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Modeling and Analysis of Nanostructured Thermoelectric Power Generation and
Cooling Systems
by
Ronil Rabari
A Thesis
presented to
The University of Guelph
In partial fulfilment of requirements
for the degree of
Doctor of Philosophy
in
Engineering
Guelph, Ontario, Canada
© Ronil Rabari, October, 2015
ABSTRACT
MODELING AND ANALYSIS OF NANOSTRUCTURED THERMOELECTRIC
POWER GENERATION AND COOLING SYSTEMS
Ronil Rabari Advisors:
University of Guelph, 2015 Dr. Shohel Mahmud
Dr. Animesh Dutta
This thesis is an investigation of heat transfer processes in nanostructured thermoelectric
(TE) systems. TE systems include solid state thermoelectric generators (TEG) and thermoelectric
coolers (TEC). Current TE systems exhibit low performance (i.e., thermal efficiency and
Coefficient of Performance (COP)) compared to conventional energy conversion devices. The
higher Figure-of-Merit nanostructured TE materials can increase the performance of TE systems.
In this study, a mathematical model of a TE system was developed including the Seebeck,
Peltier, and Thomson effects, Fourier heat conduction, Joule heat, and convection heat transfer.
Numerical simulations were performed using the coupled TE constitutive equations. The
simulated results were expressed as contours of temperature and electric potential and as
streamlines of heat flow and electric current. The effective thermal conductivities, calculated
using different transport property models, were used to investigate nanostructured TE systems.
Additionally, Bismuth-Telluride based nanostructured TE materials were prepared using the
solid state synthesis method.
The study results report parameters which affect the thermal efficiency, COP, and entropy
generation in nanostructured TEG and TEC systems. These parameters include the temperature
difference, electric current, volume fraction of nanoparticles, and convection heat transfer
coefficients at different locations: the side surfaces of TE legs; between the thermal source and
the hot side of a TE system; and between the thermal sink and the cold side of a TE system. The
results show a decrease in the thermal efficiency and COP of a TEG and TEC system,
respectively, as the convection heat transfer coefficient increases. Nevertheless, a TEC system
with a higher electric potential input increases the COP with an increase in the convection heat
transfer coefficient. This study establishes that the heat conduction contribution to the total heat
input for TEG and TEC systems should remain as low as possible for maximum system
performance. The synthesized Bi2Te2.7Se0.3, using the indirect resistance heating method,
exhibited low density which may have contributed to a higher electrical resistivity and a lower
Seebeck coefficient.
The macroscopic modeling of nanostructured TE systems performed in this thesis provided
results which can be applied to the design of next generation thermal management and power
generation solutions.
v
Acknowledgements
I would like to express my sincerest gratitude to Dr. Shohel Mahmud, whose expertise,
understanding, and patience has made the graduate research experience enjoyable. I appreciate
his vast knowledge and skill in different areas and his assistance in preparing research activities
and countless revisions of different parts of this work. He always goes above and beyond typical
advisor in research and advising activities. I would also like to thank co-advisor, Dr. Animesh
Dutta who was always there to listen and give advice. Next, I would like to thank committee
member, Dr. Roydon Fraser, for insightful comments and useful suggestions. I would like to
thank Dr. Sushanta Mitra for being on my Ph.D. examination committee as an external examiner.
Additionally, I would like to thank Dr. William Lubitz and Dr. Fantahun Defersha for being on
my qualifying exam committee and providing constructive feedback. The author is also thankful
to Dr. Douglas Joy, the graduate coordinator for his fruitful suggestions during the qualifying
exam. I would also like to acknowledge the help from Dr. Mohammad Biglarbegian during the
preparation of chapters 2 and 4.
The author would like to express special thanks to the Natural Sciences and Engineering
Research Council (NSERC) and the Ontario Ministry of Agriculture and Food, and Rural Affairs
(OMAFRA) for their financial support. The author is thankful to the Mitacs for the Globalink
Research Award which established new research collaboration with the Indian Institute of
Science. I am extremely thankful to Dr. Ramesh Chandra Mallik at the Department of Physics,
Indian Institute of Science, Bangalore for the technical guidance and help to prepare
nanocomposite thermoelectric materials. Additionally, the use of instruments for surface
analytical techniques and X-ray diffraction at the Indian Institute of Science is very much
appreciated. I would also like to also thank the thermoelectric research group at the Indian
Institute of Science: Dr. Raju Chetty, Dr. Ashoka Bali, Prem Kumar, Sayan Das, and Nilanchal
Patra for their help and technical discussions.
The author would like to thank Mike Speagle for his help in laboratory activities. I would
also like to thank Joel Best, John Whiteside, Ryan Smith, Ken Graham, Nathaniel Groendyk,
Hong Ma, Phil Watson, and David Wright for their help during various stages of this research. I
would also like to thank Laurie Gallinger, Paula Newton, Izabella Onik, Martha Davies, and
Paige Clark for their administrative help. I warmly acknowledge the cooperation and fruitful
vi
discussions with research group at the Advanced Energy Conversion and Control Lab,
University of Guelph: Muath Alomair, Yazeed Alomair, Shariful Islam, Manar Al-Jethelah,
Kaswar Jamil, Rakib Hossain, and Raihan Siddique. Author is also thankful to Tijo Joseph,
Mohammad Tushar, Bimal Acharya, Dr. Poritosh Roy, Jamie Minaret, and Harpreet Kambo at
the University of Guelph and Vaibhav Patel at the Indian Institute of Science for their generous
help.
I am extremely grateful to my parents and grandparents for the unconditional support they
provided during this journey. I must acknowledge encouragement, help, and love from my wife,
Unnati, without her I would not have finished this thesis.
vii
Table of Contents
Cover page i
Abstract ii
Dedication iv
Acknowledgements v
Table of contents vii
List of figures ix
List of tables xix
Chapter 1 Introduction 1
1.1 Background 1
1.2 Objectives 5
1.3 Scope of this thesis 7
1.4 Contribution of present study 11
1.5 Publications from present study 12
Chapter 2 Effect of convection heat transfer on performance of waste heat
thermoelectric generator 13
2.1 Introduction 13
2.2 Heat transfer modeling 16
2.3 Results and discussion 21
2.4 Conclusion 53
2.5 Nomenclature 54
Chapter 3 Numerical simulation of nanostructured thermoelectric generator
considering surface to surrounding convection 56
3.1 Introduction 56
3.2 Mathematical model and boundary conditions 60
3.3 Results and discussion 62
3.4 Conclusion 76
3.5 Nomenclature 77
Chapter 4 Analytical and numerical studies of heat transfer in nanocomposite
thermoelectric cooler 79
viii
4.1 Introduction 79
4.2 Modeling 81
4.3 Results and discussion 86
4.4 Conclusions 135
4.5 Nomenclature 136
Chapter 5 Effect of thermal conductivity on performance of thermoelectric systems
based on effective medium theory 138
5.1 Introduction 138
5.2 Modeling and boundary conditions 144
5.3 Results and discussion 148
5.4 Conclusion 199
5.5 Nomenclature 200
Chapter 6 Analysis of combined solar photovoltaic-nanostructured thermoelectric
generator system 202
6.1 Introduction 202
6.2 Modeling and boundary conditions 205
6.3 Results 213
6.4 Conclusion 235
6.5 Nomenclature 236
Chapter 7 Nanostructuring of n-type Bi2Te2.7Se0.3 based on solid state synthesis
technique 239
7.1 Introduction 239
7.2 Sample preparation and results 241
7.3 Conclusion 248
Chapter 8 Overall conclusions and future work 249
8.1 Overall conclusions 249
8.2 Future work 250
References 251
ix
List of Figures
Figure
number
Title Page
number
1.1 Typical TE modules in power generation mode or cooling mode 3
1.2 Potential applications of TE systems (Pichanusakorn and Bandaru
2010)
4
2.1 Various waste heat recovery methods in context of power plant
(Stehlik 2007, Rowe 1995)
13
2.2 Schematic diagram of location of TEG in combustion system and the
schematic view of unit TEG cell
17
2.3 Temperature distribution over the length of p-type semiconductor leg
with thermal source temperature, HT 700 K and thermal sink
temperature, CT 300 K
24
2.4 Temperature distribution over the length of n-type semiconductor leg
with thermal source temperature, HT 700 K and thermal sink
temperature, CT 300 K
25
2.5 Effect of thermal source temperature on heat input with variable
convection heat transfer coefficient at constant thermal sink
temperature, CT 300 K
27
2.6 Effect of thermal sink temperature on heat input with variable
convection heat transfer coefficient at constant thermal source
temperature, HT 700 K
28
2.7 Power generation as a function of thermal source temperature at
different thermal sink temperature
30
2.8 Effect of thermal source temperature on thermal efficiency with
variable convection heat transfer coefficient at constant thermal sink
temperature, CT 300 K
32
2.9 Effect of thermal sink temperature on thermal efficiency with variable 33
x
convection heat transfer coefficient at constant thermal source
temperature, HT 700 K
2.10 Current as a function of thermal source temperature with different
thermal sink temperature
34
2.11 Effect of convection heat transfer coefficient on thermal efficiency at
constant thermal sink temperature, CT 300 K
36
2.12 Effect of convections between the thermal source and the top surface
and between the sink and the bottom surface of TEG on thermal
efficiency when HT 700 K and CT 300 K with adiabatic side wall
condition
37
2.13 Effect of convections between the thermal source and the top surface
and between the sink and the bottom surface of TEG on thermal
efficiency when HT 700 K and CT 300 K with convection from the
side walls with h 10 Wm-2
K-1
38
2.14 Entropy generation rate as a function of thermal source temperature at
different convection heat transfer coefficients with constant thermal
sink temperature, CT 300 K
42
2.15 Temperature distribution in TEG with adiabatic boundary conditions at
vertical walls of semiconductor legs
45
2.16 Electrical potential in TEG with adiabatic boundary conditions at
vertical walls of semiconductor legs
46
2.17 Temperature distribution in TEG with convective boundary conditions,
h = 20 Wm-2
K-1
at vertical walls semiconductor legs
47
2.18 Electrical potential in TEG with convective boundary conditions, h =
20 Wm-2
K-1
at vertical walls of semiconductor legs
48
2.19 Comparison of heat input, power output, and thermal efficiency
obtained from the current work with the similar results available in
(Angrist 1982)
51
2.20 Comparison of analytical and numerical results in terms of temperature 52
xi
distribution over the p-type semiconductor leg
3.1 Schematic of unit cell of TEG 60
3.2 Contours of temperature distribution and streamlines of heat flow with
adiabatic heat transfer condition (h ≈ 0 W/m2K)
65
3.3 Contours of electric potential and streamlines of electric current flow
with adiabatic heat transfer condition (h ≈ 0 W/m2K)
66
3.4 Contours of temperature distribution and streamlines of heat flow with
adiabatic heat transfer condition (h = 15 W/m2K)
67
3.5 Contours of electric potential and streamlines of electric current flow
with adiabatic heat transfer condition (h = 15 W/m2K)
68
3.6 Contours of temperature distribution and streamlines of heat flow with
adiabatic heat transfer condition (h = 35 W/m2K)
69
3.7 Contours of electric potential and streamlines of electric current flow
with adiabatic heat transfer condition (h = 35 W/m2K)
70
3.8 Contours of temperature distribution and streamlines of heat flow with
adiabatic heat transfer condition (h = 50 W/m2K)
71
3.9 Contours of electric potential and streamlines of electric current flow
with adiabatic heat transfer condition (h = 50 W/m2K)
72
3.10 Comparison of current production using numerical and analytical
techniques
74
3.11 Thermal efficiency of TEG as a function of convection heat transfer
coefficient and temperature difference
75
4.1 Schematic diagram of unit cell of TEC (drawing is not to scale) 82
4.2 The schematic of crystal structure of (a) (Bi1-xSbx)2Te3 (Zhang et al.
2011) Reprinted by permission from Macmillan Publishers Ltd: Nature
Communications from Zhang et al.2, 574 (2011), copyright 2011
83
4.3 The schematic of crystal structure of Bi2Te3 (Chen et al. 2009) From
[Chen et al. Science 325, 178 (2009)]. Reprinted with permission from
AAAS
84
4.4 Temperature distribution over the length of p-type TE leg with hot 89
xii
surface temperature 353 K and cold surface temperature 333 K
4.5 Temperature distribution over the length of n-type TE leg with hot
surface temperature 353 K and cold surface temperature 333 K 90
4.6 Electrical resistivity of p- and n- type legs of nanocomposite TEC 91
4.7 Heat absorbed as a function of current considering hot surface
temperature 353 K with cold surface temperature 333 K
93
4.8 Heat absorbed as a function of current considering hot surface
temperature 353 K with cold surface temperature 343 K
94
4.9 COP of TEC as a function of current considering hot surface
temperature 353 K with cold surface temperature 333 K
96
4.10 COP of TEC as a function of current considering hot surface
temperature 353 K with cold surface temperature 343 K
97
4.11 Heat absorbed as a function of temperature difference with different
electric current input and hot surface temperature 353 K considering
adiabatic side wall condition
99
4.12 COP as a function of temperature difference with different electric
current input and hot surface temperature 353 K considering adiabatic
side wall condition
100
4.13 Maximum heat absorbed of TEC considering variable convection heat
transfer coefficient and variable TE leg heights with hot surface
temperature 353 K and cold surface temperature 333 K
101
4.14 Optimum electric current for maximum heat absorption of TEC
considering variable convection heat transfer coefficient and variable
TE leg heights with hot surface temperature 353 K and cold surface
temperature 333 K
104
4.15 Maximum COP of TEC considering variable convection heat transfer
coefficient and variable TE leg heights with hot surface temperature
353 K and cold surface temperature 333 K
105
4.16 Optimum electric current for maximum COP of TEC considering
variable convection heat transfer coefficient and variable TE leg
106
xiii
heights with hot surface temperature 353 K and cold surface
temperature 333 K
4.17 Internal resistance of TEC unit cell as a function of TE leg height 108
4.18 Maximum heat absorbed as a function of TE leg height by unit cell of
TEC with hot surface temperature 353 K, cold surface temperature 333
K, and adiabatic side wall condition
109
4.19 Electric scalar potential and current flow in nanocomposite TEC with
electric potential 0.02 V
111
4.20 Electric scalar potential and current flow in nanocomposite TEC with
electric potential 0.06 V
112
4.21 Heat flow and temperature distribution in nanocomposite TEC for h ≈
0 Wm-2
K-1
at vertical walls with electric potential 0.02 V
114
4.22 Heat flow and temperature distribution in nanocomposite TEC for h =
20 Wm-2
K-1
at vertical walls with electric potential 0.02 V
115
4.23 Heat flow and temperature distribution in nanocomposite TEC for h =
40 Wm-2
K-1
at vertical walls with electric potential 0.02 V
116
4.24 Heat flow and temperature distribution in nanocomposite TEC for h =
60 Wm-2
K-1
at vertical walls with electric potential 0.02 V
117
4.25 Heat flow and temperature distribution in nanocomposite TEC for h ≈
0 Wm-2
K-1
at vertical walls with electric potential 0.06 V
119
4.26 Heat flow and temperature distribution in nanocomposite TEC for h =
60 Wm-2
K-1
at vertical walls with electric potential 0.06 V
120
4.27 Heat absorbed by nanocomposite TEC as a function of convection heat
transfer coefficient with hot surface temperature 353 K and electric
potential 0.02 V
121
4.28 COP of nanocomposite TEC as a function of convection heat transfer
coefficient with hot surface temperature 353 K and electric potential
0.02 V
123
4.29 Heat absorbed by nanocomposite TEC as a function of convection heat
transfer coefficient with hot surface temperature 353 K and electric
125
xiv
potential 0.06 V
4.30 COP of nanocomposite TEC as a function of convection heat transfer
coefficient with hot surface temperature 353 K and electric potential
0.06 V
126
4.31 Comparison of analytical and numerical simulation results in terms of
heat absorbed considering variable convection heat transfer coefficient
with hot surface temperature 353 K, cold surface temperature 333 K,
and electric potential 0.02 V
128
4.32 Comparison of analytical and numerical simulation results in terms of
COP considering variable convection heat transfer coefficient with hot
surface temperature 353 K, cold surface temperature 333 K, and
electric potential 0.02 V
129
4.33 Comparison of COP using conventional (no nanostructuring) and
nanocomposite TE material considering h ≈ 0 Wm-2
K-1
, hot surface
temperature 353 K, and electric current input of 1 A
131
4.34 Thermal conductivity of conventional (no nanostructuring) and
nanocomposite TE materials
132
4.35 Comparison of results between current work and Poudel et al. (2008) 134
5.1 Different approaches to increase ZT of TE materials (Martin-Gonzalez
2013)
139
5.2 Graphical representation of (a) Maxwell model (b) Hasselman and
Johnson model (c) Minnich and Chen model
142
5.3 Schematic diagram of typical (a) TEC and (b) TEG system 144
5.4 Effective thermal conductivity of p-type and n-type thermoelectric
material based on Maxwell model
150
5.5 Effective thermal conductivity of p-type and n-type thermoelectric
material based on Hasselman-Johnson model
152
5.6 Effect of thermal boundary conductance on effective thermal
conductivity of p-type using Hasselman-Johnson model
154
5.7 Effect of thermal boundary conductance on effective thermal 155
xv
conductivity of n-type using Hasselman-Johnson model
5.8 Effective thermal conductivity of p-type thermoelectric material using
Minnich-Chen model
157
5.9 Effective thermal conductivity of n-type thermoelectric material using
Minnich-Chen model
158
5.10 COP of TEC considering various amount of volume fraction with
Maxwell model
161
5.11 Efficiency of TEG with different volume fraction with Maxwell model 162
5.12 COP of TEC considering different amount of volume fraction with
Hasselman model
164
5.13 Efficiency of TEG with different amount of volume fraction with
Hasselman model
165
5.14 Effect of boundary conductance on performance of TEC based on
Hasselman-Johnson model
166
5.15 Effect of boundary conductance on performance of TEG based on
Hasselman-Johnson model
167
5.16 COP of TEC considering different amount of volume fraction with
Minnich-Chen model
169
5.17 Efficiency of TEG with different volume fraction with Minnich-Chen
model
170
5.18 Effect of nanoparticle size on performance of TEC considering
Minnich-Chen model
171
5.19 Effect of nanoparticle size on performance of TEG considering
Minnich-Chen model
172
5.20 Performance of TEC with variable volume fractions and convection
heat transfer coefficients through side walls of TE legs
174
5.21 Performance of TEG with variable volume fractions and convection
heat transfer coefficients through side walls of TE legs
175
5.22 Influence of effective thermal conductivity on heat conduction in TEC 177
5.23 Influence of effective thermal conductivity on heat conduction in TEG 178
xvi
5.24 Contours of electric potential and streamlines of electric current flow
in TEC
180
5.25 Contours of temperature and streamlines of heat flow in TEC with cold
surface temperature 290 K, hot surface temperature 300 K, and electric
potential 0.02 V with NO particles
182
5.26 Contours of temperature and streamlines of heat flow in TEC with cold
surface temperature 290 K, hot surface temperature 300 K, and electric
potential 0.02 V with 0.8 volume fraction with Maxwell model
183
5.27 Contours of temperature and streamlines of heat flow in TEC with cold
surface temperature 290 K, hot surface temperature 300 K, and electric
potential 0.02 V with 0.8 volume fraction with Hasselman-Johnson
model
184
5.28 Contours of temperature and streamlines of heat flow in TEC with cold
surface temperature 290 K, hot surface temperature 300 K, and electric
potential 0.02 V with 0.8 volume fraction with Minnich-Chen model
185
5.29 Contours of temperature and streamlines of heat flow in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with NO
particles
187
5.30 Contours of temperature and streamlines of heat flow in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with 0.8
volume fraction with Maxwell model
188
5.31 Contours of temperature and streamlines of heat flow in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with 0.8
volume fraction Hasselman-Johnson model
189
5.32 Contours of temperature and streamlines of heat flow in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with 0.8
volume fraction with Minnich-Chen model
190
5.33 Contours of electric potential and streamlines of electric current in
TEG with cold surface temperature 300 K and hot surface temperature
350 K with NO particles
192
xvii
5.34 Contours of electric potential and streamlines of electric current in
TEG with cold surface temperature 300 K and hot surface temperature
350 K with 0.8 volume fraction with Maxwell model
193
5.35 Contours of electric potential and streamlines of electric current in
TEG with cold surface temperature 300 K and hot surface temperature
350 K with 0.8 volume fraction Hasselman-Johnson model
194
5.36 Contours of electric potential and streamlines of electric current in
TEG with cold surface temperature 300 K and hot surface temperature
350 K with 0.8 volume fraction with Minnich-Chen model
195
5.37 Comparison of analytical and numerical simulation results for TEC 197
5.38 Comparison of analytical and numerical simulation results for TEG 198
6.1 Schematic diagram of (a) photovoltaic – thermoelectric (PVTE)
system and (b) unit thermoelectric generator
207
6.2 Exploded view of Solar PV panel layers (Amrani 2007) 207
6.3 Solar PV panel back surface temperature with variable solar radiation
and ambient temperature
218
6.4 Temperature distribution over the length of nanostructured p type
and n type semiconductor leg
219
6.5 Heat input to nanostructured TE generator with different solar radiation
and variable convection heat transfer coefficient
222
6.6 Heat input comparison of TE generator using traditional and
nanostructured material thermoelectric material
223
6.7 Thermal conductivity of traditional and nanostructured TE material as
a function of temperature
224
6.8 Power output from TE generator as a function of solar radiation 226
6.9 Thermal efficiency of nanostructured TE generator with different solar
radiation and variable convection heat transfer coefficient
228
6.10 Thermal efficiency comparison of TE generator with traditional and
nanostructured TE material 229
xviii
6.11 Power output comparison of solar PV panel and TE generator 231
6.12 Solar panel conversion efficiency Vs. Solar Radiation 233
6.13 Combined efficiency of solar PVTE system Vs. Solar Radiation 234
7.1 ZT improvements in low dimensional and bulk TE materials 240
7.2 Rietveld refinement power XRD pattern for Bi2Te2.7Se0.3 242
7.3 Seebeck coefficient and Electrical resistivity of sample Bi2Te2.7Se0.3 243
7.4 Power factor of sample Bi2Te2.7Se0.3 244
7.5 Comparison of power factor between Bi2Te2.7Se0.3 manufactured via
direct current hot press and indirect resistance heating
246
7.6 SEM image of fractured surfaces of hot pressed sample 247
xix
List of Tables
Table
number
Title Page
number
1.1 Comparison of various waste heat recovery methods (BCS 2008) 2
1.2 Contribution of present study 11
2.1 Temperature dependent TE properties of n-type 75% Bi2Te3 25%
Bi2Se3 and p-type 25% Bi2Te375% Sb2Te3 with 1.75% excess Se
(Reddy et al. 2013, Angrist 1982)
22
2.2 Thermal efficiency of single unit of TEG with cold surface
temperature, CT = 300 K
50
3.1 Polynomial functions of Seebeck coefficient, electrical conductivity,
and thermal conductivity as a function of temperature for BiSbTe
nanostructured bulk alloys and Bi2Te3 with SiC nanoparticles (Poudel
et al. 2008, Zhao et al. 2008)
63
4.1 Polynomial functions of Seebeck coefficient, electrical conductivity,
and thermal conductivity as a function of temperature for BiSbTe
nanostructured bulk alloys and nanocomposite Bi2Te3 (Poudel et al.
2008, Fan et al. 2011)
87
4.2 Polynomial functions of Seebeck coefficient, electrical conductivity,
and thermal conductivity as a function of temperature for BiSbTe bulk
alloys and conventional Bi2Te3 (Poudel et al. 2008, Fan et al. 2011)
87
4.3 Height of TE legs and internal resistance of TEC for different cases
considered in Figs. 4.13, 4.14, 4.15, and 4.16
101
5.1 Comparison of different effective medium theories 143
5.2 Material properties (Pattamatta and Madnia 2009) 149
6.1 Operating conditions and dimensional parameters of combined solar
PVTE system
214
6.2 Polynomial functions of Seebeck coefficient, electrical conductivity,
thermal conductivity, and figure of merit with respect to temperature
for nanostructured BiSbTe bulk alloys (Poudel et al. 2008)
215
xx
6.3 Polynomial functions of Seebeck coefficient, electrical conductivity,
thermal conductivity, and figure of merit with respect to temperature
for BiSbTe bulk alloys (Poudel et al. 2008)
216
1
CHAPTER 1: INTRODUCTION
1.1 Background
Forty percent of the world’s energy demand is met by energy conversion systems (e.g., coal
power stations) which convert low-grade energy (e.g., coal) into high-grade energy (e.g.,
electricity) (Zhang et al. 2010). Most of the energy conversion systems produce waste heat
which leads to a lower efficiency of the overall energy conversion process. For example, 60% of
the energy from a power station is lost as a waste heat (EEA 2008). In a similar manner, internal
combustion engines which are used in most of the current transport vehicles waste 60-70% of the
input energy (Nolas et al. 2006; Bottner et al. 2006). The U.S. Department of Energy (DOE
2011) considers increment in the efficiency of energy conversion systems as one of the strategies
to address the energy demand. Waste heat recovery is one of the methods to increase the
efficiency of energy conversion systems which in turns answers the energy challenge.
Waste heat recovery solutions can be applied in different ways depending upon the applications
(BCS 2008). Table 1.1 shows different waste heat recovery solutions in terms of temperature
ranges, advantages, and disadvantages. Waste heat recovery solutions can be broadly classified
into two categories; temperature increment of input fluid and electricity generation. Further,
electricity generation can be classified into two categories: using mechanical systems (i.e.,
moving parts) and direct electricity generation (i.e., without moving parts). An industrial or
power station operation applies an air preheater and economizer to improve the temperature of
the input fluid. Some industries utilize modified versions of the basic Rankine cycle such as the
Organic Rankine Cycle (ORC) and Kalina cycle to generate electricity. The Organic Rankine
Cycle (ORC) and the Kalina cycle use fluids with low boiling temperatures such as propane, iso-
pentane, toluene, and ammonia (BCS 2008). The ORC and Kalina cycle, which use hazardous
chemicals, suffer from low efficiency, and requires continuous maintenance due to rotating parts.
On the other hand, direct electricity generation methods convert thermal energy to electricity
without using any harmful chemicals and moving parts. Some of the techniques include
piezoelectric, thermal-photovoltaic, thermionic, and thermoelectric energy conversion. In
2
comparison with mechanical systems, direct energy conversion methods are simple in structure
and require less maintenance.
Table 1.1 Comparison of various waste heat recovery methods (BCS 2008)
Conversion Technology Temperature
Range Advantages Disadvantages
Use of waste
heat to improve
thermodynamic
properties of
input load
Air Preheater 200-500°C Widely used
in industry.
Increases
overall
efficiency
Unable to
recover all
waste heat; can
only change
state of input
load
Economizer 200-500°C
Waste heat to
electricity
Electricity
generation
through
mechanical
systems
ORC >66 °C
Suitable as a
waste heat
recovery tool
in power
plant
Low efficiency
and uses
hazardous
chemicals
Kalina cycle > 100°C
Uses
ammonia,
can increase
power output
by 20-50%
Use of toxic
fluid and
storage
challenge
Direct
Electricity
Generation
Thermoelectric
generators
(TEG)
50 °C to 1100
°C
No moving
parts, green
and clean
technology
Low efficiency
Piezoelectric
Generators
100°C to
150°C
Converts
vibrations to
electricity
Very low
efficiency and
high cost
Thermionic
Generators > 1000 °C -
Very high
temperature
required
Thermal
photovoltaic
generators
100 °C to
1500 °C
No moving
parts
Low
efficiency,
requires
vacuum
3
Current work focuses on thermoelectric (TE) systems which are fabricated with an equal number
of p-type and n-type semiconductor materials. Semiconductor materials are connected
electrically in series and thermally in parallel as shown in Fig. 1.1.
Figure 1.1 Typical TE modules in power generation mode or cooling mode
TE systems can largely be classified into either thermoelectric generator (TEG) or thermoelectric
cooler (TEC). TEG converts heat into electricity and TEC uses electricity to produce
heating/cooling effect based on the Seebeck and Peltier effects, respectively. TEG works on the
principle of the Seebeck effect. A presence of temperature gradient across TE system establishes
an electric potential due to the Seebeck effect. Electrons in n-type and holes in p-type TE legs
move from hot side to cold side while carrying heat and electrical charge (Terasaki 2011).
Therefore, temperature gradient causes the flow of the electrical current inside a TEG. TEG has a
range of potential applications in military, deep space vehicles, remote power sources for
inhabitable places, and solar and waste heat generators. TEC works on the principle of the Peltier
effect. When the electric current passes through the junction of two materials then it will either
emit or absorb heat. The absorption or emission of heat can be attributed to the difference in the
Heat Rejected Substrates
Metal
connectors
p- type
semiconductor
-
+
n- type
semiconductor
Heat Absorbed
4
chemical potential of the two materials (Terasaki 2011). TEC also has a wide range of potential
applications in electronics, laser diodes, military garments, laboratory cold plates, and
transportation systems. Figure 1.2 illustrates a brief glimpse of potential applications of TE
systems in terms of power usage and generation. TE systems have many advantages such as
simple structure, no moving parts, and quiet in operation (Silva and Kaviany 2004). Moreover,
they have no adverse effect on the environment and have a long service life. TE systems can
easily work in tandem with conventional technologies under a wide temperature range which
makes them an ideal candidate for the waste heat recovery. However, because of their low
conversion efficiency, current TE systems have limited applications (Gonzalez et al. 2013).
