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SYSTEM DESIGN OF COMPOSITE THERMOELECTRICS FOR
AIRCRAFT ENERGY HARVESTING
Dissertation
Submitted to
The School of Engineering of the
UNIVERSITY OF DAYTON
In Partial Fulfillment of the Requirements for
The Degree of
Doctor of Philosophy in Engineering
By
John Mutungi Mativo
UNIVERSITY OF DAYTON
Dayton, Ohio
December 2020
ii
SYSTEM DESIGN OF COMPOSITE THERMOELECTRICS
FOR AIRCRAFT ENERGY HARVESTING
Name: Mativo, John Mutungi
APPROVED BY:
_______________________________ ________________________________
Kevin P. Hallinan, PhD Gregory W. Reich, PhD
Advisory Committee Chairman Committee Member Professor Principal Scientist/Engineer AVD
Mechanical, Aerospace, and Renewable and Aerospace Systems Directorate
Clean Energy Engineering Air Force Research Laboratory, WP
_________________________________ ________________________________
Khalid Lafdi, PhD, D.Sc James J. Joo, PhD
Committee Member Committee Member Professor Adv. Str. Concepts Team Lead, STB
Chemical and Materials Engineering Air Force Research Laboratory, WP
________________________ Robin Steininger, PhD
Committee Member
Tranter, Inc.
___________________________________ ________________________________
Robert J. Wilkens, PhD, P.E Eddy M. Rojas, PhD., M.A., P.E Associate Dean for Research and Innovation Dean, School of Engineering
Professor
School of Engineering
iii
ABSTRACT
SYSTEM DESIGN OF COMPOSITE THERMOELECTRICS
FOR AIRCRAFT ENERGY HARVESTING
Name: Mativo, John Mutungi
University of Dayton
Advisor: Dr. Kevin P. Hallinan
Thermoelectric generator (TEG) elements typically made of Bismuth Telluride
(Bi2Te3) have good thermoelectric properties but are very brittle. In practice, however,
TEG elements often are subject to both mechanical and thermal loading. Although
clamping is the main source of mechanical loading in TEGs, other loadings such as from
vibrations can occur. These can induce shear stresses in the TEGs. When these occur,
failure is far more likely. Therefore, TEG shape and orientation relative to the thermal
and structural loading are critical. In this context, a topology optimization approach is
posed to develop a compliant TEG, capable of maintaining thermoelectric functioning
and sustaining mechanical loadings.
This approach builds on previous research on topology optimization for
multifunctional materials, but uniquely deals with multifunctional design of a composite
TEG. First a tool is developed and validated to study the unique compliant structure and
second a composite 3-D unit cell comprised of structural and thermoelectric materials is
created. The volume fractions and orientation of the two materials are optimized to
support applied structural shear, bending, and axial structural loads and thermal loads. An
optimal structural model was shown to have equal shear and adjoint loads that resulted in
iv
a 1.42% increase in lateral displacement while using 20% less material. A greater void
fraction in the TEG lends to greater compliance. The implication of this research is that it
could help to inform 3-D printing of more compliant TEGs optimized for a particular
application.
However, the tailoring of the TEG for compliance does not come without cost.
The loss of effective cross-sectional area as a result of the voids, increases the thermal
resistance to heat flow. Thus, for an imposed temperature difference, the heat flow
decreases and the power decreases. Optimization is employed to tailor design of the TEG
capable of maximizing power production, while sustaining the applied shear and
vibratory loads. As a specific example, results are presented for optimized TEG legs with
voids, with about 20% in voids to achieve compliance of shear displacement of 0.0636
(from a range of 0.0504 to 0.6079) is only able to generate 80% of the power generated
by a homogeneous TEG construction.
v
I dedicate this work to my family Vivian, Thomas, Daniel, and Deborah, for being
supportive in this journey. We did it together.
vi
ACKNOWLEDGMENTS
To my supervising professor, Dr. Kevin Hallinan, thank you for providing your
time, support, patience, guidance, and encouragement through this work. I am glad you
introduced me to thermoelectrics. Dr. Greg Reich, I appreciate your consistent and firm
advice about the way forward in this endeavor. Thank you for the opportunity to work in
your lab. Dr. Khalid Lafdi, thank you for being supportive and allowing me to work in
your lab. Dr. James Joo, your guidance in structural design is acknowledged. Dr. Robin
McCarthy, thank you for your willingness to share your expertise in thermoelectrics.
