Modeling and Analysis of Nanostructured Thermoelectric Power Generation and Cooling Systems

Transcription

1 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, 05

2 ABSTRACT MODELING AND ANALYSIS OF NANOSTRUCTURED THERMOELECTRIC POWER GENERATION AND COOLING SYSTEMS Ronil Rabari University of Guelph, 05 Advisors: 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

3 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 Bi Te.7 Se 0.3, 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.

4 To all my teachers To my loving daughters - Aarna and Archa iv

5 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 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 v

6 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. vi

7 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 Introduction. Background. Objectives 5.3 Scope of this thesis 7.4 Contribution of present study.5 Publications from present study Chapter Effect of convection heat transfer on performance of waste heat thermoelectric generator 3. Introduction 3. Heat transfer modeling 6.3 Results and discussion.4 Conclusion 53.5 Nomenclature 54 Chapter 3 Numerical simulation of nanostructured thermoelectric generator considering surface to surrounding convection Introduction Mathematical model and boundary conditions Results and discussion Conclusion Nomenclature 77 Chapter 4 Analytical and numerical studies of heat transfer in nanocomposite thermoelectric cooler 79 vii

8 4. Introduction Modeling Results and discussion Conclusions Nomenclature 36 Chapter 5 Effect of thermal conductivity on performance of thermoelectric systems based on effective medium theory Introduction Modeling and boundary conditions Results and discussion Conclusion Nomenclature 00 Chapter 6 Analysis of combined solar photovoltaic-nanostructured thermoelectric generator system 0 6. Introduction 0 6. Modeling and boundary conditions Results Conclusion Nomenclature 36 Chapter 7 Nanostructuring of n-type Bi Te.7 Se 0.3 based on solid state synthesis technique Introduction Sample preparation and results Conclusion 48 Chapter 8 Overall conclusions and future work Overall conclusions Future work 50 References 5 viii

9 List of Figures Figure Title Page number number. Typical TE modules in power generation mode or cooling mode 3. Potential applications of TE systems (Pichanusakorn and Bandaru 00). Various waste heat recovery methods in context of power plant (Stehlik 007, Rowe 995). Schematic diagram of location of TEG in combustion system and the schematic view of unit TEG cell.3 Temperature distribution over the length of p-type semiconductor leg with thermal source temperature, T 700 K and thermal sink H temperature, TC 300 K.4 Temperature distribution over the length of n-type semiconductor leg with thermal source temperature, T 700 K and thermal sink H 5 temperature, TC 300 K.5 Effect of thermal source temperature on heat input with variable convection heat transfer coefficient at constant thermal sink temperature, TC 300 K.6 Effect of thermal sink temperature on heat input with variable convection heat transfer coefficient at constant thermal source temperature, TH 700 K.7 Power generation as a function of thermal source temperature at different thermal sink temperature.8 Effect of thermal source temperature on thermal efficiency with variable convection heat transfer coefficient at constant thermal sink temperature, TC 300 K.9 Effect of thermal sink temperature on thermal efficiency with variable ix

10 convection heat transfer coefficient at constant thermal source temperature, TH 700 K.0 Current as a function of thermal source temperature with different thermal sink temperature. Effect of convection heat transfer coefficient on thermal efficiency at constant thermal sink temperature, TC 300 K. 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 T 700 K and T 300 K with adiabatic side wall H C condition.3 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 T 700 K and T 300 K with convection from the side walls with H h 0 Wm - K -.4 Entropy generation rate as a function of thermal source temperature at different convection heat transfer coefficients with constant thermal sink temperature, TC 300 K.5 Temperature distribution in TEG with adiabatic boundary conditions at vertical walls of semiconductor legs.6 Electrical potential in TEG with adiabatic boundary conditions at vertical walls of semiconductor legs.7 Temperature distribution in TEG with convective boundary conditions, h = 0 Wm - K - at vertical walls semiconductor legs.8 Electrical potential in TEG with convective boundary conditions, h = 0 Wm - K - at vertical walls of semiconductor legs.9 Comparison of heat input, power output, and thermal efficiency obtained from the current work with the similar results available in (Angrist 98) C.0 Comparison of analytical and numerical results in terms of temperature 5 x

11 distribution over the p-type semiconductor leg 3. Schematic of unit cell of TEG Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h 0 W/m K) 3.3 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h 0 W/m K) 3.4 Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h = 5 W/m K) 3.5 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h = 5 W/m K) 3.6 Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h = 35 W/m K) 3.7 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h = 35 W/m K) 3.8 Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h = 50 W/m K) 3.9 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h = 50 W/m K) 3.0 Comparison of current production using numerical and analytical techniques 3. Thermal efficiency of TEG as a function of convection heat transfer coefficient and temperature difference 4. Schematic diagram of unit cell of TEC (drawing is not to scale) 8 4. The schematic of crystal structure of (a) (Bi -x Sb x ) Te 3 (Zhang et al. 0) Reprinted by permission from Macmillan Publishers Ltd: Nature Communications from Zhang et al., 574 (0), copyright The schematic of crystal structure of Bi Te 3 (Chen et al. 009) From [Chen et al. Science 35, 78 (009)]. Reprinted with permission from AAAS 4.4 Temperature distribution over the length of p-type TE leg with hot 89 xi

12 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 4.6 Electrical resistivity of p- and n- type legs of nanocomposite TEC Heat absorbed as a function of current considering hot surface temperature 353 K with cold surface temperature 333 K 4.8 Heat absorbed as a function of current considering hot surface temperature 353 K with cold surface temperature 343 K 4.9 COP of TEC as a function of current considering hot surface temperature 353 K with cold surface temperature 333 K 4.0 COP of TEC as a function of current considering hot surface temperature 353 K with cold surface temperature 343 K 4. Heat absorbed as a function of temperature difference with different electric current input and hot surface temperature 353 K considering adiabatic side wall condition 4. COP as a function of temperature difference with different electric current input and hot surface temperature 353 K considering adiabatic side wall condition 4.3 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 4.4 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 4.5 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 4.6 Optimum electric current for maximum COP of TEC considering variable convection heat transfer coefficient and variable TE leg xii

13 heights with hot surface temperature 353 K and cold surface temperature 333 K 4.7 Internal resistance of TEC unit cell as a function of TE leg height 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 4.9 Electric scalar potential and current flow in nanocomposite TEC with electric potential 0.0 V 4.0 Electric scalar potential and current flow in nanocomposite TEC with electric potential 0.06 V 4. Heat flow and temperature distribution in nanocomposite TEC for h 0 Wm - K - at vertical walls with electric potential 0.0 V 4. Heat flow and temperature distribution in nanocomposite TEC for h = 0 Wm - K - at vertical walls with electric potential 0.0 V 4.3 Heat flow and temperature distribution in nanocomposite TEC for h = 40 Wm - K - at vertical walls with electric potential 0.0 V 4.4 Heat flow and temperature distribution in nanocomposite TEC for h = 60 Wm - K - at vertical walls with electric potential 0.0 V 4.5 Heat flow and temperature distribution in nanocomposite TEC for h 0 Wm - K - at vertical walls with electric potential 0.06 V 4.6 Heat flow and temperature distribution in nanocomposite TEC for h = 60 Wm - K - at vertical walls with electric potential 0.06 V 4.7 Heat absorbed by nanocomposite TEC as a function of convection heat transfer coefficient with hot surface temperature 353 K and electric potential 0.0 V 4.8 COP of nanocomposite TEC as a function of convection heat transfer coefficient with hot surface temperature 353 K and electric potential 0.0 V 4.9 Heat absorbed by nanocomposite TEC as a function of convection heat transfer coefficient with hot surface temperature 353 K and electric xiii

14 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 4.3 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 0.0 V 4.3 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 0.0 V 4.33 Comparison of COP using conventional (no nanostructuring) and nanocomposite TE material considering h 0 Wm - K -, hot surface temperature 353 K, and electric current input of A 4.34 Thermal conductivity of conventional (no nanostructuring) and nanocomposite TE materials xiv Comparison of results between current work and Poudel et al. (008) Different approaches to increase ZT of TE materials (Martin-Gonzalez 03) 5. Graphical representation of (a) Maxwell model (b) Hasselman and Johnson model (c) Minnich and Chen model Schematic diagram of typical (a) TEC and (b) TEG system Effective thermal conductivity of p-type and n-type thermoelectric material based on Maxwell model 5.5 Effective thermal conductivity of p-type and n-type thermoelectric material based on Hasselman-Johnson model 5.6 Effect of thermal boundary conductance on effective thermal conductivity of p-type using Hasselman-Johnson model Effect of thermal boundary conductance on effective thermal 55

15 conductivity of n-type using Hasselman-Johnson model 5.8 Effective thermal conductivity of p-type thermoelectric material using Minnich-Chen model 5.9 Effective thermal conductivity of n-type thermoelectric material using Minnich-Chen model 5.0 COP of TEC considering various amount of volume fraction with Maxwell model xv Efficiency of TEG with different volume fraction with Maxwell model 6 5. COP of TEC considering different amount of volume fraction with Hasselman model 5.3 Efficiency of TEG with different amount of volume fraction with Hasselman model 5.4 Effect of boundary conductance on performance of TEC based on Hasselman-Johnson model 5.5 Effect of boundary conductance on performance of TEG based on Hasselman-Johnson model 5.6 COP of TEC considering different amount of volume fraction with Minnich-Chen model 5.7 Efficiency of TEG with different volume fraction with Minnich-Chen model 5.8 Effect of nanoparticle size on performance of TEC considering Minnich-Chen model 5.9 Effect of nanoparticle size on performance of TEG considering Minnich-Chen model 5.0 Performance of TEC with variable volume fractions and convection heat transfer coefficients through side walls of TE legs 5. Performance of TEG with variable volume fractions and convection heat transfer coefficients through side walls of TE legs Influence of effective thermal conductivity on heat conduction in TEC Influence of effective thermal conductivity on heat conduction in TEG 78

16 5.4 Contours of electric potential and streamlines of electric current flow in TEC 5.5 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with NO particles 5.6 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with 0.8 volume fraction with Maxwell model 5.7 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with 0.8 volume fraction with Hasselman-Johnson model 5.8 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with 0.8 volume fraction with Minnich-Chen model 5.9 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 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 5.3 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 5.3 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 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 xvi

17 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 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 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 5.37 Comparison of analytical and numerical simulation results for TEC Comparison of analytical and numerical simulation results for TEG Schematic diagram of (a) photovoltaic thermoelectric (PVTE) system and (b) unit thermoelectric generator 6. Exploded view of Solar PV panel layers (Amrani 007) Solar PV panel back surface temperature with variable solar radiation and ambient temperature 6.4 Temperature distribution over the length of nanostructured p type and n type semiconductor leg 6.5 Heat input to nanostructured TE generator with different solar radiation and variable convection heat transfer coefficient 6.6 Heat input comparison of TE generator using traditional and nanostructured material thermoelectric material 6.7 Thermal conductivity of traditional and nanostructured TE material as a function of temperature 6.8 Power output from TE generator as a function of solar radiation Thermal efficiency of nanostructured TE generator with different solar radiation and variable convection heat transfer coefficient 6.0 Thermal efficiency comparison of TE generator with traditional and nanostructured TE material xvii

18 6. Power output comparison of solar PV panel and TE generator 3 6. Solar panel conversion efficiency Vs. Solar Radiation Combined efficiency of solar PVTE system Vs. Solar Radiation ZT improvements in low dimensional and bulk TE materials Rietveld refinement power XRD pattern for Bi Te.7 Se Seebeck coefficient and Electrical resistivity of sample Bi Te.7 Se Power factor of sample Bi Te.7 Se Comparison of power factor between Bi Te.7 Se 0.3 manufactured via direct current hot press and indirect resistance heating 7.6 SEM image of fractured surfaces of hot pressed sample xviii

19 Table number List of Tables Title xix Page number. Comparison of various waste heat recovery methods (BCS 008). Contribution of present study. Temperature dependent TE properties of n-type 75% Bi Te 3 5% Bi Se 3 and p-type 5% Bi Te 3 75% Sb Te 3 with.75% excess Se (Reddy et al. 03, Angrist 98). Thermal efficiency of single unit of TEG with cold surface temperature, T C = 300 K 3. Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal conductivity as a function of temperature for BiSbTe nanostructured bulk alloys and Bi Te 3 with SiC nanoparticles (Poudel et al. 008, Zhao et al. 008) 4. Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal conductivity as a function of temperature for BiSbTe nanostructured bulk alloys and nanocomposite Bi Te 3 (Poudel et al. 008, Fan et al. 0) 4. Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal conductivity as a function of temperature for BiSbTe bulk alloys and conventional Bi Te 3 (Poudel et al. 008, Fan et al. 0) 4.3 Height of TE legs and internal resistance of TEC for different cases considered in Figs. 4.3, 4.4, 4.5, and Comparison of different effective medium theories Material properties (Pattamatta and Madnia 009) Operating conditions and dimensional parameters of combined solar PVTE system 6. 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. 008)

20 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. 008) 6 xx

21 CHAPTER : INTRODUCTION. 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. 00). 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 008). 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. 006; Bottner et al. 006). The U.S. Department of Energy (DOE 0) 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 008). Table. 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, isopentane, toluene, and ammonia (BCS 008). 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