Figure 1.2 Potential applications of TE systems (Pichanusakorn and Bandaru 2010)
In order to better understand TE systems, the coupled effects of heat transfer and electric
potential need to be understood. In most of the work, a heat transfer model does not account for
all internal and external heat transfer irreversibilities. It is well established that irreversibilities in
100 W
10 W
1 W
100 mW
10 mW
1 mW
100 µW
10 µW
1 µW
1 kW
Power 10 kW
Low power applications; wrist-watches
Biomedical devices; pacemakers
Remote wireless sensors
Deep space vehicles power and cooling
Consumer refrigerators, electronic cooling
Waste heat recovery; (e.g., powerstation, automotive,
redundant oil wells and nuclear facility)
5
energy conversion systems can increase the entropy and lead to the destruction of the exergy
(Bermejo 2013). In addition to this, TE systems suffer from low conversion efficiency due to the
poor material properties. Recent advancements in nanostructuring of TE materials have enhanced
the material properties and improved the conversion efficiency (Ma et al. 2013). One of the ways
to improve the TE properties is to lower the thermal conductivity (Bhandari 1995). A low-
dimensional (e.g., 0D, 1D, 2D) nanostructured TE material with the lower thermal conductivity
is costly to manufacture and difficult to apply in real world applications. However, a new set of
TE bulk nanostructured materials called “nanocomposites” offer an alternative to low-
dimensional TE structures. Therefore, the present research aims to study heat transfer in
nanocomposite TEG and TEC. The work develops mathematical models and numerical
simulations to study the heat transfer in nanocomposite TE systems. This study provides
parameters (e.g., temperature difference, convection heat transfer coefficient, electric current,
volume fraction of nanoparticles) affecting the performance of nanocomposite TE systems (e.g.,
thermal efficiency, Coefficient of Performance (COP)). This work also shows the effects of
combining TE systems with conventional energy conversion systems. Further, experimental
work also shows how critical parameters (e.g., temperature, pressure, heating method) affect the
properties of nanocomposite TE legs.
1.2 Objectives
The overall objective of this research is to develop heat transfer models of nanostructured TE
systems. The performance of nanostructured TE systems will be evaluated both quantitatively
and qualitatively through a 1D analytical model and a 2D numerical analysis, respectively. In
addition to this, a nanostructured TE system will be combined with conventional energy
conversion systems (solar panel) and the performance will be quantified using both
nanostructured and non-nanostructured TE materials. Furthermore, nanostructured TE materials
will be developed by the solid-state synthesis technique.
The specific objectives include:
Modeling and analysis of the heat transfer and irreversibility in TEG systems with
temperature dependent material properties.
6
o Developing a heat transfer model including the Seebeck, Peltier, and Thomson effects,
Joule heating, Fourier heat conduction, and convection heat losses through side walls of
the TE legs.
o Studying entropy generation within the TEG.
o Studying effects of heat transfer on thermal efficiency and power output of TEG.
Modeling and analysis of the heat transfer and irreversibility of TEG numerically with
temperature dependent TE properties with non-nanostructured and nanostructured materials.
o Identify the effects of internal irreversibilities on the performance of non-nanostructured
and nanostructured TE system.
Modeling and analysis of the heat transfer and irreversibilities in a nanocomposite TEC
o Developing heat transfer model including the Seebeck, Peltier, and Thomson effects,
Joule heating, Fourier heat conduction, and convection heat transfer from side walls of
TE legs in TEC
o Studying effects of convection heat transfer on the performance of TEC (COP, heat
absorbed, and optimum electric current) with temperature dependent properties
o Numerical simulations of TEC with variable convection heat transfer coefficient from
side surfaces of TE legs
Investigation of nanocomposite TE system using effective thermal conductivity based on
Effective Medium Theory (EMT)
o Investigating nanoparticles’ effect on the thermal conductivity of TE nanocomposite
materials; effects of volume fraction, size and shape of nanoparticles, and thermal
conductivities of nanoparticles and base material.
o Investigating nanoparticles’ effects on the heat transfer, thermal efficiency, and COP of
TE nanocomposite systems.
Investigation of TEG in combination with conventional energy conversion system
o Identifying performance curves of TE system using non-nanostructured and
nanostructured TE materials.
o Investigating the performance of combined energy conversion system based on common
input parameters such as energy input and temperature.
Nanostructuring of n-type Bi2Te2.7Se0.3 based on solid state synthesis technique
7
o Fabricating Bismuth-Telluride based TE legs using the indirect resistance heating method
o Measuring Seebeck coefficient and electrical resistivity of nanocomposite TE leg
1.3 Scope of this thesis
The work is divided around overall objective but separated into several chapters which can stand
on its own. Due to different system considerations, the nomenclature is different for each
chapter. The research comprises six chapters, scopes of which are explained in this section.
Chapter 2: Effect of convection heat transfer on performance of waste heat thermoelectric
generator
Efficiency of energy conversion processes can be improved if waste heat is converted to
electricity. A TEG can directly convert waste heat to electricity. The TEG typically suffers from
low efficiency due to various reasons, such as ohmic heating, surface-to-surrounding convection
losses, and unfavorable material properties. In this work, the effect of surface-to-surrounding
convection heat transfer losses on the performance of TEG is studied analytically and
numerically. A one-dimensional (1-D) analytical model is developed that includes surface
convection, conduction, ohmic heating, Peltier, Seebeck, and Thomson effects with top and
bottom surfaces of TEG exposed to convective boundary conditions. Using the analytical
solutions, different performance parameters (e.g., heat input, power output, and efficiency) are
calculated and expressed graphically as functions of thermal source and sink temperatures and
convection heat transfer coefficient. Finally, a two-dimensional (2-D) mathematical model is
solved numerically to observe qualitative results of thermal and electric fields inside the TEG.
For all calculations, temperature dependent thermal/electric properties are considered. Increase in
thermal source temperature results in an increase in the power output with adiabatic side wall
conditions. A change in boundary condition to convection heat transfer from adiabatic boundary
has a large impact on thermal efficiency.
Chapter 3: Numerical simulation of nanostructured thermoelectric generator considering
surface to surrounding convection
8
TE systems can directly convert heat to electricity and vice-versa by using semiconductor
materials. Therefore, coupling between heat transfer and electric field potential is important to
predict the performance of TEG systems. This work develops a general two-dimensional
numerical model of a TEG system using nanostructured TE semiconductor materials. A TEG
with p-type nanostructured material of Bismuth Antimony Telluride (BiSbTe) and n-type
Bismuth Telluride (Bi2Te3) with 0.1 vol% Silicon Carbide (SiC) nanoparticles is considered for
performance evaluations. Coupled TE equations with temperature dependent transport properties
are used after incorporating Fourier heat conduction, Joule heating, Seebeck effect, Peltier effect,
and Thomson effect. The effects of temperature difference between the hot and cold junctions
and surface to surrounding convection on different output parameters (e.g., thermal and electric
fields, power generation, thermal efficiency, and current) are studied. Selected results obtained
from current numerical analysis are compared with the results obtained from the analytical
model available in the literature. There is a good agreement between the numerical and analytical
results. The numerical results show that as temperature difference increases output power and
amount of current generated increases. Moreover, it is quite apparent that convective boundary
condition deteriorates the performance of TEG.
Chapter 4: Analytical and numerical studies of heat transfer in nanocomposite
thermoelectric cooler
TEC can produce a cooling effect (in refrigerator mode) or a heating effect (in heat pump mode)
using electrical energy input. Performance characteristics of typical TECs are poor when
compared to the traditional cooling system (e.g., vapor compression system). However,
nanostructuring of TE materials can generate high-performance TE materials (e.g., high Seebeck
coefficient, low thermal conductivity, and high electrical conductivity), and such materials show
the promise to improve the performance of TEC. The main objective of this work is to
investigate the effect of nanocomposite TE materials and surface to surrounding convection heat
transfer on the thermal performance of TEC. The mathematical model developed in this work
includes Fourier heat conduction, Joule heat, Seebeck effect, Peltier effect, and Thomson effect.
This model also includes temperature dependent transport properties. Governing transport
equations are solved numerically using the finite element method to identify the temperature and
9
electrical potential distributions and to calculate heat absorbed and the COP. Heat absorbed and
COP are also calculated using a simplified 1D analytical solution and compared with
numerically obtained results. An optimum electric current is also calculated for maximum heat
absorption rate and maximum COP for fixed geometric dimensions and variable convection heat
transfer coefficients. An increase in the convection heat transfer coefficient increases the
optimum electric current required for maximum heat absorption rate and maximum COP. For the
materials considered, results show that COP of TEC can be increased approximately by 13% ±
1% if nanostructured TE materials are used instead of the conventional TE materials.
Chapter 5: Effect of thermal conductivity on performance of thermoelectric systems based
on effective medium theory
Currently, TE systems have very low efficiency due to unfavorable TE properties (e.g., high
thermal conductivity and low power factor). Figure of Merit is a measure of TE material’s
performance which suggests that relatively lower thermal conductivity of TE materials can
improve the performance (e.g., efficiency and coefficient of performance) of TE systems. A bulk
composite TE material made-up of TE micro or nanoparticles and base TE materials can have
low thermal conductivity. There are various models reported in the literature based on the EMT
which can predict the thermal conductivity of composites. In this work, three different models
based on the EMT are applied to investigate the performance of TEG and TEC. These models
are Maxwell model, Hasselman - Johnson model, and Minnich - Chen model. Analytical
modeling and numerical simulations have been performed to evaluate the performance (e.g.,
COP and thermal efficiency) of TE systems. Thermal efficiency of TEG increases from 2.06% to
5.59%, which is 170% rise when thermal conductivity of composite decreases from 1.1 Wm-1
K-1
to 0.11 Wm-1
K-1
based on the Minnich – Chen model with a particle size of 100 nm. An increase
in the thermal efficiency or COP can be attributed to a reduction in Fourier heat conduction
contribution to total heat input which leads to increase in total heat input. Results also show that
the performance of TE systems significantly depends on the size and volume fraction of
particles.
10
Chapter 6: Analysis of combined solar photovoltaic-nanostructured thermoelectric
generator system
In this work, a combined solar photovoltaic and TE generator system is investigated. A TE
generator converts the temperature gradient into electricity that improves the overall
performance of the combined system. A nanostructured BiSbTe TE material is used in this
investigation and its power generation performance is compared with non-nanostructured
BiSbTe TE material. Using analytical solutions, different performance parameters (e.g. heat
input, power output, efficiency) are calculated and expressed graphically as a function of solar
radiation and convection heat transfer coefficient. In addition to this, different performance
parameters were also compared between non-nanostructured and nanostructured TE materials.
The nanostructured TE material leads to improvement in the performance of a TE generator due
to reduction in the thermal conductivity and an improvement in the electrical conductivity. The
TE generators have a large impact on the overall efficiency of a combined system at higher solar
radiation.
Chapter 7: Nanostructuring of n-type Bi2Te2.7Se0.3 based on solid state synthesis technique
In this work, nanocomposite TE legs were prepared using the solid state synthesis technique.
Bismuth-telluride based TE alloys have been doped with selenium for a sample preparation.
Bismuth telluride based alloys are currently the best TE materials at room temperature
applications in the areas of refrigeration and air-conditioning. The powder X-ray diffraction was
performed with the Rietveld refinement. The powder was hot pressed using the indirect
resistance heating method which is relatively cost effective method compared to the direct
current hot press. The temperature dependent Seebeck coefficient (α) and electrical resistivity (ρ)
were measured which showed a rise in the electrical resistivity as temperature rise. The reason
behind low power factor (α2ρ
-1) may be low-density sample and grains without preferred
orientation which were influenced by the pressure and temperature. This study shows that at this
stage direct current hot press method remains the cost-effective and easy to manufacture method
to make nanocomposite TE legs.
11
1.4 Contribution of present study
Table 1.2 Contribution of present thesis
Nanocomposite TE systems
Research
topic
Heat Transfer TE material properties
Research gap
No study performing numerical
simulations of nanostructured
TEG and TEC systems
Limited study investigating
effects of internal and external
heat transfer irreversibilities on
thermal efficiency and COP of
TEG and TEC systems
No study investigating effects of
effective thermal conductivity on the
performance of nanostructured TE
systems
Limited studies on fabrication of
nanostructured TE legs using
nanopowders based on solid state
synthesis technique (indirect
resistance heating)
Limited studies on the application of nanostructured TE systems combined with
conventional energy conversion systems (solar panels)
Contribution
First to report mathematical
model of heat transfer considering
Seebeck, Peltier, and Thomson
effects, Joule heating, Fourier
heat conduction, and convection
heat transfer with top and bottom
surfaces of TEG exposed to
convective boundary conditions
First study investigating effects of
convection heat transfer from side
surfaces of TE legs on the
performance of TEC (COP, heat
absorbed, optimum electric
current)
First study investigating effective
thermal conductivity (volume
fraction, particle size) of TE materials
derived from different transport
property models based on EMT
First study investigating effects of
effective thermal conductivity on the
performance (COP and thermal
efficiency) of nanostructured TEG
and TEC
Performed experiments which shows
effects of synthesis method
(temperature, pressure, heating rate
and method) on the material
properties and microstructure of TE
materials
First to perform numerical simulations of nanostructured TE systems and its
applications combining with conventional energy conversion systems
The overall study has increased the understanding of heat transfer in
nanocomposite TE systems which can be applied to design the next
generation thermal management and power generation solutions
12
1.5 Publications from present study
Parts of this thesis have been published in peer-reviewed international journals. Chapters 2 to 5
have been published and chapter 6 is currently under review.
1. R. Rabari, S. Mahmud, A. Dutta, M. Biglarbegian, 2015, Effect of convection heat transfer
on performance of waste heat thermoelectric generator, Heat Transfer Engineering, 36, 1458-
1471.
2. R. Rabari, S. Mahmud, A. Dutta, 2014, Numerical simulation of nanostructured
thermoelectric generator considering surface to surrounding convection, International
Communications in Heat and Mass Transfer, 56, 146-151.
3. R. Rabari, S. Mahmud, A. Dutta, M. Biglarbegian, 2015, Analytical and numerical studies of
heat transfer in nanocomposite thermoelectric cooler, Journal of Electronic Materials, 44,
2915-2929.
4. R. Rabari, S. Mahmud, A. Dutta, 2015, Effect of thermal conductivity on performance of
thermoelectric systems based on effective medium theory, International Journal of Heat and
Mass Transfer, 91, 190-204.
5. R. Rabari, S. Mahmud, A. Dutta, Analysis of combined solar photovoltaic-nanostructured
thermoelectric generator system, International Journal of Green Energy, (Under Review-paper
no: IJGE-2014-0066).
13
CHAPTER 2: EFFECT OF CONVECTION HEAT TRANSFER ON
PERFORMANCE OF WASTE HEAT THERMOELECTRIC GENERATOR
2.1 Introduction
Fossil fuel resources are very limited, and consumption of fossil fuel increases day by day. In
Canada 17% of electricity demand is satisfied by thermal power stations using fossil fuel such as
coal (NEB 2011). A typical thermal power station has efficiency of around 40-45% and rejects
about 50-60% of the input energy (White 1991). Figure 2.1 shows different methods to recover
the waste heat from power plant/ industrial facilities.
Figure 2.1 Various waste heat recovery methods in context of power plant (Stehlik 2007, Rowe
1995)
Yilmaz and Buyukalaca (2003) presented a mathematical model and numerical simulations for
the design of rotary regenerators used for energy recovery from various industrial and air-
conditioning applications. Budzianowski (2012) and Stehlik (2007) studied heat recirculation
phenomenon using gas-gas recuperation. The results proved that heat circulation was more
useful for power generators with high power, and it has enabled recovery of heat from flue gases
(Stehlik 2007). The thermoelectric (TE) effect can be used to recover the waste heat and it is
worthwhile to check the characteristics of TE modules because of their simple structure and no
moving parts (Rowe 1995). TE modules are made up of a number of p-type and n-type
Waste Heat Recovery Methods
Change State of Working Fluid Direct Conversion to Electricity
Air Preheater Economiser Through Mechanical
Work
Direct Electricity
Conversion
Rankine/Organic
Rankine Cycle
Kalina Cycle Thermoelectric
Generators
Thermionic
Generators
Thermal Photovoltaic Piezoelectricity
14
semiconductor materials connected electrically in series and thermally in parallel. A typical
thermoelectric generator (TEG) is made up of a number of TE modules electrically in series
(Rowe 1995). TE modules have many advantages, such as being environmentally friendly and
quiet in operation, and having a long service life (Rowe 1995). However, low conversion
efficiency of TE materials creates difficulty in wide usage (Rowe 1995). The TE effect was first
observed by Seebeck (Wang et al. 2009); when two dissimilar materials are joined together and
junctions are held at two different temperatures then electromotive force is produced (Wang et
al. 2009). The electromotive force depends on an intrinsic property of materials known as the
Seebeck coefficient ( )ΔTV=α . A few years after this experiment, Peltier observed the second
TE effect. When electric current is passed through a junction of two different materials then one
junction liberates the heat and the other absorbs the heat. The Peltier coefficient is defined as
Iq=π P (Wang et al. 2009). The interdependency of both effects dTd=T and T
was derived by Thomson (Lord Kelvin) (Goupil 2011). The Peltier heat is liberated or absorbed
at a junction and is given by TIPq .
During the last several years, various high-performance TE materials have been developed
(Udomsakdigool 2007). To improve the performance of TEG requires making a reasonably good
thermal design, as well as arrangement of TE modules. In early 20th
century, Altenkirch (1911)
established that a good TE material should have large Seebeck coefficient, low thermal
conductivity, and high electrical conductivity. Callen (1947) presented the thermodynamic
theory of the TE phenomenon. Joffe and Stil’bans (1959) introduced a figure of merit as a
parameter to classify different TE materials. Bell (2002) discussed the effect of convective heat
transfer medium on the performance of the TE module and demonstrated criteria for optimum
performance. Xiao et al. (2011) derived a generalised heat transfer model considering convection
heat loss through side walls of TE modules, assuming linear distribution of temperature across
the leg of the TEG. Xiao et al. (2011) concluded that convection heat loss causes a large loss of
heat exergy. One-dimensional analytical solutions of conventional, composite, and integrated
TEG between fixed temperature sources were obtained by Reddy et al. (2013), considering
adiabatic and convective side wall conditions. Regardless of design, increase in hot-side
temperature enhances the performance of TEG. Reddy et al. (2013) also concluded that the
15
composite and integrated TEG extracts more heat compared to the conventional TEG and
reduces rare-element material usage. Chen et al. (2002) studied effect of convection between a
heat source and surface of the TE module to optimize the distribution of heat transfer surface
area. Meng et al. (2012) demonstrated the effect of radiative heat transfer on the performance of
the TEG. A temperature-dependant thermodynamic model was developed by Meng et al. (2012),
considering external irreversibility. Meng et al. (2012) concluded that the temperature-dependent
properties have a large impact on power output and thermal efficiency, especially when the
temperature difference is large. Sahin and Yilbas (2013) considered different optimization
parameters to achieve maximum power output and maximum thermal efficiency. Sahin and
Yilbas (2013) concluded that an increase in thermal conductivity decreases the efficiency and
power output and increases entropy generation rate in TEG. Riffat and Ma (2003) performed a
review of current and potential applications of TE modules. Riffat and Ma (2003) concluded that
where supply of heat is free and abundant (e.g., waste heat or solar energy), efficiency of the TE
system is not a prime concern. The performance of a solar TEG was analysed experimentally by
Goldsmid et al. (1980) in 1980. At that time (Goldsmid et al. 1980), the TEG was made up of
Bi2Te3 and had efficiency of 1%, and could be increased to 3% if proper concentration system
was used. Considering the automotive waste heat recovery from exhaust pipe, Hsiao et al. (2010)
developed 1-D thermal resistance model of TE module. The authors (Hsiao et al. 2010) verified
modeling data with experimental data and calculated maximum power generation of 0.43 W at
290 ºC temperature difference. A TEG system was installed with carburizing furnace at Komatsu
plant to recover the waste heat (Kaibe et al. 2011). A TEG containing 16 TE modules made-up
of bismuth-telluride in groups of 4 was used. Experiments reported total power generation of 250
W with hot surface of temperature of 250 ºC (Kaibe et al. 2011).
The purpose of this work is to explore the performance of the TE device in generator mode
having different temperature dependent transport properties of p- and n- type TE leg with
convection heat transfer losses from the side surfaces to the surrounding environment.
Temperature-dependent Thomson heat is also considered. The junction temperatures of TEG are
function of thermal source and sink temperature and convection heat transfer coefficients
between thermal source and sink to TEG. Modeled equations are initially simplified to 1-D form
16
using appropriate approximations that include 1-D heat transfer, isotropic, and homogeneous
material properties, and decoupled electric and thermal fields. The thermal and electrical contact
resistances between contact surfaces are neglected. Closed forms of solutions are obtained after
solving the simplified 1-D governing equation. Thermal field solutions are used to calculate local
and average entropy generation rates for the TEG. Finally, coupled TE equations are solved
numerically to overcome the implicit problems of the 1-D analytical model to observe the
qualitative results of thermal and electric fields inside the TEG in two dimensions.
2.2 Heat Transfer Modeling
A schematic diagram of the proposed TE heat recovery system is shown in Figure 2.2. This
proposed heat recovery system has different potential arrangements among combustion
chamber/hot fluid pipes, ambient environment, and insulation covering the combustion chamber.
A typical TEG is made up of number of p- and n-type elements connected in series through
copper plate with thermally conductive and electrically insulated ceramic plate on both sides. In
the current study the TEG is placed in such a way that opposite surfaces of ceramic plates face
the combustion chamber and ambient environment, respectively. A unit TEG cell with copper
plate is shown in Figure 2.2 with geometric dimension, coordinate systems, directions of
different heat components, and thermal boundary conditions. The energy transport equation
inside a TEG for a steady state can be expressed as
genq q (1)
where q and genq represent heat generation rate per unit volume, and heat flux vector,
respectively. The continuity of electric charge through the TEG must satisfy
0 J (2)
where J is the electric current density vector. Equations (1) and (2) are coupled by the set of TE
constitutive equations (Antonova et al. 2005) as shown in Eqs. (3) and (4),
TkT Jq
T EJ
(3)
(4)
where is the Seebeck coefficient, k is the thermal conductivity, is the electrical
conductivity, and E is the electric field intensity vector. E can be expressed as , where is
the electric scalar potential (Landau et al. 1984).
17
Figure 2.2 Schematic diagram of location of TEG in combustion system and unit TEG cell
Combining Eqs. (1) to (4), the coupled TE equations for energy and electric charge transfers can
be expressed as
JEJ TkT (5)
0 T (6)
where JE represents Ohmic heat (Antonova and Looman 2005). Finally, the entropy generation
equation for TEG can be expressed as (Chakraborty 2006)
JJE 2
2T
T
k
TSgen (7)
where the first term represents the irreversibility due to Ohmic heating, second term is the heat
transfer irreversibility, and the third term is the dissipation due to the Thomson effect. For a
typical TEG shown in Fig. 2b the thermal boundary conditions are as follows:
At the top surface (i.e., 0x ) temperature is 1T , which is the hot junction temperature
At the bottom surface (i.e., lx ) temperature is 2T , which is the cold junction temperature.
Convection heat transfer per unit area from the side surfaces to the surrounding is
)(conv aTThq .
Water /Steam
Pipes
TEGs
Cross-section of combustion
system
x
l
w
p n
Rl
T1
T2
I
qconv
QC
QH
TH
TC
18
It is assumed that the thermal energy enters into the top surface from a thermal source having
constant temperature )( HT and leaves the bottom surface to a thermal sink having constant
temperature )( CT . Heat transfers between the source and the top surface of TEG and sink and
bottom surface of TEG are dominated by convection.
Initially, a simplified 1-D version of the preceding equations is solved to obtain close forms of
analytical solutions. For 1-D analytical heat transfer modeling, a TEG with p-type and n-type
semiconductor legs with load resistance lR (see Fig. 2b) is considered. TE elements having
length l and width w operates between hot and cold junction temperature 1T and 2T ,
respectively. The hot and cold junction temperatures, 1T and 2T , depend on the convection rates
from the surfaces and temperatures of the source )( HT and sink )( CT . A TEG absorbs 1q amount
of heat from the thermal source and rejects 2q amount of heat to the thermal sink. The main
mode of heat transfer through semiconductor leg is conduction, and it is accompanied by Ohmic
heating, Peltier heat generation/liberation at the junctions as well as Thomson heat generation.
The convection heat loss from the side walls of p-type and n-type semiconductor legs to the
ambient environment is also taken into account.
Assuming isotropic and homogeneous material properties and neglecting the thermal and
electrical contact resistances between contact surfaces, the one dimensional heat transfer
equation under steady state condition for semiconductor leg is given by
0)(2
2
2
2
dx
dT
Ak
ITT
Ak
ph
Ak
I
dx
Tda (8)
In Eq. (8), first term is the Fourier heat conduction, second term is the Ohmic heating, third term
is convection heat transfer loss, and fourth term is the Thomson effect.
The general solution to Eq. (8)
xDxDeCeCxT 21
21 (9)
where
2
42
1
D ;
2
42
2
D
(10)
19
;Ak
I ;
Ak
ph
2
2
Ak
IT
Ak
pha
(11)
To calculate 1C and 2C in Eq. (9), convective boundary conditions are applied.
At hot junction and cold junction of TEG, the energy balance equation between thermal source
and thermal sink with TEG can be written as
)( 1
0
TThdx
dTk HH
x
)( 2 CC
lx
TThdx
dTk
(12)
(13)
Substitution of Eqs. (9), (10), and (11) into Eqs. (12) and (13) results in
21
22
1212
2222
1
)(
)()()(
)(
)()(
21
122112
22
22
DDekek
hheeheDkeDkheDkeDk
hheeTT
hDTkDkheDTkeDk
C
lDlD
HC
lDlD
C
lDlD
H
lDlD
HC
lDlD
HC
CCH
lD
H
lD
21
22
1212
1111
2
)(
)()()(
)(
)()(
21
122112
11
11
DDekek
hheeheDkeDkheDkeDk
hheeTT
hDTkDkheDTkeDk
C
lDlD
HC
lDlD
C
lDlD
H
lDlD
HC
lDlD
HC
CCH
lD
H
lD
(14)
(15)
Now, combining heat transfer in semiconductor leg with Peltier heat (which occurs at the
junctions), the heat input at hot junction of TEG is given by
20
2
21
22
1212
1111
1
21
22
1212
2222
21
22
1212
1111
21
22
1212
2222
1
.
)(
)()()(
)(
)()(
)(
)()()(
)(
)()(
)(
)()()(
)(
)()(
)(
)()()(
)(
)()(
21
122112
11
11
21
122112
22
22
21
122112
11
11
21
122112
22
22
D
DDekek
hheeheDkeDkheDkeDk
hheeTT
hDTkDkheDTkeDk
D
DDekek
hheeheDkeDkheDkeDk
hheeTT
hDTkDkheDTkeDk
Ak
DDekek
hheeheDkeDkheDkeDk
hheeTT
hDTkDkheDTkeDk
DDekek
hheeheDkeDkheDkeDk
hheeTT
hDTkDkheDTkeDk
Iq
lDlD
HC
lDlD
C
lDlD
H
lDlD
HC
lDlD
HC
CCH
lD
H
lD
lDlD
HC
lDlD
C
lDlD
H
lDlD
HC
lDlD
HC
CCH
lD
H
lD
lDlD
HC
lDlD
C
lDlD
H
lDlD
HC
lDlD
HC
CCH
lD
H
lD
lDlD
HC
lDlD
C
lDlD
H
lDlD
HC
lDlD
HC
CCH
lD
H
lD
(16)
Equation (16) is the general form of heat transfer equation for a single leg of a TEG applied to
combustion chamber of power plant as a waste heat recovery tool. Consequently, a heat transfer
equation of a single pair of TEG can be obtained by considering respective properties of p-type
and n-type semiconductor legs. Equation (16) reveals that the hot junction temperature )( 1T
depends on the thermal source temperature )( HT and the convection heat transfer coefficient
)( Hh between the thermal source and the hot junction of TEG. In similar manner, the cold
junction temperature )( 2T depends on the thermal sink temperature )( CT and the convection heat
transfer coefficient )( Ch between the cold junction of TEG and the thermal sink. It is important
to note that in the limit of very large convection heat transfer coefficients between the source and
the hot junction and between the sink and the cold junction, the temperatures 1T and 2T approach
the source temperature HT and sink temperature CT .
The net power output of a single TEG is calculated as (Doolittle and Hale, 1984),
VIPo (17)
21
where
inp RITTV )()( 21 (18)
Electric current can be calculated by
li
np
RR
TTI
)()( 21 (19)
The thermal conversion efficiency can be evaluated as
1q
Po (20)
Thermal efficiency is independent of the number of couples, as power output and thermal input
increases linearly with number of modules.
In Eqs. (18) and (19), the temperature at the hot junction )( 1T and the temperature at the cold
junction )( 2T can be evaluated using the boundary conditions ),0( 1TTx and ),( 2TTlx
in Eq. (9).
2.3 Results and discussion
In this section, the performance of a TEG applied to a combustion system as a waste heat
recovery tool is investigated based on the one dimensional analytical solution obtained in the
previous section. The bulk crystalline semiconductor p-type material 25% Bi2Te3 75% Sb2Te3
with 1.75% excess Se and n-type material 75% Bi2Te3 25% Bi2Se3 with copper as a connector
material are used to analyze the performance. The TEG performance characteristics in terms of
thermal efficiency, power output, heat input, and produced electrical current has been studied in
detail. Different operating parameters considered in the current analysis are as follows: Thermal
source temperature (300 K HT 700 K), Thermal sink temperature (260 K CT 320 K), and
surface to surrounding heat transfer coefficient (0 W/m2K h 100 W/m
2K). The dimensions
of the TEG are as follows: length 0.03 m, width 0.01 m, and thickness 0.03 m. The Seebeck
coefficient )( , electrical resistivity )( , and thermal conductivity )(k are specified as
polynomial functions of temperatures as shown in Table 1 (Reddy et al. 2013, Angrist 1982).
These properties are evaluated at average temperature of working range. Load resistance lR is
same as internal resistance iR to get maximum power output.
