university of southampton research repository eprints soton · ﬁnite element model of the thermal...

University of Southampton Research Repository

ePrints Soton

Copyright © and Moral Rights for this thesis are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder/s. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g.

AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination

http://eprints.soton.ac.uk

http://eprints.soton.ac.uk/

UNIVERSITY OF SOUTHAMPTON

Faculty of Engineering and the Environment

Thermal conductivity of soils forenergy foundation applications

by

Jasmine Eve Low

Thesis for the degree of Doctor of Philosophy

March 2016

UNIVERSITY OF SOUTHAMPTON

ABSTRACTFACULTY OF ENGINEERING AND THE ENVIRONMENT

Civil, Maritime, and Environmental Engineering and Science

Doctor of Philosophy

THERMAL CONDUCTIVITY OF SOILS FOR ENERGY FOUNDATION APPLICATIONS

by Jasmine Eve Low

Ground source heat pumps are a low-carbon method of providing space heating. Thermal energy is

extracted by means of a heat transfer fluid pumped through a series of pipes buried in the ground.

For new builds, construction costs can be minimised by installing the pipes within the building

foundations, eliminating the need for further excavations. These are known as energy foundations.

Designing such a system requires knowledge of the ground thermal properties, in particular the

thermal conductivity. This can be determined by conducting a field thermal response test, or by

laboratory tests on soil samples. In this thesis, the thermal response test was compared to the needle

probe and thermal cell laboratory methods.

For each method, the main sources of error were investigated. Previously, the needle probe transient

temperature data was analysed by visual inspection or rules of thumb. A new analysis method was

developed and trialled on agar-kaolin samples, which reduces errors associated with the previous

methods. The greatest source of error in the thermal cell method was identified as heat losses. A

finite element model of the thermal cell showed that it overestimates the thermal conductivity by at

least 35% due to heat losses. The needle probe was found to be the more reliable method. Both

laboratory methods gave significantly lower values of thermal conductivity than the thermal

response test. Possible reasons for this include differences in scale and sampling disturbances.

The final stage of this research considered the required accuracy in soil thermal conductivity

measurement for a well-designed energy foundation system. A numerical model of an energy

foundation system was used to simulate different thermal loading scenarios. Variations in thermal

conductivity had little effect on balanced systems, but had a significant impact on heating only or

cooling only systems.

ii

Contents

Abstract ii

Contents iii

List of Figures xi

List of Tables xvii

Declaration of Authorship xix

Acknowledgements xxi

Nomenclature xxiii

1 Introduction 1

1.1 Purpose of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Ground source heat pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Current status of GSHPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Energy foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Research overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Background 7

iii

2.1 Basic soil properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Principles of heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.4 Phase change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.5 Lumped Capacitance Method . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Soil thermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Specific heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 Thermal diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Factors affecting thermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Water content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.3 Bulk density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.4 Soil structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.5 Groundwater flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Predictive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.1 De Vries model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.2 Johansen model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Laboratory test methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.1 Needle probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.1.1 Method and Apparatus . . . . . . . . . . . . . . . . . . . . . . . 22

iv

2.6.1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6.1.3 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.1.4 Evaluation of test method . . . . . . . . . . . . . . . . . . . . . . 25

2.6.2 Dual-probe heat-pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25


2.6.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.2.3 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27


2.6.3 Transient plane source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


2.6.3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.3.3 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


2.6.4 Guarded hot plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


2.6.4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.4.3 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


2.6.5 Thermal cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


2.6.5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6.5.3 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


v

2.6.6 Summary of test methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6.7 Chosen test methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 Thermal response test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7.1 Method and Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7.4 Comparison to laboratory methods . . . . . . . . . . . . . . . . . . . . . . . 38

3 Needle Probe 41

3.1 Current analysis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Specimen preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Development of analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Data preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Cut-off section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.3 Second order polynomial fit method . . . . . . . . . . . . . . . . . . . . . . 49

3.3.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.4.1 Boundary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.5.1 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.5.2 Repeatability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.5.3 Varying heating time and power . . . . . . . . . . . . . . . . . . . 55

3.3.5.4 Heating vs recovery . . . . . . . . . . . . . . . . . . . . . . . . . 56

vi

3.3.5.5 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Alternative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Comparison with other analysis methods . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Thermal Cell 65

4.1 Comparison to Clarke thermal cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Laboratory work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.2 Lumped capacitance method for determining power . . . . . . . . . . . . . . 68

4.2.3 Measuring the power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Numerical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1 Modelling heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.2 UoS thermal cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.2.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.2.2 Mesh sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.2.3 Heat transfer coefficient . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.2.4 Heat losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.2.5 Time-dependent response and recovery curve . . . . . . . . . . . . 80

4.3.3 Clarke thermal cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.4 Ideal thermal cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

vii

5 Comparing laboratory methods with the Thermal Response Test 93

5.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1.1 Laboratory tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1.2 Thermal Response Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.1 Needle probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.2 Thermal cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.3 Thermal response test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.4 Comparison of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 The effect of soil thermal conductivity on energy foundation design 105

6.1 PILESIM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Design Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.1 Effect of different soil thermal conductivity measurement methods . . . . . . 116

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7 Conclusions and further research 119

7.1 Needle Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.1.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 Thermal Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

viii

7.3 Comparison to the TRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.4 Energy foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Appendices 123

A Datasheet for kaolin 125

B Matlab code 127

C Needle probe error analysis method used by Hukseflux 141

D Results 145

E Paper for the 32nd International Thermal Conductivity Conference & 20th InternationalThermal Expansion Symposium 157

F Paper published in Acta Geotechnica, October 2014 165

G Paper for the 18th International Conference on Soil Mechanics and Geotechnical Engi-neering 167

H Soil classifications 173

H.1 Particle size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

I Calculating the effect of void ratio on soil thermal conductivity 179

J PILESIM2 parameters 183

References 186

ix

List of Figures

1.1 GSHP system with energy foundations. . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Types of GSHP system: (a) closed loop vertical, (b) closed loop horizontal, (c) open

loop groundwater with extraction and injection wells, and (d) open loop surface water

(Phetteplace, 2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 GSHP global installed capacity from 1995 to 2010 (Lund et al., 2010). . . . . . . . . 4

2.1 Heat flow through a block of material. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The apparent thermal conductivity due to vapour diffusion kvs at 1 atm and the thermal

conductivities kw and ka of water and air respectively, as a function of temperature

(Farouki, 1986). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Effect of Biot number on steady state temperature distribution in a plane wall with

surface convection (Incropera et al., 2007). . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Moisture transfer through a liquid island with arrow indicating direction of heat trans-

fer (Philip and de Vries, 1957). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Thermal conductivity as a function of water content for Toyoura sand (Sakaguchi

et al., 2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Variation of soil thermal conductivity with moisture tension (Al Nakshabandi and

Kohnke, 1965). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 The three types of soil structure (Cote and Konrad, 2009). . . . . . . . . . . . . . . . 19

2.8 Diagram of a needle probe (Hukseflux Thermal Sensors, 2011). . . . . . . . . . . . . 23

2.9 Diagram of a dual-probe sensor (Ham and Benson, 2004). . . . . . . . . . . . . . . 26

xi

2.10 Comparison of temperature-time graphs for an instantaneous pulse and a finite length

pulse (Bristow et al., 1994). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.11 Diagram of a transient plane source, or hot disc, with a diameter D (British Standards

Institution, 2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.12 Cross-section through two specimen and single specimen guarded hot plate apparatus

(British Standards Institution, 2001b). . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.13 Thermal cell apparatus (Clarke et al., 2008). . . . . . . . . . . . . . . . . . . . . . . 34

3.1 Graphs of typical raw needle probe data showing (a) temperature against time (mea-

sured at the mid point of the heating wire) and temperature against logarithmic time

to calculate the thermal conductivity for (b) heating and (c) recovery. . . . . . . . . . 43

3.2 The exponential integral plotted against the natural logarithm of time. This shows the

theoretical shape of the needle probe graph from Equation 2.40. . . . . . . . . . . . 43

3.3 Photo of the needle probe secured within an agar-kaolin specimen. . . . . . . . . . . 45

3.4 A section of the needle probe data showing the stepping resulting from thermocouple

sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Smoothing of the needle probe data. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Different sections of the needle probe temperature against logarithmic time graph: (a)

initial steeper section, (b) decreasing gradient section, (c) constant gradient section

and (d) fluctuating end section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7 Determination of the cut-off point for (a) heating and (b) recovery. . . . . . . . . . . 48

3.8 Graph of p1 on the left y-axis for sections beginning at different values of logarithmic

time ln(t)begin for heating, where the section is fitted to the equation y = p1∗x2+ p2∗x+ p3. The circled points are the two consecutive points which are closest to having

a p1 value of zero, which identifies the section of graph which is most linear. The

calculated thermal conductivity using data from each section is plotted on the right

y-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9 Needle probe graphs of temperature against logarithmic time for (a) heating and (b)

recovery for an agar-kaolin sample, showing the linear section as determined by the

polynomial fit method, and the resulting gradient. Average thermal conductivity is

1.31 Wm−1K−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xii


recovery for a London Clay sample, showing the linear section as determined by the

polynomial fit method and the resulting gradient. Average thermal conductivity is

1.00 Wm−1K−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


recovery for an agar-kaolin sample, showing the linear section as determined by the

polynomial fit method and the resulting gradient. Average thermal conductivity is

0.75 Wm−1K−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.12 Theoretical temperature at different radial distances from the needle probe during a

test. (λ=1.5 Wm−1K−1, cp=1381 Jkg−1K−1, ρ=2008 kgm−3, q=2.43 W) . . . . . . 53

3.13 Thermal conductivities for a range of heating times and heating powers, for (a) Sam-

ple 1, (b) Sample 2, (c) Sample 3, and (d) Sample 4 (in order of increasing density). . 57

3.14 Thermal conductivity calculated using heating and recovery phases for all needle

probe tests. The line shows where the two values are the same. . . . . . . . . . . . . 59

3.15 Average thermal conductivity of 12 needle probe tests, for agar-kaolin samples of

different densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.16 Box plots showing the range in needle probe results using the six different analysis

methods for (a) Sample 1, (b) Sample 2, (c) Sample 3 and (d) Sample 4. Each box plot

represents the 12 tests varying both heater power (Low, Medium, High) and heating

time (100, 300, 500 and 700 seconds). . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Thermal cell by Clarke et al. (2008). . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 UoS thermal cell with dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Cross-sectional diagrams of the (a) Clarke and (b) UoS thermal cells. Both are drawn

to the same scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Photos of the thermal cell, showing the entire cell with insulation (left), and the base

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Biot number over recovery period for a typical thermal cell test on a London Clay

sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Thermal cell result for the top half of the 8.00–8.45 m depth sample. . . . . . . . . . 70

4.7 UoS thermal cell COMSOL model cross section, showing boundary conditions and

materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xiii

4.8 Example vertical heat flux profile for the top surface of the soil. . . . . . . . . . . . . 76

4.9 COMSOL meshes for UoS thermal cell model. . . . . . . . . . . . . . . . . . . . . 77

4.10 The total heat flux at COMSOL UoS thermal cell boundaries as a percentage of the

cartridge heater power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.11 Temperature at the top of the specimen at steady state, as determined experimentally

and by the COMSOL model of the UoS thermal cell, for the 8.00-8.45 m top specimen

of London Clay. The heat transfer coefficient is varied in the COMSOL model, and

the value that gives the same top temperature as the experimental thermal cell is 13.6

Wm−2K−1 (where the lines intersect). . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.12 Temperature at the top of the specimen at steady state, as determined experimentally

and by the COMSOL model of the UoS thermal cell, for the 19.00-19.45 m top spec-

imen of London Clay. The heat transfer coefficient is varied in the COMSOL model,

and the value that gives the same top temperature as the experimental thermal cell is

15.7 Wm−2K−1 (where the lines intersect). . . . . . . . . . . . . . . . . . . . . . . 79

4.13 UoS thermal cell COMSOL model power loss for different soil thermal conductivities. 80

4.14 Calculated thermal conductivity against the COMSOL UoS thermal cell model ther-

mal conductivity. Lines showing when the calculated value is +0%, +40% and +80%

is shown for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.15 Isothermal contour plot (in C) of UoS thermal cell model for a soil thermal conduc-

tivity of 2.75 Wm−1K−1. The direction of heat flux is shown by arrows with lengths

proportional to the heat flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.16 Percentages of heat flow leaving the boundaries of the COMSOL UoS thermal cell,

for a soil thermal conductivity of 2.75 Wm−1K−1. The total heat is supplied by the

cartridge heater. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.17 Calculated thermal conductivity for specimens of different thicknesses, for the UoS

thermal cell COMSOL model. The soil thermal conductivity specified in the model

is 2.75 Wm−1K−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.18 Temperature variation with time for (a) the COMSOL model of the UoS thermal cell

(λ=1.32Wm−1K−1) and (b) the laboratory thermal cell results for the 8.00-8.45 m

depth top specimen of London Clay, showing both heating and recovery phases. . . . 84

4.19 Biot number during the recovery phase for (a) the COMSOL model of the UoS ther-

mal cell (λ=1.32Wm−1K−1) and (b) the laboratory thermal cell results for the 8.00-

8.45 m depth top specimen of London Clay. . . . . . . . . . . . . . . . . . . . . . . 85

xiv

4.20 Clarke thermal cell COMSOL model cross section, showing boundary conditions and

materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.21 Thermal cell experimental result for Leighton Buzzard sand from Clarke et al. (2008),

compared to the COMSOL Clarke model results, for different model values of soil

thermal conductivity and heat transfer coefficient. . . . . . . . . . . . . . . . . . . . 87

4.22 Fitting the COMSOL Clarke model results to the Leighton Buzzard sand experimental

results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.23 The Biot number during recovery for the COMSOL Clarke model results (λ=1.4

Wm−1K−1, h=4.4 Wm−2K−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.24 Isothermal contour plot (in C) of Clarke thermal cell model for a soil thermal con-

ductivity of 2.75 Wm−1K−1. The direction of heat flux is shown by arrows with

lengths proportional to the heat flux. . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.25 COMSOL ideal thermal cell model cross section, showing boundary conditions and

materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.26 Recovery curves generated by the COMSOL ideal thermal cell model compared to

the theoretical fit curve for increasing values of thermal conductivity. . . . . . . . . . 90

4.27 Calculated values of thermal conductivity using the lumped capacitance method, as a

percentage of the actual COMSOL ideal thermal cell model value. . . . . . . . . . . 91

5.1 Thermal conductivity with depth. For the needle probe results, on each box, the

central mark is the median, the edges of the box are the 25th and 75th percentiles, the

whiskers extend to the most extreme data points not considered outliers, and outliers

are plotted individually. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Density and moisture content with depth. . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Moisture content distribution with specimen depth before and after the thermal cell

test. For depth 2.00–2.45 m, top half. . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Moisture content at the top of the soil specimen before and after each thermal cell test.

‘Top’ and ‘Bottom’ refer to the top half and bottom half of the sample, respectively,

as the sample at each depth was cut in half for the thermal cell tests. . . . . . . . . . 99

5.5 Average fluid temperature during the TRT heat injection and heat extraction phases. . 100

5.6 Average fluid temperature against logarithmic time during the TRT (a) heat injection

and (b) heat extraction phases. The vertical lines show the points at which t = 5r2b/α . 100

xv

6.1 Schematic view of an energy pile system. PILESIM2 simulates the section within the

dashed line (Pahud, 2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2 Monthly heating demand for PILESIM2 model, showing space heating and hot water

contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3 Monthly cooling demand for PILESIM2 model. . . . . . . . . . . . . . . . . . . . . 108

6.4 Winter normalised hourly power demand for PILESIM2 model. . . . . . . . . . . . 109

6.5 Summer normalised hourly power demand for PILESIM2 model. . . . . . . . . . . . 109

6.6 Mid-season normalised hourly power demand for PILESIM2 model. . . . . . . . . . 110

6.7 Outdoor air and fluid temperatures for PILESIM2 model. . . . . . . . . . . . . . . . 110

6.8 Diagram of energy fluxes for the PILESIM2 model of the base case. The red and blue

boxes show the share of the heating and cooling load respectively that is covered by

renewable energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.9 Rate of useful energy exchanged with the pile heat exchangers, per unit length of pile,

with varying soil thermal conductivity, for a system with both heating and cooling loads.114

6.10 Rate of useful energy exchanged with the pile heat exchangers, per unit length of pile,

with varying soil thermal conductivity, for a system with only (a) heating loads and

(b) cooling loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.11 Thermal energy extracted/injected per year over a period of 20 years for a system with

both heating and cooling loads, for high and low soil thermal conductivities. Injected

thermal energies are plotted as negative values. . . . . . . . . . . . . . . . . . . . . 115

6.12 Thermal energy extracted per year over a period of 20 years for a system with only

heating loads, for high and low soil thermal conductivities. . . . . . . . . . . . . . . 115

6.13 Thermal energy injected per year over a period of 20 years for a system with only

cooling loads, for high and low soil thermal conductivities. . . . . . . . . . . . . . . 116

6.14 Estimated rate of useful energy extracted/injected for an energy foundation system

simulated in PILESIM2, based on the measured soil thermal conductivity using the

needle probe, thermal cell and TRT methods. . . . . . . . . . . . . . . . . . . . . . 118

xvi

List of Tables

1.1 Cost and GHG emissions for domestic space heating systems. . . . . . . . . . . . . 4

2.1 Typical values for thermal conductivity (Banks, 2008). . . . . . . . . . . . . . . . . 13

2.2 Comparison of test methods for soil thermal conductivity measurement. . . . . . . . 36

3.1 Densities of the agar-kaolin specimens. . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Parameter values used in the second order polynomial fit method. . . . . . . . . . . . 52

4.1 Example results for thermal cell test. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Measurement errors for thermal cell variables . . . . . . . . . . . . . . . . . . . . . 72

4.3 COMSOL model material properties. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1 Summary of laboratory test results. (Thermal cell results given here do not account

for heat losses.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1 Key parameters for PILESIM2 model. . . . . . . . . . . . . . . . . . . . . . . . . . 108

xvii

Declaration of Authorship

I, Jasmine Eve Low, declare that this thesis and the work presented in it are my own and has been

generated by me as the result of my own original research.

Thermal conductivity of soils for energy foundation applications.

I confirm that:

[1] This work was done wholly or mainly while in candidature for a research degree at this

University;

[2] Where any part of this thesis has previously been submitted for a degree or any other

qualification at this University or any other institution, this has been clearly stated;

[3] Where I have consulted the published work of others, this is always clearly attributed;

[4] Where I have quoted from the work of others, the source is always given. With the exception

of such quotations, this thesis is entirely my own work;

[5] I have acknowledged all main sources of help;

[6] Where the thesis is based on work done by myself jointly with others, I have made clear

exactly what was done by others and what I have contributed myself;

[7] Either none of this work has been published before submission, or parts of this work have

been published as:

Low, J. E., Loveridge, F. A. & Powrie, W., 2013. Measuring soil thermal properties for use in energy

foundation design. Proceedings of the 18th International Conference on Soil Mechanics and

Geotechnical Engineering, Paris 2013.

Low, J. E., Loveridge, F. A., Powrie, W. & Nicholson, D., 2015. A comparison of laboratory and in

situ methods to determine soil thermal conductivity for energy foundations and other ground heat

exchanger applications, Acta Geotechnica 10(2), 209–218.

Low, J. E., Loveridge, F. A., & Powrie, W., 2014. Thermal conductivity of simulated soils by the

needle probe method for energy foundation applications. 32nd International Thermal Conductivity

xix

Conference and 20th International Thermal Expansion Symposium, Purdue University, West

Lafayette, IN, USA 2014.

Signed:

Date:

xx

Acknowledgements

First and foremost, I would like to thank my supervisors William Powrie and Fleur Loveridge. They

have been a constant source of support, inspiration and lively discussions throughout my research.

My thanks go to Harvey Skinner for his invaluable help with the laboratory work for this project. No

question, however trivial remained unanswered. Thanks also to Charlie Thompson for her help with

the Coulter LS 130 for particle size distributions. Thanks to Concept Engineering Consultants Ltd

and Arup for providing me with soil samples, and to GECCO2 who carried out the thermal response

test. We are also grateful for the site support from Canary Wharf Contractors Ltd and Marton

Geotechnical Services Ltd.

This work forms part of a larger project funded by EPSRC (ref EP/H0490101/1) and supported by

Mott MacDonald Group Ltd, Cementation Skanska Ltd, WJ Groundwater Ltd and Golder

Associates. Thanks go to them without which none of this would be possible.

Thanks to my parents for encouraging me to be the best I can be. Thanks to Matthew Pratt for his

unwavering support.

Last but by no means least (although littlest), thanks go to Mighty the Minnow for entertaining me

during long days in the geology laboratory. RIP.

xxi

Nomenclature

Abbreviations

BHE Borehole Heat Exchanger

COP Coefficient Of Performance

GHG Greenhouse gas

GSHP Ground Source Heat Pump

SPF Seasonal Performance Factor1

TCR Temperature Coefficient of Resistance

TRT Thermal Response Test

UoS University of Southampton

Latin Letters

A area m2

Bi Biot number

C lumped thermal capacitance JK−1

CF calibration factor

cp specific heat capacity Jkg−1K−1

d thickness m

e void ratio

Ei exponential integral

1 The COP varies throughout the year as it is weather dependent. SPF is the average COP over the year.

xxiii

F factor used in de Vries method for calculating thermal con-

ductivity

Gs specific gravity of solids

h heat transfer coefficient Wm−2K−1

I0 modified Bessel function

Ke Kersten’s number for normalised thermal conductivity

L length m

m mass kg

n porosity

Q heat input J

Q power input, rate of heat flow W

q heat input per unit length Jm−1

q power input per unit length Wm−1

R thermal resistance m2KW−1 or KW−1

r radial distance m

S slope of graph

Sr degree of saturation

T temperature K

t time s

V volume m3

x distance coordinate m

x volume fraction

Greek Letters

α thermal diffusivity m2s−1

γ Euler’s constant = 0.5772...

λ thermal conductivity Wm−1K−1

ρ density kgm−2

xxiv

ρ thermal resistivity mKW−1

σ integration variable

τ time constant s

Subscripts

amb ambient (air)

b borehole

c cold

d dry soil

f fluid

g ground

h hot

heat heating period

i initial condition

m maximum

material known value for calibration material

measured measured value

s surface

s solids, soil

sat saturated soil

start start of recovery phase

th thermal

v voids

w water

xxv

Chapter 1

Introduction

1.1 Purpose of research

Fossil fuels have long been the main source of energy in the world. Unfortunately, they produce

greenhouse gas (GHG) emissions, which contribute towards climate change and are a finite resource.

Space heating accounted for 31% of the UK’s overall energy consumption in 2013 (Department of

Energy & Climate Change, 2014). To heat buildings using low carbon sources forms part of the

government’s plan to reduce the UK’s greenhouse gas emissions by at least 80% by 2050

(Department of Energy & Climate Change, 2011). Currently the majority of space heating is

generated by natural gas, but competitive low-carbon alternatives exist.

1.1.1 Ground source heat pumps

A ground source heat pump (GSHP) system is one way in which heating emissions can be reduced.

A heat pump is a device that transfers thermal energy up a temperature gradient. In a GSHP system,

heat is transferred to and from the ground. The ground makes a reliable heat store due to its high

heat capacity and fairly low thermal conductivity, with ground temperatures below a few metres

depth staying relatively constant throughout the year. This means that the ground is warmer than the

air during winter, and cooler than the air during summer. In the winter when indoor temperatures are

cold, heat is pumped from the ground into the building; in the summer, heat is pumped from the

building into the ground.

Figure 1.1 shows a typical GSHP system in heating mode (cycle is reversed in cooling mode). The

basic steps in the cycle are (Banks, 2008):

[1] An anti-freeze-based transfer fluid is pumped through a series of pipes in the ground. In the

evaporator, the ground heat absorbed by this fluid is exchanged with refrigerant circulating

1

through the heat pump. The heat causes the refrigerant to boil, absorbing latent heat and

becoming a vapour.

[2] The vapour passes through a compressor, which causes the temperature to rise.

[3] The pressurised vapour passes through a condenser (heat exchanger). As the vapour

condenses, heat is rejected to the environment (i.e. the building) via radiators or an underfloor

heating system.

[4] The refrigerant then passes through an expansion valve, cooling back to its original

temperature.

There are several types of GSHP system. These fall into two categories: closed loop, and open loop.

Closed loop uses an anti-freeze-based transfer fluid to exchange heat with the ground through pipes,

such as described in the previous cycle. Open loop uses the existing groundwater as the transfer

fluid, which makes it suitable only for areas where there is an aquifer. Closed loop systems can

either have their pipes run vertically down a borehole, or horizontally in dug trenches which are then

backfilled. The vertical boreholes can range from 40 to 180 metres deep; horizontal trenches tend to

be 1.2 to 2 metres deep (Banks, 2008). Open loop systems can extract water from a borehole and

reject it to another, or extract water from some other source such as a spring or lake. Determining the

most suitable GSHP system design is site specific. Some examples of common GSHP system

configurations are shown in Figure 1.2.

GSHPs could have environmental and financial benefits over conventional heating systems. Table

1.1 compares different heating systems with regards to cost and CO2 emissions. An average four

bedroom detached house with a GSHP system would save 1.8 to 2.9 tonnes of CO2 and £395 to

£590 per year compared with a typical non-condensing gas boiler (Energy Saving Trust, 2014a).

GSHPs run on mains electricity, so the CO2 emissions depend on the electricity source. If a higher

percentage of the electricity came from renewable sources, this would further cut emissions. The UK

Government’s Renewable Heat Incentive (RHI) could result in additional payments of £2,325 to

£3,690 per year (Energy Saving Trust, 2014a).

1.1.2 Current status of GSHPs

GSHPs are a rapidly growing renewable energy technology, with a global installed capacity of

35,236MWth in 2010 (Lund et al., 2010). Figure 1.3 shows the GSHP global installed capacity from

1995 to 2010, which is increasing at an increasing rate. The countries with the most installations are

USA, China, Sweden, Norway and Germany. Although the UK’s installed capacity is still minimal

(181.5MWth in 2010) it had the largest growth from 2005 to 2010 compared with other countries and

continues to increase rapidly (Lund et al., 2010).

The total cost (capital plus running costs) over the 20-25 year lifetime of a GSHP system is

comparable to that of a gas heating system (Banks, 2008). However, the high initial cost of design

2

Figure 1.1: GSHP system with energy foundations.

Figure 1.2: Types of GSHP system: (a) closed loop vertical, (b) closed loop horizontal, (c)open loop groundwater with extraction and injection wells, and (d) open loopsurface water (Phetteplace, 2007).

3

Table 1.1: Cost and GHG emissions for domestic space heating systems.

Heating system Cost (pence/kWh)1 GHG emissions (kgCO2e/kWh)2

Natural gas 4.21 0.185

Liquid petroleum gas (LPG) 8.59 0.215

Oil 6.43 0.247

Domestic coal 3.92 0.340

Electricity 13.52 (standard rate) 0.494

7.09 (off peak economy 7)

GSHP (SPF=4)3 3.38 (standard rate) 0.1241 Values are taken from Energy Saving Trust (2014b).2 Kilograms of carbon dioxide equivalent. Values are taken from Department for Environment Food & Rural Affairs

(2014).3 A seasonal performance factor (SPF) of 4 means that every kW of electricity supplied will produce 4kW of

heating.

and installation is dissuading people from considering this as an option. A typical domestic system

would cost around £11,000 to £15,000 (Energy Saving Trust, 2014a). Installing the ground loops

would require excavating large areas of land, or drilling deep boreholes in addition to the cost of the

heat pump itself. To achieve the best efficiency, each GSHP system requires a site specific design

taking into account the ground conditions such as thermal properties and the presence of

groundwater. This all adds to the overall cost.

1.1.3 Energy foundations

For new builds, installation costs could be minimised by installing the pipes within the building

foundations. These are known as energy foundations and most commonly take the form of pile heat

exchangers (Figure 1.1), although any structure in contact with the ground could potentially be used

(Adam and Markiewicz, 2009). This would eliminate the need to install a separate ground heat

exchanger, hence cutting costs and carbon emissions related to the installation and additional

Figure 1.3: GSHP global installed capacity from 1995 to 2010 (Lund et al., 2010).

4

materials. Energy foundations would require different considerations to typical GSHP systems. For

example, pile heat exchangers have a greater diameter and are shorter in length than boreholes for

vertical GSHP configurations, so there are differences in the way in which heat is transferred

(Loveridge and Powrie, 2012).

1.2 Research overview

To design any GSHP or energy foundation system, it is important to model accurately the heat

transfer process between the ground and the pipes, which are governed by the pile and ground

thermal properties. A key input parameter for such analyses is the soil thermal conductivity, which is

the focus of this thesis. Much research has been carried out into methods of measuring the thermal

conductivity and determining factors which influence this parameter. However, significant

improvements can be made to existing experimental procedures, some of which do not give

sufficiently accurate measurements, or are only applicable to certain soils and/or conditions.

The aim of this research is to improve the way in which GSHP and energy foundation systems are

designed, by better understanding the role of soil thermal conductivity. This is split into three main

objectives:

[1] To compare established methods of laboratory thermal conductivity measurement, determine

what improvements can be made and in what conditions the different methods are suitable;

[2] To compare laboratory test results with full-scale field measurements;

[3] To determine how the thermal conductivity affects the design of energy foundation systems

and make suitable recommendations for practice.

This thesis is presented in several chapters. Chapter 2 gives an overview of what is involved in the

measurement of soil thermal conductivity, and compares the available methods. After detailed

consideration, the needle probe and thermal cell methods were chosen for further investigation.

Chapter 3 looks more closely into the needle probe method and how best to interpret the results of a

test. Limitations are identified, and some key issues are raised requiring further research. Chapter 4

discusses the thermal cell method, and in particular addresses the key issue of heat loss through the

insulation. Chapter 5 looks at a particular case study of a central London development site, where

samples were tested using the needle probe and thermal cell methods, and the results were compared

with full-scale field measurements. In Chapter 6, a numerical model of an energy foundation system

is used to quantify the effects of varying the soil thermal conductivity on system performance.

Finally, Chapter 7 summarises the main conclusions and outlines areas requiring further research.

Recommendations for practice are given on the needle probe and thermal cell methods.

5

Chapter 2

Background

In the field of engineering, soil is defined as a mixture of mineral particles, water and air. The

particles can vary in size, shape and mineralogy. Thermal properties of soil are dependent on the

relative proportions of the constituents, and how these are packed together i.e. the soil structure.

This chapter covers the main aspects of soil thermal behaviour. Predictive and experimental methods

for determining soil thermal conductivity are discussed.

2.1 Basic soil properties

Definitions of some key soil properties are given here, as they will be referred to in this thesis. The

total volume of soil is:

V =Vs +Vw +Va (2.1)

where subscripts ‘s’, ‘w’ and ‘a’ refer to the volumes of solids, water and air respectively. Similarly,

the total mass is:

M = Ms +Mw (2.2)

The mass of air is zero. Water and air occupies the space in between solid particles, which are

known as ‘pores’ or ‘voids’. It is useful to express the volume of voids Vv compared with the volume

of solids as a ratio, known as the void ratio e:

e =Vv

Vs=

Vw +Va

Vs(2.3)

7

Porosity is the ratio of the volume of voids to the total volume, and is related to the void ratio:

n =Vv

V=

e1+ e

(2.4)

Degree of saturation is the ratio of the volume of water to the volume of voids:

Sr =Vw

Vv(2.5)

The proportion of water in a soil is commonly expressed as the moisture content, or water content:

w =Mw

Ms(2.6)

The specific gravity of the solids Gs is the ratio of the density of the solids to the density of water:

Gs =ρs

ρw(2.7)

The void ratio is related to the specific gravity, water content and saturation ratio:

e =wGs

Sr(2.8)

2.2 Principles of heat flow

Thermal energy is the intrinsic energy of a body, and is made up of the potential and kinetic energies

of the particles within it. It is determined by the body’s temperature and heat capacity (see Section

2.3.2). Heat is the term for thermal energy that is transferred from one body to another. As with all

forms of energy, it obeys the law of energy conservation and cannot be created nor destroyed but is

transferred between bodies or into different forms of energy. Heat can be transferred principally by

the mechanisms of conduction, convection, and radiation.

2.2.1 Conduction

Heat conduction is the transfer of heat when vibrating particles of higher energy interact with

adjacent atoms and in doing so transfer some of their energy but with minimal displacement of the

particles. This occurs when there is a temperature gradient. It is the main form of heat transfer in

8

solids, but is not as significant in fluids. In soils, conduction is the predominant method of heat

transfer. The heat transfer is governed by Fourier’s Law:

Q =−λAdTdx

(2.9)

where Q is the rate of heat flow, λ is the thermal conductivity, A is the cross-sectional area, and

dT/dx is the temperature gradient in the direction of heat flow. This is illustrated in Figure 2.1.

2.2.2 Convection

Convection is a mechanism of heat transfer in fluids, and involves different portions of a fluid mixing

together. Natural convection is where the fluid movement is caused by differences in temperature

resulting in differences in density. Warmer and hence lighter fluid rises while the cooler, denser fluid

sinks. In forced convection, the fluid movement is caused by an external source such as a fan.

Convection at a solid surface is defined by Newton’s Law of cooling:

Q = hA(Ts−Tf) (2.10)

where Q is the rate of heat flow, h is the heat transfer coefficient, A is the surface area, and Ts and Tf

are the temperatures of the surface and fluid respectively. The heat transfer coefficient depends on

the fluid properties and velocity, and also the properties of the surface. Convection becomes

significant in soils with groundwater flow or air movement.

2.2.3 Radiation

Electromagnetic radiation is emitted by all matter that is not at absolute zero temperature. It requires

no medium and can therefore travel through a vacuum. Heat transfer by radiation is most significant

between solid surfaces. In soils, radiation can usually be neglected unless it is surface soil (Mitchell

and Soga, 2005).

Figure 2.1: Heat flow through a block of material.

9

2.2.4 Phase change

In unsaturated soils, a temperature gradient can cause moisture migration. The increase in

temperature causes water to change phase from a liquid to a vapour i.e. evaporate, and in doing so,

the water absorbs latent heat. This causes an increase in local vapour pressure, causing water vapour

to diffuse through the pores to areas of lower vapour pressure. The vapour may then condense in

other areas, transferring the latent heat.

As water has a high latent heat of evaporation, the process of evaporation-condensation can be a

significant contribution to heat transfer through soils. The effective thermal conductivity of the soil

should include the contribution from vapour diffusion, which is shown in Figure 2.2 as a function of

temperature.

2.2.5 Lumped Capacitance Method

The lumped capacitance method is a simple method for solving transient heat transfer problems

(Incropera et al., 2007). The thermal cell method in Section 2.6.5 requires an understanding of

transient heat transfer of a solid in a fluid environment. It considers a lump of solid at an initial

uniform temperature Ti, which is suddenly immersed in a fluid of temperature Tamb at time t = 0,

where Tamb < Ti. The solid cools with time until its temperature eventually reaches Tamb. In the

lumped capacitance method, temperature gradients within the solid are assumed to be negligible.

This is impossible given Fourier’s Law, which would require the solid to have an infinite thermal

conductivity for heat conduction to occur with no temperature gradient. However, this is closely

approximated when the resistance to heat transfer within the solid is much lower than the resistance

to heat transfer between the solid and the surrounding fluid. If this is assumed, then by conservation

of energy:

Heat flow into solid = Energy change within solid (2.11)

−hA(T −Tamb) = mcpdTdt

(2.12)

where m is the mass of the solid and cp is its specific heat capacity (Jkg−1K−1). This can be

integrated using separation of variables from initial conditions:

−∫ t

0dt =

mcp

hA

∫ T

Ti

1T −Tamb

dT (2.13)

Evaluating the integrals, the equation can then be arranged to give an expression for the solid

10

Figure 2.2: The apparent thermal conductivity due to vapour diffusion kvs at 1 atm and thethermal conductivities kw and ka of water and air respectively, as a function oftemperature (Farouki, 1986).

temperature at time t:

T = Tamb +(Ti−Tamb)e−(

hAmcp

)t (2.14)

A thermal time constant for the system can be defined as:

τth =

(1

hA

)(mcp) = RthCth (2.15)

where Rth is the resistance to convection heat transfer, and Cth is known as the lumped thermal

capacitance of the solid.

