ARISTOTLE UNIVERSITY OF THESSALONIKI

SCHOOL OF ENGINEERING – CHEMICAL ENGINEERING DEPARTMENT SECTION OF UNIT OPERATIONS AND APPLIED THERMODYNAMICS

LABORATORY OF THERMOPHYSICAL PROPERTIES & ENVIRONMENTAL PROCESSES

MEASUREMENT OF THE THERMAL CONDUCTIVITY OF

COMPOSITE SOLID MATERIALS

by

KONSTANTINOS D. ANTONIADIS

DIPL. CHEMICAL ENGINEER

Thesis submitted for the degree of

Doctor of Philosophy

in Chemical Engineering

THESSALONIKI 2011

ΚΟΝΣΤΑΝΤΙΝΟΣ Δ. ΑΝΤΩΝΙΑΔΗΣ

ΜΕΤΡΗΣΗ ΤΗΣ ΘΕΡΜΙΚΗΣ ΑΓΩΓΙΜΟΤΗΤΑΣ

ΣΥΝΘΕΤΩΝ ΣΤΕΡΕΩΝ ΥΛΙΚΩΝ

ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ

Υποβλήθηκε στο Τμήμα Χημικών Μηχανικών, Πολυτεχνική Σχολή,

Τομέας Τεχνικής των Φυσικών Διεργασιών & Εφαρμοσμένης Θερμοδυναμικής

Ημερομηνία Προφορικής Εξέτασης: 13/04/2011

Εξεταστική Επιτροπή

Καθηγητής Μ.Ι. Ασσαέλ, Επιβλέπων Καθηγητής Κ. Παναγιώτου, Μέλος Τριμελούς Συμβουλευτικής Επιτροπής Αν. Καθηγητής Ε. Γ. Καστρινάκης, Μέλος Τριμελούς Συμβουλευτικής Επιτροπής Καθηγητής Γ. Σακελλαρόπουλος, Εξεταστής Καθηγητής Μ. Σταματούδης, Εξεταστής Καθηγητής Π. Γιαννακουδάκης, Εξεταστής Επικ. Καθηγητής, Ε. Βουτσάς, Εξεταστής

© Κωνσταντίνος Δ. Αντωνιάδης © Α.Π.Θ. Μέτρηση της Θερμικής Αγωγιμότητας Σύνθετων Στερεών ISBN: «Η έγκριση της παρούσης Διδακτορικής Διατριβής από το Τμήμα Χημικών Μηχανικών της Πολυτεχνικής Σχολής του Αριστοτελείου Πανεπιστημίου Θεσσαλονίκης δεν υποδηλώνει αποδοχή των γνωμών του συγγραφέως» (Ν. 5343/1932, άρθρο 202, παρ. 2)

to my parents

Abstract

The scope of the present work was the performance of very low uncertainty measurements of the thermal conductivity of solid and composite materials to be employed in thermal conductivity reference correlations. The technique employed for the measurements is the transient hot–wire method, which is established as an accurate absolute technique for thermal conductivity measurements of solids.

A new experimental device was designed and several refinements were made in relation to the already existing one. More specifically, a new silicone paste, of higher thermal conductivity was used in the sensor and two stainless steel–spacers were employed to define exactly the distance between the solid specimens that were being measured. Moreover, Comsol Multiphysics finite element software was employed to accurately describe the geometry of the sensor and reproduce the temporal temperature rise of the hot wire from the beginning of the time scale. The refinements applied to the hot–wire instrument allowed the thermal conductivity measurement of solids with an uncertainty better than 1 %.

In details, in the first chapter, the importance of the knowledge of material properties and the accurate measurement of them is pointed out. Moreover, the international terminology for reference materials is analysed and the major worldwide material certification institutions are presented.

In Chapter B, a brief overview of the most significant and widely used methods for the measurement of the thermal conductivity of solids is presented. The Fourier’s heat conduction law is presented and the dominating equations are analysed. Following that, a brief description of the fundamental principles of the available steady–state and transient methods for measuring the thermal conductivity of solids is shown. A critical

ii Abstract

evaluation of the methods is made by presenting and discussing three interlaboratory measurement programs for Pyroceram 9606. The chapter ends with a detailed historical review, starting from 1780, of the development of the transient hot–wire method.

The third chapter presents and analyses the fundamental theoretical and computational background needed for performing measurements of thermal conductivity with the transient hot–wire technique. Firstly, the “ideal” theoretical model on which the transient hot–wire technique is based on, and the equations for its analytical solution are given. In addition, the most significant corrections that have to be applied to this “ideal” solution are briefly mentioned. Continuously, the finite element method and its use together with the transient hot–wire technique is presented. The characteristics of Comsol Multiphysics finite element software are analysed, and an evaluation model representing a simple geometry, i.e. the experimental setup for measuring the thermal conductivity of liquids, is created. The comparison of the simulation results with the analytical solution of the model are in perfect agreement. Consequently, the finite–element model created for representing the actual experimental setup for solids is described analytically, and the procedure followed for obtaining the thermal conductivity of both liquid and solid samples from experimental data is shown in details. The chapter ends with a sensitivity analysis of the effect that the finite–element model’s parameters have on the calculated thermal conductivity value of solids.

In Chapter D the newly designed and constructed transient hot–wire sensor is analytically described and its main improvements compared to the older one are illustrated. A large section of the chapter is attributed to the electronic components consisting the experimental setup. An historical review of previously used Wheatstone bridge circuits is presented and the current configuration together with the data acquisition system and the working equations for calculating the experimental temperature rise of the hot wire are analysed in detail. Moreover, there is a description of the procedure followed to derive the temperature rise of the hot wire from the acquired voltage response from the sensor. The calculated temperature rise is therefore employed together with Comsol Multiphysics software to calculate the thermal conductivity value of a material. Consequently, the validation of the sensor is performed by using it for measuring the thermal conductivity of toluene, which is a reference liquid. The last part of the

Abstract iii

chapter is dedicated to the analysis of the sources of uncertainty introduced during the experiment and the overall uncertainty of the thermal conductivity is calculated to be better than 0.5 % for liquids, and better than 1 % for solids.

The following chapter is dedicated to the presentation of the thermal conductivity measurements of solid and composite materials. Firstly, the developed transient hot wire sensor is employed for the measurement of the thermal conductivity of the silicone pastes used in the sensor. Following that, the technique is used for measuring the thermal conductivity of the reference materials Pyroceram 9606 (designated as glass ceramic BCR–724) and Pyrex 7740 (designated as BCR–039) over a temperature range from 315 K to 440 K. Subsequently, the sensor is used for studying candidate reference materials for thermal conductivity. The candidate glassy materials measured from ambient temperature up to 440 K were Polymethyl Methacrylate (PMMA) and Borosilicate Crown Glass (BK7). The present thermal conductivity measurements are compared with the data of other investigators, and in all cases it was found that the deviations are within the mutual uncertainty of the measurements.

In the last part of Chapter E, the hot wire sensor is used for measuring the enhancement of the thermal conductivity of an epoxy–resin reinforced initially with plies of plain weave glass fabric, then by carbon multi–walled nanotubes (C–MWNT), and finally with both these two macroscopic and nanoscopic reinforcements at hand. The uncertainty of the measurement increases to ±2 % due to inhomogeneities in the samples preparation. The results reveal that for the epoxy reinforced with glass fibres, with volume fraction of 28 %, the thermal conductivity increase was 27 % compared to plain epoxy resin. When reinforced with C–MWNTs the maximum enhancement observed was about 20 % at a concentration of 1.2 % by weight of C–MWNT. Similarly, when reinforced with both the C–MWNTs and glass fibres the maximum thermal conductivity enhancement observed was about 60 % at a concentration of 1.2 % by weight of C–MWNTs.

The present thesis ends with a summary of the innovations achieved and present the future research challenges on the transient hot wire technique.

Περίληψη

Σκοπός της παρούσας εργασίας ήταν η πραγματοποίηση μετρήσεων θερμικής αγωγιμότητας στερεών και σύνθετων υλικών, πολύ χαμηλής αβεβαιότητας, για χρήση σε συσχετίσεις αναφοράς θερμικής αγωγιμότητας. Η μέθοδος που εφαρμόστηκε για τις μετρήσεις είναι η μέθοδος του θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση, η οποία καθιερώνεται πλέον ως μια απόλυτη και ακριβής τεχνική για τη μέτρηση της θερμικής αγωγιμότητας στερεών.