22
Table 2.1 Temperature dependent TE properties of n-type 75% Bi2Te3 25% Bi2Se3 and p-type
25% Bi2Te375% Sb2Te3 with 1.75% excess Se (Reddy et al. 2013, Angrist 1982)
Property Temperature
Range (ºC)
Polynomial functions of different TE properties in terms of
temperature
n 55025 avgT 517414311
2974
106143.11054021.2103005.1
10574.11026.210517414.1
avgavgavg
avgavg
TTT
TT
p 17020 avgT 515412310
2874
106125.5102189.210265.3
103556.2102663.710084305.2
avgavgavg
avgavg
TTT
TT
450170 avgT 312
2964
1013146.4
108969.4104027.210123379.5
avg
avgavg
T
TT
n 55025 avgT
620
517414312
21086
1049.2
103202.4107176.2102921.7
106044.710324.4103562.9
avg
avgavgavg
avgavg
T
TTT
TT
p 45020 avgT 518414312
2985
109007.81033902.110526.7
1069084.1109785.71014586.1
avgavgavg
avgavg
TTT
TT
nk 10025 avgT 37253 102988.4108823.8103139.42979.1 avgavgavg TTT
400100 avgT 41037
2521
103833.1108004.1
10454.7100936.1101235.8
avgavg
avgavg
TT
TT
550400 avgT 5104734
23
1015322.210950144.410548094.4
0.208700782405.471037663.4
avgavgavg
avgavg
TTT
TT
pk 17020 avgT 5114836
253
100425.410345.11016321.1
1063173.310085.8874746.0
avgavgavg
avgavg
TTT
TT
370170 avgT 38253 10033.710415.110511.484097.1 avgavgavg TTT
450370 avgT 3623 1074356.21069338.36305.194675.234 avgavgavg TTT
23
It is assumed that the thermal energy enters into the top surface of TEG from a thermal source
)( HTT by convection with convection heat transfer coefficient Hh and leaves the bottom
surface of TEG to a thermal sink )( CTT also by convection with convection heat transfer
coefficient Ch . In the special case of very large convection heat transfer coefficients, Hh
and Ch , the top and bottom surface temperatures, 1T and 2T , approach HT and CT (i.e.,
isothermal boundary conditions). The majority of the results presented in this work consider the
influence of convection heat transfer from the side surfaces to the surrounding while the top and
bottom surfaces are exposed to a high convective environment (i.e., nearly isothermal).
However, the effect of convection from the source to the top surface and from the bottom surface
to the sink is considered for limited cases, presented at the end of this work.
Temperature Distribution
Figures 2.3 and 2.4 show the temperature distribution along the longitudinal directions of a p-
type and n-type semiconductor leg at different values of the surface to surrounding convection
heat transfer coefficients. The thermal source and the thermal sink temperatures are kept constant
at 700 K and 300 K, respectively. For a given amount of the surface to surrounding convection
loss, it is observed that the difference in the temperature gradients of p-type and n-type legs is
negligible. This negligible difference is due to the minimal difference in the TE properties
between p- and n-type TE legs. However, it is observed from these plots that the convection
losses from the side surfaces to the surrounding have larger impact on the temperature
distribution. At higher values of the convection heat transfer coefficients, a larger amount of heat
removal occurs from the side surfaces; this, in turn, causes a rapid temperature drop along the leg
when compared to the nearly adiabatic side surface temperature profile (i.e., h = 0.1 W/m2K). As
shown later, convection heat losses affect the heat input to the TEG and thermal efficiency of the
TEG significantly.
24
x (m)
T(K
)
0 0.005 0.01 0.015 0.02 0.025 0.03300
400
500
600
700
h = 0.1 Wm-2K
-1
h = 20 Wm-2K
-1
h = 40 Wm-2K
-1
h = 60 Wm-2K
-1
h = 80 Wm-2K
-1
h = 100 Wm-2
K-1
Figure 2.3 Temperature distribution over the length of p-type semiconductor leg with thermal
source temperature, HT 700 K and thermal sink temperature, CT 300 K
25
x (m)
T(K
)
0 0.005 0.01 0.015 0.02 0.025 0.03300
400
500
600
700
h = 0.1 Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2
K-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
h = 100 Wm-2
K-1
Figure 2.4 Temperature distribution over the length of n-type semiconductor leg with thermal
source temperature, HT 700 K and thermal sink temperature, CT 300 K
26
Heat Input
Heat input to the TEG can be analysed from Figs. 2.5 and 2.6. Heat input to the TEG is plotted as
a function of the source temperature )( HT in Fig. 2.5 at different values of the convection heat
transfer coefficient for a constant sink temperature (300 K). Figure 2.5 shows that as the hot
surface temperature increase, the TEG absorbs more heat due to the larger temperature
difference. In contrast, the heat input to the TEG decreases with increase in the cold surface
temperature as shown in Fig. 2.6, where heat input to the TEG is plotted as a function of the sink
temperature )( CT at different values of convection heat transfer coefficient. For a given
temperature difference between hot and cold surfaces, with higher convection heat transfer
coefficient, heat input to the TEG increases. This establishes that due to higher convection losses
more heat is drawn from heat source to the hot surface.
27
TH
(K)
q1
(W)
300 400 500 600 7000
5
10
15
20
25
30
35
40
45
50
h = 0.1 Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2
K-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
h = 100 Wm-2
K-1
Figure 2.5 Effect of thermal source temperature on heat input with variable convection heat
transfer coefficient at constant thermal sink temperature, CT 300 K
28
TC
(K)
q1
(W)
260 270 280 290 300 310 32010
15
20
25
30
35
40
45
50
55
60
65
h = 0.1 Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2K
-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
h = 100 Wm-2K
-1
Figure 2.6 Effect of thermal sink temperature on heat input with variable convection heat
transfer coefficient at constant thermal source temperature, HT 700 K
29
Power Output
Figure 2.7 presents the power output as functions of source and sink temperatures. As source
temperature increases for a constant sink temperature, power output also increases. In contrast,
the power output decreases as cold surface temperature increases for a fixed value of the source
temperature. One can observe from Eqs. (17) and (18) that the temperature difference has
significant impact on power output. As temperature difference increases, power output also
increases. The same equation verifies that power output is independent of the convection heat
transfer losses from the side surfaces to the surrounding.
30
TH
(K)
Po
(W)
300 400 500 600 7000
0.5
1
1.5
2
TC=270 K
TC=280 K
TC=290 K
TC=300 K
Figure 2.7 Power generation as a function of thermal source temperature at different thermal
sink temperature
31
Thermal Efficiency
Thermal efficiency is plotted as a function of hot surface temperature with constant cold surface
temperature in Fig. 2.8 at different values of convection heat transfer coefficient. The thermal
efficiency plot, as shown in Fig. 2.8, demonstrates that for a constant sink temperature (300 K)
with increment in the hot surface temperature the thermal efficiency of TEG increases. In
contrast, for a constant source temperature, the thermal efficiency of the TEG increases with
decrease in the cold surface temperature as shown in Fig. 2.9, where thermal efficiency is plotted
as a function of the cold surface temperature at different convection heat transfer coefficient and
constant source temperature (700 K). A larger magnitude of the T gives greater thermal
efficiency, provided that material used in TEG can withstand upper limits of temperature
exposure. In combustion system, the hot surface temperature can be easily maintained at constant
values due to the constant heat generation in the chamber. Figures 2.8 and 2.9 also establish the
effect of the surface to surrounding convection heat losses on the thermal efficiency. For a given
temperature difference between hot and cold surfaces, an increase in the convection heat transfer
coefficient decreases the thermal efficiency of the TEG. The irreversible convection process
causes larger amount of heat loss to ambient environment, so it suggests that less heat is
available to convert into electricity and this leads to low thermal efficiency.
32
TH
(K)
300 400 500 600 7000
2
4
6
8
10
12
h = 0.1 Wm-2K
-1
h = 20 Wm-2K
-1
h = 40 Wm-2K
-1
h = 60 Wm-2K
-1
h = 80 Wm-2K
-1
h = 100 Wm-2
K-1
Figure 2.8 Effect of thermal source temperature on thermal efficiency with variable convection
heat transfer coefficient at constant thermal sink temperature, CT 300 K
33
TC
(K)
260 270 280 290 300 310 3202
4
6
8
10
12
h = 0.1 Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2K
-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
h = 100 Wm-2
K-1
Figure 2.9 Effect of thermal sink temperature on thermal efficiency with variable convection
heat transfer coefficient at constant thermal source temperature, HT 700 K
34
Output Current
Figure 10 presents the variation in the produced electric current as functions of the source and
sinks temperatures. The electric current increases linearly with increase in the source temperature
when sink temperature is constant, while the electric current decreases linearly with increase in
the sink temperature for a constant source temperature. Equation (19) shows that electric current
has a linear relation with temperature difference, and plots reflect the same phenomenon.
TH
(K)
I(A
)
300 400 500 600 7000
5
10
15
20
25
TC=270 K
TC=280 K
TC=290 K
TC=300 K
Figure 2.10 Current as a function of thermal source temperature with different thermal sink
temperature
35
Irreversible Convection Heat Transfer
The effect of the surface to surrounding convection heat transfer coefficient on thermal
efficiency is demonstrated in Fig. 2.11. It is observed from the plot that the convection heat
losses from the side surfaces have more impact on thermal efficiency. For a given temperature
difference between the source and sink, thermal efficiency decreases as convection heat transfer
coefficient increases. Heat input increases with increasing rate of side wall convection (Fig. 2.5);
however, the power output remains nearly invariant (Eqs. (17) and (18)) with increase in side
wall convection heat transfer coefficient. Therefore, efficiency decreases as side wall convection
heat transfer coefficient increases for a constant temperature difference.
The effect of convections between the thermal source and the top surface and between the sink
and the bottom surface of TEG on thermal efficiency is shown in Figs. 2.12 and 2.13 for the
adiabatic side wall condition (Fig. 2.12) and convection from the side walls with h = 10 W/m2K
(Fig. 2.13). The temperatures of the source and sink are 700 K and 300 K, respectively. A lower
convection heat transfer coefficient between thermal source and sink to TEG leads to low
thermal efficiency. An introduction of convection resistances between the thermal source and the
top surface and between the sink and the bottom surface of TEG create more irreversibility to the
TEG, which causes an efficiency reduction. Isothermal top and bottom surfaces represent a
special case of zero convection resistance and the thermal efficiency is maximum for such case,
as can be observed from Fig. 2.12 ( 410 CH hh W/m2K). Both heat input and power output
decrease with decreasing Hh and Ch (higher convection resistances), which, in turn, lower the
efficiency of the TEG. An introduction of the convection losses from the side surfaces lower the
efficiency further, as can be seen from Fig. 2.13.
36
h (Wm-2
K-1
)
20 40 60 80 1000
2
4
6
8
10
12
TH
= 400 K
TH
= 500 K
TH
= 600 K
TH
= 700 K
Figure 2.11 Effect of convection heat transfer coefficient on thermal efficiency at constant
thermal sink temperature, CT 300 K
37
I (A)
5 10 15 200
2
4
6
8
10
12
hC
= hH
= 50 Wm-2
K-1
hC
= hH
= 100 Wm-2
K-1
hC
= hH
= 103
Wm-2K
-1
hC
= hH
= 104
Wm-2
K-1
Figure 2.12 Effect of convections between the thermal source and the top surface and between
the sink and the bottom surface of TEG on thermal efficiency when HT 700 K and CT 300 K
with adiabatic side wall condition
38
I (A)
5 10 15 200
1
2
3
4
5
6
7
8
hC
= hH
= 50 Wm-2K
-1
hC
= hH
= 100 Wm-2
K-1
hC
= hH
= 103
Wm-2K
-1
hC
= hH
= 104
Wm-2
K-1
Figure 2.13 Effect of convections between the thermal source and the top surface and between
the sink and the bottom surface of TEG on thermal efficiency when HT 700 K and CT 300 K
with convection from the side walls with h=10 Wm-2
K-1
39
Irreversibility Analysis
Entropy is produced by the irreversible processes in TE devices (Yilbas and Pakdemirli 2005,
Sekulica 1986), and, in this respect a typical TEG is no exception. If these irreversible processes
could be eliminated, entropy production would be reduced to zero (Bermejo et al. 2013). In such
cases the limiting value of the Carnot efficiency for a TEG would be obtained. Unfortunately, it
is impossible to reduce the irreversibilities of a system to zero. Therefore, during the operation,
the performance of the TEG can be further improved through the minimization of the
thermodynamic losses. One of the methods to maximize the thermal efficiency of the TEG is to
minimize the entropy generation rate. Therefore, entropy generation analysis is a very important
tool to understand the performance of the TEG.
The general expression of the local entropy generation rate, given by Eq. (7), can be simplified to
obtain a 1-D entropy generation rate equation for the present problem as shown here:
2
2
,
2
2
)( TT
k
TA
IS
np
gen
(21)
Expression of the temperature distribution (Eq. (9)) is used to obtain the preceding expression of
the local entropy generation rate. Note that the spatial dependency of the Seebeck coefficient
( ) in the last term of Eq. (7) is neglected to obtain Eq. (21), assuming a homogeneous
material. Equation (21) is the volumetric local entropy generation rate (W/m3K), where first term
represents the irreversibility due to the Ohmic heating and the second term represents
irreversibility due to the temperature gradient. The volume averaged entropy generation rate
( genS ) can be obtained from the following equation:
dVdx
dT
T
k
TA
I
VS
np
gen
2
2
,
2
21 (22)
For the constant cross-sectional area of TE legs, Eq. (22) can be further simplified to
dxdx
dT
lT
k
TA
IS
Lnp
gen
0
2
2
,
2
2
. (23)
Above equation can be written in the dimensionless form,
dxTlT
k
TA
I
k
lS
L
np
np
gen
0
2
2
,
2
2
,
2
(24)
40
wher
2
,
lk
SS
np
gengen (25)
In Eq. (24), the temperature gradient contained in the second term on the right side can be
evaluated by using Eqs. (9), (14), and (15). Finally average entropy generation over the entire
volume of a TEG leg can be evaluated in non-dimensional form as shown here:
21
22
2
2
21
2
2
2
22121
2
21
2
1
22
1
2
1
2
2
2
2
21
2
2212121
2
1
2
1
2
1
2
,
2
2
,
2
2
4
4
2221
11
DD
eDCDeDCeDDCC
eDDCeDCDC
DDCDDCCDDCDC
lT
k
TA
I
k
lS
lDlDlDlD
lDlD
np
np
gen (26)
The simplified expressions for 1C and 2C are defined already in Eqs. (14) and (15). Similarly,
simplified expressions for 1D and 2D are defined in Eq. (10).
The volume averaged dimensionless entropy generation rate, as presented in Eq. (26), is plotted
as a function of the source temperature for different values of the convection heat transfer
coefficient in Fig. 2.14. The sink temperature is assumed constant and equal to
300 K. For a small temperature difference between the source and sink, the magnitude of the heat
transfer contribution to the entropy generation is negligible. For this special case, the irreversible
Ohmic heating dominates the overall entropy generation rate, which is relatively small in
magnitude when compared with the heat transfer irreversibility. Therefore, small values of the
entropy generation rate are observed at values of the source temperatures. Note that for a very
special case of isothermal system ( 0T ) heat transfer irreversibility is zero. An increase in
the source temperature increases the entropy generation rate as observed from Fig. 2.14. Higher
temperature difference between the source and sink brings more heat to the TEG along the larger
finite temperature difference, which is naturally irreversible. An introduction of the surface to
surrounding convection increases this irreversibility further. Therefore, a higher entropy
generation rate is observed in Fig. 2.14 at higher values of the convection heat transfer
coefficient for a given temperature difference between the source and sink. For small
temperature difference between the source and sink, the variation in the magnitude of the entropy
41
generation rate is insignificant with increasing values of the convection heat transfer coefficient.
In contrast, a larger variation in the entropy generation rate is observed with convection heat
transfer coefficient when temperature difference is relatively large.
42
TH
(K)
S* g
en
400 500 600 7000
5
10
15h = 0.1 Wm
-2K
-1
h = 20 Wm-2
K-1
h = 40 Wm-2K
-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
h = 100 Wm-2
K-1
||
Figure 2.14 Entropy generation rate as a function of thermal source temperature at different
convection heat transfer coefficients with constant thermal sink temperature, CT 300 K
43
2-D Numerical Results
Equations (5) and (6) are the modeled differential equations for the coupled thermal and electric
fields. Until this point, a simplified 1-D version of the energy equation (Eq. (5)) has been used to
describe the characteristic features of the TEG used in this work. In this section, some selected
parameters of the TEG are evaluated by solving the 2-D coupled thermal-electric equations (Eqs.
(5) and (6)). Equations (5) and (6) are solved using a finite element method. A description of the
discretization and solution techniques is available in Mahmud and Pop (2006). Properties of p-
and n-type semiconductor legs were approximated at the average temperature of working range
using relations given in Table 2. For the numerical analysis, the hot junction temperature (= 700
K) and the cold junction temperature (= 300 K) are kept constants. Computations are carried out
for two different cases of TEG side wall heat transfer boundary conditions: adiabatic and
convective heat transfer. All geometrical parameters are same as 1-D analysis. For adiabatic side
walls of p and n-type semiconductor legs, Figures 11(a) and 11(b) present the thermal and
electric potential field results. Figure 11(a) presents the temperature and heat flow results while
Figure 11(b) reveals the electric scalar potential and current flow. Surface to surrounding
convection is introduced next, and the entire field results are repeated for h = 20 W/m2K and
presented graphically in Figs. 2.17 and 2.18. In order to carry out the numerical simulation, both
terminals (the bottom ends of the p- and n-leg) are connected directly using a strip of material
having known electrical resistance to approximate the external load resistance. For a particular
TE leg, temperature remains nearly constant at a given location of distance x when 0h
W/m2K. However, the location of the same isothermal line is slightly different at p- and n-leg
due to the dissimilar properties. In the absence of the surface to surrounding convection, the heat
flux lines are parallel in both p - and n- leg and uniformly distributed over the cross-section of
the legs. An introduction of the surface to the surrounding convection introduces the non-
linearity in the temperature distribution, as observed from Fig. 2.17. At a given x location,
surface of a TE leg is cooler than the core due to the heat removal by convection. The heat flux
lines are no longer parallel in both p- and n-leg and a non-uniform distribution is observed. A
certain portion of the heat, entering the top surface of the TEG, is leaving through the side
surfaces by convection, as evidenced by the heat flux lines terminated at the vertical side
surfaces of the legs. Due to the coupled TE effect, potential difference is established in the leg
44
which drives an electric current through the TEG. For a particular TE leg, electric potential
remains nearly constant at a given location of distance x when h = 0 W/m2K. However, non-
linear potential distribution is observed when h = 20 W/m2K.
45
320320
340
360360
380
400
420
440
460
480
500
520
540
560
580580
600
620620
640
660
680
320
340
360
380
400
420
440
460 460
480
500
520
540 540
560
580
600
620 620
640
660
680
Figure 2.15 Temperature distribution in TEG with adiabatic boundary conditions at vertical
walls of semiconductor legs
46
-0.009
-0.008
-0.007
-0.006-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.0020.003
0.0040.005
0.0060.007
-0.009-0.008
-0.007
-0.006 -0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003 0.003
0.004
0.005
0.006
0.007 0.007
Figure 2.16 Electrical potential in TEG with adiabatic boundary conditions at vertical walls of
semiconductor legs
47
320
340340
360
380
400
420
440
460
480
500
520
540
560
580
600600
620
640
660
680
320
340
360 360
380
400
420
440
460
480
500
520
540 540
560 560
580
600 600620
640
660
680
Figure 2.17 Temperature distribution in TEG with convective boundary conditions, h = 20
Wm-2
K-1
at vertical walls semiconductor legs
48
-0.009
-0.008
-0.007
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.0010.001
0.002
0.003
0.004
0.005
0.006
0.007
-0.009 -0.009
-0.008
-0.0
08
-0.007
-0.006
-0.005
-0.004-0.004-0.003
-0.002
-0.001
00.001
0.0020.0030.004
0.0050.006
0.007
Figure 2.18 Electrical potential in TEG with convective boundary conditions, h = 20 Wm-2
K-1
at
vertical walls of semiconductor legs
49
Comparison and Validation
In this section, a comparison of current results with the similar type of results (available in the
literature) is presented. Reddy et al. (2013) performed a theoretical performance study of
conventional, composite, and integrated TE devices applicable to waste heat recovery system.
Performance results in Reddy et al. (2013) are presented for h = 0 - 1000 W/m2K with hot
surface temperature 450 K and cold surface temperature 300 K. By using the conventional TE
device dimensions, variable material properties, and thermal/electric boundary conditions
(Reddy et al. 2013), thermal efficiencies are calculated for five different hot to cold surface
temperature differences and presented in Table 2.2. Results presented in Table 2.2 are for a
single unit of TEG. Thermal efficiencies obtained using the efficiency equation of current work
show good agreement with the results obtained by Reddy et al. (2013). Note that the numerical
values available in the third column in Table 2.2 are extracted manually from Figure 4 of Reddy
et al. (2013) which is for h = 0 W/m2K with hot surface temperature 300-550 K and cold surface
temperature 300 K. An additional comparison is presented in Fig. 2.19, where heat input, power
output, and thermal efficiency results are obtained from section 4.4.3 of Angrist (1982) for an
optimized TEG and compared with the results obtained from equations derived in the current
work. Angrist (1982) used an adiabatic side boundary condition. Figure 2.19 shows good
agreement between current results and results obtained by Angrist (1982). Figure 2.20 presents
the comparison between analytical results and numerical simulation carried out in this work. The
plot shows good agreement of results between the developed analytical model and numerical
simulation at lower convection heat transfer coefficient. In contrary, discrepancy grows as the
convection heat transfer coefficient increases.
50
Table 2.2 Thermal efficiency of single unit of TEG with cold surface temperature 300 K
Hot Surface
Temperature (K)
Thermal Efficiency
(current work)
Thermal Efficiency
(Reddy et al. 2013)
350 2.07% 2.28%
400 4.02% 4.32%
450 5.88% 6.19%
500 7.63% 7.92%
550 9.3% 9.25%
51
T (oC)
50 100 150 200 250 300 350 4000
3
6
9
12
15
Heat input (current work)
Power output (current work)
50 100 150 200 250 300 350 4000
3
6
9
12
Efficiency (current work)
50 100 150 200 250 300 350 4000
3
6
9
12
Efficiency (Angrist 1982)
q1
an
dP
o(W
)
0
3
6
9
12
15
Heat input (Angrist 1982)
Power output (Angrist 1982)
Figure 2.19 Comparison of heat input, power output, and thermal efficiency obtained from the
current work with the similar results available in (Angrist 1982)
52
x (m)
T(K
)
0 0.005 0.01 0.015 0.02 0.025 0.03300
400
500
600
700
h = 0.1 Wm-2
K-1
(analytical)
h = 25 Wm-2
K-1
(analytical)
h = 50 Wm-2
K-1
(analytical)
h = 0.1 Wm-2
K-1(numerical)
h = 50 Wm-2K
-1(numerical)
h = 25 Wm-2K
-1(numerical)
Figure 2.20 Comparison of analytical and numerical results in terms of temperature distribution
over the p-type semiconductor leg
53
2.4 Conclusion
This research has developed 1-D analytical and numerical 2-D numerical analyses of TEG
applied to waste heat recovery from combustion system in power plants. Based on fundamental
theories of TE phenomenon and energy balance, detailed 1-D heat transfer modeling is derived
involving Fourier heat conduction, ohmic heating, and convection heat transfer losses, and
Peltier, Seeback and Thomson effects. In addition to this, convective boundary conditions have
been considered between thermal source and sink to TEG. The influences of thermal source and
sink temperatures and convection heat transfer coefficient on various performance parameters of
TEG such as power output, heat input, thermal efficiency, and electric current have been studied.
An increase or decrease in thermal source and sink temperature has a considerable effect on the
performance of TEG. As temperature differential T increases, power output and thermal
efficiency increases. It is also found that the convection heat transfer coefficient has extensive
impact on the performance of TEG. Escalation in heat input and drop in thermal efficiency are
observed with increment in convection heat transfer coefficient. The results also show that
increment in convection heat transfer coefficient increases entropy generation and thus destroy
the exergy. Finally, a 2-D mathematical model is solved numerically to observe qualitative
results of thermal and electric fields inside the TEG. Field results of numerical analysis match to
that of 1-D analytical results. Numerical results also prove that the presence of an irreversible
heat convection process does cause a large amount of heat loss which matches with 1-D
analytical result. A waste heat TEG needs to be designed carefully, considering the effect of
internal and external irreversible convection losses.
54
2.5 Nomenclature
A cross-sectional area (m2)
1C a parameter (see Eq.(14))
2C a parameter (see Eq.(15))
1D a parameter (see Eq.(10))
2D a parameter (see Eq.(10))
E electric field intensity vector (Vm-1
)
h convection heat transfer coefficient (Wm-2
K-1
)
I electric current (A)
J electric current density (Am-2
)
k thermal conductivity (Wm-1
K-1
)
l length of p type and n type semiconductor material respectively (m)
p perimeter (m)
P electric power output (W)
q heat or energy for TEG (W)
q heat flux (Wm-2
)
Q heat or energy for thermal source and sink (W)
.
q heat generation rate per unit volume (Wm-3
)
R electrical resistance (Ω)
S entropy (Wm-3
K-1
)
S volume averaged entropy (WK-1
)
T temperature (K)
V voltage (V), volume (m3)
w width (m)
x coordinate (m)
Greek Symbols
Seebeck coefficient (VK-1
)
a parameter (see Eq. (11))
a parameter (see Eq.(11)))
55
efficiency (%)
Peltier coefficient (V)
electrical resistivity (Ωm)
electrical conductivity (Sm-1
)
Thomson coefficient (VK-1
)
a parameter (see Eq.(11))
electric scalar potential (V)
Subscripts
1 hot junction of TEG
2 cold junction of TEG
a atmospheric condition
avg average temperature
C thermal/heat sink
conv convection
gen generation
H thermal/heat source
i internal
l external load
o output
n n-type semiconductor material
P Peltier effect
p p-type semiconductor material
Superscripts
* dimensionless form
56
CHAPTER 3: NUMERICAL SIMULATION OF NANOSTRUCTURED
THERMOELECTRIC GENERATOR CONSIDERING SURFACE TO
SURROUNDING CONVECTION
3.1 Introduction
The research and development in nanostructured thermoelectric (TE) systems have gathered
considerable attention due to their potential applications in direct electricity generation,
refrigeration, and air-conditioning. TE systems can largely be classified as thermoelectric
generator (TEG) and thermoelectric cooler (TEC). The TEG converts heat into electricity and
TEC converts electricity into heating/cooling based on Seebeck and Peltier effects, respectively.
TE systems are solid state heat engines/refrigerators which are robust, silent, compact, and
environment friendly. TE systems are made up of numbers of p-type and n-type semiconductor
elements connected electrically in series and thermally in parallel. The TEC has a wide range of
applications; for example, electronic cooling, laser diode cooling, military garment, laboratory
cold plates, and automobile seat cooler. In a similar manner, TEG has various applications in
military, deep space vehicles, remote power sources for inhabitable places, solar and waste heat
power generator. Liquid cooling of CPU using TE was proposed and experimentally investigated
considering different material and size of the heat sinks (Naphon and Wiriyasart 2009). Huang et
al. (2010) have discussed TE cooling of electronic equipment experimentally and analytically.
The results determined that the integration of water-cooling with TE is helpful to increase the
performance of electronic equipment. In order to address the site specific on-demand cooling of
hot spot in microprocessor (Sullivan et al. 2012), a numerical simulation that includes heat
spreader, thermal interface material, chip, and nine TECs was carried out considering steady
state and transient analysis. Sullivan et al. (2012) concluded that transient cooling with square
root current pulse is most effective with 10°C cooling. Wang (2013) has proposed and
investigated experimentally the TEG using waste heat of the Light-Emitting Diodes. Results
reported by Wang (2013) investigated power output of TEG using waste heat of the Light-
Emitting Diodes (LED). Experiments of Wang (2013) showed 160 mW of power output from
TEG with 6 W of input power to LED. Recently, Hsiao et al. (2010) studied the performance of
TE modules as a waste heat recovery tool from an automobile engine using a one-dimensional
57
thermal resistance model and compared their model with experimental data. Results (Hsiao et al.
2010) showed that the performance of a TE module on the exhaust pipe performs better
compared to a TE module on the radiator system. Rezania et al. (2012) have studied the effect of
cooling power on the performance of a TEG. Rezania et al. (2012) determined the optimum
coolant flow rate for maximum power output for the TEG. For example, temperature difference
of 10 K gives maximum power output of 0.035 W with coolant flow rate of 0.07 l/min (Rezania
et al. 2012). One-dimensional analytical solutions of conventional, composite, and integrated
TEG have been carried out by Reddy et al. (2013) considering adiabatic and convective side wall
conditions. Regardless of TEG design, an increment in the hot side temperature enhances the
performance of TEG systems (Reddy et al.2013). Reddy et al. (2013) concluded that composites
and integrated TEG extracts more heat compared to conventional TEG and reduces rare-element
material usage. Gou et al. (2010) investigated the performance of low temperature waste heat
TEG using one-dimensional analytical simulations and experiments. They (Gou et al. 2010)
concluded that in addition to increasing the waste heat temperature and number of modules in
series, expanding the heat sink surface area and enhancing the cold side heat transfer in proper
ranges can have dramatic effects on TEG’s performance. A three dimensional numerical model
of TEG applied to fluid power systems is developed by Chen et al. (2011) and their numerical
simulation performed with ANSYS shows fairly good match with experimental results. In
addition to this, Chen et al. (2011) concluded that convection heat transfer losses increase the
heat input to TEG thus reduces the thermal efficiency of TEG. A three-dimensional coupled
numerical simulation of integrated TE device was carried out by Reddy et al. (2012) to check the
effects of Reynolds number and fluid temperature on performance of an integrated TEG system.
Reddy et al. (2012) found that higher Reynolds number enhances the heat transfer and thus leads
to higher power output of TEG. Zhou et al. (2013) have developed simple and coupled field
model with former considering Navier-stokes and energy equations with continuity equation and
later with different TE effects such as Seebeck, Peltier, and Thomson effect. Zhou et al. (2013)
reported overall TEG efficiency of 3.5% with temperature difference of 80 K. Baranowski et al.
(2012) have developed mathematical model for solar TEG which can provide analytical solutions
of device efficiency with temperature dependant properties. They (Baranowski et al. 2012) have
also showed that considering currently available materials, total efficiency of 14.1% is possible
58
with cold and hot side temperature of 100 °C and 1000 °C, respectively. Baranowski et al.