22 comparison with mechanical systems, direct energy conversion methods are simple in structure and require less maintenance. Table. Comparison of various waste heat recovery methods (BCS 008) Use of waste heat to improve thermodynamic properties of input load Waste heat to electricity Conversion Technology Electricity generation through mechanical systems Direct Electricity Generation Temperature Range Advantages Air Preheater Economizer C C Widely used in industry. Increases overall efficiency ORC >66 C Kalina cycle > 00 C Thermoelectric generators (TEG) Piezoelectric Generators Thermionic Generators Thermal photovoltaic generators 50 C to 00 C 00 C to 50 C Suitable as a waste heat recovery tool in power plant Uses ammonia, can increase power output by 0-50% No moving parts, green and clean technology Converts vibrations to electricity > 000 C - 00 C to 500 C No moving parts Disadvantages Unable to recover all waste heat; can only change state of input load Low efficiency and uses hazardous chemicals Use of toxic fluid and storage challenge Low efficiency Very low efficiency and high cost Very high temperature required Low efficiency, requires vacuum

23 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... Heat Absorbed + Metal connectors Substrates n- type semiconductor Heat Rejected p- type semiconductor - Figure. 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 0). 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 3

24 chemical potential of the two materials (Terasaki 0). TEC also has a wide range of potential applications in electronics, laser diodes, military garments, laboratory cold plates, and transportation systems. Figure. 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 004). 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. 03). Power 0 kw kw 00 W Waste heat recovery; (e.g., powerstation, automotive, redundant oil wells and nuclear facility) Consumer refrigerators, electronic cooling 0 W W Deep space vehicles power and cooling 00 mw 0 mw Remote wireless sensors mw 00 µw Biomedical devices; pacemakers 0 µw µw Low power applications; wrist-watches Figure. Potential applications of TE systems (Pichanusakorn and Bandaru 00) 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 4

25 energy conversion systems can increase the entropy and lead to the destruction of the exergy (Bermejo 03). 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. 03). One of the ways to improve the TE properties is to lower the thermal conductivity (Bhandari 995). A lowdimensional (e.g., 0D, D, D) 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 lowdimensional 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.. 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 D analytical model and a D 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. 5

26 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 Bi Te.7 Se 0.3 based on solid state synthesis technique 6

27 o Fabricating Bismuth-Telluride based TE legs using the indirect resistance heating method o Measuring Seebeck coefficient and electrical resistivity of nanocomposite TE leg.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 : 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 (-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 (-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 7

28 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 (Bi Te 3 ) with 0. 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 8

29 electrical potential distributions and to calculate heat absorbed and the COP. Heat absorbed and COP are also calculated using a simplified D 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 3% ± % 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.06% to 5.59%, which is 70% rise when thermal conductivity of composite decreases from. Wm - K - to 0. Wm - K - based on the Minnich Chen model with a particle size of 00 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. 9

30 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 Bi Te.7 Se 0.3 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 (α ρ - ) 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. 0

31 .4 Contribution of present study Table. Contribution of present thesis Nanocomposite TE systems Research topic Research gap Contribution Heat Transfer 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 TE material properties 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) 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

32 .5 Publications from present study Parts of this thesis have been published in peer-reviewed international journals. Chapters to 5 have been published and chapter 6 is currently under review.. R. Rabari, S. Mahmud, A. Dutta, M. Biglarbegian, 05, Effect of convection heat transfer on performance of waste heat thermoelectric generator, Heat Transfer Engineering, 36, R. Rabari, S. Mahmud, A. Dutta, 04, Numerical simulation of nanostructured thermoelectric generator considering surface to surrounding convection, International Communications in Heat and Mass Transfer, 56, R. Rabari, S. Mahmud, A. Dutta, M. Biglarbegian, 05, Analytical and numerical studies of heat transfer in nanocomposite thermoelectric cooler, Journal of Electronic Materials, 44, R. Rabari, S. Mahmud, A. Dutta, 05, Effect of thermal conductivity on performance of thermoelectric systems based on effective medium theory, International Journal of Heat and Mass Transfer, 9, 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 ).

33 CHAPTER : EFFECT OF CONVECTION HEAT TRANSFER ON PERFORMANCE OF WASTE HEAT THERMOELECTRIC GENERATOR. Introduction Fossil fuel resources are very limited, and consumption of fossil fuel increases day by day. In Canada 7% of electricity demand is satisfied by thermal power stations using fossil fuel such as coal (NEB 0). A typical thermal power station has efficiency of around 40-45% and rejects about 50-60% of the input energy (White 99). Figure. shows different methods to recover the waste heat from power plant/ industrial facilities. 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 Piezoelectricity Thermal Photovoltaic Figure. Various waste heat recovery methods in context of power plant (Stehlik 007, Rowe 995) Yilmaz and Buyukalaca (003) presented a mathematical model and numerical simulations for the design of rotary regenerators used for energy recovery from various industrial and airconditioning applications. Budzianowski (0) and Stehlik (007) 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 007). 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 995). TE modules are made up of a number of p-type and n-type 3

34 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 995). TE modules have many advantages, such as being environmentally friendly and quiet in operation, and having a long service life (Rowe 995). However, low conversion efficiency of TE materials creates difficulty in wide usage (Rowe 995). The TE effect was first observed by Seebeck (Wang et al. 009); when two dissimilar materials are joined together and junctions are held at two different temperatures then electromotive force is produced (Wang et al. 009). The electromotive force depends on an intrinsic property of materials known as the Seebeck coefficient ( α = V ΔT). 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 π = qp I (Wang et al. 009). The interdependency of both effects =T d dt and T was derived by Thomson (Lord Kelvin) (Goupil 0). The Peltier heat is liberated or absorbed at a junction and is given by q I T. P During the last several years, various high-performance TE materials have been developed (Udomsakdigool 007). To improve the performance of TEG requires making a reasonably good thermal design, as well as arrangement of TE modules. In early 0 th century, Altenkirch (9) established that a good TE material should have large Seebeck coefficient, low thermal conductivity, and high electrical conductivity. Callen (947) presented the thermodynamic theory of the TE phenomenon. Joffe and Stil bans (959) introduced a figure of merit as a parameter to classify different TE materials. Bell (00) discussed the effect of convective heat transfer medium on the performance of the TE module and demonstrated criteria for optimum performance. Xiao et al. (0) 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. (0) 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. (03), considering adiabatic and convective side wall conditions. Regardless of design, increase in hot-side temperature enhances the performance of TEG. Reddy et al. (03) also concluded that the 4

35 composite and integrated TEG extracts more heat compared to the conventional TEG and reduces rare-element material usage. Chen et al. (00) 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. (0) demonstrated the effect of radiative heat transfer on the performance of the TEG. A temperature-dependant thermodynamic model was developed by Meng et al. (0), considering external irreversibility. Meng et al. (0) 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 (03) considered different optimization parameters to achieve maximum power output and maximum thermal efficiency. Sahin and Yilbas (03) concluded that an increase in thermal conductivity decreases the efficiency and power output and increases entropy generation rate in TEG. Riffat and Ma (003) performed a review of current and potential applications of TE modules. Riffat and Ma (003) 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. (980) in 980. At that time (Goldsmid et al. 980), the TEG was made up of Bi Te 3 and had efficiency of %, and could be increased to 3% if proper concentration system was used. Considering the automotive waste heat recovery from exhaust pipe, Hsiao et al. (00) developed -D thermal resistance model of TE module. The authors (Hsiao et al. 00) verified modeling data with experimental data and calculated maximum power generation of 0.43 W at 90 ºC temperature difference. A TEG system was installed with carburizing furnace at Komatsu plant to recover the waste heat (Kaibe et al. 0). A TEG containing 6 TE modules made-up of bismuth-telluride in groups of 4 was used. Experiments reported total power generation of 50 W with hot surface of temperature of 50 ºC (Kaibe et al. 0). 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 -D form 5

36 using appropriate approximations that include -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 -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 -D analytical model to observe the qualitative results of thermal and electric fields inside the TEG in two dimensions.. Heat Transfer Modeling A schematic diagram of the proposed TE heat recovery system is shown in Figure.. 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. 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 the electric scalar potential (Landau et al. 984). 6 q q gen () where q and q gen represent heat generation rate per unit volume, and heat flux vector, respectively. The continuity of electric charge through the TEG must satisfy J 0 () where J is the electric current density vector. Equations () and () are coupled by the set of TE constitutive equations (Antonova et al. 005) as shown in Eqs. (3) and (4), q T J k T J E T 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 (3) (4), where is

37 Water /Steam Pipes Q H T H T q conv x TEGs p I n l T w Cross-section of combustion system Q C T C R l Figure. Schematic diagram of location of TEG in combustion system and unit TEG cell Combining Eqs. () to (4), the coupled TE equations for energy and electric charge transfers can be expressed as where Jk T EJ T (5) 0 T (6) E J represents Ohmic heat (Antonova and Looman 005). Finally, the entropy generation equation for TEG can be expressed as (Chakraborty 006) EJ k S gen T J (7) T T 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. b the thermal boundary conditions are as follows: At the top surface (i.e., x 0 ) temperature is T, which is the hot junction temperature At the bottom surface (i.e., x l ) temperature is T, which is the cold junction temperature. Convection heat transfer per unit area from the side surfaces to the surrounding is q conv h( T T ). a 7

38 It is assumed that the thermal energy enters into the top surface from a thermal source having constant temperature ( T H ) and leaves the bottom surface to a thermal sink having constant temperature ( T C ). 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 -D version of the preceding equations is solved to obtain close forms of analytical solutions. For -D analytical heat transfer modeling, a TEG with p-type and n-type semiconductor legs with load resistance R l (see Fig. b) is considered. TE elements having length l and width w operates between hot and cold junction temperature T and T, respectively. The hot and cold junction temperatures, T and T, depend on the convection rates from the surfaces and temperatures of the source ( T H ) and sink ( T C ). A TEG absorbs q amount of heat from the thermal source and rejects q 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 d T dx I k A h p I dt ( T Ta ) 0 (8) k A k A dx 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) where x D x D x C e C e (9) T 4 D ; D 8 4 (0)

39 9 ; A k I ; A k h p A k I T A k h p a () To calculate C and C 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 ( ) 0 T T h dx dt k H H x ( C ) C l x T T h dx dt k () (3) Substitution of Eqs. (9), (0), and () into Eqs. () and (3) results in ) ( ) ( ) ( ) ( ) ( ) ( ) ( D D e k e k h h e e h e D k e D k h e D k e D k h h e e T T h D k T D k h e D k T e D k C D l D l H C D l D l C D l D l H D l D l H C D l D l H C C C H D l H D l ) ( ) ( ) ( ) ( ) ( ) ( ) ( D D e k e k h h e e h e D k e D k h e D k e D k h h e e T T h D k T D k h e D k T e D k C D l D l H C D l D l C D l D l H D l D l H C D l D l H C C C H D l H D l (4) (5) 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

40 0. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( D D D e k e k h h e e h e D k e D k h e D k e D k h h e e T T h D k T D k h e D k T e D k D D D e k e k h h e e h e D k e D k h e D k e D k h h e e T T h D k T D k h e D k T e D k A k D D e k e k h h e e h e D k e D k h e D k e D k h h e e T T h D k T D k h e D k T e D k D D e k e k h h e e h e D k e D k h e D k e D k h h e e T T h D k T D k h e D k T e D k I q D l D l H C D l D l C D l D l H D l D l H C D l D l H C C C H D l H D l D l D l H C D l D l C D l D l H D l D l H C D l D l H C C C H D l H D l D l D l H C D l D l C D l D l H D l D l H C D l D l H C C C H D l H D l D l D l H C D l D l C D l D l H D l D l H C D l D l H C C C H D l H D l (6) Equation (6) 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 (6) reveals that the hot junction temperature ) ( T depends on the thermal source temperature ) ( H T and the convection heat transfer coefficient ( H ) h between the thermal source and the hot junction of TEG. In similar manner, the cold junction temperature ) ( T depends on the thermal sink temperature ) ( C T and the convection heat transfer coefficient ) ( C h 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 T and T approach the source temperature H T and sink temperature C T. The net power output of a single TEG is calculated as (Doolittle and Hale, 984), V I P o (7)

41 where V ) ( p n)( T T I Ri (8) Electric current can be calculated by I ( p n)( T T R R The thermal conversion efficiency can be evaluated as i l ) (9) P o q (0) Thermal efficiency is independent of the number of couples, as power output and thermal input increases linearly with number of modules. In Eqs. (8) and (9), the temperature at the hot junction ( T ) and the temperature at the cold junction ( T ) can be evaluated using the boundary conditions ( x 0, T T ) and ( x l, T T ) in Eq. (9)..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 5% Bi Te 3 75% Sb Te 3 with.75% excess Se and n-type material 75% Bi Te 3 5% Bi Se 3 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 T 700 K), Thermal sink temperature (60 K T 30 K), and H surface to surrounding heat transfer coefficient (0 W/m K C h 00 W/m K). The dimensions of the TEG are as follows: length 0.03 m, width 0.0 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 (Reddy et al. 03, Angrist 98). These properties are evaluated at average temperature of working range. Load resistance same as internal resistance R i to get maximum power output. R l is