Στο πλαίσιο της διατριβής, σχεδιάστηκε μία νέα πειραματική διάταξη και έγιναν αρκετές βελτιώσεις–τροποποιήσεις σε σχέση με την ήδη υπάρχουσα διάταξη. Πιο συγκεκριμένα, στον αισθητήρα χρησιμοποιήθηκε μια νέα σιλικόνη υψηλότερης θερμικής αγωγιμότητας, καθώς και δύο μεταλλικά στηρίγματα από ανοξείδωτο χάλυβα για τον πλήρη καθορισμό του πάχους της. Ακόμη, το λογισμικό πεπερασμένων στοιχείων Comsol Multiphysics χρησιμοποιήθηκε για την ακριβή μοντελοποίηση της γεωμετρίας του αισθητήρα και την πλήρη αναπαραγωγή της θερμοκρασιακής αύξησης του θερμαινόμενου σύρματος από πραγματικό χρόνο μηδέν. Οι βελτιώσεις που έγιναν στη διάταξη του θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση επιτρέπουν τη μέτρηση της θερμικής αγωγιμότητας στερεών με αβεβαιότητα καλύτερη του 1%.

Αναλυτικά, στο πρώτο κεφάλαιο επισημαίνεται η σημασία που έχει η γνώση της θερμικής αγωγιμότητας των υλικών και η αξία της μέτρησής της με χαμηλή αβεβαιότητα. Ακόμη, αναλύεται η διεθνής ορολογία που χρησιμοποιείται για τα πρότυπα υλικά αναφοράς και παρουσιάζονται οι κυριότεροι διεθνείς φορείς πιστοποίησης πρότυπων υλικών.

Στο Κεφάλαιο Β γίνεται μία σύντομη επισκόπηση των πιο σημαντικών και ευρέως χρησιμοποιούμενων μεθόδων για τη μέτρηση της θερμικής

vi Περίληψη

αγωγιμότητας στερεών. Παρουσιάζεται ο νόμος του Fourier για τη μεταφορά θερμότητας με αγωγή και αναλύονται οι βασικές εξισώσεις. Στη συνέχεια γίνεται μία σύντομη περιγραφή των θεμελιωδών αρχών των υπαρχόντων μεθόδων μέτρησης θερμικής αγωγιμότητας στερεών υλικών σε μόνιμη και μη μόνιμη κατάσταση. Ακόμη, γίνεται κριτική αξιολόγηση των παραπάνω μεθόδων μέσω της παρουσίασης και του σχολιασμού τριών διεργαστηριακών προγραμμάτων μέτρησης της θερμικής αγωγιμότητας του στερεού κεραμικού υλικού Pyroceram 9606. Το κεφάλαιο ολοκληρώνεται με μια λεπτομερή ιστορική αναδρομή από το 1780, της εξέλιξης της μεθόδου του θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση.

Στο τρίτο κεφάλαιο παρουσιάζεται και αναλύεται το θεωρητικό και υπολογιστικό υπόβαθρο που απαιτείται για την πραγματοποίηση μετρήσεων θερμικής αγωγιμότητας με τη μέθοδο του θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση. Αρχικά περιγράφεται το “ιδανικό” θεωρητικό μοντέλο στο οποίο στηρίζεται η μέθοδος και δίνονται οι εξισώσεις για την αναλυτική του επίλυση. Ακόμη, αναφέρονται εν συντομία οι πιο σημαντικές διορθώσεις που πρέπει να εφαρμοστούν στην “ιδανική” αναλυτική λύση. Ακολούθως, παρουσιάζεται η μέθοδος των πεπερασμένων στοιχείων και η ταυτόχρονη χρήση της με τη τεχνική του θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση. Αναλύονται τα χαρακτηριστικά του λογισμικού πεπερασμένων στοιχείων Comsol Multiphysics και κατασκευάζεται ένα μοντέλο αξιολόγησης, το οποίο αντιπροσωπεύει τη γεωμετρία της πειραματικής διάταξης στη περίπτωση μέτρησης της θερμικής αγωγιμότητας υγρών. Τα αποτελέσματα της προσομοίωσης είναι σε πλήρη συμφωνία με αυτά που προκύπτουν από την αναλυτική επίλυση των εξισώσεων του μοντέλου. Στη συνέχεια περιγράφεται αναλυτικά η εφαρμογή του μοντέλου πεπερασμένων στοιχείων στη νέα πειραματική διάταξη για τη μέτρηση στερεών και αναλύεται η μεθοδολογία που ακολουθείται για τον υπολογισμό της τιμής της θερμικής αγωγιμότητας από τα πειραματικά δεδομένα, τόσο στη περίπτωση μέτρησης υγρών όσο και στη περίπτωση μέτρησης στερεών. Το κεφάλαιο τελειώνει με την ανάλυση ευαισθησίας της επίδρασης που έχουν οι παράμετροι του μοντέλου πεπερασμένων στοιχείων στην υπολογιζόμενη τιμή θερμικής αγωγιμότητας των στερεών.

Στο επόμενο κεφάλαιο δίνεται η λεπτομερής περιγραφή της νέας πειραματικής διάταξης και αναλύονται οι βελτιώσεις–τροποποιήσεις που πραγματοποιήθηκαν σε σχέση με την προηγούμενη υπάρχουσα διάταξη. Ένα μεγάλο μέρος του κεφαλαίου είναι αφιερωμένο στην ηλεκτρονική διάταξη που

Περίληψη vii

χρησιμοποιείται στις μετρήσεις. Γίνεται ιστορική ανασκόπηση των ηλεκτρονικών διατάξεων που έχουν χρησιμοποιηθεί μέχρι σήμερα και παρουσιάζεται το κύκλωμα της παρούσας αυτοματοποιημένης γέφυρας τύπου Wheatstone, καθώς και οι εξισώσεις που χρησιμοποιούνται για τον υπολογισμό της πειραματικής θερμοκρασιακής αύξησης του θερμαινόμενου σύρματος. Επιπρόσθετα, στο κεφάλαιο περιέχεται και η μεθοδολογία που ακολουθείται για τον υπολογισμό της πειραματικής θερμοκρασιακής αύξησης του σύρματος από τη μετρούμενη μεταβολή της τάσης στα άκρα του. Η πειραματική αυτή θερμοκρασιακή αύξηση χρησιμοποιείται σε συνδυασμό με το λογισμικό Comsol Multipyhysics για τον υπολογισμό της θερμικής αγωγιμότητας του υλικού. Ακολούθως, ο έλεγχος της ορθής λειτουργίας του νέου αισθητήρα πραγματοποιείται με τη μέτρηση της θερμικής αγωγιμότητας τολουολίου, το οποίο είναι ένα πρότυπο ρευστό. Το τελευταίο τμήμα του κεφαλαίου είναι αφιερωμένο στην ανάλυση των πηγών αβεβαιότητας που υπεισέρχονται κατά τη διάρκεια του πειράματος και η συνολική αβεβαιότητα των μετρήσεων της θερμικής αγωγιμότητας είναι καλύτερη από 0.5 % για τα υγρά και καλύτερη από 1 % για τα στερεά.

Το Κεφάλαιο Ε είναι αφιερωμένο στην παρουσίαση των μετρήσεων θερμικής αγωγιμότητας στερεών και σύνθετων υλικών. Αρχικά ο νέος αισθητήρας θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση χρησιμοποιείται για τη μέτρηση της θερμικής αγωγιμότητας των σιλικόνων που χρησιμοποιήθηκαν στον αισθητήρα. Έπειτα, η τεχνική χρησιμοποιείται για τη μέτρηση της θερμικής αγωγιμότητας των πιστοποιημένων πρότυπων υλικών αναφοράς θερμικής αγωγιμότητας Pyroceram 9606 (γνωστό με την ονομασία BCR-724) και του Pyrex 7740 (γνωστό με την ονομασία BCR-039) στη θερμοκρασιακή περιοχή από 315 Κ έως 440 Κ. Στη συνέχεια, ο αισθητήρας χρησιμοποιείται για τη μέτρηση υποψήφιων υλικών αναφοράς θερμικής αγωγιμότητας. Τα υποψήφια υαλώδη υλικά που μετρήθηκαν από θερμοκρασία περιβάλλοντος μέχρι τους 440 Κ είναι το Polymethyl Methacrylate (PMMA) και το Borosilicate Crown Glass (BK7). Τα αποτελέσματα των μετρήσεων θερμικής αγωγιμότητας που έγιναν στην παρούσα εργασία συγκρίθηκαν με δεδομένα άλλων ερευνητών και σε όλες τις περιπτώσεις οι αποκλίσεις ήταν μέσα στα αμοιβαία όρια αβεβαιότητας των χρησιμοποιούμενων μεθόδων.