(2012) also concluded that if figure of merit (ZT) reaches to 2 then TEG efficiency of 25% can
be obtained, for the cold and hot side temperature of 100 °C and 1000 °C, respectively.
Altenkirch (1911) introduced figure of merit, ZT as a parameter to classify different materials.
The performance of TE materials are characterised by a dimensionless parameter ‘figure of
merit’, TZT )( 2 , where is the Seebeck coefficient, is the electrical conductivity, and
is the thermal conductivity. The current ZT value of the best available TE materials is 1 at
room temperature (Vineis et al. 2010). Slack (1995) has described that good TE materials need
to have low thermal conductivity. In addition to this, Slack (1995) described that the best TE
material would behave as “Phonon Glass Electron Crystal (PGEC)”; that is, it would have
thermal properties of glass like material and electrical properties of crystalline material. Recent
advancements in the field of nanotechnology (Vineis et al. 2010) have opened the door for
further improvements of ZT for the TE materials. The expression of the figure of merit,
TZT )( 2 , evidently indicates that one of the methods to increase figure of merit is to
reduce the thermal conductivity of the TE material. Thermal conductivity is the sum of two
contributions: electrons and hole transporting the heat )( E and phonons traveling through the
lattice )( L (Tritt 2001). The electronic part of thermal conductivity )( E is related to the
electrical conductivity as par the Wiedemann-Franz law (Tritt 2001) as shown in Eq. (1),
TLoE (1)
where, oL is the Lorentz number and for metals it is equal to (Tritt 2001),
8
2
22
1045.23
e
kLo V
2K
-2 (2)
where, k is the Boltzmann constant ( 231038.1 JK-1
) (Tritt 2001) and e is the electron charge
( C1060.1 19 ) (Godart et al. 2009). The expression of the figure of merit can be written in terms
of Lattice conductivity )( L and electronic conductivity )( E as shown below (Tritt 2001),
LE
E
oLZT
2
. (3)
One method to enhance ZT is the inclusion of nanoparticles into the bulk TE materials which can
lead to low lattice thermal conductivity (Ma et al. 2013). Poudel et al. (2008) achieved ZT value
59
of 1.4 at 373 K by hot-pressing of nanopowders of Bi2Te3 and Sb2Te3 under Argon )(Ar
atmosphere. The enhanced ZT was due to the significant decrease in the lattice thermal
conductivity of material. Zhao et al. (2008) fabricated Bi2Te3 with various amount of nano SiC
particles using mechanical alloying and spark plasma sintering and tested the TE and mechanical
properties. Their results showed an improvement in ZT from 0.99 to 1.04 with an inclusion of
0.1% vol% SiC particles. Li et al. (2009) obtained ZT of 1.43 for double-doped 124SbCo
skutturedites using Indium )(In and Cerium )(Ce doping. The attractive results achieved using
nanotechnology has encouraged researchers to include nanoparticle-doped TE materials for
various low potential heat recovery applications; for example, solar TEG and automobile exhaust
heat recovery. Kraemer et al. (2011) have proposed novel solar TEG with glass vacuum
enclosure considering nanostructured TE materials. The developed solar TEG achieved
maximum efficiency of 4.6% with solar flux of 1000 W/m2 condition. McEnaney et al. (2011)
developed a novel of TEG. They (McEnaney et al. 2011) placed high performance
nanostructured material in evacuated tube with selective absorber and achieved an efficiency of
5.2%. It is quite evident that nanostructured TE materials can increase the performance of TE
systems.
It is quite clear from the above discussion that in future nanostructured TE materials will play a
significant role as a direct energy conversion tool from low potential sources. It is necessary to
understand wide range of characteristic features of newly developed TE materials. Such
characteristic features include heat transport and electric potential. The existing literature on the
numerical simulation of nanostructured TEG is very limited. In this work, performance of
nanostructured TEG is evaluated using 2D numerical simulation. The temperature dependent
thermophysical and electrical properties of nanostructured TE material, surface to surrounding
convection heat transfer losses, and Thomson effect are included in the current model. The field
plots of heat and current are presented with different convection heat transfer coefficients.
Numerical results are compared with that of one dimensional analytical results in terms of
current produced.
60
3.2 Mathematical model and boundary conditions
The two-dimensional schematic diagram of the TEG being investigated is shown in Fig. 3.1. The
TEG is mainly comprised of two vertical p-type and n-type semiconductor legs connected
electrically in series and thermally in parallel. Both legs are connected through electrically
conductive copper strip. Each leg has cross-sectional area of L×W, height of H, and separated by
distance Lg as shown in Fig. 3.1 below. The QH and QC are amount of heat available at heat
source and heat sink respectively. The Qconv is convection heat loss through the side walls of the
TEG. During the analysis following assumptions are made:
TE materials are homogeneous and isotropic with temperature dependent properties.
Contact resistances at interface of copper and TE materials are neglected.
Figure 3.1 Schematic of unit cell of TEG
L
p n
RL
TH
TC
I Qconv
QH
QC
H
Lg
x
y
61
In the TEG, the energy transport and current flow are governed by energy equation and
continuity of current density as per below,
genqt
TC
q (1)
where symbols , pC , T , q , and genq represent material density, specific heat, temperature,
heat generation rate per unit volume, and heat flux vector, respectively. The continuity of electric
charge through the system must satisfy,
0
t
DJ (2)
where J is the electric current density vector and D is the electric flux density vector,
respectively. Equation (1) and Eq. (2) are coupled by the set of TE constitutive equations (Perez-
Aparicio 2012) as shown in Eq. (3) and Eq. (4) below,
TT ][][ Jq
)][(][ T EJ
(3)
(4)
where ][ is the Seebeck coefficient matrix, ][ is the thermal conductivity matrix, ][ is the
electrical conductivity matrix, E is the electric field intensity vector, respectively. E can be
expressed as , where is the electric scalar potential (Landau 1984). Rearrangement of
Eq. (1) to Eq. (4) gives the coupled TE equations for heat transfer and electric potential as,
JEJ
TT
t
TC ][][ (5)
and
0][][][][
Tt
(6)
where ][ is the dielectric permittivity matrix (Landau 1984) and JE represents Joule heat
(Perez-Aparicio 2012).
In Eq. (5), the second term represents the Thomson heat and the third term represents heat
transfer due to conduction. In Eq. (6) first term represents electric current density due to the
Seebeck effect and standard voltage driven electric current.
Thermal boundary conditions for Fig. 3.1 are as follows;
62
The top surface of the TEG experiences constant hot temperature (TH)
The bottom surface experiences constant cold temperature (TC)
The vertical surfaces of p type and n type are considered with two different
conditions: convective heat transfer condition and adiabatic condition (special case).
3.3 Results and discussion
In this section, the performance of unit cell of TEG is investigated based on the results obtained
by numerically solving governing equations presented in previous section. For p-type material,
the nanostructured semiconductor Bismuth Antimony Telluride (BiSbTe) is considered to
analyze the performance (Poudel et al. 2008). While for n-type material, Bismuth Telluride
(Bi2Te3) with nano-particles of silicon carbide (SiC) is considered to analyze the performance
(Zhao et al. 2008). The temperature dependent transport parameters Seebeck coefficient )( ,
electrical conductivity )( , and thermal conductivity )( are specified as polynomial functions
of temperature as shown in Table 3.1. These properties are evaluated at an average temperature
of working temperature range. The semiconductor leg of unit cell of TEG has dimensions of
mmmm 5.15.1 and height of mm5 . The gap between two consecutive legs is mm3.0 . The
terminals of both p-type and n-type semiconductor legs are connected with external load (RL)
which is matched with total internal resistance of the TEG. Figures 3.2, 3.4, 3.6, and 3.8 show
field plots of temperature and heat flow. Temperature is presented by marked isothermal lines
with multi-colored background, while the heat flow is presented by the vertical lines with arrows.
Similarly, Figs. 3.3, 3.5, 3.7, and 3.9 present the field results of electric scalar potential and
current flow. The electric potential is indicated by the marked iso-potential lines with multi-
colored background, while the current flow is indicated by the lines with arrows. Surface to
surrounding convection heat losses are also considered from vertical walls of both semiconductor
legs (h=15-50 W/m2K).
63
Table 3.1 Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal
conductivity as a function of temperature for BiSbTe nanostructured bulk alloys and Bi2Te3 with
SiC nanoparticles (Poudel et al. 2008, Zhao et al. 2008)
Properties Polynomial Expressions
p 6493623 1010738.710656.310732.2656.050.172 TTTT
p 541138253 1010354.110093.210451.210575.8450.1 TTTT
p 41037254 10875.710809.310193.410946.2136.1 TTTT
n 4734213 1059.11073.21073.18.471098.4 TTTT
n 41037242 1011.31005.51003.31020.879.9 TTTT
n 41037242 1085.11022.31016.21053.625.8 TTTT
The temperature remains nearly constant at a given location of distance x when convection is
absent (i.e., h=0), irrespective of the p-type or n-type semiconductor leg. However, the location
of the same isothermal line is changed in p-type and n-type leg due to the different transport
properties; more specifically, the thermal conductivities. Due to the absence of the surface to
surrounding convection, the heat flux lines are parallel in both p-type and n-type leg and
uniformly distributed over the cross-section of the legs. An introduction of the surface to the
surrounding convection makes the temperature distribution non-linear as observed in Figs. 3.4,
3.6, and 3.8. An increment in convection heat transfer coefficient increases the irreversible
convection losses through the side walls of semiconductor legs. Higher the convection heat
transfer coefficient, more heat is carried away without being converted to electricity. A certain
portion of the heat, entering the top surface of the TEG, is leaving through the side surfaces by
convection as evidenced by the heat flux lines terminated at the vertical side surfaces of the
semiconductor legs. The surface of a semiconductor leg is cooler than the core due to the heat
removal by convection for a given x location. The heat flux lines are no longer parallel in both
p-type and n-type legs and a non-uniform distribution of heat flux line is observed. Due to the
coupled thermo-electric effect, potential difference is established in the TEG which drives an
electric current through the system. The electric potential remains nearly constant at a given
64
location of distance x when h=0 for a particular semiconductor leg. However, non-linear
electric potential distribution is observed with convective heat transfer boundary condition.
65
320 320
340
360 360
380
400 400
420
440
460
480480
500 500
320
340
360
380 380
400
420420
440
460
480
500
Figure 3.2 Contours of temperature distribution and streamlines of heat flow with adiabatic heat
transfer condition (h ≈ 0 W/m2K)
66
0.002
0.004
0.0060.006
0.008 0.008
0.01
0.012 0.012
0.014 0.014
0.016 0.016
-0.018
-0.016-0.016
-0.014-0.014
-0.012
-0.01
-0.008
-0.006-0.006
-0.004-0.004
-0.002
0
-0.0
14
-0.0
12
0.0
14
Figure 3.3 Contours of electric potential and streamlines of electric current flow with adiabatic
heat transfer condition (h ≈ 0 W/m2K)
67
320 320
340
360 360
380
400 400
420
440
460
480480
500 500
320
340
360
380380
400
420420
440
460460
480
500
Figure 3.4 Contours of temperature distribution and streamlines of heat flow with adiabatic heat
transfer condition (h = 15 W/m2K)
68
0.002
0.004
0.0060.006
0.008 0.008
0.01
0.012 0.012
0.014
-0.014-0.014
-0.012
-0.01
-0.008
-0.006-0.006
-0.004-0.004
-0.002
0
0.0
06 0
.01
Figure 3.5 Contours of electric potential and streamlines of electric current flow with adiabatic
heat transfer condition (h = 15 W/m2K)
69
320 320
340
360 360
380
400 400
420
440
460
480480
500 500
320
340
360
380380
400
420420
440
460460
480
500
Figure 3.6 Contours of temperature distribution and streamlines of heat flow with adiabatic heat
transfer condition (h = 35 W/m2K)
70
0.002
0.004
0.0060.006
0.008
0.010.01
0.012
0.014
-0.014-0.014
-0.012
-0.01-0.01
-0.008
-0.006
-0.004
-0.002
00
-0.0
14
00.0
1
Figure 3.7 Contours of electric potential and streamlines of electric current flow with adiabatic
heat transfer condition (h = 35 W/m2K)
71
320 320
340
360 360
380
400 400
420
440
460
480 480
500
320
340
360
380380
400
420420
440
460
480
500
Figure 3.8 Contours of temperature distribution and streamlines of heat flow with adiabatic heat
transfer condition (h = 50 W/m2K)
72
0.002
0.004
0.0060.006
0.008
0.010.01
0.012
0.014-0.014-0.014
-0.012
-0.01-0.01
-0.008
-0.006
-0.004
-0.002
00
-0.0
14
00.0
1
Figure 3.9 Contours of electric potential and streamlines of electric current flow with adiabatic
heat transfer condition (h = 50 W/m2K)
73
In addition to the numerical simulation, a comparison between analytical result and numerical
simulation is presented in terms of the current produced. The analytical results are obtained from
the mathematical model developed by Reddy et al. (2013). The comparison is presented in Fig.
3.10 where produced current is plotted as a function of temperature difference between hot and
cold surface of the TEG. The electric current increases as temperature difference increases.
Figure 3.10 establishes a good agreement between the current result and result available in the
literature. The proposed nanostructured TEG’s thermal efficiency is demonstrated in Fig. 3.11
where thermal efficiency is plotted as a function of temperature difference between hot and cold
surface of the TEG at different values of convection heat transfer coefficients. TEG has highest
efficiency with larger temperature difference and adiabatic boundary condition. As convection
heat transfer coefficient increases, the thermal efficiency drops and this can be attributed to heat
loss shown in temperature field plots from Figs. 3.4, 3.6, and 3.8.
74
Temperature Difference,
Cu
rren
t,I
(A)
200 250 300 350 4000.9
0.95
1
1.05
1.1
Analytical Results (Reddy et al. 2013)
Numerical Simulation Results
Figure 3.10 Comparison of current production using numerical and analytical techniques
75
h = 15 W/m2K
Temperature Difference,
Th
erm
al
Eff
icie
ncy
,
200 250 300 350 4003.5
4
4.5
5
5.5
6
6.5
7
h = 0 W/m2K
h = 35 W/m2K
h = 50 W/m2K
Figure 3.11 Thermal efficiency of TEG as a function of convection heat transfer coefficient and
temperature difference
76
3.4 Conclusion
In this research work, a numerical simulation of nanostructured TEG is carried out. The
nanostructured TE materials have low thermal conductivity and higher power factor ( 2 )
which improves performance of the TEG. Current numerical simulation considers the Seebeck
effect, Peltier effect, Thomson effect, Fourier heat conduction, and convection heat transfer
losses. The influences of hot surface temperature and convection heat transfer coefficient on the
performance parameters of TEG such as thermal efficiency and electric current have been
studied. Electric current generation using numerical simulation and analytical simulation shows a
good match. An increase in hot surface temperature leads to increase in electric current
generation and eventually the thermal efficiency. Numerical results prove that presence of
irreversible convection heat transfer causes a large amount of heat loss thus reduces the thermal
efficiency. In future, more detailed three-dimensional numerical simulation of TEG will be
carried out to observe the above mentioned effects in more detail.
77
3.5 Nomenclature
C specific heat capacity (kJkg-1
K-1
)
D electric flux density vector (Cm-2
)
E electric field intensity vector (Vm-1
)
e electron charge (C)
h convection heat transfer coefficient (Wm-2
K-1
)
H height of TE leg (m)
I current (A)
J electric current density vector (Am-2
)
k Boltzmann constant (JK-1
)
L length (m)
oL Lorentz number (V2K
-2)
R resistance (Ω)
q heat flux vector (Wm-2
)
.
q heat generation rate per unit volume (Wm-3
)
Q heat (W)
t time (S)
T temperature (K)
W width of the TE leg (m)
x coordinate (m)
y coordinate (m)
ZT dimensionless figure of merit
Greek symbols
Seebeck coefficient (VK-1
)
ε dielectric permittivity matrix (Fm-1
)
thermal conductivity (Wm-1
K-1
)
density (kgm-3
)
electrical conductivity (Sm-1
)
electric scalar potential (V)
78
Subscripts
C cold surface
conv convection
E electronic
g gap
H hot surface
l external load
L lattice
79
CHAPTER 4: ANALYTICAL AND NUMERICAL STUDIES OF HEAT
TRANSFER IN NANOCOMPOSITE THERMOELECTRIC COOLER
4.1 Introduction
Thermoelectric coolers (TEC) create a cooling or heating effect based on the Peltier effect
without any moving parts. TEC is a solid state system made up of p- and n-type semiconductor
materials. Typically in a TEC system, p- and n-type semiconductor materials are connected
electrically in series and thermally in parallel. TEC has many advantages such as silent in
operation, compact, robust, long service life, and environmentally friendly. TECs can be used to
precisely control the temperature and have potential as a cooling system for electronics, data
centers, military devices, laboratory apparatuses, and transportation vehicles.
Chein and Huang (2004) studied the cold surface temperature and temperature differences
between the hot and cold surface of TEC to analyze the cooling capacity, coefficient of
performance (COP), and heat sink thermal resistance for electronic cooling. Their analysis
showed high TEC cold surface temperature or low temperature difference between the hot and
cold surface of TEC, which increases the cooling capacity and COP of TEC. Semenyuk and
Dekhtiaruk (2013) presented experimental results of thermal management of Light-Emitting
Diode (LED) using TEC. Their TEC cooling experiments showed improvement in terms of
reduction in LED operating temperature and increased light output. They also concluded that
TEC can provide an extra 12 ºC temperature reduction compared to a metal substrate-printed
circuit board with heat sink system. Cheng et al. (2010) developed a 3D theoretical model of
TEC and concluded that COP decreases rapidly as the amount of current increases. Lee et al.
(2010) studied the effect of Seebeck coefficient and electrical conductivity on the performance of
micro TEC. Lee et al. (2010) concluded that COP decreases because of the reduction in Seebeck
coefficient and electrical conductivity. Chen et al. (2012) performed numerical simulations with
different TECs in pairs and investigated the effect of Thomson heat. They concluded that cooling
power can be improved by a factor of 5%-7% considering the Thomson heat. Gould et al. (2011)
performed TEC cooling experiments on a desktop computer and showed improved results
compared to a standard cooling system. In addition to this, they combined TEC with a
80
thermoelectric (TE) generator and obtained 4.2 mW. McCarty (2010) performed 1D analytical,
3D numerical, and experimental investigation to check temperature dependency effects of TE
properties. McCarty (2010) concluded that a 1D analytical and 3D numerical model with
temperature dependant terms give accurate results of maximum temperature drop and also
modeling results matched with experimental results. Tipsaenporm et al. (2012) performed
experiments on a TE cooling system in combination with a direct evaporative cooling (DEC)
system. Their experiments found improvement in COP of TEC from 0.43 (without DEC) to 0.52
(with DEC) with electric current input of 4.5 A. Maneewan et al.(2010) performed experiments
on a TE air-conditioner made up of three TE modules. Using 1 A electric current, they removed
29.2 W of heat at 28 °C with a COP of 0.34. Melero et al.(2003) investigated a TE air-
conditioning system with 48 TE modules combined with photovoltaic solar panels for a domestic
air-conditioning system. FLUENT numerical simulation of Melero et al. (2003) showed that a
TEC can provide the minimum temperature required for human comfort. Sullivan et al. (2012)
performed steady-state and transient numerical simulations of a TEC system in order to address
the site specific cooling of the hot spot in a microprocessor. Sullivan et al. (2012) concluded that
transient cooling with square root current pulse is most effective with 10°C cooling. Yang and
Stabler (2009) reviewed potential applications of TE materials in an automobile as a cooler and
generator. Yang and Stabler (2009) concluded that applications of TE materials will expand, as
availability of high performance TE materials increase.
From the above studies, it is clear that TEC can be used in different ways for cooling purposes.
TEC typically suffers from low conversion efficiency due to poor TE material properties which
include low Seebeck coefficient, low electrical, and high thermal conductivities. Any
performance improvement can make TEC applicable to a wide range of applications.
Performance of TE materials depends on a parameter called ‘Figure of Merit’ ZT= Tk)( 2 .
Current ZT of the state-of-the-art TE material is around 1(Ma et al. 2013). As observed from the
ZT expression, the magnitude of ZT can be improved by lowering the thermal conductivity. TE
nanocomposite, a TE material prepared using TE nanoparticles and TE base/host material, can
yield a low thermal conductivity (Slack 1995). Poudel et al. (2008) achieved a ZT value of 1.4 at
373 K by hot-pressing nanopowders of Bismuth-Telluride (Bi2Te3) and Antimony-Telluride
81
(Sb2Te3) under Argon (Ar) atmosphere. The enhanced ZT was due to the significant reduction in
the thermal conductivity of material (Poudel 2008). Fan et al. (2011) prepared a n-type Bi2Te3
nanocomposite which exhibited a ZT of 1.18 due to reduction in thermal conductivity. Tang et
al.(2011) prepared double- filled Cobalt Triantimonide (CoSb3) using Indium (In) and Lutetium
(Lu) resulting into In0.13Lu0.05Co4.02Sb12 through a high-pressure synthesis method.
In0.13Lu0.05Co4.02Sb12 yielded a ZT of 0.27 which was greater by one order of magnitude than that
of CoSb3 (Tang el al. 2011). Nagami et al. (2014) prepared a Bismuth-Antimony-Telluride
(Bi0.4Sb1.6Te3) bulk TE material using mechanical alloying and hot extrusion. Their material
exhibited a ZT of 1.2 due to high electrical and low thermal conductivity.
Nanocomposite TE materials have the potential to be used as a viable tool for cooling system in
future. In order to study a nanostructured TE generator, Rabari et al. (2014) performed 2D
numerical simulations of a nanostructured TE generator and concluded that convection heat
transfer from side surfaces of a TE generator lowers the efficiency. It is also very important to
investigate the effects of convection heat transfer on the performance of a nanocomposite TEC
which is very limited in the current literature. In this work, an analytical heat transfer model of a
nanocomposite TEC is derived. All TE effects (Seebeck, Peltier, and Thomson effect), heat
conduction, and convection heat transfer are included in both analytical modeling and numerical
simulations. Analytical results are presented with different electric currents, cold surface
temperatures, and convection heat transfer coefficients. The field plots of heat and temperature
distributions are presented with different convection heat transfer coefficients. Numerical results
are compared with that of 1D analytical results in terms of heat absorbed and COP. The structure
of the work is in the following order: derivation of an analytical model, results and discussions
on analytical model, and field plots of numerical simulations.
4.2 Modeling
Figure 4.1 shows a 2D schematic diagram of the nanocomposite TEC considered in this work. A
unit cell of a TEC is made-up of one p-type and one n-type leg of semiconductor materials
connected electrically in series and thermally in parallel. An electrically conductive copper strip
connects both semiconductor legs. Materials we have considered for the current analyses are
82
nanostructured p-type Bi0.5Sb1.5Te3 (Poudel et al. 2008) and n-type Bi2Te3
(Fan et al. 2011).
Figures 4.2 and 4.3 show crystal structures of Bi0.5Sb1.5Te3 and Bi2Te3 (Zhang et al. 2011, Chen
et al. 2009). Both materials possess the tetradymite type crystal structure formed by quintuple
layers made up of 2 sheets of Bi and 3 sheets of Te (Zhang et al. 2011). For Bi0.5Sb1.5Te3, Bi
atoms are substituted by Sb (Chen et al. 2009). P-type Bi0.5Sb1.5Te3
(Poudel et al. 2008) and n-
type Bi2Te3 (Fan et al. 2011) were prepared by hot pressing the nano-powder of materials. P-type
Bi0.5Sb1.5Te3 was prepared via direct current hot press method using nanopowders of
Bi0.5Sb1.5Te3 which was prepared from ball milling of Bi0.5Sb1.5Te3 alloy ingots (Poudel et al.
2008). Nanopowders of p-type Bi0.5Sb1.5Te3 had an average size of 20 nm (Poudel et al. 2008).
The microstructure of the p-type nanostructured Bi0.5Sb1.5Te3 shows highly crystalline structure
and nanosize grains as observed from the Scanning Electron Microscope (SEM) images available
in Poudel et al. (2008). The n-type Bi2Te3 was prepared by melt spinning and hot pressing
different amounts of nanoparticles of Bi2Te3 (Fan et al. 2011). The microstructure of the
nanostructured n-type Bi2Te3 possesses a wire like structure with a mixture of micron and nano
size particles as observed from the SEM images available in Fan et al. (2011).
Figure 4.1 Schematic diagram of unit cell of TEC (drawing is not to scale)
n
W
p
TC
TH
I
Qab
Qre
L
Wg
x
Qc
83
As shown in Fig. 4.1 each semiconductor leg has a height of L, a cross-sectional area of D×W,
and the two legs are separated by a distance of Wg. TEC absorbs Qab amount of heat from the
system and rejects Qre amount of heat to the surroundings. There is also a convection heat
transfer (Qc) through the side walls of the TEC. For the analysis, the following assumptions are
made:
TE nanocomposite materials are homogeneous and isotropic.
Contact resistances at interface of copper and TE materials are neglected.
Energy transport and current flow in nanocomposite TEC considering steady state system can be
expressed as
gen
.q q (1)
0 J (2)
where q ,gen
.q , and J are heat flux, heat generation rate per unit volume, and electric current
density, respectively.
Figure 4.2 The schematic of crystal structure of (a) (Bi1-xSbx)2Te3 (Zhang et al. 2011) Reprinted
by permission from Macmillan Publishers Ltd: Nature Communications from Zhang et al. 2, 574
(2011), copyright 2011
One
quintuple
layer
84
Figure 4.3 The schematic of crystal structure of Bi2Te3 (Chen et al. 2009) From [Chen et al.
Science 325, 178 (2009)]. Reprinted with permission from AAAS.
Equations (1) and (2) are coupled by set of TE constitutive equations (Yang et al. 2011) as
shown in Eqs. (3) and (4):
TkT effeff Jq
Teffeffeff EJ
(3)
(4)
where eff , effk , and eff are the effective Seebeck coefficient, effective thermal conductivity,
and effective electrical conductivity of TE nanocomposite, respectively. E is the electric field
intensity. E can be expressed as , where is the electric scalar potential (Yang et al.
2011). For a steady state analysis, rearrangement of Eqs. (1) to (4) give coupled TE equations for
the heat transfer (Eq. (5)) and electric potential (Eq. (6)):
JEJ TkT effeff (5)
0 effeffeff T (6)
In Eq. (5), the first term represents Thomson heat, the second term is heat transfer due to
conduction, and the third term is Joule heat. In Eq. (6), the first term is electric current density
due to Seebeck effect and the second term is standard voltage driven electric current.
One
quintuple
layer
85
In order to analyze a nanocomposite TEC analytically, Eq. (5) is simplified to obtain 1D form as
shown in Eq. (7) below and solved.
0)(
heat Thomsonferheat trans Convectionheat Joule
2
2
conductionheatFourier
2
2
dx
dT
Ak
ITT
Ak
ph
Ak
I
dx
Td
eff
a
effeff
eff
(7)
Solution of Eq. (7) in terms of temperature distribution using boundary conditions; (x = 0, T =
TC; x = L, T = TH) is given by
)(
)()(12
112221
LCLC
LC
C
LC
H
xCLC
C
LC
H
xC
ee
eTeTeeTeTexT (8)
Different terms used in Eq. (8) can be defined as
2
42
1
C ;
2
42
2
C
;Ak
I
eff
;
Ak
ph
eff
2
2
Ak
IT
Ak
ph
eff
eff
a
eff
.
(9)
(10)
Heat absorbed at the cold surface of a TEC can be written by combining Peltier heat which
occurs at junctions only
)(
)()(12
112221
21
LCLC
LC
C
LC
H
xCLC
C
LC
H
xC
eff
Ceffab
ee
eTeTeCeTeTeCAk
TIQ
(11)
An electric power input to TEC is given by
iRIP 2 (12)
where Ri can be calculated by
n
nneff,
p
ppeff,
A
L
A
LRi
(13)
Moreover, Seebeck coefficient and thermal conductivity of unit cell of a TE system can be
calculated by
n
nneff,
p
ppeff,
L
Ak
L
AkK (14)
neffpeff ,, (15)
86
To measure the performance of a TEC, the COP can be written as
COPP
Qab (16)
It is important to note that Eq. (11) is the heat absorbed by one TE leg only. Heat absorbed by
unit cell of a TEC as shown in Figure 4.1 can be calculated by considering respective properties
of both p- and n-type TE legs.
4.3 Results and discussions
In this section, the performance of a unit cell of TEC is presented based on results obtained by
analytical modeling and numerical simulations based on modeling section. For p-type material, a
nanostructured semiconductor BiSbTe is considered to analyze the performance (Poudel et al.
2008). while for n-type material nanocomposite Bi2Te3 is considered to analyze the performance
(Fan et al. 2011). The temperature dependent transport properties of nanocomposite and
conventional TE materials such as the Seebeck coefficient, electrical conductivity, and thermal
conductivity are specified as polynomial functions of temperature as shown in Table 4.1 and 4.2.
These properties are evaluated at an average temperature of a working temperature range. Two
different cases of cold surface temperatures (TC) = 333 K and (TC) = 343 K are considered with
hot surface temperature (TH) = 353 K. It is assumed that proposed TEC is used in cooling data
center processors which has a temperature limit of 85 ºC
(Ebrahimi et al. 2014). A
semiconductor leg of a unit cell of TEC has a width and thickness (W×D) of 2 mm × 2 mm and a
height (L) of 5 mm. The gap between two consecutive legs (Wg) is 0.5 mm.
