection in a finite cylindrical geometry is not amenable to rigorous analysis. It is important to notice the time at which convective motion exerts a significant effect on the observed temperature rise of the wire. Typically, these limiting times are of the order of several seconds for gases and liquids. In practice, as has already been noted, the occurrence of a significant effect from natural convection in a measurement is easily discerned by a departure from the linearity of the ∆Tid vs ln t plot. Measurements in which such a curvature exists must be discarded. A further dynamic effect arises as a result of the temperature dependence of the fluid. As the heated layer of the fluid near the wire expands, it performs compression work on the remainder of the fluid in a container of fixed volume, V, and so modifies the temperature history of the wire. An approximate analysis of this effect in gases has shown that modification to the ideal temperature rise of the wires takes the form ∆Tid = ∆TW + δTC eq. 26 and δTC = q.l.R.T ρ .cP .cV .V eq. 27 Where R is the universal gas constant and l is the length of the wire. The correction may be rendered negligible by employing a sufficiently large container for the gas. Aside from the effects brought by the variable density of the fluid, it is necessary to account separately for the variation of the thermal conductivity, λ, and the product, ρ.cp. Since the temperature rises employed in the measurements are only a few degrees Kelvin, an analysis based upon a linear expansion of these properties about their values at the equilibrium state of the fluid may be employed. Then it results the following equation. ∆Tid = q 4kt ln 2 4πλ (Tr , ρ r ) a C eq. 28 The thermal conductivity, λ(Tr), obtained from the slope of the line relating ∆tid to ln t refers to a temperature Tr and a density ρr which differ from those of the equilibrium state. In fact, for measurements carried out in the time interval t1 to t2, 12 Thermal Conductivity of Gases Tr = T0 + [∆Tid (t1 ) + ∆Tid (t 2 )] 2 eq. 29 And ρ r = ρ (Tr , P ) eq. 30 Since the pressure P is essentially unaltered during the measurement. In case of measurements in fluid mixtures, a further phenomenon occurs. The imposition of a temperature gradient in a fluid mixture gives rise, in general, to a diffusive flux of mass driven by one or more composition gradients. In a transient experiment the composition of the mixture is initially uniform, but as the heating proceeds, the conductive heat flux is combined with a diffusive flux of mass tending to establish a composition gradient. The temperature rise for a mixture is given by: ∆Tid = 4k ' t ln 2 4πλ a C q eq. 31 Here, λ is still the thermal conductivity of the mixture in the absence of a net diffusive flux. On the other hand, k’, although time independent is not simply the thermal diffusivity of the mixture at equilibrium, but also includes thermal diffusion coefficients and mass diffusion coefficients. The effect of the non-zero mass flux is to introduce a small, constant shift of the temperature rises vs ln t line along the temperature axis relative to that for a pure gas with the same physical properties. Although the composition on the gas mixture at the wire varies during the measurement, owing to thermal diffusion, the temperature raises employed in practice are so small that the composition changes are insignificant. The measured thermal conductivity, therefore, refers to the equilibrium composition of the mixture. Corrections due to Radiation Effects In addition to the heat conducted away from the hot-wire through the fluid, it is inevitable that a small amount of the energy will be transmitted as electromagnetic radiation through the fluid in the cell. In the case when the fluid is essentially transparent to radiation of all wavelengths, the fluid plays no part in the radiation process. Assuming that all the cell surfaces act as black bodies, the radiative heat flux is give by 13 Thermal Conductivity of Gases ( ) qr = 2.π .a.σ B . TW4 − T04 ≈ 8.π .a.σ B .T03 .∆TW eq. 32 Where σB is the Stefan-Boltzman constant. This radioactive heat loss is equivalent to a reduction in the temperature rise of the wire of 8.π .a.σ B .T03 (∆TW )2 q δTrad ≈ eq. 33 Which amounts to not more than 0.002% of the wire temperature rises in practice and is negligible. For fluids that absorb radiation the effects are more significant, because the energy radiated from the wire is absorbed by an element of the fluid, increasing its temperature and causing it to emit radiation isotropically to other fluid elements. These processes, which occur at the speed of light, interact with the conduction process and modify the temperature history of the fluid and, thereby, of the heating wire. The presence or absence of radiation can be observed on ∆Tid vs ln t plot. If it is a straight line the radiation effects are not significant, but if it is a small curve the following correction must be done. ∆Tid = ∆TW + δTrad eq. 34 And 8.π .a.T03 .(∆TW ) = 1 a 1 1 + E − 1 + b E − 1q 2 δTrad eq. 35 Where E is the emissivity. Corrections due to the Finite Length of the Wire The wire in a practical thermal conductivity cell must be supported in the test fluid by relatively massive connections at either end. Because the heat flux is generated by electrical dissipation in the wire itself, there will have a longitudinal, conductive heat flux in both the wire and the fluid. As a result, the longitudinal temperature profile in the wire At any instant will not be uniform along its length. The resistance of the entire wire is not then an accurate measure of the temperature in a central section far removed from its ends. It is not possible to analyze this problem rigorously, although approximately 14 Thermal Conductivity of Gases calculations have been performed. These calculations yield the minimum length of wire necessary to ensure that at least a central section of the wire behaves as if it were a finite section of an infinitely long wire within a specific tolerance. Typically, for wires with a radius of several microns, the minimum length amounts to a few centimetres. It is then necessary to remove from the measurement the effects at the ends of the wire by experimental means and to observe the temperature rise of only the central section. In summary, in a transient hot-wire experiment; the thermal conductivity of a fluid is obtained from measurements of the temperature history, ∆TW, of a central section of a wire of radius a, which acts as a source of heat flux, q, per unit length. The thermal conductivity at a thermodynamic state (Tr, ρr, x) is derived from such measurements by application of the working equation, where x is the composition vector. ∆Tid = ∆TW + ∑ δTi = i q 4k t ln 2 0 4πλ (Tr , ρ r , x ) a C eq. 36 Where Tr is given by eq. 29 and ρr is the corresponding density at the equilibrium pressure, P. in a properly designed instrument, operating under well chosen conditions, the corrections to be applied to the measured temperature rise can be reduced to just two, δT1 for the heat capacity of the wire and δT2 for the finite outer boundary of the cell. These two corrections may themselves be rendered small by design. The range of thermodynamic states and the operational zone for which the working eq. 36 is appropriate illustrated schematically in Figure 1, which shows the exclusion of low densities by temperature-jump effects, long times by the influence of natural convection, and short times by virtue of the excessive magnitude of the heat capacity correction. Figure 1. The operating range of the transient hot-wire instrument. 15 Thermal Conductivity of Gases In principle, according to eq. 36, the thermal conductivity could be deduced from just one measurement of a pair of temperature versus time coordinates. However, an evaluation in this way would require an accurate knowledge of the wire radius and the thermal diffusivity of the fluid as well as of all of the time dependent and time independent corrections mentioned earlier, since they contribute to the absolute value of the temperature rise. Moreover, because eq. 36 represents only an asymptotic form of the full solution for the temperature rise, the complete solution in the form of the exponential integral solution would have to be employed. On the other hand, if the thermal conductivity is determined from the slope of the line constructed from many pairs of temperature rise-time points, the only additional information required to evaluate the thermal conductivity is the heat flux from the wire. Moreover, the observation of the evolution of the temperature rise provides the opportunity to establish that the instrument operates in accord with the mathematical model for it, since only in this case will the time dependence of the temperature rise in eq. 36 be preserved. 2.2. The Empirical Models 2.2.1. Pure Compounds Fitting Equation (LPUR Equation) The LPUR Equation makes a prediction of the thermal conductivity values for pure compounds. This equation was proposed and developed by José Fonseca [13] . The LPUR Equation is the following: λ = A + B.T + C.P + D.T .P eq. 37 This semi empirical equation has a linear dependence with the pressure and temperature. The four parameters A, B, C and D are obtained after a fitting process with the experimental data. The results obtained with this modeling are compared with the experimental data to check its realibility for the calculation of the thermal conductibility for pure components. 16 Thermal Conductivity of Gases 2.2.2. Gas Mixtures Models The thermal conductivity of a gas mixture is not usually a linear function of mole fraction. Generally, if the constituent molecules differ greatly in polarity, the thermal conductivity of the mixture is larger than would be predicted; for non-polar molecules the opposite trend is noted. The experimental results for the mixtures will be treated with the Wassiljeva Equation modified by Maxon and Saxena and with the Extended Wassiljeva Model. 2.2.2.1 Wassiljeva Equation modified by Mason and Saxena Wassiljeva Equation The Wassiljeva equation is used to predict the thermal conductivity values of gas [21] . mixtures for low pressure, it is presented bellow yi λi n λm = ∑ i =1 n ∑y j eq. 38 Aij j =1 Where λm is the thermal conductivity of the mixture, λi is the thermal conductivity of pure component i, (yi,yj) are the moles fractions of components i and j and Aij is a function of the binary system that is equal to 1. Wassiljeva Model modified by Mason and Saxena Maxon and Saxena suggested that Aij could be expressed as: Aij = λ ε 1 + tri λtrj 1/ 2 Mi M j M 81 + i M j 1/ 4 1/ 2 2 eq. 39 17 Thermal Conductivity of Gases Where M is the molecular weight (g/mol), λtr is the monatomic value of the thermal conductivity and ε is a numerical constant close to unit. And λtri Γ j [exp(0.0464Tri ) − exp(− 0.2412Tri )] = λtrj Γi [exp(0.0464Trj ) − exp(− 0.2412Trj )] eq. 40 Where Tri is the reduced temperature for pure component i and is equal to the reason between the temperature measured for the mixture (T) and the critical temperature of the component i (Tci); and Γi is defined by Tci M i 3 Γi = 210 4 P ci 1/ 6 eq. 41 PCi is the critical pressure of the component i. 2.2.2.2. Extended Wassiljeva Model In the last works was made an effort to understand how the parameter ε changes with the pressure, temperature and composition of the mixture. In almost all the literature ε is assumed to be a constant value, equal to 1, for all mixtures, temperatures and pressures. Once more José Fonseca developed a simple mathematical model that calculates the value of ε for any conditions [13] . He verified that for fixed values of pressure ε could be expressed as a function of the temperature by a two-parameter allometric equation. ε = a.T b eq. 42 After he verified that the parameters of the equation (a, b) could be written by two parameters equations dependents of the pressure. Where a is an exponential function of pressure and b is changing linearly with the pressure. Replacing those expressions on eq. 42 is obtained a four parameter equation (eq. 43). ε = A1.e A P .T A P + A 2 3 4 eq. 43 The values of the parameters are obtained after a fitting process with the experimental points. After is verified if the ε values calculated by this model are similar to the ε experimental points. 18 Thermal Conductivity of Gases The experimental points of ε are obtained by an individually fitting for each temperature from the experimental data, using the Wassiljeva Model modified by Mason and Saxena. It is also made a comparison between the values of thermal conductivity predicted by this model and the experimental ones for the same values of temperature and pressure. After is possible to take conclusions about the model validation. 19 Thermal Conductivity of Gases 3. Apparatus and Experimental Procedure 3.1. Description of the Apparatus A scheme of the apparatus used in this work is represented bellow. Figure 2. Scheme of the thermal conductivity apparatus used. Several changes have been made by other authors to improve the operation and the performance of the apparatus. The apparatus was constructed in the Thermophysical Property’s Laboratory of Bayer AG in Leverkusen in co-operation with the University of Stuttgart. This apparatus work in a temperature range between 300 and 500 K and at pressures from 0.1 MPa to 1.8 MPa. As a safety precaution the apparatus has a diaphragm that breaks for pressures above 2 MPa. It can be divided into two main parts: the electronic part, that includes the automation systems, temperature and pressure controllers, the platinum wires connected to a multimeter, to a power supply and to a Wheatstone bridge; and the other part corresponds to the apparatus it self, that includes the measuring cell, the tubing system and the heating system. 20 Thermal Conductivity of Gases Figure 3. Picture of the apparatus itself. Figure 4. Picture with some of the electronic part. As illustrated in figure 3 the apparatus is composed of two concentric cylinders made of stainless steel, each one has 39 cm of diameter and has an approximately total height of 82 cm. 21 Thermal Conductivity of Gases In the cylinder of bass there is a support for the measuring cell and around the measuring cell there is an electrical resistance used as heat source. Also in the upper part exists another electrical resistance for the same purpose. The apparatus is provided with a cooling system of copper tubes, which can use cold water or liquid nitrogen, for the case of a cold source is needed. The thermal equilibrium is reached with the help of a fan, situated in the bottom part of the apparatus. The measuring cell inside the cylinder of bass has a cylindrical shape and is also made of stainless steel. It has 48 mm of external diameter and 200 mm of length. The measuring cell has two holes in the bottom and on the top, with 16 mm of diameter, where the supports for the platinum wires were built. The cell has inside two platinum wires with different lengths, 0.04366 m and 0.12189 m, in order to account the end effects as well as other possible sources of errors. They both have a diameter of 10 µm. The upper edge of each thin platinum wire is welded with gold to a rigid and fixed tick wire made of platinum and to assure that the wire is on the middle of the cell, was placed a ‘guide’ made of Teflon®. In the end each thin wire is also welded with gold to a rigid part made of platinum that can only move axially along the cell. This measurement cell described above is new and was implemented by José Fonseca [13] . This new structure brought some benefits, like for example, the welding of new wires, when needed, is now much easier than before. It was also possible to overcome ambiguities related to the stretch, position and stability of the wires, during transportation and placement of the cell inside the apparatus, and also during the measurements. The wires inside the cell should not be completely stretched, when they are welded, because of the thermal expansion effects. It means that a raise in the temperature during an experiment causes the dilation of the stainless steel cell that is superior to the expansion in the length of the platinum wires, and this can break the wires. The thermal expansion coefficient of platinum and stainless steel at 25° C are respectively 8.8 µm·m−1·K−1 and 17.3 µm·m−1·K−1. However the thermal expansion affects the wire at maximum of only half a millimeter, which means that after the wire is stretched at room temperature, it is possible to move the part in its lower end by 0.5 mm. Although is better to give to the wires a security margin so they don’t brake. 