Στο τελευταίο τμήμα του Κεφαλαίου Ε, η συσκευή θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση χρησιμοποιείται για τη μέτρηση της αύξησης της θερμικής αγωγιμότητας μίας εποξικής ρητίνης ενισχυμένη

viii Περίληψη

αρχικά με ύφασμα από ίνες γυαλιού, έπειτα με πολυστρωματικούς νανοσωλήνες άνθρακα (C–MWNTs), και τέλος και με τις δύο αυτές μακροσκοπικές και νανοσκοπικές ενισχύσεις μαζί. Η αβεβαιότητα των μετρήσεων αυξάνεται στο ±2 % λόγω ανομοιογενειών που υπάρχουν στα δείγματα που παρασκευάζονται. Τα αποτελέσματα δείχνουν ότι η εποξική ρητίνη ενισχυμένη με ίνες γυαλιού σε ποσοστό 28 % κατ’ όγκο έχει θερμική αγωγιμότητα 27 % μεγαλύτερη από αυτή της ρητίνης. Ακόμη, όταν η ρητίνη ενισχύεται με C–MWNTs η παρατηρούμενη αύξηση της θερμικής αγωγιμότητας είναι 20 % για συγκέντρωση C–MWNTs 1.2 % κατά βάρος. Ομοίως, όταν η ρητίνη ενισχύεται και με τα δύο, ίνες γυαλιού και C–MWNTs, η αύξηση στη θερμική αγωγιμότητα είναι περίπου 60 % για συγκέντρωση C–MWNTs 1.2 % κατά βάρος.

Η παρούσα διατριβή ολοκληρώνεται με μία σύνοψη των καινοτομιών που επιτεύχθηκαν και παρουσιάζονται οι μελλοντικές ερευνητικές προκλήσεις πάνω στη μέθοδο του θερμαινόμενου σύρματος σε μη μόνιμη κατάσταση.

Acknowledgements

Behind each postgraduate student, there are always few persons that contribute to the successful completion of the research. I would like to express my deepest thanks to my supervisor, Professor Marc J. Assael, for his guidance, cooperation and kind encouragement throughout the course of my postgraduate research. His advices were vital for solving problems and surpassing obstacles that appeared during my research. His experience and expertise on the field of thermophysical properties essentially contributed on the completion of my thesis. Especially, I would like to thank him for his friendship and the confidence that he has showed on me all these years.

I would like also to express my thanks to Dr. Ulf Hammerschmidt, for hosting me for three months in his laboratory at the Physikalisch–Technische Budesanstalt (PTB), in Braunschweig, Germany. His help and cooperation were valuable all along my staying there. He gave me the opportunity to get in contact with several thermal conductivity instruments of his laboratory, and especially to operate the Transient Hot Bridge sensor. Moreover, I would like to thank Dipl.-Ing. Vladislav Meier from PTB for his precious guidance and collaboration on the operating aspects of the Transient Hot Bridge sensor and for the endless hours we have spent on productive discussions over it.

Moreover, I would like to thank the members of my supervising committee, Professor Costas Panayiotou and Associate Professor Eleftherios Kastrinakis, from the Chemical Engineering Department of Aristotle University of Thessaloniki (A.U.Th.), for their cooperation and help all along my PhD project.

In addition, I would like to thank the Assistant Professor Eleni Pavlidou from the Physics Department of A.U.Th., for the measurement of

x Acknowledgements

the tantalum wire diameter with Scanning Electron Microscope, and Dr. Dimitrios Tzetzis for his essential help in the preparation of the reinforced epoxy–resin specimens.

Furthermore, I would like to mention the valuable help provided on construction issues by Mr. Vassileios Goutsios and Mr. Triantafyllos Tsilipiras from the Chemical Engineering Department of A.U.Th. Moreover, I would like to thank Mr. Michael Mprintakis for his precious advices on electronic matters.

I also cannot miss out to thank my friends and colleagues within the Laboratory of Thermophysical Properties and Environmental Processes for their cooperation and the productive discussions that we had. In particular, I want to thank Dr. Katerina E. Gialou and Dr. Ifigeneia N. Metaxa for their help on my first steps in the use of the transient hot wire technique, and Dr. Konstantinos E. Kakosimos for his guidance on the use of Comsol Multiphysics software and his help on computer’s software issues.

I want also to thank the undergraduate students that I had the pleasure to work with during their diploma thesis, Mr. Gregory Mantziaroglou and Mr. Vasileios Charalampidis.

Last but not least, I would like to thank my family and all my friends for their constant encouragement all these years. Especially, I want to deeply thank my parents for their multilateral support in my decisions and for consisting the most important motive for my personal development. A special thank to Vana for her patience and love that she showed and keeps showing to me.

Marie Curie has quoted: “I am among those who think that science

has great beauty. A scientist in his laboratory is not only a technician: he is also a child placed before natural phenomena which impress him like a fairy tale”. My personal tale is written in the following pages.

Contents

Abstract ____________________________________________________________ i 

Acknowledgements _________________________________________________ ix 

A Introduction _______________________________________________________ 1 

A1. The Need for Material Properties ______________________________________ 2 A1.1. The Importance of Accurate Measurements __________________________________ 3 

A2. Reference Materials __________________________________________________ 5 

A3. Composite Materials __________________________________________________ 8 A3.1. Effective Thermal Conductivity ____________________________________________ 9 

A4. Scope ______________________________________________________________ 10 

B Methods for Measuring the Thermal Conductivity ___________________ 11 

B1. Other methods ______________________________________________________ 12 B1.1.  Steady State Methods ____________________________________________________ 13 

B1.1.1. Guarded Hot Plate Method ___________________________________________ 14 B1.1.2. Heat Flow Method ___________________________________________________ 16 B1.1.3. Calorimeter Method __________________________________________________ 18 B1.1.4. 3ω Method __________________________________________________________ 20 

B1.2. Transient Methods _______________________________________________________ 22 B1.2.1. Transient Hot Wire technique _________________________________________ 23 B1.2.2. Transient Plane Source Method _______________________________________ 23 

xii Contents

B1.2.3. Transient Hot Strip Method ___________________________________________ 25 B1.2.4. Pulse Transient Method _______________________________________________ 26 B1.2.5. Transient Hot-Bridge Method _________________________________________ 27 B1.2.6. Dynamic Radial Heat Flow Method ____________________________________ 28 B1.2.7. Laser Flash Technique ________________________________________________ 30 

B1.3. Critical Evaluation _______________________________________________________ 31 B1.3.1. Round-Robin Tests ___________________________________________________ 32 B1.3.2. Conclusion ___________________________________________________________ 36 

B2. Transient Hot Wire Technique ________________________________________ 38 B2.1. Historical Development ___________________________________________________ 39 

B2.1.1. The period from 1780 to 1888 __________________________________________ 39 B2.1.2. The period from 1888 to 1971 __________________________________________ 43 B2.1.3. The period from 1971 to today _________________________________________ 47 B2.1.4. Brief Presentation ____________________________________________________ 53 

B3. Summary ___________________________________________________________ 55 

C Theoretical ______________________________________________________ 57 

C1.  Ideal Model _________________________________________________________ 58 C1.1. Correction to the Ideal Model ______________________________________________ 60 

C2. Finite–Element Analysis _____________________________________________ 65 C2.1. Fortran Finite Element Model _____________________________________________ 67 C2.2. Comsol Multiphysics Software _____________________________________________ 70 

C2.2.1. Evaluation of the Heat Transfer Model _________________________________ 72 C2.2.2. Finite Element Model for Solids ________________________________________ 80 

C2.3.  Sensitivity Analysis of the Finite Element Model ____________________________ 90 

C3. Summary ___________________________________________________________ 95 

D Experimental Procedure __________________________________________ 97 

D1. Experimental Configuration __________________________________________ 98 D1.1. Sensor Design ____________________________________________________________ 99 

D1.1.1. Sensor Assembly/Construction ________________________________________ 103 D1.1.2. Temperature Coefficient of Resistance _________________________________ 104 

Contents xiii

D1.2. Temperature Controller _________________________________________________ 106 

D2. Electronic Components _____________________________________________ 107 D2.1. Historical Development __________________________________________________ 108 D2.2. Present Bridge Circuit and Data Acquisition System _______________________ 111 

D3. Working Equations _________________________________________________ 116 D3.1. The Bridge Equations ___________________________________________________ 116 

D3.1.1. Constant Resistance Fraction [RLo(0)/RSo(0)] _________________________ 121 D3.1.2. Calculation of Initial Resistances’ Values at zero time __________________ 122 

D3.2. The Temperature Rise Equations _________________________________________ 124 D3.3. The Heat Flux Equations ________________________________________________ 126 

D4. Calculation of the Thermal Conductivity Value ________________________ 127 

D5. Validation of the Technique _________________________________________ 132 D5.1. Measurements of Toluene ________________________________________________ 132 