I acknowledge the support of Dr. Siddharth Savadatti for his review and guidance
in topology optimization modelling. Dr. Bill Tollner, thank you for reviewing my work.
Dr. Sid Thompson, thank you for checking with me on progress of this work.
Lastly, I acknowledge the support of my late mother Esther K. Mativo who kept
the faith that I will complete this work. To her, and all great mothers, I say thank you for
your unconditional love!
vii
TABLE OF CONTENTS
ABSTRACT ……………………….….…………………………..….…………..…….. iii
DEDICATION ……………………….………….………………………..…………….. v
ACKNOWLEDGMENTS …………….………………………………………………... vi
LIST OF FIGURES …………….……………..……………………………..………… ix
LIST OF TABLES ……………………….…………………………..…….……..…..…. x
CHAPTER I INTRODUCTION: PROBLEM CONSIDERED ………..……… 1
1.1 Motivation .…………………...…………….…………………………….. 1
1.2 Energy Harvesting ………………………………………………………... 1
1.3 Structural Loaded Environment ………………………………………….. 3
1.4 Aerospace Environment …….……………………………………….…… 4
CHAPTER II BACKGROUND………………………………………………… 6
2.1 Need For Flexible Thermoelectric Generators …………………………… 6
2.2 Bismuth Telluride Limitations ……………….…..……………………… 8
2.3 Flexible Thermoelectric Generators ………………….……...………….. 9
2.4 TEGs In Aerospace………………….…………………………………… 12
2.5 Potential Benefits………... ……………………………………………… 17
2.6 Other TEG Applications ………………………………………………… 18
2.7 Prior TEG Research….…………….. …………………………………… 19
2.8 Multifunctional Optimization Process …………..………………………. 20
2.9 Summary ...………………………………………………………….…… 22
CHAPTER III METHODOLOGY……………………………………………. 23
3.1 Goals ….………..………………………………………………………... 23
3.2 General Framework ...……..………………………………………………24
3.2.1 Design Domain .………………………..………….. …………. 24
3.2.2 Defining A Unit Cell For TEG Optimization …………………. 26
viii
3.2.2.1 Tool Description …………………….………..……. 27
3.2.2.2 Physical Model………………………………..……. 30
3.3 Optimizing For Structure Alone …………………………………………. 33
3.3.1 Exploring New Structure……….………………..………..……. 33
3.3.2 Process …...……………………….……………..………..……. 33
3.3.3 Finite Elements .………..………………………..………..……. 35
3.3.4 Structural Model …..…...………………………..………..……. 37
3.3.5 Objective Function ..……………………………..………..……. 38
3.3.6 Exploring Acceptable Topologies …...…………..………..……. 40
3.4 Optimizing For Thermal Only ..…….……………………………………. 46
3.4.1 Thermal Model …....……………………………..………..……. 46
3.4.2 Objective Function .……….……………………..………..……. 49
3.4.3 Cases Examined ..…...………….………………..………..……. 50
3.5 Multi-functional Optimization For Combined Structure and Thermal …...…..
Loads …. 51
3.6 Optimization of TEG Power for Combined Structural, and Thermal Loads 54
3.6.1 TEG Heat Conversion To Power……..…………………………. 54
3.6.2 Examining One Dimension Heat Transfer……....………..…….. 57
3.6.3 Integrated Model ..……...………………………..………..……. 60
3.6.4 Results ………..……………………...…………………………. 65
CHAPTER IV CONCLUSION AND DISCUSSION ………...……………….. 70
4.1 Flexible TEGs Enhancement Of Energy Harvesting ………...……..…… 70
4.2 Recommendation ……………………………………….…………….…. 71
4.3 Summary ……………………………………………..……………..…… 71
REFERENCES ……………………………………………..………………………….. 73
ix
LIST OF FIGURES
1.1 Schematic of a basic TEG with materials A and B, and junctions T1 and T2 ……...3
1.2 Vibratory effects on brittle TEGs ………………….…………….………………….4
1.3 Potential aerospace environments for energy harvesting ……….…………….…….5
2.1 Wireless sensor powered by a TEG installed in an aircraft fuselage ……….……... 6
2.2 Application of TEG in an automobile ……...….……………………………………7
2.3 Typical fatigue stress cycles …….……………………………………………….….7
2.4 Performance of Thermoelectric materials at various Temperatures ….……………. 9
2.5 Bell’s thermoelectric device ……………….……………………………………... 11
3.1 Discretized form elements (e) in a design domain ..……………………………….25
3.2 Illustration of a TEG Unit Cell …………………………………………………… 26
3.3 Illustrated dimensions of a TEG unit cell subjected to structural loads …………...31
3.4 Exploring new structure ……………………………………………………..…… 33
3.5 Standard topology optimization flowchart …………………………………..…… 34
3.6 Element showing 8 dof …………………………….………………………………36