It is necessary to determine what conditions are required for the lumped capacitance method to be

applicable. A criterion is developed by considering steady state conduction through a slab of area A,

as shown in Figure 2.3. By conservation of energy:

λAL

(T1−T2) = hA(T2−Tamb) (2.16)

This can be rearranged to get:

T1−T2

T2−Tamb=

hLλ≡ Bi (2.17)

Bi is a dimensionless parameter known as the Biot number. When Bi 1 then the temperature

11

Figure 2.3: Effect of Biot number on steady state temperature distribution in a plane wallwith surface convection (Incropera et al., 2007).

difference across the slab is much smaller than the temperature difference between the slab and the

fluid. In these cases the slab can be assumed to be at uniform temperature. Therefore, if Bi is small,

then the error associated with this assumption is small. The criterion used for determining the

validity of the lumped capacitance method is (Incropera et al., 2007):

Bi < 0.1 (2.18)

2.3 Soil thermal properties

2.3.1 Thermal conductivity

The soil thermal property of most interest in the design of a GSHP system is the thermal

conductivity. Thermal conductivity is a material property defining its ability to conduct heat, and is

defined by Fourier’s Law (Equation 2.9). Typically, soil thermal conductivities can range from 0.2 to

5Wm−1K−1 (Ground Source Heat Pump Association, 2012).

In some documentation, the thermal resistivity of a material, rather than thermal conductivity is

quoted (e.g. Institute of Electrical and Electronics Engineers, Inc (1996)). The definition of thermal

resistivity is the reciprocal of the thermal conductivity:

ρ =1λ

(2.19)

Frequently, this is multiplied by the thickness (d) of the material and expressed as the thermal

resistance (units of m2KW−1):

12

R =dλ

(2.20)

Confusingly, when this is divided by the area it can also be known as the thermal resistance, with

units of KW−1. Table 2.1 gives some typical thermal conductivity values.

Table 2.1: Typical values for thermal conductivity (Banks, 2008).

Material Thermal conductivity (Wm−1K−1)

Rocks and sediments

Coal 0.3

Limestone 1.5–3.0

Shale 1.5–3.5

Wet clay 0.9–2.2

Basalt 1.3–2.3

Diorite 1.7–3.0

Sandstone 2.0–6.5

Gneiss 2.5–4.5

Arkose 2.3–3.7

Granite 3.0–4.0

Quartzite 5.5–7.5

Minerals

Plagioclase 1.5–2.3

Mica 2.0–2.3

K-feldspar 2.3–2.5

Olivine 3.1–5.1

Quartz 7.7

Calcite 3.6

Pyrite 19.2–23.2

Galena 2.3–2.8

Continued on next page

13

Table 2.1 – Continued from previous page

Material Thermal conductivity (Wm−1K−1)

Haematite 11.3–12.4

Diamond 545

Halite 5.9–6.5

Other

Air 0.024

Glass 0.8–1.3

Concrete 0.8

Ice 1.7–2.0

Water 0.6

Copper 390

Freon-12 at 7C (liquid) 0.073

Oak 0.1–0.4

Polypropene 0.17–0.20

Expanded polystyrene 0.035

2.3.2 Specific heat capacity

The specific heat capacity describes a material’s ability to store heat. It is defined as:

Q = mcp∆T (2.21)

where Q is the heat input, m is the mass of material, cp is the specific heat capacity (units of

Jkg−1K−1), and ∆T is the change in temperature. The heat capacity is sometimes expressed per unit

volume rather than per unit mass, called the volumetric heat capacity (units of Jm−3K−1), which is

the specific heat capacity multiplied by the density (ρcp). For determining the thermal conductivity,

some of the laboratory tests require the heat capacity to be known or assumed.

14

2.3.3 Thermal diffusivity

Another thermal property often quoted is the thermal diffusivity. The relationship between thermal

conductivity, volumetric heat capacity and thermal diffusivity is:

λ = αρcp (2.22)

where α is the thermal diffusivity. As a ratio between the thermal conductivity and the heat capacity,

the thermal diffusivity gives an indication of the rate of heat transfer. A material with high thermal

diffusivity will react quickly to temperature changes, whereas a low thermal diffusivity material will

have a delayed response (Hermansson et al., 2009).

2.4 Factors affecting thermal properties

Soil thermal conductivity is difficult to measure in part due to the number of controlling factors.

Aside from the soil mineralogy, the thermal conductivity is also influenced by temperature, water

content, bulk density, soil structure, and ground water flow.

2.4.1 Temperature

Much research has been conducted to determine the effect of temperature on the thermal properties

of soil. Hiraiwa and Kasubuchi (2000) found that for the temperature range 5 - 75C, the thermal

conductivity increased with increasing temperature. However, the relationship is not a simple one.

The dependence of the thermal conductivity on temperature has been ascribed to latent heat transfer

described by the liquid-island theory (Philip and de Vries, 1957). This suggests that in fairly dry

media, the water is deposited in isolated pockets or ‘islands’, either filling small pores or attaching

themselves between soil grains. When a temperature gradient is applied, there is a vapour flux, as

shown in Figure 2.4. The solid lines are the locations of the initial menisci. Condensation at A and

evaporation at B moves the location of the menisci to where the dashed lines are. In this way,

moisture moves through regions of vapour and liquid. More on this was given in Section 2.2.4.

Sakaguchi et al. (2007) found that for sand with a volumetric water content less than around 0.07, an

increase in temperature actually led to a decrease in thermal conductivity, as shown in Figure 2.5.

This is because there are two governing heat transfer processes: the latent heat liquid-island process

and conduction. Water bridges connecting soil particles (Figure 2.4) facilitate conduction heat

transfer. As the temperature increases, the volume of water bridges decreases as the water

evaporates, causing the heat transfer via conduction to decrease. Temperature rise also causes the

transfer of latent heat to increase. This explains the reversal of the relationship between temperature

15

Figure 2.4: Moisture transfer through a liquid island with arrow indicating direction of heattransfer (Philip and de Vries, 1957).

and thermal conductivity at low water contents. However, it should be noted that this may not be the

case for other types of soil, as also demonstrated by Sakaguchi et al. (2007).

2.4.2 Water content

The effect of volumetric water content on the thermal conductivity and its association with

temperature was mentioned in Section 2.4.1. In general, the thermal conductivity increases with

increasing water content. Chen (2008) varied both the saturation ratio and porosity for sands, and

found that thermal conductivity increased with increasing saturation ratio, and decreasing porosity.

However, for some soils at higher temperatures the thermal conductivity increases with water

content until it reaches a peak, and in fact decreases beyond this point, although the reasons for this

are uncertain (Hiraiwa and Kasubuchi, 2000).

Al Nakshabandi and Kohnke (1965) investigated a possible link between thermal conductivity and

moisture tension. Moisture tension is measured using a tensiometer, which measures the pressure

related to how strongly water is held within a soil. For the three very different soils tested (clay, silt

loam, and fine sand), it was found that for the same soil moisture content their thermal conductivities

varied significantly. However, at the same moisture tension, the thermal conductivities were similar,

despite their different mineralogies (Figure 2.6). This suggests that the thermal conductivity is

actually dependent on the soil structure, and how the water arranges itself around the soil particles.

2.4.3 Bulk density

The soil thermal conductivity increases with bulk density, due to improved particle contact and

decrease in air voids as the porosity decreases. However, the influence of the bulk density varies

between different soils, Abu-Hamdeh and Reeder (2000) found that increasing bulk density for clay

loam and loam soils did not increase the thermal conductivity significantly beyond a certain bulk

density. For sandy and sandy loam soils the thermal conductivity continued to increase with

increasing bulk density.

16

Figure 2.5: Thermal conductivity as a function of water content for Toyoura sand (Sakaguchiet al., 2007).

Figure 2.6: Variation of soil thermal conductivity with moisture tension (Al Nakshabandi andKohnke, 1965).

17

2.4.4 Soil structure

Cote and Konrad (2009) looked into the way soil thermal conductivity is affected by the soil

structure, for two-phase materials (i.e. only solids and water, or only solids and air). They identify

three categories of soil structure with differing thermal characteristics: (a) unbound

rounded/sub-rounded particles, (b) unbound angular/sub-angular particles, and (c) bound/cemented

particles (see Figure 2.7). The soil structures in order of increasing thermal conductivity were found

to be (a), (b), and (c). The structure with the highest thermal conductivity had the most contact area

between particles, as expected. This is because the solid component of a soil has the highest thermal

conductivity, therefore heat is transferred most easily from particle to particle without having to go

through water or air.

2.4.5 Groundwater flow

At some sites where GSHP systems are proposed there could be groundwater flowing. This

introduces forced convection as another heat transfer mechanism, and can increase the rate of heat

transfer significantly, even at low flow rates. Research into the effect of water flow on laboratory soil

thermal conductivity measurements is limited, although there has been some attempt by Katsura

et al. (2006), who found that water flow increased the measured value of thermal conductivity.

Groundwater flow has been shown to influence the results of large-scale in situ testing, which is

discussed further in Section 2.7.3.

2.5 Predictive models

Several models have been suggested for predicting the thermal properties of a soil based on

knowledge of the constituents. One prominent paper on soil thermal properties was written by De

Vries (1963). It outlines a method of determining the thermal conductivity of a soil using the

thermal conductivities and volume fractions of the constituents, as well as making assumptions

regarding the shapes of the particles. Another method by Johansen (1975) is semi-empirical. It gives

formulae for calculating the thermal conductivities of dry and saturated soils, and then uses a linear

interpolation between the two to calculate the thermal conductivity of an unsaturated soil. The main

problem with using these models is that it requires prior knowledge of the thermal conductivities of

the soil constituents, which for the soil solids in particular may be difficult to determine.

2.5.1 De Vries model

De Vries’ method is based on Maxwell’s equation for the electrical conductivity of a mixture of

uniform spheres dispersed at random in a continuous fluid Farouki (1986). Applying this to the

18

Figure 2.7: The three types of soil structure (Cote and Konrad, 2009).

thermal conductivity of ellipsoidal soil particles in a continuous medium of air and water leads to:

λ =xfλf +Fxsλs

xf +Fxs(2.23)

where x is the volume fraction in a unit volume of soil, with subscripts ‘f’ and ‘s’ referring to the

fluid and solid components respectively. F is a factor given by:

F =13 ∑

a,b,c

[1+(

λs

λf−1)

ga

]−1

(2.24)

ga +gb +gc = 1 (2.25)

ga, gb and gc are intended to be shape factors. However, De Vries (1952) found that ga = gb = 0.125

agreed well with experimental data for saturated soils with a low λs/λf ratio, despite this implying

unrealistic needle-shaped soil particles. Therefore, this could be seen as a semi-empirical method.

However, for dry soils (λs/λf ∼ 100) the calculated values using Equation 2.23 were too low, and

should be increased by 25%. For an unsaturated soil with both air and water components, Equation

2.23 becomes:

λ =xwλw +Faxaλa +Fsxsλs

xw +Faxa +Fsxs(2.26)

where

19

Fs =13

2

1+[(λs/λw)−1]0.125+

11+[(λs/λw)−1]0.75

(2.27)

and

Fa =13

2

1+[(λa/λw)−1]ga+

11+[(λa/λw)−1]gc

(2.28)

To try and account for moisture migration, an effective thermal conductivity is used for the pore air,

which adds a vapour diffusion component λvs. The effective thermal conductivity is then λa +λvs.

λvs is temperature dependent, as given by Figure 2.2 in Section 2.2.4.

The air pore-shape factors are approximated by:

ga = gb = 0.333− (xa/n)(0.333−0.035) (2.29)

for 0.09 < xw < n. For xw < 0.09:

λa = 0.0615+1.96xw (2.30)

The unit for this is mcalcm−1s−1C−1. In SI units (Wm−1K−1), this becomes (1 calorie = 4.184

joules):

λa = 0.0257+0.820xw (2.31)

The shape factor for xw < 0.090 is:

ga = gb = 0.013+0.944xw (2.32)

2.5.2 Johansen model

Johansen (1975) developed a method to calculate the thermal conductivity of a frozen or unfrozen

unsaturated soil, which introduces a normalised thermal conductivity known as Kersten’s number:

Ke =λ −λdry

λsat−λdry(2.33)

20

where subscripts ‘dry’ and ‘sat’ refer to the thermal conductivity of the soil in a dry and saturated

state respectively. To determine the dry soil thermal conductivity, Johansen developed this

semi-empirical equation (assuming λs/λa = 120):

λdry =0.135ρdry +64.72700−0.947ρdry

±20% (2.34)

where ρdry is the dry density in kgm−3 and the solids unit weight is taken as 2700 kgm−3. For

saturated soils, frozen and unfrozen soils are treated differently. For saturated unfrozen soils:

λsat = λ(1−n)s λ

nw (2.35)

where n is the porosity and λw is the thermal conductivity of water. For saturated frozen soils:

λsat = λ(1−n)s λ

(n−wu)i λ

wuw (2.36)

where wu is the fractional volume of unfrozen water, λi is the thermal conductivity of ice (= 2.2

Wm−1K−1), and λw is the thermal conductivity of water (=0.57 Wm−1K−1according to Johansen

(1975), but 0.6 Wm−1K−1is now more widely accepted). Finally, the Kersten number needs to be

determined to calculate the thermal conductivity using Equation 2.33. It is related to the degree of

saturation Sr:

Ke ∼= 0.7logSr +1.0 (2.37)

for coarse unfrozen soil, where Sr > 0.05. For fine unfrozen soil:

Ke ∼= logSr +1.0 (2.38)

where Sr > 0.1. For a frozen soil, Ke = Sr.

2.6 Laboratory test methods

Several laboratory methods exist for measuring the thermal properties of soils. They fall into two

categories: steady state or transient. Steady state methods are more time consuming as they involve

applying a constant heat flux to the sample and waiting until a constant temperature distribution

through the sample is reached. The steady state methods detailed here are the guarded hot plate and

21

the thermal cell. With transient methods, thermal properties are determined by heating or cooling the

sample for a set period of time and monitoring the subsequent temperature variations. The transient

methods detailed here are the needle probe, dual-probe heat-pulse and the transient plane source

methods.

2.6.1 Needle probe

2.6.1.1 Method and Apparatus

The Institute of Electrical and Electronics Engineers, Inc (1996) have published a guide outlining the

needle probe method of measuring soil thermal resistivity. The device consists of a long, thin probe,

containing a heater along its length and a thermocouple located midway down the heater (Figure

2.8).

To take a measurement, the needle is first inserted into the prepared sample. Power is supplied to the

needle, heating it. For a low thermal conductivity soil, a low heat input is necessary; a high thermal

conductivity soil would require a higher heat input. This is so an acceptable temperature change can

be measured. According to the Institute of Electrical and Electronics Engineers, Inc (1996) standard,

the temperature of the thermocouple should then be recorded at 15 second intervals for 10 minutes.

However, it is suggested by Hukseflux Thermal Sensors (2003) that this time period is long enough

to introduce inaccuracies caused by water evaporation around the heater. They recommend a shorter

heating time of 100 to 300 seconds. ASTM International (2014) suggests an even shorter heating

time of 30 to 60 seconds for a probe with a diameter of less than 2.54 mm.

The needle probe is an example of an absolute measurement method (Hukseflux Thermal Sensors,

2003). This means that the thermal conductivity can be calculated by directly taking power, time and

temperature measurements, and does not rely on knowing the thermal conductivity of a reference

material. However, if the probe heater resistance is not accurately known, this may give an incorrect

value for the heating power output. Therefore, it is necessary to calibrate the probe against a material

of known thermal properties before using the probe for the first time. Standard materials

recommended by ASTM International (2014) include dry Ottawa sand, glycerol, or water made into

jelly using agar (see Section 2.6.1.3). The calibration factor is defined as (ASTM International,

2014):

CF =λmaterial

λmeasured(2.39)

where λmaterial is the known thermal conductivity and λmeasured the measured thermal conductivity of

the calibration material. Subsequent measurements of the thermal conductivity of other materials

would then be multiplied by CF to give the correct value.

22

Figure 2.8: Diagram of a needle probe (Hukseflux Thermal Sensors, 2011).

2.6.1.2 Theory

The measurement of soil thermal properties using the needle probe method is based on the theory for

an infinitely long, infinitely thin line source. This can be found in Section 10.4 of Carslaw and

Jaeger (1959). For a constant heat input to the heater, the temperature rise at a radial distance r from

the heat source is given by:

∆T =− q4πλ

Ei(− r2

4αt

)0 < t ≤ theat (2.40)

where t is the time from beginning of heating, ∆T is the temperature rise after time t, q is the power

input per unit length of heater, and theat is the time the probe is heated for. Ei is the exponential

integral. This is defined by Abramowitz and Stegun (1972) as:

−Ei(−x) =∫

∞

x

e−u

udu (2.41)

For measurements taken during the recovery phase i.e. after the heater is turned off, the temperature

rise is (ASTM International, 2014):

∆T =− q4πλ

[−Ei

(− r2

4αt

)+Ei

(− r2

4α(t− theat)

)]t > theat (2.42)

Equations 2.40 and 2.42 cannot be solved for λ and α explicitly, so a simplified analysis

approximating the exponential integral is used:

23

Ei(−x)≈ γ + ln(x) (2.43)

for small values of x. This is the case when t is large. ln is the natural logarithm and γ is Euler’s

constant. This approximation leads to the following expressions for the heating and recovery periods

respectively:

∆T ∼= q4πλ

ln(t)+B 0 < t < theat (2.44)

∆T ∼= q4πλ

ln(

tt− theat

)+B t > theat (2.45)

where B is a constant. To calculate the thermal conductivity, a graph is plotted of ∆T against ln(t)

for the heating data, and ∆T against ln(t/(t− theat)) for the recovery data. From the slope of the

linear section of the graph, the thermal conductivity is calculated using:

λ =q

4πS(2.46)

where S is the slope of the graph. Due to the simplification of the exponential integral using

Equation 2.43 and the finite dimensions of the probe, an initial non-linear section of the graph

should be excluded from the analysis. Data towards the end of the test could potentially be affected

by the boundaries of the specimen and also become non-linear. Therefore, care must be taken when

selecting the linear section.

2.6.1.3 Test conditions

The needle probe is a versatile method of thermal conductivity measurement. It can be adopted for

either an in situ or laboratory environment, with a longer probe for in situ measurement. The

working temperature range depends on the specific probe. ASTM International (2014) defines a

temperature range of 0 to 100C. Hukseflux Thermal Sensors (2003) has designed a needle probe

(model TP02) which is suitable for use in the temperature range -55 to 250C. Care must be taken

for measurements below freezing as water phase changes could produce significant errors in the

measured thermal conductivity. These errors can be reduced by minimising the amount of heat the

probe applies to the soil, either by reducing the power supplied to the probe, or reducing the heating

time (ASTM International, 2014).

A wide variety of soils can be tested using the needle probe. The TP02 manual (Hukseflux Thermal

Sensors, 2003) lists materials that the probe is suitable for, from powders to soft rock, with thermal

24

conductivities in the range of 0.1 to 6 Wm−1K−1. For harder materials, insertion of the probe can be

aided by pre-drilling a hole, or the use of a guiding tube with an interior diameter slightly greater

than that of the probe, which is inserted into the material before the probe. The probe can also be

used to measure the thermal conductivity of liquids, but they would first need to be made into a gel

so as to eliminate convection. This is done by adding a gelling agent, such as agar.

2.6.1.4 Evaluation of test method

The needle probe is a transient method and therefore has the advantage over steady-state methods of

a short test duration. The equipment is also simple and easy to set up. However, particular attention

needs to be paid to some aspects in order to achieve accurate measurements. Poor thermal contact

between the probe and the surrounding soil could introduce measurement errors (Abu-Hamdeh,

2001). This can be reduced by coating the probe with thermal grease with a thermal conductivity

greater than 4 Wm−1K−1 (ASTM International, 2014; Tang et al., 2007). The heating could also

cause moisture migration, but for short test intervals this should not be an issue (Farouki, 1986). As

needle probes can have a large length to diameter ratio (100 for the TP02 probe), care must be taken

when inserting the probe so that it is not damaged.

ASTM International (2014) found that this method gave higher thermal conductivity values than

expected. For Ottawa sand and paraffin wax the results indicated a measurement precision of ±10%

and ±15%, respectively, which can be improved with careful calibration (ASTM International,

2014). However, there are other sources that claim better precisions e.g. Hukseflux Thermal Sensors

(2003) states that for their TP02 needle probe the expected accuracy is ±(3% + 0.02) Wm−1K−1for

homogeneous media in good contact with the needle.

2.6.2 Dual-probe heat-pulse


The main components of the dual-probe heat-pulse method are a heater probe and a second probe

containing a thermocouple (see Figure 2.9). These are parallel and spaced at a distance r of about 5

to 6 mm from each other (Campbell et al., 1991; Nusier and Abu-Hamdeh, 2003). To perform a

measurement, the probes are inserted into the soil, and power is supplied to the heater to produce a

heat pulse that generally has a duration of up to 8 seconds (Abu-Hamdeh, 2001; Campbell et al.,

1991). After the heater is switched on, temperature readings in the thermocouple are taken at regular

intervals.

25

Figure 2.9: Diagram of a dual-probe sensor (Ham and Benson, 2004).

2.6.2.2 Theory

The measurement of thermal properties using the dual-probe heat-pulse method is based on the

theory for an instantaneous release of heat from an infinite line source. This is similar to the needle

probe (Carslaw and Jaeger, 1959):

∆T (r, t) =q

4πλ texp(− r2

4αt

)(2.47)

where q is the heat input per unit length of heater. By differentiating Equation 2.47 with respect to

time and determining the peak from the temperature-time measurements (Figure 2.10), the thermal

diffusivity and volumetric heat capacity can be found by the derived formulae (Bristow et al., 1994):

α =r2

4tm(2.48)

ρcp =q

eπr2∆Tm(2.49)

where ∆Tm is the maximum temperature change, tm is the time at which this occurs, and e is the

exponential of unity (2.718...). In practice, the heat pulse is applied for a finite length of time, which

gives an increase in tm but has little effect on ∆Tm (Figure 2.10). If the finite length of the pulse is to

Figure 2.10: Comparison of temperature-time graphs for an instantaneous pulse and a finitelength pulse (Bristow et al., 1994).

26

be taken into account, then (Bristow et al., 1994):

T (r, t) =

T1(r, t); 0 < t ≤ t0

T2(r, t); t > t0(2.50)

where t0 is the heat pulse duration, and

T1(r, t) =−q

4παρcpEi(−r2

4αt

)(2.51)

T2(r, t) =q

4παρcp

[Ei( −r2

4α(t− t0)

)−Ei

(−r2

4αt

)](2.52)

where q is the power input per unit length of heater. Differentiating Equation 2.52 and finding the

temperature and time values at the peak leads to:

α =r2

4

1/(tm− t0)−1/tm

ln[tm/(tm− t0)]

(2.53)

ρcp =q

4πα∆Tm

[Ei( −r2

4α(tm− t0)

)−Ei

( −r2

4αtm

)](2.54)

r is now the distance between the two probes as this is the distance from the heat source at which the

temperature is being measured. Here, the exponential integral can be evaluated using polynomial

approximations 5.1.53 and 5.1.56 presented by Abramowitz and Stegun (1972) (Bristow, 1998). The

thermal conductivity can then be found using Equation 2.22. An analysis by Kluitenberg et al.

(1993) looks into the errors introduced by the approximation of the heat pulse to an instantaneous

release of heat. The article outlines other theories that take these factors into account, but the

decision on which one to use would depend on the dimensions of the probe.


The dual-probe is suitable for the same test conditions as the needle probe. The working temperature

range may be differ between probes, e.g. Decagon Devices, Inc. (2011) gives a range of -50 to

150C for their KD2 Pro dual-probe sensor.

27


Inaccuracies may arise from the sensitivity of measurements to the probe spacing, which is

susceptible to small changes during insertion into the soil. A 2% error in r, for example, would lead

to a 4% error in heat capacity (Campbell et al., 1991). Attempts have been made to correct for this

error (Wen et al., 2015). The aforementioned issues with the needle probe would also apply to the

dual probe.

The dual-probe heat-pulse method has the advantage of being able to determine the heat capacity as

well as the thermal conductivity.

2.6.3 Transient plane source


The transient plane source, or hot disc, is made of a thin bifilar spiral made of metal foil and covered

on both sides with a thin insulating film, as shown in Figure 2.11 (British Standards Institution,

2012). It acts as both a heat source and temperature sensor. To perform a measurement, the disc is

sandwiched between two identical specimens and temperatures allowed to equilibrate. Electric

current supplied to the disc creates a heat pulse, and the increase in temperature of the disc is

measured with time.

2.6.3.2 Theory

The hot disc can measure both thermal conductivity and thermal diffusivity, and hence also

determine the specific heat capacity from Equation 2.22. The resistance of the disc increases with

temperature (British Standards Institution, 2012):

Figure 2.11: Diagram of a transient plane source, or hot disc, with a diameter D (BritishStandards Institution, 2012).

28

R(t) = R0[1+TCR·∆T (t)] (2.55)

where ∆T (t) is the mean temperature increase of the disc, R0 is the resistance of the probe at the

initial temperature T0 and TCR is the temperature coefficient of resistance of the disc. The

temperature increase consists of two components, the increase across the insulating layers ∆Ti(t),

and the increase of the specimen surface ∆Ts(t):

∆T (t) = ∆Ti(t)+∆Ts(t) (2.56)

The bifilar spiral can be approximated as a series of concentric and equally spaced circular line

sources, which gives the following equation for calculating the thermal conductivity:

∆Ts(τ) = P0

(π

32 rλ

)−1D(τ) (2.57)

where P0 is the power output of the disc and r is the radius of the outermost ring source. τ is defined

as:

τ =

√α · tr2 (2.58)

where α is the thermal diffusivity. D(τ) is the dimensionless specific time function:

D(τ) = [m(m+1)]−2∫

τ

0σ−2

[m

∑l=1

lm

∑k=1

k exp

(−(l2 + k2

)4m2σ2

)I0

(lk

2m2σ2

)]dσ (2.59)

where m is the number of concentric ring sources, σ is the integration variable and I0 is a modified

Bessel function. Due to inevitable hardware and software delays, t in Equation 2.58 should be

replaced by t− tc, where tc is a time correction.

The thermal conductivity and diffusivity are calculated through an iterative process, where α and tcare varied until a linear relationship is found between ∆Ts(t) and D(τ). λ can then be determined

using Equation 2.57. Variations on this method depending on the shape of the specimens are

described in British Standards Institution (2012).

29


For a good contact, the area of the specimen in contact with the disc should be flat and smooth. To

minimise boundary effects, the distance between any part of the bifilar spiral to the outside boundary

of the specimen must be greater than the mean radius of the bifilar spiral. Aside from satisfying

these criteria, the sample can be any size and shape.

British Standards Institution (2012) suggests that the method is suitable for testing homogeneous

and isotropic materials, as well as anisotropic materials with a uniaxial structure. It is suitable for a

wide range of thermal conductivities, from 0.01 to 500 Wm−1K−1, and can also measure thermal

diffusivities from 5×10−8 to 10−4 m2s−1. The temperature range of operation is 50 K to 1000 K.

Previous research has shown that the transient plane source method can be used for measuring the

thermal conductivity of soils (Bouguerra et al., 1997).


For measurements of plastics around room temperature, British Standards Institution (2012)

estimates the accuracy as 2% to 5% for thermal conductivity, and 5% to 10% for thermal diffusivity.

The transient plane source method has the advantage over the other transient methods of less sample

disturbance as it does not require making holes in the soil. However, the theory is based on the two

specimens on either side of the disc being identical, which may not be possible in a natural soil. This

could reduce the accuracy of measurements. Variations on the transient plane source method exist,

such as the commercially available TCi Thermal Conductivity Analyzer from C-Therm

Technologies (2015), which only requires one specimen rather than two identical specimens. The

manufacturer’s quoted accuracy is <5%, and it is suitable for testing on materials similar to soils.

However, it is more costly than other measurement methods at around US$30,000 for the complete

setup (Nickerson, J. 2014 pers. comm.).

2.6.4 Guarded hot plate


British Standards Institution (2001a,b) give guidance on measuring the thermal resistance by the

guarded hot plate method, which is a steady-state method. The apparatus can hold either one or two

specimens. A cross-section through each of these configurations (the plan of which can be either

circular or square) is shown in Figure 2.12. The two specimen apparatus consists of a central heating

unit sandwiched between two specimens. This fits between two cooling units. When in operation,

heat flows through the specimens from the heating unit to the cooling units. The single specimen

apparatus replaces one of the specimens with insulation and a guard plate. The two specimen

arrangement has the advantage of reducing heat loss from the heating unit, but requires the

30

specimens to be identical such that the power flowing through each specimen is the same. Guard

sections surround the central heating unit, and are heated to the same temperature but using a

different heat source. They are separated from the central unit by a small gap which acts as a thermal

barrier (Salmon, 2001).

2.6.4.2 Theory

The heat flow rate is obtained by measuring the power input to the heating unit in the metering

section of the apparatus. The thermal conductivity is calculated by rearranging Fourier’s Law

(Equation 2.9):

λ =Qd

A(Th−Tc)(2.60)

where Q is the power input, d is the specimen thickness, A is the metering area, and Th and Tc are the

specimen temperatures on the hot and cold sides respectively. For the double specimen arrangement,

the power input should be halved as it is split between the two specimens.


The guarded hot plate method is mainly used for testing building materials such as insulation with a

suggested width or diameter of 0.3 or 0.5 metres (British Standards Institution, 2001a). It has also

been evaluated for use with soils (Mitchell and Kao, 1978). The apparatus outlined in British

Standards Institution (2001a) have a maximum operating temperature for the heating unit of 100C

and a minimum operating temperature for the cooling unit of -100C, but the limit is only imposed

by the materials used.


The British Standards outline two similar methods, one which is intended for use with dry and moist

products of medium and low thermal resistance, and the other for products of high and medium

thermal resistance. For these samples, thermal resistance measurements should be accurate to within

±2% when the test is conducted under room temperature conditions (±5% for full temperature range

(International Organization for Standardization, 1991)). Repeatability is typically better than ±0.5%

when the specimen stays in the apparatus and there are no changes to the equipment, and better than

±1% when the same specimen is used but removed from the apparatus and mounted again.

Soil samples taken from site investigations are usually 100 mm diameter cylinders. The guarded hot

plate may not be the best method for soils as it requires a larger specimen (Clarke et al., 2008). As a

31

Figure 2.12: Cross-section through two specimen and single specimen guarded hot plate ap-paratus (British Standards Institution, 2001b).

32

steady state method, the test takes a long time to conduct, which could lead to moisture migration

affecting the results (Mitchell and Kao, 1978).

2.6.5 Thermal cell


The thermal cell determines the thermal conductivity in a similar way to the guarded hot plate

method. A cylindrical specimen is sandwiched between two platens and surrounded with insulation

to minimise radial heat loss (Figure 2.13). The temperature of each platen can be controlled as each

has a cartridge heater embedded inside it. It is only necessary to use the cartridge heater in the top

platen if a constant ambient air temperature cannot be maintained. The base platen is kept at a higher

temperature than the top platen, which causes heat to flow upwards through the soil specimen.

Thermistors monitor the temperatures at each end of the soil. Additional thermistors can be used to

monitor the temperature at intervals along the length of the soil. Steady state is reached when the

power input to the base platen reaches a constant value. The power to the base platen is then

switched off, and the temperatures monitored during the recovery phase. (Clarke et al., 2008)

2.6.5.2 Theory

The thermal conductivity is calculated based on the same theory as the guarded hot plate method,

using Equation 2.60 with d as the length of the specimen and Q as the power input to the base platen.

If Q cannot be measured directly, it can be determined using the lumped capacitance method detailed

in Section 2.2.5, provided that the temperature difference across the specimen is small compared

with the temperature difference between the specimen and the ambient air (see Equation 2.18). The

recovery curve can be fitted to the theoretical decay curve (Clarke et al., 2008):

T = Tamb +(Tstart−Tamb)e−(

hA(mcp)total

)t (2.61)

where t is the time after the start of recovery and h is the heat transfer coefficient between the air and

the top platen. T is the temperature at the base of the specimen, with subscripts ‘amb’ and ‘start’

denoting the ambient air temperature and temperature at the start of the recovery curve respectively.

Before being able to determine h using Equation 2.61, the heat capacity for the specimen needs to be

calculated from its constituents:

(mcp)total = (mcp)soil +(mcp)water (2.62)

h is adjusted until a good fit between the data and the theoretical decay curve is obtained. From the

33

Figure 2.13: Thermal cell apparatus (Clarke et al., 2008).

principle of conservation of energy, assuming the specimen is well insulated, the energy transmitted

through the specimen is equal to the energy dissipated from the top of the specimen. The thermal

conductivity can hence be determined from:

hA(Tc−Tamb) =λA(Th−Tc)

d(2.63)

where subscripts ‘c’ and ‘h’ denote the top and bottom of the specimen respectively.


This test method has been used on a wide variety of soils obtained from routine site investigations

(Clarke et al., 2008). Hemmingway and Long (2012) also used apparatus based on the thermal cell

by Clarke et al. (2008) to successfully test a number of backfill materials. The temperature range at

which tests can be conducted would depend upon the operating range for the thermistors and

cartridge heaters, but generally tests are conducted at room temperature with the base platen heated

to around 40C.


The thermal cell method has the advantage of requiring very little sample preparation as it is

designed for U100 (undisturbed, 100 mm diameter) samples taken from routine site investigations.

However, it is a steady state method which requires a significant amount of time for a test to be

34

conducted (around 24 hours). This could introduce moisture migration within the specimen. If the

power input is known, the calculation of thermal conductivity is straight forward. However, if this is

not the case, then the calculations rely on estimating the total heat capacity of the specimen, and

determining the heat transfer coefficient using the cooling curve, which could introduce errors into

the calculated thermal conductivity.

2.6.6 Summary of test methods

A summary of the advantages and disadvantages of each test method is shown in Table 2.2.

2.6.7 Chosen test methods

One transient and one steady state method were chosen for further investigation. The needle probe

and thermal cell were the preferred methods. Both are currently industry recommended laboratory

methods (ASTM International, 2014; Ground Source Heat Pump Association, 2012; Institute of

Electrical and Electronics Engineers, Inc, 1996). The needle probe was chosen for its simplicity and

short test duration. It also has the advantage over the dual-probe of not being sensitive to probe

spacing. The thermal cell was chosen for its compatibility with standard soil sample sizes, which

means less sample preparation is required compared with the other steady state methods and hence

disturbance to the soil is minimised. It also tests a larger volume of soil than the needle probe.

2.7 Thermal response test

2.7.1 Method and Apparatus

The thermal response test (TRT) is currently the most widely used method for the determination of

the in situ thermal conductivity for a GSHP system. In theory the value of thermal conductivity

obtained using this method is closest to what would be experienced by the GSHP system, as it tests

the largest volume of soil and also takes into account other ground characteristics such as

groundwater flow and large-scale soil layering.

To perform a TRT requires the construction of a borehole heat exchanger (BHE). This consists of a

heat exchanger usually made from two or four pipes connected at the base using U-tubes. This is

installed in a borehole that is backfilled with grout. During a TRT, a constant power is supplied to

heat a fluid which is circulated through the pipes for a specified period. Temperatures at the inlet and

outlet to the BHE are recorded.

35

Table 2.2: Comparison of test methods for soil thermal conductivity measurement.

