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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


I

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.


II

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.


III

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.


IV

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


V

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


VI

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


nitrogen for P = 1 bar, relatively to correlation of José Fonseca..................................34






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


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


Blowing Agent 2 for P = 1 bar, relatively to correlation of the NIST reference data. ......43


Blowing Agent 2 for P = 1 bar, relatively to correlation of the DIPPR data....................43


VII

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


BA1. ....................................................................................................................48


BA1. ....................................................................................................................49


BA1. ....................................................................................................................50


BA1. ....................................................................................................................51

Figure 32. Relative error between the experimental values of thermal conductivity for

nitrogen with the predicted values with the LPUR equation. .......................................54


Blowing Agent 1 with the predicted values with the LPUR equation. ............................54


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 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 44. ε function of the pressure at some temperatures for the system BA3 – BA1. The

lines are the values obtained by eq. 43. ..................................................................62


VIII

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


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 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












IX



Figure 62. λ function of the molar composition at 1 bar for the mixture BA2 – BA1........74









Figure 68. ε function of the pressure at some temperatures for the system BA2 – BA1. The


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


Figure 71. Deviations of ε represented as a function of the temperature, for the system

BA2 – BA1. ...........................................................................................................79













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


X










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


XI

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


XII

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


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


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


1

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 [5].

Due to their high thermal 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].


2

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).


3

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,

TQC ∇×−= λ 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.


4

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:

( ) vSvPQDt

DU ρρρ∇−∇−−∇= :..ρ eq. 2

Where U is the internal energy, t the time, P the hydrostatic pressure, vr

the

hydrodynamic velocity of the fluid, S the tensor stress, Qr 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 local-

equilibrium thermodynamic state exists, eq. 2 can be transformed to

φαα

ρ +−∇=

+−

− Q

Dt

DT

DT

DPk

kT

Dt

DTc TP

T

P

v

ρ.... eq. 3

Where cv is the heat capacity at constant volume, Pα the isobaric expansion coefficient,

Tk the isothermal compressibility and φ = vSr

∇: is the rate of internal energy increase

owing to viscous dissipation. Transient techniques are operated so that DT

DPkT


5

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 near-

equilibrium state is maintained. Secondly, any fluid movement must be avoided so that

vr=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

RQTQrr

+∇−= .λ eq. 5

In which λ is the thermal conductivity and RQr is the heat flux arising radiation. Although

there is always some contributions from radioactive transport there are some

circumstances under which it is negligible so that to formulate an ideal theory RQr 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:

Tt

TCP ...

2∇=∂

∂λρ 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


6

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,

( ) ( ) 0,, TtrTtrT −=∆ 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,

( ) ranyttrT ,0for 0, ≤=∆ eq. 8

( ) ∞=>=∆→ rttrT ,0for 0,lim 0r eq. 9

0,0for constant 2

qlim 0r =≥=−=

∂

∂→ rt

r

Tr

πλ eq. 10

With the additional condition that the thermal diffusivity, k =λ/ρ.CP, is constant. The

solution of eq. 6 is,

( )

−=∆

kt

rE

qtrTid

44,

2

1πλ

eq. 11

Where E1(x) is the exponential integral with the expansion,

( ) ( )∫ ++−−==−0

2

1 lnx

y

xOxdyy

exE γ eq. 12


7

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.

( ) ( )

++

=−=∆ ...

4

4ln

4,,

2

0

2

0

000kt

r

Cr

ktqTtrTtrTid

πλ eq. 13

Where C=exp(γ ). If the wire radius is chosen such that the second term on the right-

hand 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:

( )

=∆

Cr

ktqtrTid 2

0

0

4ln

4,

πλ 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.


8

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 non-

zero radius alone is readily found by solving eq. 6 subject to the new condition, which

replaces eq. 10, that

02

0 ≥=−

=∂

∂tanyforarat

a

q

r

T

πλ eq. 15

At large values of 4kt/r2, the solution for the temperature rise of the fluid is

( )

+

=∆

kt

a

Cr

ktqtrT

2

2

0

4ln

4,

πλ 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


9

Where the correction δT1 is

( ) ( )[ ]

+−−

−

=

WW

pWPtk

a

kt

aqcc

t

a

Ca

ktqT

λ

λ

πλρρ

λπλδ

2424..

2

4ln

4

222

21 eq. 18

and kw that is the thermal diffusivity of the wire material, is equal to

( )WP

WW

ck

.ρ

λ= 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

( ) ( )ar

SWr

TgtaTtaT

=

∂

∂−=− ,, 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

a

gqT SK

2

4

=

πλδ 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


10

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

( ) 122lim .....210−

≈ taNq A λσρ 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

( ) 00, 00 ≥==∆ tanyandbrfortrT 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,

( ) ( ) 2,, TtaTtaT Wid δ+∆=∆ eq. 24

Here, the ‘outer-boundary correction’, δT2, is given by the expression


11

( )[ ]

−+

= ∑

∞

=1

2

02

2

22

..exp

4ln

4 vv

v gYb

tkg

Cb

ktqT π

πλδ 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


12

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

CWid TTT δ+∆=∆ eq. 26

and

Vcc

TRlqT

VP

C...

...

ρδ = 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.

( )

=∆

Ca

kt

T

qT

rr

id 2

4ln

,4 ρπλ 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,


13

( ) ( )[ ]2

210

tTtTTT ididr

∆+∆+= eq. 29

And

( )PTrr ,ρρ = 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:

=∆

Ca

tkqTid 2

'4ln

4πλ 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


14

( ) WBWBr TTaTTaq ∆≈−= .....8....2 30404 σπσπ 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

( )23

0....8W

Brad T

q

TaT ∆

≈

σπδ 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.

radWid TTT δ+∆=∆ eq. 34

And

( )

−+

−+

∆=

qEb

a

E

TTaT Wrad

11

11

1

....823

0πδ 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


15

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.

( )∑

=+∆=∆

i rr

iWidCa

tk

xT

qTTT

2

04ln,,4 ρπλ

δ 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.


16

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:

PTDPCTBA .... +++=λ 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.


17

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

mixtures for low pressure, it is presented bellow [21].

∑∑=

=

=n

in

j

ijj

iim

Ay

y

1

1

λλ

eq. 38

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:

2/1

24/12/1

18

1

+

+

=

j

i

j

i

trj

tri

ij

M

M

M

M

A

λ

λε

eq. 39


18

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

( ) ( )[ ]( ) ( )[ ]rjrji

ririj

trj

tri

TT

TT

2412.0exp0464.0exp

2412.0exp0464.0exp

−−Γ

−−Γ=

λ

λ 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

6/1

4

3

210

=Γ

ci

ici

iP

MT 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.

bTa.=ε 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).

432 ..1APAPA

TeA+=ε 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.


19

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.


20

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.


21

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.


22

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.


23

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.


24

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.


25

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


26

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 start

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