3.7a Suitable leg with two distinct points [1 & 2] touching the top cover ………..…... 42
3.7b Unsuitable leg with one point touching the top cover …………………………..... 42
3.8 Plot of shear displacement versus volume fraction for structurally optimized TEG 45
3.9 TEG unit cell subject thermal loading …………………………….………..…..….. 49
3.10 Optimal topologies for volume fraction a). 100%, b). 80%, c). 60% …………….. 51
3.11 Illustration of equations used to evaluate heat conduction through a TEG leg …... 57
3.12 Detailed flowchart of integrated TEG model with power extraction …………….. 64
3.13 Chart display power generation versus displacement ………..………….……….. 68
x
LIST OF TABLES
2.1 Maximum temperatures for different aircraft systems………………….……..……14
2.2 Vibratory environment in aircraft …………………….…………………..……..….17
3.1 Structural design parameters ………………………………………………......……37
3.2 Optimal TEG considering only structural loadings alone for varied leg size …..…. 42
3.3 Percent material, displacement, and topology …………………………….……..…44
3.4 Structurally optimized topologies from volume fraction near 80% …………….…. 46
3.5 Thermal design parameters ……………………………………………………..…. 49
3.6 Search for best combination of heat flux and displacement ……………..…..……. 52
3.7 Topology differences of structural only and thermal only loadings ……..…..……. 53
3.8 Design parameters for integrated model……..……………………..……………….60
3.9 Effects of Weighting (Ws) on the combined model ………………………………..66
3.10 Displacement, power, and shape of an integrated TEG for vibratory environment 67
3.11 Maximum power generation for the baseline and the integrated model ….....…… 68
1
CHAPTER I
INTRODUCTION: PROBLEM CONSIDERED
1.1 Motivation
Thermal management has increasingly been identified as a significant challenge
for the aircraft by both military and civilian users. The Defense Advanced Research
Projects Agency (DARPA) points out that thermal management of Department of
Defense (DOD) systems often imposes the main obstacle to further enhancements of its
capacity (Cohen, 2014; Keller, 2013). Commercial aircraft have experienced increased
large amounts of heat generation through advancement in avionics, passengers, and
passenger service expectations (Martinez, 2014). Efficiency in thermal management in
both military and civilian aircraft is expected. One of the ways to improve efficiency is
by using thermoelectric devices that capture waste energy and turn it into electricity.
Materials technology used in most designs of thermoelectric devices results to rigid
structures. The devises can be readily used on flat surfaces but not easily adaptable to
non-planar surfaces. Moreover, device failure is imminent when they are subjected to
shear and bending loads. To this end, this study investigates a method of transforming
the rigid thermoelectric design to a flexible one that would enable its use in vibratory
environments.
1.2 Energy Harvesting
Energy harvesting is a method of recovering waste energy and converting it into
usable electrical energy. Many potential sources for energy harvesting exist such as light,
2
vibration, electromagnetic energy, and heat (Torres & Rincon-Mora, 2005). In military
aircraft, heat is increasingly realized in avionics, navigation systems, and precision
deployment systems. DARPA observes that “significant enhancements in fundamental
device materials, technologies and system integration have led to rapid increases in the
total power consumption of DoD system. In many cases, power consumption has
increased while system size has decreased, leading to an even greater problem of heat
density” (Cohen, 2014; Keller, 2013). Commercial aircraft echoes the same sentiment
with their heat concern being from avionics, passengers, kitchen, lighting and
entertainment systems, solar heat, and indirect heat from engines through inside piping
(Martinez, 2014)
In thermal applications, energy harvesting involves the conversion of heat energy
into electricity using mainly Thermoelectric Generators (TEGs) or thermionic devices.