42 Table. Temperature dependent TE properties of n-type 75% Bi Te 3 5% Bi Se 3 and p-type 5% Bi Te 3 75% Sb Te 3 with.75% excess Se (Reddy et al. 03, Angrist 98) Property Temperature Range (ºC) Polynomial functions of different TE properties in terms of temperature 5 T 550 n p avg 0 T 70 avg T.60 3 avg 3 avg 7 T avg T 7 T.890 avg T 4 avg 4 avg T T avg T avg T T 5 avg 5 avg 70 T avg T 3 avg 6 T avg T avg 5 T 550 n p avg 0 T 450 n avg k 5 T 00 avg 00 T avg T avg 550 k 0 T 70 p avg 70 T avg T avg T T 6 avg 3 avg 3 avg T T avg T avg T 4 avg 4 avg T T avg T avg T T 5 avg 5 avg Tavg Tavg Tavg T 4 3 avg T avg T T T 3 avg 3 3 avg avg T T avg avg 7 5 T T T 4 avg T 5 4 avg T avg avg.530 avg T 0 5 avg Tavg.450 Tavg Tavg T 5 avg Tavg Tavg Tavg

43 It is assumed that the thermal energy enters into the top surface of TEG from a thermal source ( T TH ) by convection with convection heat transfer coefficient h H and leaves the bottom surface of TEG to a thermal sink T T ) also by convection with convection heat transfer ( C coefficient h C. In the special case of very large convection heat transfer coefficients, and h C, the top and bottom surface temperatures, T and T, approach h H T H and T C (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.3 and.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. W/m K). As shown later, convection heat losses affect the heat input to the TEG and thermal efficiency of the TEG significantly. 3

44 T (K) h = 0. Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - h = 00 Wm - K x (m) Figure.3 Temperature distribution over the length of p-type semiconductor leg with thermal source temperature, T 700 K and thermal sink temperature, T 300 K H C 4

45 T (K) h = 0. Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - h = 00 Wm - K x (m) Figure.4 Temperature distribution over the length of n-type semiconductor leg with thermal source temperature, T 700 K and thermal sink temperature, T 300 K H C 5

46 Heat Input Heat input to the TEG can be analysed from Figs..5 and.6. Heat input to the TEG is plotted as a function of the source temperature ( T H ) in Fig..5 at different values of the convection heat transfer coefficient for a constant sink temperature (300 K). Figure.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..6, where heat input to the TEG is plotted as a function of the sink temperature ( T C ) 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. 6

47 q (W) h = 0. Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - h = 00 Wm - K T H (K) Figure.5 Effect of thermal source temperature on heat input with variable convection heat transfer coefficient at constant thermal sink temperature, TC 300 K 7

48 q (W) h = 0. Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - h = 00 Wm - K T C (K) Figure.6 Effect of thermal sink temperature on heat input with variable convection heat transfer coefficient at constant thermal source temperature, TH 700 K 8

49 Power Output Figure.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. (7) and (8) 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. 9

50 P o (W).5 T C =70 K T C =80 K T C =90 K T C =300 K T H (K) Figure.7 Power generation as a function of thermal source temperature at different thermal sink temperature 30

51 Thermal Efficiency Thermal efficiency is plotted as a function of hot surface temperature with constant cold surface temperature in Fig..8 at different values of convection heat transfer coefficient. The thermal efficiency plot, as shown in Fig..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..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.8 and.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. 3

52 0 8 h = 0. Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - h = 00 Wm - K T H (K) Figure.8 Effect of thermal source temperature on thermal efficiency with variable convection heat transfer coefficient at constant thermal sink temperature, TC 300 K 3

53 0 8 h = 0. Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - h = 00 Wm - K T C (K) Figure.9 Effect of thermal sink temperature on thermal efficiency with variable convection heat transfer coefficient at constant thermal source temperature, TH 700 K 33

54 I (A) Output Current Figure 0 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 (9) shows that electric current has a linear relation with temperature difference, and plots reflect the same phenomenon. 5 0 T C =70 K T C =80 K T C =90 K T C =300 K T H (K) Figure.0 Current as a function of thermal source temperature with different thermal sink temperature 34

55 Irreversible Convection Heat Transfer The effect of the surface to surrounding convection heat transfer coefficient on thermal efficiency is demonstrated in Fig... 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..5); however, the power output remains nearly invariant (Eqs. (7) and (8)) 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.. and.3 for the adiabatic side wall condition (Fig..) and convection from the side walls with h = 0 W/m K (Fig..3). 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.. ( h H h C 4 0 W/m K). Both heat input and power output decrease with decreasing h H and h C (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

56 0 8 T H = 400 K T H = 500 K T H = 600 K T H = 700 K h (Wm - K - ) Figure. Effect of convection heat transfer coefficient on thermal efficiency at constant thermal sink temperature, TC 300 K 36

57 h C = h H = 50 Wm - K - h C = h H = 00 Wm - K - h C = h H = 0 3 Wm - K - h C = h H = 0 4 Wm - K I (A) Figure. 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 T 700 K and T 300 K with adiabatic side wall condition H C 37

58 h C = h H = 50 Wm - K - h C = h H = 00 Wm - K - h C = h H = 0 3 Wm - K - h C = h H = 0 4 Wm - K I (A) Figure.3 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 T 700 K and T 300 K with convection from the side walls with h=0 Wm - K - H C 38

59 Irreversibility Analysis Entropy is produced by the irreversible processes in TE devices (Yilbas and Pakdemirli 005, Sekulica 986), 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. 03). 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 -D entropy generation rate equation for the present problem as shown here: S I TA k p, n gen ( T) () T 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. (), assuming a homogeneous material. Equation () is the volumetric local entropy generation rate (W/m 3 K), 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 ( S gen ) can be obtained from the following equation: I k p n dt, Sgen dv V () A T T dx For the constant cross-sectional area of TE legs, Eq. () can be further simplified to Above equation can be written in the dimensionless form, L I k p n dt Sgen dx TA, T l. (3) dx 0 S gen l k p, n I k p, n TA T l L 0 T dx (4) 39

60 wher S S gen (5) k gen p, n l In Eq. (4), the temperature gradient contained in the second term on the right side can be evaluated by using Eqs. (9), (4), and (5). Finally average entropy generation over the entire volume of a TEG leg can be evaluated in non-dimensional form as shown here: C D C DD 4CC DD C DD D l Dl C D C D e C DD e D l Dl Dl Dl l I k p, n 4CC DD e C De D C D e S gen (6) k p, n TA T l D D The simplified expressions for C and C are defined already in Eqs. (4) and (5). Similarly, simplified expressions for D and D are defined in Eq. (0). The volume averaged dimensionless entropy generation rate, as presented in Eq. (6), is plotted as a function of the source temperature for different values of the convection heat transfer coefficient in Fig..4. 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 ( T 0) heat transfer irreversibility is zero. An increase in the source temperature increases the entropy generation rate as observed from Fig..4. 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..4 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 40

61 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. 4

62 5 0 h = 0. Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - h = 00 Wm - K - S * gen T H (K) Figure.4 Entropy generation rate as a function of thermal source temperature at different convection heat transfer coefficients with constant thermal sink temperature, TC 300 K 4

63 -D Numerical Results Equations (5) and (6) are the modeled differential equations for the coupled thermal and electric fields. Until this point, a simplified -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 -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 (006). Properties of p- and n-type semiconductor legs were approximated at the average temperature of working range using relations given in Table. 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 -D analysis. For adiabatic side walls of p and n-type semiconductor legs, Figures (a) and (b) present the thermal and electric potential field results. Figure (a) presents the temperature and heat flow results while Figure (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 = 0 W/m K and presented graphically in Figs..7 and.8. 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 h 0 W/m K. 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 nonlinearity in the temperature distribution, as observed from Fig..7. 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 43

64 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/m K. However, nonlinear potential distribution is observed when h = 0 W/m K. 44

65 Figure.5 Temperature distribution in TEG with adiabatic boundary conditions at vertical walls of semiconductor legs 45

66 Figure.6 Electrical potential in TEG with adiabatic boundary conditions at vertical walls of semiconductor legs 46

67 Figure.7 Temperature distribution in TEG with convective boundary conditions, h = 0 Wm - K - at vertical walls semiconductor legs 47

68 Figure.8 Electrical potential in TEG with convective boundary conditions, h = 0 Wm-K- at vertical walls of semiconductor legs 48

69 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. (03) performed a theoretical performance study of conventional, composite, and integrated TE devices applicable to waste heat recovery system. Performance results in Reddy et al. (03) are presented for h = W/m K 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. 03), thermal efficiencies are calculated for five different hot to cold surface temperature differences and presented in Table.. Results presented in Table. 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. (03). Note that the numerical values available in the third column in Table. are extracted manually from Figure 4 of Reddy et al. (03) which is for h = 0 W/m K with hot surface temperature K and cold surface temperature 300 K. An additional comparison is presented in Fig..9, where heat input, power output, and thermal efficiency results are obtained from section of Angrist (98) for an optimized TEG and compared with the results obtained from equations derived in the current work. Angrist (98) used an adiabatic side boundary condition. Figure.9 shows good agreement between current results and results obtained by Angrist (98). Figure.0 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. 49

70 Table. 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. 03) %.8% % 4.3% % 6.9% % 7.9% % 9.5% 50

71 q and P o (W) 5 Efficiency (Angrist 98) Efficiency (current work) Heat input (Angrist 98) Power output (Angrist 98) Heat input (current work) Power output (current work) T ( o C) Figure.9 Comparison of heat input, power output, and thermal efficiency obtained from the current work with the similar results available in (Angrist 98) 5

72 T (K) h = 0. Wm - K - (numerical) h = 5 Wm - K - (numerical) h = 50 Wm - K - (numerical) h = 0. Wm - K - (analytical) h = 5 Wm - K - (analytical) h = 50 Wm - K - (analytical) x (m) Figure.0 Comparison of analytical and numerical results in terms of temperature distribution over the p-type semiconductor leg 5

73 .4 Conclusion This research has developed -D analytical and numerical -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 -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 -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 -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 -D analytical result. A waste heat TEG needs to be designed carefully, considering the effect of internal and external irreversible convection losses. 53

74 .5 Nomenclature A cross-sectional area (m ) C a parameter (see Eq.(4)) C D a parameter (see Eq.(5)) a parameter (see Eq.(0)) D a parameter (see Eq.(0)) E electric field intensity vector (Vm - ) h convection heat transfer coefficient (Wm - K - ) I electric current (A) J electric current density (Am - ) k thermal conductivity (Wm - K - ) l length of p type and n type semiconductor material respectively (m) p P q perimeter (m) electric power output (W) heat or energy for TEG (W) q heat flux (Wm - ) 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 - ) S volume averaged entropy (WK - ) T temperature (K) V voltage (V), volume (m 3 ) w width (m) x coordinate (m) Greek Symbols Seebeck coefficient (VK - ) a parameter (see Eq. ()) a parameter (see Eq.())) 54

75 efficiency (%) Peltier coefficient (V) electrical resistivity (Ωm) electrical conductivity (Sm - ) Thomson coefficient (VK - ) a parameter (see Eq.()) electric scalar potential (V) Subscripts hot junction of TEG 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 55

76 CHAPTER 3: NUMERICAL SIMULATION OF NANOSTRUCTURED THERMOELECTRIC GENERATOR CONSIDERING SURFACE TO SURROUNDING CONVECTION 3. 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 009). Huang et al. (00) 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. 0), 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. (0) concluded that transient cooling with square root current pulse is most effective with 0 C cooling. Wang (03) has proposed and investigated experimentally the TEG using waste heat of the Light-Emitting Diodes. Results reported by Wang (03) investigated power output of TEG using waste heat of the Light- Emitting Diodes (LED). Experiments of Wang (03) showed 60 mw of power output from TEG with 6 W of input power to LED. Recently, Hsiao et al. (00) studied the performance of TE modules as a waste heat recovery tool from an automobile engine using a one-dimensional 56

77 thermal resistance model and compared their model with experimental data. Results (Hsiao et al. 00) 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. (0) have studied the effect of cooling power on the performance of a TEG. Rezania et al. (0) determined the optimum coolant flow rate for maximum power output for the TEG. For example, temperature difference of 0 K gives maximum power output of W with coolant flow rate of 0.07 l/min (Rezania et al. 0). One-dimensional analytical solutions of conventional, composite, and integrated TEG have been carried out by Reddy et al. (03) 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.03). Reddy et al. (03) concluded that composites and integrated TEG extracts more heat compared to conventional TEG and reduces rare-element material usage. Gou et al. (00) investigated the performance of low temperature waste heat TEG using one-dimensional analytical simulations and experiments. They (Gou et al. 00) 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. (0) and their numerical simulation performed with ANSYS shows fairly good match with experimental results. In addition to this, Chen et al. (0) 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. (0) to check the effects of Reynolds number and fluid temperature on performance of an integrated TEG system. Reddy et al. (0) found that higher Reynolds number enhances the heat transfer and thus leads to higher power output of TEG. Zhou et al. (03) 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. (03) reported overall TEG efficiency of 3.5% with temperature difference of 80 K. Baranowski et al. (0) have developed mathematical model for solar TEG which can provide analytical solutions of device efficiency with temperature dependant properties. They (Baranowski et al. 0) have also showed that considering currently available materials, total efficiency of 4.% is possible 57