87
Table 4.1 Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal
conductivity as a function of temperature for BiSbTe nanostructured bulk alloys and
nanocomposite Bi2Te3 (Poudel et al. 2008, Fan et al. 2011)
TE
Properties
Polynomial Expressions
)p(eff 6493623 10)10738.710656.310732.2656.050.172( TTTT
)p(eff 541138253 10)10354.110093.210451.210575.8450.1( TTTT
)p(effk 41037254 10875.710809.310193.410946.2136.1 TTTT
)n(eff
648
362312
10)10246.1
10003.51043.110884.21035.1(
T
TTT
)n(eff 34936232 10)10075.710907.110815.1003.11024.2( TTTT
)n(effk 41037254 T10079.4T10991.1T10696.1T10907.1035.1
Table 4.2 Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal
conductivity as a function of temperature for BiSbTe bulk alloys and conventional Bi2Te3
(Poudel et al. 2008, Fan et al. 2011)
TE
Properties
Polynomial Expressions
p 6483524 10)10281.210497.110637.9504.090.201( TTTT
p 541138253 10)10323.910165.810464.310053.8165.1( TTTT
pk 4937253 T10009.1T10477.4T10920.2T10023.1415.1
n 64734222 10)1033.21020.11015.262.11004.1( TTTT
n 34936232 10)1062.81025.71057.315.11036.2( TTTT
nk 41037253 T1076.1T1059.1T1025.6T1021.531.1
88
Analytical results
This section presents analytical results of a nanocomposite TEC. A temperature distribution
across a TE leg is plotted with adiabatic and convective boundary conditions. The heat absorbed
and COP of a TEC are also plotted and discussed as functions of convection heat transfer
coefficients, electric current inputs, and temperature differences. At the end of this section, the
maximum heat absorption and COP results are plotted with an optimum electric current,
convection heat transfer coefficients and TE leg heights.
Figures 4.4 and 4.5 show temperature distribution over the length of a TE leg with convection
heat transfer coefficients from 10-7
Wm-2
K-1
(adiabatic) to 80 Wm-2
K-1
. The surrounding
temperature is considered to be an average working range of hot and cold surface temperatures.
Temperature distribution remains distinct for p- and n-type leg because of their different TE
properties. P-type leg exhibits more Joule heat generation compared to n-type leg which can be
observed by temperature profile with an adiabatic side wall condition in Figs. 4.4 and 4.5. This
can be attributed to high electrical resistivity of p-type material compared to n-type material as
shown in Fig. 4.6.
89
Length of p-type TE leg, L (m)
Tem
per
atu
re,
T(K
)
0 0.001 0.002 0.003 0.004 0.005330
335
340
345
350
355
h = 10-7
Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2
K-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
Figure 4.4 Temperature distribution over the length of p-type TE leg with hot surface
temperature 353 K and cold surface temperature 333 K
90
Length of n-type TE leg, L (m)
Tem
per
atu
re,
T(K
)
0 0.001 0.002 0.003 0.004 0.005330
335
340
345
350
355
h = 10-7
Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2K
-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
Figure 4.5 Temperature distribution over the length of n-type TE leg with hot surface
temperature 353 K and cold surface temperature 333 K
91
n-type material
Temperature, T (C)
Eff
ecti
ve
ele
ctr
ical
resi
stiv
ity
,
eff
(m
)
0 50 100 150 200 2500
5E-06
1E-05
1.5E-05
2E-05
2.5E-05
p-type material
Figure 4.6 Electrical resistivity of p- and n- type legs of nanocomposite TEC
92
Figures 4.7 and 4.8 show heat absorbed from the cold surface using a nanocomposite TEC with
electric current inputs from 0 to 10 A with adiabatic and convective boundary conditions.
Convective boundary condition at side walls of a TEC is considered with convection heat
transfer coefficients from h = 10-7
Wm-2
K-1
(adiabatic) to 80 Wm-2
K-1
. Plots in Figs. 4.7 and 4.8
show amount of heat absorbed increases with increase in an electric current input and reaches a
peak value. An electric current input corresponds to the maximum heat absorbed can be refer as
an optimum current input for a given condition. Further, increase in an electric current input
leads to decrease in amount of heat absorbed. The reason behind this phenomenon is Joule heat;
higher electric current input generates more heat within a TE leg which leads to low heat
absorption from the cold surface. In addition to this, plot also shows effect of convection heat
transfer coefficient on heat absorption. The effect of convection heat transfer coefficient can be
divided into two parts; effect with a low electric current input and high electric current input. It is
observed that with a low electric current input convection heat transfer coefficient reduces the
heat absorbed by a TEC. However, at relatively higher electric current input convection heat
transfer coefficient increases the amount of heat absorbed by a TEC. It is important to note here
that effect of an electric current and side wall convection heat transfer depend on intrinsic
properties of TE materials such as thermal and electrical conductivities. The performance of a
TEC is highly dependent on TE material properties so effect of electric current and convection
heat transfer coefficient remains distinct for different TE materials.
93
Current, I (A)
Hea
tab
sorb
ed,
Qab
(W)
2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
h = 10-7
Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2
K-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
Figure 4.7 Heat absorbed as a function of current considering hot surface temperature 353 K
with cold surface temperature 333 K
94
Current, I (A)
Hea
tab
sorb
ed,
Qab
(W)
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
h = 10-7
Wm-2K
-1
h = 20 Wm-2K
-1
h = 40 Wm-2
K-1
h = 60 Wm-2K
-1
h = 80 Wm-2K
-1
Figure 4.8 Heat absorbed as a function of current considering hot surface temperature 353 K
with cold surface temperature 343 K
COP is plotted as a function of an electric current with different convection heat transfer
coefficients in Figs. 4.9 and 4.10. A hot surface temperature is considered 353 K and cold
surface temperatures are considered 333 K and 343 K, respectively for Figs. 4.9 and 4.10. At
very low electric current, COP remains very low because of no heat absorption. As electric
current increases, COP also rises and reaches its maximum. After that more input current leads to
reduction in COP. Reasons behind the reduction in COP at high electric current are Joule
95
heating, increase in power input, and relatively low improvement in the heat absorption. The
same plot, also investigates the effect of convection heat transfer coefficient. Effect of
convection heat transfer coefficient can be divided into two distinct categories same as heat
absorption. For relatively low electric current input, COP decreases as convection heat transfer
coefficient increases. In contrary, COP increases with an increase in convection heat transfer
coefficient with relatively high electric current input. The reason behind relatively higher COP at
higher convection heat transfer coefficient is heat transfer through side walls of TE legs with
surroundings which help escapes the higher amount of Joule heat and absorbed heat.
96
Current, I (A)
CO
P
0 2 4 6 8 100
1
2
3
4
5
h = 10-7
Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2
K-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
Figure 4.9 COP of TEC as a function of current considering hot surface temperature 353 K with
cold surface temperature 333 K
97
Current, I (A)
CO
P
0 2 4 6 8 100
2
4
6
8
10
12
h = 10-7
Wm-2
K-1
h = 20 Wm-2
K-1
h = 40 Wm-2
K-1
h = 60 Wm-2
K-1
h = 80 Wm-2
K-1
Figure 4.10 COP of TEC as a function of current considering hot surface temperature 353 K
with cold surface temperature 343 K
Figures 4.11 and 4.12 show the amount of heat absorbed and COP as a function of temperature
difference between hot and cold surface temperatures with variable electric current input and
adiabatic boundary condition. Plot in Fig. 4.11 shows the amount of heat absorbed decreases
with increase in temperature difference considering a hot surface temperature 353 K. A TEC can
remove maximum heat when a temperature difference is zero. In other words temperature
difference can be maintained zero, if TEC removes all of the heat generated by a system and
98
maintains the hot surface temperature (TH) = cold surface temperature (TC). Additionally, plot
also demonstrates that higher amount of an electric current help absorb more heat but amount of
heat absorption decreases with increase in an electric current due to Joule heat. Plot in Fig. 4.12
shows variation in COP as temperature difference increases with a hot surface temperature 353 K
and adiabatic side wall condition. With variable electric current input, one can conclude that
higher amount of an electric current leads to low COP. The reason behind low COP at higher
electric current is relatively low amount of heat absorbed in comparison to power input. Also,
COP remains highest when temperature difference is zero irrespective of electric current input.
99
Temperature difference,
Hea
tab
sorb
ed,
Qab
(W)
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
I = 1 A
I = 2 A
I = 3 A
I = 4 A
I = 5 A
Figure 4.11 Heat absorbed as a function of temperature difference with different electric current
input and hot surface temperature 353 K considering adiabatic side wall condition
100
Temperature difference,
CO
P
0 5 10 15 200
1
2
3
4
5
6
I = 1 A
I = 2 A
I = 3 A
I = 4 A
I = 5 A
Figure 4.12 COP as a function of temperature difference with different electric current input and
hot surface temperature 353 K considering adiabatic side wall condition
101
From Fig. 4.7 it is clear that a TEC has an optimum electric current at which a TEC can absorb
maximum amount of heat. Moreover, an optimum electric current is also different to operate
TEC with maximum COP. It would be interesting to observe the effect of convection heat
transfer coefficient on an optimum electric current with maximum amount of heat absorbed and
maximum COP. Figures 4.13, 4.14, 4.15, and 4.16 reflect effect of convection heat transfer
coefficients on an optimum electric current with the maximum amount of heat absorbed and
maximum COP, respectively. It is important to note here that each optimum electric current is
for a fixed geometry as shown in Table 4.3 and is calculated using the Maple’s nonlinear
optimization solver (Cybernet 2012). Maple’s nonlinear optimization solver uses the Karush-
Kuhn-Tucker theorem to solve the Lagrangian function for constrained nonlinear problems
(Fishback 2009). Table 4.3 shows different TE leg heights and their corresponding internal
resistances which can be referred to locate the appropriate legend in Figs. 4.13, 4.14, 4.15, and
4.16. As observed by the trend in Fig. 4.13 increment in convection heat transfer coefficient
increases the amount of heat absorbed.
Table 4.3 Height of TE legs and internal resistance of TEC for different cases considered in
Figs. 4.13, 4.14, 4.15, and 4.16
Cases in Figs. 4.13, 4.14,
4.15, and 4.16
Height of TE legs,
L (m)
Internal resistance of TEC,
Ri (Ω)
Case 1 0.0025 0.0106
Case 2 0.005 0.0212
Case 3 0.0075 0.0318
Case 4 0.0100 0.0424
102
Convection heat transfer coefficient, h (Wm-2
K-1)
Max
imu
mh
eat
ab
sorb
ed,
Qab
max
(W)
0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Case 1
Case 2
Case 3
Case 4
Figure 4.13 Maximum heat absorbed of TEC considering variable convection heat transfer
coefficient and variable TE leg heights with hot surface temperature 353 K and cold surface
temperature 333 K
103
With adiabatic side wall conditions, the smallest TEC and longest TEC in height show maximum
and minimum heat absorption, respectively. The reason behind this is change in internal
resistance of a nanocomposite TEC as shown in Table 4.3. The smallest TEC has lowest internal
resistance which generates low Joule heat whereas longest TEC has a highest internal resistance
which generates high Joule heat. It is also clear that TEC’s optimum electric current requirement
increases with increase in a convection heat transfer coefficient. Effect of convection heat
transfer coefficient on COP can be analysed from Fig. 4.15, where COP follows decreasing trend
as convection heat transfer coefficient increases. The same plot also strengthens previous result
of higher optimum current with increment in convection heat transfer coefficient. A TEC’s
requirement of higher optimum electric current with increase in convection heat transfer
coefficient can be verified from Figs. 4.14 and 4.16. Moreover, plot also reveals that maximum
COP of a TEC is independent of height of TE legs and this same phenomenon can also verified
from Fig. 12 of Hodes (2007). Figures 4.14 and 4.16 also show that optimum electric current
remains different for maximum heat absorption and maximum COP.
104
Convection heat transfer coefficient, h (Wm-2
K-1)
Op
tim
um
elec
tric
cu
rren
t,I o
pt(A
)
0 20 40 60 800
2
4
6
8
10
12
14
16
Case 1
Case 2
Case 3
Case 4
Figure 4.14 Optimum electric current for maximum heat absorption of TEC considering variable
convection heat transfer coefficient and variable TE leg heights with hot surface temperature
353 K and cold surface temperature 333 K
105
Convection heat transfer coefficient, h (Wm-2
K-1)
CO
Pm
ax
0 20 40 60 800
1
2
3
4
5
Case 1
Case 2
Case 3
Case 4
Figure 4.15 Maximum COP of TEC considering variable convection heat transfer coefficient
and variable TE leg heights with hot surface temperature 353 K and cold surface temperature
333 K
106
Convection heat transfer coefficient, h (Wm-2
K-1)
Op
tim
um
elec
tric
cu
rren
t,I o
pt(A
)
0 20 40 60 800
0.2
0.4
0.6
0.8
1
1.2
1.4
Case 1
Case 2
Case 3
Case 4
Figure 4.16 Optimum electric current for maximum COP of TEC considering variable
convection heat transfer coefficient and variable TE leg heights with hot surface temperature
353 K and cold surface temperature 333 K
107
Additionally, internal resistance and maximum heat absorption are plotted as a function of TE
leg height. Plots in Figs.4.17 and 4.18 show performance of a unit cell of TEC as height of TE
leg vary. As seen in Fig. 4.17, an increase in internal resistance of a unit cell of TEC is consistent
with rise in height of TE leg which is quite evident from Eq. (13) as well. The rise in internal
resistance of TEC leads to the reduction in maximum amount of heat absorbed by TEC as shown
in Fig. 4.18.
108
Height of TE leg, L(m)
Inte
rnal
Res
ista
nce,
R(
)
0 0.005 0.01 0.015 0.020
0.02
0.04
0.06
0.08
0.1
Figure 4.17 Internal resistance of TEC unit cell as a function of TE leg height
109
Height of TE leg, L(m)
Max
imu
mh
eat
ab
sorb
ed,
Qab
max
(W)
0 0.005 0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
1.2
Figure 4.18 Maximum heat absorbed as a function of TE leg height by unit cell of TEC with hot
surface temperature 353 K, cold surface temperature 333 K, and adiabatic side wall condition
110
Numerical Simulation Results and Comparison
In this section, results achieved using numerical simulation is presented. Initially, the results
obtained are expressed using field plots of temperature contours with heat flow and later results
are expressed graphically. A nanocomposite TEC with width (W = 2 mm) and height (L = 5 mm)
is considered for analysis. Figures 4.19 to 4.26 present field plots of an electric potential,
temperature distribution, and heat flow with a cold surface temperature 333 K and a hot surface
temperature 353 K. Figures 4.19 and 4.20 show field results of an electric potential and electric
current flow streamlines. The terminal of TEC is applied with electric potentials of 0.02 V and
0.06 V for Figs. 4.19 and 4.20, respectively. An electric potential of 0.02 V and 0.06 V
corresponds to low and high electrical current inputs, respectively. An electric potential is
indicated by marked iso-potential lines with a multi-colored background, while current flow is
indicated by lines with arrows. Figures 4.21, 4.22, 4.23 and 4.24 show field plots of temperature
and heat flow with an electric potential of 0.02 V. At the end, Figs. 4.25 and 4.26 show field
plots of temperature and heat flow with an electric potential of 0.06 V to demonstrate Joule
heating. Temperature is presented by marked isothermal lines with multi-colored background,
while heat flow is presented by vertical lines with arrows. Vertical walls of semiconductor legs
are considered under different convective heat transfer conditions from h ≈ 0 Wm-2
K-1
to h = 60
Wm-2
K-1
. The surrounding temperature is considered to be an average temperature of a working
range. Figures 4.19 and 4.20 show distribution of an electric potential and flow of electric
current. Field plot shows electric potential distribution and current flow according to applied
electric boundary condition.
111
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
Figure 4.19 Electric scalar potential and current flow in nanocomposite TEC with electric
potential 0.02 V
112
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
Figure 4.20 Electric scalar potential and current flow in nanocomposite TEC with electric
potential 0.06 V
113
Figures 4.21, 4.22, 4.23 and 4.24 present heat flow and temperature distribution inside a
nanocomposite TEC with an electric potential 0.02 V. A nanocomposite TEC is under different
convection boundary conditions at vertical walls of semiconductor legs ranging from h ≈ 0
Wm-2
K-1
(adiabatic) to h = 60 Wm-2
K-1
. When vertical walls of semiconductor legs are adiabatic,
temperature remains nearly constant at a given location of a distance x. This remains true for
both p- and n-type semiconductor legs. Though, different thermal conductivities change the
location of the same isothermal line in p- and n-type leg. Due to the adiabatic side wall
condition, both p- and n-type legs have parallel heat flux lines with uniform distribution over the
cross-section of legs. The temperature distribution non-linearity increases as surface to the
surrounding convection heat transfer coefficient increases from h = 20 Wm-2
K-1
to h = 60
Wm-2
K-1
as observed in Figs. 4.22, 4.23, and 4.24. An introduction of a convective heat transfer
boundary condition to side walls brings heat from the surrounding because part of the
semiconductor leg is cooler than the surrounding. Higher the convection heat transfer coefficient,
more heat is being brought inside TE leg. This phenomenon reduces amount of heat absorbed
from cold surface of the TEC when low electric potential is applied.
114
334
336
338
340
342
344
346
348
350
352
334
336
338
340
342
344
346
348
350
352
Figure 4.21 Heat flow and temperature distribution in nanocomposite TEC for h ≈ 0 Wm-2
K-1
at
vertical walls with electric potential 0.02 V
115
334
336
338
340
342
344
346
348
350
352
334
336
338
340
342
344
346
348
350
352
Figure 4.22 Heat flow and temperature distribution in nanocomposite TEC for h = 20 Wm-2
K-1
at vertical walls with electric potential 0.02 V
116
334
336
338
340
342
344
346
348
350
352
334
336
338
340
342
344
346
348
350
352
Figure 4.23 Heat flow and temperature distribution in nanocomposite TEC for h = 40 Wm-2
K-1
at vertical walls with electric potential 0.02 V
117
334
336
338
340
342
344
346
348
350
352
334
336
338
340
342
344
346
348
350
352
Figure 4.24 Heat flow and temperature distribution in nanocomposite TEC for h = 60 Wm-2
K-1
at vertical walls with electric potential 0.02 V
118
Figures 4.25 and 4.26 present streamlines and contours of heat flow and temperature distribution
with a higher electric potential of 0.06 V. Due to higher electric potential, nanocomposite TEC
draws more current which eventually generates more Joule heat as one can see from Figs. 4.25
and 4.26. Nonetheless, this Joule heat keeps TE leg at a higher temperature compared to
surroundings which eventually transfers the heat to the surroundings from side surfaces as well.
A difference in the location of heat streamlines in Figs. 4.25 and 4.26 show that higher
temperature inside the TE leg (Fig. 4.26) transfers the heat to surroundings. Ultimately, this
phenomenon drags more heat from the cold surface of TEC.
119
334334
338
342
346
350350
354
354354
334
338 338
342 342
346346
350
Figure 4.25 Heat flow and temperature distribution in nanocomposite TEC for h ≈ 0 Wm-2
K-1
at
vertical walls with electric potential 0.06 V
120
334
338 338
342
346
350
354 354
354354
334
338 338
342
346
350 350
Figure 4.26 Heat flow and temperature distribution in nanocomposite TEC for h = 60 Wm-2
K-1
at vertical walls with electric potential 0.06 V
121
Figures 4.27 and 4.28 present heat absorbed and COP of a TEC with variable convection heat
transfer coefficients, a fixed hot surface temperature 353 K, and an electric potential of 0.02 V.
Plot revels that as convection heat transfer coefficient increases, heat absorbed and COP of a
TEC decreases. This phenomenon can be observed with analytical results in Fig. 4.7, where heat
absorption decreases as convection heat transfer coefficient increases with low electric potential.
Plot also shows that a TEC absorbs more heat with small temperature difference; the same
phenomenon can be seen in Fig. 4.11 where heat absorbed is plotted against temperature
differences using analytical model.
122
TC
= 343 K
Convection heat transfer coefficient, h (Wm-2
K-1)
Hea
tab
sorb
ed,
Qab
(W)
0 20 40 60 80
15
20
25
30
35
40
TC
= 333 K
Figure 4.27 Heat absorbed by nanocomposite TEC as a function of convection heat transfer
coefficient with hot surface temperature 353 K and electric potential 0.02 V
123
TC
= 343 K
Convection heat transfer coefficient, h (Wm-2
K-1)
CO
P
0 20 40 60 802
2.5
3
3.5
4
4.5
5
TC
= 333 K
Figure 4.28 COP of nanocomposite TEC as a function of convection heat transfer coefficient
with hot surface temperature 353 K and electric potential 0.02 V
124
In order to study the same phenomenon with large electric potential, heat absorbed and COP are
plotted as a function of convection heat transfer coefficient with a hot surface temperature 353 K
and an electric potential of 0.06 V in Figs. 4.29 and 4.30. Due to large electric potential TEC
produces more Joule heat and a leg of the TEC remains at higher temperature compared to
surroundings. This phenomenon is shown using contours and heat streamlines in Figs. 4.29 and
4.30. At higher convection heat transfer coefficients, a TE leg removes more heat to
surroundings which eventually assists more absorption of heat from the cold surface. It is
important to note here that COP of a TEC relatively remains same because of increment in power
input. Such large heat absorption can be helpful where amount of heat generation is large such as
large data centers.
125
TC
= 343 K
Convection heat transfer coefficient, h (Wm-2
K-1)
Heat
ab
sorb
ed
,Q
ab
(W)
0 20 40 60 80100
105
110
115
120
125
TC
= 333 K
Figure 4.29 Heat absorbed by nanocomposite TEC as a function of convection heat transfer
coefficient with hot surface temperature 353 K and electric potential 0.06 V
126
TC
= 343 K
Convection heat transfer coefficient, h (Wm-2
K-1)
CO
P
0 20 40 60 803.9
4
4.1
4.2
4.3
4.4
4.5
4.6
TC
= 333 K
Figure 4.30 COP of nanocomposite TEC as a function of convection heat transfer coefficient
with hot surface temperature 353 K and electric potential 0.06 V
127
Figures 4.31 and 4.32 present comparison between analytical and numerical simulation results.
Due to approximations such as decoupling between TE constitutive equations in analytical
modeling, analytical results over predict the heat absorbed and COP compared to numerical
simulations. Nevertheless, results show a fair agreement between analytical and numerical
simulations.
128
Numerical simulation results
Convection heat transfer coefficient, h (Wm-2
K-1)
Hea
tab
sorb
ed,
Qab
(W)
0 20 40 60 8014
15
16
17
18
19
20
Analytical results
Figure 4.31 Comparison of analytical and numerical simulation results in terms of heat absorbed
considering variable convection heat transfer coefficient with hot surface temperature 353 K,
cold surface temperature 333 K, and electric potential 0.02 V
129
Numerical simulation results
Convection heat transfer coefficient, h (Wm-2
K-1)
CO
P
0 20 40 60 802.4
2.5
2.6
2.7
2.8
2.9
3
Analytical results
Figure 4.32 Comparison of analytical and numerical simulation results in terms of COP
considering variable convection heat transfer coefficient with hot surface temperature 353 K,
cold surface temperature 333 K, and electric potential 0.02 V
130
Figure 4.33 shows COP of a TEC using conventional (no nanostructuring) and nanocomposite
TE materials. The trend in a plot is quite evident about rise in the COP using nanocomposite TE
materials. COP of TEC increases from 4.92 to 5.58 if nanocomposite materials are used, which
gives approximately 13% rise for the materials considered. The rise in COP can be attributed to a
reduction in the thermal conductivity of TE materials due to nanostructuring. As observed from
Fig. 4.34, thermal conductivity of nanocomposite is lower than conventional TE materials. For
example, thermal conductivity of p-type BiSbTe decreases from 1.36 Wm-1
K-1
to 0.98 Wm-1
K-1
at 100 ºC.
131
Temperature difference,
CO
P
0 5 10 15 202.5
3
3.5
4
4.5
5
5.5
6
Conventional TE material
Nanocomposite TE material
Figure 4.33 Comparison of COP using conventional (no nanostructuring) and nanocomposite TE
material considering h ≈ 0 Wm-2
K-1
, hot surface temperature 353 K, and electric current input of
1 A
132
p-type conventional TE material
n-type conventional TE material
Temperature, T (C)
Th
erm
al
co
nd
ucti
vit
y,
k(W
m-1
K-1
)
0 50 100 150 200 2500.5
1
1.5
2
2.5
3
3.5
p-type nanocomposite TE material
n-type nanocomposite TE material
Figure 4.34 Thermal conductivity of conventional (no nanostructuring) and nanocomposite TE
materials
133
Figure 4.35 shows a comparison in terms of a temperature difference created by unit cell of a
TEC with different levels of electric current input. Poudel et al. (2008) prepared a p-type BiSbTe
nanostructured TE leg via hot pressing the ball-milled nanopowders of BiSbTe. They (Poudel et
al. 2008) prepared a unit cell of TEC using one p-type nanostructured TE leg and one n-type
commercially available (non-nanostructured) TE leg. Authors (Poudel et al. 2008) performed an
experiment under vacuum condition with a hot surface temperature 373 K to verify the
performance of a unit cell of a TEC. Results obtained from current work shows fair agreement
with an experimental work as presented in Fig. 4.35. The current results also show good
agreement with the theoretical predictions given in Poudel et al. (2008).
134
Electric current, I(A)
Tem
per
atu
red
iffe
ren
ce,
T(o
C)
0 2 4 6 8 100
20
40
60
80
100
120Poudel et al. (2008) experimental results
Current work
Poudel et al. (2008) Theoretical results
Figure 4.35 Comparison of results between current work and Poudel et al. (2008)
135
4.4 Conclusions
In this research work, a 1D analytical heat transfer model of a TEC was derived considering the
Seebeck effect, Peltier effect, Thomson effect, heat conduction, and convection heat transfer.
TEC performance parameters, such as the heat absorbed and COP, were analyzed as functions of
the electric current and convection heat transfer coefficient. In addition, an optimum electric
current is calculated considering different convection heat transfer coefficients and TE leg
heights for two different cases: maximum heat absorption, and maximum COP. A numerical
simulation was also performed to investigate heat transfer and temperature distribution in a
nanocomposite TEC.
The following conclusions are based on conducted studies:
1. Electric current plays a significant role in the performance of a TEC. As TE materials are
temperature-dependent and as the electric current can play a significant role in internal heat
generation, an optimum electric current input is an important factor.
2. It is observed that convection heat transfer has different effects on the performance of a TEC
depending on the amount of an electric current. At a low electric current input, it was
observed that convection heat transfer deteriorates the performance of a TEC, but at high
electric current, convection heat transfer can help remove large amount of heat from the cold
surface of a TEC.
3. This study demonstrates selection criteria to optimize TEC performance. For example, a
TEC can remove large amounts of heat but cannot exhibit higher COP, which can be useful
with high-end mainframes and large data centers where large heat generation is a major
problem.
In future, a more detailed study will be performed to experimentally study the effects of
convection heat transfer on the performance of a TEC.
136
4.5 Nomenclature
A cross-sectional area (m2)
C1 a parameter (refer Eq.(9))
C2 a parameter (refer Eq.(9))
D thickness of TE leg (m)
E electric field intensity (Vm-1
)
h convection heat transfer coefficient (Wm-2
K-1
)
I electric current (A)
J electric current density vector (Am-2
)
k thermal conductivity (Wm-1
K-1
)
L height of TE leg (m)
p perimeter (m)
P electrical power output (W)
q heat flux vector (Wm-2
)
.q
heat generation rate (Wm-3
)
Q heat (W)
R electrical resistance (Ω)
T temperature (K)
W width of TE leg (m)
x co-ordinate system (m)
ZT dimensionless figure of merit
Greek Symbols
Seebeck coefficient (VK-1
)
ε a parameter (refer Eq.(10))
κ a parameter (refer Eq.(10))
ρ electrical resistivity (Ωm)
electrical conductivity (Sm-1
)
τ Thomson coefficient (VK-1
)
electric scalar potential (V)
ψ a parameter (refer Eq.(10))
137
Subscripts
a atmospheric condition
ab heat absorbed
c convection heat loss
C cold surface temperature
eff effective transport properties
g gap between TE legs
gen heat generation
H hot surface temperature
i internal resistance
max maximum value
n n-type material
opt optimum value
p p-type material
re heat rejected
138
CHAPTER 5: EFFECT OF THERMAL CONDUCTIVITY ON
PERFORMANCE OF THERMOELECTRIC SYSTEMS BASED ON
EFFECTIVE MEDIUM THEORY
5.1 Introduction
Thermoelectric (TE) systems are typically made up of multiple pairs of p-type and n-type
semiconductor materials which are connected electrically in series and thermally in parallel. A
TE system working as a generator based on the Seebeck effect is called a thermoelectric
generator (TEG). Alternatively, a TE system working as a cooler/heater based on the Peltier
effect is called a thermoelectric cooler (TEC). TE systems have potential applications in
electronics, military, laboratory equipment, and transportation systems. TE systems offer many
advantages such as silent operation, compact design, robust operation, long service life, and
environmentally friendly. The performance of TE materials is measured using a parameter
called figure of merit (ZT = α2σT/k). Due to low ZT of TE materials, TE systems are redundant in
many real world applications. A poor electrical conductivity and Seebeck coefficient and higher
thermal conductivity leads to a poor ZT. Although, ZT can be improved in different ways and
two of the ways to improve ZT are illustrated in Fig. 5.1.
139
Figure 5.1 Different approaches to increase ZT of TE materials (Martin-Gonzalez 2013)
The first way employs an increase in the Seebeck coefficient and electrical conductivity,
collectively called the ‘power factor’, while the second way employs a decrease in the thermal
conductivity. Hicks and Dresselhaus (1993) discussed the concept of quantum wire (one
dimensional) for the TE materials. ZT increases because power factor (α2σ) improves due to one
dimensional structure but not significantly lowering the thermal conductivity (Hicks and
Dresselhaus 1993). A method to increase ZT was demonstrated experimentally by
Venkatasubramanian et al. (2001) where ZT of a p-type superlattice of Bi2Te3/Sb2Te3 (Bismuth-
Telluride-Antimony) improved from 1 to 2.4 due to the reduction in the thermal conductivity.
Improvements offered by Hicks and Dresselhaus (1993) and Venkatasubramanian et al. (2001)
were primarily for low dimensional structures such as quantum dots, wires, and superlattice
structures. Such structures can be employed limitedly in the real world applications due to the
complicated physical/chemical vapor deposition method and cost to manufacturing (Ma et al.
2013). There is another route to improve the ZT in bulk TE materials called ‘nanocomposite bulk
materials’ (Bottner and Konig 2013). Nanocomposite bulk materials, which can also be called as
‘composites’ are bulk materials with nanostructured features inside it (Ma et al. 2013).