Test method Advantages Disadvantages

Needle probe Short test duration Difficult inserting into hard soils

Suitable for a wide variety ofsoils

Delicate apparatus

Standardised method Not suitable for coarse soils

Dual-probe heat-pulse Short test duration Difficult inserting into hard soils

Suitable for a wide variety ofsoils

Sensitive to variations in probespacing

Can also determine thermal dif-fusivity and heat capacity

Delicate apparatus

Not suitable for coarse soils

Transient plane source Short test duration Requires two identical speci-mens

Can also determine thermal dif-fusivity and heat capacity

Limited experience with testingsoils

Can take a range of sample sizes

Minimal sample disturbanceduring preparation

Guarded hot plate Standardised method Long test duration

May not be suitable for soils

Moisture migration

Thermal cell Suitable for standard soil samplesizes

Long test duration

Minimal sample disturbanceduring preparation

Moisture migration

36

2.7.2 Theory

As with the needle probe, the TRT data can be interpreted by assuming the BHE behaves like an

infinite line heat source. From Equations 2.40 and 2.43 the change in ground temperature can be

expressed by (Carslaw and Jaeger, 1959):

∆Tg ∼=q

4πλ

(ln(

4αtr2

)− γ

)(2.64)

The fluid temperature is not the same as the ground temperature, as heat is first transferred from the

fluid to the grout before reaching the ground. To account for this, a constant thermal resistance Rb is

assumed for the borehole, with radius rb (Loveridge, 2012). Hence the temperature change in the

fluid is given by:

∆Tf = qRb +∆Tg = qRb +q

4πλ

(ln(

4αtr2

b

)− γ

)(2.65)

As there is a difference between the inlet and outlet fluid temperatures, the average of these is taken

in the calculation. In the same way as with the needle probe, the thermal conductivity is determined

from the gradient of the linear section of a graph of ∆Tf against ln(t). The initial part of the graph

should be ignored as it is influenced by the simplification of the exponential integral using Equation

2.43, the finite diameter of the pile, and the time it takes for the heat exchanger itself to reach a

thermal steady state. As a general rule, the time which should be used in calculations is:

t > 5r2b/α (2.66)

where rb is the borehole radius and α is the thermal diffusivity, calculated by estimating the thermal

conductivity from the gradient of the graph. Hellstrom (1991) estimated that using this rule would

give model fit errors of less than 10%.

2.7.3 Limitations

Despite being the most widely used method for determining soil thermal conductivity for GSHP

applications, the TRT has its limitations. To begin with, the data from a TRT can be analysed in

different ways. The line source method described here is widely used due to its simplicity, but it

assumes that the borehole is infinitely long and infinitely thin. For the case of pile heat exchangers,

TRTs have been performed on piles which are shorter and have a greater diameter, making these

assumptions even less applicable. The greater the diameter of the pile, the longer the initial time

period of data that would need to be ignored in calculations. Other methods of interpreting the data,

37

such as using a cylindrical source model, have been attempted (Loveridge et al., 2014).

In Equation 2.65, the power q is assumed to be constant. However, in practice the power may

fluctuate (International Energy Agency, 2013; Austin III, 1998). Different approaches can minimise

this effect, such as directly controlling the temperature difference whilst maintaining a constant flow

rate, controlling the flow rate whilst maintaining a constant temperature difference, or using

electricity stabilisation (International Energy Agency, 2013; Witte et al., 2002). Heat losses or gains

from the environment can also introduce errors, so it is necessary to properly insulate the piping and

the experimental apparatus. Temperature sensors installed in the piping could also give a more

accurate calculation of the power (Witte et al., 2002).

Before beginning a TRT, the borehole and the surrounding ground are assumed to be at a constant

temperature. During the installation of the borehole or pile, there is a localised temperature rise and

soil disturbance caused by drilling and heat generated by the grout setting. Some research has been

done into the minimum time required between installation and testing to allow the ground to

equilibrate (Kavanaugh, 2000). Even when enough time has elapsed, the ground will never be at the

same temperature due to solar radiation warming the top few metres of soil, and also seasonal

variations in temperature.

The ground is naturally inhomogeneous, with variations in thermal conductivity with depth and

possible fractures. Numerical simulations by Eskilson (1987) on a TRT in rock with a top layer of

lower thermal conductivity soil led to the assumption that the top layer only has a significant effect

on results for thicknesses greater than 10 m. The presence of groundwater flow could also have an

impact, resulting in the TRT giving an ‘effective thermal conductivity’. Gehlin (1998) showed that

groundwater flow through fractured rock, or increased groundwater flow due to melting snow, led to

higher thermal conductivity measurements from a TRT than would be expected. Chiasson et al.

(2000) used numerical modelling to show that groundwater flow has a significant effect on heat

transfer in soils with high hydraulic conductivity.Wang et al. (2009) performed several thermal

response tests, where the strata contained a layer with a groundwater flow velocity of 0.96×10−6

ms−1. They found that the heat transfer rate is greatest at the start of the test, and decreases with

time towards a steady state value. On average, the performance of the borehole heat exchanger was

improved by 9.8% and 12.9% for heat injection and heat extraction modes respectively.

2.7.4 Comparison to laboratory methods

Previous research has compared the measured soil thermal conductivity from a TRT and laboratory

methods. Brettmann and Amis (2011) performed a group TRT test on three piles (0.3, 0.3 and 0.45

m diameters), which gave an average soil thermal conductivity of 2.66 Wm−1K−1. Laboratory

needle probe tests following the ASTM D5334 Standard (2008) gave an average thermal

conductivity of 2.98 Wm−1K−1.

To determine the ground thermal characteristics in Harbin, China, Zhang et al. (2014) conducted 22

38

TRTs on 11 boreholes (two heating powers tested at each), and collected a total of 337 rock and soil

samples from the boreholes. The soil was characterised as a combination of silty clay, silt, fine sand,

and mudstone. The soil samples were tested using a Quick Thermal Conductivity Meter (QTM), a

transient line source method similar to the needle probe. The laboratory tests gave a thermal

conductivity range of 1.0 to 1.7 Wm−1K−1, while the TRTs gave a range of 1.42 to 1.92 Wm−1K−1.

With the exception of one borehole, the laboratory values were consistently lower than the TRT

values of thermal conductivity.

39

Chapter 3

Needle Probe

The needle probe used was a Hukseflux TP02 Non-Steady-State Probe for Thermal Conductivity

Measurement (Figure 2.8). This was chosen based on its compatibility with the Campbell Scientific

CR1000 data logger, the flexibility that can be achieved by altering the program, and the ability to

extract the raw temperature data for analysis. Of the commercially available needle probes, it also

has one of the smallest diameter to length ratios (1.5 mm diameter, 100 mm heater length), which

means it most closely approximates an infinite line heat source. The probe is connected to the data

logger which is controlled by a Campbell Scientific LoggerNet program.

To measure the thermal conductivity, the probe is inserted into the soil. It is clamped in place at the

base and left unpowered for a period of time to allow the probe to reach the same initial (ambient)

temperature as the soil. Power is supplied to the probe for a specified heating time and then switched

off, allowing the soil to return to ambient temperature. Temperature is recorded over time for the

duration of the heating and recovery periods.

Analysis and experimental methods have been developed and are described in this chapter. The

method by which data from a needle probe test are analysed can significantly affect the calculated

thermal conductivity. There are several standards relating to the needle probe, but they do not

elaborate on the data analysis, which relies mainly on a visual interpretation of the data (ASTM

International, 2014; Institute of Electrical and Electronics Engineers, Inc, 1996). In this chapter, a

more rigorous method of analysing the data is outlined, which aims to minimise the potential for

human error associated with current methods. The new analysis method is compared with other

methods.

3.1 Current analysis methods

The equations used to determine the thermal conductivity for heating and recovery respectively, are:

41

∆T ∼= q4πλ

ln(t)+B 0 < t < theat (3.1)

∆T ∼= q4πλ

ln(

tt− theat

)+B t > theat (3.2)

The derivations were given in Section 2.6.1. Graphs are plotted of change in temperature against

ln(t) and ln(t/(t− theat)) (both hereafter referred to as logarithmic time), for the heating and

recovery phases respectively. The time is measured in seconds. A typical result is shown in Figure

3.1. During the initial part of each phase, the contact resistance and thermal capacity of the probe are

overcome. The temperature change is also a function of the exponential integral, which only

becomes linear after a certain time (see Figure 3.2). After this, the logarithmic graphs become linear

and the gradient can be used to calculate the thermal conductivity. The time it takes for linearity to

occur depends on the quality of the contact between the probe and the soil. The better the contact, the

shorter the time taken to reach linearity. The last part of the graph for each phase can also become

non-linear, as boundary conditions at the outer surfaces of the sample may start to have an effect.

Current standards suggest selecting the linear section of the graph by visual inspection (ASTM

International, 2014; Institute of Electrical and Electronics Engineers, Inc, 1996), or excluding the

first 10 to 30 seconds from the analysis for smaller diameter probes (ASTM International, 2014).

Both methods can be subjective and introduce significant errors. Commercial needle probes may

have built in programs for calculating the thermal conductivity, e.g. the KD2 Pro Thermal Properties

Analyzer by Decagon Devices excludes the first third of data from the analysis (Decagon Devices,

Inc., 2011). King et al. (2012) calculated the thermal conductivity for different intervals during the

heating time to then find the average. They suggest that a reliable value is obtained when the

standard deviation is <0.1 Wm−1K−1 or <10%. To determine the best approach, an extensive range

of needle probe data must be collected. The needle probe was used to test both agar-kaolin mixtures

produced in the laboratory as described in this chapter, and also natural soil samples collected from a

site investigation, details of which are given in Chapter 5. A MATLAB program was developed to

try and better select the linear section automatically, and tested on the needle probe data gathered.

3.2 Experimental procedure

3.2.1 Specimen preparation

Four agar-kaolin specimens resembling a simple two-phase soil were prepared as follows. (Agar is a

gelling agent and is used to solidify the water, preventing moisture migration when the specimens

are heated.) The datasheet for kaolin is given in Appendix A. De-aired water was heated in a conical

flask over a hot plate. The temperature of the hot plate was set at 370C, and the water was gently

stirred using a magnetic stirrer. A thermometer was used to measure the temperature of the water

42

Figure 3.1: Graphs of typical raw needle probe data showing (a) temperature against time(measured at the mid point of the heating wire) and temperature against logarith-mic time to calculate the thermal conductivity for (b) heating and (c) recovery.

Figure 3.2: The exponential integral plotted against the natural logarithm of time. This showsthe theoretical shape of the needle probe graph from Equation 2.40.

43

every few minutes. When the water reached 85C (the melting temperature of agar) the hot plate

temperature was reduced to 200C, and the stirrer speed was increased slightly to prevent agar from

sticking to the bottom of the flask. The agar was added to the water, with 4 grams of agar to every

litre of water (Hukseflux Thermal Sensors, 2003). When the agar had dissolved (which took

approximately 20 minutes) the hot plate was switched off. The mixture was poured into a large tray,

and the stir bar removed. Kaolin was mixed in gradually using palette knives. When a smooth

consistency with minimal air bubbles had been reached, the mixture was poured into a 100 mm

internal diameter, 220 mm long cylinder.

Different water to kaolin ratios were used for each specimen to achieve a range of thermal

conductivities. The densities are given in Table 3.1. The specimens were left overnight in a 20C

temperature controlled room to equilibrate. To ensure good contact between the probe and the

specimen, the probe was inserted into the mixture while it was still liquid. The base of the probe was

secured by clamping it so that the probe stood vertically through the center of the sample. Figure 3.3

shows a photo of the experimental setup.

3.2.2 Test procedure

To prevent the specimens from drying out, thermal conductivity measurements were taken the day

after the specimen was made, when the specimens had cooled to form a gel. Measurements were

taken for heating times of 100, 300, 500, and 700 seconds, at low, medium, and high power (0.82,

2.43, and 4.13 Wm−1 respectively). Each measurement had three phases, and lasted for a total of

four times the heating time. In the first phase (the same length as the heating time) the power was

off, and the thermocouple measured the initial temperature of the soil to ensure that the temperature

was not drifting. The second phase was the heating phase. The final phase was recovery, which was

twice as long as the heating time. A total of twelve measurements (4 heating times × 3 heating

powers) were taken per specimen.

The repeatability of the needle probe was also assessed, by taking eight needle probe readings in the

agar gel (with no added kaolin) for 300 seconds of heating at medium power.

Table 3.1: Densities of the agar-kaolin specimens.

Specimen Density (kgm−3)

1 1000

2 1181

3 1275

4 1444

44

Figure 3.3: Photo of the needle probe secured within an agar-kaolin specimen.

3.3 Development of analysis method

3.3.1 Data preparation

When a measurement is taken using the needle probe, the total change in temperature of the probe is

at most around 4C. Temperature readings are taken every 0.5 seconds, which means that the change

in temperature readings can be very small. Due to the sensitivity of the thermocouples in the probe,

sometimes the small changes in temperature cannot be detected, leading to the data increasing in

steps, an example of which is shown in Figure 3.4. This effect is problematic when attempting to

find the linear section of the logarithmic time graph, as it essentially introduces noise into the data.

To eliminate this effect, some form of data smoothing is required.

A MATLAB function was written to take needle probe data and eliminate the stepped effect. Taking

heating and recovery data separately, all data points having the same value of temperature were

represented by one point, and the time was the averaged time of the points. For example, if there

were three data points (t1,T1), (t2,T2), and (t3,T3), where T1 = T2 = T3 = T , then these three points

can be represented by one point (t,T), where t = (t1 + t2 + t3)/3. This is illustrated in Figure 3.5. The

resulting smoothed data was much less noisy and therefore easier to analyse.

One issue with this method was that the final temperature of the heating and recovery phases could

potentially be over represented, which is easiest to illustrate using an example. In Figure 3.5, the

temperature of the final step is around 23.5C and lasts for 20.5 s, beginning at 243.5 s and ending at

264 s. Averaging all the points at 23.5C, the step is represented by a point at 254.2 s. However, if

the heating period were to last for 250 s, this final step would be cut in the middle, and the

representative point would instead be at 247.6 s, which would increase the end gradient. To prevent

45

this from happening, after removing the steps from the data the final point for heating and recovery

was discarded from the analysis.

Even after the steps have been eliminated, the data can still sometimes fluctuate. The final step in

data preparation was to perform the ‘smooth’ MATLAB function, which smooths the data using a

moving average filter with a span of 5. If T is a vector of temperature, and Ts is the smoothed vector,

then the first few elements are

Ts(1) = T (1)

Ts(2) = (T (1)+T (2)+T (3))/3

Ts(3) = (T (1)+T (2)+T (3)+T (4)+T (5))/5

Ts(4) = (T (2)+T (3)+T (4)+T (5)+T (6))/5

(3.3)

This smooths any remaining fluctuations. Once the dataset has been prepared, it is then ready for

analysis.

3.3.2 Cut-off section

During the development of a robust analysis method, occasionally a section of graph that was clearly

incorrect would be selected as being the most linear section. The graph of temperature against

logarithmic time follows a similar shape in each test. A typical graph is shown in Figure 3.6. During

heating, there is (a) an initial steeper gradient, followed by (b) a decrease in gradient until (c) a

constant more moderate slope is reached, and sometimes there would also be (d) a fluctuating end

section. The recovery curve follows a similar pattern in reverse. When trying to identify the most

linear section of the graph, occasionally section (a) rather than (c) would be chosen by the MATLAB

program. The initial section is affected by contact resistances between the probe and the soil and the

approximation of the exponential integral, therefore calculating the thermal conductivity from this

data would be completely inaccurate.

To prevent this from happening, the location of the transition section (b) was identified and data

before it was ignored in the analysis. This was achieved by determining the straight line equation for

start and end sections of the graph. The length that was chosen for each section was 2 in logarithmic

time. The intersection of these two straight lines would be around (b). Figure 3.7 shows this process,

where the straight line equations were determined for sections 0≤ ln(t)≤ 2, and 4≤ ln(t)≤ 6 for

heating. Where these two lines intersect is the cut-off.

46

Figure 3.4: A section of the needle probe data showing the stepping resulting from thermo-couple sensitivity.

Figure 3.5: Smoothing of the needle probe data.

47

Figure 3.6: Different sections of the needle probe temperature against logarithmic timegraph: (a) initial steeper section, (b) decreasing gradient section, (c) constantgradient section and (d) fluctuating end section.

Figure 3.7: Determination of the cut-off point for (a) heating and (b) recovery.

48

3.3.3 Second order polynomial fit method

After the cut-off section has been removed from the analysis, the remaining data may still have a

varying gradient so selecting the linear section of the graph is a challenge. Several different methods

were trialled, and a method using a second order polynomial fit was found to be the most successful.

Here, the method is outlined.

A section of graph of length 1 in logarithmic time is taken, and the MATLAB ‘fit’ function is used to

fit a second order polynomial equation of form y = p1∗ x2 + p2∗ x+ p3 to the data. The second

order coefficient p1 is stored. This is repeated for different sections of graph, where the starting

point of the section, ln(t)begin, increases by 0.1 each time until all the data after the cut-off have been

analysed in this way. An example of p1 values and the corresponding thermal conductivity

calculated for different sections is shown in Figure 3.8. If a particular section is linear, p1 = 0 so that

the equation of the section becomes the equation of a straight line of form y = p2∗ x+ p3. To find

the most linear part of the graph, the p1 values were sorted in ascending order, and the consecutive

points with the smallest absolute p1 values were selected (circled in red in Figure 3.8). The full

MATLAB code is included in Appendix B.

This method was tested on the agar-kaolin samples described previously, as well as on U100 samples

of London Clay (see Chapter 5). Although the two soil types produced graphs that had significantly

different shapes, the method was found to work well for both. An example for each is shown in

Figures 3.9 and 3.10. The agar-kaolin graphs have longer and more well defined linear sections

which are reached after a short period of time. The London Clay graphs have shorter linear sections

and longer initial non-linear sections due to additional contact resistances, which are later discussed

in Chapter 5. The final value of the thermal conductivity was the average of the heating and recovery

values.

3.3.4 Limitations

The polynomial fit method appears to pick out a suitable linear graph section for the agar-kaolin and

London Clay data. It is always worth inspecting the graph to make sure that the section of graph

chosen is sensible. In some cases, there could be some uncertainty in how the graph should be

interpreted, such as in Figure 3.11. The MATLAB program has selected what appears to be the most

linear section of graph, but it is possible for the heating phase that the gradient at an earlier section

of graph is more representative of the sample. A section with ln(t)begin = 2.7 also gave a small value

for p1, and the gradient would be closer to the recovery section gradient. In the case of Figure 3.11,

it is suggested that the section with ln(t)begin = 2.7 be used instead of the section chosen by the

MATLAB program. The calculated thermal conductivities for heating are 0.71 and 0.81

Wm−1K−1for ln(t)begin = 5 (as chosen by the program) and 2.7 respectively. For the purposes of this

research and to enable a fair comparison between the different methods, the linear section was

always chosen by the program.

49

Figure 3.8: Graph of p1 on the left y-axis for sections beginning at different values of log-arithmic time ln(t)begin for heating, where the section is fitted to the equationy = p1∗x2+ p2∗x+ p3. The circled points are the two consecutive points whichare closest to having a p1 value of zero, which identifies the section of graphwhich is most linear. The calculated thermal conductivity using data from eachsection is plotted on the right y-axis.

Figure 3.9: Needle probe graphs of temperature against logarithmic time for (a) heating and(b) recovery for an agar-kaolin sample, showing the linear section as determinedby the polynomial fit method, and the resulting gradient. Average thermal con-ductivity is 1.31 Wm−1K−1.

50

Figure 3.10: Needle probe graphs of temperature against logarithmic time for (a) heating and(b) recovery for a London Clay sample, showing the linear section as deter-mined by the polynomial fit method and the resulting gradient. Average thermalconductivity is 1.00 Wm−1K−1.

Figure 3.11: Needle probe graphs of temperature against logarithmic time for (a) heating and(b) recovery for an agar-kaolin sample, showing the linear section as determinedby the polynomial fit method and the resulting gradient. Average thermal con-ductivity is 0.75 Wm−1K−1.

51

The polynomial method requires several parameters to be chosen:

• smoothing span

• section length for determining the cut-off

• section length for implementing the second order polynomial fit method

• interval between starting logarithmic times for the section

Parameter values were chosen to work well with the available data, but this only covers a limited

thermal conductivity range of around 0.6 to 1.6 Wm−1K−1. The chosen values are summarised in

Table 3.2. For soils outside this range, it is uncertain as to whether the same values for these

parameters are suitable, and some trial and error may be necessary to determine the best values to

use.

Another disadvantage of this method is that it selects a short section of the graph. Calculating the

thermal conductivity based on a longer section of the graph should give a more representative value

for the sample. One possibility is to increase the section length, currently set at 1 in logarithmic

time. Further research should be conducted to determine the effects of varying the section length

along with the other parameters.

3.3.4.1 Boundary effects

An assumption that boundary conditions affect the latter part of the data has been made previously.

It is worth considering the theoretical significance of this. Due to the location in which the five

needle probe measurements are taken in each sample, the shortest radial distance from the needle

probe to the edge of the sample is 25 mm. Boundary effects are more likely to appear for a

measurement taken in high thermal conductivity soil, with the needle probe set to high power and

having a long heating time, as this is the situation where it is most likely for the boundary

temperature to increase. With this in mind, the theoretical temperature difference at the needle probe

governed by Equation 2.40 was plotted for different radial distances in Figure 3.12. At the sample

boundary 25 mm away from the needle probe, the theoretical rise in temperature at the end of the

test is 0.07 C. This is 5% of the temperature change at 0.1 mm i.e. at the probe, which indicates that

boundary effects could potentially influence the results.

Table 3.2: Parameter values used in the second order polynomial fit method.

Smoothing span 5

Section length (in logarithmic time)Cut-off 2

Analysis method 1

Logarithmic time interval 0.1

52

Figure 3.12: Theoretical temperature at different radial distances from the needle probe dur-ing a test. (λ=1.5 Wm−1K−1, cp=1381 Jkg−1K−1, ρ=2008 kgm−3, q=2.43W)

3.3.5 Results and discussion

3.3.5.1 Error analysis

In all scientific experiments, there will be uncertainties associated with the results. Error analysis is

the process of studying and quantifying uncertainties, and is a critical part of determining the

validity of experimental results and conclusions. There are uncertainties in all measurements, which

can propagate to a result that is a function of other measured quantities. Another method of

assessing uncertainties is to take repeat measurements and analyse them statistically, i.e. finding the

standard deviation is a good measure of uncertainty (Taylor, 1997).

An assumption has been made that all the errors encountered are independent and random, rather

than systematic. Unlike random errors, systematic errors cannot be determined by repeat

measurements, and are therefore difficult to detect. Agar gel has been used to determine

repeatability, and has an accepted value of 0.6 Wm−1K−1. When performing a calibration using agar

gel, Hukseflux Thermal Sensors (2003) suggests that if the deviation from the accepted value is

within (6%+0.04) Wm−1K−1, the calibration information should remain unchanged. This is the

case for the agar gel tests, so systematic errors can be assumed to be minimal.

Measuring the thermal conductivity using the needle probe method involves the measurement of

various quantities such as time, temperature, and heater power. Hukseflux uses propagation of

uncertainties to determine a combined standard uncertainty for the final result. This method is

described in Appendix C. However, this method calculates the uncertainty when using the raw data

in calculation, and does not account for the data smoothing as described in Section 3.3.1. Therefore,

it is more suitable to analyse the results of this thesis using the following statistical approach (Taylor,

53

1997).

If N number of points (x1,y1), ...,(xN ,yN) follow a straight line y = A+Bx, the constants A and B are

calculated using linear regression, such that

A =∑x2

∑y−∑x∑xy∆

(3.4)

B =N ∑xy−∑x∑y

∆(3.5)

where the denominator ∆ is

∆ = N ∑x2−(∑x

)2 (3.6)

The uncertainty in the y values and constants A and B are then

uy =

√1

N−2

N

∑i=1

(yi−A−Bxi)2 (3.7)

uA = uy

√∑x2

∆(3.8)

uB = uy

√N∆

(3.9)

For calculating the thermal conductivity,

y =4π∆T

q(3.10)

x = ln(t) (3.11)

B =1λ

(3.12)

When the uncertainty in B is calculated, the uncertainty in λ can be calculated by taking the upper

and lower bounds such that

54

λupper =1

B−uB(3.13)

λlower =1

B+uB(3.14)

Finally, the uncertainty in λ is

uλ =λupper−λlower

2(3.15)

For consistency, the error in this thesis will always be expressed as a percentage uncertainty, such

that the final result is of the form

λ ± uλ

λ×100% (3.16)

If a number of repeat measurements are taken, the standard deviation is a measure of uncertainty.

For N separate measurements x1,...,xN of the quantity x, the standard deviation is

σx =

√1

N−1

N

∑i=1

(xi− x)2 (3.17)

3.3.5.2 Repeatability

For the eight tests on the agar gel sample, the repeatability was found by calculating the standard

deviation of the results. The thermal conductivity was 0.62±0.7% Wm−1K−1. This is close to the

manufacturer’s stated repeatability of ±1% (Hukseflux Thermal Sensors, 2003). The repeatability

could possibly be improved if the time between measurements was longer, the temperature of the

room could be more accurately controlled, and the sample was a dry solid so no evaporation or

moisture migration could occur. Most natural soils contain moisture, so the same repeatability as

using agar gel is unlikely to be achieved.

3.3.5.3 Varying heating time and power

The results from the agar-kaolin samples are shown in Figure 3.13; the values are given in Appendix

D. The average thermal conductivity of the 12 readings for each sample were 0.63±4% Wm−1K−1,

0.81±3% Wm−1K−1, 0.91±5% Wm−1K−1, and 1.33±6% Wm−1K−1, for Samples 1, 2, 3, and 4

respectively, with the uncertainty given as the standard deviation. The absolute range in results

55

increases with increasing density for the four samples, from a range of 0.10 Wm−1K−1for Sample 1

to 0.28 Wm−1K−1for Sample 4. It appears that increasing the density led to an increase in the range

in results. This may have been because the higher density samples also had a higher thermal

conductivity. The heat would be conducted away from the probe more rapidly and temperature

differences at the probe would be smaller. Small temperature fluctuations would therefore have a

greater influence on the gradient.

A heating time of 700 s gave the greatest range in results, and 300 s the smallest. With a longer

heating time there is a longer section of graph to analyse so this is not altogether unexpected.

Considering the samples separately, a heating time of 100 s also gave a small range. Therefore, for

these samples a heating time between 100 and 300 s appears to be the most suitable. For higher

thermal conductivity soils, the heating time may need to be increased for the data to give a suitably

long linear section. If there is significant contact resistance between the probe and the soil e.g. in the

London Clay samples, the heating time should also be increased. For all the samples, analysis with a

section length of 1 in logarithmic time gave satisfactory results; it is recommended that this be the

minimum length of the linear section.

The range in results for the different heating powers was greatest at low power. This is because the

higher powers gave a greater temperature difference between readings which led to a better defined

gradient in the graph. However, there is a difference between the samples of lower thermal

conductivity and higher thermal conductivity. For the lower thermal conductivity samples (Samples

1 and 2), Low and Medium powers gave a smaller range in results. For the higher thermal

conductivity samples (Samples 3 and 4), High power gave a smaller range in results. This is due to

the fact that heat is transferred more slowly for low thermal conductivity materials, thus giving a

more rapid temperature increase at the probe. When plotting the data, this would give a well-defined

gradient even at lower powers. However, for samples of higher thermal conductivity, the temperature

increase at the probe is smaller due to more rapid heat transfer away from the probe. For the plot of

temperature data to give as well-defined a gradient i.e. more rapid temperature increase, this would

require a higher heating power. Based on these results, the recommended heating power for soils

with an estimated thermal conductivity of <0.85 Wm−1K−1is around 0.82 to 2.43 Wm−1, and

around 4.13 Wm−1 for soils with an estimated thermal conductivity >0.85 Wm−1K−1. As most

soils have a higher thermal conductivity than 0.85 Wm−1K−1, the high power setting is

recommended in most cases.

3.3.5.4 Heating vs recovery

For each test, the thermal conductivity can be calculated using both the heating and recovery phases.

These should ideally give similar values. The results are plotted in Figure 3.14. The greatest

difference in results for the agar-kaolin samples was 0.37 Wm−1K−1for a test on Sample 4; the

greatest difference in results for the London Clay samples was 0.46 Wm−1K−1for a test on the

8.00-8.45 m depth sample. The undisturbed London Clay samples showed more scatter between the

heating and recovery thermal conductivities than the agar-kaolin samples. One reason could be that

56

Figure 3.13: Thermal conductivities for a range of heating times and heating powers, for (a)Sample 1, (b) Sample 2, (c) Sample 3, and (d) Sample 4 (in order of increasingdensity).

57

the contact between the probe and the soil was poorer for the London Clay, as it was a hard soil

requiring a larger diameter hole to be pre-drilled and filled with contact fluid. The additional time

required to overcome this contact resistance could mean that the linear section of graph was shorter

and therefore the calculated thermal conductivity would fluctuate more.

The average difference between the heating value and recovery value was 9%, as a percentage of the

heating value. From Figure 3.14, it can be seen that heating tends to give a higher value than

recovery. 76% of the results have a higher value with heating than recovery. This could be due to

moisture migration during the heating phase, which results in a slightly drier soil being tested during

the recovery phase. If this were the case, then the heating phase would slightly overestimate the

thermal conductivity, and the recovery phase would slightly underestimate the thermal conductivity,

so that an average of the two values should lead to the most accurate value. However, Hukseflux

Thermal Sensors (2003) does not mention the use of the recovery data for calculating the thermal

conductivity. From these results, it is recommended that an error threshold be set, such that if the

difference in results is greater than 10%, the recovery value is neglected and only the heating value

taken as the thermal conductivity, provided that the heating data gives a clear linear section.

3.3.5.5 Density

The average thermal conductivity of the 12 tests for the each of the four samples increased with

increasing density, as shown in Figure 3.15. The density was increased by increasing the proportion

of kaolin in the sample.

3.4 Alternative methods

Several alternative methods were considered before the second order polynomial fit method was

developed. One promising method was based on minimising the difference between the heating and

recovery thermal conductivity values. This was found to work reasonably well for the agar-kaolin

samples, but was unsuitable for data from the undisturbed London Clay samples, which produced

graphs with shorter linear sections. The method is described in a conference paper accepted for

publication (Appendix E).

3.5 Comparison with other analysis methods

This new method was compared with the other standard methods. Data from the agar-kaolin

specimens were analysed and the linear section determined using the following six methods1:

1 The London Clay data was not used in this comparison as each location at which the probe was inserted was onlytested once, so would give no indication as to how robust each method is with varying heating time and heating power.

58

Figure 3.14: Thermal conductivity calculated using heating and recovery phases for all nee-dle probe tests. The line shows where the two values are the same.

Figure 3.15: Average thermal conductivity of 12 needle probe tests, for agar-kaolin samplesof different densities.

59

• Method 1: visual inspection (ASTM International, 2014; Institute of Electrical and Electronics

Engineers, Inc, 1996)

• Method 2: first 10 seconds of data are ignored (ASTM International, 2014)

• Method 3: first 30 seconds of data are ignored (ASTM International, 2014)

• Method 4: first third of data is ignored (Decagon Devices, Inc., 2011)

• Method 5: t < 5r2/α data is ignored (Ground Source Heat Pump Association, 2012)

• Method 6: second order polynomial fit method

Prior to analysis, the data was prepared as described in Section 3.3.1. The visual inspection for

Method 1 was carried out by the author. Method 5 is based on the method described in Section 2.7

used to interpret the thermal response test, where r is the radius of the needle probe. The results

using these methods are illustrated in Figure 3.16.

The average and standard deviation from results using all the methods was 0.62±4% Wm−1K−1,

0.81±5% Wm−1K−1, 0.91±5% Wm−1K−1, and 1.27±7% Wm−1K−1for Samples 1, 2, 3 and 4

respectively. This suggests that the needle probe method, coupled with the data preparation as

detailed in Sections 3.3.1 and 3.3.2 is a fairly robust method.

Using the averaged thermal conductivity from heating and recovery, the methods all gave a similar

range of results except for Method 4. The visual inspection method has the advantage of being able

to adjust the linear section location. If end data were to be affected by boundary conditions, this

could also be spotted visually. However, it is time consuming and the interpretation of data may vary

from person to person.

For Methods 2 and 3, ignoring the first 10 to 30 seconds is simple to implement, but depending on

the sample this might not eliminate all the data affected by contact resistance at the probe. It also

does not allow for the possibility of boundary effects for longer heating times.

Cutting off the first third of data in Method 4 gave the greatest range in results. The results also

differed significantly from the other methods. This may be because ignoring the first third of data

results in only a small section of the graph being analysed, particularly when the data is plotted on a

logarithmic scale, and this may not be representative of the linear section. Of all the methods,

Method 4 is thought to give the least accurate results, and is therefore not recommended.

Method 5 relies on an estimate of the thermal diffusivity. This was made by using Method 1 to

estimate the thermal conductivity, estimating the heat capacity from the properties of the constituents

using Equation 2.62, and hence calculating the thermal diffusivity using Equation 2.22. The heat

capacity at 20C of kaolin and water used in the estimations was 940.5 Jkg−1K−1 (Robie and

Hemingway, 1991) and 4182 Jkg−1K−1 (Howatson et al., 1991) respectively. The reliance on

estimating the thermal diffusivity adds time and complexity to this method, but it has the advantage

60

Figure 3.16: Box plots showing the range in needle probe results using the six different anal-ysis methods for (a) Sample 1, (b) Sample 2, (c) Sample 3 and (d) Sample 4.Each box plot represents the 12 tests varying both heater power (Low, Medium,High) and heating time (100, 300, 500 and 700 seconds).

61

of being tailored to different types of soil. However, Method 5 still does not account for the

possibility of additional contact resistance, for example in a hard soil requiring a pre-drilled hole.

This was not an issue for the agar-kaolin samples, but would apply to the London Clay samples. For

this case, the value of r used to determine the data to ignore should perhaps be the hole radius to try

and account for this.

The newly developed Method 6 has the advantage of specifically finding the most linear section of

graph instead of assuming a section as with Methods 2 to 5, and has the advantage over Method 1 of

being automated. Sometimes the results differ significantly from the other methods, but this may be

because end fluctuations are being excluded. As previously discussed in Section 3.3.4, Method 6 still

has limitations which require further research.

3.6 Conclusions

The needle probe has been successfully used to measure the thermal conductivity of agar-kaolin and

London Clay samples. It had a repeatability of ±0.7% when there were identical test conditions,

heating time and heating power. When performing a test, it is important to determine the most

suitable heating time and heating power to use. For the agar-kaolin samples, a heating time of 100 to

300 s was preferred. Low and Medium heating were most suitable for the samples of <0.85

Wm−1K−1, and High heating most suitable for the samples of >0.85 Wm−1K−1.

For each test, the thermal conductivity was calculated for both heating and recovery phases. In

general, the heating phase gave higher values of thermal conductivity than the recovery phase. This

is most likely due to moisture migration during the heating phase, causing the recovery phase to test

a slightly drier soil. For tests where the difference in results is greater than 10%, it is recommended

that only the heating phase be used for calculations.

An analysis method was developed for automatic interpretation of the needle probe data. To first

collect a range of data, specimens were made using kaolin-agar mixtures of varying densities.

Needle probe tests were carried out at different heating times and heating powers.

The analysis method consists of three stages:

• Data is smoothed to eliminate stepping.

• A cut-off point is determined and data before this point eliminated from further analysis.

• A second order polynomial is fitted to sections of the remaining data, and the most linear

section is chosen for the thermal conductivity calculation.

The new method gave comparable thermal conductivity values to the other methods, with the added

benefit of being automated and considering boundary effects. A visual inspection of the data is

62

recommended to validate the result.

63

Chapter 4

Thermal Cell

The thermal cell was chosen for further investigation, as a steady state method of soil thermal

conductivity measurement. The thermal cell design was loosely based on Clarke et al. (2008), shown

in Figure 4.1. In this chapter, the experimental method is described and the limitations discussed.

One area of particular concern was the heat losses through the apparatus. This was considered

further by modelling the thermal cell using finite element analysis. The results from this were

compared to models of the Clarke thermal cell and an ideal lossless thermal cell.

4.1 Comparison to Clarke thermal cell

For clarity, the thermal cell of this research will hereafter be referred to as the UoS (University of

Southampton) thermal cell, and the thermal cell from Clarke et al. (2008) as the Clarke thermal cell.

A diagram of the UoS thermal cell is shown in Figure 4.2. The thermal conductivity of a 100 mm

diameter, 100 mm length cylinder of soil was measured by generating one-directional heat flow

along the axis of the specimen. The heat was supplied by a cartridge heater embedded in the

aluminium platen. A cartridge heater is a small cylindrical heating element, comprising a heating

wire typically encased in stainless steel. Radial heat flow was minimised by surrounding the sample

with insulation.

The UoS thermal cell design had some differences to the Clarke thermal cell (Figure 4.3). The

insulation layer and acrylic base were made thicker in the UoS thermal cell to minimise heat losses.