22 Thermal Conductivity of Gases Figure 5. A scheme and a photo with the interior system of the measuring cell. The head of the cell contains the electric connections between the platinum wires and the electronic part of the apparatus. The sealing around the electrical connections in the head of the cell, was made from Teflon®. However Nelson Oliveira [12] proposed a new model based on ceramic sealing due to some problems with the Teflon® sealing, but this new cell was built from the spare measuring cell that contained the Teflon® sealing. Figure 6. Schematic representation of the Teflon® sealing for the connections in the head of the cell. 23 Thermal Conductivity of Gases The most relevant part of the electronic devices is the automatic Wheatstone bridge, that determines the variation of the wires potential with time and calculates the temperature increasing during the transient heating, from which the thermal conductivity is calculated. Figure 7. Schematic representation of the Wheatstone bridge. The program Lambda 2000 was written in the programming language C++ for these experiments and all the electronic parts of the apparatus are controlled by this software. The program needs of some properties of the studied gas and of the equipment, among other things, to run. A PID controller connected to a Pt 100 thermometer does the temperature control. There are other thermometers placed in the apparatus, but the most important ones are placed inside the apparatus, one on the top and one on the bottom. The temperature measured by these two thermometers should not differ more than 0.1 K. 3.2. Experimental Procedure The first step, before starting any measurement, is to clean the apparatus inside. For that vacuum must be done during some time (one hour or more) at 100°C. This cleaning is long because of the connections tubes are very thin. Afterwards the measuring cell must be cleaned with the gas under study. For that, the cell is filled at least twice with the gas and is made a good vacuum again. 24 Thermal Conductivity of Gases The next step is to program the desired temperature in the apparatus and after it is stable, fill the cell with the gas until the desired pressure is obtained. The program Lambda 2000 can be started and it is possible to check a more precise value of the temperature and pressure target, because the program gives a pressure value with three decimal places and the corresponding temperature is known with greater accuracy, since the value recorded by the computer is given by two PT 100 placed in the top and in the bottom of the cell. The measurements of the thermal conductivity start when both pressure and temperature are stable. One point is considered valid, in the software, when ∆T = 2.000+/-0.025 K and 0.03<∆ (Kurz/Lang) <0.10. If the gas to be studied is in a high pressure bottle the measuring cell if filled directly from the gas bottle. However if the working gas is liquid at room temperature is necessary to use a pressure vessel (bomb). For the liquid is necessary to calculate, using the liquid density, the maximum amount of liquid that can be placed inside the bomb. For safety reasons the bomb is filled at 90% of its volume and is necessary to calculate the maximum volume, i. e., the volume at the highest working temperature. The maximum mass is obtained using the density and this is the maximum mass that should be inside the bomb at room temperature. These calculations are very important, because if the vessel is completely full at room temperature the heating will lead to an overpressure in the vessel that can damage the vessel or even cause an accident. To fill the pressure vessel with the component is necessary first to do vacuum and then fill it, by suction, with the desired component. The bomb must be weighted after the vacuum and after the filling. After the compounds need to be degassed, i.e. to be freed from all the dissolved gases. This is done through 7 or 8 cycles of cooling-evacuating-warming, with a final evacuation for some seconds at room temperature. The cooling is made with dry ice (CO2) during half an hour and the warming is made with hot water. After each cycle the bomb must be weighted. If at the end of the cycles the weight of the liquid inside bomb is bigger than the calculated maximum mass, some liquid must be removed doing a vacuum of some minutes. The bomb is then ready to be placed inside the apparatus that was already set with the desired temperature for the measurement. Before starting measuring the program Lambda 2000 is used to check the equilibrium inside the cell. The time that takes to reach the equilibrium inside the cell is a long, even if the pressure and temperature are stable already; the gas inside the cell may not be yet at equilibrium. 25 Thermal Conductivity of Gases To know how is the gas equilibrium inside the cell is sufficient to observe the ∆T = f (ln t) graph, that in equilibrium is a straight line. The next pictures show the cell in non-equilibrium. Figure 8. Graphs of ∆T = f (ln t). On the first picture the set temperature was not yet stable and that it’s able to see by the points that are very unstable; the second picture shows occurrence of convection that corresponds to the curvature of the points. In total equilibrium none of these situations should happen. 3.2.1. Experimental Procedure for the Measurement of Pure Components To guarantee that a small amount of sample is used during the measurements, due to economic reasons, the procedure described bellow should be followed. First of all is important to plan the range of pressures and temperatures to the experiment. The measurement of the thermal conductivity starts at the lowest temperature and pressure, then for this temperature the pressure is increased, by putting more gas into the cell, and a new thermal conductivity measurement is made. The next isotherm will be at a higher temperature and the different pressures are obtained releasing some pressure of the cell, until a lower pressure is reached. The procedure for the next isotherms is equal to the one described above. For pure components in liquid phase at the room temperature the maximum pressure for each temperature must be around 80% of the saturation pressure. At higher 26 Thermal Conductivity of Gases temperatures the range of pressures is larger because the saturated vapour increases with the temperature. The pure liquid component is put in a pressure vessel bigger (150 cm3) than the ones used for the liquids to make the gas mixtures (50 cm3), this change is less time consuming because the smaller bomb runs out of component very quickly with the procedure described. As will be seen in the next sub-chapter the use of the small bomb in the experimental procedure for gas mixtures is not as time consuming. 3.2.2 Preparation and Experimental Procedure for the Measurement of Gas Mixtures The preparation of a mixture, at one desired composition, depends essentially of the temperature, the partial pressure for each component and of the vapour pressure of the mixture at the highest temperature. The partial pressures for each component are achieved using the program Aspen Plus 2004.1, that calculates the volume flow, and consequently it is calculated the molar density of the mixture. The necessaries steps for these calculations are explained below. For a given amount of the component j and for a desired composition the molar density of the mixture is calculated using the volume flow given by the program Aspen Plus 2004.1, calculated from the maximum temperature and partial pressure of j. After the temperature and the composition are kept constant and the total pressure of the mixture is varied until the density of the mixture matches with the calculated molar density of the mixture. At the end of this steps the values for the partial pressure for each component were achieved, however is very difficult to put exactly that values of pressures in the apparatus, so the calculations were repeated with the real values of pressure obtained to know the exact molar density and real composition of the mixture. After the mixture is inside the measuring cell it takes around half an hour to reach the equilibrium. Once the equilibrium is achieved it is possible to start measuring values of thermal conductivity. The mixture was prepared for the maximum pressure and temperature desired, so the measurements are performed fixing the pressure and changing the temperatures between the values of the several isotherms. After the measurements for this pressure at all the temperatures finishes, some pressure is released from the cell, repeating the same process to all the temperatures again. 27 Thermal Conductivity of Gases This procedure is very different from the one for pure components. It is not possible to measure continuous isotherms but continuous isobars; because the composition would be lost and it is very difficult to achieve the same composition again, as explained before. This procedure is more time consuming since it takes three hours or more to establish the new set value of temperature. The apparatus is not in a temperature-controlled room and then the same settings for the temperature controllers may lead to different temperatures of the apparatus. This can be overcome by changing the input on the temperature controllers; however this correction means more time consumed. To release pressure from the cell it is necessary to take account of the Joule-Thompson effect. The pressure decrease leads to an expansion of the gases and consequently their cooling, this may provoke the condensation of the less volatile component, changing the composition of the mixture that is released from the cell and the one that is inside the cell. For volatile gases or mixtures that contain these components this is a problem, but for pure gases like nitrogen this constitutes no problem. In the figure below a scheme of the tubing and valve system of the apparatus is shown. Figure 9. Representation of the valves and tubing system of the apparatus. To avoid this problem the following procedure should be done. Before taking out pressure from the cell, valve 7 must be opened and it is done vacuum to the tubing system. Valve 8 is always closed for this vacuum and also during the measurements. After valve 7 is closed and valve 8 is opened promoting an expansion of the gas mixture that is smaller that the one mentioned before. Even if some condensation occurs it is instantaneous, because the system will re-establish the equilibrium. After more or less 28 Thermal Conductivity of Gases than one minute a homogeneous composition is achieved, so valve 8 is closed and the space between this valve and valve 7 is evacuated again. This process is repeated until the desire value of pressure is achieved. With this procedure the composition of the gas mixture is conserved for a pressure decrease. Valve 4 can be used as security valve if valve 7 fails, because the procedure described above can also be done with this valve; and also to protect the sensor P1 from overpressures in case of failure of valves 1 to 3. Is important to mentioned that valve 4 can be used in normal conditions when it is required to have big steps in the pressure, however the time to guarantee a homogeneous composition must be respected and it is a little bit more that one minute. One of the input properties for running the program Lambda 2000 is the critical temperature (TC), and for a binary mixture TCM is given by: 2 2 TCM = ∑∑ xi .x j .Tcij eq. 44 i =1 j =1 And TC11 = TC1 ; TC 22 = TC 2 ; eq. 45 TC12 = TC1 .TC 2 = TC 21 Where x1 and x2 are the molar compositions of component 1 and 2; and TC1 and TC2 are the critical temperatures of the pure components 1 and 2, respectively. 29 Thermal Conductivity of Gases 4. Experimental Results 4.1. Calibration of the Apparatus To calibrate the apparatus some isotherms of the thermal conductivity for nitrogen were measured. After this experimental values were compared with referenced values from NIST. Before starting any measurements the wires were welded inside the measuring cell, because they were broken. The length of the wires was measured with a cathetometer. The first measurements for the calibration are showed in the graphic bellow and were carried out at temperatures between 304 K and 363 K and pressures from 2 bar to 8 bar. 31 T = 304.16 K T = 322.87 K 30 NIST (T = 300K) NIST (T = 305K) -1 -1 λ (mW.m .K ) 29 NIST (T = 320K) NIST (T = 325K) 28 NIST (T = 340K) NIST (T = 345K) 27 T = 344.02K NIST (T = 360K) 26 NIST (T = 365K) T = 362,61K 25 1 2 3 4 5 6 7 8 9 P (bar) Figure 10. Thermal conductivity function of pressure for nitrogen; first calibration. As can be seen in Figure 10 the experimental isotherms are distant from the NIST isotherms, this happened because the length of the wires must be corrected by a factor. When the wires are fixed they are not completely straight and the cathetometer doesn’t account for that, so the length measured must be corrected with a factor between 0 – 5%. So to achieve the true values for the length of the wires several measurements were 30 Thermal Conductivity of Gases made with different input values of the lengths until the results were in agreement with the NIST reference data. After these first measurements the long wire broke several times and also the short wire once, this happened because both wires were too much stretched. To solve this problem the wires were welded with some ‘little waves’, because, as it was mentioned before, it is good to give a security margin to the wires and after this welding the wires did not broke again. One of the two pressure sensors became damaged, consequently a new one was installed in the apparatus and a calibration of the pressure sensor was done. This calibration was made using the Lambda program (that has an option to calibrate the read pressure on the computer) and with a pressure controller/calibrator. The experimental isotherms for nitrogen obtained with the new wires and with the new pressure sensor are in next figure and were carried out at temperatures between 312 K and 392 K and pressures from 2 bar to 8 bar. 33 NIST (T = 310K) 32 NIST (T = 315K) NIST (T=350K) -1 -1 λ (mW.m .K ) 31 NIST (T=355K) NIST (T = 390K) 30 NIST (T = 395 K) 29 T = 391.29K T = 353.84K 28 T = 312.62K 27 26 1 2 3 4 5 6 7 8 9 P (bar) Figure 11. Thermal conductivity function of pressure for nitrogen; final calibration. In this graphic is possible to check the good accuracy between the experimental points with the NIST reference data. All the experiments were performed with the temperature inside the Wheatstone bridge being constantly monitored with a resistance thermometer PT 100 inside the instrument, and with a help of a tube with cold compressed air, positioned directly inside the 31 Thermal Conductivity of Gases instrument, the temperature was always kept between 21 C and 26 C. It is important to keep this temperatures inside the instrument, otherwise the values of λ starts to increase very fast for one isotherm. The λ values have the following relation with the pressure, for one temperature: λ (mW .m −1 .K −1 ) = a + b.P (bar ) eq. 46 Where a and b are the parameters of a linear equation and is expected that b should be always positive (positive slope), because when the pressure rises the thermal conductivity is also rising. In the next table are presented the parameters values for each isotherm. Table I. Parameters values, a and b, for each isotherm for nitrogen. T (K) 312.62 a 26.334 b 0.0652 353.84 29.000 0.0653 391.29 31.718 0.0336 After was made an extrapolation of the experimental isotherms and an interpolation of the NIST reference data for a pressure of 1 bar; this allows to make an evaluation of the viability of calibration. The results were also compared with the extrapolated isotherms for nitrogen of José Fonseca [13] . In the next table is presented the thermal conductivity values for nitrogen at 1 bar for each temperature. Table II. Thermal conductivity values at the pressure of 1 bar for the present work. 1 bar Present Work (Nitrogen) T (K) 312.62 λ (mW.m-1.K-1) 31.752 353.84 29.094 391.29 26.399 The figure bellow shows the behaviour of the thermal conductivity with the temperature at 1 bar, for the different data available and for the experimental results obtained in this work. 32 Thermal Conductivity of Gases 33 -1 -1 λ (mW.m .K ) 31 Present Work Jose Fonseca NIST 29 27 25 300 320 340 360 380 400 T (K) Figure 12. Thermal conductivity function of temperature for nitrogen at 1 bar. As can be seen in Figure 12 the temperature has also a linear dependence with the thermal conductivity with a positive slope. λ (mW .m −1 .K −1 ) = c + d .T (K ) eq. 47 At 1 bar the equation parameters, c and d, for nitrogen were determined for this present work, NIST data and José Fonseca values. Table III. Values of parameters, c and d, for nitrogen at 1 bar. 1 bar