Composites are made up of a base material and macro/nano particles which can be manufactured
140
via wet-chemical and mechanical synthesis techniques (e.g., solvothermal, extrusion, and high
energy milling). Scoville et al. (1995) reduced the thermal conductivity by 40% by adding Boron
Nitride and Boron Carbide particles into Silicon-Germanium composite. A bulk p-type BixSb2-
xTe3 composite prepared by hot-pressing nanopowders gave ZT of 1.4 due to the low thermal
conductivity (Poudel et al. 2008). In the similar manner, a Bi2Te3 with SiC (Silica Carbide)
nanoparticles (Zhao et al. 2008), Co4Sb12 (Skutterudites) (Li et al. 2009), and AgPbxSbTe2+x
(Lead Antimony Silver Telluride) (Hsu et al. 2004) reported improvement in ZT due to low
thermal conductivities of composites. Bulk composites with the low thermal conductivity
demonstrate promise of improved ZT and, more importantly, routes to synthesize them are more
cost-effective compared to low-dimensional structures (Ma et al. 2013). Thermal conductivities
of composites depend on various parameters, such as particle size, volume fraction, particle
shape, and thermal conductivities of base and particle materials. Thermal conductivities of
composites can be predicted by analytical models based on the Effective Medium Theory
(EMT). The EMT is a method of treating a macroscopically inhomogeneous medium in which
transport properties varies in space (Stroud 1998). EMTs have been applied to different
situations, such as Yu et al. (2015) applied the EMT to calculate the effective permittivity, Gong
et al. (2014) derived modified EMT for a porous media to model the thermal conductivity, Hou
et al. (2015) applied EMT to calculate the effective thermal conductivity of porous thin films,
and Chen et al. (2014) developed the effective thermal conductivity model for bentonites which
is considered engineered barrier material for radioactive wastes. The classical work of Rayleigh
(1892) and Maxwell (1954) can predict the transport properties of a mixture which can also be
applied to thermal conductivity of a composite as being one of the prime transport properties.
Maxwell (1954) considered a heterogeneous mixture with the spherical particles with thermal
conductivities of base material, particles, and composite as kb, kp, and keff, respectively. In
addition to this, is a volume fraction of particle inclusions. Equation (1) is an expression of the
effective thermal conductivity of a composite in terms of thermal conductivities of base and
particle materials and volume fraction of particles.
)(2
)(3
bpbp
bp
beffkkkk
kkkk
(1)
141
Maxwell model does not consider an interaction between particles and base material as one can
observe from Eq. (1) which only includes thermal conductivities of base and particle materials
and volume fraction terms. A model derived by Hasselman and Johnson (1987) can predict the
thermal conductivity of composites considering the ‘Thermal Barrier Resistance’ (TBR). A TBR
arises due to interfacial gap between the particles and base material. Due to TBR, thermal
conductivity of a composite not only depends on particle shape and volume fraction but also on
particle size (Hasselman and Johnson 1987). Equation (2) is an expression of Hasselman and
Johnson (1987) model of the effective thermal conductivity for a composite with spherical
particles.
22
1
22
12
Kr
k
k
k
Kr
k
k
k
Kr
k
k
k
Kr
k
k
k
kkp
b
pp
b
p
p
b
pp
b
p
beff
(2)
In the above equations, r is the radius of spherical particle, K is the TBR expressed in terms of
boundary conductance, and is the volume fraction of particles. Benveniste (1987) also
proposed a thermal conductivity model for composites considering a TBR using the Mori-
Tanaka theory which is also a method to calculate the effective transport properties of
composites. Similarly, Nan et al. (1997) derived a thermal conductivity model for composites
considering different geometries. Thermal conductivity model for a composite can be reduced to
Eq. (3) for spherical particles (Nan et al. 1997).
))1((2)21(
))1((22)21(
bpbp
bpbp
beffkkkk
kkkkkk
(3)
In Eq. (3), thermal conductivity of composite )( effk is expressed in terms of thermal
conductivities of base material )( bk and material of particle )( pk , volume fraction )( , and a
dimensionless parameter for thermal barrier resistance )( . In order to predict the thermal
conductivity of nanocomposites, one has to consider a phonon mean free path (MFP) which is
greater than size of the nanoparticles (Yang et al. 2005). Several methods were also applied to
142
calculate the effective thermal conductivity, such as Boltzmann equation (Yang et al. 2005) and
Monte Carlo (MC) simulations (Jeng et al. 2008). Methods used in (Yang et al. 2005) and (Jeng
et al. 2008) require significant computational resources and time (Minnich and Chen 2007).
Minnich and Chen (2007) has developed a thermal conductivity model based on the EMT for
nanocomposites as shown in Eq. (4).
))1((62)21(
))1((622)21(
)4(3
4
bpbp
bpbp
b
bbbeff
kkdkk
kkdkkCk
(4)
Equation (4) combines base TE material properties, interface density )( , and Eq. (3) which is
also similar to Hasselman-Johnson model. Figure 5.2 shows schematics of internal-structure
representation for different transport property models. Figure 5.2a shows the Maxwell model
where base material and embedded spherical particles are shown. As Maxwell model does not
include interaction between base material and spherical particles, there is no connection between
base material and spherical particles. In similar way, the Hasselman-Johnson model is presented
in Fig. 5.2b which shows base material and spherical particles. The interconnecting dark solid
lines in Fig. 5.2b represent a TBR. Figure 5.2c represents Minnich-Chen model which considers
base material, spherical particles, boundary resistance, and base material with interface scattering
of phonons due to the interface density.
(c)
(a) (b)
143
Figure 5.2 Graphical representation of (a) Maxwell model (b) Hasselman and Johnson model (c)
Minnich and Chen model
Table 5.1 shows the comparison between the effective transport property models for macro and
nanocomposites in terms of the applicability and limitations.
Table 5.1 Comparison of different effective medium theories
Effective Medium Theories Applicability and Limitations
Maxwell model (Maxwell
1954)
Spherical particles embedded in spherical region, No
consideration thermal barrier resistance
Hasselman and
Johnson model (Hasselman
and Johnson 1987)
Spherical, cylindrical, and flat plate dispersions into host
material and thermal barrier resistance between particles and
base material, Valid for macro size particles
Minnich and Chen model
(Minnich and Chen 2007)
Spherical and cylindrical nanoparticle embedded in cube,
Valid for nano size particles
Yang and Stabler (2009) pointed out that high performance TE materials such as bulk
composites can expand the applications of TE systems. Thus, it is very important to study the
effects of a low thermal conductivity of TE bulk composites on the performance of TE systems.
To our best knowledge, authors are not aware of studies investigating the effect of low thermal
conductivities based on the EMT on the performance of TE systems. In this work, performance
of TE systems in a generator and cooler mode is investigated with three different effective
transport property models. Authors have considered Maxwell model, Hasselman-Johnson model,
and Minnich-Chen model. It is important to emphasize here that each model presents
modifications to their predecessor, for e.g., Maxwell model was first to consider the spherical
particles to calculate transport properties in a composite structure followed by Hasselman-
Johnson model, and last Minnich-Chen model. In addition to quantitative results, qualitative
results are also presented for a TEG and TEC system.
144
5.2 Modeling and boundary conditions
A schematic diagram of TE systems being investigated is shown in Fig. 5.3 with p-type and n-
type composite semiconductor legs. A copper strip connects semiconductor legs together. Each
semiconductor leg has a cross-sectional area of DW , height of H , and separated by a distance
dL . Qin and Qout are amounts of heat available at the heat source and heat sink, respectively.
During the subsequent analysis following assumptions are made:
Contact resistances at the interface of copper and TE legs are negligible.
TE system operates under a steady-state condition.
Figure 5.3 Schematic diagram of typical (a) TEC and (b) TEG system
The energy transport in TE system considering steady state can be expressed as (Antonova and
Looman 2005)
genpm qt
TC
q . (5)
where m , pC , T , q , and genq represent material density, specific heat, temperature, heat flux
vector, and volumetric heat generation, respectively. The continuity of electric charge through
the system must satisfy
Qin
(a)
n
W
p
T1
T2
I
Qout
H
Ld
x
T1 < T2 Qin
n
W
p
T1
T2
I
Qout
H
Ld
x
(b)
T1 > T2
qconv qconv
145
0
t
DJ . (6)
where J is the electric current density vector and D is the electric flux density vector,
respectively. Equation (5) and Eq. (6) are coupled by the set of TE constitutive equations
(Antonova and Looman 2005) as shown in Eq. 7(a) and Eq. 7(b) below
TkT ][][ Jq
)][(][ T EJ
7(a)
7(b)
where ][ is the Seebeck coefficient matrix, ][k is the thermal conductivity matrix, ][ is the
electrical conductivity matrix, E is the electric field intensity vector, respectively. E can be
expressed as , where is the electric scalar potential (Landau 1984).
Combining Eq. (5) to Eq. 7(b), the coupled TE equations for energy and charge transfers can be
expressed as
JEJ
TkT
t
TCpm ][][ (8)
0][][][][
Tt
(9)
where ][ is the dielectric permittivity matrix (Antonova and Looman 2005) and JE represents
Joule heat (Antonova and Looman 2005).
For a typical TE system shown in Fig. 5.3 thermal boundary conditions are as follow:
At the top surface ( 0x ) temperature is constant ( 1TT )
At the bottom surface ( Hx ) temperature is constant ( 2TT )
Convection heat transfer from the side surfaces to the surrounding, )(conv aTThq
where Ta is atmospheric temperature and h is convection heat transfer coefficient.
Note that the material property matrices (e.g., ][ and ][k , etc.) are suitable for non-
homogeneous materials. These material property matrices become a single-valued property in the
case of homogeneous materials. Initially, a simplified 1-D version of above equations will be
solved to obtain the close forms of analytical solutions.
146
For 1-D analytical heat transfer modeling, a TEC system with p-type and n-type semiconductor
legs with an electrical power input is considered as shown in Fig. 5.3a. In the similar manner, a
TEG system with p-type and n-type semiconductor legs and external load with resistance Rl
connected across it is considered as shown in Fig. 5.3b. TE elements have height H, width W,
and works between the temperature limits of 1T and 2T , respectively. A TE system absorbs inQ
amount of heat from the heat source and rejects outQ amount of heat to the heat sink. The main
mode of heat transfer in a semiconductor leg is conduction. Conduction is supplemented by the
Joule heating, Peltier heat generation/liberation at the junctions, and Thomson heat. Additionally,
side walls of semiconductor legs are in contact with the surrounding air which enables
convection heat losses from the side walls.
Assuming isotropic material properties and neglecting the thermal and electrical contact
resistances between the contact surfaces a one dimensional steady state heat transfer equation for
semiconductor legs is given by
0
2
2
hom
2
2
heatingJoule
eff
transferheatConvection
a
eff
heatsonT
eff
conductionHeat
Ak
ITT
Ak
Ph
dx
dT
Ak
I
dx
Td .
(10)
In Eq. (10), τ is Thomson coefficient, ρ is electrical resistivity, and keff is effective thermal
conductivity of composite. Thomson coefficient represents temperature dependency of Seebeck
coefficient and can be calculated using )/( dTdT .
Applying thermal boundary conditions at the top surface ( 1,0 TTx ) and the bottom surface
( 2, TTHx ) temperature distribution inside semiconductor leg can be given by
147
APh
ITAPh
eeAPh
APhTIeAPhTeAPhTeIAPhTe
eeAPh
APhTIeAPhTeAPhTeIAPhTe
xT
a
Ak
Hm
Ak
Hm
a
Ak
Hm
a
Ak
Hm
Ak
Hm
Ak
xm
Ak
Hm
Ak
Hm
a
Ak
Hm
a
Ak
Hm
Ak
Hm
Ak
xm
effeff
effeffeffeff
effeff
effeffeffeff
2
22
222
1
22
2
2
22
222
1
22
2
2
21
1112
21
2221
(11)
where
AkPhIIm eff422
1 ; AkPhIIm eff422
2 . (12)
Now heat transfer rate in semiconductor legs can be given by combining heat transfer within a
semiconductor leg with the Peltier heat which occurs only at the junctions,
Ak
Hm
Ak
Hm
eff
a
Ak
Hm
a
Ak
Hm
Ak
Hm
Ak
xm
Ak
Hm
Ak
Hm
eff
a
Ak
Hm
a
Ak
Hm
Ak
Hm
Ak
xm
effnp
effeff
effeffeffeff
effeff
effeffeffeff
eePhAk
TPAh
IeTAPheAPhTeIAPhTem
eePhAk
TPAh
IeTAPheAPhTeIAPhTem
AkTIq
222
222
1
22
22
2
222
222
1
22
22
1
,
12
1112
12
2221
5.0
5.0
(13)
Total heat input from heat source in TE system can be given by
00 xqxqQ npin . (14)
Total heat output to heat sink from TE system can be given by
148
HxqHxqQ npout . (15)
COP of TEC (Fig. 5.3a) can be calculated from the following equation:
i
in
RI
QCOP
2 , (16)
while the thermal efficiency of TEG (Fig. 5.3b) can be calculated from the following equation:
in
outin
Q
QQ . (17)
It is important to note here that when convection heat transfer from the side walls of
semiconductor legs are extremely low ( 0h ) and if temperature dependency of TE materials is
negligible then heat input to TE system (Eq. (14)) reduces to
H
T
Ak
HI
H
TAkTIQ
eff
effnpin1
2
2
21,
2
. (18)
In similar manner, heat output to heat sink (Eq. (15)) becomes
H
T
H
T
Ak
HIAkTIQ
eff
effnpout12
2
2
2,2
. (19)
5.3 Results and discussion
In this section, analytical and numerical simulation results are presented which are obtained
considering three different models of the effective transport properties based on the EMT.
Maxwell model, Hasselaman - Johnson model, and Minnich-Chen model are applied to calculate
effective thermal conductivities. Results are presented in terms of the effective thermal
conductivity of TE materials, COP, and thermal efficiency of TE systems. For a p-type TE
material, Bi2Te3 as a base material and Sb2Te3 particles are selected. The size of the Sb2Te3
particles varies from micrometer to nanometer. For n-type TE material, Bi2Te3 is considered
which can be used as a base material and nanoparticles. Table 5.2 (Pattamatta and Madnia 2009)
presents some properties (bulk thermal conductivity, volumetric specific heat, bulk MFP, and
phonon group velocity) of Bi2Te3 and Sb2Te3 which can be used to calculate the effective thermal
conductivity based on the Minnich-Chen model. Note that the bulk MFP can be defined as the
averaged distance travelled by an energy carrier per collision over a sufficient number of
collisions (Tzou 2014), while the phonon group velocity represents velocity of phonons, a quasi-
149
particle which represents quantization of the modes of lattice vibrations which exchanges energy
(Wang 2012).
Table 5.2 Material properties (Pattamatta and Madnia 2009)
Material
Thermal
conductivity
(Wm-1
K-1
)
Volumetric
specific heat
(MJm-3
K-1
)
Phonon group
velocity (ms-1
)
Mean Free Path
(Å)
Bi2Te3 1.1 0.5 212 310
Sb2Te3 0.9 0.53 200 254
Effective thermal conductivity
Effective thermal conductivities of p-type and n-type TE materials are calculated using different
effective transport property models. Figures 5.4 to 5.9 present the effective thermal conductivity
results with different amounts of volume fractions for Maxwell, Hasselman-Johnson, and
Minnich-Chen models. Maxwell model considers thermal conductivities of base material and
particles, and volume fraction to calculate the effective thermal conductivity. Figure 5.4 shows
that the effective thermal conductivity decreases with an increase in the volume fraction. Volume
fraction for Maxwell model can be defined as 3
2
3
1 rrn , where n is the number of spherical
particles, r1 and r2 are radii of particle sphere and base sphere (Maxwell 1954). Figure 5.4 shows
that thermal conductivity decreases from 1.1 Wm-1
K-1
to 0.95 Wm-1
K-1
as volume fraction
increases from 0 to 0.8. Figure 5.4 also shows that the effective thermal conductivity of
composite TE material slowly approaches that of thermal conductivity of particles as volume
fraction increases. Additionally, Fig. 5.4 shows that effective thermal conductivity of composite
remains between thermal conductivities of base and particle materials. This can be observed in
Fig. 5.4 where the effective thermal conductivity of n-type material stays at 1.1 Wm-1
K-1
which
is a thermal conductivities for both base and particle material. The reason behind this is Maxwell
model neglects the interaction between a base material and particles; therefore, the effective
thermal conductivity of composite remains same as base material and particle thermal
conductivity.
150
kef
f(W
m-1
K-1
)
0 0.2 0.4 0.6 0.80.9
0.95
1
1.05
1.1
1.15
n-type material
p-type material
Figure 5.4 Effective thermal conductivity of p-type and n-type thermoelectric material based on
Maxwell model
151
Figure 5.5 shows results of the effective thermal conductivity using Hasselman-Johnson model.
Figure 5.5 shows significant reduction in the effective thermal conductivity as volume fraction
increases. The effective thermal conductivity predicted by Hasselman-Johnson model is much
lower than Maxwell model for similar amount of volume fraction. For example, the effective
thermal conductivity using Hasselman – Johnson model with volume fraction 0.5 is 0.44
Wm-1
K-1
, whereas with similar volume fraction Maxwell model predicts effective thermal
conductivity of 0.99 Wm-1
K-1
. The drop in the magnitude of effective thermal conductivity is due
to the TBR which is represented by boundary conductance (K). The effective thermal
conductivity of TE material drops to 0.15 Wm-1
K-1
with volume fraction 0.8 and extremely low
(10-15
) boundary conductance considering Hasselman-Johnson model. If boundary conductance
remains very high (→∞) then effective thermal conductivity based on Hasselman – Johnson
model approaches to that of Maxwell model. This can be verified using Figs. 5.6 and 5.7 where
the effective thermal conductivity is plotted against boundary conductance at higher boundary
conductance. It is important to note here that in order to show effects of TBR, very low boundary
conductance (=10-15
) and very high boundary conductance (→∞) are taken arbitrarily.
152
kef
f(W
m-1
K-1
)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
1.2
n-type material
p-type material
Figure 5.5 Effective thermal conductivity of p-type and n-type thermoelectric material based on
Hasselman-Johnson model
153
As shown in Figs. 5.6 and 5.7, the effective thermal conductivity remains same as the thermal
conductivity of the base material when volume fraction is zero. With an increase in the volume
fraction, the effective thermal conductivity decreases. Moreover, an increase in the boundary
conductance leads to an increase in the effective thermal conductivity. Figures 5.6 and 5.7 also
show that the trend for p-type and n-type remains different as thermal conductivity of particles is
different for both materials.
154
K (Wm-2
K-1)
keff
(Wm
-1K
-1)
0 20000 40000 60000 80000 1000000
0.2
0.4
0.6
0.8
1
1.2
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.6 Effect of thermal boundary conductance on effective thermal conductivity of p-type
using Hasselman-Johnson model
155
K (Wm-2
K-1)
keff
(Wm
-1K
-1)
0 20000 40000 60000 80000 1000000
0.2
0.4
0.6
0.8
1
1.2
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.7 Effect of thermal boundary conductance on effective thermal conductivity of n-type
using Hasselman-Johnson model
156
Figures 5.8 and 5.9 demonstrate effective thermal conductivities using Minnich-Chen model,
where particle size and thermal boundary resistance contribute in calculation of the effective
thermal conductivity. An observation from Figs. 5.8 and 5.9 reveals that the effective thermal
conductivity decreases with an increase in the volume fraction. Additionally, sizes of particles
also have significant impact on the effective thermal conductivity. The smallest particle size
yields to the lowest thermal conductivity and as particle size increases the effective thermal
conductivity also increases. The effective thermal conductivity decreases from 1.1 Wm-1
K-1
to
0.089 Wm-1
K-1
for the p-type material with particle size of 50 nm. Similarly, effective thermal
conductivity decreases from 0.13 Wm-1
K-1
to 0.018 Wm-1
K-1
when particle size decreases from
250 nm to 5 nm. The reason behind this reduction is the term called ‘interface density’ which
takes care of interface scattering due to particle size effects. The interface density (Φ) term is
available in the first term of Eq. (7) which is the ratio of surface area of particle to unit volume of
composite housing the particle. As interface density increases, the effective thermal conductivity
decreases. In Minnich – Chen model, TBR is functions of size of particles and interface density
(Φ), whereas, in Hasselman – Johnson model TBR is a function of boundary conductance only.
157
kef
f(W
m-1
K-1
)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
1.2d = 5 nm
d = 15 nm
d = 50 nm
d = 100 nm
d = 250 nm
Figure 5.8 Effective thermal conductivity of p-type thermoelectric material using Minnich-Chen
model
158
kef
f(W
m-1
K-1
)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
1.2d = 5 nm
d = 15 nm
d = 50 nm
d = 100 nm
d = 250 nm
Figure 5.9 Effective thermal conductivity of n-type thermoelectric material using Minnich-Chen
model
159
Performance of TE systems
In this section, different effective transport property models are applied to investigate the
performance of TE systems. Figures 5.10 to 5.23 show the performance of TEC and TEG with
Maxwell model, Hasselman-Johnson model, and Minnich-Chen model. TEC and TEG systems
considered for the investigation have dimensions of 1.5 mm × 1.5 mm × 1.5 mm which is chosen
arbitrarily. Thermoelectric properties other than thermal conductivity are assumed to be constant
to observe the performance change in TE systems due to the change in thermal conductivity.
Some experimental observations conclude that the percentage reduction in thermal conductivity
is much higher than the reduction in power factor for the case of nanocomposite. For example,
nanocomposite BiSbTe exhibited reduction in power factor by 3%, while, thermal conductivity
exhibited reduction by 20% when compared to non-nanocomposite BiSbTe at room temperature
(Poudel et al. 2008). The Seebeck coefficient and electrical conductivity for p-type material are
114.44 μVK-1
and 130.33×103 Sm
-1. Similarly, the Seebeck coefficient and electrical
conductivity for n-type material are -137.23 μVK-1
and 202.44 ×103 Sm
-1. TEC creates
temperature difference of 20 K with surrounding temperature at 300 K. An electric power input
to TEC is calculated using I2Ri. In the similar manner, TEG works with constant cold surface
temperature of 300 K. An external load connected to TEG is matched with that of an internal
resistance of TEG. Performance of TE systems is evaluated with respect to input parameters; for
example, TEC requires an input electric current and TEG requires temperature difference to
generate an electric potential. Figures 5.10, 5.12, and 5.16 show COP of a TEC as a function of
an electric current input with different amounts of volume fraction of particles. Different
amounts of volume fraction of particles are considered for Maxwell model, Hasselman-Johnson
model, and Minnich-Chen model, respectively. All effective transport property models agree on
increase in the performance of TEC and TEG with increase in the volume fraction of particles.
Nevertheless, performance improvement varies with different effective transport property models
as each model’s prediction of thermal conductivity differs. Figures 5.10 and 5.11 show
performance of TEC and TEG based on Maxwell model. Figure 5.10 shows COP improvement
from 0.03 to 0.61 when volume fraction increases from 0 to 0.8 with an electric current input of
1 A considering the Maxwell model. Figure 5.10 also reveals that the variation in the COP with
increasing electric current input is small. This small variation in the COP can be attributed from
the dominance of Joule heat which contributes significantly to the irreversible losses. The
160
variation in COP suggests that TEC can exhibit maximum COP only at certain amount of an
electric current input. Thermal efficiency of a TEG increases with increase in the volume
fraction as shown in Fig. 5.11. Thermal efficiency increases from 2.06% to 2.17% with rise in
the volume fraction from 0 to 0.8 with temperature difference of 50 K considering Maxwell
model. The trend in Fig. 5.11 also shows a rise in the thermal efficiency with increment in the
temperature difference between hot and cold surface temperatures.
161
I(A)
CO
P
1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
=0
=0.2
=0.4
=0.6
=0.8
Figure 10 COP of TEC considering various amount of volume fraction with Maxwell model
162
(%
)
0 10 20 30 40 500
0.5
1
1.5
2
2.5
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.11 Efficiency of TEG with different volume fraction with Maxwell model
163
Figures 5.12 and 5.13 show COP and thermal efficiency of TEC and TEG with Hasselman-
Johnson model. There is a steep rise in COP from 0.03 to 6.75 with 1 A electric current input
when volume fraction increases from 0 to 0.8 due to high TBR. A very high TBR leads to a low
effective thermal conductivity as shown in Figs. 5.6 and 5.7. Low TBR creates scenario similar
to the Maxwell model which can be verified from Figs. 5.14 and 5.15 at very high boundary
conductance (K). Figures 5.14 and 5.15 show COP of TEC and thermal efficiency of TEG
system as a function of boundary conductance. At very high boundary conductance, COP and
thermal efficiency of TE system matches with that of Maxwell model (marked by ellipse in Figs.
5.14 and 5.15) which also can be verified with Figs. 5.10 and 5.11. Low boundary conductance
between particles and base TE material can improve the performance of a TE system. For
example, thermal efficiency of TEG increases from 2.06% to 5.4% which is 162% increment
when boundary conductance changes from very high (105) to very low (10
-5). Range of boundary
conductance is chosen arbitrarily (also see Figs. 5.6 and 5.7) to demonstrate effects of TBR on
the performance of TEG and TEC. In reality, boundary conductance can be a function of size,
shape, and surface area of particles (Nan et al. 1997, Minnich and Chen 2007).
164
I(A)
CO
P
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.12 COP of TEC considering different amount of volume fraction with Hasselman
model
165
(%
)
0 10 20 30 40 500
1
2
3
4
5
6 = 0
= 0.2
= 0.4
= 0.6
= 0.8
Figure 5.13 Efficiency of TEG with different amount of volume fraction with Hasselman model
166
K (Wm-2
K-1)
CO
P
0 20000 40000 60000 80000 100000
0
1
2
3
4
5
6
7
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.14 Effect of boundary conductance on performance of TEC based on Hasselman-
Johnson model
167
K (Wm-2
K-1)
(%
)
0 20000 40000 60000 80000 1000001.5
2
2.5
3
3.5
4
4.5
5
5.5
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.15 Effect of boundary conductance on performance of TEG based on Hasselman-
Johnson model
168
Figures 5.16 and 5.17 show performance of a TE system based on Minnich-Chen model with
radius of spherical particle as 5 nm, which is chosen arbitrarily. Similar to other models
Minnich-Chen model also predicts increase in the performance of TE system with increase in
volume fraction as shown in Figs. 5.16 and 5.17. For example, thermal efficiency of TEG
increases from 2.06 % to 7.03 % (Fig. 5.17). Additionally, size of particles can affect the
effective thermal conductivity (Figs. 5.8 and 5.9) which eventually influences the performance of
TEG and TEC. Figures 5.18 and 5.19 show the effect of particle size on the performance of a
TEG and TEC. COP decreases as size of nanoparticle increases which is consistent with rise in
the effective thermal conductivity as shown in Figs. 5.8 and 5.9.
169
I (A)
CO
P
1 2 3 4 5 6 7 8 9 100
2
4
6
8
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.16 COP of TEC considering different amount of volume fraction with Minnich-Chen
model
170
(%
)
0 10 20 30 40 500
1
2
3
4
5
6
7 = 0
= 0.2
= 0.4
= 0.6
= 0.8
Figure 5.17 Efficiency of TEG with different volume fraction with Minnich-Chen model
171
d (m)
CO
P
0 2E-08 4E-08 6E-08 8E-08 1E-07-1
0
1
2
3
4
5
6
7
8
9
10
11
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.18 Effect of nanoparticle size on performance of TEC considering Minnich-Chen
model
172
d (m)
(%
)
0 2E-08 4E-08 6E-08 8E-08 1E-071
2
3
4
5
6
7
8
9
10
=0
=0.2
=0.4
=0.6
=0.8
Figure 5.19 Effect of nanoparticle size on performance of TEG considering Minnich-Chen
model
173
A TEG and TEC considered so far for analysis have adiabatic side walls but in reality there is
always a heat loss through side walls of the TEG and TEC. Figures 5.20 and 5.21 show COP and
thermal efficiency as functions of volume fraction and convection heat transfer coefficient. TEC
considered for Fig. 5.20 creates a temperature difference of 10 K with the surrounding
temperature at 300 K with 1 A electric current input. Figure 5.20 demonstrates that with the
increase in convection heat transfer coefficient COP decreases. COP increases with increase in
volume fraction which can also be verified from Figs. 5.10, 5.12, and 5.16. In similar way, Fig.
5.21 shows the performance of a TEG working with 50 K temperature gradient and the cold
surface temperature 300 K. Thermal efficiency of TEG drops as convection heat transfer
coefficient increases from 10-4
to 100 Wm-2
K-1
. Additionally, Figs. 5.20 and 5.21 show COP and
thermal efficiency remains same for each transport property models at volume fraction zero.
174
h = 10-4
Wm-2K
-1
h = 100 Wm-2K
-1
Minnich-Chen model
CO
P
0 0.2 0.4 0.6 0.82
3
4
5
6
7
8
9
10
h = 10-4
Wm-2K
-1
h = 100 Wm-2K
-1
Maxwell model
h = 10-4
Wm-2K
-1
h = 100 Wm-2
K-1
Hasselman-Johnson model
Figure 5.20 Performance of TEC with variable volume fractions and convection heat transfer
coefficients through side walls of TE legs
175
h = 10-4
Wm-2
K-1
h = 100 Wm-2K
-1
Minnich-Chen model
(%
)
0 0.2 0.4 0.6 0.81
2
3
4
5
6
h = 10-4
Wm-2K
-1
h = 100 Wm-2K
-1Maxwell model
h = 10-4
Wm-2
K-1
h = 100 Wm-2
K-1
Hasselman - Johnson model
Figure 5.21 Performance of TEG with variable volume fractions and convection heat transfer
coefficients through side walls of TE legs
176
The results reported so far indicate improved performances of a TEC and TEG using different
effective transport property models; however, predicted results are different for different
transport property models. For example, Maxwell model, Hasselman-Johnson model, and
Minnich-Chen model show COP of 0.61, 6.76, and 7.69, respectively, with an electric current
input of 1 A and volume fraction of 0.8. The reason behind such difference is the reduction in the
effective thermal conductivity in each transport property model with highest thermal
conductivity in Maxwell model and lowest in Minnich-Chen model. Higher or lower thermal
conductivity significantly effects the heat conduction part of total heat transfer in a TE system.