Expanded polystyrene was used for the insulation as it has a low thermal conductivity, and could be

easily wrapped around the soil specimen. The type of insulation used in the Clarke thermal cell was

not identified in the paper.

The top platen in the Clarke thermal cell was used to maintain a constant temperature at the top of

the specimen if a constant ambient air temperature could not be maintained. This platen was

removed from the UoS thermal cell as testing was conducted in a temperature controlled room. The

65

Figure 4.1: Thermal cell by Clarke et al. (2008).

top platen could still be used to seal in the specimen and prevent it from drying out during testing.

However, during initial testing, a significant temperature difference was measured between the top

and bottom of the platen, which would affect the calculation of the soil thermal conductivity.

Therefore, the top platen was removed from the UoS thermal cell, and aluminium foil was used

instead to cover the top of the specimen to prevent drying. To measure the temperature at the top of

the soil, a thermistor was mounted inside the tip of a hypodermic needle and inserted a couple

millimetres deep into the top of the soil, at the centre of the specimen cross-section.

The Clarke thermal cell monitored the temperature gradient within the specimen by pushing two

hypodermic needle thermistors radially into the specimen at a height of one third and two thirds of

the total height. The UoS thermal cell did not have these additional thermistors as some soils were

too hard for the hypodermic needles to be inserted. Where this was not the case, the needles would

still cause additional disturbance to the soil. This would also require holes in the insulation for

inserting the needles, which may have compromised the integrity of the insulation.

4.2 Laboratory work

4.2.1 Method

Photos of the thermal cell are shown in Figure 4.4. Tests were done on six samples of London Clay

taken at different depths from a TRT borehole at a central London development site (details in

66

Figure 4.2: UoS thermal cell with dimensions.

Figure 4.3: Cross-sectional diagrams of the (a) Clarke and (b) UoS thermal cells. Both aredrawn to the same scale.

67

Figure 4.4: Photos of the thermal cell, showing the entire cell with insulation (left), and thebase (right).

Chapter 5). Two 100 mm length specimens were cut from each sample, which are hereafter referred

to as ‘top half’ and ‘bottom half’ for each depth. To perform a test, the cartridge heater was turned

on, and the power controlled so that the platen remained at a constant temperature of 40C.

Temperatures were monitored until steady state was reached and then maintained for at least 2 h.

The power to the cartridge heater was switched off, and the recovery period monitored.

4.2.2 Lumped capacitance method for determining power

Clarke et al. (2008) suggested applying the lumped capacitance method to determine the power input

using the recovery data (see Sections 2.2.5 and 2.6.5). The method assumes that the temperature

difference across the length of the soil is small compared to the temperature difference between the

soil and the ambient air. This is the case when the Biot number is small:

Th−Tc

Tc−Tamb= Bi < 0.1 (4.1)

where subscripts ‘c’, ‘h’ and ‘amb’ refer to the temperatures at the top of the specimen, bottom of

the specimen, and ambient air respectively. If this criterion is satisfied, the theoretical decay curve

during recovery is:

T = Tamb +(Tstart−Tamb)e−(

hA(mcp)total

)t (4.2)

where t is the time since the start of recovery, h is the heat transfer coefficient between the top of the

specimen and the air, and (mcp)total is the heat capacity of the specimen, determined from its

constituents using Equation 2.62. T is the temperature of the specimen, which can be taken as an

average of the temperatures at the top and bottom of the specimen (temperatures should be similar

68

due to Biot number criterion). h is adjusted until a good fit between the data and the theoretical

decay curve is obtained. From the principle of conservation of energy, assuming the specimen is

well insulated, the energy transmitted through the specimen is equal to the energy dissipated from

the top of the specimen. At steady state, and having determined the value of h, the thermal

conductivity can hence be calculated from:

Q = hA(Tc−Tamb) =λA(Th−Tc)

d(4.3)

In all twelve tests, the temperatures at the top and bottom of the soil were never that similar, so the

Biot number never fell below 0.1 (for example, see Figure 4.5). Therefore the power could not be

calculated using the lumped capacitance method but instead had to be measured directly. This was

done using a MuRata ACM20-5-AC1-R-C wattmeter. When steady state was reached, the power

supplied to the cartridge heater was recorded and averaged over 10 minutes.

A typical test result is shown in Figure 4.6. Table 4.1 shows the sample properties at steady state and

the calculated thermal conductivity. The experiment was fairly straight forward to perform, and the

data was simple to interpret. However, there were concerns about the accuracy of the wattmeter

measurements.

4.2.3 Measuring the power

Initially, the power was measured using a wattmeter. A digital display would show the power, and by

timing the changes it was shown to update itself every 1.16 seconds. This could be introducing an

error into the power measurement if the length of the heat pulses is not measured accurately. Human

error is another issue, as the power is read off the wattmeter display, so there is the potential for

pulses to be missed. An alternative method was introduced to give a better power measurement. The

data logger was programmed to log when the cartridge heater switches on and off. As the cartridge

heater rating is known to be 50 W, the average power over a time period can be calculated.

For two of the thermal cell tests, the power was measured using both methods. Calculating the

Table 4.1: Example results for thermal cell test.

Depth (m) 8.00-8.45 (top half)

Diameter (mm) 103

Length (mm) 106

Base temperature (C) 40.1

Top temperature (C) 28.4

Power (W) 1.85

Thermal conductivity (Wm−1K−1) 2.01

69

15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (h)

Bio

t num

ber

Figure 4.5: Biot number over recovery period for a typical thermal cell test on a London Claysample.

0 5 10 15 20 25 30 35 4015

20

25

30

35

40

45

50

Tem

pera

ture

(o C)

Time (h)

Base of sample

Top of sample

Ambient

Steady state

Figure 4.6: Thermal cell result for the top half of the 8.00–8.45 m depth sample.

70

average power over 20 minutes, the power measured from the wattmeter was on average about 4%

lower than that of the data logger. This is probably due to the wattmeter being unable to register

some of the shorter pulses. Therefore, the preferred method of calculating the power was using the

data logger to record the heat pulses.

4.2.4 Error analysis

The thermal conductivity is calculated by

λ =QL

A(Th−Tc)=

4QLπD2 (Th−Tc)

(4.4)

where Q is the power input, L is the specimen length, A is the cross-sectional area, D is the specimen

diameter, and Th and Tc are the specimen base and top temperatures respectively. The individual

measurement errors for each variable are known, and hence the final error in thermal conductivity

can be calculated using error propagation.

During the test, temperature measurements were taken every 10 seconds. Th and Tc were calculated

by averaging the temperature readings at steady state. The standard deviation of the temperature

readings gives an estimate of the associated uncertainty.

As previously discussed, the power was measured using either the wattmeter or data logger. The

error was determined differently for the two methods. Using the wattmeter, the heat pulses had a

fixed length of 1.16 seconds each, and the power of each pulse was read off the wattmeter display.

Using data from the datalogger, the power rating of the cartridge heater is known at 50 W, and the

time taken for each pulse is recorded by the data logger. The error in power and pulse length is

different for each. Table 4.2 gives the errors associated with each variable.

The general equation for calculating the error in q, which is a function of several variables x,...,z with

associated uncertainties ux,...uz, is

uq =

√(∂q∂x

ux

)2

+ ...+

(∂q∂ z

uz

)2

(4.5)

Power is calculated by summing up the individual pulses and dividing it by the total time

Q =

N

∑i=1

qi · ti

ttotal(4.6)

71

Table 4.2: Measurement errors for thermal cell variables

VariableUncertainty, u

Absolute Percentage

Length, L 2 mm -

Diameter, D 2 mm -

Temperature, Th,Tc Standard deviation

Power by data logger

Power, Q 0.5 W 1%

Pulse length, t 0.005 s -

Power by wattmeter

Power, Q 0.5 W -

Pulse length, t 0.1 s 9%

where N is the number of pulses. From Equation 4.5, the uncertainty in Q is then

uQ =

√N

∑i=1

((ti ·upi)

2 +(pi ·uti)2)

ttotal(4.7)

Partial derivatives for each of the variables are required for Equation 4.5. These are

∂λ

∂ Q=

4QLπD2 (Th−Tc)

(4.8)

∂λ

∂L=

4QπD2 (Th−Tc)

(4.9)

∂λ

∂D=− 8QL

πD3 (Th−Tc)(4.10)

∂λ

∂Th=− 4QL

πD2 (Th−Tc)2 (4.11)

∂λ

∂Tc=

4QL

πD2 (Th−Tc)2 (4.12)

As with the needle probe results, the uncertainty in the thermal cell results will be expressed as

percentages. The errors were assumed to be random, with no systematic errors. This was a valid

72

assumption for the length and diameter readings, measured using a standard ruler with 1 mm as the

smallest division (uncertainty was estimated as ± 2 mm due to slight variations in the dimensions).

The tests were carried out in a temperature controlled room set to 20 C, which was compared to the

initial temperatures measured by the thermal cell thermistors, and any systematic errors were shown

to be insignificant. The determination of power input is assumed to have no associated systematic

error. This is a valid assumption as two methods were used to determine the power (see Section

4.2.3) and gave similar results. Error due to heat loss has been treated separately, and covered in the

following sections.

4.3 Numerical modelling

The greatest source of error in the thermal conductivity method is in determining the power input.

The method detailed here does not take into account any heat losses that may occur through the base

and insulation. Numerical modelling using the finite element software COMSOL was used to

determine the significance of this. Three models were made: (a) UoS thermal cell, (b) Clarke

thermal cell, and (c) lossless idealised thermal cell. As the thermal cells have rotational symmetry

about the axis of the cylindrical specimen, 2D axisymmetric models are used. This generates a mesh

over the area of a radial slice, which is rotated 360to obtain the 3D results. It is assumed that the

thermal properties of the materials do not change with temperature. These models were used to

determine the heat losses, and to discover why the decay curve does not satisfy the criterion for

using the lumped capacitance method.

4.3.1 Modelling heat transfer

COMSOL has a selection of physics models. As the focus was on heat transfer, the ‘Heat Transfer in

Solids’ model was chosen. This uses the heat equation:

ρcp∂T∂ t−∇·(λ∇T ) = Q (4.13)

where ρ is the density, cp is the heat capacity, T is temperature, t is time, λ is the thermal

conductivity, and Q is the heat flux per unit volume.

4.3.2 UoS thermal cell

The COMSOL model of the UoS thermal cell is shown in Figure 4.7. The following sections outline

how the properties of the model were chosen, such as determining the appropriate mesh sizing and

heat transfer coefficients at the boundaries. ‘Stationary’ i.e. steady state and ‘time-dependent’

73

studies were run, depending on the required output. For the stationary study, a constant power was

specified for a heat source in the aluminium platen, simulating the cartridge heater. A constant

temperature condition at the top of the platen could also be used. For the time-dependent study, a

constant heat source would be unrepresentative of the actual thermal cell, because the power varies

initially to maintain a constant temperature of 40C. Therefore, the heat source was replaced with a

constant temperature condition at the top of the platen during the heating phase, which was disabled

during the recovery phase. Heat losses and the lumped capacitance method were investigated using

these studies.

4.3.2.1 Material properties

Unless stated otherwise, the material properties used in the models are shown in Table 4.3. The

ambient temperature and the initial temperature of the thermal cell was 20C.

4.3.2.2 Mesh sizing

The model mesh had triangular elements automatically generated by COMSOL. The mesh could

also be defined by the user, but for this model the automatic meshing was suitable. The element size

could be specified from ‘Extremely fine’ to ‘Extremely coarse’. A comparison of meshes is shown in

Figure 4.9. To determine which mesh size to use, the cartridge heater power was specified as 4 W

(using the stationary study), and compared to the total heat flux at the outer boundaries of the

thermal cell, which was calculated as follows.

For each node, the heat flux in radial and vertical directions at steady state was given in Wm−2. An

example profile of the vertical heat flux over the top surface of the soil is shown in Figure 4.8. If the

profile is rotated 360about the vertical axis, the volume underneath gives the total heat flux across

the surface. This requires integration in cylindrical coordinates.

The heat flux magnitude qz(r) is discrete and would follow a profile as in Figure 4.8. Through linear

interpolation between two nodes (r1,z) and (r2,z) where r is the radial distance and z the vertical

distance, qz(r) can be expressed as:

Table 4.3: COMSOL model material properties.

Soil1 Aluminium2 Acrylic2 Insulation3

Thermal conductivity (Wm−1K−1) 2.75 160 0.18 0.03

Specific heat capacity (Jkg−1K−1) 1632 900 1470 1130

Density (kgm−3) 2010 2700 1190 231 Values for saturated fine sand, taken from Clarke et al. (2008).2 Values taken from COMSOL’s Material Library.3 Thermal conductivity from Hukseflux Thermal Sensors (2003) and specific heat capacity from

Jablite Intelligent Insulation (2014). Density was calculated from mass and volume measure-ments.

74

Figure 4.7: UoS thermal cell COMSOL model cross section, showing boundary conditionsand materials.

qz(r) =qz2− qz1

r2− r1·r+

(qz1−

qz2− qz1

r2− r1·r)= mr+ c (4.14)

where m is the gradient and c is the vertical axis intercept. The surface can be divided up into a

series of concentric rings. Using equation 4.14, the heat flux over a circular ring (the area between

the two nodes) can be calculated by the following integration:

dQz =∫ 2π

0

∫ r2

r1

(mr+ c)drdθ = π

[23

m(r3

2− r31)+ c(r2

2− r21)]

(4.15)

where θ is the angle about the vertical axis. Summing the ring heat fluxes would give the total

vertical heat flux Qz over the surface.

For a ring element of radial heat flux, the calculation is simpler as the radial distance r is constant.

The radial heat flux between nodes (r,z1) and (r,z2) is:

dQr = 2πr(z2− z1) ·qr1 + qr2

2(4.16)

As before, calculating the total radial heat flux Qr over an area would be to sum the ring elements.

The appropriate boundary conditions had yet to be determined (see Section 4.3.2.3, so assumptions

were made. Convective boundaries with a heat transfer coefficient of 25 Wm−2K−1, as used by

Clarke et al. (2008), were assumed for the boundaries in contact with the air. The boundary in

75

Figure 4.8: Example vertical heat flux profile for the top surface of the soil.

contact with the laboratory bench was also modelled as a convective boundary with a heat transfer

coefficient of 5 Wm−2K−1. Figure 4.10 shows the result of the mesh comparison. The advantage to

using a coarser mesh would be that it takes less time for each simulation to run. However, even with

an ’Extremely fine’ mesh, a simulation took no more than a few seconds due to it being a simple 2D

model. This mesh sizing was therefore chosen for subsequent simulations as it would model the heat

flow with greater accuracy.

4.3.2.3 Heat transfer coefficient

The heat transfer coefficient between the UoS thermal cell and the ambient air had to be determined.

Clarke et al. (2008) gave a typical value of 25 Wm−2K−1 which was used in the Clarke model. To

determine a suitable heat transfer coefficient for the UoS model, COMSOL stationary studies were

run for heat transfer coefficients from 10 to 35 Wm−2K−1, and the temperature at the top of the

platen set to 40C. The temperature at the top of the specimen during steady state was obtained from

the output. This was compared to the experimentally measured top temperature for the 8.00-8.45 m

top specimen, and the 19.00-19.45 m top specimen. The soil thermal conductivity in the models

were set to the values measured at the two depths by the needle probe. Figures 4.11 and 4.12 show

the results of these simulations, where the intersection gives the value of heat transfer coefficient that

produces the same top temperature as the experimental thermal cell. The average heat transfer

coefficient was 15 Wm−2K−1; this was used in subsequent simulations.

During these simulations, the boundary between the thermal cell base and the laboratory bench on

which it stood was approximated by a heat transfer coefficient of 5 Wm−2K−1. Similarly, this was

then varied between 0 and 25 Wm−2K−1, while the heat transfer coefficient at the other boundaries

were set at 15 Wm−2K−1, as determined previously. It was found that varying this base heat transfer

coefficient had a negligible effect on the model, perhaps due to the relatively small surface area of

76

Figure 4.9: COMSOL meshes for UoS thermal cell model.

Figure 4.10: The total heat flux at COMSOL UoS thermal cell boundaries as a percentage ofthe cartridge heater power.

77

the thermal cell in contact with the bench. Therefore, for subsequent simulations, the initial

assumption of 5 Wm−2K−1 was maintained.

4.3.2.4 Heat losses

The main objective of the numerical modelling was to determine the significance of heat losses in

the thermal cell laboratory tests. In the UoS thermal cell model, a constant power to the cartridge

heater was specified (located at the centre of the platen) and adjusted until a soil base temperature of

40C was achieved, and the heat flux at the top and bottom of the soil at steady state were used to

calculate an average heat flux through the soil. The thermal conductivity of soils tends to lie in the

range of 0.2 to 5Wm−1K−1 (Ground Source Heat Pump Association, 2012). For this range, the

power loss was calculated:

Power loss (%) =Heater power - (Heat flux at top of soil + Heat flux at bottom of soil)/2

Heater power×100

(4.17)

The results are plotted in Figure 4.13. Figure 4.14 shows the difference between the model thermal

conductivity, and the calculated thermal conductivity using the total power. For a typical soil of

thermal conductivity 2 Wm−1K−1, the UoS thermal cell is likely to overestimate the thermal

conductivity by 70%. It can be seen that the power loss is significant, particularly for soils of low

thermal conductivity. The power loss varies from 35% to 75%. This makes it difficult to determine

what value of power to use in thermal conductivity calculations for the actual thermal cell. Above

3.5 Wm−1K−1 it may be possible to estimate the power going through the soil as 35% less than the

power to the cartridge heater. Below 3.5 Wm−1K−1 the power going through the soil is much more

dependent on the soil thermal conductivity. For any soil, estimating the power going through the soil

as 35% less than the power to the cartridge heater would be an improvement to the thermal

conductivity calculation.

Figure 4.15 aids visualisation of the temperatures in the thermal cell using an isothermal contour

plot. The boundary heat fluxes were also aggregated to determine where the heat was being lost, and

this is shown in Figure 4.16 for a soil specimen of 2.75 Wm−1K−1. This shows that heat is lost

through both the insulation and the base. Although it may be possible to reduce heat loss through the

insulation by using better materials or a vacuum, the base must provide a stable platform to support

the other components, whilst having as low a thermal conductivity as possible. It may be difficult to

find a material that would perform better than acrylic, so it is unlikely for base heat losses to be

further reduced.

One possible method of reducing the radial heat loss is to reduce the thickness of the specimen. For

a range of thicknesses, the power to the cartridge heater was varied until a base temperature of 40C

was reached. The thermal conductivity was calculated using the total power and temperature

78

Figure 4.11: Temperature at the top of the specimen at steady state, as determined experi-mentally and by the COMSOL model of the UoS thermal cell, for the 8.00-8.45m top specimen of London Clay. The heat transfer coefficient is varied in theCOMSOL model, and the value that gives the same top temperature as the ex-perimental thermal cell is 13.6 Wm−2K−1 (where the lines intersect).

Figure 4.12: Temperature at the top of the specimen at steady state, as determined experimen-tally and by the COMSOL model of the UoS thermal cell, for the 19.00-19.45m top specimen of London Clay. The heat transfer coefficient is varied in theCOMSOL model, and the value that gives the same top temperature as the ex-perimental thermal cell is 15.7 Wm−2K−1 (where the lines intersect).

79

Figure 4.13: UoS thermal cell COMSOL model power loss for different soil thermal conduc-tivities.

difference across the specimen. Figure 4.17 shows the results of different thicknesses. Even for a

thickness of 10 mm, the calculated thermal conductivity was significantly different from the model

thermal conductivity. Smaller thicknesses would be infeasible from an experimental point of view.

For the thermal cell, heat losses cannot be entirely eliminated using this method, as there are base

losses as well as radial losses. Alrtimi et al. (2013) used a similar method of calculating the thermal

conductivity at different specimen lengths, and extrapolating from these values to find the thermal

conductivity at zero length. This reduces the influence of radial heat losses on the calculated thermal

conductivity. The heater platen was sandwiched between two specimens, so base losses would be

less significant.

4.3.2.5 Time-dependent response and recovery curve

The UoS thermal cell model was able to produce a time-dependent result giving the temperatures

over time for both heating and recovery periods, as shown in comparison with the experimental

results in Figure 4.18. The soil thermal conductivity in the COMSOL model was set to the average

value measured by the needle probe. During the heating phase, the power to the cartridge heater was

not constant but varies to keep the temperature at the base of the soil constant. Therefore, to get a

time-dependent result similar to the experimental setup, the constant power condition was replaced

with a constant temperature boundary condition of 40C applied at the base of the soil during the

heating phase, which was then disabled during the recovery phase.

80

Figure 4.14: Calculated thermal conductivity against the COMSOL UoS thermal cell modelthermal conductivity. Lines showing when the calculated value is +0%, +40%and +80% is shown for comparison.

Figure 4.15: Isothermal contour plot (in C) of UoS thermal cell model for a soil thermalconductivity of 2.75 Wm−1K−1. The direction of heat flux is shown by arrowswith lengths proportional to the heat flux.

81

Figure 4.16: Percentages of heat flow leaving the boundaries of the COMSOL UoS thermalcell, for a soil thermal conductivity of 2.75 Wm−1K−1. The total heat is sup-plied by the cartridge heater.

Figure 4.17: Calculated thermal conductivity for specimens of different thicknesses, for theUoS thermal cell COMSOL model. The soil thermal conductivity specified inthe model is 2.75 Wm−1K−1.

82

For the thermal cell tests on the London Clay samples, the recovery curve could not be used to

estimate the power as the temperature difference in the soil was too big for the lumped capacitance

method to apply. The Biot number during recovery was plotted for both the UoS thermal cell model

and for the laboratory test on the 8.00-8.45 m depth top specimen of London Clay, as shown in

Figure 4.19. The Biot number in the COMSOL model was higher than in the laboratory test, and did

not gradually decrease. This is because the recovery temperatures in the model decreased more

rapidly than in the laboratory test, which could be due to imperfect contacts at boundaries in the test,

such as between the soil and the insulation. This would slow the rate of temperature decrease. This

effect can also be seen during the heating phase, with the top temperature rising more rapidly in the

model. Despite the difference in shape of the two graphs, the Biot number never fell below 0.1 in

either case. Therefore, the lumped capacitance method is not recommended for calculating the

power in a UoS thermal cell test. The power should instead be measured directly.

4.3.3 Clarke thermal cell

In Clarke et al. (2008) a theoretical decay curve was fit to the recovery curve to determine the power

through the soil specimen. The temperature difference through the soil was small during recovery,

and allowed the lumped capacitance method to be used. A COMSOL model of the Clarke cell was

made (Figure 4.20) to see if this could be the case, and how the recovery curve may differ from the

UoS thermal cell model. The model dimensions were the same as the Clarke cell (Figure 4.1),

except for the removal of the top aluminium plate and platen. This was the configuration for a test on

saturated fine Leighton Buzzard sand (Clarke, B. 2015 pers. comm.), the result of which was given

in Clarke et al. (2008) and used here for comparison. The material properties were assumed to be the

same as the UoS thermal cell, which were given in Table 4.3.

The thermal conductivity of the soil in the model was set to 2.75 Wm−1K−1, which was measured in

the Clarke cell test on Leighton Buzzard sand. Clarke et al. (2008) assumed a heat transfer

coefficient of 25 Wm−2K−1. However, when these values were used in the COMSOL model, the

shape of the temperature-time graph was significantly different from the Clarke et al. (2008) graph.

The heat transfer coefficient and thermal conductivity were individually varied until the

experimental steady-state temperature of 30.6C at the top of the specimen was reached, to see if a

closer fit could be achieved, as shown in Figure 4.21. None of this produced a graph that was similar

to the experimental result. However, when both soil thermal conductivity and heat transfer

coefficient were varied, the graph was able to be reproduced, as in Figure 4.22. Using this model, a

significant portion of the recovery curve had a Biot number just above 0.1, as in Figure 4.23. It could

be argued that based on this, it may be inappropriate to use the lumped capacitance method to

determine the power input. The graph was only achieved by a model thermal conductivity of 1.4

Wm−1K−1, compared to the calculated thermal conductivity of 2.75 Wm−1K−1. This follows the

trend of Figure 4.14, with calculated values greater than the model values of thermal conductivity.

Compared to the recovery of the UoS thermal cell results, the top and base temperatures of the

Clarke cell converge much more rapidly. This is possibly due to greater heat losses through the

83

Figure 4.18: Temperature variation with time for (a) the COMSOL model of the UoS thermalcell (λ=1.32Wm−1K−1) and (b) the laboratory thermal cell results for the 8.00-8.45 m depth top specimen of London Clay, showing both heating and recoveryphases.

84

Figure 4.19: Biot number during the recovery phase for (a) the COMSOL model of the UoSthermal cell (λ=1.32Wm−1K−1) and (b) the laboratory thermal cell results forthe 8.00-8.45 m depth top specimen of London Clay.

85

insulation and base, resulting in a more rapid decrease in temperature at the specimen base.

The alternative to the lumped capacitance method would be to directly measure the power. Figure

4.24 shows the isothermal contour plot of the Clarke thermal cell during steady state. It can be seen

that the temperature of the acrylic base has been raised significantly by the power supplied to the

cartridge heater. The significant heat losses through the insulation and base would lead to an

incorrect calculation of the thermal conductivity, if the power was measured directly.

4.3.4 Ideal thermal cell

The lumped capacitance method was unsuitable for use with the UoS thermal cell. The Biot number

for the recovery phase of the COMSOL Clarke model was significantly smaller, but still did not

satisfy the criteria for the lumped capacitance method to be valid. To determine whether the lumped

capacitance method can ever be used, an ideal thermal cell model was produced. The ideal model

had perfectly insulated sides and base, with only the top of the soil cylinder open to convection, as

shown in Figure 4.25. This model was used to see whether the lumped capacitance method could be

used in the case where there are no heat losses. At the base was a thin disk (1 mm thickness), made

of the same soil material. This is because COMSOL could not model a boundary with a constant

temperature condition that was also perfectly insulating. Therefore, a constant temperature condition

was defined at the top of the disk, and the base was perfectly insulated.

The model was run for different values of thermal conductivity. For each value, a theoretical curve

(see Equation 2.61) based on the lumped capacitance method was fit to the numerical model

recovery curve, and then the thermal conductivity was calculated based on the theoretical curve. The

results from this analysis are in Figure 4.26. It can be seen that the higher the thermal conductivity,

the more closely the theoretical curve resembles the model curve. Hence, the thermal conductivity

calculated using the theoretical curve is closer to the model value as the thermal conductivity

increases, as shown in Figure 4.27.

Within the range of soil thermal conductivities, the lumped capacitance method would give a

significant underestimate of the thermal conductivity for an ideal thermal cell, according to the

numerical analysis. Interestingly, the reason why the lumped capacitance method is not applicable

may be because of the lack of heat losses. The significant difference in temperature between the base

and top of the soil specimen is due to the heat being dissipated much more rapidly from the top by

convection, while the base has insulation on all sides slowing the rate of temperature decrease. One

could argue that for a poorly insulated thermal cell, the power would be better estimated using the

lumped capacitance method, and for a well insulated thermal cell the power should be measured

directly. This demonstrates the complexity of determining the correct power, and calls into question

the validity of the thermal cell method.

86

Figure 4.20: Clarke thermal cell COMSOL model cross section, showing boundary condi-tions and materials.

Figure 4.21: Thermal cell experimental result for Leighton Buzzard sand from Clarke et al.(2008), compared to the COMSOL Clarke model results, for different modelvalues of soil thermal conductivity and heat transfer coefficient.

87

Figure 4.22: Fitting the COMSOL Clarke model results to the Leighton Buzzard sand exper-imental results.

Figure 4.23: The Biot number during recovery for the COMSOL Clarke model results (λ=1.4Wm−1K−1, h=4.4 Wm−2K−1).

88

Figure 4.24: Isothermal contour plot (in C) of Clarke thermal cell model for a soil thermalconductivity of 2.75 Wm−1K−1. The direction of heat flux is shown by arrowswith lengths proportional to the heat flux.

Figure 4.25: COMSOL ideal thermal cell model cross section, showing boundary conditionsand materials.

89

Figure 4.26: Recovery curves generated by the COMSOL ideal thermal cell model comparedto the theoretical fit curve for increasing values of thermal conductivity.

90

Figure 4.27: Calculated values of thermal conductivity using the lumped capacitance method,as a percentage of the actual COMSOL ideal thermal cell model value.

4.4 Conclusions

Laboratory testing showed that the temperature difference across the soil is too great for the lumped

capacitance method to be used during the recovery phase as a means of calculating the power. The

power was measured directly instead, using a wattmeter. It was found that the wattmeter

underestimates the power by about 4%, and a better method was introduced where the data logger

records the times when the cartridge heater is switched on and off.

Numerical modelling showed that the UoS thermal cell has significant heat losses of at least 30%

over the range of typical soil thermal conductivities. This would impact the thermal conductivity

calculation from the laboratory work. As with the laboratory testing, the model showed a significant

temperature difference across the soil during recovery, which prevents the lumped capacity method

from being used to calculate the power. With the Clarke thermal cell, the temperature difference was

less significant.

To determine whether the recovery curve can ever be used, an ideal thermal cell model was made

with perfect insulation. This showed that for the range of soil thermal conductivities, the lumped

capacitance method would give a significant error in the calculated thermal conductivity. Only for

thermal conductivities above 15 Wm−1K−1 did the error fall below 10%. It was noted that heat

losses would actually cause the temperature difference across the specimen during recovery to

decrease.

This research showed that neither the UoS or Clarke thermal cells gave accurate results. The heat

losses vary depending on the type of soil, and the lumped capacitance method is seldom applicable.

91

If the power could be measured directly, eliminating the need for the lumped capacitance method,

and heat losses were be drastically reduced, a more accurate variation on the thermal cell could be

developed. However, the current setup potentially leads to significant errors in thermal conductivity

determination. A modified version of the thermal cell has since been explored by Alrtimi et al.

(2013), which attempts to minimise heat losses by having the heating platen sandwiched between

two identical specimens (similar to the double specimen guarded hot plate described in Section

2.6.4), and insulating it radially with a temperature-controlled thermal jacket. However, even if heat

losses were eliminated, it is probable that any steady state method testing a wet soil will experience

some amount of moisture migration error, due to the long heating times required.

92

Chapter 5

Comparing laboratory methods with theThermal Response Test

An opportunity to compare the laboratory tests to a field TRT presented itself at a central London

development site. The TRT was done on a test pile constructed by Concept Consultants Limited, as

part of a site investigation to determine the geotechnical and thermal properties of the ground and

hence evaluate the ground source energy potential of the site using energy foundations. The pile was

0.3 m in diameter and 26 m deep. It was constructed by reaming out the site investigation borehole

to a diameter of 0.3 m. The soil description is very stiff fissured dark brown CLAY (London Clay).

Results from this research have been published in Acta Geotechnica (Low et al., 2015) and in the

Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering

(Low et al., 2013) (Appendices F and G). It is worth noting that the results in these papers were

made prior to the development of the second order polynomial fit method for analysing the needle

probe data. The results will therefore be slightly different from those presented in this chapter,

although the conclusions remain unchanged.

5.1 Method

5.1.1 Laboratory tests

Six U100 samples were taken from the pile bore during the site investigation. Each sample is stored

in a metal tube which is sealed with wax at both ends to prevent the samples from drying out. These

were tested several months later, using the needle probe and thermal cell methods. Before any

measurements were taken, the samples were left in a temperature controlled room overnight to

equilibrate. The samples were extruded from the tubes before testing. Each sample was treated as

follows.

93

To accommodate the needle probe, a 200 mm length specimen was prepared and secured in a rubber

membrane. The specimen was taken from the middle of the U100 sample as the ends are more likely

to have experienced drying. Shavings taken from the top of the specimen were used to determine the

initial moisture content at the top. The soil was too hard to directly insert the probe. Therefore, a 5

mm diameter hole had to be pre-drilled, and the hole filled with a high thermal conductivity contact

fluid (toothpaste as suggested by the manufacturer Hukseflux Thermal Sensors (2003)) to reduce the

contact resistance between the probe and the soil. The probe was inserted into the hole, and secured

with a clamp stand. It was left for 20 min to equilibrate with the soil. A constant power was supplied

to the needle probe heater for 300–600 s, and then switched off. The heating time had to be

increased from 300 s if the results showed a long initial period and hence had yet to display a linear

relationship. The temperatures during the heating and recovery periods were recorded. Using this

procedure, five measurements were taken over the cross-sectional area of the specimen. One

measurement was taken at the centre of the cross-section, and the other four were equally spaced at a

radial distance of 25 mm from the centre.

To reduce the time it takes to reach steady state in the thermal cell, the specimen was then cut in half

and the top 100 mm weighed and secured to the platen of the thermal cell (see Figure 4.2). The

specimen was sealed at the top using aluminium foil to prevent moisture from leaving the top of the

sample. Shavings taken from the bottom of the top half were used to determine the initial moisture

content at the bottom. Insulation was wrapped around the specimen. The temperature difference

across the specimen was measured by two thermistors, one secured to the top of the platen and the

other embedded at the top of the soil. The cartridge heater was turned on, and the power controlled

so that the platen remained at a constant temperature of 40C. Temperatures were monitored until

steady state was reached and then maintained for at least 2 h. The power to the cartridge heater was

switched off, and the recovery period monitored. At the end of the test, shavings were taken from the

top, middle and bottom of the specimen to determine the final moisture contents.

The holes drilled into the specimen, and the contact fluid could potentially affect the thermal

conductivity measurement using the thermal cell. To verify the result, the bottom half of the sample

was also tested in the thermal cell, where these effects would be less significant. This is because the

hole was 150 mm deep, which would go through the length of the top 100 mm specimen, but only

through 50 mm of the bottom 100 mm specimen. Following testing, the specimens were cut up to

confirm that the contact fluid had remained inside the drilled holes and did not seep into the

surrounding soil.

A full soil classification was then conducted based on the British Standard 1377 (British Standards

Institution, 1990), to determine the soil density, moisture content, liquid limit, plastic limit, particle

density, and particle size distribution. Details and results of the soil classifications are given in

Appendix H.

To determine the thermal conductivity, the test data was analysed using the methods described in

Chapters 3 and 4.

94

5.1.2 Thermal Response Test

The TRT was conducted 10 days after grouting the pile. The test was carried out by GECCO2 Ltd

using their test rig. Water was used as the circulating fluid. The fluid flow rate and temperature were

recorded at 5 min intervals, using an electromagnetic flow meter and iron-constantan (J type)

thermocouples, respectively. After an initial circulation phase lasting 4.5 days, a 3 day heat injection

test was performed, followed by a 3 day recovery period. The next stage was a 3 day heat extraction

test, followed by a 4 day recovery period. The average power supplied to the heat exchanger was 2.2

and -2.1 kW during the heat injection and heat extraction phases respectively. Cyclic testing was

then commenced comprising two heat injection phases separated by heat extraction phases. Here,

only the results from the first heat injection and heat extraction phases are compared to the

laboratory tests, as these are considered to be the most reliable. The thermal conductivity was

calculated using the procedure described in Section 2.7, assuming α = 1.16×10−6m2s−1. Details of

the TRT analysis are given in Loveridge et al. (2014).

5.2 Results and Discussion

The results are summarised in Table 5.1. The thermal cell error stated in the table is calculated using

propagation of errors as detailed in Section 4.2.4, and does not include the contribution from heat

losses. The heat losses are analysed in detail in Section 4.3.2.4. The needle probe results are an

average of the five measurements for each sample. The full range of results is represented in Figure

5.1. Figure 5.2 shows the variation in density and moisture content with depth. There is a general

trend for a decrease in the moisture content of the samples with depth, which would be typical for

London Clay. However, this is not reflected in the thermal conductivity values which show no

significant variation with depth. The exception is the sample from 19.00 to 19.45 m depth which has

a lower thermal conductivity despite having a high moisture content, perhaps reflecting the lower

density of this sample. In general, the results show reasonable correlation between density and

thermal conductivity, while variations in moisture content have less of an effect.

5.2.1 Needle probe

The measured thermal conductivity ranges from 1.03 to 1.52 Wm−1K−1for heating and 0.89 to 1.42

Wm−1K−1for recovery. The variation in the five needle probe readings within the same sample was

on average about ±12% for both heating and recovery, but this varied significantly. The sample at

depths 19.00-19.45 m and 21.50-21.95 m had less variation. The needle probe repeatability was

previously determined using agar gel to be ±0.7%, so most of the differences in results should be

due to natural variability in thermal conductivity over the cross-section of the soil. Previous research

on measuring the thermal conductivity of London Clay using the needle probe gave a range of

values. Midttømme et al. (1998) determined values of 0.84 and 0.94 Wm−1K−1, for the thermal

95

Table 5.1: Summary of laboratory test results. (Thermal cell results given here do not accountfor heat losses.)