Present Work c 5.713 d 0.0661 NIST data 6.882 0.0634 José Fonseca 6.995 0.0631 The next pictures shows the relative errors obtained with the correlation of the experimental results with nitrogen relatively to the correlation of NIST reference data and to the correlation of José Fonseca, at 1 bar. 33 Thermal Conductivity of Gases 0,0 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 -0,2 Relative error (%) -0,4 -0,6 -0,8 -1,0 -1,2 -1,4 T (K) Figure 13. Relative error of the correlation obtained from the experimental points with nitrogen for P = 1 bar, relatively to correlation of the NIST reference data. 0,0 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 -0,2 Relative error (%) -0,4 -0,6 -0,8 -1,0 -1,2 -1,4 T (K) Figure 14. Relative error of the correlation obtained from the experimental points with nitrogen for P = 1 bar, relatively to correlation of José Fonseca. At lower temperatures the relative error is bigger than for higher temperatures, however these values are in the range of acceptable values, being the calibration viable and 34 Thermal Conductivity of Gases workable. It is always expected to obtained deviations due to the measurements themselves, that’s why the same point is measured at least 10 times, and due to the temperature of the apparatus that is not always the same, because, as it was mentioned before, the apparatus is in a laboratory that does not have a control of the room temperature. The same extrapolation was made for pressures of 2, 5 and 10 bar and the obtained results were compared with the interpolation of the NIST reference data. The parameters values of eq. 45 and the relative error are shown below. Table IV. Values of parameters, c and d, for nitrogen at 2 bar. 2 bar Present Work c 5.904 d 0.0658 NIST data 6.933 0.0633 2 bar 0,2 5 0 5 0 0 40 39 39 38 0 5 5 38 37 37 36 5 0 0 36 35 5 35 34 5 0 34 5 0 0 33 33 32 32 0 5 31 31 Relative error (%) 0,0 -0,2 -0,4 -0,6 -0,8 T (K) Figure 15. Relative error of the correlation obtained from the experimental points with nitrogen for P = 2 bar, relatively to correlation of the NIST reference data. Table V. Values of parameters, c and d, for nitrogen at 5 bar. 5 bar Present Work c 6.478 d 0.0646 NIST data 7.010 0.0631 35 Thermal Conductivity of Gases 5 bar Relative error (%) 0,4 0,2 5 0 5 0 0 40 39 39 38 5 38 5 0 37 37 0 36 0 5 36 35 0 5 5 35 34 34 33 5 0 5 0 33 32 32 31 31 0 0,0 T (K) Figure 16. Relative error of the correlation obtained from the experimental points with nitrogen for P = 5 bar, relatively to correlation of the NIST reference data. Table VI. Values of parameters, c and d, for nitrogen at 10 bar. 10 bar Present Work c 7.435 d 0.0626 NIST data 7.388 0.0627 10 bar Relative error (%) 1,2 1,0 0,8 0,6 0,4 0,2 5 0 0 40 39 39 0 5 5 38 38 5 0 37 37 36 5 0 5 0 0 36 35 35 34 34 0 5 0 5 5 33 33 32 32 31 31 0 0,0 T (K) Figure 17. Relative error of the correlation obtained from the experimental points with nitrogen for P = 10 bar, relatively to correlation of the NIST reference data. As illustrated in the figures above the relative error continues to be tolerable, even for the highest pressure. 36 Thermal Conductivity of Gases Once more it is possible to conclude that the calibration is valid and that the apparatus is working well and so giving good experimental results. 4.2. Pure Components 4.2.1. Blowing Agent 1 (BA1) The measurements of the thermal conductivity for Blowing Agent 1 (BA1) were carried out at temperatures between 353 K and 394 K. The Blowing Agent 1 (BA1) used is from Sigma-Aldrich with a purity exceeding 99%. Carlos Antunes made the last measurements of this compound; however his results were not very consistent and did not present at some temperatures the expectable behaviour. Therefore was decided to carry out new measurements. The experimental isotherms and the extrapolation of the obtained results at 1 bar for Blowing Agent 1 are presented bellow. The extrapolation at 1 bar was compared with other existent data and is possible to observe that the experimental points are in good agreement with the points obtained by Nelson Oliveira (2000). The other data available is from Philips (ZF-TVG5, DI. R. Treckmann and Dr. C. Braden, Report 1594207-1 from 1995-01-05) and from EC at 1995 (ZF-TVG5, DI. R. Treckmann and Dr. C. Braden, Report 1594207-1 from 1995-01-05) and at 1998 (ZT-TE 5.3 Dr. R. Dohrn, Report DHR 60 November 1998). 37 Thermal Conductivity of Gases 22 21 λ (mW.m -1 .K -1) 20 T = 361.63K 19 T = 372.29K T = 353.71K 18 T = 394.00K 17 16 15 0 1 2 3 4 5 6 P (bar) Figure 18. Thermal conductivity function of pressure for Blowing Agent 1. 27 26 25 24 -1 -1 λ (mW.m .K ) 23 22 21 Present Work Oliveira EC 1995 EC 1998 Philips 1995 20 19 18 17 16 15 14 320 340 360 380 400 420 T (K) Figure 19. Thermal conductivity function of temperature for Blowing Agent 1 at 1 bar. The coefficients for eq. 44 and eq. 45 are presented in the next tables and also the values of the thermal conductivity for Blowing Agent 1 at 1 bar for each isotherm. 38 Thermal Conductivity of Gases Table VII. Parameters values, a and b, for each isotherm for Blowing Agent 1. T (K) 353.71 a 16.050 b 0.0713 361.63 17.006 0.0329 372.29 18.279 0.0360 394.00 20.926 0.0909 Table VIII. Thermal conductivity values at the pressure of 1 bar for the present work. 1 bar T (K) 353.71 λ (mW.m-1.K-1) 16.121 Present Work 361.63 17.039 (Blowing Agent 1) 372.29 18.315 394.00 21.017 Table IX. Parameters values, c and d, for Blowing Agent 1 at 1 bar. 1 bar Present Work c - 26.992 d 0.1218 Nelson Oliveira - 28.200 0.1254 Philips - 28.986 0.1329 EC 1995 - 24.392 0.1162 EC 1998 - 32.701 0.1398 The next picture shows the relative error obtained with the correlation of the experimental results with Blowing Agent 1 relatively to the correlation of Nelson Oliveira [12] , at 1 bar, since Oliveira results are the closer and the most recent ones. 39 Thermal Conductivity of Gases 0,0 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 Relative error (%) -0,2 -0,4 -0,6 -0,8 -1,0 -1,2 T (K) Figure 20. Relative error of the correlation obtained from the experimental points with Blowing Agent 1 for P = 1 bar, relatively to correlation of Nelson Oliveira. The results from this measurements show a good behaviour with the pressure and the calculated relative error of the extrapolation is in a tolerable range. It is possible to conclude that these experimental results are valid. 4.2.2. Blowing Agent 2 (BA2) The Blowing Agent 2 (BA2) used for the measurements was from Sigma Aldrich with a purity of 99%. The last measurements for this component were made a long time ago, so it was decided to make new measurements with the new cell implemented by José Fonseca. The experimental data was collected at temperatures between 350K and 374K and at pressures from 1.7 bar to 7.6 bar. The four experimental isotherms are showed in the next figure. 40 Thermal Conductivity of Gases 31 29 -1 -1 λ (mW.m .K ) 27 25 23 T = 373.84K 21 T = 322.26 K T = 393.84K 19 T = 350.93K 17 15 0 1 2 3 4 5 6 7 8 P (bar) Figure 21. Thermal conductivity function of pressure for Blowing Agent 2. The extrapolation of the obtained results at 1 bar for Blowing Agent 2 was compared with other existent data (DIPPR - Database of the American Institute of Chemical Engineers) and with the NIST reference values. 30 29 28 27 NIST BA2 -1 -1 λ (mW.m .K ) 26 Present Work 25 DIPPR 1987 24 23 22 21 20 19 18 17 300 320 340 360 380 400 T (K) Figure 22. Thermal conductivity function of temperature for Blowing Agent 2 at 1 bar. 41 Thermal Conductivity of Gases Is possible to observe that the experimental points are concordant with the other points, they have a bigger inclination compared with the NIST reference data and a slower one compared with the data from DIPPR. The coefficients for eq. 44 and eq. 45 are presented in the next tables and also the values of the thermal conductivity for Blowing Agent 2 at 1 bar for each temperature. Table X. Parameters values, a and b, for each isotherm for Blowing Agent 2. T (K) 322.26 a 18.870 b 0.2110 350.93 22.370 0.2233 373.84 25.648 0.1792 393.84 28.708 0.1344 Table XI. Thermal conductivity values at the pressure of 1 bar for the present work. 1 bar T (K) 322.26 λ (mW.m-1.K-1) Present Work 350.93 22.593 (Blowing Agent 2) 373.84 25.827 393.84 28.842 19.081 Table XII. Parameters values, c and d, for Blowing Agent 2 at 1 bar. 1 bar Present Work c - 25.012 d 0.1363 NIST data - 20.576 0.1236 DIPPR 1987 - 28.886 0.1473 The next pictures shows the relative errors obtained with the correlation of the experimental results with Blowing Agent 2 relatively to the correlation of NIST reference data and to the correlation of DIPPR data, at 1 bar. 42 Thermal Conductivity of Gases 2,5 Relative error (%) 1,5 0,5 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 -0,5 -1,5 -2,5 -3,5 T (K) Figure 23. Relative error of the correlation obtained from the experimental points with Blowing Agent 2 for P = 1 bar, relatively to correlation of the NIST reference data. 3,0 Relative error (%) 2,0 1,0 0,0 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 -1,0 -2,0 T (K) Figure 24. Relative error of the correlation obtained from the experimental points with Blowing Agent 2 for P = 1 bar, relatively to correlation of the DIPPR data. 43 Thermal Conductivity of Gases The relative errors obtained are higher in a range of temperatures that was not measured in this present work. However the thermal conductivity values measured are very closer of the NIST reference data and they have small relative errors at the measured isotherms, consequently is possible to conclude that the experimental results are suitable. 4.3. Gas Mixtures 4.3.1. Blowing Agent 3 (BA3) and Blowing Agent 1 (BA1) The experimental measurements for pure Blowing Agent 3 (BA3) were carried out by José Fonseca and these measures and the measures for pure Blowing Agent 1 (BA1) will be used further in the treatment of the obtained results for the mixture. The Blowing Agent 3 used for the preparation of the mixtures is from Sigma-Aldrich with a purity of 99.5%. A study of the relationship between the molar composition and the thermal conductivity was made, for this purpose were carried out measurements with a molar composition of 25%, 50% and 75% of Blowing Agent 3. The experimental measures were not carried out at very high pressures due to the low vapour pressure of these mixtures. 4.3.1.1. 26% BA3 – 74% BA1 The experimental measurements for the mixture 26% BA3 and 74% BA1 were performed at temperatures between 395K and 354K, four isotherms were measured, and at pressures from 1 bar to 5 bar. The exact composition achieved for the mixture was 25.67% of Blowing Agent 3 and 74.33% of Blowing Agent 1. In the figure below is presented the experimental isotherms and the next tables shows the values for the coefficients of eq. 44 and eq. 45. 44 Thermal Conductivity of Gases 24 23 22 λ (mW.m -1 .K -1) T = 394.51K 21 T = 373.76K 20 T = 362.64K T = 354.44K 19 18 17 0 1 2 3 4 5 P (bar) Figure 25. Thermal conductivity function of pressure for the mixture 26% BA3 - 74% BA1. Table XIII. Parameters values, a and b, for each isotherm for the mixture 26% BA3 - 74% BA1. T (K) 394.51 a 22.539 b 0.1790 373.76 19.839 0.1968 362.64 18.457 0.2060 354.44 17.551 0.1743 Table XIV. Parameters values, c and d, for mixture 26% BA3 - 74% BA1 at 1 bar. 1 bar Present Work c - 26.682 d 0.1251 4.3.1.2. 52% BA3 – 48% BA1 For this mixture the measurements were executed at temperatures between 395K and 354K, once more four isotherms were measured, and at pressures from 1 bar to 5 bar. The exact mixture composition was 51.63% of Blowing Agent 3 and 48.37% of Blowing Agent 1. The next figure shows the experimental isotherms for this mixture and the coefficients of eq. 44 and eq. 45 are presented in the next tables. 45 Thermal Conductivity of Gases 25 24 23 T = 394.84K λ (mW.m -1 .K -1) 22 T = 374.55K 21 T = 362.68K 20 T = 354.43K 19 18 17 0 1 2 3 4 5 6 P (bar) Figure 26. Thermal conductivity function of pressure for the mixture 52% BA3 – 48% BA1. Table XV. Parameters values, a and b, for each isotherm for the mixture 52% BA3 – 48% BA1. T (K) 394.84 a 23.051 b 0.2569 374.55 20.555 0.2329 362.68 18.960 0.2932 354.43 18.161 0.2055 Table XVI. Parameters values, c and d, for mixture 52% BA3 – 48% BA1 at 1 bar. 1 bar Present Work c - 25.401 d 0.1233 These experimental results were a second measurement for this composition, because the first measurements for this composition showed that the measuring cell had a leak and was discovered that it was in one of the teflon sealing on the head of the cell. The screw from the teflon was tight a little bit and the leak stopped. The next figure shows the behaviour of the thermal conductivity for one temperature when exists a big leak in the apparatus. 46 Thermal Conductivity of Gases 23 372.78K λ (mW.m-1.K-1) 22 21 20 19 0 1 2 3 4 P (bar) Figure 27. Expectable behaviour of the thermal conductivity function of pressure when there is a leak in the apparatus. 4.3.1.3. 75% BA3 – 25% BA1 The exact molar composition for this mixture was 74.68% of Blowing Agent 3 and 25.32% of Blowing Agent 1. Four isotherms were measured between 353 K and 394 K and at pressures from 1 bar to 7 bar. The experimental measurements of λ function of the pressure are shown in the figure below and in the next tables are the values for the coefficients of eq. 44 and eq. 45. 47 Thermal Conductivity of Gases 27 26 25 λ (mW.m -1 .K -1) 24 23 T = 393.91K 22 T = 373.80K T = 362.16K 21 T = 353.98K 20 19 18 0 1 2 3 4 5 6 7 8 P (bar) Figure 28. Thermal conductivity function of pressure for the mixture 75% BA3 – 25% BA1. Table XVII. Parameters values, a and b, for each isotherm for the mixture 75% BA3 – 25% BA1. T (K) 393.91 a 24.402 b 0.2706 373.80 21.777 0.2620 362.16 20.273 0.2868 353.98 19.364 0.2397 Table XVIII. Parameters values, c and d, for mixture 75% BA3 – 25% BA1 at 1 bar. 1 bar Present Work c - 25.589 d 0.1275 4.3.2. Blowing Agent 2 (BA2) and Blowing Agent 1 (BA1) The next binary system studied in this work was the Blowing Agent 2 plus Blowing Agent 1. Were carried out thermal conductivity measurements for three different molar compositions (25%, 50% and 75% of Blowing Agent 2) for several temperatures, four isotherms for each composition, and pressures. 48 Thermal Conductivity of Gases Later with the experimental results obtained for the mixtures and for the pure components was made an analysis of the system. 4.3.1.1. 32% BA2 – 68% BA1 The exact molar composition achieved for this mixture was 31.57% of Blowing Agent 2 and 68.43% of Blowing Agent 1. Four isotherms were measured at temperatures between 352K and 394K and at pressures between 7 and 1.3 bar. In the next figure are the isotherms obtained for this composition. 27 26 T = 393.79K T = 352.25K 25 T = 361.47K T = 373.08K -1 -1 λ (mW.m .K ) 24 23 22 21 20 19 18 0 1 2 3 4 5 6 7 8 P (bar) Figure 29. Thermal conductivity function of pressure for the mixture 32% BA2 – 68% BA1. In the next tables are presented the coefficients values of eq. 44 and eq. 45 for this mixture. Table XIX. Parameters values, a and b, for each isotherm for the mixture 32% BA2 – 68% BA1. T (K) 393.79 a 23.318 b 0.2666 373.08 20.439 0.3412 361.68 19.126 0.3337 352.25 18.113 0.3090 49 Thermal Conductivity of Gases Table XX. Parameters values, c and d, for mixture 32% BA2 – 68% BA1 at 1 bar. 1 bar Present Work c - 25.636 d 0.1248 4.3.1.1. 55% BA2 – 45% BA1 The thermal conductivity for this mixture was measured for four isotherms between 352K and 374K and at pressures from 1.7 bar to 7.4 bar. The exact molar composition obtained was 54.63% of Blowing Agent 2 and 45.37% of Blowing Agent 1. In the next picture are presented the experimental results for the thermal conductivity as a function of the pressure. 28 27 T = 393.79K T = 352.13K 26 T = 361.47K -1 -1 λ (mW.m .K ) 25 T = 373.25K 24 23 22 21 20 19 0 1 2 3 4 5 6 7 8 9 P (bar) Figure 30. Thermal conductivity function of pressure for the mixture 55% BA2 – 45% BA1. In the tables below are presented the coefficients values of eq. 44 and eq. 45 for this molar composition. Table XXI. Parameters values, a and b, for each isotherm for the mixture 55% BA2 – 45% BA1. T (K) 393.79 a 24.365 b 0.2775 373.25 21.691 0.2816 361.47 19.953 0.3890 50 Thermal Conductivity of Gases 352.13 18.852 0.3994 Table XXII. Parameters values, c and d, for mixture 55% BA2 – 45% BA1 at 1 bar. 1 bar Present Work c - 26.730 d 0.1304 4.3.1.1. 73% BA2 – 27% BA1 For this last mixture the exact molar composition determinated was of 73.48% of Blowing Agent 2 and 26.52% of Blowing Agent 1. The experimental measurements were carried out at pressures between 2 and 7.5 bar and temperatures from 352 K to 373 K, four isotherms were considered. In the next graphic is possible to observe the experimental results obtained for this mixture. 