Figures 5.22 and 5.23 explain how effective thermal conductivity influences the heat conduction
portion of total heat input to TE system. Figure 5.22 shows heat conduction portion to total heat
input (Qin) in TEC as a function of volume fraction. TEC creates temperature difference of 20 K
with surrounding temperature 300 K and an electric current input of 1 A. As one can be observed
from Fig. 5.22 that increase in the volume fraction decreases the amount of heat conduction
which eventually contributes to rise in COP. A rise in COP can be verified from Figs. 5.10, 5.12,
and 5.16 which is due to the drop in effective thermal conductivity as already presented in Figs.
5.4, 5.5 and 5.8. In the similar way, heat conduction through TEG with temperature gradient of
50 K with cold surface temperature 300 K is plotted in Fig. 5.23. Heat conduction in TEG also
follows similar trend as that of TEC showing decreasing trend as volume fraction increases due
to the drop in the effective thermal conductivity. For a TEC system, heat removed can be higher
if heat conduction and Joule heating remain as low as possible. For a TEG system, it is very
important to maintain high temperature gradient to generate higher electric potential. Heat
conduction and Joule heating should remain as low as possible to maintain higher temperature
gradient and thus higher electric potential. It is important to note here that Peltier heat and Joule
heat remains unchanged because change in the effective thermal conductivity only influences
heat conduction through TE system.
177
Qin
(Co
nd
ucti
on
)(W
)
0 0.2 0.4 0.6 0.80
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Hasselman-Johnson model
Maxwell model
Minnich-Chen model
Figure 5.22 Influence of effective thermal conductivity on heat conduction in TEC
178
Qin
(Co
nd
ucti
on
)(W
)
0 0.2 0.4 0.6 0.80
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Hasselman-Johnson model
Maxwell model
Minnich-Chen model
Figure 5.23 Influence of effective thermal conductivity on heat conduction in TEG
179
Numerical simulation results
In this section, results achieved by solving Eq. (8) and (9) using the Finite Element Method are
presented based on discretization and solution techniques available in Mahmud and Pop (2006).
Initially, results obtained from numerical solutions are expressed using the field plots of
temperature contours, heat flow lines, electric potential contours, and electric current flow lines.
A composite TE system with width (W = 1.5 mm) and height (H = 1.5 mm) is considered for the
analysis. Both legs are separated by distance Ld = 0.2 mm. Maxwell model, Hasselman-Johnson
model, and Minnich-Chen model are applied to calculate the effective thermal conductivity. The
performance variation in TEG and TEC due to change in effective thermal conductivity is
observed. The Seebeck coefficient and electrical conductivity are assumed to be constant for
each simulation. Figures 5.24 to 5.28 represent field plots of temperature and electric potential
for a composite TEC. An electric potential is indicated by marked iso-potential lines with a
multi-colored background, while electric current flow is indicated by lines with arrows in Fig.
5.24. Similarly, temperature is presented by marked isothermal lines with multi-colored
background, while heat flow is presented by lines with arrows in Fig. 5.25 to 5.28. Composite
TEC maintains temperature difference of 10 K with surrounding temperature at 300 K and
electric potential of 0.02 V. Similarly, a TEG works between temperature gradient of 50 K and
hot surface temperature at 350 K. An applied electric potential creates a potential distribution
inside the TEC system which can be observed from the distribution of iso-potential lines in Fig.
5.24. Also, the current flow direction can be visualized from the streamlines in Fig. 5.24.
180
0.001 0.001
0.002 0.002
0.003 0.003
0.004
0.005
0.006 0.006
0.007
0.008
0.009
0.01 0.01
0.011 0.011 0.012
0.013
0.014
0.015 0.015
0.016
0.017
0.018 0.018
0.019 0.019
Figure 5.24 Contours of electric potential and streamlines of electric current flow in TEC
181
Field plots in Figs. 5.25 to 5.28 show temperature profile and streamlines of heat flow. Figures
5.25, 5.26, 5.27, and 5.28 represent four different cases of the effective thermal conductivities
with no particles, particle volume fraction 0.8 with Maxwell model, particle volume fraction 0.8
with Hasseman-Johnson model, and particle volume fraction 0.8 with Minnich-Chen model,
respectively. There is also a difference between the field plots in Figs. 5.26, 5.27, and 5.28 as
each field plot represents different effective transport property models. It can also be observed
that the temperature drop in a TE leg increases as the effective thermal conductivity decreases. A
low thermal conductivity decreases the heat conduction, which improves heat removal from the
cold surface of a TEC as demonstrated in Figs. 5.22 and 5.23. Temperature contours are linear in
Figs. 5.25, 5.26, and 5.27 but temperature contours show some non-linearity in Fig. 5.28. The
reason behind this could be very low thermal conductivity based on Minnich-Chen model and
generation of Joule heat.
182
290 290 290
291 291 291
292 292 292
293 293
294 294 294 294
295 295 295
296 296 296
297 297 297
298 298 298
299 299 299
Figure 5.25 Contours of temperature and streamlines of heat flow in TEC with cold surface
temperature 290 K, hot surface temperature 300 K, and electric potential 0.02 V with NO
particles
183
290 290 290
291 291 291
292 292 292
293 293
294 294 294 294
295 295 295
296 296 296
297 297 297
298 298 298
299 299 299
Figure 5.26 Contours of temperature and streamlines of heat flow in TEC with cold surface
temperature 290 K, hot surface temperature 300 K, and electric potential 0.02 V with 0.8
volume fraction with Maxwell model
184
290 290 290 290
291 291291
292292 292
293293
294294 294
295 295295
296 296 296
297 297 297
298 298
299 299 299 299
Figure 5.27 Contours of temperature and streamlines of heat flow in TEC with cold surface
temperature 290 K, hot surface temperature 300 K, and electric potential 0.02 V with 0.8 volume
fraction with Hasselman-Johnson model
185
290 290
290
291 291
291292 292
292293293294 294
294 294295 295 295296 296 296297 297 297298 298 298299 299
Figure 5.28 Contours of temperature and streamlines of heat flow in TEC with cold surface
temperature 290 K, hot surface temperature 300 K, and electric potential 0.02 V with 0.8 volume
fraction with Minnich-Chen model
186
Figures 5.29 to 5.36 are for a composite TEG showing the temperature and electric potential
field plots. Figures 5.29 to 5.32 represent temperature which is marked by isothermal lines with
multi-colored background, while heat flow is presented by lines with arrows. Similarly, Figs.
5.33 to 5.36 represent electric potential which is indicated by marked iso-potential lines with a
multi-colored background, while electric current flow is indicated by lines with arrows.
Additionally, an external load with an electrical resistance similar to an internal resistance of
TEG is attached. Temperature contours show applied temperature gradient condition and the heat
flow direction. A change in temperature contour location is quite evident in the field plots as
shown in Figs. 5.29 to 5.32. Figures 5.29, 5.30, 5.31, and 5.32 represent TEG composite with no
particles, particle volume fraction of 0.8 with Maxwell model, particle volume fraction of 0.8
with Hasselman-Johnson model, and particle volume fraction of 0.8 with Minnich-Chen model,
respectively.
187
305 305 305 305
315 315 315
325 325 325
335 335
345 345 345
Figure 5.29 Contours of temperature and streamlines of heat flow in TEG with cold surface
temperature 300 K and hot surface temperature 350 K with NO particles
188
305 305 305
315 315 315
325 325 325
335 335 335
345 345 345
Figure 5.30 Contours of temperature and streamlines of heat flow in TEG with cold surface
temperature 300 K and hot surface temperature 350 K with 0.8 volume fraction with Maxwell
model
189
305 305 305
315 315 315
325 325 325
335 335 335
345 345 345
Figure 5.31 Contours of temperature and streamlines of heat flow in TEG with cold surface
temperature 300 K and hot surface temperature 350 K with 0.8 volume fraction Hasselman-
Johnson model
190
305 305 305 305
315 315 315
325325 325
335
335
345
345 345
Figure 5.32 Contours of temperature and streamlines of heat flow in TEG with cold surface
temperature 300 K and hot surface temperature 350 K with 0.8 volume fraction with Minnich-
Chen model
191
Figures 5.33, 5.34, 5.35, and 5.36 presents the electric potential and current flow in TEG
composite with no particles, particle volume fraction of 0.8 with Maxwell model, particle
volume fraction of 0.8 with Hasselman-Johnson model, and particle volume fraction of 0.8 with
Minnich-Chen model, respectively. For a TEG, a low thermal conductivity can lower the amount
of heat absorbed from the heat source which brings down the heat input to TEG. A temperature
gradient to TEG remains unchanged so there is no change in the electric potential generation but
lower heat input increases the thermal efficiency of a TEG. Figures 5.34 to 5.36 show the
electrical potential generated due to the temperature gradient and flow of electric current. A
change in thermal conductivity shows no influence on electric potential because the effective
transport property models are only applied to thermal conductivity which only influences the
heat input.
192
-0.003 -0.003
-0.0025 -0.0025
-0.002 -0.002
-0.0015
-0.001
-0.0005
0
0
0.0005 0.00050.001
0.0010.001
0.0015
0.002 0.002
0.0
02
0.0025
Figure 5.33 Contours of electric potential and streamlines of electric current in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with NO particles
193
-0.003
-0.0025 -0.0025
-0.002
-0.0015
-0.001 -0.001
-0.0005
0 0
0.0005
0.001
0.0
01
0.001
0.0015
0.002 0.002
0.0025
Figure 5.34 Contours of electric potential and streamlines of electric current in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with 0.8 volume fraction with
Maxwell model
194
-0.003
-0.0025 -0.0025
-0.002
-0.0015
-0.001 -0.001
-0.0005
0 0
0.00050.001
0.0
01
0.001
0.0015
0.002 0.002
0.0025
Figure 5.35 Contours of electric potential and streamlines of electric current in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with 0.8 volume fraction
Hasselman-Johnson model
195
-0.0
03
5
-0.0035-0.003 -0.003-0.0025 -0.0025
-0.002-0.0015
-0.0
01
-0.001 -0.001-0.0005 -0.0005
00
0 0
00
0.00050.0005
0.0005
0.0005
0.001 0.001
0.0
01
0.001
0.001
0.00150.002 0.002
0.0
020.0025
Figure 5.36 Contours of electric potential and streamlines of electric current in TEG with cold
surface temperature 300 K and hot surface temperature 350 K with 0.8 volume fraction with
Minnich-Chen model
196
Figures 5.37 and 5.38 are the comparisons between the analytical and numerical simulation
results. It can be observed from the plots that analytical results and numerical simulation show a
fair agreement. Nevertheless, analytical results overestimate results compared to the numerical
simulation due to decoupling between TE constitutive equations in analytical modeling. One can
also observe from plots that all effective transport property models predict same results when
particle volume fraction is zero.
197
Analytical results - Hasselman-Johnson model
Numerical simulation - Hasselman-Johnson model
Numerical simulation - Maxwell model
Analytical results - Maxwell model
CO
P
0 0.2 0.4 0.6 0.81.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
Analytical results - Minnich-Chen model
Numerical simulalation - Minnich-Chen model
Figure 5.37 Comparison of analytical and numerical simulation results for TEC
198
Analytical results - Minnich-Chen model
Numerical simulation - Minnich-Chen model
(%
)
0 0.2 0.4 0.6 0.80
1
2
3
4
5
6
7
8
9Analytical results - Maxwell model
Numerical simulation -- Maxwell model
Numerical simulation - Hasselman-Johnson model
Analytical results - Hasselman-Johnson model
Figure 5.38 Comparison of analytical and numerical simulation results for TEG
199
5.4 Conclusion
In this work, different effective transport property models based on the EMT were applied to
investigate the performance of a TEG and TEC. A one-dimensional (1-D) analytical heat transfer
model of a TEG and TEC was derived considering the Seebeck, Peltier, and Thomson effects,
Fourier heat conduction, Joule heating, and convection heat transfer. The performance of a TEC
and TEG are evaluated in terms of COP and thermal efficiency as a function of input electric
currents and temperature gradients, respectively. Additionally, the performance of a TEC and
TEG are also evaluated with respect to volume fractions of particles inside a composite TE leg.
The following conclusions are based on the studies:
1. The effective transport property models predict the decrease in the effective thermal
conductivity of composite TE materials. The composite TE materials are bulk materials and
easy to integrate in the real world applications.
2. A reduction in the effective thermal conductivity is favorable to the performance of a TEG
and TEC. COP and thermal efficiency increases as the effective thermal conductivity of
composite TE leg decreases.
3. This study shows that the heat conduction contribution to total heat input to a TEG and TEC
should remain as low as possible to increase the performance of a TEG and TEC. At this
stage, only way to decrease the heat conduction is reduction in thermal conductivity of TE
materials. Bulk TE composites provide economical and practical solution to decrease the
thermal conductivity.
200
5.5 Nomenclature
A cross-sectional area (m2)
C volumetric specific heat (MJm-3
K-1
)
Cp specific heat (Jkg-1
K-1
)
D electric flux density (Nm2C
-1)
d depth (m)
E electric field intensity vector, (NC-1
)
h convection heat transfer coefficient, (Wm-2
K-1
)
H height (m)
I electric current (A)
J electric current density (Am-2
)
k thermal conductivity (Wm-1
K-1
)
K thermal boundary conductance (WK-1
m-2
)
L length (m)
1m A parameter (See Eq. (12))
2m A parameter (See Eq. (12))
n n-type material, number of particles
p p-type material
P Power input (W)
q heat flux vector (Wm-2
)
q heat (W)
.
q heat generation (Wm-3
)
Q heat (W)
r radius of sphere (m)
t time (s)
T temperature (K)
V electric potential (V)
W width (m)
ZT Figure of merit
201
Greek symbols
Seebeck coefficient (VK-1
)
ε dielectric permittivity matrix (Fm-1
)
efficiency
thermal barrier resistance (KW-1
m-2
)
mean free path (Å)
electrical resistivity (Ω m), density (kgm-3
)
electrical conductivity (Sm-1
)
τ Thomson coefficient (VK-1
)
phonon group velocity (ms-1
)
volume fraction of particles
interface density (m-1
)
Subscripts
1 heat source, particle sphere
2 heat sink, base sphere
a atmospheric condition
b base material
eff effective property
c composite, characteristic
conv convection
d gap distance
h hot temperature side
in input
l low temperature side
L Lattice contribution
m material
n n-type
out output
p particles, p-type
x co-ordinate system
202
CHAPTER 6: ANALYSIS OF COMBINED SOLAR PHOTOVOLTAIC-
NANOSTRUCTURED THERMOELECTRIC GENERATOR SYSTEM
6.1 Introduction
Currently, most of the power generation technologies use fossil fuels. Harmful emissions of
fossil fuels have forced to develop cost-effective power generation systems based on the
renewable energy conversion devices such as fuel cell, solar photovoltaic (PV) panel, solar-
thermal power generator, geothermal heat engine, and wind turbine. A direct power generation
technique using photovoltaic (PV) panels has been studied widely due to the huge availability of
solar energy. Solar PV panels are still less efficient and energy generated is expensive compared
to conventional energy conversion technologies. Although, different techniques have been used
(e.g., solar concentrators) to increase the efficiency of PV panels which increases the intensity of
solar radiation. The use of solar concentrators generates the higher PV cell temperature due to
higher irradiation on a solar PV panel which decreases the efficiency of solar panels (Dincer and
Meral 2010). Nevertheless, passive and active cooling methods (e.g., PV-thermal collectors) can
overcome the problem of higher cell temperature where waste heat is rejected to the environment
(Royne, Dey and Mills 2005). An option that remains largely unexplored is the use of
thermoelectric (TE) generator as a waste heat recovery tool in solar panels. A typical TE module
is made up of a number of p type and n type semiconductor legs connected electrically in
series and thermally in parallel. A TE module generates a voltage potential in the presence of an
applied temperature gradient across the module (Hoods 2005). The solar TE energy conversion
systems, where solar energy creates the temperature difference and produces electrical power
directly from the heat without moving parts has been discussed widely (Lenoir et al. 2003; Omer
and Infield 1998; Xi, Luo, and Fraisse 2007; Sahin et al. 2011). Shanmugam et al. (2011) have
developed the mathematical model of a TE generator system driven by the solar parabolic dish
collector. Their experiments found a maximum power output of 14.7 W with the solar radiation
ranges from 300 W/m2 to 1200 W/m
2. Chen (2011) has developed a mathematical model of a
solar TE generator considering an optical concentrator and selective surface. Chen (2011)
proposed a novel TE generator with an evacuated environment and concluded that such
arrangement can have attractive efficiency with 32TeBi TE materials and temperature ranges
203
from 150 ˚C to 250 ˚C. Khattab and Shenaway (2006) have proposed the use of a TE generator
to drive the TE cooler and optimized number of TE modules to achieve the maximum cooling
from one TE couple. Baranowski, Snyder and Toberer (2012) have developed a model for a solar
TE generator which can provide the analytical solutions of the device efficiency with
temperature dependant properties. They have also showed that with currently available materials,
total efficiency of 14.1% is possible for the cold and hot side temperature settings of 100 ˚C and
1000 ˚C, respectively. They also observed that the system efficiency can reach up to 25% if
figure of merit ( ZT ) reaches to 2 for the cold and hot side temperature settings of 100 ˚C and
1000 ˚C, respectively. Vatcharasathien et al. (2005) have developed a design methodology for a
solar TE power generation plant using TRANSYS software. Their simulation and experimental
work did not show the high performance but it demonstrated feasibility of power generation. Li
et al. (2010) have carried out experiments on three different types of TE generators (Bismuth-
Telluride, Skutterudite, and Silver-Antimony-Lead-Telluride) using the concentrated solar
energy. Li et al. (2010) have concluded that the conversion efficiency of a TE generator
increases with the increase in solar concentration ratio. Xiao et al. (2012) proposed three-stage
TE generator modules with medium temperature material (e.g., Skutterudite) and low
temperature material (e.g, Bismuth-Telluride). They achieved the efficiency of 10.52 %.
Vorobiev et al. (2006) have proposed a thermal-PV-solar hybrid system consisting of
concentrator, PV cell, and TE module. They concluded that TE modules can have considerable
effect on the overall efficiency of the thermal-PV-solar hybrid system. Rockendorf et al. (1999)
have studied a solar-PV-TE hybrid system with liquid heat-transfer medium and showed that
efficient back side cooling and low radiative losses can help achieving electrical conversion
efficiency up to 30% of the Carnot efficiency. Muhtaroglu, Yokochi, and Von Jouanne (2008)
demonstrated the use of PV and TE as a power source for mobile computing devices. They
showed an effective management of PV and TE simultaneously for onboard power generation
and concluded that such arrangement can extend battery life of mobile computing devices. Najafi
and Woodbury (2013) have developed combined PV-TE generator heat transfer model using
MATLAB. The hot surface of the TE module is considered in contact with air channel which is
in contact with the back surface of the PV panel. Najafi and Woodbury (2013) concluded that the
efficient TE modules can lead to a better power output of a combined system.
204
Performance of TE materials are characterised by a dimensionless parameter ‘figure of
merit’, TkZT )( 2 , where is the Seebeck coefficient, is the electrical resistivity, and k
is the thermal conductivity. Based on the available literature (Vineis et al. 2010), the ZT value of
the best available TE materials reach around 1 at room temperature. Recent advancement in
nanotechnology (Vineis et al. 2010) opens the door for further improvements of ZT for the TE
materials. The expression of the figure of merit, TkZT )( 2 , clearly indicates that one of the
methods to increase figure of merit is to reduce the thermal conductivity of the TE material.
Thermal conductivity has two components: the lattice conductivity Lk and electronic
conductivity ek (Godart et al. 2009). The electronic part of thermal conductivity ek is related
to the electrical conductivity using Wiedemann-Franz law (Godart et al. 2009) as shown in Eq.
(1),
TLke 0 (1)
where, 0L is the Lorentz number and for metals it is equal to (Godart et al. 2009),
8
2
22
0 1045.23
e
L
V2K
-2 (2)
where, is the Boltzmann constant (= 231038.1 JK-1
) and e is the electron charge
(= 191060.1 C)(Godart et al. 2009). The expression of figure of merit can be written in terms
of Lattice conductivity Lk and electronic conductivity ek as shown below (Godart et al.
2009),
Le
e
kk
k
LZT
0
2
. (3)
One method to enhance ZT is the inclusion of nanoparticles into the bulk TE materials which
can lead to low lattice thermal conductivity (Ma, Heijl, and Palmqvist 2013). Poudel et al. (2008)
achieved ZT value of 1.4 at 373 K by hot-pressing of nanopowders of 32TeBi and 32TeSb under
Argon Ar atmosphere. The enhanced ZT was attributed to significant decrease in lattice
thermal conductivity of material. Li et al. (2009) obtained ZT of 1.43 for double-doped 124SbCo
skutturedites using Indium In and Cerium Ce doping. The attractive results achieved using
nanotechnology has encouraged researchers to include nanoparticle-doped TE materials for
205
various applications, such as, solar TE generator and waste heat recovery. Kraemer et al. (2011)
have proposed novel solar TE generator with glass vacuum enclosure considering nanostructured
TE materials. The developed solar TE generator achieved maximum efficiency of 4.6% with
solar flux of 1000 Wm-2
condition. McEnaney et al. (2011) developed a novel of TE generator.
They placed high performance nanostructured material in evacuated tube with selective absorber
and achieved an efficiency of 5.2%.
It can be seen from the above discussion that the existing literature on the solar PV–nanomaterial
doped TE generator is very limited which is the major motivating factor to conduct the current
research. A combined system, which includes a solar PV and nano-particle doped TE modules, is
analyzed in this work. The temperature dependent thermophysical and electrical properties of TE
material, surface to surrounding convection heat transfer losses, and Thomson effect are included
in the current model. The energy transport in the solar PV panel and nanomaterial doped TE
module are performed separately and then combined to obtain a general expression of the overall
system.
6.2 Modeling and boundary conditions
The proposed photovoltaic – thermoelectric (PVTE) system is shown in Fig. 6.1a. A portion of
the heat rejected by PV panel will act as a heat source for the TE module. A number of
nanostructured p type and n type elements of the TE module are connected in series through
a copper plate with thermally conductive and electrically insulated ceramic plate on both sides.
In the current study, TE modules are placed in such a way that the ceramic plates of the modules
are attached to the back surface of the PV panel and exposed to the ambient environment,
respectively. For simplicity, a unit TE module with the copper plate is shown in Fig. 6.1b with
geometric dimension, a co-ordinate system, directions of different heat components, and thermal
boundary conditions. The PV panel has length PVl , width PVw and thickness PVt . A TE module
has length TEl , width TEw , and operates between the high and low temperature reservoirs bsT and
ambT , respectively. The TE module absorbs hQ amount of heat from the back surface of the PV
panel and rejects cQ amount of heat to the surrounding environment. The main mode of heat
206
transfer through PV-TE system is conduction. In addition to this, it is accompanied by
convection, radiation losses to surroundings from PV panel and internal heat generation, Peltier
heat generation/liberation at the junctions as well as Thomson heat generation in the TE module.
Convection heat loss from the side walls of a TE module to the surrounding environment is also
taken into account. Following section presents a mathematical model of the heat transfer for a
PV panel and TE module.
Modeling of PV system
Following assumptions are considered during the heat transfer modeling of a PV panel (Tiwari et
al. 2006):
The system is in quasi-steady state.
Transmittivity of ethylene vinyl acetate material (EVA) is nearly 100%.
Thermal resistance assumed to be negligible along the width of the PV panel considering
various layers such as glass, EVA, solar cells and tedlar.
The ethylene vinyl acetate (EVA) is used for encapsulation of photovoltaic modules due to their
good optical transmissivity, good electric insulator, and low water absorption ratio and tedlar is
polyvinyl fluoride film used for the back surface protection (Stark and Jaunich, 2011). The
different layers of solar PV panel are as shown in Fig. 6.2.
207
Figure 6.1 Schematic diagram of (a) photovoltaic – thermoelectric (PVTE) system and (b) unit
thermoelectric generator
Figure 6.2 Exploded view of Solar PV panel layers (Amrani 2007)
x
wTE
p n
Rl
Tbs
Tamb
I
Qconv
(a) (b)
Qh
Qc
Solar PV panel
TE Generator
Solar Radiation
Glass
Cover
Ethylene
Vinyl
Acetate
(EVA)
Ethylene
Vinyl
Acetate
(EVA)
Tedlar
Silicon Cells
208
In order to calculate the back surface temperature of PV panel an energy balance is applied
across the PV panel and step by step procedure is presented next.
The rate of solar energy available on the PV the panel (Tiwari et al. 2006),
GAGAQ cTPVgccPVgs 1 (4)
where, g is the transmissivity of glass cover, c and T are the absorptivity of cell and tedlar,
c is the packing factor , G is the solar radiation, and PVA is surface area of PV panel,
respectively. In Eq. (4), the first term represents the rate of solar energy received by solar cell
after transmission from EVA and the second term is the rate of solar energy absorbed by tedlar
after transmission from EVA.
Heat loss from top surface of the PV panel to the ambient by convection is (Najafi and
Woodbury 2013)
PVambcconvconv ATTUQ (5)
where, convU is the overall heat transfer coefficient from the solar cell to the ambient air through
glass cover which includes conduction and convection losses (Sarhaddi 2010). convU can be
expressed as (Najafi and Woodbury 2013)
1
1
PVg
g
convhk
lU . (6)
where, gl and gk are the length and thermal conductivity of glass cover and PVh is the
convection heat transfer coefficient for heat loss from solar cell to the ambient through glass
cover.
Heat loss from the top surface of the PV panel to the ambient by radiation is (Najafi and
Woodbury 2013)
44
skycpvrad TTAQ (7)
where is the emissivity of the PV panel and is the Stefan-Boltzmann’s constant. The
effective temperature of sky )( skyT can be written as follow (Wong and Chow 2001),
5.10552.0 ambsky TT . (8)
Now, the heat conduction from the solar cell to the tedlar (Najafi and Woodbury 2013),
209
pvbsccondcond ATTUQ (9)
where condU is the overall conductive heat transfer coefficient from the solar cell to the ambient
air through tedlar and can be expressed as (Najafi and Woodbury 2013)
1
T
T
si
sicond
k
l
k
lU . (10)
In Eq. (10), sil and sik are length and thermal conductivity of the silicon layer. Tl and Tk are
length and thermal conductivity of the tedlar.
The electrical power output from the PV panel can be expressed as (Najafi and Woodbury 2013)
PVelcgPV AGP (11)
where, el is PV the panel conversion efficiency.
Combining and rearranging Eq. (4) to Eq. (11) and by applying the assumption, bsc TT , results
into,
pvelcg
skycpvpvambctpvcTccg
AG
TTAATTUAGG
44
1. (12)
Eq. (12) gives the cell temperature which is same as back surface temperature of PV panel due to
the negligible thermal resistance of PV panel assumption.
The solar cell temperature is important parameter to estimate power output and thermal
efficiency of the PV panel. The power output and efficiency of the PV panel in terms of the cell
temperature is given by (Skoplaki and Palyvos 2009)
refCrefrefPVPV TT 1, (13)
refCrefPVrefPVgPV TTAGP 1, (14)
In Eq. (13), the reference efficiency and temperature coefficient are provided by manufacturers.
The thermal efficiency will be used to calculate overall efficiency of combined system in later
stage.
210
Modeling of TE system
Heat transfer modeling of the TE effect has been carried out in this section. Following
assumption were made during the derivation of the heat transfer model for a TE system:
Isotropic and homogeneous material properties.
Thermal and electrical contact resistances were assumed negligible.
The energy transport equation inside a nanostructured TE module can be expressed as (Antonova
and Looman 2005):
genm qt
TC
q (15)
where symbols m , pC , T , q , and genq represent material density, specific heat, temperature,
heat generation rate per unit volume, and heat flux vector, respectively. The continuity of the
electric charge through the system must satisfy (Antonova and Looman 2005)
0
t
DJ (16)
where J is the electric current density vector and D is the electric flux density vector,
respectively. Equation (15) and Eq. (16) are coupled by the set of TE constitutive equations
(Antonova and Looman 2005) as shown in Eq. (17) and Eq. (18) below
TkT ][][ Jq
)][(][ T EJ
(17)
(18)
where ][ is the Seebeck coefficient matrix, ][k is the thermal conductivity matrix, ][ is the
electrical conductivity matrix, E is the electric field intensity vector, respectively. E can be
expressed as , where is the electric scalar potential (Landau, Lifshitz, and Pitaevskii
1984). Combining Eq. (15) to Eq. (18), the coupled TE equations for energy and charge transfers
can be expressed as
JEJ
TkT
t
TCm ][][ (19)
and
0][][][][
Tt
(20)
211
where ][ is the dielectric permittivity matrix and JE represents Joule heat (Antonova and
Looman 2005). Note that the material-property matrices (e.g., ][ and ][k , etc.) are suitable for
non-homogeneous materials. These, material-property matrices become single-valued property in
case of homogeneous materials. Initially, a simplified 1-D version of above equations will be
solved to obtain close forms of analytical solutions.
For 1-D analytical heat transfer modeling, a TE generator with N number of nanostructured
p type and n type semiconductor modules are connected electrically in series and thermally
in parallel and the end terminals are connected with the load resistance lR as shown in Fig. 1b.