Depth (m)

Thermal conductivity (Wm−1K−1)

Needle Probe Thermal Cell

Heating Recovery Top Bottom

2.00–2.45 1.25±6% 1.30±10% 1.86±4% 1.72±4%

8.00–8.45 1.45±9% 1.19±6% 2.01±5% 1.88±5%

10.00–10.45 1.25±9% 1.06±17% 1.85±5% 1.91±5%

17.00–17.45 1.32±20% 1.32±11% 1.92±4% 1.88±4%

19.00–19.45 1.03±3% 0.89±6% 1.65±4% 1.75±4%

21.50–21.95 1.52±3% 1.42±0.9% 2.19±4% 1.84±4%

Figure 5.1: Thermal conductivity with depth. For the needle probe results, on each box, thecentral mark is the median, the edges of the box are the 25th and 75th percentiles,the whiskers extend to the most extreme data points not considered outliers, andoutliers are plotted individually.

96

Figure 5.2: Density and moisture content with depth.

conductivity perpendicular and parallel to the soil layering respectively. Bloomer (1981) gives a

thermal conductivity range of 2.45±0.07 Wm−1K−1. London Clay can exhibit a variable coarse

grain content, as well as moisture content and density (Pantelidou and Simpson, 2007). In addition,

moisture content variation can be introduced during the sampling process (see Section 5.2.4).

A factor which may affect the results is the use of contact fluid. In theory, the contact fluid should

only decrease the time it takes to reach the straight line section of the graph, i.e. it should have no

effect on the calculated thermal conductivity. However, the fluid could potentially seep into cracks in

the soil, and in doing so alter the thermal conductivity. After testing, the specimens were cut up to

see if this was the case. The soil at depths of 8.00-8.45m and 10.00-10.45m did not contain many

fissures, and the contact fluid seemed to have stayed within the drilled holes. It can therefore be

assumed that the contact fluid had little effect on the needle probe results. However, for the sample

at depth 19.00-19.45m there were a significant number of fissures, which contact fluid had seeped

into. This could affect both needle probe and thermal cell measurements, giving higher thermal

conductivity results than otherwise.

5.2.2 Thermal cell

The difference in thermal conductivity values between the top and bottom sections was between 2

and 17%. If the holes for the needle probe were to have a significant effect on the thermal

97

conductivity values, the measurement for the top section would be expected to always be higher than

for the bottom section, or vice versa. This is not the case, and as the area of the holes was only

1.25% of the total cross-sectional area, it can be assumed that the differences between the top and

bottom sections were mainly due to the soil’s natural variability.

Moisture contents were taken before an after the thermal cell test, and a typical distribution through

a specimen is shown in Figure 5.3. The moisture content at the top of the specimen after the test was

consistently higher than before the test, as shown in Figure 5.4. The greatest increase in moisture

content was 5.2%. This shows that over the long heating period, moisture migration occurs in the

direction of heat flow.

5.2.3 Thermal response test

The average fluid temperature during the TRT is shown in Figure 5.5. Temperature versus

logarithmic time graphs are also plotted in Figure 5.6, showing the points at which t = 5r2b/α (see

Equation 2.66). The TRT gave thermal conductivities of 2.5 and 2.7 Wm−1K−1for heating and

recovery phases respectively (Loveridge et al., 2014). As these results are higher than the laboratory

test results reported in Table 5.1, it is worth considering the accuracy of the in situ test. Various

sources of uncertainty affect TRTs, those relevant for this test will include variability in the applied

power, the larger diameter and relatively short length of the heat exchanger and any variability in the

initial undisturbed temperature condition. Nevertheless, studies of errors in TRTs suggest that

well-conducted tests should be accurate to within 10% (Signorelli et al., 2007; Witte, 2013; Javed

and Fahlen, 2011). However, the error may be a little larger in this case as the pile is of greater

diameter than usually recommended (Austin III, 1998). The Ground Source Heat Pump Association

(2012) recommends that at least 21 days lapse between pouring the pile concrete and starting a TRT,

unless ground temperature conditions are well understood. The TRT was conducted after only 10

days, which could have the effect of raising the average ground temperature and result in a higher

thermal conductivity measurement.

5.2.4 Comparison of methods

The measured thermal conductivity obtained using the thermal cell is consistently higher than that

using the needle probe by around 40 to 50%. This could be explained by a number of factors. In the

thermal cell calculations, the total power is used and any losses neglected. However, in Chapter 4 the

heat losses were shown to be significant using finite element analysis. Ideally these should be taken

into account. This is difficult to do experimentally, although some attempts have been made (Alrtimi

et al., 2013). There have been suggestions that heat losses for this type of cell could be in excess of

20% (Hemmingway, P. 2013 pers. comm.), and if this were the case it could explain much of the

variation between the thermal cell and the needle probe. The COMSOL model in Chapter 4 showed

the heat losses to be at least 30%. Consequently, heat losses are most likely the greatest source of

98

24 25 26 27 28 290

20

40

60

80

100

120

Moisture content (%)

Dis

tanc

e fr

om p

late

n (m

m)

BeforeAfter

Figure 5.3: Moisture content distribution with specimen depth before and after the thermalcell test. For depth 2.00–2.45 m, top half.

0 5 10 15 20 25 30 35

Moisture content (%)

Before

After

Top

Top

2.00-2.45

Top

Top

Top

Top

Bottom

Bottom

Bottom

Bottom

Bottom

Bottom

8.00-8.45

10.00-10.45

17.00-17.45

19.00-19.45

21.50-21.95

De

pth

(m

)

Figure 5.4: Moisture content at the top of the soil specimen before and after each thermalcell test. ‘Top’ and ‘Bottom’ refer to the top half and bottom half of the sample,respectively, as the sample at each depth was cut in half for the thermal cell tests.

99

Figure 5.5: Average fluid temperature during the TRT heat injection and heat extractionphases.

Figure 5.6: Average fluid temperature against logarithmic time during the TRT (a) heat in-jection and (b) heat extraction phases. The vertical lines show the points at whicht = 5r2

b/α .

100

error in the thermal cell calculations.

Other factors could also be contributing to the difference in results. The needle probe and thermal

cell measure the thermal conductivity in the radial and axial directions respectively. It could be that

the soil is anisotropic, and naturally has a higher thermal conductivity in the axial direction.

However, the layers in the soil sample were horizontal i.e. perpendicular to the cylinder axis. The

thermal conductivity measured parallel to layering is in general found to be higher than that

measured perpendicular to the layering (Midttømme and Roaldset, 1998). If anisotropy was the

reason behind the difference between needle probe and thermal cell values, then the needle probe

would be expected to give higher values of thermal conductivity than the thermal cell. Anisotropy

can be investigated by taking larger block samples and trimming specimens to the required sizes in

both orientations. However, such large high quality samples were not available in this investigation.

In any case, it is unlikely that anisotropy is the reason behind these differences.

The thermal cell test follows the needle probe tests, in which contact fluid was used to fill the holes.

The contact fluid could have potentially been aiding heat transfer in the thermal cell test. However,

this should not be the main reason for higher thermal conductivity values, as the volume of contact

fluid is comparatively small (at most only 1.25% of the sample volume).

As previously mentioned, moisture migration occurs in the thermal cell owing to the large

temperature gradient applied. As an additional mechanism for heat transfer, this may lead to higher

measured values of thermal conductivity (Farouki, 1986). With the needle probe method, moisture

migration should be insignificant as the power applied (and hence the temperature gradient) and the

heating time are much smaller.

In summary, the main reasons why the measured thermal conductivity from the thermal cell is higher

than that of the needle probe is that heat losses in the thermal cell have not been accounted for, and

that there was moisture migration during the thermal cell test.

Both laboratory methods gave significantly lower values of thermal conductivity than the TRT. This

is consistent with research by Zhang et al. (2014), but another comparison by Brettmann and Amis

(2011) resulted in similar needle probe and TRT results (see Section 2.7.4). The TRT thermal

conductivity value was about twice the needle probe value, and 40% higher than the thermal cell

value. One possible reason is that after the soil samples are taken, the soil no longer experiences the

same stresses as when it was in the ground; the laboratory tests were undertaken without any

confining pressure. This could give a looser soil with diminished contact between particles

(Abu-Hamdeh and Reeder, 2000). Results from oedometer tests on London Clay samples taken from

similar depths have previously been documented by Gasparre (2005), and indicate an increase in

void ratio of approximately 0.15 after sampling. The effect this has on the thermal conductivity can

be estimated from the De Vries equations for calculating the thermal conductivity of soils based on

their constituents (Farouki, 1986). The calculated thermal conductivity was 1.37 and 1.20

Wm−1K−1 for the in situ and sample soil respectively, which is a 12% decrease (see Appendix I for

full calculation). This cannot entirely explain the difference between TRT and laboratory results, but

101

could be a contributing factor.

The process of taking samples also causes disturbance and it has been observed that U100 type

samples in over consolidated clay will have a reduced moisture content in the middle of the sample

compared with the circumference (Graham, 2006; Vaughan et al., 1993). This could mean that the

needle probe is testing drier soil (expected to have lower thermal conductivity on average) than the

thermal cell. The samples were also tested some months after being taken from site, and despite

being contained in a metal tube and sealed with wax on both ends, there could still have been some

drying of the sample before testing, particularly as it was observed that the wax became brittle and

pulled away from the tube edges over time.

Another issue is differences in scale. The laboratory tests are carried out on samples that are much

smaller than the volume of soil tested in a TRT. This in itself would cause differences in results, as

the TRT would take into account large scale soil layering. Properties such as moisture content and

density vary with depth, so whereas a localised change in soil property at a depth at which a sample

is taken would significantly effect the sample thermal conductivity, it would have a much smaller

effect on the TRT which mirrors the averaged property over the length of the pile. There could also

be localised laminations affecting the samples more than the TRT. The pile is located within the

units near the base of the London Clay, which are known to exhibit greater grain size and

mineralogical variations than other parts of the formation (Pantelidou and Simpson, 2007). TRTs

could potentially overestimate the thermal conductivity, as the power used in calculations neglects

any heat lost between the measurement rig and the ground. The other issue which would lead to an

overestimate is if the ground temperature was elevated due to the curing process of the grout after it

has been poured. As the TRT was conducted 10 days after the grout was poured instead of the

recommended 21 days (Ground Source Heat Pump Association, 2012), the temperatures during the

TRT could be higher, leading to a higher measurement of the thermal conductivity.

5.3 Conclusions

The needle probe and thermal cell methods were compared. The needle probe took less time to

conduct and the soil was only heated slightly and for a short period, which means the effect of

moisture migration on the results should be minimal. However, hard soil samples may require

pre-drilling and back-filling with a contact fluid, which may increase the contact resistance. The

thermal cell tested a larger soil sample, but raised accuracy issues regarding heat losses. The long

heating time also meant that moisture would migrate towards the top of the specimen. The thermal

cell gave higher thermal conductivity values than the needle probe, which was mainly due to the

significant heat losses investigated in Section 4.3.2.4. As a consequence of these errors, the needle

probe is the preferred laboratory method.

The laboratory test methods gave consistently lower values of thermal conductivity than the TRT.

Possible reasons for this are the loss in confining pressure after the sample is taken, sample

102

disturbance including drying during the sampling process, further drying of the sample after

extraction, and the difference in the volume of soil tested. Some of these effects could be eliminated

by only using high quality truly undisturbed samples for laboratory testing, and this is recommended

wherever possible. While overall the TRT appears to give a better measurement of the in situ thermal

conductivity, it is a more expensive and time consuming approach and does include other sources of

error which require a better understanding.

103

Chapter 6

The effect of soil thermal conductivity onenergy foundation design

The fundamental aim of this research is to improve the way in which ground source heat pump and

energy foundations systems are designed. In previous chapters, the soil thermal conductivity was

selected as an area of interest, and measurement methods were compared. For the case study in

Chapter 5, each method gave a slightly different value for soil thermal conductivity. One aspect

which has yet to be quantified is how differences in soil thermal conductivity will affect the

performance of an energy foundations system i.e. the energy and financial impact that an incorrect

estimate of the ground thermal characteristics could have.

To answer this question, an accurate model of an energy foundation system was required. Pahud

(2007) developed PILESIM2, a simulation tool for heating/cooling systems with pile or bore heat

exchangers. Simulations were run for three different thermal loading scenarios, and the ground

thermal conductivity was varied to determine what effect this would have on the performance of an

energy foundation system.

6.1 PILESIM2

PILESIM2 is a fairly comprehensive simulation tool. It models how the heat pump operates, heat

transfer between the horizontal connecting sections of pipework and the surroundings, and the

variation in pipe fluid temperatures over time. The system boundary limits to the PILESIM2

simulations are shown in Figure 6.1.

The fluid temperatures are determined by the thermal response of the piles and the ground to the

applied loads. To simplify the problem, PILESIM2 makes a number of assumptions. The main

assumptions are:

105

Figure 6.1: Schematic view of an energy pile system. PILESIM2 simulates the section withinthe dashed line (Pahud, 2007).

• The number of piles is relatively large.

• The piles are evenly spaced in an approximate circle or square formation.

• Each pile has about the same active length.

PILESIM2 uses the transient system simulation program TRNSYS to model the ground by

implementing the ‘duct storage model’ (DST) for ground energy storage systems (Hellstrom, 1989;

Klein, 1998). The model assumes a large number of vertical heat exchangers (ducts), which when

packed close together act as one cylindrical energy store. Local and global problems are considered

separately and their results superimposed to give the overall ground thermal response.

The local problem looks at the thermal response for the area immediately surrounding each duct.

The solution is based on short duration heat pulses from an infinite line heat source, with

temperatures calculated using a 1D finite difference model (Adam, 2010). This is valid up until the

time when the thermal response reaches the midpoint between ducts. After this, the DST model

assumes steady state has been reached within the store, with further heat input equally affecting the

entire store temperatures. The global problem uses a 2D finite difference model to simulate the

thermal response between the store as a whole and the surrounding ground. Field data has been used

to validate the DST and there is general agreement to around 1C (Hellstrom, 1983).

There are four different modes of operation that can be simulated:

• Heating onlySome or all of the heating demand is covered by the pile heat exchangers.

• Heating and GeocoolingSome or all of the heating demand is covered by the pile heat exchangers, and some or all of

the cooling demand is covered by geocooling. Geocooling, also known as direct cooling, is

where the pile loop is used directly for cooling without the use of a heat pump.

106

• Heating and Cooling with Geocooling or a Cooling MachineSome or all of the heating demand is covered by the pile heat exchangers. The cooling

demand is covered by geocooling where possible, and when the demand exceeds geocooling

capacity then the cooling machine (a heat pump working in reverse) is used.

• Heating and Cooling with a Cooling MachineSome or all of the heating demand is covered by the pile heat exchangers. Some or all of the

cooling demand is covered by the cooling machine, with no geocooling.

To make the most of the energy foundation system in supplying the building demands, ‘heating and

cooling with geocooling or a cooling machine’ was the selected mode.

6.2 Design Input Parameters

The input parameters and energy demand profiles were adapted from Adam (2010), who estimated

the requirements for a 19 storey building comprising mainly office space, with 70 pile heat

exchangers and an energy base slab. The total yearly heating and cooling demands were estimated as

1508.5 and 1536.0 MWh/year respectively, which is a fairly balanced system i.e. heating and cooling

demands are almost equal. The heating demand also covers the year-round provision of hot water.

PILESIM2 requires an input file containing information on the hourly heating and cooling demand.

Monthly heating and cooling demands were estimated, with hot water included in the heating

demand. These are shown in Figures 6.2 and 6.3. PILESIM2 is not capable of modelling this

separately, so as a consequence the hot water in the simulation only reaches the temperature as

required for space heating. An additional heat source would be required to further raise the hot water

temperature. To get an hourly demand, normalised daily demand profiles were estimated, with

different profiles for Winter (November to February inclusive), Summer (June to August inclusive),

and Mid-season (remaining months). These are shown in Figures 6.4, 6.5 and 6.6.

Temperatures for the fluid and outdoor air temperatures are also required, as shown in Figure 6.7.

The temperature of the fluid for heating has been set as a reflection of the outdoor air temperature

with a minimum temperature of 20 C, and the fluid for cooling has been set at a constant 20 C.

Other parameter inputs defining the ground, pile heat exchangers, ground-building interface, heat

pump, cooling machine, and the loading conditions for heating and cooling are also required. Table

6.1 summarises the key parameters, with the full list of parameters given in Appendix J.

107

Figure 6.2: Monthly heating demand for PILESIM2 model, showing space heating and hotwater contributions.

Figure 6.3: Monthly cooling demand for PILESIM2 model.

Table 6.1: Key parameters for PILESIM2 model.

Average pile length 42 m

Average pile diameter 1.3 m

Distance between piles 5.1 m

Heat pump power 105 kW

Ground thermal conductivity 1.5 Wm−1K−1

Ground volumetric heat capacity 2.2 MJm−3K−1

Initial ground temperature 12 C

Room temperature 25 C

108

Figure 6.4: Winter normalised hourly power demand for PILESIM2 model.

Figure 6.5: Summer normalised hourly power demand for PILESIM2 model.

109

Figure 6.6: Mid-season normalised hourly power demand for PILESIM2 model.

Figure 6.7: Outdoor air and fluid temperatures for PILESIM2 model.

110

6.3 Outputs

After a simulation is run, PILESIM2 returns output files giving detailed information on temperatures

and energy fluxes over the number of years simulated. Figure 6.8 is a diagram showing the energy

fluxes for the base case. The base case has both heating and cooling demands and a ground thermal

conductivity of 1.5 Wm−1K−1. The load covered by renewable energy has been defined as:

Heating load covered by renewable energy = Heat extracted from the ground+Cooling for heating

(6.1)

Cooling load covered by renewable energy = Cooling for heating

+Heat injected to the ground through direct cooling

+Heat injected to the ground using cooling machine

(6.2)

The ‘cooling for heating’ term counts towards both heating and cooling cover as it is the energy

extracted from parts of the building which require cooling and used to heat other parts which

simultaneously require heating.

The results of most interest are the useful energy extracted or injected to the ground i.e. excluding

waste heat. PILESIM2 gives the following outputs:

• total energy extracted from the piles per year (QHextGrnd)

• energy injected to the piles via direct cooling per year (QFreeCool)

• energy injected to the piles via cooling machine per year (QCoolMach)

The total energy injected is a sum of the direct cooling and cooling machine components. These

outputs are more usefully expressed as the rate of useful energy extracted/injected per unit length of

pile, in Wm−1, by dividing the useful energy by the number of hours in a year and the total length of

the piles.

6.4 Simulations

Three different energy demand scenarios were considered:

111

Figure 6.8: Diagram of energy fluxes for the PILESIM2 model of the base case. The redand blue boxes show the share of the heating and cooling load respectively thatis covered by renewable energy.

• space heating, hot water and space cooling provided

• only space heating and hot water provided

• only space cooling provided

The first scenario is a fairly balanced system i.e. heating and cooling demands are similar, whereas

the other two scenarios are imbalanced as heat is only injected or extracted from the ground. To

determine how sensitive each energy demand scenario is to changes in ground thermal conductivity,

simulations were run with the ground thermal conductivity varied between 1.5 and 6

Wm−1K−1while all other parameters stayed the same (lower thermal conductivities could not be

simulated due to PILESIM2 limitations). This was repeated for each energy demand scenario.

6.5 Results and Discussion

Figures 6.9 and 6.10 show the affect of changing the ground thermal conductivity on the useful

energy extracted from or injected to the ground, for the three different energy demand scenarios. For

the system with both heating and cooling demands, increasing the soil thermal conductivity had a

minimal effect on the fraction of the heating demand that can be covered by the energy foundations.

Between a thermal conductivity of 1.5 and 6 Wm−1K−1there was an increase in extracted energy of

1%, and an increase in injected energy of 7%. Changing the soil thermal conductivity had a much

greater effect on the imbalanced systems with an increase in extracted energy of 40% for the heating

only case, and an increase in injected energy of 93% for the cooling only case. The significance of

112

increasing the ground thermal conductivity was greater at lower thermal conductivities than at higher

thermal conductivities, as can be seen by the gradients of the graphs in Figure 6.10.

These results are compelling, as it suggests that for a balanced system it is much less important to

accurately determine the ground thermal conductivity. If this is the case, balanced systems may not

require a costly TRT, particularly for smaller scale systems. A quick and simple laboratory test such

as the needle probe test could be enough for design purposes, particularly for higher thermal

conductivity soils. However, this relies on an accurate estimate of what the building demands would

be and that these do not become imbalanced over the years due to changes in weather conditions or

building usage, for instance.

For a heating or cooling dominated system the soil thermal conductivity has a greater effect on the

share of the demand that can be covered by renewable energy. This implies that such a system would

require a more accurate determination of the average soil thermal conductivity for the correct sizing

of the system. Perhaps this is an argument for balancing the heating and cooling loads as much as

possible. For example, most buildings that do not require space heating (such as in a tropical climate)

would still require hot water which could be provided by the ‘cooling for heating’ mechanism. To

provide the same amount of cooling, less heat injection into the ground would be required.

Over the 20 year simulation period, Figures 6.11, 6.12 and 6.13 show the change in the amount of

thermal energy extracted from/injected to the ground, for the two extremes of soil thermal

conductivity. Values do not include the exchange of thermal energy between the ground and the

sections of horizontal piping i.e. energy base slab, which is only a small contribution. The graphs

show that the system with both heating and cooling loads exchange roughly the same amount of

thermal energy with the ground each year. This means that the average ground temperature should

not experience significant changes, and the performance of the energy foundations system will

remain stable over time. For the heating only and cooling only systems, the amount of thermal

energy that can be extracted/injected rapidly decreases over the first few years of operation and then

levels off. The value at which the graphs level off is dependent on the soil thermal conductivity, with

the higher thermal conductivity allowing a greater amount of thermal energy to be transferred

between the ground and the building.

This supports the previous observation that the soil thermal conductivity plays a much more

important role in the design of an imbalanced system. Fewer energy piles would be required to cover

the same share of the building load for a soil with higher thermal conductivity, as shown in Figures

6.12 and 6.13. In the design of an imbalanced system, it is important to analyse the performance

over time. For the low thermal conductivity case of Figure 6.12, the energy foundation system starts

out in year one delivering 39% of the heating load, but by year five this has already decreased to only

12%. This implies that for a viable long-term energy solution, the energy foundations system should

be sized according to the required life-span.

Rules of thumb have previously been suggested for the design of pile or bore heat exchangers, for

estimating the rate of energy extraction/injection. It is worth putting the PILESIM2 results within

113

Figure 6.9: Rate of useful energy exchanged with the pile heat exchangers, per unit length ofpile, with varying soil thermal conductivity, for a system with both heating andcooling loads.

Figure 6.10: Rate of useful energy exchanged with the pile heat exchangers, per unit lengthof pile, with varying soil thermal conductivity, for a system with only (a) heatingloads and (b) cooling loads.

114

Figure 6.11: Thermal energy extracted/injected per year over a period of 20 years for a sys-tem with both heating and cooling loads, for high and low soil thermal conduc-tivities. Injected thermal energies are plotted as negative values.

Figure 6.12: Thermal energy extracted per year over a period of 20 years for a system withonly heating loads, for high and low soil thermal conductivities.

115

Figure 6.13: Thermal energy injected per year over a period of 20 years for a system withonly cooling loads, for high and low soil thermal conductivities.

the context of previous research. For moist clay, British Standards Institution (2007) gives a specific

heat extraction rate of 30 to 40 Wm−1 for an operation period 100 days, which is greater than the

simulation results. However, this is for vertical bore heat exchangers with a heat pump power up to

30 kW, which may not be applicable to the PILESIM2 model. For piles with a diameter ≥0.6 m,

Brandl (2006) assumes a heat extraction rate of 35 W per m−1 earth contact area. Given the

PILESIM2 piles have an average diameter of 1.3 m, this would give 8.57 Wm−1, which is

significantly less than the simulation results.

Pahud and Hubbuch (2007) used measurements from the energy pile system at Zurich Airport to

inform the development of the PILESIM2 programme. The piles were 30 m in length and 0.9 to 1.5

m in diameter, and gave heat transfer rates of 45 and 16 Wm−1 for extraction and injection

respectively. The rate of heat extraction is greater than that of the PILESIM2 model, but the rate of

heat injection is similar. The range in quoted values for rate of energy extraction/injection show that

there still remain many unknowns on the subject of energy foundation design.

6.5.1 Effect of different soil thermal conductivity measurement methods

As shown in Chapter 5, the three different test methods led to different values for the soil thermal

conductivity. The average measured thermal conductivities were 1.34, 1.87 and 2.6 Wm−1K−1, for

the needle probe, thermal cell, and TRT methods respectively. The PILESIM2 simulation results in

Figure 6.10 were used (with a slight extrapolation at the lower end) to estimate how much energy

could be provided by the piles for different soil thermal conductivities. The results are shown in

116

Figure 6.14. For the balanced system, the differences in measured soil thermal conductivity made

little difference to the rates of energy transfer. For a heating only system, the TRT value for soil

thermal conductivity would provide 8% (1 Wm−1) and 15% (2 Wm−1) more useful energy than the

value obtained using the thermal cell and needle probe respectively. Similarly, for a cooling only

system, the TRT value would allow 16% (1 Wm−1) and 33% (2 Wm−1) more useful energy to be

extracted than the value obtained using the thermal cell and needle probe respectively.

Using a measured soil thermal conductivity in the design stage that is lower than the actual soil

thermal conductivity could lead to a more costly system being built than is necessary. Conversely, if

the measured thermal conductivity is higher than the actual soil thermal conductivity, the system

could be under-designed. This highlights the importance of accurately determining the soil thermal

conductivity when designing GSHP and energy foundation systems with imbalanced thermal loads.

6.6 Conclusions

PILESIM2 was used to simulate three different load cases for an energy foundations system: both

heating and cooling, heating only and cooling only. For a fairly balanced system with both heating

and cooling loads, significant changes in the soil thermal conductivity have little effect on the

amount of renewable ground energy that can be utilised. For a system with only heating or cooling

loads, the soil thermal conductivity has a much greater effect, with more renewable ground energy

utilised with increasing thermal conductivity. The efficiency of the system also decreases over time

as heat is only being transferred in one direction, which means the temperature of the ground will

change over time. To summarise, the results imply that it is less important to accurately determine

the soil thermal conductivity for a balanced system than for an imbalanced system.

117

Figure 6.14: Estimated rate of useful energy extracted/injected for an energy foundation sys-tem simulated in PILESIM2, based on the measured soil thermal conductivityusing the needle probe, thermal cell and TRT methods.

118

Chapter 7

Conclusions and further research

7.1 Needle Probe

Tests were done on agar-kaolin and London Clay samples. The needle probe proved to be a simple

and quick method of soil thermal conductivity measurement. However, for hard soils and rock it

may require pre-drilled holes and a high thermal conductivity contact fluid, or guiding tubes.

Although the needle probe was found to have a repeatability of ±1.2%, the results were found to

vary more significantly when the heating time and heating power was varied.

Current analysis methods require visual inspection or rules of thumb, which could incorrectly select

which part of data to use for calculating the thermal conductivity. A new analysis method was

developed which minimises the potential for human error, and would also be able to eliminate any

data influenced by boundary effects. This ‘second order polynomial fit’ method was successfully

used to analyse needle probe tests on agar-kaolin and London Clay samples with thermal

conductivities in the range 0.6 to 1.6 Wm−1K−1, with a range of heating times and heating powers.

Shorter heating times gave better repeatability. Higher thermal conductivity samples gave better

repeatability using a higher heating power. The value of thermal conductivity calculated for the

heating phase was generally higher than for the recovery phase.

Further research should be done to determine whether the second order polynomial analysis method

is suitable for soils with a wider range of thermal conductivities. Several parameters such as the

length for the selected linear section require manual selection, and varying these parameters could

potentially lead to an inaccurate analysis. A sensitivity study could be done to determine the effects

of varying the parameters on the calculated thermal conductivity. Improvements could be made to

the programme such that the appropriate parameter values are automatically selected, or parameter

values that lie outside a given range are flagged up as unsuitable.

Selecting an appropriate heating time and heating power are difficult without some understanding of

what the soil thermal conductivity is likely to be. Time can be wasted in repeating tests, with each

119

test increasing the likelihood of introducing moisture migration errors. The water content of the soil

surrounding the probe could be measured before and after a test to determine whether significant

moisture migration occurs. It might be possible to perform a short initial test which gives a good

enough estimate of the thermal conductivity to select the appropriate heating time and heating power

for subsequent tests. Further research can be done to determine what the initial part of the test can

reveal about the nature of the soil.

The needle probe method is currently not recommended for use with coarse soils. This is due to poor

contact between the probe and larger grains. Further research should be done on the influence of

grain size on the measured thermal conductivity, and whether there is a method of improving the

contact between the probe and the soil by means of a thermal grease or otherwise.

Groundwater flow can greatly improve the heat transfer within a soil, increasing its effective thermal

conductivity. Research into studying this effect within a laboratory setting has been limited. Katsura

et al. (2006) has designed laboratory apparatus that allows the thermal conductivity to be measured

using a needle probe, for a soil with water flowing through it. The range of water flow velocities

tested is higher than what would be expected in groundwater flow, so further research at lower flow

rates would give the experiments more significance in a ground heat exchanger context.

7.1.1 Recommendations

To analyse the needle probe data, it is recommended to use a combination of the second order

polynomial fit method and visual inspection. Visual inspection is necessary to see whether the linear

section has been reached. In most cases where the second order polynomial fit method gives

erroneous results, this is due to the heating time being too short. If this is the case, a further test

should be performed with increased heating time.

If good contact between the probe and the soil can be established i.e. soft soils requiring no

pre-drilled holes, a heating time of 100 to 300 s is recommended. The heating power for soils with

an estimated thermal conductivity of <0.85 Wm−1K−1should be around 0.82 to 2.43 Wm−1, and

around 4.13 Wm−1 for soils with an estimated thermal conductivity >0.85 Wm−1K−1. As most

soils have a higher thermal conductivity than 0.85 Wm−1K−1, the high power setting is

recommended in most cases. The thermal conductivity values calculated using the heating and

recovery phases should not differ by more than 10%. If the difference is greater than this, the value

for the heating phase should be used, provided that the heating data has a well-defined linear section.

7.2 Thermal Cell

The UoS thermal cell was loosely based on the thermal cell by Clarke et al. (2008), but had a thicker

acrylic base and thicker insulation to minimise heat losses. It was used to test London Clay samples.

120

The experimental method was simple, but took a long time to perform due to being a steady state

method.

To quantify and compare the heat losses, COMSOL models of the UoS thermal cell and the Clarke

thermal cell were made. Heat losses were greatest for soils of low thermal conductivity, but even for

higher thermal conductivities, at least 30% of the heat was lost through the base and insulation. This

was found to be even greater for the Clarke thermal cell, which conflicted with the results presented

in Clarke et al. (2008). Reducing the thickness of the soil specimen also reduced heat losses, but not

enough to give a good calculated thermal conductivity. The heat losses meant measuring the power

directly was unreliable and lead to overestimation of thermal conductivity. An alternative method to

determine the power involves using the lumped capacitance method to interpret the recovery curve.

However, the criterion for using the lumped capacitance method was also never satisfied. A model of

an ideal thermal cell showed that even when perfectly insulated, the lumped capacitance method

could not reliably be used.

Due to the unreliability of the method, the thermal cell is not a recommended method. A modified

version of the thermal cell has since been explored by Alrtimi et al. (2013), which further minimises

heat losses by having the platen sandwiched between two identical specimens (similar to the double

specimen guarded hot plate described in Section 2.6.4), and insulating it radially with a

temperature-controlled thermal jacket. However, even if heat losses were eliminated, any steady

state method testing a wet soil will experience some amount of moisture migration error, due to the

long heating times required. Moisture migration was prevented in the agar-kaolin samples due to the

agar solidifying the water. This could also be done for thermal cell samples, but the samples would

have to be reconstituted. The agar water also sets within a shorter time frame than it would take to

make a typical soil sample requiring consolidation, so this method would only work to produce

samples of high water content.

7.2.1 Recommendations

Heat losses and moisture migration are two sources of error that are difficult to eliminate from a

steady state method. It has the advantage over the needle probe of being able to test a larger volume

of soil, but this could be compensated for by taking more needle probe measurements. As the needle

probe method is much quicker to perform, this would be preferable to using a steady state method. It

is advisable that steady state methods be limited to use on dry soils and solid materials, and that the

method used minimises heat losses by means of a thermal barrier such as in the guarded hot plate.

7.3 Comparison to the TRT

The needle probe and thermal cell tests were compared to a TRT. The thermal cell results are

disregarded due to the unreliability. The TRT gave a value of thermal conductivity almost twice that

121

of the needle probe. Possible reasons for this include degradation during the sampling process such

as drying of the soil samples, removal of confining pressures, and sample disturbance causing

fissures. The TRT could also potentially overestimate the thermal conductivity, as the power used in

calculations neglects any heat lost between the measurement rig and the ground.

Modifications could be made to how samples are prepared in the laboratory to give a better

approximation to the in situ soil, such as applying a confining pressure whilst performing a needle

probe measurement. Other aspects such as groundwater flow could also be modelled in the

laboratory.

7.4 Energy foundations

The effect of ground thermal conductivity on an energy foundation system was investigated using a

PILESIM2 numerical model. For a balanced heating and cooling system, the thermal conductivity

did not significantly affect the performance of the system. However, for an imbalanced system with

only heating or only cooling loads, varying the thermal conductivity influenced the amount of heat

that could be transferred to and from the ground. The results imply that it is less important to

accurately determine the soil thermal conductivity for a balanced system than for an imbalanced

system.

122

Appendices

123

Appendix A

Datasheet for kaolin

125

SPECIFICATION

Brightness (ISO R457) 85.5 ± 1.0

+ 300 mesh (mass % max.) 0.02

+ 10 µm (mass % max.) 0.5

- 2 µm (mass %) 76 - 83

Moisture (mass % max.) 1.5

TYPICAL PROPERTIES

Yellowness 4.7

Specific gravity 2.6

pH 5.5

Surface area (BET; m2/g) 14

Oil absorption (g/100g) 42

Aerated powder density (kg/m3) 360

Tapped powder density (kg/m3) 620

Water soluble salt content (mass %) 0.20

Chemical analysis by X-ray fluorescence

SiO2 (mass %) 47

Al2O3 (mass %) 38

CAS No. 1332-58-7

TYPICAL PARTICLE SIZE DISTRIBUTION

SpeswhiteTM

Speswhite is a highly refined kaolin of ultrafine particle size and high brightness fromdeposits in the South West of England.

IMERYS PERFORMANCE &

FILTRATION MINERALSPar Moor Centre,Par Moor Road, ParCornwall, PL24 2SQ - UKTel: +44 1726 818000Fax: +44 1726 811200

Kaolin does not appear inEINECS as an individual entrybut is classified as “naturallyOccurring Substance” with theEINECS No. 310-127-6.

DAT002KMarch 2008 - Eighth Edition. This data sheet supersedes the datasheet dated January 2007.

The data quoted are determined by the

use of IMERYS Minerals Ltd Standard Test

Methods, copies of which will be

supplied on request. Every precaution is

taken to ensure the products conform to

our published data, but since the

products are based on naturally

occurring raw materials, we reserve the

right to change these data should it

become necessary. Sales are in

accordance with our ‘Conditions of

Sale’, copies of which will be supplied on

request.

ISO 9001FM 14752

100

80

60

40

20

0100 10 1 0.1

% f

iner

by

wei

ght

Equivalent spherical diameter µm

Speswhite

Appendix B

Matlab code

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− START −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−% This programme is used to analyse the needle probe data in

% order to calculate the thermal conductivity from the linear section of

% the graph.

%

% Last update: Jasmine Low, 03/11/2014

%

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Start with blank slate

clc

close all

clear all

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− FILL THIS IN −−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% File name

file = '.dat';

% Experiment number

ExpID = 1;

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−− VARIABLES −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% The variables here have been used to analyse soils with thermal

% conductivities in the range 0.6 to 1.6 W/mK. These values may not be

% suitable for all test situations and may require alterations.