28 27 T = 393.49K T = 352.31K 26 T = 362.16K T = 373.20K -1 -1 λ (mW.m .K ) 25 24 23 22 21 20 19 0 1 2 3 4 5 6 7 8 P (bar) Figure 31. Thermal conductivity function of pressure for the mixture 73% BA2 – 27% BA1. The coefficients values of eq. 44 and eq. 45 for this molar composition are presented below. 51 Thermal Conductivity of Gases Table XXIII. Parameters values, a and b, for each isotherm for the mixture 73% BA2 – 27% BA1. T (K) 393.49 a 25.424 b 0.2290 373.20 22.368 0.3444 362.16 20.880 0.3182 352.31 19.910 0.2490 Table XXIV. Parameters values, c and d, for mixture 73% BA2 – 27% BA1 at 1 bar. 1 bar Present Work c - 27.595 d 0.1351 52 Thermal Conductivity of Gases 5. Analysis of the Experimental Results 5.1. Analysis of Pure Components For pure Blowing Agent 1, Blowing Agent 2 and nitrogen the LPUR Equation was applied to predict the values of the thermal conductivity. The next table shows the values for the parameters A, B, C and D of the LPUR Equation (eq. 37), after a fitting process with the experimental points for Blowing Agent 1, Blowing Agent 2 and nitrogen. Table XXV. Parameters values of the eq. 37 after a fitting process with the experimental data obtained for pure components in this work. A B C D Nitrogen 5.70678 0.06591 0.16153 -2.94×10-4 Blowing Agent 1 -26.58844 0.12038 -0.23906 8.59×10-4 Blowing Agent 2 -23.96158 0.13291 0.26205 -2.23×10-4 Was calculated the relative error between the experimental values of the thermal conductivity and the values estimated by the LPUR Equation (eq. 37) using the parameters above. As can be observed in the figures below the error is small, being smaller for nitrogen than for Blowing Agent 1 and Blowing Agent 2. However for Blowing Agent 1 the maximum relative error is of -1.67% and for Blowing Agent 2 is 1.89% that are lower error values. Is important mention that these are individual data points where the scattering of the data has to be counted. 53 Thermal Conductivity of Gases 1,00% 0,80% 0,60% 0,40% 0,20% 0,00% -0,20% -0,40% -0,60% -0,80% 25 26 27 28 29 30 31 32 33 -1 -1 λ (mW .m .K ) Figure 32. Relative error between the experimental values of thermal conductivity for nitrogen with the predicted values with the LPUR equation. 1,50% 1,00% 0,50% 0,00% -0,50% -1,00% -1,50% -2,00% 15 16 17 18 19 20 -1 21 22 23 -1 λ (mW . m .K ) Figure 33. Relative error between the experimental values of thermal conductivity for Blowing Agent 1 with the predicted values with the LPUR equation. 54 Thermal Conductivity of Gases 2,50% 2,00% 1,50% 1,00% 0,50% 0,00% -0,50% -1,00% -1,50% -2,00% 17 19 21 23 25 -1 27 29 31 -1 λ (mW . m .K ) Figure 34. Relative error between the experimental values of thermal conductivity for Blowing Agent 2 with the predicted values with the LPUR equation. The average error for nitrogen is 0.321% from 64 experimental points, for Blowing Agent 1 is 0.586% from 251 experimental points and for Blowing Agent 2 is 0.656% from 229 experimental points. It is possible to conclude that the LPUR Equation makes a good estimation of the thermal conductivity values for the pure components studied in this present work and as a result is valid for these components. 5.2. Analysis of Gas Mixtures 5.2.1. The Blowing Agent 1 and Blowing Agent 3 Mixture With the experimental isotherms obtained for the three mixtures and for the pure components was made a study of the dependence of the thermal conductivity with the molar composition. For this purpose extrapolations of the experimental results for 55 Thermal Conductivity of Gases constant values of pressure were made. The next figures show the dependence of the λ with the temperature for 1 bar, 2 bar and 5 bar. 1 bar 28 26 -1 -1 λ (mW.m .K ) 24 22 20 52% BA3 - 48% BA1 BA3 18 BA1 26% BA3 - 74% BA1 16 14 340 75% BA3 - 25 % BA1 360 380 400 T (K) Figure 35. Thermal conductivity function of the temperature for 1 bar for the mixture BA3 – BA1 and pure components. 2 bar 28 26 -1 -1 λ (mW.m .K ) 24 22 20 52% BA3 - 48% BA1 BA3 18 BA1 26% BA3 - 74% BA1 16 75% BA3 - 25% BA1 14 340 360 380 400 T (K) Figure 36. Thermal conductivity function of the temperature for 2 bar for the mixture BA3 – BA1 and pure components. 56 Thermal Conductivity of Gases 5 bar 28 26 -1 -1 λ (mW.m .K ) 24 22 20 52% BA3 - 48% BA1 BA3 18 BA1 26% BA3 - 74% BA1 16 14 340 75% BA3 - 25% BA1 360 380 400 T (K) Figure 37. Thermal conductivity function of the temperature for 5 bar for the mixture BA3 – BA1 and pure components. By the figures above is possible to observe that for all the pressures the thermal conductivity has a linear dependence with the temperature, which is the expectable behaviour (eq. 47); and is also possible to see that the slope of the isobaric lines and the value of the thermal conductivity is increasing with the pressure. It is evident that the thermal conductivity values for 52% and 26% of Blowing Agent 3 are very close to each other compared to the mixture of 75% of Blowing Agent 3. This can be explained by the several leaks that were find and corrected during the measurements, because is quite probable that the leaks changed the composition of the mixture inside the cell. For the last composition measured, 75% of Blowing Agent 3, there was no leak and as it will be possible to see below these results have a very small error with the empirical models. In the next figures λ is represented as a function of the molar composition for each isotherm at 1, 2 and 5 bar. The lines in the graphics match to the Wassiljewa Model with the Maxon and Saxena modifications, where the parameter ε was fitted for each temperature from the experimental points. Is also represented the deviations between the experimental points and the λ values estimated by the model. 57 Thermal Conductivity of Gases 31 Blowing Agent 3 + Blowing Agent 1 (1 bar) 29 27 -1 -1 λ (mW.m .K ) 25 23 21 19 393.15K 17 373.15K 363.15K 15 353.15K 13 413.15K 11 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA3) Figure 38. λ function of the molar composition at 1 bar for the mixture BA3 – BA1. Blowing Agent 3 + Blowing Agent 1 (1 bar) 2,10% 1,80% 353.15K 1,50% 363.15K 1,20% 373.15K 393.15K 0,90% 413.15K 0,60% 0,30% 0,00% -0,30% -0,60% -0,90% -1,20% -1,50% 1,00 0,75 0,52 0,26 0,00 mole fraction (BA3) Figure 39. Relative deviations between the experimental data and the Wassiljewa Model at 1 bar, for the mixture BA3 – BA1. 58 Thermal Conductivity of Gases 31 Blowing Agent 3 + Blowing Agent 1 (2 bar) 29 27 -1 -1 λ ( mW.m .K ) 25 23 21 19 17 393.15K 373.15K 15 363.15K 353.15K 13 413.15K 11 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA3) Figure 40. λ function of the molar composition at 2 bar for the mixture BA3 – BA1. Blowing Agent 3 + Blowing Agent 1 (2 bar) 1,80% 1,50% 353.15K 1,20% 363.15K 373.15K 0,90% 393.15K 0,60% 413.15K 0,30% 0,00% -0,30% -0,60% -0,90% -1,20% -1,50% 1,00 0,75 0,52 0,26 0,00 mole fraction (BA3) Figure 41. Relative deviations between the experimental data and the Wassiljewa Model at 2 bar, for the mixture BA3 – BA1. 59 Thermal Conductivity of Gases 31 Blowing Agent 3 + Blowing Agent 1 (5 bar) 29 27 -1 -1 λ (mW.m .K ) 25 23 21 19 17 393.15K 373.15K 15 363.15K 353.15K 13 413.15K 11 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA3) Figure 42. λ function of the molar composition at 5 bar for the mixture BA3 – BA1. 1,50% Blowing Agent 3 + Blowing Agent 1 (5 bar) 1,20% 353.15K 0,90% 0,60% 0,30% 363.15K 373.15K 393.15K 413.15K 0,00% -0,30% -0,60% -0,90% -1,20% -1,50% -1,80% 1,00 0,75 0,52 0,26 0,00 mole fraction (BA3) Figure 43. Relative deviations between the experimental data and the Wassiljewa Model at 5 bar, for the mixture BA3 – BA1. 60 Thermal Conductivity of Gases By the analyse of the figures is possible to observe that for 2 and 5 bar the Wassiljewa Model changed its behaviour compared to 1 bar, where it can be seen a curve. This exchange has to do with the ε value, that for 2 and 5 bar is under the unit and for 1 bar the ε is bigger than 1. It is also possible to check that for the composition of the 75% of Blowing Agent 3 the relative error is very small and as a result the experimental results are of good quality. For the other compositions the relative error is bigger as it was mentioned before. Other cause for these higher errors has to do with the uncertainty of some experimental results; however they are in a range of tolerable errors. On the other hand the model seems to predict well the λ values and so is possible to admit that these estimated values are more reliable for those compositions. In the next tables is presented the ε values obtained from the experimental results for each isotherm and different pressures and the parameters values of eq. 43 after a fitting process. Table XXVI. ε values obtained after a fitting process for the different pressures and temperatures for the system BA3 – BA1. P (bar) ε (353.15 K) ε (363.15 K) ε (373.15 K) ε (393.15 K) ε (413.15 K) 1 1.015657 1.015476 1.015311 1.015024 1.014782 2 0.992209 0.993809 0.995211 0.997553 0.999430 5 0.932043 0.938300 0.943819 0.953109 0.960626 Table XXVII. Parameters values for the Extended Wassiljewa Model (eq. 43) after a fitting process with the experimental ε. A1 A2 A3 A4 1.41836 -0.30872 0.04903 -0.05349 The next figures illustrate how the experimental ε and the predicted ε by eq. 43 change with pressure and temperature for this system and the respective deviations between them. 61 Thermal Conductivity of Gases 1,05 1 353.15K 0,95 ε 363.15K 373.15K 393.15K 413.15K 0,9 0,85 0 1 2 3 4 5 6 7 8 9 P (bar) Figure 44. ε function of the pressure at some temperatures for the system BA3 – BA1. The lines are the values obtained by eq. 43. 0,15% 353.15 K 363.15 K 0,10% 373.15 K 393.15 K 0,05% 413.15 K 0,00% -0,05% -0,10% -0,15% 1 2 5 P (bar) Figure 45. Deviations of ε represented as a function of the pressure, for the system BA3 – BA1. 62 Thermal Conductivity of Gases 1,05 ε 1 0,95 1 bar 2 bar 5 bar 0,9 0,85 320 340 360 380 400 420 440 Temperature (K) Figure 46. ε function of the temperature at some pressures for the system BA3 – BA1. The lines are the values obtained by eq. 43. 0,15% 1 bar 2 bar 0,10% 5 bar 0,05% 0,00% -0,05% -0,10% -0,15% 353,15 363,15 373,15 393,15 413,15 Temperature (K) Figure 47. Deviations of ε represented as a function of the temperature, for the system BA3 – BA1. The ε values for 5 bar changes much more with the temperature compared with the other pressures, which almost don’t change with the temperature. 63 Thermal Conductivity of Gases However how it can be observed the ε values predicted by the Extended Wassiljewa Model fit very well the experimental results, the maximum relative error is of 0.14% which is a very small value and consequently is possible to start admitting that eq. 43 is applicable for this mixture. After the thermal conductivity values by the Extended Wassiljewa Model were calculated and compared with the experimental values for the thermal conductivity. These calculations were made for 1, 2 and 5 bar; the results and respectively relative errors can be seen in the next figures. 31 Blowing Agent 3 + Blowing Agent 1 (1 bar) 29 27 -1 -1 λ (mW.m .K ) 25 23 21 19 393.15K 17 373.15K 353.15K 15 353.15K 413.15K 13 11 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA3) Figure 48. λ function of the molar composition at 1 bar for the system BA3 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. 64 Thermal Conductivity of Gases 2,10% Blowing Agent 3 + Blowing Agent 1 (1 bar) 1,80% 353.15K 1,50% 363.15K 1,20% 373.15K 393.15K 0,90% 413.15K 0,60% 0,30% 0,00% -0,30% -0,60% -0,90% -1,20% -1,50% 1,00 0,75 0,52 0,26 0,00 mole fraction (BA3) Figure 49. Relative deviations between the experimental data and the Extended Wassiljewa Model at 1 bar, for the mixture BA3 – BA1. 31 Blowing Agent 3 + Blowing Agent 1 (2 bar) 29 27 -1 -1 λ ( mW.m .K ) 25 23 21 19 393.15K 17 373.15K 363.15K 15 353.15K 413.15K 13 11 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA3) Figure 50. λ function of the molar composition at 2 bar for the system BA3 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. 65 Thermal Conductivity of Gases 2,10% Blowing Agent 3 + Blowing Agent 1 (2 bar) 1,80% 353.15K 1,50% 363.15K 1,20% 373.15K 393.15K 0,90% 413.15K 0,60% 0,30% 0,00% -0,30% -0,60% -0,90% -1,20% -1,50% 1,00 0,75 0,52 0,26 0,00 mole fraction (BA3) Figure 51. Relative deviations between the experimental data and the Extended Wassiljewa Model at 2 bar, for the mixture BA3 – BA1. 31 Blowing Agent 3 + Blowing Agent 1 (5 bar) 29 27 -1 -1 λ (mW.m .K ) 25 23 21 19 413.15K 17 393.15K 373.15K 15 363.15K 353.15K 13 11 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA3) Figure 52. λ function of the molar composition at 5 bar for the system BA3 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. 66 Thermal Conductivity of Gases 1,80% 1,50% 1,20% Blowing Agent 3 + Blowing Agent 1 (5 bar) 353.15K 363.15K 373.15K 0,90% 0,60% 393.15K 413.15K 0,30% 0,00% -0,30% -0,60% -0,90% -1,20% -1,50% -1,80% 1,00 0,75 0,52 0,26 0,00 mole fraction (BA3) Figure 53. Relative deviations between the experimental data and the Extended Wassiljewa Model at 5 bar, for the mixture BA3 – BA1. The thermal conductivity values obtained by the Extended Wassiljewa Model and the normal Wassiljewa Model are not especially different, for 1 bar the average error with the experimental results was of 1.16% and 1.15% respectively. At 2 bar it was 1.02% for both models and for 5 bar was 1.13% and 1.14%. The values predicted by both models are very closer and is possible to verify that eq. 43 is adequate for this binary system. The last analyses performed for this mixture were made using the LPUR Equation and the Extended Wassiljewa Model. For this all the experimental λ values were compared with the λ predict values by using the both models mentioned, for the same conditions of pressure and temperature of all the data obtained for the mixtures. 67 Thermal Conductivity of Gases 27 26 25 -1 -1 λ predicted (mW.m .K ) 24 23 22 26% BA3 21 52% BA3 75% BA3 20 19 18 17 17 18 19 20 21 22 23 -1 24 25 26 27 -1 λ experimental (mW.m .K ) Figure 54. Comparison of the individual data points measured with the values obtained for the same conditions of pressure and temperature using the LPUR Equation and the Extended Wassiljewa Model, for the system BA3 – BA1. 68 Thermal Conductivity of Gases 4,0% (λ experimental − λ predicted) / λ predicted 3,0% 2,0% 1,0% 0,0% -1,0% 26% BA3 52% BA3 75% BA3 -2,0% -3,0% 17 19 21 23 25 -1 27 -1 λ experimental (mW.m .K ) Figure 55. Relative errors between the experimental thermal conductivity and the data obtained for the same conditions of pressure and temperature using the LPUR equation and the Extended Wassiljewa Model, for the system BA3 – BA1. The better results are for the 75% molar composition of Blowing Agent 3 were the results are more consistent, however for the others composition, how it can be seen by the figures, the results are also reliable. The average deviation for all the data is of 1.20% from 673 data points. It is important to remember again that the dispersal of the experimental results has a strong influence and can be a source of error. Finally the next figures show the experimental isotherms for each molar composition and the isotherms estimated by the junction of both models. 69 Thermal Conductivity of Gases 24 26 % Blowing Agent 3 + 74% Blowing Agent 1 23 T = 394.51K T = 373.76K -1 -1 λ (mW.m .K ) 22 T = 354.44K T = 362.64K 21 20 19 18 17 1 2 3 4 5 P(bar) Figure 56. Experimental results for the mixture 26% of Blowing Agent 3 and 74% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. 26 52% Blowing Agent 3 + 48% Blowing Agent 1 25 -1 -1 λ (mW.m .K ) 24 T = 394.84K 23 T = 374.55K T = 362.68K 22 T = 354.43K 21 20 19 18 1 2 3 4 5 P (bar) Figure 57. Experimental results for the mixture 52% of Blowing Agent 3 and 48% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. 70 Thermal Conductivity of Gases 27 75% Blowing Agent 3 + 25% Blowing Agent 1 26 -1 -1 λ (mW.m .K ) 25 T = 393.91K T = 373.80K 24 T = 362.16K T = 353.98K 23 22 21 20 19 1 2 3 4 5 6 7 P (bar) Figure 58. Experimental results for the mixture 75% of Blowing Agent 3 and 25% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. Analysing the figures is possible to take the same conclusions than before about the agreement between the experimental data and the predicted values. Finally after all this analysis is possible to conclude that both models are suitable for this system and furthermore they are able to identify experimental errors. 5.2.2. The Blowing Agent 1 and Blowing Agent 2 Mixture The same treatment will be followed for the Blowing Agent 1 and Blowing Agent 2 mixture. To study the dependence of the thermal conductivity with the molar composition it was made an extrapolation of the experimental thermal conductivity values for the mixtures and pure components for 1 bar, 2 bar and 3 bar. The dependence of λ with the temperature, at these constant values of pressure, can be observed below. 