78 with cold and hot side temperature of 00 C and 000 C, respectively. Baranowski et al. (0) also concluded that if figure of merit (ZT) reaches to then TEG efficiency of 5% can be obtained, for the cold and hot side temperature of 00 C and 000 C, respectively. Altenkirch (9) 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, ZT ( ) T, 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 at room temperature (Vineis et al. 00). Slack (995) has described that good TE materials need to have low thermal conductivity. In addition to this, Slack (995) 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. 00) have opened the door for further improvements of ZT for the TE materials. The expression of the figure of merit, ZT ( )T, 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 00). The electronic part of thermal conductivity ( E ) is related to the electrical conductivity as par the Wiedemann-Franz law (Tritt 00) as shown in Eq. (), E L o T () where, L o is the Lorentz number and for metals it is equal to (Tritt 00), k 3 e where, k is the Boltzmann constant ( L o.450 V K - () 3 58 JK - ) (Tritt 00) and e is the electron charge 9 (.60 0 C ) (Godart et al. 009). The expression of the figure of merit can be written in terms of Lattice conductivity ( L ) and electronic conductivity ( E ) as shown below (Tritt 00), ZT L o E E L. (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. 03). Poudel et al. (008) achieved ZT value

79 of.4 at 373 K by hot-pressing of nanopowders of Bi Te 3 and Sb Te 3 under Argon (Ar) atmosphere. The enhanced ZT was due to the significant decrease in the lattice thermal conductivity of material. Zhao et al. (008) fabricated Bi Te 3 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.04 with an inclusion of 0.% vol% SiC particles. Li et al. (009) obtained ZT of.43 for double-doped Co 4Sb 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. (0) 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 000 W/m condition. McEnaney et al. (0) developed a novel of TEG. They (McEnaney et al. 0) placed high performance nanostructured material in evacuated tube with selective absorber and achieved an efficiency of 5.%. 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 D 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. 59

80 3. Mathematical model and boundary conditions The two-dimensional schematic diagram of the TEG being investigated is shown in Fig. 3.. 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 L g as shown in Fig. 3. below. The Q H and Q C are amount of heat available at heat source and heat sink respectively. The Q conv 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. Q H T H Q conv p I n H y x L L g T C Q C R L Figure 3. Schematic of unit cell of TEG 60

81 In the TEG, the energy transport and current flow are governed by energy equation and continuity of current density as per below, where symbols, C p, T, q, and T C q q () gen t q gen 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, D J 0 () t where J is the electric current density vector and D is the electric flux density vector, respectively. Equation () and Eq. () are coupled by the set of TE constitutive equations (Perez- Aparicio 0) as shown in Eq. (3) and Eq. (4) below, q T[ ] J [ ] T (3) J [ ] ( E [ ] T ) (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 984). Rearrangement of Eq. () to Eq. (4) gives the coupled TE equations for heat transfer and electric potential as, and T C T [ ] J [ ] T E J (5) t [ ] t [ ] [ ] T [ ] 0 where [ ] is the dielectric permittivity matrix (Landau 984) and E J represents Joule heat (Perez-Aparicio 0). 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. are as follows; (6) 6

82 The top surface of the TEG experiences constant hot temperature (T H ) The bottom surface experiences constant cold temperature (T C ) 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. 008). While for n-type material, Bismuth Telluride (Bi Te 3 ) with nano-particles of silicon carbide (SiC) is considered to analyze the performance (Zhao et al. 008). The temperature dependent transport parameters Seebeck coefficient (), electrical conductivity (), and thermal conductivity () are specified as polynomial functions of temperature as shown in Table 3.. These properties are evaluated at an average temperature of working temperature range. The semiconductor leg of unit cell of TEG has dimensions of.5 mm. 5 mm and height of 5 mm. The gap between two consecutive legs is 0.3 mm. The terminals of both p-type and n-type semiconductor legs are connected with external load (R L ) which is matched with total internal resistance of the TEG. Figures 3., 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 multicolored 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=5-50 W/m K). 6

83 Table 3. Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal conductivity as a function of temperature for BiSbTe nanostructured bulk alloys and Bi Te 3 with SiC nanoparticles (Poudel et al. 008, Zhao et al. 008) Properties Polynomial Expressions p p T.73 0 T T T T.450 T.0930 T T p T T T T n T.730 T.730 T.590 T n T T T 3.0 T n T.60 T 3.0 T.850 T 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 63

84 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. 64

85 Figure 3. Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h 0 W/m K) 65

86 Figure 3.3 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h 0 W/m K) 66

87 Figure 3.4 Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h = 5 W/m K) 67

88 Figure 3.5 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h = 5 W/m K) 68

89 Figure 3.6 Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h = 35 W/m K) 69

90 Figure 3.7 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h = 35 W/m K) 70

91 Figure 3.8 Contours of temperature distribution and streamlines of heat flow with adiabatic heat transfer condition (h = 50 W/m K) 7

92 Figure 3.9 Contours of electric potential and streamlines of electric current flow with adiabatic heat transfer condition (h = 50 W/m K) 7

93 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. (03). The comparison is presented in Fig. 3.0 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 3.0 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. 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

94 Current, I (A). Analytical Results (Reddy et al. 03) Numerical Simulation Results Temperature Difference, Figure 3.0 Comparison of current production using numerical and analytical techniques 74

95 Thermal Efficiency, h = 0 W/m K h = 5 W/m K h = 35 W/m K h = 50 W/m K Temperature Difference, Figure 3. Thermal efficiency of TEG as a function of convection heat transfer coefficient and temperature difference 75

96 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 ( ) 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. 76

97 3.5 Nomenclature C specific heat capacity (kjkg - K - ) D electric flux density vector (Cm - ) E electric field intensity vector (Vm - ) e electron charge (C) h convection heat transfer coefficient (Wm - K - ) H height of TE leg (m) I current (A) J electric current density vector (Am - ) k Boltzmann constant (JK - ) L length (m) L o Lorentz number (V K - ) R resistance (Ω) q heat flux vector (Wm - ). q heat generation rate per unit volume (Wm -3 ) Q t T W x y ZT heat (W) time (S) temperature (K) width of the TE leg (m) coordinate (m) coordinate (m) dimensionless figure of merit Greek symbols Seebeck coefficient (VK - ) ε dielectric permittivity matrix (Fm - ) thermal conductivity (Wm - K - ) density (kgm -3 ) electrical conductivity (Sm - ) electric scalar potential (V) 77

98 Subscripts C conv E g H l L cold surface convection electronic gap hot surface external load lattice 78

99 CHAPTER 4: ANALYTICAL AND NUMERICAL STUDIES OF HEAT TRANSFER IN NANOCOMPOSITE THERMOELECTRIC COOLER 4. 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 (004) 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 (03) 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 ºC temperature reduction compared to a metal substrate-printed circuit board with heat sink system. Cheng et al. (00) developed a 3D theoretical model of TEC and concluded that COP decreases rapidly as the amount of current increases. Lee et al. (00) studied the effect of Seebeck coefficient and electrical conductivity on the performance of micro TEC. Lee et al. (00) concluded that COP decreases because of the reduction in Seebeck coefficient and electrical conductivity. Chen et al. (0) 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. (0) 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 79

100 thermoelectric (TE) generator and obtained 4. mw. McCarty (00) performed D analytical, 3D numerical, and experimental investigation to check temperature dependency effects of TE properties. McCarty (00) concluded that a D 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. (0) 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 0.5 (with DEC) with electric current input of 4.5 A. Maneewan et al.(00) performed experiments on a TE air-conditioner made up of three TE modules. Using A electric current, they removed 9. W of heat at 8 C with a COP of Melero et al.(003) investigated a TE airconditioning system with 48 TE modules combined with photovoltaic solar panels for a domestic air-conditioning system. FLUENT numerical simulation of Melero et al. (003) showed that a TEC can provide the minimum temperature required for human comfort. Sullivan et al. (0) 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. (0) concluded that transient cooling with square root current pulse is most effective with 0 C cooling. Yang and Stabler (009) reviewed potential applications of TE materials in an automobile as a cooler and generator. Yang and Stabler (009) 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= ( k) T. Current ZT of the state-of-the-art TE material is around (Ma et al. 03). 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 995). Poudel et al. (008) achieved a ZT value of.4 at 373 K by hot-pressing nanopowders of Bismuth-Telluride (Bi Te 3 ) and Antimony-Telluride 80

101 (Sb Te 3 ) under Argon (Ar) atmosphere. The enhanced ZT was due to the significant reduction in the thermal conductivity of material (Poudel 008). Fan et al. (0) prepared a n-type Bi Te 3 nanocomposite which exhibited a ZT of.8 due to reduction in thermal conductivity. Tang et al.(0) prepared double- filled Cobalt Triantimonide (CoSb 3 ) using Indium (In) and Lutetium (Lu) resulting into In 0.3 Lu 0.05 Co 4.0 Sb through a high-pressure synthesis method. In 0.3 Lu 0.05 Co 4.0 Sb yielded a ZT of 0.7 which was greater by one order of magnitude than that of CoSb 3 (Tang el al. 0). Nagami et al. (04) prepared a Bismuth-Antimony-Telluride (Bi 0.4 Sb.6 Te 3 ) bulk TE material using mechanical alloying and hot extrusion. Their material exhibited a ZT of. 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. (04) performed D 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 D 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. Modeling Figure 4. shows a D 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 8

102 nanostructured p-type Bi 0.5 Sb.5 Te 3 (Poudel et al. 008) and n-type Bi Te 3 (Fan et al. 0). Figures 4. and 4.3 show crystal structures of Bi 0.5 Sb.5 Te 3 and Bi Te 3 (Zhang et al. 0, Chen et al. 009). Both materials possess the tetradymite type crystal structure formed by quintuple layers made up of sheets of Bi and 3 sheets of Te (Zhang et al. 0). For Bi 0.5 Sb.5 Te 3, Bi atoms are substituted by Sb (Chen et al. 009). P-type Bi 0.5 Sb.5 Te 3 (Poudel et al. 008) and n- type Bi Te 3 (Fan et al. 0) were prepared by hot pressing the nano-powder of materials. P-type Bi 0.5 Sb.5 Te 3 was prepared via direct current hot press method using nanopowders of Bi 0.5 Sb.5 Te 3 which was prepared from ball milling of Bi 0.5 Sb.5 Te 3 alloy ingots (Poudel et al. 008). Nanopowders of p-type Bi 0.5 Sb.5 Te 3 had an average size of 0 nm (Poudel et al. 008). The microstructure of the p-type nanostructured Bi 0.5 Sb.5 Te 3 shows highly crystalline structure and nanosize grains as observed from the Scanning Electron Microscope (SEM) images available in Poudel et al. (008). The n-type Bi Te 3 was prepared by melt spinning and hot pressing different amounts of nanoparticles of Bi Te 3 (Fan et al. 0). The microstructure of the nanostructured n-type Bi Te 3 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. (0). Q ab T C I x Q c p n L W W g T H Q re Figure 4. Schematic diagram of unit cell of TEC (drawing is not to scale) 8

103 As shown in Fig. 4. 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 W g. TEC absorbs Q ab amount of heat from the system and rejects Q re amount of heat to the surroundings. There is also a convection heat transfer (Q c ) 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. q q () gen J 0 (). where q, q gen, and J are heat flux, heat generation rate per unit volume, and electric current density, respectively. One quintuple layer Figure 4. The schematic of crystal structure of (a) (Bi -x Sb x ) Te 3 (Zhang et al. 0) Reprinted by permission from Macmillan Publishers Ltd: Nature Communications from Zhang et al., 574 (0), copyright 0 83

104 One quintuple layer Figure 4.3 The schematic of crystal structure of Bi Te 3 (Chen et al. 009) From [Chen et al. Science 35, 78 (009)]. Reprinted with permission from AAAS. Equations () and () are coupled by set of TE constitutive equations (Yang et al. 0) as shown in Eqs. (3) and (4): q T J k eff eff T (3) J eff E eff T eff (4) where eff, k eff, 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. 0). For a steady state analysis, rearrangement of Eqs. () to (4) give coupled TE equations for the heat transfer (Eq. (5)) and electric potential (Eq. (6)): T J k T EJ (5) eff eff 0 eff eff T eff (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. 84

105 85 In order to analyze a nanocomposite TEC analytically, Eq. (5) is simplified to obtain D form as shown in Eq. (7) below and solved. 0 ) ( Thomsonheat Convectionheat transfer Jouleheat conduction Fourier heat dx dt A k I T T A k h p A k I dx T d eff a eff eff eff (7) Solution of Eq. (7) in terms of temperature distribution using boundary conditions; (x = 0, T = T C ; x = L, T = T H ) is given by ) ( ) ( ) ( L C L C L C C L C H x C L C C L C H x C e e e T e T e e T e T e x T (8) Different terms used in Eq. (8) can be defined as 4 C ; 4 C ; A k I eff ; A k h p eff A k I T A k h p eff eff a eff. (9) (0) Heat absorbed at the cold surface of a TEC can be written by combining Peltier heat which occurs at junctions only ) ( ) ( ) ( L C L C L C C L C H x C L C C L C H x C eff C eff ab e e e T e T e C e T e T e C A k T I Q () An electric power input to TEC is given by R i I P () where R i can be calculated by n n n eff, p p p eff, A L A L R i (3) Moreover, Seebeck coefficient and thermal conductivity of unit cell of a TE system can be calculated by n n n eff, p p p eff, L A k L A k K (4) n eff p eff,, (5)