D6. Uncertainty Analysis _______________________________________________ 136 

D7. Summary __________________________________________________________ 139 

E Thermal Conductivity Measurements ______________________________ 141 

E1. Solid Materials _____________________________________________________ 142 E1.1.  Silicone Pastes __________________________________________________________ 142 

E1.1.1. Thermal Conductivity Measurements of Silicone Paste AS1803 _________ 143 E1.1.2. Thermal Conductivity Measurements of Silicone Paste BORO, Type 650 _ 145 

E1.2. Pyroceram 9606 _________________________________________________________ 147 E1.2.1. Thermal Conductivity Measurements _________________________________ 148 

E1.3. Pyrex 7740 _____________________________________________________________ 156 E1.3.1. Thermal Conductivity Measurements _________________________________ 156 

E1.4. Polymethyl Methacrylate (PMMA) ________________________________________ 164 E1.4.1. Thermal Conductivity Measurements _________________________________ 164 

E1.5. Borosilicate Crown Glass BK7 ____________________________________________ 172 E1.5.1. Thermal Conductivity Measurements _________________________________ 172 

E2. Composite Materials ________________________________________________ 179 E2.1. Epoxy Resin ____________________________________________________________ 180 

xiv Contents

E2.1.1. Samples Preparation ________________________________________________ 180 E2.1.2. Thermal Conductivity Measurements _________________________________ 180 

E2.2. Epoxy Resin Reinforced with Glass Fibers _________________________________ 182 E2.2.1. Samples Preparation ________________________________________________ 182 E2.2.2. Thermal Conductivity Measurements _________________________________ 183 

E2.3. Epoxy Resin Reinforced with Carbon Multi-Walled Nanotubes _______________ 185 E2.3.1. Samples Preparation ________________________________________________ 185 E2.3.2. Thermal Conductivity Measurements _________________________________ 188 

E2.4. Epoxy Resin Reinforced with Glass Fibres and Carbon Multi-Walled Nanotubes189 E2.4.1. Samples Preparation ________________________________________________ 189 E2.4.2. Thermal Conductivity Measurements _________________________________ 191 

E3. Summary __________________________________________________________ 193 

F Innovation – Future Work ________________________________________ 195 

F1.  Innovations ________________________________________________________ 196 

F2.  Future Work _______________________________________________________ 197 

References ________________________________________________________ 199 

List of Publications ________________________________________________ 219 

A Introduction

The chapter constitutes a brief introduction on the thermal conductivity. In the beginning of the chapter the need for knowledge of the material properties is described while the importance that accurate measurements plays in many aspects of daily life is shown. Consequently, international terms employed for defining reference materials are explained and the major suppliers of reference materials are mentioned. The chapter ends with the presentation of the scope of this thesis.

2 Introduction

A1. The Need for Material Properties The measurements of material properties take place in many

occasions of the daily lives, e.g. during weighting of food and raw materials, measuring the time to go from one place to other, monitoring the air quality and ambient temperature, e.t.c. Almost, everything needs the performance of a measurement in order to be accurately described. A measurement can range from huge (distance between earth and sun) to tiny (nanoparticles detected by microscopy).

Lord Kelvin in 1889 quoted [Thompson, 1889]: “When you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind: It may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be”.

The measurement of material properties has a major impact on the well being of people, the security of national infrastructure and the effectiveness of world trade, as without accurate testing there cannot be any guarantee of the ability to deliver the specified products to meet all the individual needs. Moreover, governments require accurate and reliable data in order to create regulations and laws that will protect the public health and welfare, as well as the environment.

Furthermore, the material properties are crucial for the optimum design of industrial equipment or of a simple product. Nowadays, the production of new products includes extensive design and computational modelling, in order to reassure that it will have the desired properties. This procedure is facilitated by the continuous development on computer software’s capabilities. The design consists in predicting the behaviour of the product, during and after the manufacture process, by solving mathematical models. However, in order to get a realistic model of the product, the precise material properties (at specified conditions) should be included in the models.

A recent survey in the United Kingdom investigated the need for material property data on engineering materials. The survey covered all material types, all applications and industrial sectors, and all sizes of companies [Bennet & Sims, 2010]. In Figure A.1, the sources of material data are shown. It is notable that no source dominates over the others but

Introduction 3

on the contrary there are numerous and varied sources.

*MSDS: Material Safety Data Sheet

Figure A.1 Sources of material data [Bennet & Sims, 2010].

It should be noted that the thermal properties of materials are among the most important material properties governing industrial heat transport processes. More specific, the thermal conductivity and thermal diffusivity are the main thermal properties in the optimum design of industrial equipment used for heat transfer applications. Moreover, the thermophysical properties of a material determine its use, i.e materials with low thermal conductivity can be used for insulation purposes (in heat exchanger for avoiding heat losses), whereas materials with high thermal conductivity, such as metals, can be employed in electronic applications.

A1.1. The Importance of Accurate Measurements

The available materials’ property data should be characterised by low uncertainty in order to be useful. In the same survey mentioned before

4 Introduction

[Bennet & Sims, 2010], all the participants declared that the required data in all phases, from design to manufacture of the final product, is expected to have an accuracy of better than 10 %, while 50 % of the companies that participated in the survey declared that they need data with an accuracyof lower than 5 %. However, these requirements are far away from the already available data, and therefore it is necessary the newly available data to rely on best metrology practice, that will reassure the high accuracy demanded by industry.

From an economic point of view, the accurate measurement of material properties can have a considerable effect on the production efficiency of a company and on the price of the final product. The lack of quality material data makes a company liable to accidents due to the destruction of the equipment (equipment breakdown due to fatigue, corrosion, operation at extreme conditions, et.c.), or/and to the production of defective products. Thus, the company reduce its production capacity in order to avoid the destruction of its equipment. This company’s inefficiency to fully exploit its equipment capacity has impact on the final price of the product, which is higher. As an example, it is mentioned that the economic loss in Japan due to accidents and defects of industrial products, is estimated to be as high as 4 % its gross domestic product (GDP) [Bennet & Sims, 2010].

Therefore, the need for accurate measurements is evident and an international metrology system is crucial for ameliorating the quality of the provided material data. This worldwide metrology network is established through the National Metrology Institutes of each country. The central coordination of the network is undertaken by the Bureau International des Poids et Mesures (BIPM) in Sevres, Cedex France.

In order to be comparable across borders and over time, measurements need be traceable to appropriate and stated references. Therefore, there are certified reference materials that enable metrological traceability, allow the calibration of test equipment and ensure the confidence in measuring and testing data. The metrological traceability is defined in the International Vocabulary of Metrology as “the property of a measurement result whereby the result can be related to a stated reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty” [International Organization for Standardization, 2007].

Introduction 5

A2. Reference Materials According to the ISO Guide 35: 2006 the following definitions are

given [International Organization for Standardization, 2006]:

a) Reference Material (RM) It is a material, sufficiently homogeneous and stable with respect to

one or more specified properties, which has been established to be fit for its intended use in a measurement process. The properties can be quantitative or qualitative, e.g. identity of substances or species. Moreover, uses may include the calibration of a measurement system, assessment of a measurement procedure, assigning values to other materials, and quality control. However, a single RM cannot be used for both calibration and validation of results in the same measurement procedure.

b) Certified Reference Material (CRM) It is a reference material characterized by a metrologically valid

procedure for one or more specified properties, accompanied by a certificate that provides the value of the specified property, its associated uncertainty, and a statement of metrological traceability.

Except from the above two terms, there are two more defined by the National Institute of Standards and Technology (NIST) [National Institute of Standards and Technology, 2000]:

c) Standard Reference Material (SRM) It is a CRM issued by NIST that also meets additional NIST–specified

certification criteria. NIST SRMs are issued with Certificates of Analysis or Certificates that report the results of their characterizations and provide information regarding the appropriate use(s) of the material.

d) NIST Traceable Reference Material (NTRM) It is a commercially produced reference material with a well–defined

traceability linkage to existing NIST standards. This traceability linkage is established via criteria and protocols defined by NIST.

The ISO Committee on Reference Materials (ISO/REMCO) has

pusblished a series of Guides related to the reference materials (Table A.1).

6 Introduction

The objective of this committee is to carry out and encourage a broad international effort for the harmonization and promotion of CRMs, their production and application.

Table A.1 ISO reference material guides [International Organization for Standardization, 1997, 2000a, 2000b, 2006, 2009].

