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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 − 1q 


 
 
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
81 + 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