106 To measure the performance of a TEC, the COP can be written as Q COP ab (6) P It is important to note that Eq. () is the heat absorbed by one TE leg only. Heat absorbed by unit cell of a TEC as shown in Figure 4. 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. 008). while for n-type material nanocomposite Bi Te 3 is considered to analyze the performance (Fan et al. 0). 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. and 4.. These properties are evaluated at an average temperature of a working temperature range. Two different cases of cold surface temperatures (T C ) = 333 K and (T C ) = 343 K are considered with hot surface temperature (T H ) = 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. 04). A semiconductor leg of a unit cell of TEC has a width and thickness (W D) of mm mm and a height (L) of 5 mm. The gap between two consecutive legs (W g ) is 0.5 mm. 86

107 Table 4. Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal conductivity as a function of temperature for BiSbTe nanostructured bulk alloys and nanocomposite Bi Te 3 (Poudel et al. 008, Fan et al. 0) TE Properties Polynomial Expressions eff (p) ( T.730 T T T ) eff (p) ( T.450 T.0930 T.3540 T ) k eff (p) T T T T eff (n) ( T ) 0 6 T T T eff (n) ( T.850 T.9070 T T ) k eff (n) T.6960 T.990 T T Table 4. Polynomial functions of Seebeck coefficient, electrical conductivity, and thermal conductivity as a function of temperature for BiSbTe bulk alloys and conventional Bi Te 3 (Poudel et al. 008, Fan et al. 0) TE Properties Polynomial Expressions p ( T T T.80 T ) p ( T T T T ) k p T.900 T T.0090 T n (.040.6T.50 T.00 T.330 T ) n (.360.5T T 7.50 T 8.60 T ) k n T 6.50 T.590 T.760 T 87

108 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 0-7 Wm - K - (adiabatic) to 80 Wm - K -. 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

109 Temperature, T (K) h = 0-7 Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K Length of p-type TE leg, L (m) 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 89

110 Temperature, T (K) h = 0-7 Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K Length of n-type TE leg, L (m) 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 90

111 Effective electrical resistivity, eff ( m).5e-05 E-05 p-type material n-type material.5e-05 E-05 5E Temperature, T ( C) Figure 4.6 Electrical resistivity of p- and n- type legs of nanocomposite TEC 9

112 Figures 4.7 and 4.8 show heat absorbed from the cold surface using a nanocomposite TEC with electric current inputs from 0 to 0 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 = 0-7 Wm - K - (adiabatic) to 80 Wm - K -. 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. 9

113 Heat absorbed, Q ab (W) Current, I (A) h = 0-7 Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K - Figure 4.7 Heat absorbed as a function of current considering hot surface temperature 353 K with cold surface temperature 333 K 93

114 Heat absorbed, Q ab (W) h = 0-7 Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K Current, I (A) 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 4.0. 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 4.0. 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 94

115 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. 95

116 COP h = 0-7 Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K Current, I (A) Figure 4.9 COP of TEC as a function of current considering hot surface temperature 353 K with cold surface temperature 333 K 96

117 COP 0 8 h = 0-7 Wm - K - h = 0 Wm - K - h = 40 Wm - K - h = 60 Wm - K - h = 80 Wm - K Current, I (A) Figure 4.0 COP of TEC as a function of current considering hot surface temperature 353 K with cold surface temperature 343 K Figures 4. and 4. 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. 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 97

118 maintains the hot surface temperature (T H ) = cold surface temperature (T C ). 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. 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. 98

119 Heat absorbed, Q ab (W) I = A I = A I = 3 A I = 4 A I = 5 A Temperature difference, Figure 4. 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

120 COP I = A I = A I = 3 A I = 4 A I = 5 A Temperature difference, Figure 4. COP as a function of temperature difference with different electric current input and hot surface temperature 353 K considering adiabatic side wall condition 00

121 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 4.3, 4.4, 4.5, and 4.6 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 0). Maple s nonlinear optimization solver uses the Karush- Kuhn-Tucker theorem to solve the Lagrangian function for constrained nonlinear problems (Fishback 009). Table 4.3 shows different TE leg heights and their corresponding internal resistances which can be referred to locate the appropriate legend in Figs. 4.3, 4.4, 4.5, and 4.6. As observed by the trend in Fig. 4.3 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. 4.3, 4.4, 4.5, and 4.6 Cases in Figs. 4.3, 4.4, 4.5, and 4.6 Height of TE legs, L (m) Internal resistance of TEC, Ri (Ω) Case Case Case Case

122 Maximum heat absorbed, Q ab max (W) Case Case Case 3 Case Convection heat transfer coefficient, h (Wm - K - ) Figure 4.3 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 0

123 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. 4.5, 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. 4.4 and 4.6. 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. of Hodes (007). Figures 4.4 and 4.6 also show that optimum electric current remains different for maximum heat absorption and maximum COP. 03

124 Optimum electric current, I opt (A) Case Case Case 3 Case Convection heat transfer coefficient, h (Wm - K - ) Figure 4.4 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 04

125 5 4 COP max 3 Case Case Case 3 Case Convection heat transfer coefficient, h (Wm - K - ) Figure 4.5 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 05

126 Optimum electric current, I opt (A) Case Case Case 3 Case Convection heat transfer coefficient, h (Wm - K - ) Figure 4.6 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 06

127 Additionally, internal resistance and maximum heat absorption are plotted as a function of TE leg height. Plots in Figs.4.7 and 4.8 show performance of a unit cell of TEC as height of TE leg vary. As seen in Fig. 4.7, 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. (3) 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

128 Internal Resistance, R () Height of TE leg, L(m) Figure 4.7 Internal resistance of TEC unit cell as a function of TE leg height 08

129 Maximum heat absorbed, Q ab max (W) Height of TE leg, L(m) Figure 4.8 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 09

130 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 = mm) and height (L = 5 mm) is considered for analysis. Figures 4.9 to 4.6 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 4.9 and 4.0 show field results of an electric potential and electric current flow streamlines. The terminal of TEC is applied with electric potentials of 0.0 V and 0.06 V for Figs. 4.9 and 4.0, respectively. An electric potential of 0.0 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., 4., 4.3 and 4.4 show field plots of temperature and heat flow with an electric potential of 0.0 V. At the end, Figs. 4.5 and 4.6 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 - K - to h = 60 Wm - K -. The surrounding temperature is considered to be an average temperature of a working range. Figures 4.9 and 4.0 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. 0

131 Figure 4.9 Electric scalar potential and current flow in nanocomposite TEC with electric potential 0.0 V

132 Figure 4.0 Electric scalar potential and current flow in nanocomposite TEC with electric potential 0.06 V

133 Figures 4., 4., 4.3 and 4.4 present heat flow and temperature distribution inside a nanocomposite TEC with an electric potential 0.0 V. A nanocomposite TEC is under different convection boundary conditions at vertical walls of semiconductor legs ranging from h 0 Wm - K - (adiabatic) to h = 60 Wm - K -. 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 = 0 Wm - K - to h = 60 Wm - K - as observed in Figs. 4., 4.3, and 4.4. 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. 3

134 Figure 4. Heat flow and temperature distribution in nanocomposite TEC for h 0 Wm - K - at vertical walls with electric potential 0.0 V 4

135 Figure 4. Heat flow and temperature distribution in nanocomposite TEC for h = 0 Wm - K - at vertical walls with electric potential 0.0 V 5

136 Figure 4.3 Heat flow and temperature distribution in nanocomposite TEC for h = 40 Wm - K - at vertical walls with electric potential 0.0 V 6

137 Figure 4.4 Heat flow and temperature distribution in nanocomposite TEC for h = 60 Wm - K - at vertical walls with electric potential 0.0 V 7

138 Figures 4.5 and 4.6 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. 4.5 and 4.6. 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. 4.5 and 4.6 show that higher temperature inside the TE leg (Fig. 4.6) transfers the heat to surroundings. Ultimately, this phenomenon drags more heat from the cold surface of TEC. 8

139 Figure 4.5 Heat flow and temperature distribution in nanocomposite TEC for h 0 Wm - K - at vertical walls with electric potential 0.06 V 9

140 Figure 4.6 Heat flow and temperature distribution in nanocomposite TEC for h = 60 Wm - K - at vertical walls with electric potential 0.06 V 0

141 Figures 4.7 and 4.8 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 0.0 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. where heat absorbed is plotted against temperature differences using analytical model.

142 Heat absorbed, Q ab (W) T C = 343 K T C = 333 K Convection heat transfer coefficient, h (Wm - K - ) Figure 4.7 Heat absorbed by nanocomposite TEC as a function of convection heat transfer coefficient with hot surface temperature 353 K and electric potential 0.0 V

143 COP T C = 343 K T C = 333 K Convection heat transfer coefficient, h (Wm - K - ) Figure 4.8 COP of nanocomposite TEC as a function of convection heat transfer coefficient with hot surface temperature 353 K and electric potential 0.0 V 3

144 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. 4.9 and 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. 4.9 and 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. 4

145 Heat absorbed, Q ab (W) T C = 343 K T C = 333 K Convection heat transfer coefficient, h (Wm - K - ) Figure 4.9 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 5

146 COP T C = 343 K T C = 333 K Convection heat transfer coefficient, h (Wm - 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 6

147 Figures 4.3 and 4.3 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. 7

148 Heat absorbed, Q ab (W) 0 9 Analytical results Numerical simulation results Convection heat transfer coefficient, h (Wm - K - ) Figure 4.3 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 0.0 V 8

149 COP 3.9 Analytical results Numerical simulation results Convection heat transfer coefficient, h (Wm - K - ) Figure 4.3 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 0.0 V 9

150 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 4.9 to 5.58 if nanocomposite materials are used, which gives approximately 3% 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.36 Wm - K - to 0.98 Wm - K - at 00 ºC. 30

151 COP Conventional TE material Nanocomposite TE material Temperature difference, Figure 4.33 Comparison of COP using conventional (no nanostructuring) and nanocomposite TE material considering h 0 Wm - K -, hot surface temperature 353 K, and electric current input of A 3

152 Thermal conductivity, k (Wm - K - ) n-type conventional TE material p-type conventional TE material n-type nanocomposite TE material p-type nanocomposite TE material Temperature, T ( C) Figure 4.34 Thermal conductivity of conventional (no nanostructuring) and nanocomposite TE materials 3

153 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. (008) prepared a p-type BiSbTe nanostructured TE leg via hot pressing the ball-milled nanopowders of BiSbTe. They (Poudel et al. 008) 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. 008) 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 The current results also show good agreement with the theoretical predictions given in Poudel et al. (008). 33

154 Temperature difference, T( o C) 0 00 Poudel et al. (008) experimental results Poudel et al. (008) Theoretical results Current work Electric current, I(A) Figure 4.35 Comparison of results between current work and Poudel et al. (008) 34

155 4.4 Conclusions In this research work, a D 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:. 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.. 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. 35

156 4.5 Nomenclature A cross-sectional area (m ) C C D a parameter (refer Eq.(9)) a parameter (refer Eq.(9)) thickness of TE leg (m) E electric field intensity (Vm - ) h convection heat transfer coefficient (Wm - K - ) I electric current (A) J electric current density vector (Am - ) k thermal conductivity (Wm - K - ) L p P height of TE leg (m) perimeter (m) electrical power output (W) q heat flux vector (Wm - ). q Q R T W x ZT Greek Symbols heat generation rate (Wm -3 ) heat (W) electrical resistance (Ω) temperature (K) width of TE leg (m) co-ordinate system (m) dimensionless figure of merit Seebeck coefficient (VK - ) ε κ ρ a parameter (refer Eq.(0)) a parameter (refer Eq.(0)) electrical resistivity (Ωm) electrical conductivity (Sm - ) τ Thomson coefficient (VK - ) electric scalar potential (V) ψ a parameter (refer Eq.(0)) 36

157 Subscripts a ab c C eff g gen H i max n opt p re atmospheric condition heat absorbed convection heat loss cold surface temperature effective transport properties gap between TE legs heat generation hot surface temperature internal resistance maximum value n-type material optimum value p-type material heat rejected 37

158 CHAPTER 5: EFFECT OF THERMAL CONDUCTIVITY ON PERFORMANCE OF THERMOELECTRIC SYSTEMS BASED ON EFFECTIVE MEDIUM THEORY 5. 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 = α σt/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

159 Figure 5. Different approaches to increase ZT of TE materials (Martin-Gonzalez 03) 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 (993) discussed the concept of quantum wire (one dimensional) for the TE materials. ZT increases because power factor (α σ) improves due to one dimensional structure but not significantly lowering the thermal conductivity (Hicks and Dresselhaus 993). A method to increase ZT was demonstrated experimentally by Venkatasubramanian et al. (00) where ZT of a p-type superlattice of Bi Te 3 /Sb Te 3 (Bismuth- Telluride-Antimony) improved from to.4 due to the reduction in the thermal conductivity. Improvements offered by Hicks and Dresselhaus (993) and Venkatasubramanian et al. (00) 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. 03). There is another route to improve the ZT in bulk TE materials called nanocomposite bulk materials (Bottner and Konig 03). Nanocomposite bulk materials, which can also be called as composites are bulk materials with nanostructured features inside it (Ma et al. 03). Composites are made up of a base material and macro/nano particles which can be manufactured 39