Guide Title

ISO Guide 30 Terms and definitions used in connection with reference materials ISO Guide 31 Reference materials – Contents of certificates and labels

ISO Guide 32 Terms and definitions used in connection with reference materials ISO Guide 33 Uses of certified reference materials

ISO Guide 34 General requirements for the competence of reference material producers

ISO Guide 35 Reference materials – General and statistical principles for certification Reference materials with certified material properties are

commercially available from several sources. The most known suppliers are:

i. The Institute for Reference Materials and Measurements (IRMM) IRMM was found in 1960, and its mission is to promote a common and reliable European measurement system in support of EU policies. IRMM offers over 600 certified reference materials for applications in the fields of food and feed analysis, environmental analysis, engineering and health. IRMM reference materials laboratory is one of the largest and most sophisticated in the world. It hosts a unique multi–functional and flexible production laboratory, with cryo–grinding and freeze drying equipment, high purity milling and levitation melting devices. The Reference Materials unit of IRMM was the first European reference material producer to obtain accreditation according to ISO Guide 34 [International Organization for Standardization, 2009]. Today, IRMM has teamed up with other two European reference material producers, Bundesanstalt für Materialforschung und prüfung (BAM) and LGC (UK), in a European Reference Material (ERM®) partnership. The ERM concept guarantee to use the most advanced principles currently available, described in ISO Guides 34 and 35 [International Organization for Standardization, 2006, 2009], for the

Introduction 7

production of certified reference materials. Moreover, they assure to demonstrate rigorously homogeneity and stability for all materials and guarantee the certified value for every single unit over the complete shelf life of the materials. In other words, the ERM is a trademark for certified reference materials, similar to the SRM trademark used by NIST.

ii. The National Institute of Standards and Technology (NIST) NIST, founded in 1901, is a non–regulatory federal agency within the U.S. Department of Commerce. It supplies industry, academia, government and other users with over 1300 Standard Reference Materials of the highest quality and metrological value. These materials are used to perform instrument calibrations in units as part of overall quality assurance programs, to verify the accuracy of specific measurements and to support the development of new measurement methods.

iii. The Korea Research Institute of Standards and Science (KRISS) KRISS was founded in 1975 and its mission is to promote the industrial competitiveness of Korea by advancing measurement standards, science, and technologies in ways that enhance the nation’s economic performance and secure a better quality of life for all. It supplies domestic industry with more than 500 Certified Reference Materials.

Concluding, the importance of the Certified Reference Materials should be pointed out once more. CRMs consist an essential part of the international metrology science. They are used in the calibration of instruments, establishing traceability of measurement results, method development, method validation and quality control of measurements and laboratories. Without the use of CRMs it would not be possible to compare material data measurements acquired with different measuring techniques. Although there are CRMs for every property, there is still a long way to go until having materials with certified properties for every possible measuring conditions. Moreover, there is still work to do in order to ameliorate the uncertainty of the provided certified material properties.

As far as thermal conductivity of solid material is concerned, there are already two CRMs, Pyroceram 9606 and Pyrex 7740. Thermal conductivity measurements of these two materials are presented in Chapter

8 Introduction

E, along with measurements of two candidate reference materials, Polymethyl Methacrylate (PMMA) and Borosilicate Crown Glass BK7.

A3. Composite Materials In recent years, there have been an increasing number of applications

requiring more efficient and lightweight thermal management such as communication satellites, high–density electronics, and advanced aircraft. The fibre–reinforced composites are considered as ideal candidates for many of these applications and knowledge of the mechanical, as well as physical properties, pertinent to these materials, has become a very crucial design input.

Moreover, fibre–reinforced composites are commonly used as a replacement for metals, in cases where they have better mechanical properties. Composites have the advantage of higher corrosion resistance and being lighter than metals. However, one significant drawback compared to metals is the anisotropic low thermal conductivity, which in some applications may prove to be significant, e.g in the case of electronic applications. Generally, the thermal conductivity of polymer composites is about 10 to 100 times lower than the thermal conductivity of metals. In order to overcome this hindrance, carbon nanotubes are employed as reinforcement of the fibre–reinforced composites. Carbon nanotubes have an exceptionally high axial thermal conductivity and therefore contribute to the increase of the effective thermal conductivity of the composite. The production of nanocomposites represent a new frontier in material science, as the reinforcement scale has change from micrometers, with traditional carbon and glass fibres, to nanometers.

Despite the fact that there is a plethora of established standards and/or procedures for the measurement of various mechanical properties of composites, the same does not apply to the measurement of a number of physical properties, including the effective thermal conductivity. As a result, the lack of reliable thermophysical data may hinder the full utilization of composites. It should be mentioned, that the thermal conductivities of a fibre–reinforced composites can be calculated from various theoretical, empirical, as well as numerical methods. However, each of these methods have certain assumptions applicable for certain specific cases and condition

Introduction 9

ranges [Hasselman et al., 1980; Agari et al., 1991; Gorthala et al., 1994; Ramani & Vaidyanathan, 1995; Amazouz & Gauvin, 1997; Ning & Chou, 1998; Pantano & Averill, 2002]. Thus, the determination of reliable experimental data for validation purposes is of paramount importance.

Nowadays, the use of nanoparticles for the polymer matrix reinforcement becomes a highly desirable procedure in order to enhance both the physical and the mechanical properties of the composite [Paradise & Goswami, 2007; Zhou et al., 2008; Wang & Qiu, 2010]. It should be noted that the numerical and the analytical methods and assumptions regarding the calculation of the effective thermal conductivity, developed for conventional composites, have to be reanalysed in order to be applied for nanocomposites. Clearly, such analyses also necessitate very accurate and reliable experimental data for comparison.

A3.1. Effective Thermal Conductivity

It should be noted that the term “thermal conductivity” is referred only for homogeneous materials. However, it is conventional to speak of the “thermal conductivity” of various types of composite materials, glass–fibre insulation, carbon–fibre composites, or polymer blends. In the case of these materials, the thermal conductivity is taken to be the empirical constant of proportionality in the linear relationship between a measured heat transport per unit area and the temperature difference over a prescribed distance in the material. The “thermal conductivity” is not then, strictly, a property of the material, since it can often depend on a large number of parameters, including the history of the material, its method of manufacture, and even the character of its surface [Wakeham & Assael, 2000]. In other words, the measured property is the “effective thermal conductivity” of the material.

Hence, when performing thermal conductivity measurements, it is important always to keep in mind the distinction between homogeneous and inhomogeneous materials and the terms used in reporting the results. Therefore, in the present work the term “effective thermal conductivity” should strictly be employed, when it is used for measurements of reinforced polymers. However, for simplicity, the term “thermal conductivity” is also used for the measurements of composite materials, meaning though the

10 Introduction

effective thermal conductivity of the specimens.

A4. Scope It is well known that there are significant differences in the measured

thermal conductivities values of a solid material obtained by different techniques. Therefore, the aim of the present work is to measure the thermal conductivity of solid materials and composite polymers with very low uncertainty. The measuring method employed for the measurement of the thermal conductivity of the specimens is the transient hot wire technique, which is described in details in the following two chapters. The experimental device designed and employed here consist an improvement of the already available transient hot–wire instrument for the measurement of the thermal conductivity of solids of Assael et al. [Assael et al., 2002]. The refinements made on the new improved thermal conductivity sensor confront the drawbacks of the older one and allowed the performance of low uncertainty measurements. More specifically the lower uncertainty is achieved by:

i. The use of stainless–steel spacers to define exactly the distance between the two solid samples.

ii. The use of a new silicone paste of higher thermal conductivity. iii. The use of Comsol Multiphysics FEM software to accurately describe

the geometry of the sensor (for measurements from 20 μs to 10 s).

The developed transient hot wire sensor is employed for the low uncertainty measurement of the thermal conductivity of Pyroceram 9606 and Pyrex 7740, which are certified reference materials. Moreover, the thermal conductivity of two candidate reference materials, Polymethyl methacrylate (PMMA) and Borosilicate Crown Glass BK7, is also measured.

The last part of the work is devoted to employment of the transient hot wire method in the measurement of the thermal conductivity of composite materials. Specifically, the hot wire sensor is used for measuring the enhancement of the thermal conductivity of an epoxy–resin reinforced initially with plies of plain weave glass fabric, then by carbon multi–walled nanotubes (C–MWNT), and finally with both these two macroscopic and nanoscopic reinforcements at hand.

11 Methods for Measuring the Thermal Conductivity

B Methods for Measuring

the Thermal Conductivity

In this chapter, a brief overview of the most significant and widely used methods for the measurement of the thermal conductivity of solids is presented. The chapter begins by presenting the Fourier’s heat conduction law. Consequently, the thermal conductivity measurement methods are divided into two main sections, the steady state methods and the transient ones, and a brief description and the fundamental principles of each method are given. Following this, a critical evaluation of the methods is made by presenting and discussing three interlaboratory measurement programs for Pyroceram 9606. The last section of the chapter is focused on the transient hot wire technique, which is established as an accurate technique for thermal conductivity measurements of solids. As this technique is used in this thesis, a detailed historical review of its development, starting from 1780, is presented and the main modifications of the technique are described.