127

cut sec = 2; % Section length for calculating the cut−offsec = 1; % Section length for finding the linear section

dln = 0.1; % Interval between starting log times

span = 5; % Smoothing span

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− IMPORT DATA −−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Import data

A = importdata(file,',',4);

Raw = A.data;

% Print experiment number

fprintf('\nExperiment %.0f:\n\n',ExpID)

% Extract data for this particular ExpID

data index = find(Raw(:,2) == ExpID);

data = Raw(data index(1):data index(end),4:9);

t heat = −data(1,1); % heating time

dt = 0.5; % time interval between readings

Re = Raw(data index(1),3);

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−− CALCULATE TEMPERATURES −−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Calculate log graph data

m = size(data,1); % index for end of test

n = find(data(:,1)==0); % index for time = 0 sec

h = n+t heat/dt; % index for end of heating time

time = 0:dt:data(end,1); % time vector

time heat = 0:dt:t heat; % time vector for heating

time cool = t heat+dt:dt:data(end,1); % time vector for recovery

ln time = log(time heat); % ln(t) for heating

ln time cool = log(time cool./(time cool−t heat));

% ln(t/(t−t heat) for recovery

Theat = data(n:m,6); % hot joint temperature

V = data(n:m,3); % voltage

C = mean(data(n+1:2*n−1,2)); % current

Q = Cˆ2*Re; % power

% Take into account temperature dependency of thermocouple sensitivity

Tdiff = zeros(size(time)); % change in temperature between readings

Tdiff sum = zeros(size(time)); % temperature difference from start heating

Theat calc = zeros(size(time)); % calculated hot joint temperature

Esen = zeros(size(time)); % sensitivity of thermocouple in V/K

Theat calc(1) = Theat(1);

Esen(1) = 39.455+4.8034e−2*Theat calc(1)−2.8925e−4*Theat calc(1)ˆ2 ...

+1.3823e−6*Theat calc(1)ˆ3−2.8357e−8*Theat calc(1)ˆ4 ...

+1.7839e−10*Theat calc(1)ˆ5−3.1836e−13*Theat calc(1)ˆ6;

128

for i = 2:length(time)

Tdiff(i) = (V(i)−V(i−1))*1000/Esen(i−1);Tdiff sum(i) = Tdiff sum(i−1)+Tdiff(i);Theat calc(i) = Theat calc(i−1)+Tdiff(i);Esen(i) = 39.455+4.8034e−2*Theat calc(i)−2.8925e−4*Theat calc(i)ˆ2 ...

+1.3823e−6*Theat calc(i)ˆ3−2.8357e−8*Theat calc(i)ˆ4 ...

+1.7839e−10*Theat calc(i)ˆ5−3.1836e−13*Theat calc(i)ˆ6;

end

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− PREPARE DATA −−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Use function "needle dataprep.m" to make data nice i.e. get rid of steps

nice = needle dataprep(time,Theat calc,t heat);

xy heat = nice.xy heat;

xy cool = nice.xy cool;

% Make data ready for log graphs

ln xy heat = [log(xy heat(:,1)) 4*pi*(xy heat(:,2)−xy heat(1,2))/Q];

ln xy heat = ln xy heat(2:end,:); % Delete first point which is "Inf"

ln xy cool = [log(xy cool(:,1)./(xy cool(:,1)−t heat)) ...

4*pi*(xy cool(:,2)−xy heat(1,2))/Q];

ln xy cool = sortrows(ln xy cool,1); % Sort ln xy cool in ascending order

% Smooth data

ln xy heat(:,2) = smooth(ln xy heat(:,2),span);

ln xy cool(:,2) = smooth(ln xy cool(:,2),span);

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− FIND CUT−OFF −−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Calculate straight line equation at beginning and end. Where these

% intersect is the cut−off.

% Vector for storing cut−offs for heating and recovery respectively

cut x = zeros(2,1);

for i = 1:2

if i == 1

% Heating

ln cut = ln xy heat;

else

% Recovery

ln cut = ln xy cool;

end

% Get points

range1 = find(ln cut(:,1) <= ln cut(1,1) + cut sec);

cut xy1 = ln cut(1:range1(end),:);

129

range2 = find(ln cut(:,1) >= ln cut(end,1) − cut sec);

cut xy2 = ln cut(range2(1):end,:);

% Calculate equation of lines

cut line1 = fit(cut xy1(:,1),cut xy1(:,2),'poly1');

cut line2 = fit(cut xy2(:,1),cut xy2(:,2),'poly1');

% Save for heating and recovery

if i == 1

cut line1h = cut line1;

cut line2h = cut line2;

else

cut line1c = cut line1;

cut line2c = cut line2;

end

% Calculate point of intersection

cut x(i) = (cut line2.p2 − cut line1.p2)/(cut line1.p1 − cut line2.p1);

end

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−− FIND LINEAR SECTION −−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Calculate vectors of starting logarithmic time

% Cut−offcut = cut x(1);

cut cool = cut x(2);

% Find limits

ln range = log(t heat)−cut;ln range cool = cut cool−log(data(end,1)/(data(end,1)−t heat));

lim sec = min([ln range ln range cool]);

% Heating − vector of starting ln(t)

ln tbegin heat = ceil(10*cut)/10:dln:log(t heat)−sec;% Recovery − vector of starting ln(t/(t−theat))ln tbegin cool = ceil(10*log(data(end,1)/(data(end,1)−t heat)))/ ...

10:dln:cut cool−sec;

% Loop for heating and recovery

for hc = 1:2

% Vector of starting logarithmic time

if hc == 1

ln tbegin = ln tbegin heat;

else

ln tbegin = ln tbegin cool;

end

% Vector for curve

curvy = zeros(2,length(ln tbegin));

curvy(2,:) = 1:length(curvy); % indices

130

for l = 1:length(ln tbegin)

% Pick out points in section

if hc == 1

ind = find(ln xy heat(:,1)>=ln tbegin(l) & ln xy heat(:,1)< ...

ln tbegin(l)+sec);

x = ln xy heat(ind(1):ind(end),1);

y = ln xy heat(ind(1):ind(end),2);

else

ind = find(ln xy cool(:,1)>=ln tbegin(l) & ln xy cool(:,1)< ...

ln tbegin(l)+sec);

x = ln xy cool(ind(1):ind(end),1);

y = ln xy cool(ind(1):ind(end),2);

end

% If there are not at least three points in section, move on to

% next section

if length(x)>2

% Fit curve

[curve,gof2] = fit(x,y,'poly2');

% Store curve coefficient of xˆ2

curvy(1,l) = curve.p1;

else

curvy(1,l) = NaN;

end

end

% Get rid of NaN columns

curvy2 = curvy(:,˜any(isnan(curvy),1));

% Find smallest curvy next to each other and this should be straight

[˜,Icurve] = sort(abs(curvy2(1,:)));

I = [];

i = 1;

a = 0;

while a == 0

i = i + 1;

diff = abs(Icurve(i)−Icurve(1:i));[˜,I] = find(diff == 1);

if length(I) >= 1

a = 1;

end

end

% Indices for the straight line sections

point1 = curvy2(2,Icurve(length(diff)));

131

point2 = curvy2(2,Icurve(I));

points = sort([point1 point2]);

% Straight line section range

line start = ln tbegin(points(1));

line end = ln tbegin(points(end)) + sec;

% Now calculate final straight line!

if hc == 1

ln xy = ln xy heat;

else

ln xy = ln xy cool;

end

ind = find(ln xy(:,1)>=line start & ln xy(:,1)<=line end);

xfinal = ln xy(ind(1):ind(end),1);

yfinal = ln xy(ind(1):ind(end),2);

% Fit straight line

[pfinal,gof] = fit(xfinal,yfinal,'poly1');

% Store results

if hc == 1

curvy heat = curvy;

xfinal heat = xfinal;

yfinal heat = yfinal;

pfinal heat = pfinal;

gof heat = gof;

points heat = points;

range heat = [line start line end];

else

curvy cool = curvy;

xfinal cool = xfinal;

yfinal cool = yfinal;

pfinal cool = pfinal;

gof cool = gof;

points cool = points;

range cool = [line start line end];

end

end

% Finally, calculate thermal conductivity for heating and recovery

lambda heat = 1/pfinal heat.p1;

lambda cool = 1/pfinal cool.p1;

lambda mean = mean([lambda heat lambda cool]);

lambda = [lambda heat lambda cool lambda mean];

fprintf('Thermal conductivity (heating) = %.3f W/mK \n',lambda heat)

fprintf('Thermal conductivity (recovery) = %.3f W/mK \n',lambda cool)

fprintf('Thermal conductivity = %.3f W/mK \n',lambda mean)

% Print other stuff

fprintf('Initial temperature = %.3f degC \n',Theat calc(1))

fprintf('Heating time = %.0f s \n',t heat)

132

fprintf('Heating power, Q = %.3f W/m \n',Q)

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−− PLOT GRAPHS −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Plot raw data

figure(1)

plot(nice.x,nice.y,'.',xy heat(:,1),xy heat(:,2),'ro', ...

xy cool(:,1),xy cool(:,2),'ro')

xlabel('Time (s)')

ylabel('Temperature (ˆoC)')

legend('Raw','Smoothed')

% Plot log data with cut−offsfigure(2)

% Heating

subplot(1,2,1), plot(ln xy heat(:,1),ln xy heat(:,2),'.')

hold on

subplot(1,2,1), plot([cut cut],[0 ln xy heat(end,2)],'r')

z = [ln xy heat(1,1) ln xy heat(end,1)];

m = cut line1h.p1;

c = cut line1h.p2;

subplot(1,2,1), plot(z,[m*z(1)+c m*z(2)+c],'r');

m = cut line2h.p1;

c = cut line2h.p2;


ylim([0 ln xy heat(end,2)])

title('Heating')

xlabel('ln(t)')

ylabel('4\pi\DeltaT/q')% Recovery

subplot(1,2,2), plot(ln xy cool(:,1),ln xy cool(:,2),'.')

hold on

subplot(1,2,2), plot([cut cool cut cool],[0 ln xy cool(end,2)],'r')

z = [ln xy cool(1,1) ln xy cool(end,1)];

m = cut line1c.p1;

c = cut line1c.p2;


m = cut line2c.p1;

c = cut line2c.p2;


ylim([0 ln xy cool(end,2)])

title('Recovery')

xlabel('ln(t/(t−t heat))')ylabel('4\pi\DeltaT/q')

% Final log graphs showing straight line section

figure(3)

% Heating

z = [ln xy heat(1,1) ln xy heat(end,1)];

m = pfinal heat.p1;

133

c = pfinal heat.p2;

subplot(1,2,1), plot(ln xy heat(:,1),ln xy heat(:,2),'.', ...

[xfinal heat(1) xfinal heat(1)], ...

[yfinal heat(1)−0.5 yfinal heat(1)+0.5],'r', ...

[xfinal heat(end) xfinal heat(end)], ...

[yfinal heat(end)−0.5 yfinal heat(end)+0.5],'r', ...

z,[m*z(1)+c m*z(2)+c],'r')

title('Heating')

xlabel('ln(t)')

ylabel('4\pi\DeltaT/q')% Recovery

z = [ln xy cool(1,1) ln xy cool(end,1)];

m = pfinal cool.p1;

c = pfinal cool.p2;

subplot(1,2,2), plot(ln xy cool(:,1),ln xy cool(:,2),'.', ...

[xfinal cool(1) xfinal cool(1)], ...

[yfinal cool(1)−0.5 yfinal cool(1)+0.5],'r', ...

[xfinal cool(end) xfinal cool(end)], ...

[yfinal cool(end)−0.5 yfinal cool(end)+0.5],'r', ...

z,[m*z(1)+c m*z(2)+c],'r')

title('Recovery')

xlabel('ln(t/(t−t heat))')ylabel('4\pi\DeltaT/q')

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− END −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− START −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−% Function prepares the raw needle probe data such that fluctuations and

% stepping due to instrument sensitivity are reduced.

%

% For each temperature value, all the points that have this value are

% represented by one point.

%

% Last update: Jasmine Low, 05/11/2014

%

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

function data = needle dataprep(x,y,t heat)

% x = time

% y = temperature

% t heat = heating time

% Sometimes the data can have tiny differences which cause havoc. Remove.

y = round(y*1e3)/1e3;

% Do different things for heating and recovery using for loop

for split = 1:2

134

if split == 1

% Select only heating data

xsplit = x(x<=t heat+1)';

ysplit = y(1:length(xsplit))';

else

% Select only recovery data

xsplit = x(x>t heat+1)';

ysplit = y(length(x)+1−length(xsplit):end)';end

% Make vector of just the unique temperature values

yuni = unique(ysplit);

xuni = zeros(size(yuni));

% Sort data

xy = [xsplit ysplit];

xy = sortrows(xy,2);

% For each value of yuni, find corresponding xuni value using mean

for i = 1:length(yuni)

range = find(xy(:,2)==yuni(i));

xuni(i) = mean(xy(range(1):range(end),1));

end

% Final chronological sort

xy = sortrows([xuni yuni],1);

% Store heating and recovery data separately

if split == 1

% Get rid of end point as may skew data

xy heat = xy(1:end−1,:);else

xy cool = xy(1:end−1,:);end

end

% Results

data.xy heat = xy heat;

data.xy cool = xy cool;

data.x = x;

data.y = y;

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− END −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−− START −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

135

% This programme is used to process the thermal cell data and plot a graph

% of temperature against time for the test.

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−


clc

close all

clear all

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− FILL THIS IN −−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

file = ''; % file name

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−− 1. Import data −−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Import data


Raw = A.data;

% Time between readings

date1 = datevec(A.textdata(5,1));


dt = date2(4)*60 + date2(5) − date1(4)*60 − date1(5); % minutes

% Ambient temperature

T amb = Raw(:,3);

% Temperatures from thermistor readings

a = 3.3540154e−3;b = 2.5627725e−4;c = 2.082921e−6;d = 7.3003206e−8;T3000 = Raw(:,4:end)/3000;

T = 1./(a+b*log(T3000)+c*(log(T3000)).ˆ2+d*(log(T3000)).ˆ3)−273.15;% Check limits

lim max = 3.274;

lim min = 0.36036;

if ((max(max(T3000))<lim max)&&(min(min(T3000))>lim min))

fprintf('Thermistors operating within bounds.')

else

fprintf('WARNING: Thermistors operating out of bounds.')

end

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− 2. Plot graph −−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Make time vector

136

time = 0:dt:dt*(size(T,1)−1);

% Plot graph

plot(time,T)

xlabel('Time (minutes)')

ylabel('Temperature (ˆoC)')

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− END −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− START −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−% Function to calculate the power from heater data for thermal cell test.

% Input the file name, duration for which power is to be calculated, and

% index of the row from which the calculation should begin.

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

function power = thermalcell heater(file,duration,index)

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−− 1. Import Data −−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Power rating for heater

rating = 50; %W

% Import data


pulse = A.data(:,2)/1000;

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−− 2. Get start time for pulses −−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

t start = zeros(size(pulse));

time = zeros(size(pulse));

for i = 1:length(pulse)

% Get time in seconds

date = datevec(A.textdata(i+5,1));

time(i) = date(4)*3600+date(5)*60+date(6);

% Starting time of pulse

t start(i) = time(i) − pulse(i);

end

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

137

%−−−−−−−−−−−−−−−−−−−−−−−−−−− 3. Get Power −−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% End time for calculation

t end = t start(index) + duration*60;

% Add up pulses

i = index;

total pulse = 0;

while t start(i)+pulse(i)<t end

total pulse = total pulse + pulse(i);

i = i + 1;

end

% If end pulse is cut

i = i + 1;

if ((t start(i)<t end)&&(t end<t start(i)+pulse(i)))

pulse bit = t end−t start(i);

total pulse = total pulse + pulse bit;

end

% Calculate the power

power = rating*total pulse/duration/60;

fprintf('Power = %f W\n',power)

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− 4. Plot graphs −−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Graph of pulse length vs start time

figure

plot(t start(index:end),pulse(index:end),'.')

ylabel('Pulse length (s)')

xlabel('Start time (s)')

% Graph of pulses over time

time onoff = zeros(4*length(t start(index:end)),1);

pulse onoff = time onoff;

a = 1;

for i = index:length(t start(index:end))+index−1time onoff(a)= t start(i);

time onoff(a+1) = t start(i);

time onoff(a+2) = time(i);

time onoff(a+3) = time(i);

pulse onoff(a+1) = rating;

pulse onoff(a+2) = rating;

a = a + 4;

end

figure

plot(time onoff,pulse onoff)

xlabel('Time (s)')

ylabel('Power (W)')

ylim([0 60])

138

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− END −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−− START −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−% This programme is used to calculate the thermal conductivity from the

% thermal cell data.

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−


clc

close all

clear all

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−− FILL THIS IN −−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% For thermal cell data

file = ''; % file name

duration = ; % minutes of time period to calculate for

index = ; % index of starting time

% For heater data

heatfile = ''; % file name

heatduration = ; % minutes of time period to calculate for

heatindex = ; % index of starting time

% Sample dimensions

D = ; % diameter (mm)

L = ; % length (mm)

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−− 1. Import data −−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

% Import data


Raw = A.data;

% Time between readings



dt = date2(4)*60 + date2(5) − date1(4)*60 − date1(5); % minutes

% Ambient temperature

T amb = Raw(:,3);

139

% Temperatures from thermistor readings

a = 3.3540154e−3;b = 2.5627725e−4;c = 2.082921e−6;d = 7.3003206e−8;T3000 = Raw(:,4:end)/3000;

T = 1./(a+b*log(T3000)+c*(log(T3000)).ˆ2+d*(log(T3000)).ˆ3)−273.15;

% Make time vector

time = 0:dt:dt*(size(T,1)−1);

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−− 2. Calculate average temperatures −−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

T base = mean(T(index:index+duration/dt,1));

T top = mean(T(index:index+duration/dt,4));

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−− 2. Calculate thermal conductivity −−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Q = thermalcell heater(heatfile,heatduration,heatindex);

Area = pi*Dˆ2/4;

lambda = Q*L/Area/(T base−T top)*1000;

fprintf('Length = %f mm\n',L)fprintf('Diameter = %f mm\n',D)fprintf('Base temperature = %f ˆoC\n',T base)

fprintf('Top temperature = %f ˆoC\n',T top)

fprintf('Thermal conductivity = %f W/mK\n',lambda)

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− END −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

140

Appendix C

Needle probe error analysis method usedby Hukseflux

Hukseflux used a simplified method to find the uncertainty in measured thermal conductivity due to

propagation of errors. This uses only two points at times t1 and t2 to calculate the gradient, instead of

fitting a line to a series of points (Hukseflux, 2015 pers. comm.). The thermal conductivity is given

by

λ =Q ln(t2/t1)

4π(T2−T1)(C.1)

where

Q = I2Re (C.2)

T2−T1 =U2−U1

S(C.3)

I is the current across the shunt resistor, and Re is the heater resistance in Ωm−1. T1 and T2 are the

temperatures at the hot joint at times t1 and t2 respectively, calculated by the thermocouple voltages

U1 and U2 and sensitivity S.

The uncertainty calculations are based on the ”Guide to the Expression of Uncertainty in

Measurement” (JCGM, 2008). The possible values for the variables are assumed to have a

rectangular distribution, with a half-width a. These are given in Table C.1. For a rectangular

distribution, the variance is given by

141

u2 = a2/3 (C.4)

and hence

u = a/√

3 (C.5)

u is the standard uncertainty, i.e. the expression of uncertainty of a measurement as a standard

deviation.

The general equation for calculating the error in q, which is a function of several variables x,...,z with

associated uncertainties δx,...δ z, is

δq =

√(∂q∂x

δx)2

+ ...+

(∂q∂ z

δ z)2

(C.6)

From Equations C.1 and C.2, the partial derivatives are

∂λ

∂Re=

I2 ln(

t2t1

)4π(T2−T1)

(C.7)

∂λ

∂ (T2−T1)=−ReI2 ln

(t2t1

)4π(T2−T1)2 (C.8)

∂λ

∂ I=

2IRe ln(

t2t1

)4π(T2−T1)

(C.9)

Table C.1: Measurement errors for needle probe variables

VariableHalf-width, a

Absolute Percentage

t 0.1 s -

I - 0.1%

Re 0.2 Ωm−1 0.25%

U1,U2 0.001 mV -

S - 1%

142

∂λ

∂ t1=−ReI2

(1t1

)4π(T2−T1)

(C.10)

∂λ

∂ t2=

ReI2(

1t2

)4π(T2−T1)

(C.11)

To find the uncertainty in (T2−T1), the following partial derivatives are required

∂ (T2−T1)

∂U1=−1

S(C.12)

∂ (T2−T1)

∂U2=

1S

(C.13)

∂ (T2−T1)

∂S=

U1−U2

S2 (C.14)

The combined standard uncertainty in thermal conductivity is then calculated by inserting the partial

derivatives into Equation C.6. To find the overall or expanded uncertainty, this is multiplied by a

coverage factor k of 2. An example of this calculation is shown in Table C.2, using the straight line

section limits as determined by the polynomial fit method.

143

Table C.2: Example calculation of combined standard uncertainty of thermal conductivity

Variable, x Value of xa u

∂λ

∂x∂λ

∂x ·uAbsolute %

(=√

3a)

Re 82.48 Ωm−1 0.2 0.3 1.19E-01 1.32E-02 1.58E-03

I 0.18 A 0.0002 0.1 1.03E-04 1.23E+01 1.26E-03

t1 78.5 s 0.01 0.01 5.77E-03 1.36E-02 7.83E-05

t2 219.0 s 0.01 0.005 5.77E-03 4.86E-03 2.81E-05

U1 98.2 µV 1.06 1.1 6.1E-01 2.5E-02 1.5E-02

U2 106.1 µV 1.07 1.0 6.2E-01 2.5E-02 1.5E-02

S 40.4 µVC−1 0.4 1.0 2.3E-01 4.8E-03 1.0E-03

T2−T1 0.20 C 0.02 11.0 1.25E-02 5.59E+00 6.96E-02

(= (U2−U−1)/S)

λ 1.09 Wm−1K−1 u = k ·√

∑c ·u = 2 ·√

∑c ·u = ±0.14Wm−1K−1

±12.7%

144

Appendix D

Results

Needle Probe tests on agar-kaolin samples

Here are the Tables of results from needle probe tests on four agar-kaolin samples of different

densities. Heating times and heating power were varied. The heating times were 100, 300, 500 and

700 seconds. The power could be set to Low, Medium or High (0.82, 2.43 and 4.13 Wm−1

respectively). Six different analysis methods were used to analyse the data for comparison. These

were (see Section 3.5):

• Method 1: visual inspection

• Method 2: first 10 seconds of data are ignored

• Method 3: first 30 seconds of data are ignored

• Method 4: first third of data is ignored

• Method 5: t < 5r2/α data is ignored

• Method 6: second order polynomial fit method

For Method 1, the limits to the selected linear section are also given as values of ln(t) and

ln(t/(t− theat)) for heating and recovery phases respectively, with t in seconds. For Method 5, the

cut-off t = 5r2/α is given (in seconds).

145

Table D.1: Sample 1 thermal conductivities calculated using six different analysis methods.

Heatingtime (s)

Heatingpower

Heating thermal conductivity (Wm−1K−1) Recovery thermal conductivity (Wm−1K−1)

1 2 3 4 5 6 1 2 3 4 5 6

100 Low 0.633 0.632 0.630 0.628 0.631 0.64±0.6% 0.597 0.602 0.606 0.564 0.612 0.58±0.6%

100 Medium 0.649 0.648 0.646 0.643 0.645 0.65±0.4% 0.591 0.637 0.619 0.605 0.642 0.61±0.7%

100 High 0.641 0.639 0.640 0.640 0.641 0.64±0.3% 0.608 0.613 0.627 0.651 0.617 0.61±0.2%

300 Low 0.659 0.650 0.642 0.595 0.649 0.66±0.8% 0.605 0.581 0.554 0.462 0.569 0.59±0.7%

300 Medium 0.638 0.644 0.646 0.675 0.644 0.64±0.3% 0.644 0.638 0.635 0.509 0.641 0.66±0.4%

300 High 0.654 0.641 0.637 0.579 0.641 0.64±0.2% 0.608 0.608 0.603 0.568 0.609 0.59±0.3%

500 Low 0.704 0.716 0.743 0.906 0.731 0.69±2% 0.521 0.602 0.524 0.391 0.568 0.53±0.8%

500 Medium 0.641 0.632 0.624 0.648 0.629 0.65±0.3% 0.631 0.615 0.600 0.534 0.610 0.62±0.3%

500 High 0.650 0.620 0.601 0.580 0.613 0.67±0.3% 0.603 0.597 0.597 0.562 0.596 0.63±0.2%

700 Low 0.654 0.696 0.725 0.815 0.713 0.67±0.7% 0.502 0.564 0.505 0.323 0.540 0.63±1%

700 Medium 0.639 0.631 0.620 0.561 0.625 0.67±0.3% 0.627 0.615 0.609 0.528 0.614 0.66±0.2%

700 High 0.617 0.616 0.602 0.611 0.609 0.58±0.5% 0.600 0.598 0.590 0.572 0.595 0.55±0.3%

146

Heatingtime (s)

Heatingpower

Average thermal conductivity (Wm−1K−1) Method 1 section limits Method 5cut-off (s)1 2 3 4 5 6 Heating Recovery

100 Low 0.615 0.617 0.618 0.596 0.622 0.608 2.3 4.5 1.0 2.5 19.1

100 Medium 0.620 0.642 0.633 0.624 0.644 0.628 2.4 4.5 0.5 1.2 19.0

100 High 0.624 0.626 0.633 0.646 0.629 0.623 2.5 4.3 0.8 2.3 18.8

300 Low 0.632 0.615 0.598 0.529 0.609 0.625 2.5 5.0 1.6 3.0 18.6

300 Medium 0.641 0.641 0.640 0.592 0.642 0.650 2.1 5.2 1.2 3.3 18.4

300 High 0.631 0.625 0.620 0.574 0.625 0.613 2.5 5.0 0.5 3.5 18.6

500 Low 0.612 0.659 0.633 0.649 0.650 0.607 2.5 5.4 0.8 2.3 19.2

500 Medium 0.636 0.623 0.612 0.591 0.619 0.636 2.5 5.0 1.5 4.0 18.5

500 High 0.626 0.609 0.599 0.571 0.604 0.649 2.5 5.0 0.8 3.5 18.8

700 Low 0.578 0.630 0.615 0.569 0.626 0.648 2.0 5.5 1.0 2.5 20.4

700 Medium 0.633 0.623 0.614 0.545 0.620 0.664 2.0 5.5 1.0 4.0 18.6

700 High 0.609 0.607 0.596 0.592 0.602 0.568 2.5 6.0 0.5 4.5 19.3

147


Heatingtime (s)

Heatingpower


1 2 3 4 5 6 1 2 3 4 5 6

100 Low 0.800 0.825 0.823 0.804 0.830 0.82±0.9% 0.814 0.829 0.937 0.923 0.844 0.78±2%

100 Medium 0.850 0.844 0.860 0.856 0.853 0.85±0.6% 0.812 0.853 0.842 0.801 0.877 0.74±0.6%

100 High 0.842 0.842 0.865 0.864 0.850 0.84±0.4% 0.801 0.811 0.892 0.954 0.826 0.75±0.5%

300 Low 0.843 0.855 0.869 1.007 0.864 0.85±0.5% 0.849 0.814 0.817 0.490 0.819 0.79±1%

300 Medium 0.852 0.827 0.805 0.723 0.825 0.86±0.4% 0.804 0.810 0.823 0.701 0.817 0.79±0.4%

300 High 0.809 0.807 0.793 0.754 0.806 0.84±0.3% 0.810 0.802 0.810 0.791 0.807 0.82±0.2%

500 Low 0.874 0.894 0.932 1.183 0.898 0.83±2% 0.938 0.769 0.682 0.456 0.756 0.87±2%

500 Medium 0.828 0.803 0.775 0.727 0.798 0.75±0.6% 0.811 0.827 0.839 0.700 0.833 0.83±0.5%

500 High 0.844 0.815 0.805 0.712 0.815 0.71±0.4% 0.758 0.760 0.757 0.763 0.760 0.78±0.3%

700 Low 0.813 0.812 0.810 0.920 0.812 0.82±2% 0.815 0.769 0.722 0.471 0.752 0.83±2%

700 Medium 0.838 0.829 0.826 0.915 0.828 0.82±0.6% 0.802 0.767 0.753 0.611 0.767 0.73±0.7%

700 High 0.801 0.801 0.793 0.846 0.799 0.83±0.3% 0.770 0.769 0.761 0.750 0.770 0.80±0.3%

148

Heatingtime (s)

Heatingpower


100 Low 0.807 0.827 0.880 0.864 0.837 0.800 1.8 4.5 0.5 2.5 13.9

100 Medium 0.831 0.849 0.851 0.829 0.865 0.792 2.7 4.2 0.5 1.3 13.5

100 High 0.822 0.827 0.878 0.909 0.838 0.791 2.3 4.6 0.2 2.5 13.6

300 Low 0.846 0.835 0.843 0.749 0.842 0.819 2.0 5.0 1.0 3.0 13.2

300 Medium 0.828 0.819 0.814 0.712 0.821 0.828 2.4 5.0 1.3 3.2 13.5

300 High 0.809 0.804 0.802 0.773 0.806 0.833 2.2 5.5 0.8 3.0 13.9

500 Low 0.906 0.832 0.807 0.820 0.827 0.846 2.8 5.0 1.7 3.5 12.4

500 Medium 0.820 0.815 0.807 0.714 0.815 0.790 3.0 5.0 1.5 4.0 13.7

500 High 0.801 0.787 0.781 0.738 0.788 0.748 2.6 5.0 0.8 4.0 14.0

700 Low 0.814 0.791 0.766 0.695 0.782 0.825 2.5 5.5 1.5 3.5 13.8

700 Medium 0.820 0.798 0.789 0.763 0.798 0.774 2.5 5 2 4 13.7

700 High 0.785 0.785 0.777 0.798 0.785 0.815 2.3 5.8 0.4 4.2 14.3

149


Heatingtime (s)

Heatingpower


1 2 3 4 5 6 1 2 3 4 5 6

100 Low 0.888 0.897 0.918 0.918 0.895 0.92±1% 0.801 0.799 0.764 0.960 0.800 0.79±0.8%

100 Medium 0.904 0.933 0.936 0.928 0.940 0.96±0.8% 0.824 0.913 0.830 0.797 0.917 1.00±2%

100 High 0.931 0.929 0.904 0.901 0.930 0.90±0.5% 0.903 0.883 0.982 0.936 0.892 0.82±0.3%

300 Low 1.020 1.019 1.038 0.964 1.020 0.98±0.4% 0.769 0.799 0.828 0.517 0.805 0.85±1%

300 Medium 0.930 0.976 1.020 1.044 0.985 0.94±0.5% 0.872 0.961 0.981 0.729 0.971 0.89±0.6%

300 High 0.932 0.929 0.923 0.919 0.930 0.96±0.3% 0.897 0.893 0.890 0.748 0.896 0.90±0.5%

500 Low 0.968 1.026 1.104 1.737 1.041 0.86±2% 0.914 0.869 0.808 0.449 0.863 0.88±0.2%

500 Medium 0.925 0.944 0.959 1.076 0.949 0.92±0.6% 0.853 0.904 0.916 0.866 0.906 0.86±0.8%

500 High 0.947 0.912 0.892 0.857 0.910 0.85±1% 0.897 0.896 0.914 0.763 0.899 0.93±0.3%

700 Low 0.969 0.940 0.924 1.159 0.941 1.00±0.3% 0.925 0.858 0.808 0.602 0.854 0.97±2%

700 Medium 0.915 0.936 0.944 0.782 0.938 0.92±0.6% 0.927 0.943 0.940 0.804 0.948 0.98±0.4%

700 High 0.906 0.906 0.901 0.922 0.905 0.90±0.6% 0.852 0.861 0.864 0.807 0.862 0.88±0.3%

150

Heatingtime (s)

Heatingpower


100 Low 0.844 0.848 0.841 0.939 0.848 0.856 2.2 4.5 0.5 2.5 12.9

100 Medium 0.864 0.923 0.883 0.863 0.928 0.979 2.2 3.5 0.4 1.4 12.6

100 High 0.917 0.906 0.943 0.919 0.911 0.859 2.5 4.5 0.5 2.0 11.9

300 Low 0.894 0.909 0.933 0.740 0.913 0.914 2.2 5.5 1.0 4.0 12.2

300 Medium 0.901 0.968 1.001 0.887 0.978 0.918 2.3 4.5 1.9 3.6 12.1

300 High 0.915 0.911 0.906 0.833 0.913 0.926 2.3 5.2 1.0 3.4 11.9

500 Low 0.941 0.947 0.956 1.093 0.952 0.868 2.3 5.0 1.0 4.4 11.6

500 Medium 0.889 0.924 0.937 0.971 0.927 0.892 2.2 5.2 1.9 4.2 12.3

500 High 0.922 0.904 0.903 0.810 0.904 0.894 2.5 5.0 1.0 3.3 11.8

700 Low 0.947 0.899 0.866 0.880 0.897 0.985 2.3 5.0 2.0 4.4 11.5

700 Medium 0.921 0.939 0.942 0.793 0.943 0.948 2.5 5.0 2.0 4.3 11.9

700 High 0.879 0.883 0.883 0.865 0.884 0.890 2.5 5.5 1.0 4.4 12.4

151


Heatingtime (s)

Heatingpower


1 2 3 4 5 6 1 2 3 4 5 6

100 Low 1.241 1.241 1.313 1.313 1.226 1.21±0.4% 1.177 1.169 1.079 0.783 1.150 1.28±0.7%

100 Medium 1.343 1.335 1.389 1.398 1.322 1.30±1% 1.227 1.325 1.222 1.144 1.303 1.24±0.7%

100 High 1.309 1.295 1.301 1.306 1.287 1.32±0.5% 1.163 1.281 1.365 1.145 1.248 1.39±2%

300 Low 1.786 1.554 1.715 2.121 1.459 1.52±0.2% 1.380 1.211 1.135 0.513 1.197 1.29±0.6%

300 Medium 1.357 1.354 1.349 1.352 1.346 1.37±0.4% 1.192 1.335 1.328 0.858 1.320 1.40±0.7%

300 High 1.301 1.301 1.293 1.211 1.296 1.28±0.5% 1.260 1.255 1.251 0.995 1.245 1.23±0.7%

500 Low 1.487 1.403 1.367 1.518 1.386 1.48±0.03% 1.327 1.203 1.143 0.720 1.193 1.49±1%

500 Medium 1.316 1.292 1.272 1.129 1.290 1.32±0.6% 1.182 1.269 1.286 1.051 1.258 1.22±1%

500 High 1.311 1.267 1.239 1.118 1.265 1.35±0.6% 1.233 1.199 1.172 1.034 1.196 1.26±0.5%

700 Low 1.470 1.371 1.349 1.557 1.356 1.64±0.6% 1.228 1.080 0.986 0.683 1.090 1.27±0.6%

700 Medium 1.334 1.291 1.266 1.088 1.290 1.31±1% 1.193 1.273 1.292 1.185 1.260 1.10±0.9%

700 High 1.299 1.275 1.259 1.125 1.275 1.31±0.7% 1.246 1.229 1.214 1.063 1.222 1.32±0.5%

152

Heatingtime (s)

Heatingpower


100 Low 1.209 1.205 1.196 1.048 1.188 1.242 2.3 4.5 0.6 2.6 8.6

100 Medium 1.285 1.330 1.306 1.271 1.313 1.271 2.4 4.5 0.5 1.7 8.1

100 High 1.236 1.288 1.333 1.225 1.268 1.354 2.5 4.5 1.2 2.5 8.4

300 Low 1.583 1.382 1.425 1.317 1.328 1.406 3.9 5.4 1.0 3.0 6.6

300 Medium 1.275 1.345 1.339 1.105 1.333 1.388 2.4 5.6 2.2 3.3 8.2

300 High 1.280 1.278 1.272 1.103 1.271 1.253 2.1 5.5 1.0 3.5 8.1

500 Low 1.407 1.303 1.255 1.119 1.290 1.482 2.5 5.0 1.5 3.7 7.4

500 Medium 1.249 1.281 1.279 1.090 1.274 1.268 2.3 5.5 2.2 4.0 8.3

500 High 1.272 1.233 1.205 1.076 1.230 1.304 2.5 5.4 1.3 3.5 8.2

700 Low 1.349 1.226 1.167 1.120 1.223 1.455 2.5 4.5 2.3 4.5 7.7

700 Medium 1.263 1.282 1.279 1.137 1.275 1.205 2.3 5.5 2.1 4.4 8.2

700 High 1.272 1.252 1.236 1.094 1.249 1.314 2.9 5.8 1.1 4.3 8.2

153

U100 London Clay samples

Tests were done on six U100 London Clay samples taken at different depths from a TRT borehole.