160 via wet-chemical and mechanical synthesis techniques (e.g., solvothermal, extrusion, and high energy milling). Scoville et al. (995) reduced the thermal conductivity by 40% by adding Boron Nitride and Boron Carbide particles into Silicon-Germanium composite. A bulk p-type Bi x Sb - xte 3 composite prepared by hot-pressing nanopowders gave ZT of.4 due to the low thermal conductivity (Poudel et al. 008). In the similar manner, a Bi Te 3 with SiC (Silica Carbide) nanoparticles (Zhao et al. 008), Co 4 Sb (Skutterudites) (Li et al. 009), and AgPb x SbTe +x (Lead Antimony Silver Telluride) (Hsu et al. 004) 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. 03). 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 998). EMTs have been applied to different situations, such as Yu et al. (05) applied the EMT to calculate the effective permittivity, Gong et al. (04) derived modified EMT for a porous media to model the thermal conductivity, Hou et al. (05) applied EMT to calculate the effective thermal conductivity of porous thin films, and Chen et al. (04) developed the effective thermal conductivity model for bentonites which is considered engineered barrier material for radioactive wastes. The classical work of Rayleigh (89) and Maxwell (954) 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 (954) considered a heterogeneous mixture with the spherical particles with thermal conductivities of base material, particles, and composite as k b, k p, and k eff, respectively. In addition to this, is a volume fraction of particle inclusions. Equation () 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. k eff k b k p 3 ( k k ) () k b p b ( k p k b ) 40

161 Maxwell model does not consider an interaction between particles and base material as one can observe from Eq. () which only includes thermal conductivities of base and particle materials and volume fraction terms. A model derived by Hasselman and Johnson (987) 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 987). Equation () is an expression of Hasselman and Johnson (987) model of the effective thermal conductivity for a composite with spherical particles. k eff k b k p k p k p k p kb r K kb r K k p k p k p k p kb r K kb r K 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 (987) 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. (997) 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. 997). k eff k b k p k ( ) k ( k ( ) k ) (3) p b ( ) k b ( k ( ) k In Eq. (3), thermal conductivity of composite ( k eff ) is expressed in terms of thermal conductivities of base material ( k b ) and material of particle ( k p ), 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. 005). Several methods were also applied to 4 p p b b ) ()

162 calculate the effective thermal conductivity, such as Boltzmann equation (Yang et al. 005) and Monte Carlo (MC) simulations (Jeng et al. 008). Methods used in (Yang et al. 005) and (Jeng et al. 008) require significant computational resources and time (Minnich and Chen 007). Minnich and Chen (007) has developed a thermal conductivity model based on the EMT for nanocomposites as shown in Eq. (4). k eff 4 Cb b 3 (4 b b k p ) k ( ) k p b ( ) k d b 6 ( k d 6 ( k p p ( ) k ( ) k Equation (4) combines base TE material properties, interface density (), and Eq. (3) which is also similar to Hasselman-Johnson model. Figure 5. shows schematics of internal-structure representation for different transport property models. Figure 5.a 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.b which shows base material and spherical particles. The interconnecting dark solid lines in Fig. 5.b represent a TBR. Figure 5.c 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. b b ) ) (4) (a) (b) (c) 4

163 Figure 5. Graphical representation of (a) Maxwell model (b) Hasselman and Johnson model (c) Minnich and Chen model Table 5. shows the comparison between the effective transport property models for macro and nanocomposites in terms of the applicability and limitations. Table 5. Comparison of different effective medium theories Effective Medium Theories Applicability and Limitations Maxwell model (Maxwell Spherical particles embedded in spherical region, No 954) consideration thermal barrier resistance Hasselman and Spherical, cylindrical, and flat plate dispersions into host Johnson model (Hasselman material and thermal barrier resistance between particles and and Johnson 987) base material, Valid for macro size particles Minnich and Chen model Spherical and cylindrical nanoparticle embedded in cube, (Minnich and Chen 007) Valid for nano size particles Yang and Stabler (009) 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. 43

164 5. 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 W D, height of H, and separated by a distance L d. Q in and Q out 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. T < T Q in T Q in T T > T q conv p I n x H q conv p I n x H W L d T W L d T Q out Q out (a) (b) 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 005) where m, C p, T, q, and T mcp q q gen. (5) t q gen represent material density, specific heat, temperature, heat flux vector, and volumetric heat generation, respectively. The continuity of electric charge through the system must satisfy 44

165 D J 0. (6) t 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 005) as shown in Eq. 7(a) and Eq. 7(b) below q T [ ] J [ k] T J [ ] ( E [ ] T) 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 984). Combining Eq. (5) to Eq. 7(b), the coupled TE equations for energy and charge transfers can be expressed as 7(a) 7(b) T m C p T [] J [ k] T E J (8) t [ ] t [ ] [ ] T [ ] 0 where [ ] is the dielectric permittivity matrix (Antonova and Looman 005) and EJ represents Joule heat (Antonova and Looman 005). (9) For a typical TE system shown in Fig. 5.3 thermal boundary conditions are as follow: At the top surface ( x 0 ) temperature is constant ( T T ) At the bottom surface ( x H ) temperature is constant ( T T ) Convection heat transfer from the side surfaces to the surrounding, q h( T T ) where T a is atmospheric temperature and h is convection heat transfer coefficient. conv a Note that the material property matrices (e.g., [ ] and [k], etc.) are suitable for nonhomogeneous materials. These material property matrices become a single-valued property in the case of homogeneous materials. Initially, a simplified -D version of above equations will be solved to obtain the close forms of analytical solutions. 45

166 For -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 R l 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 T and T, respectively. A TE system absorbs amount of heat from the heat source and rejects Q in Q out 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 d T dx Heat conduction I dt keff A dx T homsonheat h P a keff A I keff A T T 0 Convectionheat transfer Jouleheating. (0) In Eq. (0), τ is Thomson coefficient, ρ is electrical resistivity, and k eff is effective thermal conductivity of composite. Thomson coefficient represents temperature dependency of Seebeck coefficient and can be calculated using T ( d / dt). Applying thermal boundary conditions at the top surface ( x 0, T T ) and the bottom surface x H, T T ) temperature distribution inside semiconductor leg can be given by ( 46

167 47 h P A I h P AT e e h P A h P A T I e h P A T e h P A T e I h P A T e e e h P A h P A T I e h P A T e h P A T e I h P A T e x T a A k H m A k H m a A k H m a A k H m A k H m A k x m A k H m A k H m a A k H m a A k H m A k H m A k x m eff eff eff eff eff eff eff eff eff eff eff eff () where A h P k I I m eff 4 ; A h P k I I m eff 4. () 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, A k H m A k H m eff a A k H m a A k H m A k H m A k x m A k H m A k H m eff a A k H m a A k H m A k H m A k x m eff n p eff eff eff eff eff eff eff eff eff eff eff eff e e h P A k h PAT I e h P AT e h P A T e I h P A T e m e e h P A k h PAT I e h P AT e h P A T e I h P A T e m A k T I q, (3) Total heat input from heat source in TE system can be given by 0 0 x q x q Q n p in. (4) Total heat output to heat sink from TE system can be given by

168 Q out x H q x H q. (5) COP of TEC (Fig. 5.3a) can be calculated from the following equation: p Q COP I Ri while the thermal efficiency of TEG (Fig. 5.3b) can be calculated from the following equation: n in, (6) Q Q Q in out. (7) in It is important to note here that when convection heat transfer from the side walls of semiconductor legs are extremely low ( h 0 ) and if temperature dependency of TE materials is negligible then heat input to TE system (Eq. (4)) reduces to Q in T I H T p, n I T keff A. (8) H keff A H In similar manner, heat output to heat sink (Eq. (5)) becomes Q out I H T T p, n I T keff A. (9) keff A H H 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, Bi Te 3 as a base material and Sb Te 3 particles are selected. The size of the Sb Te 3 particles varies from micrometer to nanometer. For n-type TE material, Bi Te 3 is considered which can be used as a base material and nanoparticles. Table 5. (Pattamatta and Madnia 009) presents some properties (bulk thermal conductivity, volumetric specific heat, bulk MFP, and phonon group velocity) of Bi Te 3 and Sb Te 3 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 04), while the phonon group velocity represents velocity of phonons, a quasi- 48

169 particle which represents quantization of the modes of lattice vibrations which exchanges energy (Wang 0). Table 5. Material properties (Pattamatta and Madnia 009) Material Thermal Volumetric Phonon group Mean Free Path conductivity specific heat velocity (ms - ) (Å) (Wm - K - ) (MJm -3 K - ) Bi Te Sb Te 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 n r r 3 3, where n is the number of spherical particles, r and r are radii of particle sphere and base sphere (Maxwell 954). Figure 5.4 shows that thermal conductivity decreases from. Wm - K - to 0.95 Wm - K - 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. Wm - K - 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. 49

170 k eff (Wm - K - ).5 n-type material p-type material Figure 5.4 Effective thermal conductivity of p-type and n-type thermoelectric material based on Maxwell model 50

171 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 - K -, whereas with similar volume fraction Maxwell model predicts effective thermal conductivity of 0.99 Wm - K -. 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 0.5 Wm - K - with volume fraction 0.8 and extremely low (0-5 ) 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 (=0-5 ) and very high boundary conductance ( ) are taken arbitrarily. 5

172 k eff (Wm - K - ). 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 5

173 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. 53

174 k eff (Wm - K - ) =0 =0. =0.4 =0.6 = K (Wm - K - ) Figure 5.6 Effect of thermal boundary conductance on effective thermal conductivity of p-type using Hasselman-Johnson model 54

175 k eff (Wm - K - ) =0 =0. =0.4 =0.6 = K (Wm - K - ) Figure 5.7 Effect of thermal boundary conductance on effective thermal conductivity of n-type using Hasselman-Johnson model 55

176 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. Wm - K - to Wm - K - for the p-type material with particle size of 50 nm. Similarly, effective thermal conductivity decreases from 0.3 Wm - K - to 0.08 Wm - K - when particle size decreases from 50 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. 56

177 k eff (Wm - K - ). d = 5 nm d = 5 nm d = 50 nm d = 00 nm d = 50 nm Figure 5.8 Effective thermal conductivity of p-type thermoelectric material using Minnich-Chen model 57

178 k eff (Wm - K - ). d = 5 nm d = 5 nm d = 50 nm d = 00 nm d = 50 nm Figure 5.9 Effective thermal conductivity of n-type thermoelectric material using Minnich-Chen model 58

179 Performance of TE systems In this section, different effective transport property models are applied to investigate the performance of TE systems. Figures 5.0 to 5.3 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.5 mm.5 mm.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 0% when compared to non-nanocomposite BiSbTe at room temperature (Poudel et al. 008). The Seebeck coefficient and electrical conductivity for p-type material are 4.44 μvk - and Sm -. Similarly, the Seebeck coefficient and electrical conductivity for n-type material are μvk - and Sm -. TEC creates temperature difference of 0 K with surrounding temperature at 300 K. An electric power input to TEC is calculated using I R i. 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 5.0, 5., and 5.6 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 5.0 and 5. show performance of TEC and TEG based on Maxwell model. Figure 5.0 shows COP improvement from 0.03 to 0.6 when volume fraction increases from 0 to 0.8 with an electric current input of A considering the Maxwell model. Figure 5.0 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 59

180 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.. Thermal efficiency increases from.06% to.7% 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. also shows a rise in the thermal efficiency with increment in the temperature difference between hot and cold surface temperatures. 60

181 COP =0 =0. =0.4 =0.6 = I(A) Figure 0 COP of TEC considering various amount of volume fraction with Maxwell model 6

182 (%) =0 =0. =0.4 =0.6 = Figure 5. Efficiency of TEG with different volume fraction with Maxwell model 6

183 Figures 5. and 5.3 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 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. 5.4 and 5.5 at very high boundary conductance (K). Figures 5.4 and 5.5 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. 5.4 and 5.5) which also can be verified with Figs. 5.0 and 5.. 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.06% to 5.4% which is 6% increment when boundary conductance changes from very high (0 5 ) to very low (0-5 ). 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. 997, Minnich and Chen 007). 63

184 COP =0 =0. =0.4 =0.6 = I(A) Figure 5. COP of TEC considering different amount of volume fraction with Hasselman model 64

185 (%) = 0 = 0. = 0.4 = 0.6 = Figure 5.3 Efficiency of TEG with different amount of volume fraction with Hasselman model 65

186 COP =0 =0. =0.4 =0.6 = K (Wm - K - ) Figure 5.4 Effect of boundary conductance on performance of TEC based on Hasselman- Johnson model 66

187 (%) =0 =0. =0.4 =0.6 = K (Wm - K - ) Figure 5.5 Effect of boundary conductance on performance of TEG based on Hasselman- Johnson model 67

188 Figures 5.6 and 5.7 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. 5.6 and 5.7. For example, thermal efficiency of TEG increases from.06 % to 7.03 % (Fig. 5.7). 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 5.8 and 5.9 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

189 COP 8 6 =0 =0. =0.4 =0.6 = I (A) Figure 5.6 COP of TEC considering different amount of volume fraction with Minnich-Chen model 69

190 (%) = 0 = 0. = 0.4 = 0.6 = Figure 5.7 Efficiency of TEG with different volume fraction with Minnich-Chen model 70

191 COP =0 =0. =0.4 =0.6 = E-08 4E-08 6E-08 8E-08 E-07 d (m) Figure 5.8 Effect of nanoparticle size on performance of TEC considering Minnich-Chen model 7