71 Thermal Conductivity of Gases 1 bar 30 28 26 55% BA2 - 45%BA1 BA2 BA1 32% BA2 - 68% BA1 -1 -1 λ (mW.m .K ) 73% BA2 - 27 % BA1 24 22 20 18 16 14 310 330 350 370 390 410 T (K) Figure 59. Thermal conductivity function of the temperature for 1 bar for the mixture BA2 – BA1 and pure components. 2 bar 30 55% BA2 - 45% BA1 BA2 28 -1 -1 λ (mW.m .K ) 26 BA1 32% BA2 - 68% BA1 73% BA2 - 27% BA1 24 22 20 18 16 14 310 330 350 370 390 410 T (K) Figure 60. Thermal conductivity function of the temperature for 2 bar for the mixture BA2 – BA1 and pure components. 72 Thermal Conductivity of Gases 3 bar 30 28 -1 -1 λ (mW.m .K ) 26 55% BA2 - 45% BA1 BA2 BA1 32% BA2 - 68% BA1 73% BA2 - 27% BA1 24 22 20 18 16 14 310 330 350 370 390 410 T (K) Figure 61. Thermal conductivity function of the temperature for 3 bar for the mixture BA2 – BA1 and pure components. Once more is possible to observe that, for all the pressures, the thermal conductivity has a linear dependence with the temperature with a positive inclination; and is also possible to check that the value of the thermal conductivity is increasing with the pressure. However the slope of the isobaric lines for the mixtures and for the pure Blowing Agent 2 is decreasing with the increasing of the pressure. For all the mixtures the values of the thermal conductivity doesn’t change significantly with the molar composition, which is possible to see in the figures above where the isobaric lines for the mixtures are very close. In the next figures λ is represented as a function of the molar composition for each isotherm at 1, 2 and 3 bar. The lines in the graphics match to the Wassiljewa Model with the Maxon and Saxena modifications, where the parameter ε was fitted for each temperature from the experimental points. And are also presented the deviations between the experimental points and the λ values estimated by the model. 73 Thermal Conductivity of Gases 32 Blowing Agent 2 + Blowing Agent 1 (1 bar) 30 28 -1 -1 λ (mW.m .K ) 26 24 22 20 393.15K 373.15K 18 363.15K 353.15K 16 413.15K 14 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA2) Figure 62. λ function of the molar composition at 1 bar for the mixture BA2 – BA1. Blowing Agent 2 + Blowing Agent 1 (1 bar) 3,50% 3,00% 353.15K 2,50% 363.15K 2,00% 373.15K 1,50% 393.15K 1,00% 413.15K 0,50% 0,00% -0,50% -1,00% -1,50% -2,00% -2,50% -3,00% -3,50% 1,00 0,73 0,55 0,32 0,00 mole fraction (BA2) Figure 63. Relative deviations between the experimental data and the Wassiljewa Model at 1 bar, for the mixture BA2 – BA1. 74 Thermal Conductivity of Gases 32 Blowing Agent 2 + Blowing Agent 1 (2 bar) 30 28 -1 -1 λ ( mW.m .K ) 26 24 22 20 393.15K 373.15K 18 363.15K 353.15K 16 413.15K 14 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA2) Figure 64. λ function of the molar composition at 2 bar for the mixture BA2 – BA1. Blowing Agent 2 + Blowing Agent 1 (2 bar) 3,50% 3,00% 2,50% 2,00% 1,50% 1,00% 353.15K 363.15K 373.15K 393.15K 413.15K 0,50% 0,00% -0,50% -1,00% -1,50% -2,00% -2,50% -3,00% -3,50% -4,00% 1,00 0,73 0,55 0,32 0,00 mole fraction (BA2) Figure 65. Relative deviations between the experimental data and the Wassiljewa Model at 2 bar, for the mixture BA2 – BA1. 75 Thermal Conductivity of Gases 32 Blowing Agent 2 + Blowing Agent 1 (3 bar) 30 28 -1 -1 λ (mW.m .K ) 26 24 22 393.15K 20 373.15K 363.15K 18 353.15K 413.15K 16 14 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA2) Figure 66. λ function of the molar composition at 3 bar for the mixture BA2 – BA1. Blowing Agent 2 + Blowing Agent 1 (3 bar) 3,50% 3,00% 2,50% 2,00% 1,50% 1,00% 0,50% 353.15K 363.15K 373.15K 393.15K 413.15K 0,00% -0,50% -1,00% -1,50% -2,00% -2,50% -3,00% -3,50% -4,00% 1,00 0,73 0,55 0,32 0,00 mole fraction (BA2) Figure 67. Relative deviations between the experimental data and the Wassiljewa Model at 3 bar, for the mixture BA2 – BA1. 76 Thermal Conductivity of Gases Observing the graphics is possible to see that for 3 bar the Wassiljewa Model changed the behaviour and it seems to be a right line. Once again this change has to do with the ε value, that for 3 bar is under the unit (except for the temperature of 413.15 K) and for 1 and 2 bar is bigger than 1. For the molar composition of 55% of Blowing Agent 2 the relative error is very small compared with the other compositions. So is possible to conclude that, for the mixture in study, the experimental results for this molar composition are the ones that are closer to the values predicted by the models. The main source for errors has to do with the uncertainty of some experimental results, however the relative errors achieved are in a range of reasonable values and they are acceptable. Once again is possible to see that the model predicts well the λ values and is likely that, for the molar compositions of 32% and 73% of Blowing Agent 2, the model estimates more reliable values for the thermal conductivity. In the next tables is presented the ε values obtained from the experimental results for each isotherm and different pressures and the parameters values of eq. 43 after a fitting process. Table XXVIII. ε values obtained after a fitting process for the different pressures and temperatures for the system BA2 – BA1. P (bar) ε (353.15 K) ε (363.15 K) ε (373.15 K) ε (393.15 K) ε (413.15 K) 1 1.033592 1.031306 1.029222 1.025570 1.022486 2 1.006080 1.007448 1.008580 1.010324 1.011585 3 0.979638 0.984437 0.988611 0.995507 1.000962 Table XXIX. Parameters values for the Extended Wassiljewa Model (eq. 43) after a fitting process with the experimental ε. A1 A2 A3 A4 2.89131 -0.62736 0.10246 -0.17091 The next figures show the behaviour between the experimental ε and the predicted ε by eq. 43 with the pressure and temperature for this system and also the respective deviations. 77 Thermal Conductivity of Gases 1,05 1 353.15K 0,95 ε 363.15K 373.15K 393.15K 413.15K 0,9 0,85 0 1 2 3 4 5 6 7 8 9 P (bar) Figure 68. ε function of the pressure at some temperatures for the system BA2 – BA1. The lines are the values obtained by eq. 43. 0,08% 0,06% 0,04% 0,02% 0,00% -0,02% 353.15 K -0,04% 363.15 K -0,06% 373.15 K 393.15 K -0,08% 413.15 K -0,10% 1 2 3 P (bar) Figure 69. Deviations of ε represented as a function of the pressure, for the system BA2 – BA1. 78 Thermal Conductivity of Gases ε 1,05 1 1 bar 2 bar 3 bar 0,95 320 340 360 380 400 420 440 Temperature (K) Figure 70. ε function of the temperature at some pressures for the system BA2 – BA1. The lines are the values obtained by eq. 43. 0,08% 1 bar 0,06% 0,04% 2 bar 3 bar 0,02% 0,00% -0,02% -0,04% -0,06% -0,08% -0,10% 353,15 363,15 373,15 393,15 413,15 Temperature (K) Figure 71. Deviations of ε represented as a function of the temperature, for the system BA2 – BA1. The ε values for 3 bar are more temperature dependent, how it is possible to see in figure 45. 79 Thermal Conductivity of Gases Once more the ε values predicted by the Extended Wassiljewa Model fit very well the experimental results, for this system the maximum relative error is of 0.08%. Next it was calculated the thermal conductivity values by the Extended Wassiljewa Model and they were compared with the experimental values. The calculations were made for 1, 2 and 3 bar; the results and the respectively relative errors can be seen below. 32 Blowing Agent 2 + Blowing Agent 1 (1 bar) 393.15K 30 373.15K 353.15K 28 353.15K 413.15K -1 -1 λ (mW.m .K ) 26 24 22 20 18 16 14 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA2) Figure 72. λ function of the molar composition at 1 bar for the system BA2 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. 80 Thermal Conductivity of Gases Blowing Agent 2 + Blowing Agent 1 (1 bar) 3,50% 3,00% 2,50% 2,00% 1,50% 1,00% 0,50% 0,00% -0,50% -1,00% 353.15K -1,50% 363.15K -2,00% 373.15K -2,50% 393.15K -3,00% 413.15K -3,50% -4,00% 1,00 0,73 0,55 0,32 0,00 mole fraction (BA2) Figure 73. Relative deviations between the experimental data and the Extended Wassiljewa Model at 1 bar, for the mixture BA2 – BA1. 32 Blowing Agent 2 + Blowing Agent 1 (2 bar) 393.15K 373.15K 30 363.15K 28 353.15K 413.15K -1 -1 λ ( mW.m .K ) 26 24 22 20 18 16 14 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA2) Figure 74. λ function of the molar composition at 2 bar for the system BA2 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. 81 Thermal Conductivity of Gases 3,50% Blowing Agent 2 + Blowing Agent 1 (2 bar) 3,00% 2,50% 2,00% 1,50% 1,00% 0,50% 0,00% -0,50% 353.15K -1,00% 363.15K -1,50% 373.15K -2,00% 393.15K -2,50% 413.15K -3,00% -3,50% -4,00% 1,00 0,73 0,55 0,32 0,00 mole fraction (BA2) Figure 75. Relative deviations between the experimental data and the Extended Wassiljewa Model at 2 bar, for the mixture BA2 – BA1. 32 413.15K Blowing Agent 2 + Blowing Agent 1 (3 bar) 393.15K 30 373.15K 28 363.15K 353.15K -1 -1 λ (mW.m .K ) 26 24 22 20 18 16 14 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 mole fraction (BA2) Figure 76. λ function of the molar composition at 3 bar for the system BA2 – BA1. The lines refer to the Extended Wassiljewa Model, where ε is calculated by eq. 43. 82 Thermal Conductivity of Gases Blowing Agent 2 + Blowing Agent 1 (3 bar) 3,80% 3,30% 2,80% 2,30% 1,80% 1,30% 0,80% 0,30% -0,20% 353.15K -0,70% 363.15K -1,20% 373.15K -1,70% 393.15K -2,20% 413.15K -2,70% -3,20% -3,70% -4,20% 1,00 0,73 0,55 0,32 0,00 mole fraction (BA2) Figure 77. Relative deviations between the experimental data and the Extended Wassiljewa Model at 3 bar, for the mixture BA2 – BA1. The predictions made by the Extended Wassiljewa Model and the normal Wassiljewa Model are not very different, for 1 bar the average error with the experimental results was of 1.92% for both models. At 2 bar it was 1.96% again for both models and for 3 bar was 2.12% for both models. The values predicted by the models are very close for this binary system and is possible to conclude that eq. 43 is valid and adequate for this mixture. The final analyse carried out was done using the LPUR Equation and the Extended Wassiljewa Model. For this purpose all the experimental λ values were compared with the λ predict values using the both models mentioned, for the same conditions of pressure and temperature of all the data obtained for the mixtures. 83 Thermal Conductivity of Gases 27 26 25 -1 -1 λ predicted (mW.m .K ) 24 23 22 32% BA2 21 55% BA2 73% BA2 20 19 18 17 17 18 19 20 21 22 23 -1 24 25 26 27 -1 λ experimental (mW.m .K ) Figure 78. Comparison of the individual data points measured with the values obtained for the same conditions of pressure and temperature using the LPUR Equation and the Extended Wassiljewa Model, for the system BA2 – BA1. 84 Thermal Conductivity of Gases 6,0% 32% BA2 55% BA2 4,0% (λ experimental − λ predicted) / λ predicted 73% BA2 2,0% 0,0% -2,0% -4,0% -6,0% 17 19 21 23 25 -1 27 -1 λ experimental (mW.m .K ) Figure 79. Relative errors between the experimental thermal conductivity and the data obtained for the same conditions of pressure and temperature using the LPUR equation and the Extended Wassiljewa Model, for the system BA2 – BA1. The best results are for the 55% molar composition of Blowing Agent 2, but for the others compositions, how it can be seen by the figures, the results are not so good. The average deviation for all the data is of 2.36% from 531 data points. Once again is important to remember that the dispersal of the experimental results has a strong influence and can be a source of error. Finally the next figures show the experimental isotherms for each molar composition and the isotherms estimated by the junction of both models. 85 Thermal Conductivity of Gases 27 32 % Blowing Agent 2 + 68% Blowing Agent 1 T = 393.79K T = 373.08K 26 T = 352.25K T = 361.47K 25 -1 -1 λ (mW.m .K ) 24 23 22 21 20 19 18 1 2 3 4 5 6 7 8 P(bar) Figure 80. Experimental results for the mixture 32% of Blowing Agent 2 and 68% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. 55% Blowing Agent 2 + 45% Blowing Agent 1 27 26 25 -1 -1 λ (mW.m .K ) 24 23 22 T = 393.79K 21 T = 373.25K 20 T = 361.47K T = 352.13K 19 18 1 2 3 4 5 6 7 8 P (bar) Figure 81. Experimental results for the mixture 55% of Blowing Agent 2 and 45% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. 86 Thermal Conductivity of Gases 73% Blowing Agent 2 + 27% Blowing Agent 1 28 27 26 -1 -1 λ (mW.m .K ) 25 24 23 22 T = 393.49K 21 T = 373.08K 20 T = 362.16K T = 352.31K 19 18 1 2 3 4 5 6 7 8 P (bar) Figure 82. Experimental results for the mixture 73% of Blowing Agent 2 and 27% of Blowing Agent 1. The lines corresponds to the isotherms using the LPUR equation and the Extended Wassiljewa Model. In general, the results appear to be less reliable as those of the binary system of Blowing Agent 3 and Blowing Agent 2, where the deviations are smaller. It is possible to conclude that both models are suitable for this mixture and they are in a very good agreement with the experimental results for the 50% molar composition. Analysing the figures is possible to see that the experimental values for the 32% molar composition of Blowing Agent 2 are higher than the predicted values and for the 73% of Blowing Agent 2 the experimental values are lower. An easy explanation is not at hand, particularly since the results for each mixture are consistent concerning pressure and temperature dependence. A possible explanation might be a smaller leak in the apparatus at the time of the preparation of the mixture, so that the real composition is different from what we have calculated. In future work, this should be further investigated, e. g. by additional measurements for certain compositions. 87 Thermal Conductivity of Gases 5.3. Testing of the Extended Wassiljewa Model for other Mixture In this present work was also studied the binary mixture of nitrogen and methane. It is important to test the Extended Wassiljewa Model for other mixtures in order to learn more about their validity and accuracy. The experimental results for this mixture were obtained by other authors in other laboratory, bibliographic references [27] and [28]. The experimental measurements were carried out at temperatures between 300K and 425K and at pressures up to 16MPa; and was measured three different molar compositions of the system: 25% methane, 50% methane and 75% methane. The next figures illustrate how ε changes with pressure and temperature for the experimental results and for the ε predicted by the Extended Wassiljewa Model (represented by line) and also the respective deviations between the experimental points of ε with the ε results obtained by eq. 43, after a fitting process. The model was applied at higher pressures than usual, because the authors measured ate very high pressures; then it was possible to check if for higher pressures the model continues to give good results. 1,06 1,04 1,02 300K 325K 1 ε 350K 375K 0,98 400K 0,96 0,94 0,92 0 5 10 15 20 25 30 35 P (bar) Figure 83. ε function of the pressure at some temperatures for the system of nitrogen and methane. The lines are the values obtained by eq. 43. 88 Thermal Conductivity of Gases 0,30% 300 K 325 K 0,20% 350 K 375 K 400 K 0,10% 0,00% -0,10% -0,20% -0,30% 1 2 5 10 20 30 P (bar) Figure 84. Deviations of ε represented as a function of the pressure, for the system of nitrogen and methane. 1,05 1 bar 2 bar 5 bar 10 bar 1 20 bar ε 30 bar 0,95 0,9 300 320 340 360 380 400 420 440 Temperature (K) Figure 85. ε function of the temperature at some pressures for the system of nitrogen and methane. The lines are the values obtained by eq. 43. 89 Thermal Conductivity of Gases 0,30% 1 bar 2 bar 0,20% 5 bar 10 bar 20 bar 0,10% 30 bar 0,00% -0,10% -0,20% -0,30% 300 325 350 375 400 Temperature (K) Figure 86. Deviations of ε represented as a function of the temperature, for the system of nitrogen and methane. How it is possible to observe the ε value has a minimum variance with the pressure. With the temperature it is more dependent; however for all the pressures, ε has the same behaviour. One of the possible reason has to do with the physical and chemical properties of each component and of the mixture, and other important point is related with the quality of the results, because the measurements were done in other apparatus that was not described in the literature and by the analysis of their results was possible to observe that for the mixture of 75% methane – 25% nitrogen the accuracy of the results was not very good. However the results of the model are also very pleasing for this system, because they fit well and the relative errors are small and acceptable. Furthermore is proved that the model is suitable for high pressures. In the tables bellow are the ε values obtained after a fitting process for different temperatures and pressures and the parameters values for the Extended Wassiljewa Model after a fitting process. Table XXX. ε values obtained after a fitting process for the different pressures and temperatures for the system nitrogen and methane. P (bar) ε (300 K) ε (325 K) ε (350 K) ε (375 K) ε (400 K) 1 1.029117 0.997444 0.971299 0.949343 0.930636 2 1.030555 0.998877 0.972708 0.950718 0.931973 90 Thermal Conductivity of Gases 5 1.029883 0.998302 0.972155 0.950144 0.931351 10 1.030201 0.998731 0.972581 0.950500 0.931599 20 1.033578 1.002318 0.976149 0.953913 0.934778 30 1.034733 1.003754 0.977626 0.955285 0.935957 Table XXXI. Parameters values for the Extended Wassiljewa Model (eq. 43) after a fitting process with the experimental ε. A1 7.54310 A2 A3 -5 5.9716×10 2.43117×10 A4 -5 -0.34957 For this system ε is increasing with the pressure and decreasing with the temperature, this was also possible to observe in the figures above. It is possible to conclude that for all the systems studied so far the values for ε do not vary significantly; they have just different behaviours in the way they change with the pressure and temperature. 