The samples were tested using both the needle probe and thermal cell methods. A TRT was also

conducted. Five needle probe readings and two thermal cell readings were taken for each sample.

Details are given in Chapter 5.

154

Table D.5: Thermal conductivity results for needle probe and thermal cell tests on U100 Lon-don Clay samples.

Depth (m)


Needle probe

Heating Recovery

1 (centre) 2 3 4 5 Average 1 (centre) 2 3 4 5 Average

2.00-2.451.15 1.24 1.37 1.22 1.27 1.25 1.31 1.21 1.48 1.11 1.40 1.30

±0.9% ±0.8% ±0.8% ±0.3% ±0.3% ±6% ±2% ±0.6% ±3% ±0.8% ±2% ±10%

8.00-8.451.20 1.54 1.54 1.57 1.42 1.45 1.29 1.15 1.07 1.25 1.19 1.19

±0.6% ±1% ±2% ±0.9% ±0.5% ±9% ±1% ±0.6% ±0.9% ±0.5% ±2% ±6%

10.00-10.451.09 1.23 1.43 1.27 1.23 1.25 0.80 1.16 1.33 0.97 1.04 1.06

±0.4% ±0.9% ±2% ±0.8% ±0.6% ±9% ±0.6% ±2% ±2% ±0.7% ±0.7% ±17%

17.00-17.451.53 1.43 0.80 1.49 1.37 1.32 1.36 1.42 1.04 1.41 1.38 1.32

±0.6% ±0.3% ±0.2% ±0.4% ±0.3% ±20% ±0.4% ±0.5% ±0.3% ±0.2% ±0.3% ±11%

19.00-19.451.01 1.06 1.07 1.05 0.97 1.03 0.86 0.85 0.93 0.83 0.97 0.89

±0.6% ±0.3% ±0.5% ±0.4% ±0.4% ±3% ±0.5% ±0.4% ±0.8% ±0.5% ±0.3% ±6%

21.50-21.951.60 1.51 1.48 1.51 — 1.52 1.41 1.44 1.40 1.43 — 1.42

±0.5% ±0.5% ±0.3% ±0.4% ±3% ±0.6% ±0.4% ±0.5% ±0.4% ±0.9%

155

Depth (m)


Thermal cell

Top half Bottom half Average

2.00-2.451.859 1.719 1.789

±4% ±4%

8.00-8.452.014 1.880 1.947

±5% ±5%

10.00-10.451.846 1.913 1.880

±5% ±5%

17.00-17.451.919 1.875 1.897

±4% ±4%

19.00-19.451.649 1.750 1.699

±4% ±4%

21.50-21.952.187 1.839 2.013

±4% ±4%

156

Appendix E

Paper for the 32nd International ThermalConductivity Conference & 20thInternational Thermal ExpansionSymposium

157

1

32nd

International Thermal Conductivity Conference 20

th International Thermal Expansion Symposium

April 27– May 1, 2014

Purdue University, West Lafayette, Indiana, USA

Thermal conductivity of simulated soils by the needle probe method, for energy foundation applications

J. E. Low, [email protected], F. A. Loveridge, [email protected], and W. Powrie, [email protected]

Faculty of Engineering and the Environment, University of Southampton

ABSTRACT

Soil thermal conductivity is an important parameter in the design of ground source heat pump and energy foundation systems. One laboratory method for measuring the soil thermal conductivity is the needle probe method. Previously, analysis of the needle probe test data has been simplistic, relying heavily on human judgment and rules of thumb. This paper presents an alternative method of analyzing the needle probe data with the aid of a MATLAB program. Four agar-kaolin specimens of varying densities were prepared to resemble simple soils. These were tested using the needle probe for a range of heating times and heating powers, to see what effect these parameters would have on the results. The repeatability when keeping the heating time and heating power constant was within ±2%. When the heating time and heating power were varied, the variation in results from the average for a given specimen ranged from ±4% to +10%/-8%. This range is significantly higher than the repeatability. Possible reasons for this are discussed.

KEYWORDS

Needle probe, soil thermal conductivity, energy foundations, transient laboratory methods.

1. INTRODUCTION

Ground source heat pump (GSHP) systems provide a viable alternative to conventional heating and cooling systems in the development of sustainable building solutions (Banks, 2008). Heat is transferred between the ground and the building by means of a refrigerant pumped through a series of pipes buried in the ground, known as the ground loop. To minimize initial construction costs, the pipes can be cast into the building foundations, eliminating the need for further excavations. These are known as energy foundations. To design such a system, it is important to model accurately the heat transfer process between the foundations and the soil. An important input parameter for such analysis is the soil thermal conductivity.

Soil thermal conductivity can range between 0.2 and 5 Wm

-1K

-1 (Ground Source Heat Pump Association

[GSHPA], 2012). A typical soil has three main constituents: solid particles, water and air. The solid particles have the highest thermal conductivity, so a high soil thermal conductivity relies on good contact between particles. Water has a thermal conductivity

of 0.6 Wm-1

K-1

compared to air which has a thermal conductivity of 0.025 Wm

-1K

-1, so saturated soils

tend to have higher thermal conductivities than dry soils (Hukseflux Thermal Sensors, 2003). A low conductivity soil would require a longer ground loop than a high conductivity soil to meet the same energy load for a building.

The thermal response test (TRT) is currently the most widely used method for the determination of the in situ soil thermal conductivity for a GSHP system (GSHPA, 2012). It is a large-scale transient field test involving the construction of a ground heat exchanger. In theory, the value of thermal conductivity obtained using this method should relate directly to the heat transfer performance of a GSHP system. However, performing a TRT is both expensive and time consuming, so it may be preferable to measure the soil thermal conductivity using a laboratory method.

Laboratory methods for measuring soil thermal conductivity fall into one of two categories: steady state or transient methods (Farouki, 1981; Mitchell & Kao, 1978). At the laboratory scale, steady state

2

methods involve applying one-directional heat flow to a specimen and measuring the power input and temperature difference across it when a steady state is reached. The thermal conductivity is then calculated directly using Fourier’s Law. However, steady state methods can be difficult to implement as heat losses must be minimized for the results to be reliable.

Transient methods involve applying heat to the specimen and monitoring temperature changes over time. The transient data are used to determine the thermal conductivity, usually by application of an analytical solution to the heat diffusion equation. One transient method is the needle probe method. It is analogous to the TRT, but at a much smaller scale.

The method by which data from a needle probe test is analyzed can significantly affect the thermal conductivity. There are several standards on the needle probe, but they do not elaborate on the data analysis, which relies mainly on a visual interpretation of the data (ASTM International, 2008; Institute of Electrical and Electronics Engineers Inc. [IEEE], 1996). In this paper, a more rigorous method of analyzing the data is developed, which aims to minimize the human error associated with current methods.

2. THEORY

The calculation of thermal conductivity is based on the theory for an infinitely long, infinitely thin line heat source (Carslaw & Jaeger, 1959). If a constant power is applied to the heat source, the temperature rise ∆𝑇 at time 𝑡 after the start of heating, at a radial

distance 𝑟 from the heat source, is:

∆𝑇 = −𝑞

4𝜋𝜆Ei (−

𝑟2

4𝛼𝑡) (1)

where 𝑞 is the power per unit length of heater, 𝜆 is

the thermal conductivity of the soil, 𝛼 is the thermal diffusivity and Ei is the exponential integral (Abramowitz & Stegun, 1972):

Ei(𝑥) = − ∫𝑒−𝑢

𝑢𝑑𝑢

∞

−𝑥

(2)

After the power has been switched off (i.e. the start of the recovery phase), the temperature difference is given by:

∆𝑇 = −𝑞

4𝜋𝜆[−Ei (−

𝑟2

4𝛼𝑡)

+ Ei (−𝑟2

4𝛼(𝑡 − 𝑡heat))]

(3)

where 𝑡heat is the time at which the power is switched off. Equations (1) and (3) cannot be solved explicitly for 𝜆 and 𝛼. The exponential integral can be represented as a series expansion, and approximated using the first two terms as (Abramowitz & Stegun, 1972):

Ei(𝑥) = 𝛾 + ln|𝑥| (4)

where γ is Euler’s constant. This approximation is valid for small values of x, which is the case when t is large. Substituting Equation (4) into Equations (1) and (3) gives (ASTM International, 2008):

Δ𝑇 ≅𝑞

4𝜋𝜆ln(𝑡) −

𝑞

4𝜋𝜆(𝛾 + ln (

𝑟2

4𝛼)) (5)

Δ𝑇 ≅𝑞

4𝜋𝜆ln(𝑡) + 𝐵 0 < 𝑡 ≤ 𝑡heat (6)

Δ𝑇 ≅𝑞

4𝜋𝜆ln (

𝑡

𝑡 − 𝑡heat) 𝑡 > 𝑡heat (7)

where B is a constant, grouping together the end terms of Equation (5).

Graphs are plotted of change in temperature against ln(𝑡) and ln(𝑡 (𝑡 − 𝑡heat)⁄ ), for the heating and recovery phases respectively (Figure 2.1). During the initial part of each phase, the contact resistance and thermal capacity of the probe are overcome.

Figure 2.1 Typical needle probe results showing (a) temperature against time, and change in temperature against logarithmic time for (b) heating and (c) recovery.

3

After this, the logarithmic graphs become linear and the gradient can be used to calculate the thermal conductivity. The time it takes for linearity to occur depends on the quality of the contact between the probe and the soil. The better the contact, the shorter the time taken to reach linearity. The last part of the graph for each phase can also become non-linear, as boundary conditions at the outer surfaces of the sample may start to have an effect.

Current standards suggest selecting the linear section of the graph by visual inspection (ASTM, 2008; IEEE, 1996), or excluding the first 10 to 30 seconds from the analysis for smaller diameter probes (IEEE, 1996). Both methods can be subjective and introduce significant errors. Commercial needle probes may have built in programs for calculating the thermal conductivity, e.g. the KD2 Pro Thermal Properties Analyzer by Decagon Devices Inc. (2014). They use a similar method to the standards and exclude the first third of data in their analysis. Subsequent research has been done by King, Banks & Findlay (2012) where the thermal conductivity is calculated for different intervals during the heating time to find the average. They suggest that a reliable value is obtained when the standard deviation is <0.1 Wm

-1K

-1 or <10%.

3. METHOD

The needle probe used was the TP02 probe produced by Hukseflux Thermal Sensors (2003). This is 150 mm long with a diameter of 1.5 mm, and encloses a 100 mm long heating wire with a thermocouple located midway along its length to

measure the temperature (see Figure 3.1). The thermocouple was NiCr-NiAl type K. The radius of the soil specimen should be at least 20 mm and encompass the length of the needle. The range of thermal conductivities that can be measured by the probe is 0.1 to 6 Wm

-1K

-1.

3.1 Preparation

Four agar-kaolin specimens resembling a simple two-phase soil were prepared as follows. Agar is a gelling agent and is used to solidify the water, preventing moisture migration when the specimens are heated. Kaolin is a type of clay which comes in the form of a dry, white powder. De-aired water was heated in a conical flask over a hot plate. The temperature of the hot plate was set at 370

oC, and

the water was gently stirred using a magnetic stirrer. A thermometer was used to measure the temperature of the water every few minutes. When the water reached 85

oC (the melting temperature of

agar) the hot plate temperature was reduced to 200

oC, and the stirrer speed was increased slightly

to prevent agar from sticking to the bottom of the flask. The agar was added to the water, with 4 grams of agar to every liter of water. When the agar had dissolved (which took approximately 20 minutes) the hot plate was switched off. The mixture was poured into a large tray, and the stir bar removed. Kaolin was gradually mixed in using palette knives. When a smooth consistency with minimal air bubbles had been reached, the mixture was poured into a 100 mm internal diameter cylinder, 220 mm long.

Different water to kaolin ratios were used for each specimen to achieve a range of thermal conductivities, as summarized in Table 3.1. The specimens were left overnight in a 20

oC temperature

controlled room to equilibrate. To ensure good contact between the probe and the specimen, the probe was inserted into the mixture while it was still liquid. The base of the probe was secured by clamping it so that the probe stood vertically through the center of the sample.

Table 3.1 Specimen densities.

Specimen No. Density (kgm-3)

1 1000

2 1181

3 1275

4 1444

3.2 Measurement

To prevent the specimens from drying out, thermal conductivity measurements were taken the day after

Figure 3.1 Diagram of needle probe (after Hukseflux Thermal Sensors (2003)).

4

the specimen was made, when the specimens had cooled to form a jelly. Data was recorded using a Campbell Scientific CR1000 data logger. Measurements were taken for heating times of 100, 300, 500, and 700 seconds, at low, medium, and high power (0.82, 2.43, and 4.13 Wm

-1 respectively).

Each measurement had three phases, and lasted four times the heating time. In the first phase (the same length as the heating time) the power was off, and the thermocouple measured the initial temperature of the soil to ensure that the temperature was not drifting. The second phase was the heating phase. The final phase was recovery, which was twice as long as the heating time. There were therefore a total of twelve measurements (4 heating times × 3 heating powers) per specimen.

The repeatability of the needle probe was also determined, by taking eight needle probe measurements of the thermal conductivity of agar jelly (with no added kaolin) for 300 seconds of heating at medium power.

3.3 Analysis

The thermal conductivity was calculated from the graphs of change in temperature against ln(𝑡) and ln(𝑡 (𝑡 − 𝑡heat)⁄ ), for the heating and recovery phases respectively. The thermal conductivity is inversely proportional to the gradient of the straight line section (Equations (6) and (7)). To determine the linear section of the graph more systematically, a MATLAB program was produced to help extract and process the raw needle probe data. MATLAB is a numerical computing environment and programming language developed by MathWorks. Linear regression was used to determine the gradient, but as time is plotted on a logarithmic scale, if all data points were taken into account the best-fit line would have a bias towards the end of the line where the

points are closer together. Therefore, points evenly spaced in logarithmic time were used for the linear regression.

There are two aspects in the positioning of the straight line section: the starting time (ln(𝑡)begin and

ln(𝑡 (𝑡 − 𝑡heat)⁄ )begin for the heating and recovery

phases respectively) and the length of the section. To begin with, the section length was fixed. For different starting times, the thermal conductivity was calculated based on the gradient of that section of the graph. The two consecutive sections with the most similar gradients were identified, and the average gradient of those sections used to calculate the thermal conductivity. An example of this is shown in Figure 3.2. The graphs show an increase in calculated thermal conductivity with starting time before reaching a plateau and decreasing again. The plateaus in Figure 3.2 help identify the linear sections of Figure 2.1 (b) and (c).

This whole process was repeated for different section lengths for both heating and recovery phases. When the calculated thermal conductivities were plotted against the length of section, it was found that after an initial phase with significant scatter, the thermal conductivities for heating and recovery converged and then diverged again slightly. An example of this is shown in Figure 3.3. For small section lengths, the calculated thermal conductivity can be influenced by small fluctuations in the data, causing scatter. As the section length increases, these fluctuations have less of an effect as more data are taken into consideration. The point of convergence is where the section length reaches the length of the straight line section of the graph. After this point, increasing the section length starts to

Figure 3.3 Thermal conductivity for heating and recovery against length of section used in the calculation. The data points used in the final thermal conductivity calculation are circled.

Figure 3.2 Thermal conductivity during (a) heating and (b) recovery, for different starting times. For this example, the heating time is 700 seconds and the section length is fixed at 2.8. The consecutive points circled have the closest values and are therefore used to calculate the thermal conductivity.

5

include data that should be excluded due to contact resistance or boundary influences. Inspection of Figure 2.1 graphs (b) and (c) show that including these extra data in the linear regression would cause the gradient to increase for both heating and recovery, and the calculated thermal conductivity to decrease. This is the case in Figure 3.3 after the point of convergence.

The point of convergence is found in the MATLAB program by determining the difference between the calculated thermal conductivities for heating and recovery. The two consecutive section lengths with the smallest combined difference were then used to calculate the final thermal conductivity, which is the average of the four points (circled in Figure 3.3).

4. RESULTS AND DISCUSSION

The repeatability in the agar jelly for the same heating time and heating power was found to be within ±2%, which is slightly worse than the repeatability stated by the manufacturer of ±1% (Hukseflux Thermal Sensors, 2003). The results from the four samples with varying heating time and heating power are plotted in Figure 4.1. The deviation in results from the average of the 12 measurements ranged from ±4% for Sample 2, to +10% to -8% for Sample 1, which is within the limits set by King, Banks & Findlay (2012) discussed previously. This is significantly higher than when the heating time and heating power were kept constant, and shows that the needle probe method is not as repeatable as it may initially seem. The variation is slightly greater for the low power measurements. This may be because low power gives smaller temperature differences and the limitations in

sensitivity of the needle probe thermocouple cause the temperature data to rise in steps, making it more difficult to determine the gradient accurately.

There are several possible reasons for the greater range of results when heating time and heating power are varied. It may reasonably be assumed that moisture migration is not a heat transfer mechanism as the water is solidified into jelly using the agar. The thermal conductivity of soils can increase with temperature but this is largely attributed to latent heat transfer by moisture migration (Hiraiwa & Kasubuchi, 2000). It is possible that the agar does not eliminate moisture migration entirely, which could be a contributing factor at high power and longer heating times. The total temperature change during heating varies between 0.6

oC and 5

oC. However, if moisture migration were

a factor then a trend of measured thermal conductivity increasing with heating power and heating time would be expected; this is not the case.

Although moisture migration is not expected to be a significant factor, evidence of water evaporation at the top of the sample was seen; the specimen was weighed after preparation and after testing. After leaving a specimen in the temperature controlled room overnight, small cracks at the surface around the circumference were already observed. The total testing time for one sample was six hours, so some evaporation may have occurred during that time. This could alter the thermal conductivity close to the surface of the sample.

A further possible factor is that, at the shorter heating times, the contact resistance affects the results, or that the straight line section is too short to give an accurate gradient. It can be seen in Figure

Figure 4.1 Thermal conductivities for a range of heating times and heating powers, for (a) Specimen 1, (b) Specimen 2, (c) Specimen 3, and (d) Specimen 4 (in order of increasing density).

Figure 4.2 Average thermal conductivity against density.

6

4.1 that the calculated thermal conductivities at a heating time of 100 seconds deviate more from the mean value than for longer heating times. At longer heating times, boundary effects could also be influencing the results.

Figure 4.2 shows the variation of the average thermal conductivity of the twelve measurements with density. The thermal conductivity increases with density, in agreement with previous research (Farouki, 1981).

5 CONCLUSIONS

A detailed method for calculating the thermal conductivity using the needle probe has been proposed. In contrast to previous methods which rely heavily on human judgment, this method has been fully programmed, to reduce the potential for user error. A visual inspection of the data should always still be carried out to check that a sensible result is obtained. This method was used in subsequent tests on agar-kaolin samples.

The repeatability of the needle probe method for measuring the thermal conductivity of agar jelly was found to be within ±2% for tests using the same heating power and heating time. When the heating power and heating time were varied, the range in results was significantly greater. Surface water evaporation may be a contributing factor. Contact resistance could affect tests with shorter heating times, and boundary conditions could affect tests with longer heating times. Even in a well-controlled environment these test variables have a significant impact on the results, so it is worth choosing the heating time and heating power carefully on the basis of the properties of the soil.

When using the needle probe method, it is advisable to use a program that excludes the data affected by contact resistance or boundary conditions, while using as much of the relevant data as possible to ensure an accurate calculation of the thermal conductivity.

REFERENCES

Abramowitz, M. & Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: United States Department of Commerce.

ASTM International. (2008). D 5334-08 Standard Test Method for Determination of Thermal Conductivity of Soil and Soft Rock by Thermal Needle Probe Procedure. West Conshohocken, PA: ASTM International.

Banks, D. (2008). An introduction to thermogeology: ground source heating and cooling. Oxford, England: Blackwell Publishing Ltd.

Carslaw, H. S. & Jaeger, J. C. (1959). Conduction of heat in solids. Oxford, England: Clarendon Press.

Decagon Devices Inc. (2014). KD2 Pro Thermal Properties Analyzer: Operator’s Manual. Pullman, WA: Decagon Devices Inc.

Farouki, O. (1981). Thermal properties of soils: CRREL Monograph 81-1. Hanover, NH: United States Army Corps of Engineers, Cold Regions Research and Engineering Laboratory.

Ground Source Heat Pump Association. (2012). Thermal pile design, installation & materials standards. Milton Keynes, England: GSHPA.

Hiraiwa, Y. & Kasubuchi, T. (2000). Temperature dependence of thermal conductivity of soil over a wide range of temperature (5-75

oC). European

Journal of Soil Science, 51, 211-218.

Hukseflux Thermal Sensors. (2003). TP02 Non-Steady-State Probe for Thermal Conductivity Measurement – manual v0908. Delft, Netherlands: Hukseflux Thermal Sensors.

Institute of Electrical and Electronics Engineers Inc. (1996). IEEE Std 442-1981 Guide for Soil Thermal Resistivity Measurements. New York, NY: IEEE.

King, W., Banks, D. & Findlay, J. (2012). Field determination of shallow soil thermal conductivity using a short-duration needle probe test. Quarterly Journal of Engineering Geology and Hydrogeology, 45(4), 497-504.

Mitchell, J. K. & Kao, T. C. (1978). Measurement of soil thermal resistivity. Journal of the Geotechnical Engineering Division, 104, 1307-1320.

Appendix F

Paper published in Acta Geotechnica,October 2014

The following version of the paper is the accepted manuscript submitted to the journal. The final

publication is available at http://link.springer.com/article/10.1007%2Fs11440-014-0333-0.

165

Acta Geotechnica manuscript No.(will be inserted by the editor)

A comparison of laboratory and in situ methods to determine soilthermal conductivity for energy foundations and other ground heatexchanger applications

Jasmine E. Low · Fleur A. Loveridge · William Powrie · Duncan Nicholson

Received: date / Accepted: date

Abstract Soil thermal conductivity is an important factorin the design of energy foundations and other ground heatexchanger systems. It can be determined by a field ther-mal response test, which is both costly and time consuming,but tests a large volume of soil. Alternatively, cheaper andquicker laboratory test methods may be applied to smallersoil samples. This paper investigates two different labora-tory methods: the steady state thermal cell and the transientneedle probe. U100 soil samples were taken during the siteinvestigation for a small diameter test pile, for which a ther-mal response test was later conducted. The thermal conduc-tivities of the samples were measured using the two labo-ratory methods. The results from the thermal cell and nee-dle probe were significantly different, with the thermal cellconsistently giving higher values for thermal conductivity.The main difficulty with the thermal cell was determiningthe rate of heat flow, as the apparatus experiences significantheat losses. The needle probe was found to have fewer sig-nificant sources of error, but tests a smaller soil sample thanthe thermal cell. However, both laboratory methods gavemuch lower values of thermal conductivity compared to thein situ thermal response test. Possible reasons for these dis-crepancies are discussed, including sample size, orientationand disturbance.

Keywords thermal conductivity · thermal cell · needleprobe · ground source heat pumps · energy foundations

J. E. Low · F. A. Loveridge ·W. PowrieFaculty of Engineering and the Environment, University of Southamp-ton, University Road, Southampton SO17 1BJ, United KingdomE-mail: [email protected]

D. NicholsonOve Arup and Partners Limited, 13 Fitzroy Street London W1T 4BQ,United Kingdom

1 Introduction

Ground source heat pump (GSHP) systems provide a viablealternative to conventional heating and cooling systems inthe move towards sustainable building solutions [6]. Heat istransferred between the ground and the building by meansof a refrigerant which is pumped through a series of pipesburied in the ground. To minimise initial construction costs,the pipes can be cast into the foundations, eliminating theneed to make further excavations. These systems are knownas energy or thermal foundations. To design such a system,it is important to model accurately the heat transfer processbetween the foundations and the soil. One important inputparameter for such analysis is the soil thermal conductivity.

There are several different laboratory methods for mea-suring soil thermal conductivity [26,14]. They fall into oneof two categories: steady state or transient methods. At thelaboratory scale, steady state methods involve applying one-directional heat flow to a specimen and measuring the powerinput and temperature difference across it when a steadystate is reached. The thermal conductivity is then calculateddirectly using Fourier’s Law. Transient methods involve ap-plying heat to the specimen and monitoring temperature changesover time. The transient data is used to determine the thermalconductivity, usually by application of an analytical solutionto the heat diffusion equation. Some transient methods canalso be used to assess other thermal properties such as ther-mal diffusivity [8]. This paper compares the two approachesusing a thermal cell (steady state) and a needle probe (tran-sient) apparatus. Both the thermal cell and needle probe arecurrently industry recommended laboratory methods [4,22,18].

The thermal response test (TRT) [13] is currently themost widely used method for the determination of the in situthermal conductivity for a GSHP system. It is a large-scaletransient field test and involves construction of a ground heat

Appendix G

Paper for the 18th InternationalConference on Soil Mechanics andGeotechnical Engineering

167

Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013

1

Measuring soil thermal properties for use in energy foundation design

La mesure des caractéristiques thermiques du sol pour la conception des fondations énergie

J. E. Low, F. A. Loveridge & W. Powrie Faculty of Engineering and the Environment, University of Southampton, Southampton, UK

ABSTRACT: Energy foundations incorporated into ground source heat pump systems provide a viable alternative to conventional building temperature regulation systems in the move towards sustainable building solutions. To design such a system, it is important to accurately model the heat transfer process between the foundations and the soil, which is largely governed by the soil thermal conductivity. This paper compares two laboratory test methods for determining soil thermal conductivity: the thermal cell which is a steady state method, and the needle probe which is a transient method.

RÉSUMÉ : Pour l’orientation vers des immeubles durables, les fondations énergie incorporées dans des systèmes de pompe à chaleur géothermique fournissent une alternative viable aux systèmes conventionnels de régulation de température des immeubles. La conception d’un tel système implique le modelage précis du processus, qui est en grande partie déterminé par la conductivité thermique du sol, de transfert thermique entre les fondations et le sol. Dans le texte qui suit l’on compare deux méthodes d’essai de laboratoire pour la détermination de la conductivité thermique du sol : la cellule thermique, méthode de régime établi, et sonde à aiguille, méthode de régime transitoire.

KEYWORDS: soil thermal conductivity, thermal cell, needle probe

1 INTRODUCTION

Ground source heat pump systems provide a viable alternative to conventional heating and cooling systems in the move towards sustainable building solutions (Banks, 2008). Heat is transferred between the ground and the building by means of a refrigerant which is pumped through a series of pipes buried in the ground. To minimize initial construction costs, the pipes can be cast into the foundations, eliminating the need to make further excavations. These systems are known as energy foundations. To design such a system, it is important to accurately model the heat transfer process between the foundations and the soil. This is largely governed by the soil thermal conductivity.

There are several different methods of measuring soil thermal conductivity (Mitchell and Kao, 1978). They fall into one of two categories: steady state or transient methods. At the laboratory scale, steady state methods involve applying one-directional heat flow to a specimen and measuring the power input and temperature difference across it when a steady state is reached. The thermal conductivity is then calculated directly using Fourier’s Law of heat conduction. Transient methods involve applying heat to the specimen and monitoring temperature changes over time, and using the transient data to determine the thermal conductivity. This paper compares the two approaches using a thermal cell (steady state) and a needle probe (transient) apparatus. The tests were carried out on U100 samples of London Clay upon which a full soil classification was afterwards conducted.

2 BACKGROUND

There are several methods of measuring thermal conductivity which are considered as suitable for use with soils. For this study, the needle probe and thermal cell methods were chosen due to the simplicity of the apparatus.

2.1 Needle probe

The measurement of thermal conductivity using the needle probe method is based on the theory for an infinitely long, infinitely thin line heat source (Carslaw and Jaeger, 1959). If a constant power is applied to the heat source, the temperature rise ΔT at time t after the start of heating, at a radial distance r from the heat source, is:

t

rEi

qT

44

2

(1)

where q is the power per unit length of heater, λ is the thermal conductivity, α is the thermal diffusivity and Ei is the exponential integral. After the power is switched off, the temperature difference is given by:

heattt

rEi

t

rEi

qT

444

22

(2)

where heatt is the time at which the power is switched off.

Equations (1) and (2) cannot be solved for λ and α explicitly, so

a simplified analysis approximating the exponential integral is

used which leads to (ASTM International, 2008):

tqT ln

4

heattt 0 (3)

heattt

tqT ln

4

heattt (4)

The needle probe used is the TP02 probe produced by

Hukseflux Thermal Sensors (2003). It is 150mm long with a

diameter of 1.5mm, and encloses a 100mm long heating wire

with a thermocouple located midway along this heater

measuring the temperature (see Figure 1).


2

Figure 1. TP02 probe (Hukseflux, 2003).

2.2 Thermal cell

The thermal cell design was loosely based on Clarke et al. (2008). A diagram of the apparatus is shown in Figure 2. The thermal conductivity of a cylinder of soil is measured by generating one-directional heat flow along the axis of the specimen. The heat is generated by a cartridge heater embedded in the aluminium platen. Provided the specimen is well insulated so that radial heat losses can be neglected, the heat flow through the specimen during steady state is governed by Fourier’s Law of heat conduction:

L

TAQ

(5)

where Q is the power input, A is the cross-sectional area, ΔT is the temperature difference across the length of the specimen, and L is the length of the specimen. If Q cannot be accurately determined, measurement of the temperatures in the specimen as it cools after the power is switched off (the recovery phase) can be used to determine the heat transfer coefficient between the top of the soil and the air and hence the power. This uses the lumped capacitance method, which is valid when the temperature difference across the soil is small compared with the temperature difference between the soil surface and the ambient temperature (Incropera et al., 2007):

1.0

ambtop

topbase

TT

TT (6)

where subscripts base, top and amb refer to temperature at the base of the soil, top of the soil, and of the ambient air respectively. Where this is satisfied, the temperature of the soil at time t is (Clarke et al., 2008):

t

mc

hATTTT

p

ambamb exp0 (7)

where T0 is the temperature of the soil at time t = 0 (when Equation (6) starts to apply), h is the convection heat transfer coefficient, m is the total mass of the soil, and cp is the soil specific heat capacity. This is estimated from the properties of the soil constituents:

waterpsoilpp mcmcmc (8)

Equation (7) gives a theoretical decay curve which can be fitted to the experimental data by changing h until the two curves match. During steady state, conservation of energy dictates that the heat flow rate across the soil is equal to the heat flow rate at the top of the specimen from the soil to the air.

ambtop

topbaseTThA

L

TTAQ

(9)

This is used to calculate the thermal conductivity. It is worth noting that this method introduces an error associated with the

estimation of the specific heat capacity from constituents whose properties may not be accurately known.

Figure 2. Thermal cell.

3 METHODOLOGY

3.1 Measurement procedure

The thermal conductivity of U100 samples of London Clay taken from a thermal response test borehole were tested using both techniques described in Section 2. Before any measurements were taken, the sealed samples were left in a temperature controlled room overnight to equilibrate. Each sample was treated as follows.

To accommodate the needle probe, a 200mm length specimen was prepared and secured in a rubber membrane. Shavings taken from the top of the sample were used to determine the initial moisture content at the top. The soil was found to be too hard to directly insert the probe. Therefore, a 5mm diameter hole had to be predrilled, and the hole filled with a high thermal conductivity contact fluid (in this case toothpaste was used) to reduce the contact resistance between the probe and the soil (Hukseflux, 2003). The probe was inserted into the hole, and secured with a clamp stand. It was then left for 20min to equilibrate with the soil. A constant power was then supplied to the needle probe heater for 300s, and then turned off. The temperatures during the heating and recovery periods were recorded. Using this procedure, five measurements were taken over the cross-sectional area of the specimen. One measurement was taken at the centre of the cross-section, the other four were equally spaced at a radial distance of 25mm from the centre.

To reduce the time it takes to reach steady state, the specimen was then cut in half and the top 100mm weighed and secured to the platen of the thermal cell (see Figure 2), and sealed at the top using aluminium foil to prevent moisture from leaving the top of the sample. Shavings taken from the bottom of the top half were used to determine the initial moisture content at the bottom. Insulation was then wrapped around the specimen. The temperature difference across the specimen is measured by two thermistors, one secured to the top of the platen, the other embedded at the top of the soil. The cartridge heater was then turned on, and the power controlled so that the platen remains at a constant temperature of 40°C. The power was measured using a MuRata ACM20-5-AC1-R-C wattmeter. Temperatures were monitored until steady state was reached and then maintained for at least 2hours. The power to the cartridge heater was then switched off, and the recovery period monitored. At the end of the test, shavings were taken from the top, middle and bottom of the specimen to determine the final moisture contents.

The holes drilled into the specimen and the contact fluid could potentially affect the thermal conductivity measurement using the thermal cell. To verify the result, the bottom half of the sample was also tested in the thermal cell, where these effects would be less significant.


3

A full soil classification was then conducted based on the British Standard 1377 (British Standards Institution, 1990), to determine the soil density, moisture content, liquid limit, plastic limit, particle density, and particle size distribution.

3.2 Data analysis

For the needle probe, using Equations (3) and (4) for heating and recovery respectively, graphs were plotted of temperature against the natural logarithm of time, and the gradient of the straight line section used to determine the thermal conductivity. A typical result is shown in Figure 3.

Figure 3. Graph of needle probe data for (a) heating and (b) recovery.

For the thermal cell, average temperatures during the steady

state period were calculated for each thermistor. The average power supplied to the cartridge heater was also calculated. Equation (5) was then used to determine the thermal conductivity.

4 RESULTS AND DISCUSSION

The results of the tests are shown in Table 1, with the average value of the five needle probe readings given. The needle probe consistently gave lower values of thermal conductivity than the thermal cell. The sample properties are given in Table 2, where the moisture content given is the average of the values before and after testing. There is a decrease in thermal conductivity with depth. This may be due to a decrease in density, and also change in mineralogy. The top two samples were of firm slightly sandy clay. The bottom sample had a significant number of fissures, and a slightly greater sand content.

Table 1. Thermal conductivity measured using the needle probe for heating and recovery, and using the thermal cell.

Sample depth (m)

Thermal conductivity (Wm-1K-1)

Needle probe in heating

Needle probe in recovery

Thermal cell

8.00-8.45 1.47 1.30 2.01 (t)* 1.88 (b)

10.00-10.45 1.24 1.36 1.85 (t) 1.91 (b)

19.00-19.45 1.06 0.93 1.65 (t) 1.75 (b)

*t – top half; b – bottom half.

4.1 Needle probe

The variation in the five needle probe readings within the same sample was about ±10% for heating and ±15% for recovery. The sample at depth 19.00-19.45m had less variation. When the needle probe was previously tested using five identical agar gel samples, it gave a repeatability of ±2% for both heating and recovery, so most of the variation in results would seem to be due to natural variation in thermal conductivity of the soil.

The greatest disadvantage with the needle probe is in the interpretation of results relying on human judgement. The calculated thermal conductivity is highly sensitive to the selection of the part of the graph deemed to be a straight line. Another factor which may affect the results is the use of contact fluid. In theory, the contact fluid should only decrease the time it takes to reach the straight line section of the graph, i.e. it should have no effect on the calculated thermal conductivity. However, the fluid could potentially seep into cracks in the soil, and in doing so alter the thermal conductivity. After testing, the specimens were cut up to see if this was the case. The soil at depths of 8.00-8.45m and 10.00-10.45m did not contain many fissures, and the contact fluid seemed to have stayed within the drilled holes. It can therefore be assumed that the contact fluid had little effect on the needle probe results. However, for the sample at depth 19.00-19.45m there were a significant number of fissures, which contact fluid had seeped into. This could affect both needle probe and thermal cell measurements, giving higher thermal conductivity results than otherwise.