192 (%) =0 =0. =0.4 =0.6 = E-08 4E-08 6E-08 8E-08 E-07 d (m) Figure 5.9 Effect of nanoparticle size on performance of TEG considering Minnich-Chen model 7

193 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 5.0 and 5. show COP and thermal efficiency as functions of volume fraction and convection heat transfer coefficient. TEC considered for Fig. 5.0 creates a temperature difference of 0 K with the surrounding temperature at 300 K with A electric current input. Figure 5.0 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. 5.0, 5., and 5.6. In similar way, Fig. 5. 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 0-4 to 00 Wm - K -. Additionally, Figs. 5.0 and 5. show COP and thermal efficiency remains same for each transport property models at volume fraction zero. 73

194 COP 0 9 h = 0-4 Wm - K - h = 00 Wm - K - Minnich-Chen model 8 7 h = 0-4 Wm - K - h = 00 Wm - K - Hasselman-Johnson model h = 0-4 Wm - K - h = 00 Wm - K - Maxwell model Figure 5.0 Performance of TEC with variable volume fractions and convection heat transfer coefficients through side walls of TE legs 74

195 (%) 6 5 h = 0-4 Wm - K - h = 00 Wm - K - Minnich-Chen model 4 h = 0-4 Wm - K - h = 00 Wm - K - Hasselman - Johnson model 3 h = 0-4 Wm - K - h = 00 Wm - K Maxwell model Figure 5. Performance of TEG with variable volume fractions and convection heat transfer coefficients through side walls of TE legs 75

196 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 0.6, 6.76, and 7.69, respectively, with an electric current input of 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. and 5.3 explain how effective thermal conductivity influences the heat conduction portion of total heat input to TE system. Figure 5. shows heat conduction portion to total heat input (Q in ) in TEC as a function of volume fraction. TEC creates temperature difference of 0 K with surrounding temperature 300 K and an electric current input of A. As one can be observed from Fig. 5. 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. 5.0, 5., and 5.6 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 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. 76

197 Q in (Conduction) (W) Hasselman-Johnson model Maxwell model Minnich-Chen model Figure 5. Influence of effective thermal conductivity on heat conduction in TEC 77

198 Q in (Conduction) (W) Hasselman-Johnson model Maxwell model Minnich-Chen model Figure 5.3 Influence of effective thermal conductivity on heat conduction in TEG 78

199 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 (006). 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 =.5 mm) and height (H =.5 mm) is considered for the analysis. Both legs are separated by distance L d = 0. 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 5.4 to 5.8 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 Similarly, temperature is presented by marked isothermal lines with multi-colored background, while heat flow is presented by lines with arrows in Fig. 5.5 to 5.8. Composite TEC maintains temperature difference of 0 K with surrounding temperature at 300 K and electric potential of 0.0 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 Also, the current flow direction can be visualized from the streamlines in Fig

200 Figure 5.4 Contours of electric potential and streamlines of electric current flow in TEC 80

201 Field plots in Figs. 5.5 to 5.8 show temperature profile and streamlines of heat flow. Figures 5.5, 5.6, 5.7, and 5.8 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. 5.6, 5.7, and 5.8 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. and 5.3. Temperature contours are linear in Figs. 5.5, 5.6, and 5.7 but temperature contours show some non-linearity in Fig The reason behind this could be very low thermal conductivity based on Minnich-Chen model and generation of Joule heat. 8

202 Figure 5.5 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with NO particles 8

203 Figure 5.6 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with 0.8 volume fraction with Maxwell model 83

204 Figure 5.7 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with 0.8 volume fraction with Hasselman-Johnson model 84

205 Figure 5.8 Contours of temperature and streamlines of heat flow in TEC with cold surface temperature 90 K, hot surface temperature 300 K, and electric potential 0.0 V with 0.8 volume fraction with Minnich-Chen model 85

206 Figures 5.9 to 5.36 are for a composite TEG showing the temperature and electric potential field plots. Figures 5.9 to 5.3 represent temperature which is marked by isothermal lines with multi-colored background, while heat flow is presented by lines with arrows. Similarly, Figs 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. 5.9 to 5.3. Figures 5.9, 5.30, 5.3, and 5.3 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. 86

207 Figure 5.9 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 87

208 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 88

209 Figure 5.3 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 89

210 Figure 5.3 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 90

211 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. 9

212 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 9

213 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 93

214 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 94

215 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 95

216 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. 96

217 COP Analytical results - Minnich-Chen model Numerical simulalation - Minnich-Chen model Analytical results - Hasselman-Johnson model Numerical simulation - Hasselman-Johnson model Numerical simulation - Maxwell model Analytical results - Maxwell model Figure 5.37 Comparison of analytical and numerical simulation results for TEC 97

218 (%) Analytical results - Maxwell model Numerical simulation -- Maxwell model Numerical simulation - Hasselman-Johnson model Analytical results - Hasselman-Johnson model Analytical results - Minnich-Chen model Numerical simulation - Minnich-Chen model Figure 5.38 Comparison of analytical and numerical simulation results for TEG 98

219 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 (-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:. 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.. 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. 99

220 5.5 Nomenclature A cross-sectional area (m ) C volumetric specific heat (MJm -3 K - ) C p specific heat (Jkg - K - ) D electric flux density (Nm C - ) d depth (m) E electric field intensity vector, (NC - ) h convection heat transfer coefficient, (Wm - K - ) H I height (m) electric current (A) J electric current density (Am - ) k thermal conductivity (Wm - K - ) K thermal boundary conductance (WK - m - ) L length (m) m A parameter (See Eq. ()) m A parameter (See Eq. ()) n p P n-type material, number of particles p-type material Power input (W) q heat flux vector (Wm - ) q heat (W). q heat generation (Wm -3 ) Q r t T V W ZT heat (W) radius of sphere (m) time (s) temperature (K) electric potential (V) width (m) Figure of merit 00

221 Greek symbols Seebeck coefficient (VK - ) ε dielectric permittivity matrix (Fm - ) efficiency thermal barrier resistance (KW - m - ) mean free path (Å) electrical resistivity (Ω m), density (kgm -3 ) electrical conductivity (Sm - ) τ Thomson coefficient (VK - ) phonon group velocity (ms - ) volume fraction of particles interface density (m - ) Subscripts heat source, particle sphere 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 0

222 CHAPTER 6: ANALYSIS OF COMBINED SOLAR PHOTOVOLTAIC- NANOSTRUCTURED THERMOELECTRIC GENERATOR SYSTEM 6. 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, solarthermal 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 00). 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 005). 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 005). 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. 003; Omer and Infield 998; Xi, Luo, and Fraisse 007; Sahin et al. 0). Shanmugam et al. (0) 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 4.7 W with the solar radiation ranges from 300 W/m to 00 W/m. Chen (0) has developed a mathematical model of a solar TE generator considering an optical concentrator and selective surface. Chen (0) proposed a novel TE generator with an evacuated environment and concluded that such arrangement can have attractive efficiency with Bi Te 3 TE materials and temperature ranges 0

223 from 50 C to 50 C. Khattab and Shenaway (006) 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 (0) 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 4.% is possible for the cold and hot side temperature settings of 00 C and 000 C, respectively. They also observed that the system efficiency can reach up to 5% if figure of merit ( ZT ) reaches to for the cold and hot side temperature settings of 00 C and 000 C, respectively. Vatcharasathien et al. (005) 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. (00) 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. (00) have concluded that the conversion efficiency of a TE generator increases with the increase in solar concentration ratio. Xiao et al. (0) 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 0.5 %. Vorobiev et al. (006) 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. (999) 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 (008) 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 (03) 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 (03) concluded that the efficient TE modules can lead to a better power output of a combined system. 03

224 Performance of TE materials are characterised by a dimensionless parameter figure of merit, ZT ( k) T, where is the Seebeck coefficient, is the electrical resistivity, and k is the thermal conductivity. Based on the available literature (Vineis et al. 00), the ZT value of the best available TE materials reach around at room temperature. Recent advancement in nanotechnology (Vineis et al. 00) opens the door for further improvements of ZT for the TE materials. The expression of the figure of merit, ZT ( k) T, 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 k and electronic conductivity e k (Godart et al. 009). The electronic part of thermal conductivity L k is related to the electrical conductivity using Wiedemann-Franz law (Godart et al. 009) as shown in Eq. (), k e L 0 T () where, L 0 is the Lorentz number and for metals it is equal to (Godart et al. 009), 8 L V K - () 3e 3 where, is the Boltzmann constant (=.38 0 JK - ) and e is the electron charge 9 (=.60 0 C)(Godart et al. 009). The expression of figure of merit can be written in terms of Lattice conductivity k and electronic conductivityk as shown below (Godart et al. 009), L ZT L 0 ke k k e L e. (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 03). Poudel et al. (008) achieved ZT value of.4 at 373 K by hot-pressing of nanopowders of Bi Te and 3 Sb Te 3 under e Argon Ar atmosphere. The enhanced ZT was attributed to significant decrease in lattice thermal conductivity of material. Li et al. (009) obtained ZT of.43 for double-doped Co 4Sb skutturedites using Indium In and Cerium Ce doping. The attractive results achieved using nanotechnology has encouraged researchers to include nanoparticle-doped TE materials for 04

225 various applications, such as, solar TE generator and waste heat recovery. Kraemer et al. (0) 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 000 Wm - condition. McEnaney et al. (0) developed a novel of TE generator. They placed high performance nanostructured material in evacuated tube with selective absorber and achieved an efficiency of 5.%. 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. Modeling and boundary conditions The proposed photovoltaic thermoelectric (PVTE) system is shown in Fig. 6.a. 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.b with geometric dimension, a co-ordinate system, directions of different heat components, and thermal boundary conditions. The PV panel has length l PV, width w PV and thickness t PV. A TE module has length l TE, width w TE, and operates between the high and low temperature reservoirs T bs and T amb, respectively. The TE module absorbs Q h amount of heat from the back surface of the PV panel and rejects Q c amount of heat to the surrounding environment. The main mode of heat 05

226 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. 006): The system is in quasi-steady state. Transmittivity of ethylene vinyl acetate material (EVA) is nearly 00%. 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, 0). The different layers of solar PV panel are as shown in Fig

227 Solar Radiation Q h T bs I x Q conv p n TE Generator T amb w TE Solar PV panel (a) Q c R l (b) Figure 6. Schematic diagram of (a) photovoltaic thermoelectric (PVTE) system and (b) unit thermoelectric generator Silicon Cells Glass Ethylene Ethylene Tedlar Cover Vinyl Vinyl Acetate Acetate Figure 6. Exploded view of Solar PV panel layers (Amrani 007) 07

228 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. 006), where, g is the transmissivity of glass cover, A G A Q G (4) s g PV c c g PV T c c and T are the absorptivity of cell and tedlar, c is the packing factor, G is the solar radiation, and A PV 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 03) where, Q conv conv c amb APV U T T (5) U conv is the overall heat transfer coefficient from the solar cell to the ambient air through glass cover which includes conduction and convection losses (Sarhaddi 00). expressed as (Najafi and Woodbury 03) U conv can be where, l g and lg U conv. (6) kg hpv 08 k g are the length and thermal conductivity of glass cover and h PV 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 03) Q rad pv 4 c 4 sky A T T (7) where is the emissivity of the PV panel and is the Stefan-Boltzmann s constant. The effective temperature of sky ( T sky ) can be written as follow (Wong and Chow 00), T sky Tamb. (8) Now, the heat conduction from the solar cell to the tedlar (Najafi and Woodbury 03),

229 where Q cond cond T c Tbs Apv U (9) U cond 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 03) l si lt U cond. (0) ksi kt In Eq. (0), l si and k si are length and thermal conductivity of the silicon layer. l T and length and thermal conductivity of the tedlar. k T are The electrical power output from the PV panel can be expressed as (Najafi and Woodbury 03) where, el is PV the panel conversion efficiency. P PV G A () g Combining and rearranging Eq. (4) to Eq. () and by applying the assumption, T T into, g c 4 4 G G A U T T A A T T g c c c el G A T pv c pv t c el PV amb pv pv c sky c bs, results. () Eq. () 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 009) P PV PV g T T PV, ref ref C ref (3) G PV ref APV ref TC T (4), ref In Eq. (3), 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. 09

230 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 005): where symbols m, C p, T, q, and T mc q q gen (5) t q gen 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 005) D J 0 (6) t 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 005) as shown in Eq. (7) and Eq. (8) below q T[ ] J [ k] T J [ ] ( E [ ] T) 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 984). Combining Eq. (5) to Eq. (8), the coupled TE equations for energy and charge transfers can be expressed as and T C t T ] J[ k T EJ m ] (7) (8) [ (9) [ ] t [ ] [ ] T [ ] 0 (0) 0

231 where [ ] is the dielectric permittivity matrix and E J represents Joule heat (Antonova and Looman 005). 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 -D version of above equations will be solved to obtain close forms of analytical solutions. For -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 R l as shown in Fig. b. TE elements have length l TE, width w TE, and works between the high and low temperature reservoirs T bs and T amb respectively. A TE module absorbs Q h amount of heat from the back surface of PV panel and rejects Q c 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 semiconductor legs is given by, p type and n type d T I hte p I dt ( T T ) 0 () amb dx ka ka ka dx In Eq. () 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 () can be written in the following form d T dx dt dx T 0 (3)