12 Methods for Measuring the Thermal Conductivity

B1. Other methods The heat transfer through a material can take place with three

different mechanisms: conduction, convection and radiation. Heat conduction is the transfer of thermal energy in a molecular level, which is caused by the combination of the molecules’ vibrations in a lattice, their collisions during their random movement (in case of fluids) and the energy transfer by free electrons. On the other side, convective heat transfer, which occurs only on gases and liquids, include the heat transfer due to the circulation of currents from one region to another, while radiation is the heat transferred by the energy emitted or absorbed by solids, liquids or gases, in the form of electromagnetic waves as a result of their temperature.

In order to perform an accurate thermal conductivity measurement the heat energy should be transferred only by the conduction mechanism, and therefore the experimental setup should ensure conditions that are in accordance with the above mentioned requirement. However, in the case of heat transfer in solids, which is the research issue of the present work, there is no convection and the effect of radiation can be considered negligible for small measurement times, and thus the heat transfer depends mainly on the thermal conductivity of the material. Specifically, the thermal conductivity, λ, is an intrinsic physical property of a material and expresses the facility with which heat is propagated in passing from one internal molecule to another [Fourier, 1822], whereas thermal diffusivity, α, indicates how fast heat is conducted.

The differential conductive heat transfer equation for a viscous, isotropic, and incompressible material, can be obtained from the energy conservation equation [Wakeham et al., 1991]:

h PDTQ g = C Dt

, (B.1)

where ρ and PC represents the material’s density and heat capacity respectively, Q

denotes the heat flux vector, hg is the internal energy

increase due to viscous dissipation, and the notation D Dt represents a substantive derivative.

The first term of equation (B.1) refers to heat transferred by molecular means, the second one to the heat generation and the right hand

Methods for Measuring the Thermal Conductivity 13

term refers to the change of the internal energy of the material. A general solution of (B.1) is not possible and therefore some restrictions need to be applied. Thus, the produced temperature gradients should be small so as to have a near-equilibrium state, while material (in case fluids and gases) movement should be avoided so that 0hg . Under these conditions, the substantive derivative is replaced by the partial derivative.

Moreover, the heat flux vector, for conductive heat transfer in an isotropic material, is defined by the Fourier’s law equation [Fourier, 1822]:

Q T

, (B.2)

where T is the temperature gradient. Therefore, combining equations (B.1) and (B.2), and using the above

analysed constraints it is

2pTC = Tt

, (B.3)

which is the fundamental equation for the calculation of the thermal conductivity of a material.

The developed experimental techniques for the measurement of the thermal conductivity are divided into two broad categories: - the steady state methods, mainly based on Fourier’s law, and - the transient methods, mainly based on equation (B.3).

In the following sections, each category is analysed in detailed for the case of solid sample measurements.

B1.1. Steady State Methods

The steady state methods are based on the measurement of the heat flux necessary to maintain a temperature difference, constant in time, between two surfaces of a measured solid sample. The thermal conductivity is calculated as a function of the heat flux, the temperature gradient, and the properties and geometry of the sample. In general, the working equations of the steady state methods are much simpler than the one that describe the transient methods. In the case of solids, the difficulties in the application of a steady state technique derive from a need of extremely high

14 Methods for Measuring the Thermal Conductivity

experimental times and samples of large dimensions. Many steady state methods can be applied to solid, such as:

i. Guarded Hot Plate method ii. Heat Flow method iii. Calorimeter method

iv. 3 omega (3ω) method v. Comparative technique vi. Longitudinal Heat Flow technique .

In the following subsections the fundamental principles of the main steady state methods for the measurement of the thermal conductivity of solids will be briefly presented. It is obvious that there are steady state methods that cannot be applied to solids and therefore are not mentioned in this work.

B1.1.1. Guarded Hot Plate Method

Guarded hot plate is a versatile and widely used steady state method for measuring the thermal conductivity of solids. The guarded hot plate method is covered by ASTM test method C177 [American Society for Testing and Materials, 1997] and ISO 830 [International Organization for Standardization, 1996], and many instruments based on this method are commercially available. This method is applicable to the measurements of a wide variety of specimens, ranging from opaque solids to porous or transparent materials, and a widely range of environmental conditions [Filla, 1997]. Moreover, the guarded hot plate method can be used in apparatus operated with either vertical or horizontal heat flow.

A typical measurement arrangement for determining the thermal conductivity according to ASTM C177 is shown in Figure B.1. The main components of this arrangement are two isothermal cold surface assemblies and a guarded hot plate. The guarded hot plate is composed of a metered section thermally isolated from a concentric primary guard by a definite separation gap. This arrangement uses two almost identical specimens of the under measurement material and the hot plate is sandwiched between them. Electrical energy, which is measured accurately, is dissipated through the guarded hot plate and causes heat flow in the top and bottom sample. The primary guard is used to reduce the radial heat flow, while a temperature controlled secondary guard, in the form of edge insulation, is necessary in order to restrict heat losses from the outer edge of the primary

Methods for Measuring the Thermal Conductivity 15

guard. The secondary guard can be omitted in systems designed for ambient conditions. In some guarded hot plate apparatus [Sheffield & Schorr, 1991] optional auxiliary heaters are used between the specimens and the cold surface assembly in order to control the sample temperature, independently of the heat flux or the temperature gradient that the hot plate causes. The cold surfaces assemblies, which are adjusted to the same temperature, are isothermal heat sinks used for removing the energy generated by the heating units. The temperature gradient within the specimen is usually measured with thermocouples.

Figure B.1 Typical Guarded Hot Plate apparatus arrangement [American Society for Testing and Materials, 1997]

Under steady state conditions, Fourier’s equation (B.2) for the above geometry can be written:

Hot Plate top bottom

Q T TA x x

, (B.4)

where Q and A denote the heat flow and the total metered surface of the hot plate respectively, and ΔT and Δx represent the temperature drop across the specimens (top or bottom) and their thickness. Moreover, the final calculated thermal conductivity, λ, of the samples is the average of the two pieces and thus it is important to use nearly identical specimens.

Usually deviations from the above described configuration can be caused by samples’ inhomogeneities, or temperature differences between the

16 Methods for Measuring the Thermal Conductivity

metered section and the guard, or temperature differences between the outer edge of the assembly and the surrounding controlled environment. Other errors affecting the experimental measurements are the edge heat flow at the periphery of the specimens, and the heat flow across the gap between the hot plate and the primary guard. Moreover, it was noted that the uncertainty of a guarded hot plate apparatus depends on the thickness of the specimen being measured [Salmon, 2001].

Filla [Filla, 1997] made a typical application of the method for the measurement of the thermal conductivity of ceramics with an estimated uncertainty of 5 %, whereas Salmon et al. in their review [Salmon, 2009], claimed that the uncertainty level of the method is better than 3 %, for measurements of thermal insulations in the region of 253 K to 353 K.

B1.1.2. Heat Flow Method

The heat flow method provides a rapid mean of determining the steady-state transmission properties of thermal insulations and other materials. There is the ASTM C518 test method that describes the use and construction of a heat flow meter apparatus [American Society for Testing and Materials, 2010]. The principle of the technique is that known heat input is applied to the one end of the sample and removed at the other, through a heat sink. This method is very similar to the guarded hot plate method and their only difference is that the temperature drop along the sample is measured by thermocouples immersed in the specimens and not placed in the heating plates. The main difference between the heat flow meter instrument and the hot plate apparatus is that in the first one the thermal conductivity value depends on the heat flux reference standards, while the second one operates in an absolute way. Therefore, the uncertainty of the measurements made with ASTM C518 test method can not be better than those achieved with the guarded hot plate.

In general, heat flow meter apparatus consist of one or two specimens, two isothermal plate assemblies, one or more heat flux transducers, and equipment to control the environmental conditions (if needed). The technique can be divided into two categories according to the geometry of the used instrument, i.e. radial and axial heat flow method. Salmon and Tye made an intercomparison of seventeen heat flow meter

Methods for Measuring the Thermal Conductivity 17

apparatus used for measuring the thermal conductivity of thermal insulation materials at 283 K and 296 K [Salmon & Tye, 2000]. It shown that in a total of 154 data points, 69 % were within 3 % and 50 % were within 2 % of the values acquired with the guarded hot plate apparatus of the National Physical Laboratory, at the United Kingdom.

a) Radial Heat Flow Method In the radial heat flow method, a steady current of electricity is

passed through a long cylindrical bar of the material, establishing a zone of uniform axial temperature. A radial temperature gradient is created by allowing the surface of the sample to radiate to a cooled environment. The thermal conductivity can then be obtained from the measurement of the energy dissipation at the chosen portion of the bar and the temperature difference between the axis and the surface [Angell, 1911; Powell & Schofield, 1939]. The primary concern for accurate thermal conductivity measurements with this method is to eliminate the axial heat flow. For a resistance-heated cylindrical solid specimen the radial heat flow is [Carslaw & Jaeger, 1959]:

1 0oqd dTrr dr dr, (B.5)

where oq is heat generation rate, λ the specimen’s thermal conductivity, T the temperature and r the radius. For independent λ along the specimen radius, the solution of equation (B.5) is

2

4o

a

q rT T , (B.6)

where Ta denotes the temperature on the axis of the cylinder, and T the temperature at distance r.