5.4. All Mixtures The next figures show how ε changes with the pressure and the temperature, for all the mixtures studied at Bayer Technology Services in this present work and at the works before. 1,09 N2-BA1 1,04 N2-nC5 BA3-nC5 N2-BA3 0,99 BA3-BA1 ε BA2-BA1 0,94 0,89 0,84 353,15 363,15 373,15 393,15 413,15 Temperature (K) Figure 87. ε as a function of the temperature at 1 bar. 91 Thermal Conductivity of Gases 1,09 N2-BA1 N2-nC5 1,04 BA3-nC5 N2-BA3 BA3-BA1 0,99 ε BA2-BA1 0,94 0,89 0,84 353,15 363,15 373,15 393,15 413,15 Temperature (K) Figure 88. ε as a function of the temperature at 5 bar. 1,06 1,04 N2-BA1 1,02 N2-nC5 BA3-nC5 1 N2-BA3 0,98 BA3-BA1 BA2-BA1 ε 0,96 0,94 0,92 0,9 0,88 0,86 0 2 4 6 8 10 12 P (bar) Figure 89. ε as a function of the pressure at 353K. 92 Thermal Conductivity of Gases 1,06 1,04 N2-BA1 N2-nC5 1,02 BA3-nC5 N2-BA3 1 BA3-BA1 BA2-BA1 ε 0,98 0,96 0,94 0,92 0,9 0 2 4 6 8 10 12 P (bar) Figure 90. ε as a function of the pressure at 413K. How it is possible to observe is not possible to guess for a mixture how the ε value will change for different pressures and temperatures before knowing the parameters of equation 43. However its value is always close to the unit, which was expectable since ε was always assumed to be equal to one by some authors and in some models. 93 Thermal Conductivity of Gases 6. Conclusions The Transient Hot-Wire Method was the method used in this work for the measurement of the thermal conductivity of pure components and binary mixtures. In this work two pure components were studied and for both components good results were achieved and were proved the validity and applicability of the LPUR Equation for these components. Also two mixtures were studied and were proved that the Wassiljewa Model and the Extended Wassiljewa model are valid and suitable for these binary mixtures and that they can also show errors in the experimental measurements. However with the experimental results for the mixture Blowing Agent 2 + Blowing Agent 1, bigger relative errors with the predicted values for the thermal conductivity by the models were achieved, in comparison with the first binary system studied. More compositions for this mixture should be measured, by the reasons mentioned previously. One explanation for errors in the experimental values for the thermal conductivity has to do with the head of the measuring cell. With time the teflon sealing looses its properties and starts to let pass some gas that is inside the cell to the environment, this causes a change in the molar composition of the mixture and consequently higher errors related with the bigger dispersion of the experimental results. The leak also causes problems related with the pressure stability that was more significant at the higher pressures and the results are affected by an error, which is notable at the lower pressures. To avoid this problem is recommended to change the sealing of the head of the cell, instead of a teflon sealing the head should have a ceramic ring, like was suggested by Oliveira. This change will stop with the leak problem during the measures giving better and more consistent experimental results of the thermal conductivity. Another recommendation for further works, like was also suggest before by other students, has to do with the environment temperature surrounding the apparatus. It is suggested to place the apparatus in a temperature control room. Small temperatures differences in the room change the temperature in the measuring cell and as a result new temperatures had to be set in the temperature controllers. And when the measures had already start the change in the temperature of the cell has a big influence on the value of the thermal conductivity creating errors when the experimental values are compared with the estimated values by the models for an isotherm. Other change is related with the length of the platinum wires going from the cell to the Wheatstone bridge. These ones should be a little longer to facilitate the welding that is difficult. 94 Thermal Conductivity of Gases Like was mentioned before in this thesis, the platinum wires inside the cell, when welded, should not be completed stretched. The platinum wires welded must always have at least one ‘wave’, otherwise they will break at elevated temperatures. In conclusion new mixtures and pure components shall continue to be studied to find the system with the lowest thermal conductivity and high vapour pressure possible. And at the same time to continue to validate, improve and optimize the LPUR Equation and the Extended Wassiljewa Model. 95 Thermal Conductivity of Gases 7. Bibliography [1] - Perkins, R., Cusco, L., Howley, J., Laesecke, A. Matthes, S., Ramires, M.L.V. (2001). Thermal Conductivities of Alternatives to CFC-11 for Foam Insulation, J. Chem. Eng.Data, 46:428-432. [2] - Richard, R.G., Shankland, I.R. (1989). A Transient Hot-Wire Method for Measuring the Thermal Conductivity of Gases and Liquids, International Journal of Thermophysics, 10 (3): 673-686. [3] - Johnson, R.W., Bowman, J. (2003). The Effect of Blowing Agents on Energy Use and Climate Impact of a Refrigerator, APPLIANCE European Edition, Nov., 22-24. [4] - Wilkes K.E., Gabbard, W.A., Fred J. Weaver, F.J. (1999). Aging of Polyurethane Foam Insulation in Simulated Refrigerator Panels - One-Year Results with ThirdGeneration Blowing Agents, The Earth Technologies Forum, Washington, DC [5] - Wakeham,W.A.; Nagashima,A.; Sengers,J.V.(1991).Transient Methods for Thermal Conductivities, Measurement of the Transport-Properties of Fluids ,Blackwell, Oxford. [6] - J. Healy, J. J. de Groot, J. Kestin, The theory of the Transient Hot-Wire Method for measuring the thermal conductivity, Physica C82, 392-408. [7] - B. Taxis, K. Stephan, Application of the transient hot-wire method to gases at low pressures, Int. J. Thermophys. 15 (1) 1994. [8] - A.N. Gurova, Condutibilidade Térmica de Refrigerantes Ambientalmente Aceitáveis, PhD. Thesis, Dep. de Química - Faculdade de Ciencias da Universidade de Lisboa, 1996. [9] - H.M. Roder, R.A. Perkins, A. Laesecke, C.A. Nieto de Castro, Absolute Steady-State Thermal Conductivity Measurements by Use of a Transient Hot-Wire System, J. Res. Natl. Inst. Stand. Technol., 2000, 105 (2), 221-253. [10] - E. Takikawa, Thermal Conductivity Measurements With Transient Hot Wire Method, Bayer A.G., 2001. [11] - F. Santos, Thermal Conductivity of Fluids – An Internship Report for Graduation in 96 Thermal Conductivity of Gases Industrial Chemistry and Management, Bayer A.G / Universidade de Aveiro, 2002. [12] - N.S. Oliveira, Thermal Conductivity of Gases – An Internship Report for Graduation in Chemistry – Branch of Analytical Chemistry, Bayer A.G / Universidade de Aveiro, 2001. [13] - J. M. Fonseca, Aging of Polyurethane Rigid Foams – Thermal Conductivity of Gas Mixtures, An Internship Report for Graduation in Chemistry, Bayer A. G./ Universidade de Aveiro, December 2006. [14] C. D. Antunes, AGING OF POLYURETHANE RIGID FOAMS: Thermal Conductivity of Gas Mixtures involving Nitrogen and Hydrocarbons, An Internship Report for Graduation in Industrial Chemistry, Bayer A. G./ Instituto Superior Tecnico, September 2007. [15] – E.W. Lemmon, M.O. McLinden and D.G. Friend, Thermophysical Properties of Fluid Systems, in NIST Chemistry WebBook, NIST Standard Reference Database Number 69, Eds. P. J. Linstrom and W. G. Mallard, National Institute of Standards and Technology, June 2005 (http://webbook.nist.gov/chemistry/fluid/) [16] – N.B.Vargaftik, Tables on the Thermophysical properties of liquids and gases, 2nd ed. Hemisphere Publishing Corporation, Washington, 1975. [17] – T.E. Daubert, R.P. Danner, H.M. Sibul, C.C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801, Design Institute for Physical Property Data, AIChE, New York, 2006. [18] – A. Wassiljewa, Phys. Z., 1904, 5, 737 (cited in [17]). [19] – E.A. Mason, S.C. Saxena, Phys. Fluids, 1958, 1, 361 (cited in [17]). [20] – V. Vesovic, Prediction of the Thermal Conductivity of Gas Mixtures at LowPressures, International Journal of Thermophysics, 2001, 22 (3), 801- 828. [21] – R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, 4th ed., 1988. [22] - http://www.wikipedia.org/ (15/01/2008) 97 Thermal Conductivity of Gases [23] - W.A. Wakeham, A. Nagashima, J.V. Sengers; Measurements of the transport properties of fluids; Blackwell Scientific Publications, 1991. [24] - N. V. Tsederberg; Thermal Conductivity of Gases and Liquids; The M.I.T Press, 1965. [25] - J. Kestin, W. A. Wakeham; Transport Properties of Fluids: Thermal Conductivity, Viscosity, and Diffusion Coefficient; Cindas Data Series on Material Properties, Volume I1, 1988. [26] - Y. S. Touloukian, P. E. Liley, S. C. Saxena ; Thermal Conductivity: Nonmetallic Liquids and Gases; Thermophysical Properties of Matter, Volume 3, The TPRC Data Series, IFI/Plenum, New York-Washington, 1970. [27] - J. Patek, J. Klomfar; Measurement of the thermal conductivity of argon and methane: a test of a transient hot-wire apparatus; Fluid Phase Equilibria 198, 147-163, 2002. [28] - J. Patek, J. Klomfar, L. Capla, P. Buryan; Thermal Conductivity of NitrogenMethane Mixtures at Temperatures Between 300 and 425 K and at Pressures up to 16 MPa; International Journal of Thermophysics, Vol. 24, No. 4, July 2003. [29] – I. Marrucho, F. Santos, N. Oliveira, R. Dohrn; Aging of Rigid Polyurethane Foams: Thermal Conductivity of N2 and Cyclopentane Gas Mixtures; Journal of Cellular Plastics, Vol. 41, May 2005. 98 Thermal Conductivity of Gases 8. Appendix 8.1. Experimental Results Table A1. 1. Selected results of the calibration for Nitrogen. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 120.34 5.007 31.7497 80.72 5.013 29.35275 120.37 5.007 31.77432 80.69 5.013 29.41015 120.37 5.008 31.80753 80.72 5.013 29.35342 120.34 5.007 31.92597 80.71 5.011 29.25711 120.38 5.007 31.66317 80.64 5.009 29.10871 120.42 5.007 31.88673 80.67 2.018 29.1007 120.41 5.007 31.9642 80.70 2.019 29.32953 120.41 5.007 31.75452 80.81 2.020 29.09332 120.38 7.997 31.83495 80.85 2.020 29.07012 120.35 7.996 32.05621 80.84 2.020 29.20828 120.35 7.994 32.05409 39.44 7.979 26.84315 120.35 7.993 31.99793 39.45 7.974 26.81665 120.36 7.992 32.01993 39.44 7.966 26.74123 120.37 7.990 32.09278 39.43 7.960 26.91289 120.36 7.988 32.0527 39.43 7.952 26.90807 120.38 7.987 32.09665 39.43 7.945 26.78858 120.38 7.986 31.95947 39.49 7.931 26.92261 120.42 2.015 31.78912 39.5 7.923 26.92299 120.41 2.015 31.97319 39.51 7.916 26.8327 120.36 2.016 31.87873 39.48 5.005 26.6604 120.34 2.015 31.57055 39.49 5.003 26.64908 120.37 2.015 31.79015 39.48 4.995 26.62005 80.69 8.005 29.64955 39.49 4.989 26.68102 80.65 8.003 29.65574 39.48 4.983 26.65804 80.64 8.002 29.53812 39.48 4.981 26.69884 80.61 7.997 29.32641 39.49 4.976 26.62395 80.65 7.996 29.59446 39.47 7.979 26.84315 80.60 7.992 29.2981 39.42 2.018 26.46261 80.65 7.99 29.64647 39.41 2.018 26.42399 80.72 5.014 29.29344 39.46 2.019 26.50139 80.71 5.014 29.40718 99 Thermal Conductivity of Gases Table A1. 2. Selected results of the measurements for Blowing Agent 1. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 87.83 1.356 16.98303 88.67 2.083 17.04379 87.78 1.356 16.92648 88.66 2.081 17.14094 87.76 1.362 17.14711 88.63 2.080 17.21015 87.80 1.368 16.93016 88.61 2.080 17.0604 87.88 1.373 16.9001 88.61 2.080 16.94251 87.97 1.378 17.02622 88.63 2.079 17.07915 88.03 1.382 16.89455 88.61 2.079 16.97706 88.12 1.386 17.00291 88.63 2.077 17.28504 88.17 1.389 16.96565 88.60 2.075 17.10561 88.25 1.391 16.96884 88.60 2.074 17.00882 88.31 1.394 16.99151 88.56 2.590 17.20448 88.33 1.395 16.93667 88.54 2.590 16.91389 88.39 1.397 17.07253 88.54 2.589 17.01593 88.40 1.398 17.13054 88.53 2.589 17.10998 88.45 1.397 17.15975 88.56 2.582 17.11902 88.56 1.828 17.0528 88.56 2.581 17.24334 88.54 1.827 17.04277 88.57 2.578 16.97761 88.53 1.827 17.03863 88.59 2.576 16.93681 88.53 1.826 17.08382 88.62 2.575 17.16283 88.56 1.825 17.15876 88.61 2.574 17.17293 88.53 1.822 16.93893 88.62 2.571 16.86117 88.54 1.821 17.28027 88.58 2.570 17.01277 88.53 1.821 17.27642 88.61 2.569 16.89869 88.54 1.820 16.97473 88.60 2.567 17.1377 88.58 1.819 17.19034 88.60 2.565 16.87096 88.59 1.818 17.29608 99.24 2.690 18.52455 88.60 1.817 16.95101 99.22 2.688 18.33402 88.59 1.817 16.93777 99.22 2.686 18.48929 88.59 1.815 16.99622 99.20 2.685 18.46487 88.60 1.814 17.08491 99.17 2.683 18.41423 88.64 1.813 17.10657 99.17 2.681 18.43322 88.63 1.812 17.12676 99.16 2.679 18.46882 88.60 1.811 17.27216 99.19 2.677 18.40118 88.70 2.078 17.11002 99.15 2.675 18.34692 88.70 2.077 17.2328 99.16 2.673 18.34413 88.71 2.076 17.12965 99.11 2.696 18.34022 88.69 2.074 17.1858 99.14 2.693 18.3041 88.69 2.073 17.09662 99.10 2.691 18.49911 88.69 2.071 17.25632 99.09 2.689 18.32378 88.64 2.069 17.15331 99.07 2.687 18.45025 88.65 2.067 17.09395 99.05 2.685 18.39555 88.61 2.066 16.97287 99.03 2.684 18.39623 88.67 2.086 17.25798 99.07 2.682 18.32594 100 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 99.08 2.680 18.26533 99.20 1.700 18.34055 99.05 2.679 18.36989 99.20 1.698 18.41873 98.96 3.240 18.40667 99.20 1.696 18.27244 98.97 3.238 18.30816 99.18 1.694 18.2738 98.99 3.237 18.30901 99.17 1.690 18.24951 98.99 3.235 18.28764 99.20 1.685 18.34354 99.00 3.232 18.28814 99.21 1.682 18.32143 99.03 3.232 18.39712 99.22 1.677 18.25149 99.05 3.230 18.53771 99.21 1.674 18.35121 99.08 3.225 18.27265 99.21 1.671 18.43686 99.12 3.224 18.50967 99.20 1.669 18.36266 99.10 3.222 18.27363 99.21 1.666 18.33011 99.14 3.220 18.40281 99.12 2.148 18.41859 99.13 3.219 18.39823 99.10 2.147 18.26218 99.14 3.217 18.35652 99.10 2.146 18.3722 99.13 3.215 18.25687 99.08 2.145 18.51659 99.16 3.214 18.51279 99.10 2.143 18.2957 99.15 3.212 18.48096 99.10 2.142 18.45163 99.15 3.210 18.45821 99.12 2.140 18.37414 99.15 3.208 18.3542 80.53 1.361 16.07937 99.14 2.152 18.51105 80.54 1.361 16.0861 99.14 2.152 18.29016 80.52 1.361 16.15797 99.13 2.151 18.38688 80.51 1.359 16.15908 99.12 2.150 18.25765 80.46 1.358 16.17205 99.13 2.149 18.40892 80.50 1.357 16.2287 99.12 2.148 18.41859 80.53 1.357 16.17345 99.10 2.147 18.26218 80.55 1.356 16.18369 99.10 2.146 18.3722 80.57 1.355 16.23973 99.08 2.145 18.51659 80.60 1.353 16.19569 99.10 2.143 18.2957 80.61 1.352 16.0392 99.10 2.142 18.45163 80.62 1.352 16.12192 99.12 2.140 18.37414 80.62 1.352 16.20722 99.17 2.137 18.37945 80.62 1.352 16.01322 99.18 2.136 18.25931 80.61 1.353 16.23389 99.20 2.134 18.41235 80.64 1.353 16.25077 99.21 2.132 18.4155 80.60 1.712 16.28311 99.22 2.131 18.29352 80.61 1.710 16.18544 99.20 2.129 18.31153 80.60 1.709 16.10767 99.23 1.715 18.34832 80.61 1.708 16.12539 99.21 1.714 18.24869 80.60 1.706 16.30008 99.20 1.712 18.3144 80.55 1.705 16.07613 99.19 1.709 18.30166 80.52 1.703 16.06774 99.20 1.704 18.33244 80.52 1.703 16.05742 101 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] 80.50 1.700 16.04923 120.92 3.356 21.3967 80.50 1.700 16.02966 120.90 3.354 21.2194 80.49 1.697 16.16766 120.90 3.352 21.18932 80.47 1.696 16.2517 120.88 3.349 21.17164 80.44 1.695 16.12406 120.88 3.344 21.2578 80.45 1.694 16.16636 120.88 3.342 21.21097 80.46 1.694 16.15688 120.91 3.34 21.14908 80.49 1.693 16.12653 120.91 3.335 21.26073 80.49 1.692 16.17105 121.11 4.167 21.24131 80.57 1.983 16.12077 121.05 4.164 21.35192 80.54 1.975 16.16032 121.04 4.159 21.19221 80.57 1.972 16.16269 120.98 4.155 21.37971 80.58 1.970 16.22544 120.95 4.15 21.19109 80.58 1.97 16.17161 120.89 4.146 21.26035 80.57 1.969 16.14844 120.84 4.141 21.29339 80.60 1.968 16.2191 120.82 4.137 21.32671 80.61 1.967 16.18998 120.81 4.131 21.16077 80.63 1.966 16.2053 120.81 4.125 21.27292 80.60 1.959 16.31301 120.84 4.121 21.23567 80.56 1.959 16.25646 120.83 4.118 21.24134 80.61 1.957 16.34124 120.83 4.114 21.22415 80.59 1.956 16.20049 120.83 4.111 21.16263 P [bar] λ [mW.m-1.K-1] 80.59 1.955 16.21805 120.83 4.107 21.22931 121.01 2.454 21.15294 120.81 4.104 21.34936 121.02 2.452 21.22889 120.80 4.101 21.29928 120.99 2.45 21.09011 120.80 4.097 21.26831 120.99 2.449 21.16294 120.80 4.094 21.29282 120.97 2.448 21.26531 120.35 5.195 21.42027 120.95 2.446 21.24979 120.39 5.193 21.50312 120.91 2.444 21.0282 120.40 5.19 21.41708 120.88 2.439 21.3359 120.31 5.150 21.49284 120.89 2.437 21.16255 120.33 5.147 21.48376 120.89 2.434 21.09615 120.34 5.143 21.42776 120.88 2.432 21.06011 120.34 5.139 21.37974 120.86 2.427 21.05926 120.37 5.136 21.40867 120.88 2.425 21.16575 120.38 5.133 21.36531 120.89 2.421 21.06989 120.38 5.129 21.49247 120.96 3.388 21.35867 120.37 5.126 21.48164 120.97 3.384 21.25302 120.38 5.122 21.48218 120.91 3.370 21.26324 120.36 5.119 21.42838 120.92 3.366 21.37033 120.91 3.335 21.26073 120.91 3.363 21.13713 121.11 4.167 21.24131 120.89 3.358 21.33863 121.05 4.164 21.35192 102 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 121.04 4.159 21.19221 121.10 5.136 21.28069 121.10 5.139 21.232 121.07 5.123 21.34688 Table A1. 