4.2 Thermal cell

In Section 2.2, two methods for calculating the thermal conductivity using the thermal cell were outlined. One involves measuring the power directly, the other uses the lumped capacitance method to calculate the power. Only the first method was deemed suitable for this study, as the temperature difference across the soil after the power is switched off was too great for lumped capacitance to apply i.e. Equation (6) was not satisfied.

The difference in thermal conductivity values between the top and bottom sections was about 0.1Wm-1K-1. If the holes for the needle probe were to have a significant effect on the thermal conductivity values, the measurement for the top section would be expected to always be higher than for the bottom section, or vice versa. This is not the case, and as the area of the holes is only 1.25% of the total cross-sectional area, it can be assumed that the differences between the top and bottom sections are mainly due to the soil’s natural variability.

The moisture content at the top of the specimens were measured before and after the thermal cell tests. The values after the test were consistently higher than those before the test. The greatest increase in moisture content was 5.2%. This shows that over the long heating period, moisture migration occurs in the direction of heat flow. This is where a temperature gradient causes the water to transfer latent heat through the pores as described by the liquid-island theory (Philip and de Vries, 1957). This theory suggests that in fairly dry media, the water is deposited in isolated pockets or 'islands', either filling small pores or attaching themselves between soil grains. When a temperature gradient is applied, there is a vapour flux in the direction of heat flow. Water evaporates from one island, and condenses at the boundary of the next island, thereby transferring heat from one island to the next.

4.3 Comparing test methods

The measured thermal conductivity for the thermal cell is higher than that of the needle probe by 40%, 45%, and 71% for a depth of 8.00-8.45m, 10.00-10.45m, and 19.00-19.45m respectively. This could be explained by a number of factors. The needle probe and thermal cell measure the thermal conductivity in the


4

radial and axial directions respectively. It could be that the soil is anisotropic, and naturally has a higher thermal conductivity in the axial direction. However, the layers in the soil sample tended to be in the horizontal direction i.e. perpendicular to the cylinder axis. The thermal conductivity measured parallel to the layering should in general be higher than that measured perpendicular to the layering (Midttømme and Roaldset, 1998). If anisotropy was the reason behind the difference between needle probe and thermal cell values, then the needle probe would be expected to give higher values of thermal conductivity than the thermal cell. Therefore, it is unlikely that anisotropy is the reason behind these differences.

In the thermal cell calculations, the total power is used and any losses neglected. A simple finite element analysis was conducted, and indicated only minor losses. However, if losses are in fact significant, then the calculated thermal conductivities would be overestimates. A more thorough analysis would be necessary to determine whether this is the case.

The presence of contact fluid in the thermal cell test could potentially be aiding heat transfer. If the thermal conductivity of the contact fluid is determined, this would give a better indication as to what effect it could have. This should not be the main reason for higher thermal conductivity values, as the volume of contact fluid is comparatively small.

As previously mentioned, significant moisture migration occurs due to the large temperature gradient applied. As an additional mechanism for heat transfer, this may lead to higher measured values of thermal conductivity. Table 2. Soil properties.

Sample depth (m) Density (kgm-3) Average moisture

content (%)

8.00-8.45 Top

Bottom

2092

2142

23.4

23.3

10.00-10.45 Top

Bottom

2053

1951

26.9

27.1

19.00-19.45 Top

Bottom

1783

1787

26.3

26.4

5 FURTHER RESEARCH

This study highlights the need for further investigation into the needle probe and thermal cell methods of thermal conductivity measurement for soils. With the needle probe, it is still unclear as to why heating and recovery gave different results for the thermal conductivity. As mentioned previously, the needle probe relies on human judgement in the interpretation of the results. Further research will be carried out to find a method which eliminates this source of error.

Some possible sources of error in the thermal cell method require investigation. A more detailed finite element analysis could be used to determine what power losses might be expected, so that this could be factored into the thermal conductivity calculation. The specimens were prepared by hand, so that the surface in contact with the platen may not be entirely flat. Tests on standard materials with and without a contact fluid between the platen and the soil could determine how significant the effects of this may be on the heat transfer. From the recovery data, there was a considerable temperature difference between the top and bottom of the soil for a long time after the power had been switched off. Clarke et al. (2008) was able to use the recovery curve to determine the power input, as the temperature difference was small. The reasons behind this discrepancy are unclear, so further tests using the thermal cell

on different types of soil with a range of thermal conductivities will be beneficial.

The soil samples were taken from a borehole where a thermal response test was later conducted. Other samples were also taken to another laboratory to test for thermal conductivity using the thermal cell method. Once the results from these tests are known, a comparison will be made to the results gathered from this study.

6 CONCLUSION

Two test methods for thermal conductivity, the needle probe and thermal cell, have been compared. The needle probe takes less time to conduct, and the soil is only heated slightly and for a short period which means moisture migration is not expected to affect the results. However, hard soil samples may require predrilling, and the use of contact fluid which can seep into any existing fissures thereby potentially affecting the thermal conductivity measurements.

The thermal cell requires very little alterations to the soil sample, but raises some issues to do with power losses. The long heating time also means that moisture migrates towards the top of the specimen. Within the context of energy foundations, the thermal cell may prove more suitable for measuring the thermal conductivity of other relevant materials such as grout and concrete.

7 ACKNOWLEDGEMENTS

The authors would like to thank Harvey Skinner for his help in the design, build, and instrumentation of the apparatus. The soil samples were provided by Concept Engineering Consultants Ltd. The site work has been carried out by Arup, Canary Wharf Contractors Ltd, and Concept. This work forms part of a larger project funded by EPSRC (ref EP/H0490101/1) and supported by Mott MacDonald Group Ltd, Cementation Skanska Ltd, WJ Groundwater Ltd, and Golder Associates.

8 REFERENCES

ASTM International. 2008. D 5334-08 Standard Test Method for Determination of Thermal Conductivity of Soil and Soft Rock by Thermal Needle Probe Procedure. West Conshohocken, PA: ASTM International.

Banks D. 2008. An introduction to thermogeology: ground source heating and cooling. Oxford: Blackwell Publishing Ltd.

British Standards Institution. 1990. BSI 1377:1990 Methods of test for soils for civil engineering purposes. London: BSI.

Carslaw H.S. and Jaeger J.C. 1959. Conduction of heat in solids. 2nd ed. Oxford: Oxford University Press.

Clarke B.G., Agab A. and Nicholson D. 2008. Model specification to determine thermal conductivity of soils. Proceedings of the Institution of Civil Engineers - Geotechnical Engineering 161(3), 161-168.

Farouki O.T. 1986. Thermal properties of soils. Rockport, MA: Trans Tech.

Hukseflux Thermal Sensors. 2003. TP02 Non-Steady-State Probe for Thermal Conductivity Measurement – manual v0908. Delft: Hukseflux Thermal Sensors.

Incropera F.P., DeWitt D.P., Bergman T.L., and Lavine A.S. 2006. Fundamentals of Heat and Mass Transfer. 6th ed. Hoboken, NJ: Wiley.

Midttømme K. and Roaldset E. 1998. The effect of grain size on thermal conductivity of quartz sands and silts. Petroleum Geoscience 4, 165-172.

Mitchell J.K. and Kao T.C. 1978. Measurement of soil thermal resistivity. Journal of the Geotechnical Engineering Division 104, 1307-1320.

Philip J. R. and de Vries D. A. 1957. Moisture movement in porous materials under temperature gradients. Transactions, American Geophysical Union 38, 222-232.

Appendix H

Soil classifications

For the six London Clay samples taken from the central London development site investigation, each

was tested according to the British Standard 1377-2 (1990) to determine:

• Density

• Moisture content

• Liquid limit

• Plastic limit

• Particle density

The results are shown in Table H.1.1

Table H.1: Soil classifications for six London Clay samples.

Depth (m) Density(Mgm−3)

Moisturecontent (%)

Liquid limit(%)

Plastic limit(%)

Particledensity

(Mgm−3)

2.00-2.45 1.99 26 84 30 2.56

8.00-8.45 2.10 23 69 25 —

10.00-10.45 1.99 27 70 27 2.63

17.00-17.45 2.10 19 61 20 2.59

19.00-19.45 1.77 26 79 26 2.54

21.50-21.95 2.12 18 55 18 2.66

1 The sample at depth 8.00-8.45 m was unfortunately not tested for particle density or particle size distribution assomeone threw it out by accident.

173

H.1 Particle size distribution

For finding the particle size distribution of fine soils, the British Standards recommend using

sedimentation by the pipette method or the hydrometer method. These are both time consuming, so a

different approach was implemented using the Coulter LS 130 (Beckman Coulter, Inc., 2011). The

Coulter LS 130 uses laser diffraction to determine the particle size distribution, and can measure

particles from 0.1 to 900 µm in diameter. A small amount of soil suspended in water is contained in

a sample cell (see Figure H.1), situated between a laser source and a detector array. As a laser is

shone through the suspension, the scattered light leaving the sample cell is detected. The angle of

diffraction is related to the size of the particle using Mie theory (Mie, 1908), and thus if the intensity

of light at different angles is measured, the particle size distribution can be determined. Each method

of determining the particle size distribution gives slightly different results, but generally the laser

diffraction method can be regarded as suitable (Eschel et al., 2004).

For the sample at each depth, wet sieving was carried out according to British Standards to

determine the particle size distribution down to 63 µm. The distribution of particles <63 µm was

determined using the Coulter LS 130 on 20 g of material passing the 63 µm sieve. To separate out

the particles, the 20 g samples were placed in a beaker of sodium hexametaphosphate solution

overnight. Before a measurement, the beaker was put in an ultrasonic water bath for 10 minutes to

disperse the particles. A sub-sample was placed in a smaller beaker using a pipette, and diluted with

distilled water. The particles were put in suspension in the smaller beaker using a magnetic stirrer.

Whilst stirring, a pipette was used to take a sample of the suspension. A drop of liquid from the

pipette was added to the sample cell, which had been filled with distilled water. The sample cell

cover was screwed on, and the sample cell was slowly turned upside down a couple times to evenly

distribute the particles. It was then placed into the Coulter LS 130 for measurement. After a

measurement is conducted, the sample cell is taken out and washed thoroughly with distilled water.

Five measurements were carried out for each sample depth. An average of the five measurements

gave the particle size distribution. The results from wet sieving and the Coulter method were

combined to give the final particle size distribution, plotted in Figures H.2, H.3, H.4, H.5 and H.6.

174

Figure H.1: Sample cell for Coulter LS 130 for determining particle size distribution.

Figure H.2: Particle size distribution for London Clay sample at depth 2.00-2.45 m.

175



176



177

Appendix I

Calculating the effect of void ratio on soilthermal conductivity

In Chapter 5 the laboratory tests on soil samples yielded lower soil thermal conductivity values than

the TRT. One potential reason for this was the increase in void ratio of the sample due to the removal

of confining pressures. This appendix estimates the difference in soil thermal conductivity before

and after sampling, using the de Vries model for soil thermal conductivity (Farouki, 1986).

Estimating the change in void ratio

Gasparre (2005) conducted a number of oedometer tests on undisturbed London Clay samples,

including samples from depths of 7, 10, 17 and 25 m. These are at a similar depth to the samples in

Chapter 5. An oedometer applies one-directional compression to a soil sample whilst recording the

change in sample height. The vertical load, or total normal stress σ , is sustained by a combination of

the pore water and the soil skeleton. The component of total normal stress taken by the soil skeleton

is known as the effective stress σ ′, and is given by Terzaghi’s equation:

σ′ = σ −u (I.1)

where u is the pore water pressure. As a constant vertical load is applied, water gradually seeps out

of the pores between soil particles, which results in a closer packing of the soil particles. This

process is known as consolidation. During consolidation, the component of total normal stress taken

by the soil skeleton increases, while the component taken by the pore water pressure decreases.

When consolidation stops i.e. the sample hight remains constant, the void ratio can be recorded and

the corresponding vertical effective stress determined using Equation I.1. The sample is loaded in

increments, and for each additional load the void ratio and effective stress are recorded after

179

consolidation. A graph can then be plotted of void ratio e against the natural logarithm of vertical

effective stress ln(σ ′v), as in Figure I.1. The point at which the maximum change in gradient occurs

gives the maximum past vertical effective stress, or preconsolidation pressure σ ′c. This should be

similar to the in situ vertical effective stress. Hence, the difference in void ratio ∆e between the

sample and the in situ soil can be determined. The graphs from Gasparre (2005) gave an estimated

change in void ratio of 0.15.

Using the de Vries model

The de Vries model is outlined in Section 2.5.1. The values used in the calculation are shown in

Table I.1. The effective air thermal conductivity (λa +λvs) at 20 C was estimated from Figure 2.2.

Void ratio e and porosity n were calculated using Equations 2.8 and 2.4 respectively.

The in situ soil is assumed to be saturated, so Equations 2.23 and 2.24 were used to calculate the

thermal conductivity. The soil sample has a void ratio increase of 0.15 and is therefore unsaturated.

Equations 2.26, 2.27 and 2.28 were used to calculate the sample thermal conductivity. As the

specific volume of water xw lies in the range 0.09 < xw < n, the shape factors were calculated using

Equations 2.25 and 2.29.

This gave a final thermal conductivity of 1.37 Wm−1K−1and 1.20 Wm−1K−1for the in situ and

sample soil respectively, which is a decrease of 12%.

180

Figure I.1: Typical result of an oedometer test, with void ratio plotted against the natural log-arithm of vertical effective stress. The preconsolidation pressure and the changein void ratio are shown.

Table I.1: Values used in de Vries model calculation.

λa +λvs 0.1 Wm−1K−1

λs1 2 Wm−1K−1

λw2 0.6 Wm−1K−1

w 0.2

Gs3 2.60

In situ Sample

e 0.52 0.67

n 0.342 0.401

xa 0 0.090

xs 0.658 0.599

xw 0.342 0.311

ga,gb 0.125 0.266

gc 0.750 0.467

F 0.637 –

Fs – 0.637

Fa – 1.166

λ 1.37 Wm−1K−1 1.20 Wm−1K−1

1 Brigaud and Vasseur (1989).2 Howatson et al. (1991).3 Averaged from soil classification tests.

181

Appendix J

PILESIM2 parameters

Simulation Parameters

Month for Simulation Start January

Length of Simulation Twenty Years Simulation

Output Results

Time interval for output results Monthly results

Print hourly values for last year Yes

Weather Data and Loading Conditions

File name (look in the piledata directory): C:\PILESIM2\PILEDATA\Paddington1.pil

Get information on input file format (cursor in next

box and press F1)

0

System Type

System design Heating and geocooling or cooling ma-

chine

Annual Energies and Temperature Levels

Annual energy demand for heating (ignored if the

input data are not normalised)

0 MWh

Scaling factor for heating demand [ScaleH] 1 -

Annual energy demand for cooling (ignored if the

input data are not normalised)

0 MWh

Scaling factor for cooling demand [ScaleC] 1 -

183

Outdoor air temperature for heating design 15 C

Design forward fluid temp. for heating [TfoHea] -99 C

Design forward fluid temp. for cooling [TfoCol] -99 C

Design forward-return temp. difference in cooling

distribution [dTGeocool]

3 K

Relevant temperature level for geocooling operation forward fluid temperature to cooling

distribution

Temperature Limitations

Minimum allowed temperature of the heat carrier

fluid in the piles/boreholes [TfMin]

-1 C

Maximum allowed temperature of the heat carrier

fluid in the piles/boreholes [TfMax]

30 C

Heat Pump and Cooling Machine

Design electric power of the heat pump [Pel] 105 kW

Design performance coefficient [COPo] 4 -

Constant COP and efficiency during simulation No

Design inlet fluid temperature in evaporator 4 C

Design outlet fluid temperature from condenser 40 C

Temperature difference for COP reduction 50 K

Temperature difference for COP stagnation 150 K

Maximum possible COP (PAC and cooling ma-

chine)

4.5 -

Penalty on the COP (PAC and cooling machine) 0.5 -

Design electric power of the cooling machine [Pel-

COM]

80 kW

Design efficiency of the cooling machine [EffCOM] 3 -

Design inlet fluid temperature in evaporator 5 C

Design outlet fluid temperature from condenser 40 C

Design inlet-outlet temp. difference in evaporator

(PAC and cooling machine)

4 K

184

Design inlet-outlet temp. difference in condenser

(PAC and cooling machine)

10 K

Counterflow Heat Exchangers

Design temperature loss in cooling heat exchanger

[LossTCool]

0 K

Design temperature loss in geocooling heat ex-

changer [LossTGeo]

0.5 K

Interface Ground-Building

Room air temperature in building [TairH] 25 C

Height of the cellar between rooms and ground

[Hfloor]

4 m

Air change rate in the cellar [AchRat] 0 1/h

Global room-cellar heat transfer coefficient

[UCelBu]

10000 W/m2K

Insulation thickness between ground and cellar

[Hinsul]

0 m

Concrete thickness between ground and cellar

[Hmagco]

1 m

Length of the horizontal pipes on ground [LCOEPF] 1200 m

Energy Piles or Borehole Heat Exchangers

Diameter of pile/borehole type 1 [dp1] 1.2 m

Number of piles/boreholes for type 1 [N1] 43 -

Average active length of piles/boreholes type 1 [H1] 40 m

Thermal resistance Rb of pile/borehole type 1 [Rb1] 0.06 K/(W/m)

Internal thermal resistance Ra of pile/borehole type

1 [Ra1]

0.104 K/(W/m)

Diameter of pile/borehole type 2 1.5 m

Number of piles/boreholes for type 2 27 -

Average active length of piles/boreholes type 2 45 m

Thermal resistance Rb of pile/borehole type 2 0.06 K/(W/m)

185

Internal thermal resistance Ra of pile/borehole type

2

0.104 K/(W/m)

Average spacing between the piles/boreholes

[BPILE]

5.1 m

Number of piles/boreholes coupled in series

[NSERIE]

1 -

Pipe configuration in pile/borehole U-pipe configuration

Pipe number in a cross section of a pile/borehole 8 -

Inner diameter of one pipe 20.4 mm

Fraction of pile/borehole concrete/filling material

thermal capacity

50 %

Ground Characteristics

Initial ground temperature [TGRDIN] 12 C

Mean temperature gradient in the undisturbed

ground [dTGRND]

0 K/km

Thermal conductivity of ground layer 1 [LG1] 1.5 W/mK

Volumetric thermal capacity of ground layer 1

[CG1]

2.2 MJ/m3K

Thickness of ground layer 1 1000 m

Darcy velocity of ground water in layer 1 [DA1] 0 m/day

Simulate forced convection on global process No

Simulate forced convection on local process No

186

References

Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions With Formulas,

Graphs, and Mathematical Tables, U.S. Government Printing Office.

Abu-Hamdeh, N. H. (2001), ‘Measurement of the thermal conductivity of sandy loam and clay loam

soils using single and dual probes’, Journal of Agricultural Engineering Research 80, 209–216.

Abu-Hamdeh, N. H. and Reeder, R. C. (2000), ‘Soil thermal conductivity: effects of density,

moisture, salt concentration, and organic matter’, Soil Science Society of America Journal

64, 1285–1290.

Adam, D. (2010), Ground-source heating and cooling system for Crossrail Paddington integrated

project London, Technical report, Geotechnik Adam.

Adam, D. and Markiewicz, R. (2009), ‘Energy from earth-coupled structures, foundations, tunnels,

and sewers’, Geotechnique 59(3), 229–236.

Al Nakshabandi, G. and Kohnke, H. (1965), ‘Thermal conductivity and diffusivity of soils as related

to moisture tension and other physical properties’, Agricultural Meteorology 2, 271–279.

Alrtimi, A. A., Rouainia, M. and Manning, D. A. C. (2013), ‘Thermal enhancement of PFA–based

grout for geothermal heat exchangers’, Applied Thermal Engineering 54, 559–564.

ASTM International (2014), D 5334–14 Standard Test Method for Determination of Thermal

Conductivity of Soil and Soft Rock by Thermal Needle Probe Procedure, ASTM International,

West Conshohocken, PA.

Austin III, W. A. (1998), Development of an in situ system for measuring ground thermal properties,

Master’s thesis, Oklahoma State University.

Banks, D. (2008), An introduction to thermogeology: ground source heating and cooling, Blackwell

Publishing Ltd.

Beckman Coulter, Inc. (2011), Coulter LS Series Product Manual.

Bloomer, J. R. (1981), ‘Thermal conductivities of mudrocks in the United Kingdom’, Quarterly

Journal of Engineering Geology and Hydrogeology 14, 357–362.

187

Bouguerra, A., Laurent, J. P., Goual, M. S. and Queneudec, M. (1997), ‘The measurement of the

thermal conductivity of solid aggregates using the transient plane source technique’, Journal of

Physics D: Applied Physics 30, 2900–2904.

Brandl, H. (2006), ‘Energy foundations and other thermo-active ground structures’, Geotechnique

56, 81–122.

Brettmann, T. and Amis, T. (2011), Thermal Conductivity Evaluation of a Pile Group Using

Geothermal Energy Piles, in ‘Proceedings of the Geo-Frontiers Conference’, Dallas, Texas.

Brigaud, F. and Vasseur, G. (1989), ‘Mineralogy, porosity and fluid control on thermal conductivity

of sedimentary rocks’, Geophysical Journal 98, 525–542.

Bristow, K. L. (1998), ‘Measurement of thermal properties and water content of unsaturated sandy

soil using dual-probe heat-pulse probes’, Agricultural and Forest Meteorology 89, 75–84.

Bristow, K. L., Kluitenberg, G. J. and Horton, R. (1994), ‘Measurement of Soil Thermal Properties

with a Dual-Probe Heat-Pulse Technique’, Soil Science Society of America Journal

58, 1288–1294.

British Standards Institution (1990), BS 1377:1990 Methods of test for soils for civil engineering

purposes, BSI, London.

British Standards Institution (2001a), BS EN 12664:2001 Thermal performance of building

materials and products – Determination of thermal resistance by means of guarded hot plate and

heat flow meter methods – Dry and moist products of medium and low thermal resistance, BSI,

London.

British Standards Institution (2001b), BS EN 12667:2001 Thermal performance of building

materials and products – Determination of thermal resistance by means of guarded hot plate and

heat flow meter methods – Products of high and medium thermal resistance, BSI, London.

British Standards Institution (2007), BS EN 15450:2007 Heating systems in buildings – Design of

heat pump heating systems, BSI.

British Standards Institution (2012), BS EN ISO 22007–2:2012 Plastics – Determination of thermal

conductivity and thermal diffusivity – Part 2: Transient plane heat source (hot disc) method, BSI.

C-Therm Technologies (2015), ‘C-Therm TCi Thermal Conductivity Analyzer’.

URL: http://www.ctherm.com/products/tci_thermal_conductivity/

Campbell, G. S., Calissendorff, C. and Williams, J. H. (1991), ‘Probe for measuring soil specific

heat using a heat-pulse method’, Soil Science Society of America Journal 55, 291–293.

Carslaw, H. S. and Jaeger, J. C. (1959), Conduction of heat in solids, Oxford University Press.

Chen, S. X. (2008), ‘Thermal conductivity of sands’, Heat Mass Transfer 44, 1241–1246.

188

http://www.ctherm.com/products/tci_thermal_conductivity/

Chiasson, A. D., Rees, S. J. and Spitler, J. D. (2000), ‘A preliminary assessment of the effects of

groundwater flow on closed-loop ground-source heat pump systems’, ASHRAE Transactions

106, 380–393.

Clarke, B. G., Agab, A. and Nicholson, D. (2008), ‘Model specification to determine thermal

conductivity of soils’, Geotechnical Engineering 161, 161–168.

Cote, J. and Konrad, J.-M. (2009), ‘Assessment of structure effects on the thermal conductivity of

two-phase porous geomaterials’, International Journal of Heat and Mass Transfer 52, 796–804.

De Vries, D. A. (1952), ‘The thermal conductivity of soil’, Mededelingen van de

Landbouwhogeschool te Wageningen 52(1), 1–73.

De Vries, D. A. (1963), Thermal Properties of Soils, in W. R. van Wijk, ed., ‘Physics of Plant

Environment’, North-Holland Publishing Company, chapter 7.

Decagon Devices, Inc. (2011), KD2 Pro Thermal Properties Analyzer: Operator’s Manual Version

10.

Department for Environment Food & Rural Affairs (2014), ‘Government conversion factors for

company reporting’.

URL: http://www.ukconversionfactorscarbonsmart.co.uk/LandingPage.aspx

Department of Energy & Climate Change (2011), The Carbon Plan: Delivering our low carbon

future, HM Government.

Department of Energy & Climate Change (2014), ‘Energy consumption in the UK: All ECUK data

tables in thousand tonnes of oil equivalent (ktoe)’.

URL: https://www.gov.uk/government/statistics/energy-consumption-in-the-uk

Energy Saving Trust (2014a), ‘Ground source heat pumps’.

URL: http:

//www.energysavingtrust.org.uk/domestic/content/ground-source-heat-pumps

Energy Saving Trust (2014b), ‘Our Calculations’.

URL: http://www.energysavingtrust.org.uk/domestic/content/our-calculations

Eschel, G., Levy, G. J., Mingelgrin, U. and Singer, M. J. (2004), ‘Critical evaluation of the use of

laser diffraction for particle-size distribution analysis’, Soil Science Society of America Journal

68(3), 736–743.

Eskilson, P. (1987), Thermal Analysis of Heat Extraction Boreholes, PhD thesis, University of Lund.

Farouki, O. (1986), Thermal properties of soils, Series on rock and soil mechanics, Trans Tech.

Gasparre, A. (2005), Advanced Laboratory Characterisation of London Clay, PhD thesis, Imperial

College London.

Gehlin, S. (1998), Thermal Response Test – In Situ Measurements of Thermal Properties in Hard

Rock, Licentiate thesis, Lulea University of Technology.

189

http://www.ukconversionfactorscarbonsmart.co.uk/LandingPage.aspx

https://www.gov.uk/government/statistics/energy-consumption-in-the-uk

http://www.energysavingtrust.org.uk/domestic/content/ground-source-heat-pumps

http://www.energysavingtrust.org.uk/domestic/content/ground-source-heat-pumps

http://www.energysavingtrust.org.uk/domestic/content/our-calculations

Graham, J. (2006), ‘The 2003 R.M. Hardy Lecture: Soil parameters for numerical analysis in clay’,

Canadian Geotechnical Journal 43, 187–209.

Ground Source Heat Pump Association (2012), ‘Thermal Pile Design, Installation & Materials

Standards’.

Ham, J. M. and Benson, E. J. (2004), ‘On the Construction and Calibration of Dual-Probe Heat

Capacity Sensors’, Soil Science Society of America Journal 68, 1185–1190.

Hellstrom, G. (1983), Comparison between theoretical models and field experiments for ground heat

systems, in ‘Proceedings International Conference on Subsurface Heat Storage in Theory and

Practice’, Vol. 2, Swedish Council for Building Research.

Hellstrom, G. (1989), Duct Ground Heat Storage Model, Manual for Computer Code, Department

of Mathematical Physics, University of Lund, Lund, Sweden.

Hellstrom, G. (1991), Ground Heat Storage: Thermal Analyses of Duct Storage Systems.

Hemmingway, P. and Long, M. (2012), ‘Interpretation of In Situ and Laboratory Thermal

Measurements Resulting in Accurate Thermogeological Characterisation’, Geotechnical and

Geophysical Site Characterization 4, 1779–1787.

Hermansson, A., Charlier, R., Collin, F., Erlingsson, S., Laloui, L. and Srsen, M. (2009), ‘Heat

Transfer in Soils’, Water in Road Structures 5, 69–79.

Hiraiwa, Y. and Kasubuchi, T. (2000), ‘Temperature dependence of thermal conductivity of soil over

a wide range of temperature (5-75C)’, European Journal of Soil Science 51, 211–218.

Howatson, A. M., Lund, P. G. and Todd, J. D. (1991), Engineering Tables and Data, 2nd edn,

Chapman & Hall, Oxford.

Hukseflux Thermal Sensors (2003), TP02 Non-Steady-State Probe for Thermal Conductivity

Measurement – manual v0908, Hukseflux Thermal Sensors, Delft.

Hukseflux Thermal Sensors (2011), ‘TP02 Non-Steady-State Probe for Thermal Conductivity

Measurement’.

URL: http://www.hukseflux.com/products/thermalConductivity/tp02.html

Incropera, F. P., DeWitt, D. P., Bergman, T. L. and Lavine, A. S. (2007), Fundamentals of Heat and

Mass Transfer, 6 edn, John Wiley & Sons Inc.

Institute of Electrical and Electronics Engineers, Inc (1996), IEEE Std 442–1981 Guide for Soil

Thermal Resistivity Measurements, IEEE, New York.

International Energy Agency (2013), IEA ECES Annex 21: Thermal Response Test (TRT) Final

Report, Technical report, IEA.

International Organization for Standardization (1991), ISO 8302:1991 Thermal insulation –

Determination of steady-state thermal resistance and related properties – Guarded hot plate

apparatus, ISO, Geneva.

190

http://www.hukseflux.com/products/thermalConductivity/tp02.html

Jablite Intelligent Insulation (2014), ‘Jablite Classic Expanded Polystyrene Technical Information’.

URL: http://www.jablite.co.uk/site/datasheets/6/6.pdf

Javed, S. and Fahlen, P. (2011), ‘Thermal response testing of a multiple borehole ground heat

exchanger’, International Journal of Low–Carbon Technologies 6, 141–148.

JCGM (2008), Evaluation of measurement data - Guide to the expression of uncertainty in

measurement, JCGM, Paris.

Johansen, O. (1975), Thermal conductivity of soils, Technical report, DTIC Document.

Katsura, T., Nagano, K., Takeda, S. and Shimakura, K. (2006), Heat transfer experiment in the

ground with ground water advection, in ‘ECOSTOCK 10th international conference on thermal

energy storage’.

Kavanaugh, S. P. (2000), ‘Field Tests for Ground Thermal Properties - Methods and Impact on

Ground-Source Heat Pump Design’, ASHRAE Transactions 106(1), 851–855.

King, W., Banks, D. and Findlay, J. (2012), ‘Field determination of shallow soil thermal conductivity

using a short-duration needle probe test’, Quarterly Journal of Engineering Geology and

Hydrogeology 45(4), 497–504.

Klein, S. A. (1998), TRNSYS. A Transient System Simulation Program. Version 14.2., Solar Energy

Laboratory, University of Wisconsin, Madison, USA.

Kluitenberg, G. J., Ham, J. M. and Bristow, K. L. (1993), ‘Error analysis of the heat pulse method for

measuring soil volumetric heat capacity’, Soil Science Society of America Journal 57, 1444–1451.

Loveridge, F. A. (2012), The Thermal Performance of Foundation Piles used as Heat Exchangers in

Ground Energy Systems, PhD thesis, University of Southampton.

Loveridge, F. and Powrie, W. (2012), ‘Pile heat exchangers: thermal behaviour and interactions’,

Geotechnical Engineering 166, 178–196.

Loveridge, F., Powrie, W. and Nicholson, D. (2014), ‘Comparison of two different models for pile

thermal response test interpretation’, Acta Geotechnica 9(3), 367–384.

URL: http://dx.doi.org/10.1007/s11440-014-0306-3

Low, J. E., Loveridge, F. A. and Powrie, W. (2013), Measuring soil thermal properties for use in

energy foundation design, in ‘Proceedings of the 18th International Conference on Soil Mechanics

and Geotechnical Engineering’, Paris.

Low, J. E., Loveridge, F. A., Powrie, W. and Nicholson, D. (2015), ‘A comparison of laboratory and

in situ methods to determine soil thermal conductivity for energy foundations and other ground

heat exchanger applications’, Acta Geotechnica 10, 209–218.

Lund, J. W., Freeston, D. H. and Boyd, T. L. (2010), Direct Utilization of Geothermal Energy 2010

Worldwide Review, in ‘Proceedings World Geothermal Congress 2010’.

191

http://www.jablite.co.uk/site/datasheets/6/6.pdf

http://dx.doi.org/10.1007/s11440-014-0306-3

Midttømme, K. and Roaldset, E. (1998), ‘The effect of grain size on thermal conductivity of quartz

sands and silts’, Petroleum Geoscience 4, 165–172.

Midttømme, K., Roaldset, E. and Aagaard, P. (1998), ‘Thermal conductivity of selected claystones

and mudstones from England’, Clay Minerals 33, 131–145.

Mie, G. (1908), ‘Beitrage zur optik truber medien, speziell kolloidaler metallosungen’, Anaalen der

Physik 25(3), 377–445.

Mitchell, J. K. and Kao, T. C. (1978), ‘Measurement of soil thermal resistivity’, Journal of the

Geotechnical Engineering Division 104, 1307–1320.

Mitchell, J. K. and Soga, K. (2005), Fundamentals of Soil Behavior, 3rd edn, John Wiley & Sons Inc.

Nusier, O. K. and Abu-Hamdeh, N. H. (2003), ‘Laboratory techniques to evaluate thermal

conductivity for some soils’, Heat and Mass Transfer 39, 119–123.

Pahud, D. (2007), PILESIM2 Simulation Tool for Heating/Cooling Systems with Energy Piles or

multiple Borehole Heat Exchangers: User Manual, Istituto Sostenibilita Applicata Ambiente

Costrutivo, Dipartimento Ambiente Costruzioni e Design, Scuola Universitaria Professionale

della Svizzera Italiana, Lugano, Switzerland.

Pahud, D. and Hubbuch, M. (2007), Measured Thermal Performances of the Energy Pile System of

the Dock Midfield at Zurich Airport, in ‘Proceedings European Geothermal Congress’.

Pantelidou, H. and Simpson, B. (2007), ‘Geotechnical variation of London clay across central

London’, Geotechnique 57, 101–112.

Phetteplace, G. (2007), ‘Geothermal Heat Pumps’, Journal of Energy Engineering 133, 32–38.

Philip, J. R. and de Vries, D. A. (1957), ‘Moisture Movement in Porous Materials under

Temperature Gradients’, Transactions, American Geophysical Union 38, 222–232.

Robie, R. A. and Hemingway, B. S. (1991), ‘Heat capacities of kaolinite from 7 to 380 K and of

DMSO-intercalated kaolinite from 20 to 310 K. The entropy of kaolinite Al2Si2O5(OH)4’, Clay

and Clay Minerals 39(4), 362–368.

Sakaguchi, I., Momose, T. and Kasubuchi, T. (2007), ‘Decrease in thermal conductivity with

increasing temperature in nearly dry sandy soil’, European Journal of Soil Science 58(1), 92–97.

Salmon, D. (2001), ‘Thermal conductivity of insulations using guarded hot plates, including recent

developments and sources of reference materials’, Measurement Science and Technology

12, R89–R98.

Signorelli, S., Bassetti, S., Pahud, D. and Kohl, T. (2007), ‘Numerical evaluation of thermal response

tests’, Geothermics 36, 141–166.

Tang, A.-M., Cui, Y.-J. and Le, T.-T. (2007), ‘A study on the thermal conductivity of compacted

bentonites’, Applied Clay Science 41, 181–189.

192

Taylor, J. R. (1997), An Introduction to Error Analysis: The Study of Uncertainties in Physical

Measurements, 2nd ed. edn, University Science Books.

Vaughan, P. R., Chandler, R. J., Apted, J. P., Maguire, W. M. and Sandroni, S. S. (1993), Sampling

disturbance – with particular reference to its effect on stiff clays, in ‘Predictive Soil Mechanics:

Proceedings of the Wroth Memorial Symposium’, pp. 685–708.

Wang, H., Qi, C., Du, H. and Gu, J. (2009), ‘Thermal performance of borehole heat exchanger under

groundwater flow: A case study from Baoding’, Energy and Buildings 41, 1368–1373.

Wen, M., Liu, G., Li, B., Si, B. C. and Horton, R. (2015), ‘Evaluation of a self-correcting dual probe

heat pulse sensor’, Agricultural and Forest Meteorology 200, 203–208.

Witte, H. J. L. (2013), ‘Error analysis of thermal response tests’, Applied Energy 109, 302–311.

Witte, H. J. L., van Gelder, G. J. and Spitler, J. D. (2002), ‘In Situ Measurement of Ground Thermal

Conductivity: A Dutch Perspective’, ASHRAE Transactions 108, 263–272.

Zhang, Y., Gao, P., Yu, Z., Fang, J. and Li, C. (2014), ‘Characteristics of ground thermal properties

in Harbin, China’, Energy and Buildings 69, 251–259.

193

university of southampton research repository eprints soton · ﬁnite element model of the thermal...

Documents