A representative application of the method was done by Donaldson and Taylor [Donaldson & Taylor, 1975] for the measurement of the thermal diffusivity of Armco iron and the values found were within 5 % of the accepted ones.

b) Axial Heat Flow Method . The axial heat flow method was firstly described in a series of papers

18 Methods for Measuring the Thermal Conductivity

in 1954 [Jain & Krishnan, 1954a, 1954c, 1954d, 1954b] and since then used by many researchers for the measurement of molten and solid metals [Goldratt & Greenfield, 1978; Hemminger, 1989]. In these papers, the theoretical and the experimental part of the method was developed and applied to a thin circular rod of tungsten and Acheson graphite. In general, the thermal conductivity is computed from the axial temperature gradient and certain parameters. The differential equation, in steady state, for the thin rod, and neglecting the effects of the radial temperature gradient, can be written [Carslaw & Jaeger, 1959]:

2 2

4 402 2 0

T p IT Tx

, (B.7)

where T is the temperature at distance x of the under consideration point, To is the temperature of the walls of the chamber and therefore the temperature of the ends of the rod, p is the perimeter of the rod’s cross-section, ρ its specific resistance. Moreover, ε is the total emissivity of the surface, ω is the cross-sectional area of the rod, σ is Stefan’s constant of radiation, I is the heating electric current, and λ is the thermal conductivity of the specimen.

Following the mathematical procedure described elsewhere [Wagner, 1969], it is obtained:

4 4

2 0

2 L

T Tp L x T T

, (B.8)

where T is the temperature of a rod of infinite length and LT is the temperature at the midpoint of a rod whose length is 2L.

A good application of the method for the measurement of solid and molten lead was done by Hemminger, who claimed an estimated uncertainty of 2.5 % for the solid phase and 3 % for the liquid one [Hemminger, 1989].

B1.1.3. Calorimeter Method

The calorimeter method is a steady state method for the measurement of the thermal conductivity of refractory materials which is a

Methods for Measuring the Thermal Conductivity 19

standard test since 1945 [American Society for Testing and Materials, 1993]. The principle of the method is depicted in Figure B.2. Heat flows through the under measurement specimen, which is surrounded by guard bricks, into a water-cooled calorimeter used for measuring the quantity of heat flowing through it. Under steady state conditions, the thermal conductivity of the solid material can be obtained from the temperature gradient through the sample and from the heat flow through it.

It should be mentioned that the calorimeter and the guards are independently supplied with cooled-water from a central water-circulating system, which maintains the provided water at a constant temperature, pressure, and flow rate. Calibrated differential 10-junction thermocouples are used to measure the temperature difference between the calorimeter and the inner guard, while a second set of differential 10-junction thermocouples is used for recording the temperature of the incoming and outgoing cooling water. Finally, the temperature of the test specimen is also measured with calibrated thermocouples embedded in the sample.

Figure B.2 Typical testing section set up for a thermal conductivity apparatus using

the calorimeter method [American Society for Testing and Materials, 1993]

It is important to mention that under steady state conditions, when the hot face (the face of the sample that undergoes the heating) temperature

20 Methods for Measuring the Thermal Conductivity

is constant and the temperature difference between the calorimeter and the inner guard is around 0.03 K or lower, the thermal conductivity, λ, of the sample can be calculated from

2 1Q L

A T T

, (B.9)

where L denotes the distance between the two thermocouple junctions where temperatures T1 and T2 are measured, and A represent the calorimeter area. Moreover, T1 and T2 are respectively the lower and higher test temperatures measured in the test specimen. The heat rate flowing through the specimen, and consequently into the calorimeter, Q, can be expressed as

PC T mQ t

, (B.10)

where PC is the water heat capacity, ΔΤ is the temperature difference of the cooling water between the calorimeter’s outlet and inlet, and m denotes the mass of water flowing in the calorimeter during time t.

Therefore, combining equations (B.9) and (B.10), the specimen thermal conductivity can be obtain from experimental variables as

2 1

PC T m LA t T T

. (B.11)

The uncertainty of the method depends on the design parameters of the apparatus and it increases in temperature [Salmon et al., 2009]. A typical calorimeter apparatus, created and used by Barth et al. [Barth et al., 2007] for the measurement of thermal insulations up to 1,923 K, has a relative expanded uncertainty of 6.2 %.

B1.1.4. 3ω Method

The 3 omega (3ω) method was firstly proposed in 1987 [Cahill & Pohl, 1987] and is a reliable and accurate method widely used for the measurement of the thermal conductivity of thin films [Cahill & Pohl, 1987; Cahill, 1990; Kaul et al., 2007; Lee, 2009]. The principle of the 3ω method is

Methods for Measuring the Thermal Conductivity 21

the application of an alternating current of angular frequency ω through a thin metal heater line that has been directly deposited on an electrically insulated specimen. Due to Joule heating, this current heats the sample at an angular frequency of 2ω and produce temperature oscillations at the same frequency, with amplitude ΔΤ and phase difference φ. Since the resistance of the metal heater depends linearly on the temperature, the temperature variation causes also resistance oscillations to the metal line. It was noted that the resistance oscillation at 2ω, with the source current at angular frequency ω, generates a small oscillating voltage signal across the metal heater at 3ω [Birge & Nagel, 1987]. Therefore, the thermal conductivity of the sample is evaluated by measuring the amplitude and phase of the resistance changes of the metal film as a function of the frequency. The amplitude ΔΤ is given by the equation [Carslaw & Jaeger, 1959]:

o rPT K ql

, (B.12)

where P l is the amplitude of the power per unit length generated by a heater current of angular frequency ω passing through the narrow metal line, λ is the thermal conductivity of the sample, and o rK q is the zeroth-order modified Bessel function.

By measuring the third harmonic signal at two different frequencies, f1 and f2, the thermal conductivity of the sample can be obtained (without knowing the its density and specific heat) by the equation [Cahill & Pohl, 1987]:

3 2

12

3,1 3,2

ln

4

fV f dRdTl R V V

, (B.13)

where R denotes the average resistance of the metal line, V is the voltage across the line at frequency ω, V3,1 is the voltage at the third harmonic for frequency f1, V3,2 is the voltage at the third harmonic for frequency f2, and dR dT is the slope of the calibration of the metal line source at the measurement temperature.

In the 3ω method a single element is used both as heater and thermometer, as it is done in the hot strip and hot wire methods (see

22 Methods for Measuring the Thermal Conductivity

subsection B1.2.3 and Section B2). However, their main difference is the use of frequency dependent temperature oscillation instead of the time dependent temperature response. A typical application of the method, for the measurement of thin solid polymer specimens was presented by Gu et al. [Gu et al., 2009], in which the uncertainty for the thermal conductivity is 3 %.

B1.2. Transient Methods

Transient methods for the measurement of thermal conductivity and/or thermal diffusivity are based on the generation of a non-stationary temperature field inside the specimen. They are widely used as they have many advantages compared to the steady state methods. Some of the advantages are the simple sample configuration, the short time duration of measurements and mainly low measurement uncertainty. In addition, they can be used for the measurement of materials with significant heterogeneity or multilayer materials. In the case of liquids, the transient methods have the advantage that provided the small measurement times involved, convection does not affects significantly the measurement due to the inertia of the fluid.

There are several transient techniques, which differ on the method of heat generation in time, or/and on the geometry of the employed instruments and heat source. Moreover, there are methods in which there is contact between the heat source and the sample, and others which are non-contact methods. The most used transient methods for the measurement of solids or thin films are:

i. Transient Hot Wire technique ii. Transient Plane Source method iii. Transient Hot Strip method iv. Pulse transient method v. Step-wise Transient method vi. Transient Hot Bridge method

vii. Dynamic Radial Heat Flow methodviii. Laser Flash technique ix. Force Rayleigh Scattering method x. Thermal Waves Analysis xi. Interferometry xii. Photothermal Deflection method .