3. Selected results of the measurements for the mixture 26% BA3 – 74% BA1. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 121.32 4.722 23.44619 121.48 2.406 22.9991 121.30 4.715 23.49616 121.44 2.404 23.09458 121.31 4.708 23.46691 121.44 2.403 23.01333 121.31 4.699 23.32729 121.41 2.401 23.05526 121.30 4.689 23.33684 121.49 2.179 23.02262 121.31 4.681 23.22662 121.50 2.173 22.7916 121.20 4.667 23.37759 121.44 2.164 22.99904 121.19 4.654 23.31468 121.43 2.152 22.79171 121.17 4.647 23.42052 121.38 2.130 23.03431 121.17 4.640 23.44864 121.34 2.120 23.06015 121.16 4.632 23.27871 121.33 2.110 22.90778 121.12 4.625 23.39425 121.29 2.099 22.93993 121.15 4.619 23.28163 121.26 2.093 22.78356 121.59 3.423 23.31226 121.27 2.087 22.83198 121.51 3.421 22.98567 121.28 2.079 22.80037 121.47 3.418 23.1369 121.28 2.072 22.82036 121.47 3.414 23.07161 121.26 2.064 22.80986 121.43 3.410 23.1338 121.27 2.057 22.76804 121.40 3.407 23.02039 100.56 3.909 20.57127 121.42 3.403 23.32189 100.54 3.906 20.61995 121.41 3.398 23.12881 100.52 3.903 20.61345 121.41 3.394 22.9637 100.48 3.898 20.54136 121.41 3.390 23.1858 100.46 3.894 20.52501 121.45 3.381 23.11136 100.45 3.892 20.57612 121.43 3.379 23.24494 100.53 3.892 20.55133 121.47 3.375 23.24914 100.55 3.891 20.61293 121.71 2.432 22.99285 100.58 3.890 20.61743 121.68 2.429 22.90916 100.62 3.890 20.56897 121.67 2.428 23.06179 100.60 3.889 20.5778 121.59 2.425 23.02288 100.60 3.889 20.65502 121.55 2.423 23.0021 100.65 3.888 20.57991 121.52 2.421 23.0538 100.71 3.154 20.43683 121.51 2.419 23.0178 100.68 3.152 20.42032 121.50 2.416 22.93674 100.67 3.151 20.40443 121.49 2.414 23.03193 100.66 3.15 20.49661 121.49 2.411 22.94485 100.66 3.147 20.40548 121.47 2.409 23.01453 100.68 3.144 20.54243 103 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 100.66 3.142 20.57678 89.59 2.603 19.0184 100.63 3.140 20.4318 89.61 2.601 19.06276 100.64 3.139 20.63026 89.58 2.599 19.01116 100.66 3.138 20.43826 89.63 2.597 18.95928 100.69 3.135 20.56834 89.48 2.206 18.85867 100.65 3.131 20.61817 89.46 2.206 18.83799 100.73 2.285 20.28988 89.45 2.205 18.85039 100.72 2.283 20.22428 89.46 2.204 18.91809 100.68 2.282 20.15763 89.50 2.202 18.86682 100.66 2.280 20.13526 89.49 2.200 18.96163 100.62 2.278 20.46609 89.52 2.199 18.82013 100.64 2.276 20.22575 89.49 2.198 18.7366 100.61 2.275 20.112 89.53 2.198 18.99291 100.64 2.273 20.36449 89.51 2.196 18.91949 100.67 2.273 20.45699 89.47 2.194 18.764 100.66 2.272 20.14861 89.47 2.191 18.77098 100.66 2.271 20.32111 89.44 2.190 18.83925 100.67 2.269 20.32575 89.53 1.460 18.7379 100.65 2.268 20.24768 89.51 1.458 18.88663 100.64 2.267 20.1744 89.48 1.456 18.84827 100.62 2.265 20.44347 89.49 1.453 18.64647 100.73 1.709 20.17147 89.49 1.450 18.87538 100.68 1.704 20.35403 89.43 1.448 18.81233 100.64 1.699 20.16624 89.47 1.444 18.74027 100.58 1.688 20.21863 89.49 1.442 18.68878 100.57 1.684 20.1062 89.47 1.440 18.80355 100.55 1.679 20.07614 89.43 1.437 18.89182 100.50 1.674 20.28864 89.48 1.435 18.67509 100.51 1.669 20.35527 89.45 1.433 18.90618 100.49 1.666 20.05973 89.47 1.431 18.63426 100.48 1.661 20.05742 89.45 1.429 18.82119 100.47 1.658 20.12374 89.47 1.427 18.82644 100.51 1.655 20.14089 89.42 1.196 18.7027 100.49 1.650 20.06949 89.40 1.194 18.71912 89.67 2.624 18.96862 89.41 1.192 18.72149 89.66 2.622 18.91073 89.39 1.190 18.62116 89.64 2.620 18.95689 89.37 1.189 18.6286 89.65 2.618 18.9653 89.41 1.188 18.63495 89.60 2.611 19.10963 89.38 1.188 18.63636 89.57 2.610 19.08384 89.40 1.186 18.6927 89.57 2.607 19.07005 89.37 1.185 18.76018 89.58 2.606 19.09181 89.39 1.185 18.73881 89.58 2.605 19.13472 89.40 1.185 18.69416 104 Thermal Conductivity of Gases (Continuation) P [bar] λ [mW.m-1.K-1] T [°C] P [bar] 89.37 1.184 18.69811 81.13 2.106 17.8049 89.40 1.184 18.6853 81.43 1.299 17.88815 T [°C] λ [mW.m-1.K-1] 81.41 2.371 17.82027 81.43 1.298 17.86622 81.40 2.370 18.07708 81.40 1.296 17.83666 81.34 2.368 17.89361 81.41 1.294 17.7521 81.33 2.367 17.84341 81.39 1.292 17.71719 81.30 2.366 18.08601 81.34 1.290 17.99683 81.30 2.364 17.93762 81.37 1.288 17.93634 81.31 2.362 18.04228 81.41 1.286 17.79906 81.31 2.361 17.93919 81.43 1.284 17.65499 81.37 2.342 17.81876 81.41 1.284 17.90882 81.37 2.341 18.07036 81.45 1.281 17.68525 81.31 2.340 18.05753 81.44 1.279 17.7323 81.28 2.338 18.08581 81.46 1.277 17.81075 81.26 2.335 17.93953 81.45 1.275 17.6666 81.19 2.116 17.93944 81.34 1.162 17.60829 81.19 2.115 17.9586 81.27 1.162 17.7672 81.17 2.114 17.99286 81.25 1.160 17.63811 81.14 2.114 17.84135 81.24 1.159 17.71851 81.17 2.114 17.95543 81.22 1.160 17.81473 81.17 2.113 17.85019 81.26 1.161 17.77473 81.16 2.111 17.92784 81.24 1.160 17.6897 81.17 2.110 17.97509 81.23 1.160 17.67278 81.12 2.109 17.94393 81.25 1.158 17.73973 81.12 2.109 17.88038 81.24 1.157 17.78505 81.13 2.107 17.98447 81.25 1.157 17.65209 81.12 2.107 17.85459 81.26 1.156 17.72271 81.13 2.107 17.80303 81.28 1.156 17.79416 81.13 2.107 17.85968 81.32 1.155 17.78077 Table A1. 4. Selected results of the measurements for the mixture 52% BA3– 48% BA1. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 121.92 5.072 24.45372 121.66 4.925 24.11141 121.86 5.050 24.49828 121.65 4.917 24.51392 121.86 5.035 24.3295 121.63 4.909 24.10446 121.86 5.024 24.4868 121.62 4.901 24.20479 121.81 5.011 24.42469 121.54 3.380 23.8473 121.80 4.999 24.28544 121.49 3.378 23.66082 121.79 4.988 24.4834 121.50 3.375 23.63584 121.76 4.976 24.40898 121.46 3.371 24.09736 121.69 4.952 24.16381 121.41 3.368 23.8521 121.68 4.935 24.34044 121.44 3.365 24.01039 105 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 121.44 3.361 24.14975 101.57 4.177 21.49983 121.44 3.357 24.01777 101.59 4.174 21.54344 121.43 3.353 23.66226 101.58 4.172 21.53058 121.44 3.350 23.57755 101.54 4.169 21.44881 121.42 3.348 23.67956 101.53 4.164 21.48894 121.41 3.346 24.14593 101.47 4.159 21.34598 121.45 3.345 24.09867 101.47 4.156 21.60894 121.46 3.341 24.05654 101.47 4.154 21.33988 121.41 3.338 23.60027 101.45 4.153 21.7037 121.91 2.387 23.82041 101.47 4.150 21.5733 121.87 2.385 23.64261 101.48 4.148 21.59576 121.86 2.384 23.60981 101.63 3.189 21.44515 121.79 2.381 23.80119 101.60 3.187 21.36969 121.77 2.378 23.57277 101.61 3.179 21.17792 121.76 2.375 23.79422 101.61 3.177 21.20366 121.74 2.373 23.91035 101.58 3.175 21.16655 121.72 2.371 23.87279 101.59 3.174 21.44193 121.71 2.368 23.87548 101.57 3.171 21.39702 121.69 2.367 23.64288 101.57 3.171 21.3803 121.69 2.366 23.40311 101.58 3.169 21.20068 121.70 2.363 23.78701 101.56 3.166 21.43234 121.68 2.363 23.64541 101.57 3.163 21.30699 121.71 2.361 23.43858 101.55 3.160 21.18303 121.69 2.359 23.65592 101.52 3.156 21.08965 121.66 1.984 23.70106 101.51 3.155 21.41892 121.63 1.983 23.5119 101.04 2.216 21.1836 121.61 1.980 23.55959 101.05 2.215 21.05257 121.58 1.977 23.60201 101.03 2.213 21.14788 121.55 1.974 23.43526 101.02 2.212 21.19708 121.51 1.971 23.4049 101.00 2.212 21.14223 121.51 1.970 23.63654 101.02 2.210 21.26198 121.50 1.970 23.67762 101.03 2.209 20.9028 121.51 1.969 23.65356 101.02 2.208 20.90719 121.49 1.967 23.29706 101.04 2.206 20.96832 121.49 1.967 23.61501 101.03 2.205 21.23657 121.51 1.964 23.71603 101.02 2.203 21.21368 121.50 1.962 23.32182 101.04 2.201 20.88403 121.50 1.960 23.38128 101.05 2.200 20.91089 121.50 1.957 23.70567 101.06 2.198 21.07212 101.73 4.192 21.55831 101.05 2.196 21.09964 101.70 4.190 21.43139 101.60 1.902 21.07228 101.65 4.185 21.4002 101.55 1.901 20.89239 101.65 4.180 21.72742 101.54 1.899 20.79907 106 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 101.49 1.896 20.84816 89.44 1.823 19.58757 101.46 1.895 21.05232 89.47 1.823 19.5995 101.45 1.893 21.04161 89.43 1.823 19.60608 101.40 1.891 21.18773 89.45 1.822 19.47397 101.42 1.889 20.89226 89.43 1.822 19.45972 101.39 1.888 21.0304 89.43 1.820 19.4758 101.39 1.887 20.9333 89.42 1.819 19.44645 101.38 1.885 21.03867 89.44 1.817 19.59811 101.39 1.884 21.02247 89.44 1.816 19.57321 101.35 1.883 21.0577 89.44 1.814 19.54332 101.36 1.883 20.88283 89.46 1.813 19.51122 89.41 2.754 19.67272 89.70 1.313 19.31524 89.43 2.753 19.73806 89.68 1.310 19.28153 89.39 2.751 19.74843 89.69 1.311 19.29764 89.39 2.749 19.71852 89.68 1.310 19.25262 89.37 2.747 19.70214 89.63 1.310 19.39742 89.35 2.745 19.6634 89.67 1.309 19.28863 89.36 2.743 19.7827 89.62 1.309 19.22166 89.38 2.742 19.68763 89.66 1.309 19.47318 89.38 2.741 19.67073 89.60 1.308 19.25963 89.39 2.740 19.77466 89.61 1.307 19.26137 89.35 2.738 19.77276 89.61 1.307 19.20737 89.34 2.737 19.65691 89.62 1.306 19.43811 89.33 2.735 19.727 89.62 1.306 19.21685 89.30 2.733 19.64594 89.62 1.305 19.2 89.32 2.731 19.7616 89.61 1.303 19.26743 89.67 2.152 19.5833 81.19 2.512 18.77639 89.66 2.149 19.58496 81.19 2.511 18.68229 89.65 2.149 19.62815 81.15 2.509 18.53661 89.66 2.145 19.75475 81.10 2.508 18.67303 89.63 2.143 19.77163 81.13 2.506 18.52826 89.65 2.142 19.72655 81.16 2.505 18.55833 89.63 2.140 19.73806 81.16 2.505 18.6574 89.63 2.137 19.57589 81.20 2.504 18.78619 89.64 2.136 19.57979 81.22 2.504 18.51757 89.65 2.135 19.63306 81.26 2.504 18.59855 89.67 2.133 19.70143 81.26 2.503 18.7063 89.68 2.131 19.58285 81.28 2.502 18.59263 89.68 2.129 19.65025 81.25 2.501 18.63265 89.50 1.829 19.52829 81.26 2.501 18.70101 89.49 1.827 19.67387 81.28 2.499 18.75506 89.48 1.826 19.53292 81.42 2.095 18.60979 89.46 1.825 19.41612 81.41 2.094 18.65788 107 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 81.37 2.093 18.73101 81.26 1.789 18.46617 81.35 2.091 18.61757 81.27 1.788 18.63825 81.31 2.087 18.49287 81.25 1.788 18.60442 81.31 2.086 18.67567 81.24 1.788 18.66079 81.33 2.085 18.61911 81.31 1.279 18.38189 81.32 2,083 18.62652 81.31 1.279 18.45441 81.36 2.082 18.55683 81.29 1.278 18.45765 81.35 2.082 18.60018 81.28 1.278 18.54448 81.36 2.081 18.59484 81.28 1.278 18.35459 81.37 2.080 18.72299 81.29 1.276 18.32028 81.38 2.079 18.75955 81.28 1.277 18.38257 81.27 1.796 18.44711 81.28 1.274 18.42613 81.23 1.796 18.39683 81.29 1.276 18.30794 81.23 1.796 18.57178 81.27 1.273 18.38116 81.22 1.796 18.5693 81.28 1.274 18.43062 81.25 1.794 18.51999 81.29 1.271 18.42638 81.27 1.792 18,52401 81.27 1.272 18.3799 81.29 1.792 18.48115 81.30 1.271 18.41211 81.28 1.791 18.51967 81.32 1.791 18.55067 81.30 1.790 18.56633 Table A1. 5. Selected results of the measurements for the mixture 75% BA3 – 25% BA1. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] 120.67 6.864 26.17739 120.59 5.620 25.97721 120.64 6.857 26.21372 120.60 5.617 26.01493 120.66 6.849 26.18772 120.60 5.613 25.99839 120.59 6.830 26.2025 120.63 5.608 26.13618 120.59 6.824 26.26628 120.63 5.604 26.0934 120.56 6.812 26.15307 120.64 5.601 26.14589 120.56 6.805 26.27425 120.66 5.597 26.08963 120.55 6.793 26.15399 120.67 5.594 26.07986 120.54 6.787 26.25466 120.66 5.590 26.05241 120.52 6.781 26.17626 120.87 4.203 25.68057 120.53 6.768 26.11775 120.89 4.202 25.67269 120.52 6.762 26.10995 120.86 4.200 25.60859 120.52 6.756 26.06046 120.87 4.199 25.63419 120.50 6.751 25.99922 120.89 4.198 25.54549 120.61 5.648 26.07037 120.90 4.198 25.61265 120.59 5.640 25.98673 120.94 4.197 25.4481 120.59 5.635 25.85432 120.95 4.195 25.40904 120.57 5.630 25.9721 120.99 4.194 25.48993 120.60 5.625 25.99571 121.01 4.193 25.52971 P [bar] λ [mW.m-1.K-1] 108 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 120.59 4.190 25.4426 100.59 3.887 22.83272 120.60 4.188 25.41875 100.60 3.886 22.73553 120.60 4.186 25.41216 100.49 2.795 22.3704 120.63 2.983 25.37381 100.49 2.793 22.45749 120.63 2.974 25.07327 100.47 2.790 22.43276 120.64 2.973 25.27861 100.44 2.788 22.42875 120.66 2.971 25.17697 100.43 2.786 22.37508 120.67 2.968 25.08925 100.42 2.784 22.57391 120.66 2.966 25.14248 100.43 2.780 22.35761 120.87 2.964 25.1259 100.42 2.777 22.58393 120.89 2.962 25.18325 100.43 2.776 22.56535 120.86 2.960 25.17455 100.43 2.774 22.42461 120.87 2.959 25.1344 100.44 2.771 22.58576 120.89 2.957 25.2689 100.44 2.770 22.3514 120.90 2.954 25.11887 100.44 2.767 22.40006 120.94 2.951 25.20842 100.80 2.219 22.37583 120.95 2.949 25.08764 100.75 2.217 22.35588 100.90 4.928 23.10517 100.75 2.216 22.34154 100.85 4.926 23.07805 100.72 2.214 22.24827 100.88 4.923 23.14107 100.68 2.212 22.53529 100.85 4.920 23.191 100.69 2.210 22.53382 100.81 4.916 23.00983 100.68 2.209 22.38063 100.81 4.913 23.10148 100.69 2.208 22.43447 100.81 4.910 23.12255 100.69 2.206 22.42859 100.79 4.907 22.90811 100.69 2.205 22.34082 100.75 4.904 22.95137 100.68 2.203 22.43123 100.76 4.902 22.85703 100.70 2.202 22.25567 100.73 4.899 23.21319 100.69 2.201 22.4427 100.77 4.897 23.13472 100.70 2.200 22.40274 100.75 4.895 23.22359 100.70 2.199 22.38564 100.74 3.910 22.83719 89.20 3.504 21.18263 100.70 3.908 22.74442 89.18 3.502 21.30347 100.69 3.906 22.77597 89.12 3.495 21.27447 100.67 3.904 22.86028 89.11 3.494 21.08921 100.65 3.902 22.76277 89.16 3.492 21.40485 100.67 3.900 22.74949 89.16 3.492 21.42936 100.65 3.898 22.72298 89.12 3.489 21.37084 100.61 3.897 22.68702 89.13 3.487 21.45293 100.60 3.895 22.79026 89.14 3.485 21.29153 100.61 3.894 22.89468 89.13 3.484 21.37372 100.60 3.893 22.90803 89.08 3.481 21.08545 100.58 3.891 22.84912 89.09 3.480 21.21219 100.59 3.889 22.69946 89.11 3.478 21.28471 109 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] 89.11 3.477 21.16577 88.72 1.586 20.83351 89.09 3.474 21.26289 81.02 3.217 20.24914 89.06 2.696 21.12412 80.98 3.216 19.97695 89.08 2.695 20.9711 80.99 3.214 20.28304 89.08 2.695 20.9447 80.95 3.211 20.12055 89.08 2.694 20.92819 80.95 3.209 20.164 89.05 2.693 20.9916 80.94 3.208 20.08738 89.07 2.693 21.11628 80.95 3.206 20.21764 89.05 2.692 20.9536 80.93 3.204 20.11411 89.06 2.690 21.08146 80.92 3.202 20.02663 89.09 2.690 20.93236 80.92 3.200 20.16744 89.09 2.690 21.12866 80.93 3.200 20.11385 89.09 2.689 21.05303 80.92 3.198 19.98756 89.07 2.688 21.0884 80.91 3.197 19.9344 89.09 2.687 20.96495 80.88 2.628 20.05867 89.08 2.687 21.04454 80.88 2.628 20.08526 89.16 2.040 20.83521 80.87 2.628 19.99131 89.14 2.039 20.87164 80.87 2.629 20.05101 89.10 2.038 20.88136 80.88 2.629 19.97538 89.09 2.038 20.93306 80.86 2.629 20.13648 89.03 2.036 20.84381 80.87 2.629 20.06459 89.02 2.034 20.83917 80.90 2.628 19.96561 89.04 2.033 21.00271 80.88 2.628 19.96902 89.05 2.032 20.98801 80.90 2.627 20.1026 89.09 2.031 20.84495 80.88 2.626 19.95693 89.10 2.031 20.82353 80.86 2.626 20.10563 89.12 2.030 20.87464 80.83 2.625 20.06876 89.14 2.029 20.83677 80.83 2.624 20.12028 89.13 2.029 20.90548 80.85 2.623 19.93936 89.17 2.027 20.8817 80.85 1.907 19.8871 88.85 1.589 20.68429 80.82 1.907 19.82007 88.81 1.588 20.79662 80.83 1.907 19.82354 88.81 1.588 20.73028 80.80 1.905 19.76552 88.79 1.589 20.64804 80.75 1.904 19.78224 88.77 1.589 20.67628 80.78 1.902 19.92464 88.79 1.588 20.82715 80.77 1.902 19.80127 88.75 1.587 20.6789 80.77 1.900 19.88181 88.75 1.588 20.77929 80.75 1.900 19.69949 88.74 1.587 20.63777 80.76 1.898 19.70766 88.75 1.587 20.75836 80.76 1.897 19.77959 88.70 1.587 20.60736 80.77 1.896 19.70006 88.72 1.586 20.62575 80.79 1.895 19.6744 88.72 1.586 20.75048 80.80 1.894 19.80658 P [bar] λ [mW.m-1.K-1] 110 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 80.70 1.503 19.65877 80.72 1.518 19.69337 80.70 1.504 19.72516 80.74 1.520 19.624 80.69 1.506 19.82907 80.76 1.521 19.76494 80.71 1.509 19.71458 80.76 1.523 19.69951 80.72 1.510 19.70952 80.75 1.524 19.80069 80.72 1.513 19.74923 80.76 1.525 19.8049 80.73 1.515 19.74634 80.75 1.526 19.67895 80.72 1.517 19.80602 Table A1. 6. Selected results of the measurements for Blowing Agent 2. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 100.87 7.526 27.21267 100.67 4.164 26.34336 100.80 7.514 26.93301 100.64 4.163 26.31621 100.77 7.508 27.0084 100.66 4.163 26.31165 100.79 7.502 26.9465 100.67 4.162 26.29259 100.77 7.495 27.13199 100.67 4.161 26.34768 100.79 7.489 26.99003 100.67 4.160 26.31047 100.75 7.483 27.14661 100.71 4.159 26.30236 100.76 7.477 27.0763 100.7 4.159 26.23436 100.78 7.470 27.07891 100.72 4.158 26.3521 100.78 7.464 27.21446 100.73 4.158 26.19348 100.77 7.458 27.17509 100.73 4.157 26.37119 100.75 7.444 26.97019 100.72 4.156 26.22826 100.70 7.431 27.05233 100.68 4.155 26.22201 100.69 7.424 27.13492 100.70 2.035 26.07798 100.60 6.119 26.74141 100.71 2.036 26.20425 100.60 6.119 26.60221 100.72 2.036 26.00079 100.62 6.117 26.66297 100.69 2.037 26.13108 100.63 6.116 26.60257 100.68 2.037 26.12508 100.61 6.113 26.80309 100.66 2.037 26.12204 100.58 6.109 26.69601 100.67 2.038 26.06733 100.57 6.104 26.63357 100.65 2.038 26.14164 100.60 6.099 26.81296 100.67 2.038 26.06948 100.61 6.096 26.7132 100.69 2.038 26.09938 100.63 6.091 26.57804 100.72 2.038 26.05533 100.63 6.088 26.66829 100.72 2.038 26.09589 100.67 6.079 26.75262 100.69 2.039 26.15267 100.63 6.075 26.65983 49.46 1.718 19.33959 100.62 6.074 26.73861 49.42 1.718 19.37283 100.62 6.073 26.60719 49.43 1.717 19.43432 100.66 4.165 26.22597 49.37 1.716 19.34128 100.69 4.165 26.2718 49.36 1.716 19.29145 111 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 49.32 1.716 19.28899 49.00 4.878 19.96631 49.35 1.716 19.19563 48.99 4.876 19.99024 49.37 1.715 19.24771 48.97 4.875 19.97076 49.40 1.715 19.36996 48.95 4.872 20.15004 49.36 1.715 19.42309 48.96 4.870 20.06954 49.41 1.715 19.27345 120.81 1.991 29.21434 49.38 1.715 19.26953 120.79 1.990 29.08774 49.32 1.714 19.1891 120.80 1.990 29.02866 49.46 3.019 19.42389 120.78 1.991 28.76803 49.38 3.018 19.40037 120.80 1.991 29.11842 49.32 3.018 19.37487 120.81 1.990 28.7734 49.23 3.017 19.35269 120.83 1.990 29.10862 49.16 3.015 19.53498 120.86 1.990 29.21299 49.14 3.015 19.40783 120.87 1.990 28.74547 49.10 3.015 19.36907 120.83 1.990 28.97804 49.05 3.014 19.47313 120.85 1.990 28.91569 49.00 3.013 19.41606 120.83 1.990 29.14735 48.94 3.012 19.40271 120.86 1.990 28.81723 48.95 3.011 19.39796 120.88 1.991 29.08338 48.91 3.010 19.43166 120.86 4.252 29.26099 48.89 3.010 19.50745 120.85 4.251 29.29554 48.91 3.010 19.49483 120.87 4.251 29.34192 48.93 3.010 19.38337 120.84 4.250 29.29948 48.95 3.934 19.71693 120.84 4.250 29.14758 48.96 3.932 19.72646 120.86 4.251 29.2753 48.97 3.931 19.50033 120.87 4.251 29.28043 48.99 3.930 19.52418 120.85 4.250 29.3593 48.96 3.928 19.5185 120.86 4.250 29.06955 48.95 3.927 19.66585 120.87 4.252 29.14067 48.96 3.932 19.51565 120.84 4.252 29.3627 48.99 3.935 19.62916 120.89 4.251 29.06989 49.00 3.932 19.65774 120.88 4.251 29.33323 48.98 3.930 19.63004 120.94 4.252 29.32605 48.98 3.928 19.70897 121.19 5.969 29.29134 49.03 3.925 19.52605 121.22 5.935 29.38741 49.06 3.923 19.73944 121.27 5.926 29.51958 48.93 4.901 20.14191 121.28 5.917 29.59095 48.94 4.897 19.8593 121.27 5.907 29.50267 48.98 4.895 19.94795 121.25 5.898 29.44775 49.00 4.891 20.12866 121.26 5.894 29.30301 48.99 4.887 20.06023 121.19 5.869 29.45424 48.98 4.883 19.82272 121.14 5.864 29.39419 49.01 4.881 19.90249 121.12 5.860 29.5687 112 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 121.10 5.851 29.33876 77.63 5.455 23.50093 121.08 5.846 29.49823 77.64 5.452 23.44989 121.08 5.842 29.61785 77.61 5.448 23.48631 119.86 7.283 29.55787 77.59 5.444 23.40701 119.86 7.278 29.54804 77.59 5.441 23.45027 119.89 7.265 29.69564 77.63 5.438 23.50991 119.86 7.257 29.76477 77.63 5.435 23.48776 119.87 7.250 29.63645 77.65 5.432 23.51409 119.86 7.244 29.677 77.67 5.429 23.50012 119.86 7.237 29.71139 77.64 5.426 23.47503 119.85 7.231 29.85171 77.68 5.423 23.44892 119.86 7.225 29.79308 77.66 5.418 23.58792 119.85 7.228 29.80949 77.65 5.416 23.48621 119.84 7.220 29.62785 77.65 5.414 23.51415 119.86 7.213 29.76908 77.64 4.153 23.1118 119.86 7.206 29.67717 77.63 4.153 23.13862 119.85 7.200 29.7468 77.64 4.152 23.13813 119.86 7.194 29.79982 77.65 4.151 23.2507 119.84 7.187 29.73714 77.68 4.150 23.14681 119.85 7.179 29.65203 77.70 4.149 23.18367 119.82 7.171 29.69277 77.73 4.148 23.22702 119.82 7.165 29.74593 77.76 4.146 23.35034 77.96 6.536 23.93405 77.77 4.146 23.4166 77.94 6.529 23.9698 77.76 4.146 23.36495 77.95 6.525 23.93487 77.73 4.145 23.39265 77.96 6.519 24.01793 77.73 4.143 23.15802 77.97 6.513 23.90195 77.74 4.142 23.29939 77.92 6.507 23.87075 77.78 2.029 22.99822 77.93 6.501 23.96499 77.84 2.031 22.93095 77.92 6.496 23.80177 77.84 2.032 22.90497 77.92 6.490 23.88182 77.88 2.032 22.86906 77.93 6.484 23.90473 77.88 2.033 22.86734 77.89 6.477 23.88352 77.86 2.033 22.90276 77.87 6.472 23.82234 77.85 2.033 22.87056 77.86 6.466 23.81436 77.81 2.033 22.82351 77.88 6.460 23.96134 77.81 2.033 22.8103 77.88 6.455 23.75286 77.77 2.033 22.95144 77.88 6.450 23.9163 77.77 2.033 22.85451 77.87 6.445 23.93586 77.78 2.034 22.79838 77.90 6.439 23.85729 77.77 2.034 22.81447 77.62 5.458 23.52926 113 Thermal Conductivity of Gases Table A1. 7. Selected results of the measurements for the mixture 55% BA2 – 45% BA1. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 120.53 7.729 26.64537 120.54 7.732 26.64666 120.52 4.642 25.84614 120.51 4.640 25.62989 120.61 7.735 26.56875 120.47 4.639 25.8517 120.60 7.735 26.71721 120.47 4.638 25.51045 120.80 7.739 26.49106 120.45 4.635 25.69847 120.81 7.737 26.56945 120.44 4.633 25.655 120.77 7.737 26.47907 120.45 4.632 25.64451 120.77 7.736 26.56417 120.40 4.629 25.75346 120.76 7.735 26.66864 120.39 4.627 25.78615 120.75 7.732 26.52937 120.39 4.626 25.65055 120.74 7.731 26.59675 120.41 4.624 25.70607 120.74 7.730 26.5684 120.44 4.621 25.7222 120.74 7.729 26.49963 120.42 4.619 25.6255 120.74 7.727 26.54284 120.41 4.618 25.76176 120.74 7.726 26.49157 99.97 5.820 23.43887 120.73 7.724 26.62513 99.97 5.815 23.29915 120.64 7.735 26.52259 99.97 5.812 23.3113 120.64 7.736 26.46839 99.94 5.809 23.23905 120.62 7.734 26.47217 99.94 5.811 23.38764 120.60 7.731 26.56352 99.94 5.808 23.39314 120.62 7.730 26.42113 99.95 5.807 23.40579 120.71 6.643 26.24611 99.97 5.805 23.36431 120.67 6.642 26.10676 99.97 5.804 23.32249 120.65 6.640 26.15102 99.99 5.803 23.30483 120.59 6.633 26.06192 99.99 5.802 23.26728 120.55 6.631 26.05127 100.06 5.012 23.01014 120.53 6.628 26.06561 100.06 5.010 23.09753 120.55 6.624 26.00335 100.03 5.009 23.09625 120.54 6.622 26.27301 100.02 5.007 23.06151 120.55 6.620 26.0033 100.01 5.005 23.09495 120.54 6.613 26.07795 99.99 5.004 23.13362 120.56 6.610 26.08173 99.99 5.002 23.13791 120.54 6.603 26.06456 100.04 5.001 23.05938 120.86 5.497 25.78875 100.01 5.000 23.03362 120.83 5.494 25.79447 99.98 4.997 23.00691 120.82 5.492 25.76904 100.01 4.995 23.08487 120.81 5.490 25.99233 100.43 4.185 22.87675 120.81 5.482 25.7274 100.38 4.184 22.93062 120.82 5.478 25.91329 100.38 4.180 22.8446 120.82 5.473 25.876 100.26 4.173 22.83159 120.85 5.472 25.7239 100.25 4.170 22.84707 120.84 5.468 25.90891 100.25 4.168 22.89702 120.64 4.652 25.701 100.26 4.167 22.86776 114 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 100.25 4.165 22.94072 88.15 2.576 20.90262 100.23 4.163 22.89568 88.13 2.575 21.10072 88.74 3.861 21.23 88.15 2.574 21.03118 88.76 3.859 21.55709 88.15 2.573 21.06019 88.76 3.857 21.49353 88.13 1.877 20.75536 88.76 3.855 21.40669 88.10 1.874 20.57615 88.74 3.854 21.51033 88.07 1.878 20.55226 88.74 3.851 21.42124 88.06 1.871 20.60332 88.74 3.849 21.19847 88.05 1.875 20.63143 88.73 3.848 21.63397 88.04 1.869 20.7162 88.72 3.846 21.28199 88.02 1.871 20.49704 88.72 3.845 21.21496 88.02 1.867 20.52908 88.70 3.844 21.58273 88.01 1.871 20.57045 88.71 3.841 21.18118 88.01 1.864 20.59256 88.70 3.841 21.59958 88.04 1.869 20.68033 88.70 3.838 21.32102 88.03 1.860 20.60789 88.69 3.837 21.4347 88.01 1.865 20.60789 88.68 3.241 21.36519 79.23 3.031 20.149 88.54 3.235 21.11755 79.18 3.027 20.25495 88.47 3.232 21.29403 79.12 3.025 20.10964 88.40 3.228 21.23963 79.07 3.022 19.96795 88.37 3.225 21.20597 79.05 3.020 20.04239 88.31 3.222 21.37421 79.07 3.018 20.05449 88.28 3.221 21.14435 79.05 3.015 20.08738 88.26 3.218 21.13835 79.06 3.014 20.11693 88.22 3.216 21.35148 79.07 3.013 19.85149 88.20 3.215 21.19813 79.07 3.012 20.0428 88.18 3.214 21.19373 79.07 3.011 20.06487 88.17 3.213 21.15119 79.07 3.010 20.00466 88.13 3.212 21.16844 79.02 3.008 20.10635 88.15 3.211 21.25206 79.01 3.006 20.10055 88.14 3.209 21.20038 78.97 3.006 19.83821 88.26 2.585 21.04249 79.01 2.412 19.81199 88.24 2.584 21.15679 79.00 2.411 19.85219 88.22 2.583 20.87718 79.00 2.410 19.83177 88.22 2.584 21.17773 78.97 2.407 19.82696 88.20 2.583 21.16171 78.95 2.406 19.82785 88.18 2.582 21.1696 78.91 2.406 19.81246 88.19 2.581 20.83662 78.92 2.403 19.83176 88.18 2.580 21.08619 78.92 2.405 19.85341 88.18 2.579 20.94014 78.93 2.402 19.71988 88.16 2.577 21.03621 78.94 2.402 19.70722 88.16 2.577 21.04937 78.95 2.399 19.93362 115 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 78.96 2.402 19.82933 78.91 1.761 19.45836 78.93 2.398 19.78942 78.94 1.768 19.52603 78.96 2.400 19.70756 78.94 1.764 19.70407 78.92 2.400 19.91915 78.95 1.770 19.71907 78.82 1.741 19.39096 78.97 1.767 19.4058 78.84 1.746 19.40096 78.98 1.772 19.45845 78.86 1.749 19.58044 78.99 1.765 19.64849 78.87 1.758 19.60343 78.99 1.770 19.64067 78.88 1.765 19.72902 79.00 1.772 19.60529 78.91 1.770 19.43208 Table A1. 8. Selected results of the measurements for the mixture 32% BA2 – 68% BA1. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 120.69 6.982 25.21366 120.56 4.812 24.59953 120.70 6.983 25.28325 120.54 4.810 24.61452 120.70 6.984 25.30534 120.52 4.809 24.71962 120.71 6.985 25.05411 120.52 4.807 24.45062 120.74 6.984 25.18914 120.55 4.805 24.76845 120.71 6.985 25.23936 120.53 4.803 24.73735 120.69 6.984 25.12728 120.83 2.509 23.8682 120.67 6.984 25.13651 120.82 2.508 23.80931 120.64 6.982 25.23274 120.77 2.506 23.91735 120.63 6.982 25.09226 120.73 2.505 24.16908 120.66 6.981 25.12553 120.70 2.505 24.17929 120.59 5.963 24.71605 120.70 2.503 24.09272 120.58 5.959 24.67353 120.69 2.502 23.94531 120.55 5.957 24.80208 120.68 2.502 23.98496 120.52 5.956 24.95267 120.69 2.501 23.96787 120.51 5.955 24.97916 120.68 2.500 24.02214 120.49 5.954 24.96636 120.69 2.500 23.94356 120.49 5.952 24.98831 120.69 2.501 23.96694 120.49 5.951 25.02235 120.71 2.501 23.90939 120.48 5.951 24.6505 120.71 2.499 24.03193 120.49 5.950 24.95225 120.70 2.497 23.89762 120.49 5.949 24.97352 120.71 2.497 23.93881 120.78 4.830 24.79923 100.29 3.989 21.77887 120.77 4.829 24.76081 100.14 3.981 21.70392 120.73 4.827 24.53466 100.10 3.978 21.73972 120.61 4.818 24.5782 100.06 3.976 21.8748 120.59 4.816 24.51334 100.03 3.975 21.81317 120.59 4.815 24.60973 100.02 3.973 21.74266 120.56 4.814 24.5269 99.99 3.971 21.75158 116 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 99.96 3.969 21.56975 99.71 1.545 20.83119 99.96 3.967 21.73483 99.73 1.544 20.88903 99.93 3.966 21.86411 88.61 2.796 20.06874 99.93 3.964 21.72273 88.61 2.796 20.06981 99.92 3.963 21.69489 88.62 2.795 20.04388 99.95 3.011 21.70189 88.61 2.795 20.03737 99.94 3.009 21.49061 88.58 2.793 20.13664 99.94 3.008 21.3984 88.57 2.792 19.98522 99.96 3.006 21.41028 88.57 2.791 20.07231 99.96 3.006 21.66172 88.56 2.790 19.97628 99.96 3.005 21.53753 88.58 2.788 19.98356 99.96 3.004 21.38426 88.54 2.787 19.92975 99.96 3.003 21.51133 88.56 2.786 20.14599 99.96 3.003 21.50598 88.56 2.785 20.03622 99.94 3.002 21.6893 88.54 2.783 20.13406 99.94 3.001 21.43154 88.55 2.783 19.96413 100.08 2.307 21.31034 88.55 2.782 19.97959 100.06 2.304 21.25669 88.45 2.203 19.85312 100.05 2.302 21.28174 88.48 2.203 19.73347 100.04 2.299 21.26789 88.48 2.203 19.86973 100.07 2.298 21.30035 88.50 2.203 19.87995 100.03 2.297 21.14113 88.50 2.202 19.90099 100.01 2.296 21.26065 88.51 2.202 19.86557 99.98 2.295 21.26767 88.53 2.202 19.91822 99.94 2.294 21.35961 88.54 2.201 20.02479 99.95 2.292 21.38614 88.55 2.201 20.01253 99.94 2.291 21.29263 88.55 2.201 19.98984 99.94 2.290 21.18563 88.54 2.200 19.94422 99.96 2.290 21.22754 88.51 2.200 19.79355 99.97 2.289 21.18938 88.49 2.199 19.9371 99.95 2.289 21.16015 88.55 1.475 19.63968 99.80 1.552 21.06534 88.52 1.474 19.5852 99.79 1.550 20.92755 88.49 1.474 19.61892 99.79 1.550 20.87299 88.48 1.474 19.75754 99.76 1.549 20.82082 88.49 1.474 19.72445 99.74 1.548 20.77967 88.46 1.474 19.70746 99.73 1.548 21.08132 88.49 1.474 19.41208 99.76 1.548 21.04277 88.51 1.476 19.45907 99.77 1.547 20.79752 88.53 1.476 19.42345 99.75 1.546 20.98035 79.21 2.570 18.78563 99.77 1.546 20.88777 79.18 2.569 18.93973 99.74 1.545 20.93637 79.17 2.569 18.80178 99.72 1.545 20.99799 79.15 2.567 18.81551 117 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 79.13 2.565 18.92298 79.07 2.096 18.82099 79.13 2.564 18.93413 79.04 2.096 18.86777 79.13 2.564 18.80792 79.01 2.095 18.88431 79.11 2.562 18.8471 79.00 2.093 18.75765 79.09 2.561 18.71824 79.01 2.094 18.70774 79.11 2.561 19.00544 79.04 2.093 18.68272 79.12 2.560 18.99774 79.21 1.382 18.58449 79.17 2.559 18.94085 79.19 1.379 18.53598 79.18 2.558 18.91981 79.15 1.376 18.51632 79.19 2.558 18.91358 79.13 1.373 18.54344 79.16 2.557 18.96825 79.12 1.371 18.6078 79.16 2.556 18.87664 79.11 1.370 18.66105 79.08 2.095 18.88032 79.08 1.369 18.56623 79.09 2.096 18.86809 79.05 1.370 18.48441 79.10 2.096 18.66193 79.01 1.371 18.52395 79.12 2.096 18.70156 78.99 1.374 18.57531 79.11 2.096 18.73632 79.00 1.377 18.2914 79.09 2.095 18.90777 79.00 1.380 18.54241 79.08 2.096 18.87372 79.02 1.382 18.37035 Table A1. 9. Selected results of the measurements for the mixture 73% BA2 – 27% BA1. T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 120.35 7.509 27.12937 120.31 6.533 26.92447 120.35 7.507 27.14049 120.60 4.854 26.50373 120.37 7.504 27.15423 120.56 4.853 26.39082 120.34 7.502 27.12173 120.54 4.851 26.57297 120.34 7.501 27.05047 120.55 4.849 26.60592 120.34 7.499 27.13318 120.51 4.847 26.61817 120.36 7.498 27.1132 120.49 4.845 26.53203 120.39 7.495 27.06284 120.51 4.843 26.27794 120.41 7.494 27.17878 120.47 4.841 26.61388 120.36 7.492 27.04911 100.14 7.412 24.95174 120.40 6.551 27.02925 100.10 7.405 25.09814 120.38 6.549 26.95562 100.08 7.397 25.08089 120.35 6.547 26.98628 100.09 7.392 25.05832 120.34 6.545 26.95924 100.08 7.384 24.87767 120.34 6.543 26.97017 100.08 7.377 25.04825 120.33 6.541 26.92178 100.04 7.371 24.85522 120.31 6.539 26.91134 100.05 7.366 25.09903 120.32 6.537 27.06216 100.03 7.354 24.78313 120.33 6.536 26.94923 99.96 6.602 24.61235 118 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 99.94 6.599 24.66223 88.92 4.297 22.36337 99.93 6.597 24.5667 88.92 4.298 22.3375 99.92 6.594 24.58342 88.93 4.298 22.18738 99.88 6.591 24.69283 88.92 4.298 22.30932 99.85 6.587 24.5346 88.91 4.298 22.29981 99.85 6.585 24.69301 88.91 4.298 22.13463 99.85 6.583 24.63175 88.94 4.298 21.99488 99.86 6.580 24.50347 88.94 4.298 22.092 99.85 6.577 24.48433 88.58 2.837 21.69085 99.84 6.575 24.45684 88.92 2.841 21.82574 99.85 6.574 24.70389 89.06 2.841 21.75229 99.85 6.571 24.82818 89.07 2.849 21.84432 99.84 6.567 24.57116 89.08 2.849 21.82244 100.39 5.905 24.24918 89.07 2.849 21.71692 100.35 5.902 24.35313 89.07 2.849 21.68198 100.31 5.899 24.2574 89.04 2.847 21.82754 100.28 5.896 24.21571 89.00 2.849 21.74976 100.26 5.894 24.16876 89.02 2.845 21.67655 100.22 5.890 24.33467 89.18 2.079 21.46716 100.11 4.521 24.28054 89.16 2.073 21.41803 100.09 4.524 24.21388 89.13 2.078 21.72051 100.09 4.526 23.94716 89.12 2.072 21.3728 100.07 4.527 23.8513 89.11 2.077 21.47709 100.04 4.524 23.86562 89.12 2.073 21.59015 89.08 5.468 22.48661 89.08 2.075 21.39708 89.03 5.465 22.66683 89.07 2.074 21.41039 89.03 5.463 22.63694 89.08 2.072 21.75419 89.04 5.460 22.58718 79.30 4.908 21.15091 89.00 5.458 22.66252 79.31 4.907 21.03266 89.00 5.453 22.65112 79.33 4.907 20.96032 89.01 5.452 22.54897 79.34 4.904 21.17534 89.00 5.450 22.65735 79.32 4.902 20.95999 89.02 5.448 22.58093 79.32 4.901 21.25519 89.02 5.446 22.59573 79.31 4.899 21.14485 89.01 5.445 22.52487 79.29 4.897 20.99221 89.05 5.443 22.56823 79.31 4.896 20.99305 89.02 5.441 22.62679 79.31 4.895 20.96415 89.03 5.438 22.73171 79.34 4.894 21.05999 89.02 5.437 22.66787 79.34 4.893 21.23363 89.01 4.295 22.42897 78.89 4.003 21.01273 88.97 4.295 22.36991 78.89 4.006 20.85715 88.93 4.297 22.2449 78.93 4.008 21.07967 88.90 4.297 22.10297 79.30 4.908 21.15091 119 Thermal Conductivity of Gases (Continuation) T [°C] P [bar] λ [mW.m-1.K-1] T [°C] P [bar] λ [mW.m-1.K-1] 78.97 4.009 20.95959 78.67 2.724 20.57764 79.01 4.012 20.91914 78.65 2.722 20.6126 79.09 4.014 20.98863 79.28 1.954 20.26622 79.12 4.016 20.92399 79.27 1.951 20.28666 79.19 4.019 20.88788 79.25 1.952 20.41701 79.24 4.021 20.99253 79.23 1.951 20.40646 79.27 4.023 21.00104 79.22 1.948 20.35154 79.31 4.025 21.03131 79.23 1.950 20.46795 79.28 4.025 20.97662 79.28 1.946 20.35451 79.14 2.743 20.65174 79.30 1.952 20.32949 79.05 2.739 20.68264 79.31 1.948 20.40085 79.01 2.737 20.62138 79.27 1.943 20.48096 78.94 2.734 20.64583 79.28 1.949 20.37102 78.94 2.732 20.59764 79.29 1.946 20.13321 78.87 2.729 20.72737 79.29 1.950 20.45954 78.81 2.727 20.58908 79.32 1.956 20.1978 78.74 2.725 20.69966 79.34 1.964 20.42013 120