232 where I ; ka h TE P h ; TE ptamb I. (4) ka ka ka Equation (3) is a linear and non-homogeneous ordinary differential equation. The general solution to Eq. (3) is x D X D X C e C e. (5) T where 4 4 D ; D. (6) Applying thermal boundary conditions at the top surface ( x 0, T T ) and the bottom surface ( x lte T Tamb, ) one can determine the constants, C and C, of Eq. (5) as given below: DLt DLt DLt D Lt Tbs e e Tamb Tbs e e Tamb C ; C DLt DLt. (7) DLt DLt ( e e ) ( e e ) Finally, the temperature distribution inside the nanostructured semiconductor legs can be approximated from bs T x T bs e DLt ( e e DLt amb DLt DLt e T ) e D x T bs e DLt ( e e DLt DLt e T D Lt ) amb e Dx. (8) 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 Q h bs p p C D C D k A C D C D. I T k A (9) p p p p n n n n n n Heat rejected by TE generator to surrounding environment is given by p t Qc I Tamb k p Ap Cp Dp e C DnLt Dn Lt k A C D e C D e. n n n The power output of single TE generator can be calculated as n where ( I P p n n TE R l D L p D p e D p Lt (30) I. (3) n)( Th Tc ). (3) R R i l

233 The thermal efficiency can be evaluated as TE Power Output Heat Input P Q h TE. (33) The overall thermal efficiency of combined PV- nanostructured TE system is given by combining Eq. (3) and (33) = Solar panel efficiency O 6.3 Results Thermal efficiency of TE generator PV 3 (34) 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 TE p type material Bismuth Antimony Telluride BiSbTe is considered to analyze the performance (Poudel et al. 008). 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.. The Seebeck coefficient ( ), electrical resistivity ( ), and thermal conductivity ( k ) are specified as polynomial functions of temperatures as shown in Table 6. and Table 6.3 Poudel et al. (008). These properties are evaluated at an average temperature of working range. A load resistance R is considered equal to internal resistance R to get l maximum power output as per Eq. (3). Figures 6.4 to 6. 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 30 K to 370 K with solar radiation of 00 W/m as shown in Fig The cold side of TE generator is considered to be facing the ambient environment with a range of temperatures ( 53 T 33). The performance between nanostructured TE generator and traditional material c TE generator is also investigated in Figs. 6.6, 6.8, and 6.0. In addition to this, combined system efficiency is also investigated. i

234 Table 6. Operating conditions and dimensional parameters of combined solar PVTE system Parameter Value Solar Radiation, G 0 to 00 Convection heat transfer coefficient for TE generator, h 0 to 50 TE TE generator dimensions, Solar PV panel dimensions, l TE l PV w t TE PV TE w t Transmissivity of glass cover, 0.95 g PV Conductivity of glass cover, k g Thickness of glass cover, Absorptivity of solar cell, L g 0.85 c Packing factor, 0.83 c Absorptivity of tedlar, Reference thermal efficiency, 0.5 T % PV, ref Convection heat transfer coefficient for solar panel, h 5.8 PV Emissivity of solar PV panel, 0.88 Temperature coefficient, PV, ref Reference Temperature, T 5 ref 4

235 Table 6. 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. 008) Property (For n - type and p -type material) Seebeck Coefficient, Electrical Conductivity, Thermal Conductivity, k Temperature range, (ºC) 0 T 50 0 T 50 0 T 50 Figure of Merit, ZT 0 T 50 Polynomial functions of different thermoelectric properties in terms of temperature T.730 T 6 6 T T T.3540 T T T T T T 9 4 T.7650 T T T T 0 5 5

236 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. 008) Property (For n - type and p - type material) Seebeck Coefficient, Electrical Conductivity, Thermal Conductivity, k Figure of Merit, ZT Temperature range, (ºC) 0 T 50 0 T 50 0 T 50 0 T 50 Polynomial functions of different thermoelectric properties in terms of temperature T T 6 5 T T T T T T T T T T T T T T 5 6

237 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, T and the surrounding acts as the heat sink, T. The back surface bs temperature is calculated using Eq. (). The hot surface temperature T 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 93 K to 303 K. For surrounding temperature 303 K, the PV panel back surface temperature increases from 97 K to 358 K due to an increase in solar radiation from 0 W/m to 00 W/m. amb bs 7

238 PV Panel Back Surface Temperature, T bs (K) T amb = 53 K T amb = 63 K T amb = 73 K T amb = 83 K T amb = 93 K T amb = 303 K T amb = 33 K Solar Radiation, G (Wm - ) Figure 6.3 Solar PV panel back surface temperature with variable solar radiation and ambient temperature 8

239 Figure 6.4 shows the temperature distribution along the length of p type and n type nanostructured semiconductor legs. Equation (8) 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. 9

240 Temperature, T(K) h = 0.00 Wm - K - h = 0 Wm - K - h = 0 Wm - K - h = 30 Wm - K - h = 40 Wm - K - h = 50 Wm - K Length, x(m) Figure 6.4 Temperature distribution over the length of nanostructured semiconductor leg p type and n type 0

241 Effect of thermal energy input to the nanostructured thermoelectric system can be analysed from Fig 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 00 W/m and adiabatic side-wall conditions, the heat available to convert is.94 W for nanostructured TE generator and.3 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.

242 Heat Input, Q h (W) 3.5 h = 0-6 Wm - K - h = 0 Wm - K - h = 0 Wm - K - h = 30 Wm - K - h = 40 Wm - K - h = 50 Wm - K Solar Radiation, G (Wm - ) Figure 6.5 Heat input to nanostructured TE generator with different solar radiation and variable convection heat transfer coefficient

243 Heat Input, Q h (W).5 Nanostructured TE material Traditional TE material Solar Radiation, G (Wm - ) Figure 6.6 Heat input comparison of TE generator using traditional and nanostructured material thermoelectric material 3

244 Thermal Conductivity, (Wm - K - ).4. Nanostructured TE material Traditional TE material Temperature, (C) Figure 6.7 Thermal conductivity of traditional and nanostructured TE material as a function of temperature 4

245 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. (3) & (3), 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 00 W/m input solar radiation condition. 5

246 Power Output, P o (W) Nanostructred TE material Traditional TE material Solar Radiation, G (Wm - ) Figure 6.8 Power output from TE generator as a function of solar radiation 6

247 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 6.0 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., % increment compared to traditional TE material. 7

248 Thermal Efficiency, t (%) h = 0-6 Wm - K - h = 0 Wm - K - h = 0 Wm - K - h = 30 Wm - K - h = 40 Wm - K - h = 50 Wm - K Solar Radiation, G (Wm - ) Figure 6.9 Thermal efficiency of nanostructured TE generator with different solar radiation and variable convection heat transfer coefficient 8

249 Thermal Efficiency, t (%) Nanostructured TE material Traditional TE material Solar Radiation, G (Wm - ) Figure 6.0 Thermal efficiency comparison of TE generator with traditional and nanostructured TE material 9

250 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. a solar panel with surface area of m is considered and back surface is covered with TE generators with surface area m on its back surface. Theoretically the power output of solar PV panel remains 00 W whereas TE generators reach up to 00 W of power output at 00 W/m. 30

251 Power Output, P o (W) 00 Solar Panel Output TE Generator Output Solar Radiation, G (Wm - ) Figure 6. Power output comparison of solar PV panel and TE generator 3

252 The solar panel efficiency is investigated by Eq. (3). 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. 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. 6.3 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/m the efficiency of solar PV panel is 0.68% and it rises to 4.88% 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. 3

253 Thermal efficiency, t (%) 0 Solar Panel TE generator Solar Radiation, G (Wm - ) Figure 6. Solar panel conversion efficiency Vs. Solar Radiation 33

254 Combined efficiency, c (%) Solar Radiation, G (Wm - ) Figure 6.3 Combined efficiency of solar PVTE system Vs. Solar Radiation 34

255 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 (-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. 35

256 6.5 Nomenclature A cross-sectional area (m ) C specific heat capacity (kjkg - K - ) D electric flux density vector (Nm C - ) e electron charge (C) E electric field intensity vector (Vm - ) G solar irradiation (Wm - ) h convection heat transfer coefficient (Wm - K - ) I electric current (A) J electric current density (Am - ) k thermal conductivity (Wm - K - ) L 0 Lorentz number (V K - ) l N p P length (m) number of modules Perimeter (m) power output (W) q heat flux vector (Wm - ). q heat generation rate per unit volume (Wm -3 ) Q heat or energy (W) R resistance () S entropy (Wm -3 K - ) t thickness (m), time (s0 T temperature (K) U heat transfer coefficient (Wm - K - ) V w x ZT voltage (V) width (m) coordinate (m) dimensionless figure of merit absorptivity, Seebeck coefficient (VK - ) 36

257 packing factor, temperature coefficient emissivity of PV panel, dielectric permittivity (Fm - ) efficiency a parameter (see Eq. (4)) a parameter (see Eq. (4)), temperature coefficient (K - ) electrical resistivity () m density (kgm -3 ) Boltzmann constant (JK - ), electrical conductivity (Sm - ) Transmissivity, Thomson coefficient (VK - ) electric scalar potential (V) a parameter (see Eq. (4)) Subscripts amb bs c cond conv e el g gen h i L l mp n o p ambient Solar panel back surface tempeature cell, cold side of thermoelectric module conduction convection electronic electrical glass cover generation hot side of thermoelectric module internal Lattice External load maximum powerpoint n type semiconductor material output, overall p type semiconductor material 37

258 PV rad ref s si sky T TE solar PV panel radiation reference sun silicon sky tedlar, thermal thermoelectric module 38

259 CHAPTER 7: NANOSTRUCTURING OF n-type Bi Te.7 Se 0.3 BASED ON SOLID STATE SYNTHESIS TECHNIQUE 7. 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=α σ/k) of TE materials. For a good ZT, the power factor (α σ) 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 (993) discussed the concept of quantum wire (one dimensional) for TE materials. ZT improvement in quantum wire is due to increase in the power factor (α σ) but not significantly due to the low thermal conductivity (Hicks and Dresselhaus 993). Another method to increase ZT was demonstrated experimentally by Venkatasubramanian et al. (00), where ZT of p-type superlattice of Bi Te 3 /Sb Te 3 (bismuth-telluride-antimony) improved from to.4 due to the reduction in the thermal conductivity. Improvements offered by Hicks and Dresselhaus (993) and Venkatasubramanian et al. (00) 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. 008), Superlattice Nanowire Pattern Transfer (SNAP) (Boukai et al. 008), Chemical Vapor deposition (Venkatasubramanian et al. 997), and Molecular-Beam Epitaxy (MBE) (Hicks et al. 996). 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. 03). There is another route to improve the ZT in bulk TE materials called nanocomposite bulk materials (Bottner and Konig 03). 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, 39

260 and solid state reaction (Nolas et al. 000, Zhou et al. 008, Poudel et al. 008, He et al. 006). Figure 7. shows some of the ZT improvements in TE materials in low-dimensional and bulk structures. Figure 7. ZT improvements in low dimensional and bulk TE materials 40

261 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, D, D, 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. 008). 7. 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 Bi Te.7 Se 0.3. The mixture was loaded into the graphite die of diameter 4 mm for the hot press. Bi Te.7 Se 0.3 powders were hot pressed (Vacuum technology, Bangalore, India) under the dynamic vacuum at 500 C with 40 MPa for hours. The resulted disks of the Bi Te.7 Se 0.3 were polished and cut into the bars with the size of 3 mm to measure the transport properties. The X-ray diffraction (XRD) was performed by the Bruker D8 advance diffractometer using Cu- K α radiation with /min. For the crystallographic phase identification, the Rietveld refinement was performed using the FullProf software (Roisnel and Rodriguez-Carvajal, 00). 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 ±0%, respectively. The fractured surface of the hot pressed sample was observed by the FEI Quanta 00. 4

262 Figure 7. 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 Bi Te.7 Se 0.3. The peaks of Bi Te.7 Se 0.3 were indexed with Bi Te.4 Se 0.6 (ICSD#670). Figure 7. Rietveld refinement powder XRD pattern for Bi Te.7 Se 0.3 4

263 Seebeck Coefficient (VK - ) Electrical Resistivity (m m) Figures 7.3 to 7.5 present the transport properties of Bi Te.7 Se 0.3 and Sb.5 Bi 0.5 Te 3. Figure 7.3 show the Seebeck coefficient and electrical resistivity of Bi Te.7 Se 0.3 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 - K - at 87 C Electrical Resistivity Seebeck Coefficient Temperature ( C) Figure 7.3 Seebeck coefficient and Electrical resistivity of sample Bi Te.7 Se

264 Power Factor (mwm - K - ) Figure 7.4 Power factor of sample Bi Te.7 Se 0.3 Temperature ( C) 44

265 Figure 7.5 presents a comparison between the power factors of Bi Te.7 Se 0.3 manufactured using different hot press techniques. Yan et al. (00) used the direct current hot press to synthesize the Bi Te.7 Se 0.3 powder, whereas, the current work used the indirect resistance heating to synthesize the Bi Te.7 Se 0.3 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. 00). 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. 45

266 Power Factor (mwm - K - ) Direct Current Hot Press (Yan et al. 00) Indirect Resistance Heating (Current work) Temperature ( C) Figure 7.5 Comparison of power factor between Bi Te.7 Se 0.3 manufactured via direct current hot press and indirect resistance heating 46

267 Figure 7.6 SEM image of fractured surfaces of hot pressed sample 47