In the following subsections, the principles of the main transient methods used for the measurement of the thermal conductivity of solids will be briefly presented.

Methods for Measuring the Thermal Conductivity 23

B1.2.1. Transient Hot Wire technique

The transient hot wire method, which was used in the present work, is presented in detail in Section B2 and Chapter C, and therefore it is not discussed here.

B1.2.2. Transient Plane Source Method

The transient plane source method is one of the most commonly used methods for the measurement of the thermal conductivity of liquids and solids [Gustafsson, 1991; Gustavsson et al., 1994; Boumaza & Redgrove, 2003; Assael et al., 2004a; Malinaric, 2004; Jannot et al., 2006; Jannot & Acem, 2007; Malinaric, 2007]. Its principle is very similar to that of the transient hot wire and transient hot strip method (see subsections B1.2.3 and B2). In these three methods, a resistive element is used both as a heat source and as a temperature sensor. Moreover, the assumption that the heat source is placed among an infinite medium is made. The heat source should be thin and its electrical resistance as large as possible, in order to provide applicability to small samples and high sensitivity to temperature measurements.

In the transient plane source method, which was proposed by Gustafsson [Gustafsson et al., 1979a; Gustafsson, 1991], the planar heat source is sandwiched between two distinct specimens of the same material or fully immersed in a single specimen for powders, pastes, and liquids. A constant current is applied to it, so as to sufficiently increase the sensor temperature by 1 to 2 K. Thus, as the sensor’s temperature varies so does its resistance. Consequently, monitoring the resistance change versus time, the thermal conductivity of the sample can be obtained. Despite the fact that the resistive element of the heat source can have any form, for reason of experimental and theoretical convenience, the arrangements used are in the form of “hot plate/square” or “hot disks” (see Figure B.3). The patterns of the resistive element are usually created by deposition techniques or by using thin metal foils. In both cases, the resistive element should be enclosed in thin electrical insulating layers. Moreover, there is the Gustafsson-probe which resembles to a hot-disk with the difference that it consists of a spiral metal foil enclosed inside two Kapton electrical insulating films (see Figure B.3) and its use is defined by ISO 22007-2 [International Organization for

24 Methods for Measuring the Thermal Conductivity

Standardization, 2008]. Although the method is widely used for the measurements of a great

variety of materials, it is not an absolute technique and it does not have a theoretical analytical solution of the heat transfer model. The majority of the commercial plane source devices are accompanied with the suitable software, which numerically solves the partial differential heat transfer equations. A typical application of the method for the measurement of the thermal conductivity of solids by Gustafsson [Gustafsson, 1991], showed an uncertainty of is 3 %. The main disadvantage of such techniques is that there can be no exact theory that will correspond to a spiral disk.

(a) (b)

(c)

Figure B.3 Schematic illustration of different transient plane source heat sources: (a) Hot plate/square sensor, (b) Hot disk sensor, and (c) Gustafsson probe.

Methods for Measuring the Thermal Conductivity 25

B1.2.3. Transient Hot Strip Method

The transient hot strip method, which was firstly proposed in 1979 [Gustafsson et al., 1979b], consists the development of the transient plane source method (see subsection B1.2.2). The main difference of the two methods is the shape of the resistive element, which is in a thin metal strip. The planar metal strip used, acts both as a resistive heater and as a temperature sensor. The principle of the method is that the thin metal strip is placed between two electrical insulating specimen slabs and it is heated by supplying a constant current. By monitoring the voltage change across the strip, and therefore the variation of its electrical resistance, the temperature rise of the strip can be calculated using its premeasured temperature coefficient of resistance. The method has been widely used for the measurement of non electrical conducting solids and fluids, as well as for conducting materials by applying a thin insulating layer on each side of the metal strip [Gustafsson et al., 1979b; Log & Metallinou, 1992; Sabuga & Hammerschmidt, 1995; Hammerschmidt & Sabuga, 2000; Hammerschmidt, 2003; Stosch & Hammerschmidt, 2003; Gustavsson et al., 2006].

As no exact theory exists for such an arrangement, Gustafsson adapted the already existing theory for the hot wire to the strip equivalent and applied modifications on the working equations. The working equations of the method, in the ideal case of neglecting the end effects due to strip’s electrical contacts and finite length (the strip length to width ratio should be kept larger than 20 – 30 in order to minimize the effects), are [Gustafsson et al., 1979b]:

2

2oa V IV t fL

, (B.14)

where

22

1 2114 4

f erf e Ei

. (B.15)

Here oV denotes the voltage at time t =0, V t and I are the voltage drop and the current at time t, a is the temperature coefficient of the strip’s electrical resistance, τ is the dimensionless time, L is the strips half length and λ is the thermal conductivity of the material. The error function and the

26 Methods for Measuring the Thermal Conductivity

exponential integral are denoted by erf and Ei respectively. The transient hot strip method has the advantage that a strip makes

better contact to the solid specimen than a hot wire and it is handled much more easily. However, one of the main drawback of the method compared to the hot wire one (see subsection B2) is that a strip has smaller electrical resistance than the wire, and thus the change of the temperature dependent voltage signal is much smaller. Moreover, the working equation (B.14) is nonlinear and the measurands cannot be determined analytically, but have to be calculated numerically.

Hammerschmidt and Sabuga, in their uncertainty assessment for the transient hot strip method [Hammerschmidt & Sabuga, 2000], determined that in the case of solids the uncertainty of the method for the measurement of the thermal conductivity is 3 % and for thermal diffusivity is 11 %.

B1.2.4. Pulse Transient Method

In the pulse transient method the specimen consists of three parts (I, II and III). A planar heat source, clamped between part I and part II, produces a heat pulse due to Joule heating of its electrical resistance. A junction of a thermocouple that is placed between the second and the third part measures the temperature response to the heat pulse. An experimental setup and the principle of the method is depicted Figure B.4.

The ideal model of the pulse transient methods is [Kubicar et al., 2005b]

2

4,h

t

P

QT h t eC t

, (B.16)

where ,T h t represents the transient temperature at the thermocouple junction and Q is the energy of the heat pulse, whereas ρ, PC and α are respectively the density, the specific heat and the thermal diffusivity of the specimen. In addition, h depicts the distance between the heater and the thermocouple.

The thermophysical properties referred in equation (B.16) can be determined either by the one-point procedure, where the maximum of the temperature response is taken as the input, or by the fitting procedure, where the function of equation (B.16) is fitted to the observed data within

Methods for Measuring the Thermal Conductivity 27

the time window of the temperature response [Kubicar et al., 2005b] The uncertainty of the method depends on the measuring time during

which the temperature field is developed in the specimen, the geometry of the specimen, and the properties of the heating source. Kubicar et al. applied the method for the measurement of homogenous (BK7, PMMA, stainless steel) and heterogeneous (composites, concrete) solid materials, and estimated an uncertainty of about 4 % for thermal conductivity and about 6 % for thermal diffusivity measurements [Kubicar et al., 2005a; Kubicar et al., 2006].

Figure B.4 Experimental setup of the transient pulse method

(part I is “cut-out” to show the heat source structure) [Kubicar et al., 2005b].

B1.2.5. Transient Hot-Bridge Method

An evolution of the transient hot strip method is the transient hot bridge technique [Hammerschmidt & Meier, 2006]. In this technique, which was developed at Physikalisch-Technische Bundesanstalt in Germany, the novel thermoelectric sensor is a printed circuit of nickel foil between two polyimide sheets. Its layout consists of four tandem strips connected in parallel. Each tandem strip is segmented into two parts of different lengths. The eight strips are connected symmetrically in order to make an equal resistance Wheatstone bridge. A typical transient hot bridge sensor can be seen in Figure B.5.

At constant and homogeneous temperature profile, the bridge is initially in balance and therefore there is no need to balance it prior to a

28 Methods for Measuring the Thermal Conductivity

measurement. Applying a heating electric current to a pair of unequally spaced strips, an inhomogeneous temperature profile is created and therefore the bridge becomes unbalanced. Thus, the high sensitivity and time dependent output voltage signal, produced by the sensor, is recorded and analysed in order to provide the thermal diffusivity, the thermal conductivity, and the volumetric specific heat of the under measurement material. Except of the advantages of the transient hot strip method, the transient hot bridge method has the advantage that as the specimen surrounds all the bridge’s resistors, there is no error introduced due to the wiring of the bridge. Moreover, the use of tandem strips segmented into two parts compensates for the perturbing end effects. According to Hammerschmidt and Meier the preliminary estimated uncertainty of the hot bridge instrument, in the case of solid measurements, is determined to be about 2 % for the measurement of thermal conductivity and 8 % for the thermal diffusivity.

Figure B.5 Hot Bridge sensor [Hammerschmidt & Meier