TE elements have length TEl , width TEw , and works between the high and low temperature
reservoirs bsT and ambT respectively. A TE module absorbs hQ amount of heat from the back
surface of PV panel and rejects cQ amount of heat to the surrounding environment. The main
mode of heat transfer through a nanostructured semiconductor leg is the conduction and it is
accompanied by an internal heat generation in the form of the Joule effect, Peltier heat
generation/liberation at the junctions as well as Thomson heat generation. Convection heat loss
from the side walls of p type and n type semiconductor legs to the ambient environment is
also taken into account. Assuming isotropic and homogeneous material properties and neglecting
the thermal and electrical contact resistances between the contact surfaces a one dimensional
heat transfer equation under a steady state condition for nanostructured p type and n type
semiconductor legs is given by,
0)(2
2
2
2
dx
dT
kA
ITT
kA
ph
kA
I
dx
Tdamb
TE (22)
In Eq. (22) the first term is the Fourier heat conduction, second term is the Joule heating, third
term is the convection heat transfer loss, and fourth term is the Thomson effect. Equation (22)
can be written in the following form
02
2
Tdx
dT
dx
Td (23)
212
where
kA
I ;
kA
PhTE ; 2
2
kA
I
kA
pTh ambTE . (24)
Equation (23) is a linear and non-homogeneous ordinary differential equation. The general
solution to Eq. (23) is
XDXDeCeCxT 21
21 . (25)
where
2
42
1
D ;
2
42
2
D . (26)
Applying thermal boundary conditions at the top surface ( bsTTx ,0 ) and the bottom surface
( ambTE TTlx , ) one can determine the constants, 1C and 2C , of Eq. (25) as given below:
)( 12
22
1tt
tt
LDLD
amb
LDLD
bs
ee
TeeTC
;
)( 12
11
2tt
tt
LDLD
amb
LDLD
bs
ee
TeeTC
. (27)
Finally, the temperature distribution inside the nanostructured semiconductor legs can be
approximated from
.)()(
2
12
11
1
12
22
xD
LDLD
amb
LDLD
bsxD
LDLD
amb
LDLD
bse
ee
TeeTe
ee
TeeTxT
tt
tt
tt
tt
(28)
Now, combining heat transfer within the nanostructured semiconductor leg with Peltier heat,
which occurs at the junctions, heat transfer in nanostructured TE generator from back surface of
solar panel to surrounding environment is given by
.22112211 nnnnnnppppppbsh DCDCAkDCDCAkTIQ (29)
Heat rejected by TE generator to surrounding environment is given by
.21
21
2211
2211
tntn
tptp
LD
nn
LD
nnnn
LD
pp
LD
ppppambc
eDCeDCAk
eDCeDCAkTIQ
(30)
The power output of single TE generator can be calculated as
lTE RIP 2 . (31)
where
li
chnp
RR
TTI
)()(. (32)
213
The thermal efficiency can be evaluated as
.h
TETE
QInputHeat
POutputPower
(33)
The overall thermal efficiency of combined PV- nanostructured TE system is given by
combining Eq. (13) and (33)
O = Solar panel efficiency PV Thermal efficiency of TE generator TE (34)
6.3 Results
In this section, the performance of a nanostructured TE generator applied to a solar PV panel as a
waste heat recovery mechanism is investigated based on the one dimensional analytical solution
obtained in the previous section. The nanostructured semiconductor p type material Bismuth
Antimony Telluride BiSbTe is considered to analyze the performance (Poudel et al. 2008). For
n type material, similar properties as that of p type material Bismuth Antimony
Telluride BiSbTe is considered with copper as a connector material. A TE generator
performance characteristic in terms of the thermal efficiency, power output, and heat input have
been studied in detail. The operating parameters and dimensions considered in current analysis
are as per Table 6.1. The Seebeck coefficient ( ), electrical resistivity ( ), and thermal
conductivity ( k ) are specified as polynomial functions of temperatures as shown in Table 6.2
and Table 6.3 Poudel et al. (2008). These properties are evaluated at an average temperature of
working range. A load resistance lR is considered equal to internal resistance iR to get
maximum power output as per Eq. (32). Figures 6.4 to 6.12 show the effect of solar radiation
and convection heat transfer coefficient on the performance of a nanostructured TE generator. In
real application, hot side of TE generator is considered to be in contact with back surface of solar
PV panel with a range of temperatures. For example, back surface of solar PV panel varies from
320 K to 370 K with solar radiation of 1200 W/m2
as shown in Fig. 6.3. The cold side of TE
generator is considered to be facing the ambient environment with a range of temperatures
( 313253 cT ). The performance between nanostructured TE generator and traditional material
TE generator is also investigated in Figs. 6.6, 6.8, and 6.10. In addition to this, combined system
efficiency is also investigated.
214
Table 6.1 Operating conditions and dimensional parameters of combined solar PVTE system
Parameter Value
Solar Radiation, G 0 to 1200
Convection heat transfer coefficient for TE generator, TEh 0 to 50
TE generator dimensions, TETETE twl 01.001.001.0
Solar PV panel dimensions, PVPVPV twl 05.011
Transmissivity of glass cover, g 0.95
Conductivity of glass cover, gk 1
Thickness of glass cover, gL 0.003
Absorptivity of solar cell, c 0.85
Packing factor, c 0.83
Absorptivity of tedlar, T 0.5
Reference thermal efficiency, refPV , 12%
Convection heat transfer coefficient for solar panel, PVh 5.8
Emissivity of solar PV panel, 0.88
Temperature coefficient, refPV , 0.0045
Reference Temperature, refT 25
215
Table 6.2 Polynomial functions of Seebeck coefficient, electrical conductivity, thermal
conductivity, and figure of merit with respect to temperature for nanostructured BiSbTe bulk
alloys (Poudel et al. 2008)
Property
(For n - type and
p -type material)
Temperature
range, (ºC)
Polynomial functions of different
thermoelectric
properties in terms of temperature
Seebeck
Coefficient,
2500 T 6
4936
23
1010738.710656.3
10732.2656.050.172
TT
TT
Electrical
Conductivity,
2500 T 5
41138
253
1010354.110093.2
10451.210575.8450.1
TT
TT
Thermal
Conductivity, k
2500 T
41037
254
10875.710809.3
10193.410946.2136.1
TT
TT
Figure of Merit, ZT 2500 T
4937
253
10149.110679.4
10765.110698.5034.1
TT
TT
216
Table 6.3 Polynomial functions of Seebeck coefficient, electrical conductivity, thermal
conductivity, and figure of merit with respect to temperature for BiSbTe bulk alloys (Poudel et
al. 2008)
Property
(For n - type
and p - type
material)
Temperature
range, (ºC)
Polynomial functions of different thermoelectric
properties in terms of temperature
Seebeck
Coefficient,
2500 T 6
4835
24
1010281.210497.1
10637.9504.090.201
TT
TT
Electrical
Conductivity,
2500 T 5
41138
253
1010323.910165.8
10464.310053.8165.1
TT
TT
Thermal
Conductivity,
k
2500 T 4937
253
10009.110477.4
10920.210023.1415.1
TT
TT
Figure of
Merit, ZT 2500 T
41037
253
10321.110488.1
10021.910731.8809.0
TT
TT
217
The back surface temperature of a solar PV panel is an important parameter as TE generator is
considered to be attached directly beneath the solar PV panel. The back surface of the PV panel
acts as the heat source, bsT and the surrounding acts as the heat sink, ambT . The back surface
temperature is calculated using Eq. (12). The hot surface temperature bsT depends on the
surrounding temperature and solar radiation. The back surface temperature of a solar PV panel is
plotted as a function of solar radiation in Fig. 6.3 at different values of ambient temperature. It is
observed from Fig. 6.3 that an increase in the surrounding temperature increases the PV panel
back surface temperature. It is also observed from Fig. 6.3 that the PV panel’s back surface
temperature also increases with the increasing solar radiation. For example, PV panel back
surface temperature increases from 350 K to 358 K due to the increase in surrounding
temperature from 293 K to 303 K. For surrounding temperature 303 K, the PV panel back
surface temperature increases from 297 K to 358 K due to an increase in solar radiation from 0
W/m2 to 1200 W/m
2.
218
Solar Radiation, G (Wm-2
)
PV
Panel
Back
Su
rface
Tem
per
atu
re,
Tb
s(K
)
0 200 400 600 800 1000 1200240
260
280
300
320
340
360
380T
amb= 253 K
Tamb
= 263 K
Tamb
= 273 K
Tamb
= 283 K
Tamb
= 293 K
Tamb
= 303 K
Tamb
= 313 K
Figure 6.3 Solar PV panel back surface temperature with variable solar radiation and ambient
temperature
219
Figure 6.4 shows the temperature distribution along the length of p type and n type
nanostructured semiconductor legs. Equation (28) is used to calculate values of temperature as
presented in Fig. 6.4 for a specified hot surface temperature (356 K), surrounding temperature
(300 K), and different values of the convection heat transfer coefficients. It is observed from the
figure that the convection losses have larger impact on the temperature distribution along the
nanostructured thermoelectric legs. At higher values of the convection heat transfer coefficient
larger amount of heat was removed from the surface due to convection so that the temperature
drops more rapidly along the leg. It is shown later that convection affects the heat input to the
system and thermal efficiency of the system significantly.
220
Length, x(m)
Tem
per
atu
re,T
(K)
0 0.002 0.004 0.006 0.008 0.01300
310
320
330
340
350h = 0.001 Wm
-2K
-1
h = 10 Wm-2
K-1
h = 20 Wm-2
K-1
h = 30 Wm-2K
-1
h = 40 Wm-2K
-1
h = 50 Wm-2
K-1
Figure 6.4 Temperature distribution over the length of nanostructured p type and n type
semiconductor leg
221
Effect of thermal energy input to the nanostructured thermoelectric system can be analysed from
Fig. 6.5. Heat input to the system is plotted as a function of the solar radiation at different values
of the convection heat transfer coefficients. Plot in Fig. 6.5 shows that with increase in the solar
radiation, more heat is available to convert. In addition, Fig. 6.5 also depicts the effect of
convection heat transfer coefficient. For a given solar radiation, with higher convection heat
transfer coefficient, heat input to the system increases. This establishes that due to higher
convection losses more heat is drawn from heat reservoir to the hot surface. Figure 6.6 shows the
heat input comparison between the traditional TE generator and nanostructured TE generator.
For the given range of input solar radiation, the magnitude of heat input to the TE system is
higher for a traditional TE generator compared to nanostructured TE generator. For example,
with solar radiation of 1200 W/m2 and adiabatic side-wall conditions, the heat available to
convert is 1.94 W for nanostructured TE generator and 2.13 W for traditional TE generator. This
can be attributed to the decrease in the thermal conductivity of a nanostructured TE material. As
shown in Fig. 6.7, traditional TE material has higher thermal conductivity so more amount of
heat is available for traditional TE generator than nanostructured TE generator.
222
Solar Radiation, G (Wm-2
)
Hea
tIn
pu
t,Q
h(W
)
200 400 600 800 1000 1200
0.5
1
1.5
2
2.5
3
h = 10-6
Wm-2
K-1
h = 10 Wm-2K
-1
h = 20 Wm-2K
-1
h = 30 Wm-2
K-1
h = 40 Wm-2
K-1
h = 50 Wm-2K
-1
Figure 6.5 Heat input to nanostructured TE generator with different solar radiation and variable
convection heat transfer coefficient
223
Traditional TE material
Solar Radiation, G (Wm-2
)
Hea
tIn
pu
t,Q
h(W
)
200 400 600 800 1000 1200
0.5
1
1.5
2
2.5
Nanostructured TE material
Figure 6.6 Heat input comparison of TE generator using traditional and nanostructured material
thermoelectric material
224
Temperature, (C)
Th
erm
al
Co
nd
ucti
vit
y,
(Wm
-1K
-1) Traditional TE material
0 50 100 150 200 250
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Nanostructured TE material
Figure 6.7 Thermal conductivity of traditional and nanostructured TE material as a function of
temperature
225
Figure 6.8 demonstrates the power output of a TE generator as a function of solar radiation. As
the hot surface temperature increases due to increase in the solar radiation, power output also
increases. As one can analyse from Eq. (31) & (32), temperature difference has large impact on
the power output. In this case, the hot surface temperature depends largely on the solar radiation.
The same equation verifies that power output is independent of convection heat transfer losses. It
also shows power output comparison between the nanostructured and traditional TE generator.
An increment in the figure of merit due to the incorporation of nano-particle in the bulk material
matrix can be attributed to the surge in power output. The power output increases by 3% due to
the use of a nanostructured TE material with 1200 W/m2
input solar radiation condition.
226
Traditional TE material
Solar Radiation, G (Wm-2
)
Po
wer
Ou
tpu
t,P
o(W
)
200 400 600 800 1000 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Nanostructred TE material
Figure 6.8 Power output from TE generator as a function of solar radiation
227
Figure 6.9 also establishes the effect of the surface to surrounding convection heat transfer on the
thermal efficiency. For a given temperature difference between hot and cold surfaces, an increase
in the convection heat transfer coefficient decreases the thermal efficiency of a system.
Irreversible convection process cause the larger amount of heat loss to the ambient environment;
therefore, it suggests that less heat is available to generate electric potential and this leads to
lower thermal efficiency. Figure 6.10 shows a comparison between the thermal efficiency of a
TE generator using the traditional TE material and nanostructured TE material. The
nanostructured TE material improves the thermal efficiency of a TE generator from 3.09% to
3.47% i.e., 12% increment compared to traditional TE material.
228
Solar Radiation, G (Wm-2
)
Th
erm
al
Eff
icie
ncy
,
t(%
)
200 400 600 800 1000 1200
0.5
1
1.5
2
2.5
3
3.5
h = 10-6
Wm-2K
-1
h = 10 Wm-2
K-1
h = 20 Wm-2
K-1
h = 30 Wm-2K
-1
h = 40 Wm-2K
-1
h = 50 Wm-2
K-1
Figure 6.9 Thermal efficiency of nanostructured TE generator with different solar radiation and
variable convection heat transfer coefficient
229
Traditional TE material
Solar Radiation, G (Wm-2
)
Th
erm
al
Eff
icie
ncy
,
t(%
)
200 400 600 800 1000 1200
0.5
1
1.5
2
2.5
3
3.5
Nanostructured TE material
Figure 6.10 Thermal efficiency comparison of TE generator with traditional and nanostructured
TE material
230
In order to compare the power output between the solar PV panel and TE generator, the back
surface of a PV panel is considered to be filled with TE modules. For example, for a plot in Fig.
6.11 a solar panel with surface area of 1 m2 is considered and back surface is covered with TE
generators with surface area 0.00025 m2 on its back surface. Theoretically the power output of
solar PV panel remains 100 W whereas TE generators reach up to 200 W of power output at
1200 W/m2.
231
Solar Radiation, G (Wm-2
)
Po
wer
Ou
tpu
t,P
o(W
)Solar Panel Output
200 400 600 800 1000 1200
50
100
150
200
TE Generator Output
Figure 6.11 Power output comparison of solar PV panel and TE generator
232
The solar panel efficiency is investigated by Eq. (13). The performance of a solar PV panel
deteriorates as the temperature rises, because efficiency of solar panel is a function of cell
temperature. Fig. 6.12 shows the downward trend in the thermal efficiency as intensity of solar
radiation rises. In contrary, the efficiency of a TE generator rises as solar radiation increases. Fig.
6.13 is a combined efficiency of a solar PV-TE system. The efficiency of a combined system
remains low during the low solar radiation because it does not generate high temperature
gradient. The efficiency of a TE generator increases as solar radiation increases and that
compensates the low efficiency of solar PV panel at higher solar radiation. For example, at solar
radiation of 500 W/m2 the efficiency of solar PV panel is 10.68% and it rises to 14.88% with
combined the solar PV-TE system. A TE generator has less impact on overall efficiency of
combined system at lower solar radiation but it has a large impact on thermal efficiency at the
higher solar radiation.
233
Solar Panel
Solar Radiation, G (Wm-2)
Th
erm
al
effi
cie
ncy
,t(%
)
200 400 600 800 1000 1200
1
2
3
4
5
6
7
8
9
10
11
12
TE generator
Figure 6.12 Solar panel conversion efficiency Vs. Solar Radiation
234
Solar Radiation, G (Wm-2
)
Co
mb
ined
effi
cie
ncy
,c
(%)
200 400 600 800 1000 1200
14
16
18
20
22
24
26
Figure 6.13 Combined efficiency of solar PVTE system Vs. Solar Radiation
235
6.4 Conclusion
In the present work, a combined solar PV-nanostructured TE power generation system is
proposed. TE modules are attached to the back surface of a solar PV panel to use the excess heat
of a PV panel. A heat transfer model of a solar PV panel and TE generator has been derived. A
one-dimensional (1-D) heat transfer model is derived involving the Fourier heat conduction,
Joule and convection losses, and Peltier, Seeback and Thomson effects. The temperature
dependent nanostructured thermoelectric properties have been considered for the analysis. The
influences of solar radiation and convection heat transfer coefficients on various performance
parameters of a nanostructured TE generator such as power output, heat input, and thermal
efficiency have been studied. In addition to this, the performance between nanostructured and
traditional TE material has been compared. The improved nanostructured TE material has
enhanced TE properties which are reflected in terms of better power output and thermal
efficiency. The higher electrical conductivity and lower thermal conductivity of nanostructured
TE materials are key reasons to increase the thermal efficiency and power output of a
nanostructured TE generator. Furthermore, the effect of a TE generator on combined system is
evaluated in terms of improved combined efficiency of the system. TE modules have a large
impact on the performance of a combined solar PV-TE system at the higher solar radiation.
236
6.5 Nomenclature
A cross-sectional area (m2)
C specific heat capacity (kJkg-1
K-1
)
D electric flux density vector (Nm2C
-1)
e electron charge (C)
E electric field intensity vector (Vm-1
)
G solar irradiation (Wm-2
)
h convection heat transfer coefficient (Wm-2
K-1
)
I electric current (A)
J electric current density (Am-2
)
k thermal conductivity (Wm-1
K-1
)
0L Lorentz number (V2K
-2)
l length (m)
N number of modules
p Perimeter (m)
P power output (W)
q heat flux vector (Wm-2
)
.
q heat generation rate per unit volume (Wm-3
)
Q heat or energy (W)
R resistance )(
S entropy (Wm-3
K-1
)
t thickness (m), time (s0
T temperature (K)
U heat transfer coefficient (Wm-2
K-1
)
V voltage (V)
w width (m)
x coordinate (m)
ZT dimensionless figure of merit
absorptivity, Seebeck coefficient (VK-1
)
237
packing factor, temperature coefficient
emissivity of PV panel, dielectric permittivity (Fm-1
)
efficiency
a parameter (see Eq. (24))
a parameter (see Eq. (24)), temperature coefficient (K-1
)
electrical resistivity )(
m density (kgm-3
)
Boltzmann constant (JK-1
), electrical conductivity (Sm-1
)
Transmissivity, Thomson coefficient (VK-1
)
electric scalar potential (V)
a parameter (see Eq. (24))
Subscripts
amb ambient
bs Solar panel back surface tempeature
c cell, cold side of thermoelectric module
cond conduction
conv convection
e electronic
el electrical
g glass cover
gen generation
h hot side of thermoelectric module
i internal
L Lattice
l External load
mp maximum powerpoint
n n type semiconductor material
o output, overall
p p type semiconductor material
238
PV solar PV panel
rad radiation
ref reference
s sun
si silicon
sky sky
T tedlar, thermal
TE thermoelectric module
239
CHAPTER 7: NANOSTRUCTURING OF n-type Bi2Te2.7Se0.3 BASED ON
SOLID STATE SYNTHESIS TECHNIQUE
7.1 Introduction
Thermoelectric (TE) energy conversion systems offer unique advantages of being silent in
operation, no moving parts, no refrigerants, robust in nature, and long service life. Nevertheless,
extremely low energy conversion efficiency limits TE systems from widespread real world
applications in power generation and cooling. TE systems are first choice of interest for power
generation and cooling/heating for extreme environments, such as deep space probes. The
performance of TE systems are measured based on Figure of Merit (ZT=α2σ/k) of TE materials.
For a good ZT, the power factor (α2σ) should be high and thermal conductivity (k) should be as
low as possible. The power factor improvement and the thermal conductivity reduction have
been achieved in different materials. Hicks and Dresselhaus (1993) discussed the concept of
quantum wire (one dimensional) for TE materials. ZT improvement in quantum wire is due to
increase in the power factor (α2σ) but not significantly due to the low thermal conductivity
(Hicks and Dresselhaus 1993). Another method to increase ZT was demonstrated experimentally
by Venkatasubramanian et al. (2001), where ZT of p-type superlattice of Bi2Te3/Sb2Te3
(bismuth-telluride-antimony) improved from 1 to 2.4 due to the reduction in the thermal
conductivity. Improvements offered by Hicks and Dresselhaus (1993) and Venkatasubramanian
et al. (2001) were primarily for the low dimensional structures, such as quantum dots, wires, and
superlattice structures. The low dimensional structures can be manufactured using different
methods, such as electroless etching (Hochbaum et al. 2008), Superlattice Nanowire Pattern
Transfer (SNAP) (Boukai et al. 2008), Chemical Vapor deposition (Venkatasubramanian et al.
1997), and Molecular-Beam Epitaxy (MBE) (Hicks et al. 1996). A low-dimensional structure
can be employed limitedly in the real world applications due to the complicated
physical/chemical vapor deposition method and cost of manufacturing (Ma et al. 2013). There is
another route to improve the ZT in bulk TE materials called ‘nanocomposite bulk materials’
(Bottner and Konig 2013). Nanocomposites can be manufactured using various methods which
are relatively less complicated compared to the low-dimensional methods. A 3D bulk composite
can be made with the combination of hot press, high energy ball milling, spark plasma sintering,
240
and solid state reaction (Nolas et al. 2000, Zhou et al. 2008, Poudel et al. 2008, He et al. 2006).
Figure 7.1 shows some of the ZT improvements in TE materials in low-dimensional and bulk
structures.
Figure 7.1 ZT improvements in low dimensional and bulk TE materials
241
In previous chapters, heat transfer in nanocomposite TE systems and their applications have been
investigated. Analysis showed that the nanocomposite TE systems has promising place as a
future energy conversion system. Manufacturing methods to generate the higher ZT TE
nanostructures (0D, 1D, 2D, 3D) materials are costly (e.g., electroless etching, chemical vapor
deposition, direct current hot press). In this work, nanocomposite TE materials using the indirect
resistance heating method was produced which is relatively cheaper method compared to other
techniques (e.g., direct current hot press). In this work, bismuth telluride based alloys were
chosen for the sample preparation. Bismuth telluride and its alloys present the best TE materials
to this date at the room temperature. The improvement in ZT of bismuth-telluride alloys can
expand their application range in the areas of refrigeration, air-conditioning, and waste heat
recovery. For experimental work, ball milling and hot press (indirect resistance heating) method
for 3D bulk materials is chosen which offers one of the most economical methods to synthesize
the TE materials (Poudel et al. 2008).
7.2 Sample preparation and results
The experimental work was performed at the Department of Physics in the Indian Institute of
Science, Bangalore under the Mitacs Globalink Research Award under the supervision of Dr.
Ramesh Chadra Mallik. To make alloy powder, appropriate amounts of nanopowders of Bi
(99.999%), Te (99.999%), and Se (99.999%) were weighted based on the stoichiometry
Bi2Te2.7Se0.3. The mixture was loaded into the graphite die of diameter 14 mm for the hot press.
Bi2Te2.7Se0.3 powders were hot pressed (Vacuum technology, Bangalore, India) under the
dynamic vacuum at 500˚C with 40 MPa for 2 hours. The resulted disks of the Bi2Te2.7Se0.3 were
polished and cut into the bars with the size of 2×3×12 mm to measure the transport properties.
The X-ray diffraction (XRD) was performed by the Bruker D8 advance diffractometer using Cu-
Kα radiation with 2˚/min. For the crystallographic phase identification, the Rietveld refinement
was performed using the FullProf software (Roisnel and Rodriguez-Carvajal, 2001). The
Electron Probe Micro Analysis (EPMA) was performed using the JEOL JXA-8530F
HyperProbe. The electrical conductivity and Seebeck coefficient were measured using the
Linseis LSR-3 under the Helium atmosphere. The uncertainties of the Seebeck coefficient and
electrical resistivity measurements were ±7% and ±10%, respectively. The fractured surface of
the hot pressed sample was observed by the FEI Quanta 200.
242
Figure 7.2 presents the Rietveld refinement of the XRD pattern of powders. The XRD patterns
verify that the powder is in the single phase and well matched with Bi2Te2.7Se0.3. The peaks of
Bi2Te2.7Se0.3 were indexed with Bi2Te2.4Se0.6 (ICSD#617110).
Figure 7.2 Rietveld refinement powder XRD pattern for Bi2Te2.7Se0.3
243
Figures 7.3 to 7.5 present the transport properties of Bi2Te2.7Se0.3 and Sb1.5Bi0.5Te3. Figure 7.3
show the Seebeck coefficient and electrical resistivity of Bi2Te2.7Se0.3 as a function of
temperature. The Seebeck coefficient decreases and electrical resistivity increases as the
temperature rises. The power factor shows a decreasing trend in Figure 7.4 as temperature rises.
The highest power factor was observed 955 μWm-1
K-2
at 87 ˚C.
Temperature (C)
Seeb
eck
Co
eff
icie
nt
(V
K-1
)
50 100 150 200-176
-174
-172
-170
-168
-166
-164
Seebeck Coefficient
Ele
ctr
ical
Resi
stiv
ity
(m
m)
0.029
0.03
0.031
0.032
0.033
0.034
0.035
Electrical Resistivity
Figure 7.3 Seebeck coefficient and Electrical resistivity of sample Bi2Te2.7Se0.3
244
Temperature (C)
Po
wer
Facto
r(m
Wm
-1K
-2)
50 100 150 2000.88
0.9
0.92
0.94
0.96
Figure 7.4 Power factor of sample Bi2Te2.7Se0.3
245
Figure 7.5 presents a comparison between the power factors of Bi2Te2.7Se0.3 manufactured using
different hot press techniques. Yan et al. (2010) used the direct current hot press to synthesize
the Bi2Te2.7Se0.3 powder, whereas, the current work used the indirect resistance heating to
synthesize the Bi2Te2.7Se0.3 powder. One can observe from the plot that the power factor
increases rapidly for a similar temperature range when the direct current hot press is used. The
reason behind the low power factor can be randomness of the grains (Yan et al. 2010). The SEM
images in Figure 7.6 show those random grains without preferred crystal orientation.
Additionally, density of a sample turned out to be 84% which was measured using the
Archimedes’ principle. This lower density suggests that the sample was not compressed enough
due to the inadequate pressure and temperature.
246
Temperature (C)
Po
wer
Facto
r(m
Wm
-1K
-2)
50 100 150 200
0.0010
0.0012
0.0014
0.0016
0.0018
0.0020
0.0022Direct Current Hot Press (Yan et al. 2010)
Indirect Resistance Heating (Current work)
Figure 7.5 Comparison of power factor between Bi2Te2.7Se0.3 manufactured via direct current
hot press and indirect resistance heating
248
7.3 Conclusion
In this work, Bi2Te2.7Se0.3 samples were prepared via ball milling and an indirect resistance
heating hot press. The power factor of hot-pressed Bi2Te2.7Se0.3 sample followed a decreasing
trend as the temperature increases. The grains in Bi2Te2.7Se0.3 were observed (via SEM image)
without preferred orientation which could have contributed to the higher electrical resistivity. An
indirect resistance heating hot press method provides an alternative economical route to make TE
legs, but the synthesized bulk TE legs do not exhibit the high power factor which can reduce the
ZT. On the contrary, the direct current hot press or spark plasma sintering can produce the high
ZT materials.
249
CHAPTER 8: OVERALL CONCLUSIONS AND FUTURE WORK
8.1 Overall conclusions
This study constitutes novel work in the area of heat transfer processes in nanocomposite
thermoelectric (TE) systems. Effects of heat transfer on the performance of nanostructured
thermoelectric generators (TEG) and thermoelectric coolers (TEC) have been evaluated in terms
of thermal efficiency (η) and Coefficient of Performance (COP). Numerical simulations have
been performed to obtain qualitative results which show temperature profile and streamlines of
heat flow. In addition to that, effects of nanostructuring of TE systems have been presented in
terms of improvements in their thermal efficiency and COP. In the case of nanocomposites, the
volume fraction of nano-size or micro-size particles influences the performance of TE systems.
Therefore, effects of the volume fraction on the performance of TE systems were presented
qualitatively and quantitatively. Nanocomposite TE legs were prepared based on the solid state
synthesis technique in which an indirect resistance heating method was used. The power factor
remained low due to the lower pressure and temperature in indirect resistance heating method
which influenced grains and density of the sample. The major conclusions are:
Thermal source and sink temperatures have significant effects on the performance of a TEG
as the temperature gradient under which TEG works depends on source and sink
temperatures.
The electrical power output of TEG increases as the hot surface temperature increases due to
the increase in the electric potential which eventually increases the thermal efficiency.
An increase in the convection heat transfer coefficient at the side surfaces of the TEG legs
decreases the TEG thermal efficiency due to the increase in the heat input of the TEG.
In TEC systems, the influence of the convection heat transfer coefficient depends on the
electric current input. For low electric current input, heat absorbed decreases as the
convection heat transfer coefficient increases and thus reduces the COP. However, under
higher electric current input, the amount of heat absorbed increases with the increase in
convection heat transfer coefficient and thus increases the COP.
In nanocomposite TE systems, performance (e.g., thermal efficiency and COP) largely
depends on the particle volume fraction.
250
The study also reveals that the heat conduction part of total heat input to TE systems should
remain as low as possible to increase the performance of TE systems.
The higher solar radiation decreases the efficiency of a solar panel but the combined solar
PV-TEG system has higher efficiency compared to a solo solar panel.
The solid state synthesis technique using an indirect resistance heating method provides an
alternate route to fabricate the nanocomposite TE legs but lower temperature, pressure, and
the heating rate can adversely affect TE material properties.
8.2 Future work
One important conclusion of this research work is that the convection heat transfer from the side
surfaces of TE legs increases the heat absorbed by a TEC at the higher electrical current input.
This phenomenon can be applied to remove large amounts of heat from hot spots. A single cell
TEC prototype can be developed with air flowing through it to validate the higher amount of the
heat absorption from the cold surface of the TEC experimentally.
A multiphysics simulation can be created by coupling thermoelectric phenomenon with air flow
to observe the effects of convection heat transfer on TEG and TEC with multiple p-type and n-
type TE legs.
TE materials such as Tellurium (Te), Antimony (Sb), and Germanium (Ge) are not widely
available (0.001 – 1.4 ppm by weight) in the earth’s crust (Amatya and Ram 2012). Such low
availability translates to material price volatility. So, it is important to explore other sources of
TE materials such as agricultural biomass which contains Silica Carbide (SiC), a compound
which stays stable and exhibits higher Figure-of-Merit at higher temperatures (Fujisawa et al.
2004). Future efforts can also be directed towards manufacturing Silica based higher Figure-of-
Merit TE materials.
251
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