TEGs operate when a temperature gradient is placed across the device (Rowe, 2005).
The device is made of many couples that could be referred to as unit cells. Each unit cell
is made of two dissimilar elements which are connected electrically in series but
thermally in parallel. As depicted in figure 1.1, when junctions T1 and T2 are maintained
at different temperatures, a voltage is developed between a and b which causes a current
to flow as indicated by the arrow.
3
B
T2 T1
A A
Current
a b
Figure 1.1 Schematic of a basic TEG with materials A and B, and junctions T1 and T2
1.3 Structural Loaded Environments
In practice, TEG elements are subject to both mechanical and thermal loading.
Although clamping is the main source for mechanical loading in TEGs, other forms such
as vibration could occur and thus induce unintended stress (Global Thermoelectric, 1992;
Ferrotec, 2011; Laird Technologies, 2011). If the allowable stress is exceeded, then
device failure will result. Axial stress is predominantly found in vertically oriented
elements. Elements oriented in other positions experience both axial and bending
stresses. Therefore, shape and orientation of an element are critical to the absorption of
mechanical loading.
The brittle nature of TEG elements make them very sensitive to principal stresses
(McCarty, 2008). Brittle materials under plane-stress conditions will fail if any point
within the material experiences principal stresses exceeding the ultimate normal strength
of the material as explained by the maximum normal stress theory. Failure starts with a
brittle fracture whereby a rapid run of cracks through stressed material occurs. No plastic
V
4
deformation is visible and in many cases no special pattern on their fractured surface is
observed (see figure 1.2). The fracture could be a transgranular or intergranular in
nature. For ductile materials, the Von mises criterion is often used to estimate the yield.
It is therefore recommended that element maximum stresses should not exceed the Von
mises stresses to prevent failure (Moaveni, 2008).
Figure 1.2 Vibratory effects on brittle TEGs (Meisner, 2011)
Thermal loading to the TEG accounts for the heat from the hot to cold side of the
TEG. But these thermal loads can also affect structural loads. Solids naturally change
sizes and shape as a result of a change of temperature. The elements will experience
thermal stress and strain depending on their constraint conditions and coefficient of
expansion of the material. Non-uniform thermal loading can result in severe structural
alterations that cause premature failure of the device (Laird Technologies, 2011).
1.4 Aerospace Environment
Military aircraft, commercial aircraft, space station, satellites, and associated
infrastructure like radar stations and air traffic management qualify as aerospace
environment (see figure 1.3). Further, the characterization of aerospace applications
could include severe vibration and thermal swings; complex integrated systems; life
5
cycles of about 30 years for aircraft; high cost operations; and extremely high safety
conditions considering human life involvement (Huang, 2009).
Some of the ways the aerospace environment could benefit through thermoelectric
applications include: help reduce aircraft weight through high efficiency cooling reducing
or eliminating liquid cooling that is associated to thermal management weight; improve
system efficiency when paired with solar cells to produce more power; reduce cost such
as fuel cost by engine waste heat harvesting; and reduce carbon emissions (Huang, 2009;
Callier, 2010). Further, aircraft maintenance, the third highest aircraft expense after labor
and fuel, can be significantly be reduced if TEGs are used to power aircraft health
monitoring sensor (Samson, et al. 2010).
Further, the microair-vehicle (MAV) challenge of developing a light weight propulsion
system could be achieved by converting waste heat into useful electricity through the use
of a TEG (Fleming, et al., 2004).
(a) Boeing 787 (2018) (b) Delta_IV (2018) (c) V-22 Osprey (2016)
Figure 1.3 Potential aerospace environments for energy harvesting
6
CHAPTER II
BACKGROUND
2.1 Need For Flexible Thermoelectric Generators
Literature indicates that while a thermoelectric device is quite strong in
compression loading, it is relatively weak in shear that is induced by shock and vibrations
(Laird Technologies, 2011; Global Thermoelectric, 1992; Ferrotec, 2011).
Unfortunately, prime targets for high energy waste heat sources such as aircraft wings
(Figure 2.1), automobile exhaust systems (Figure 2.2), and braking systems experience a
high degree of vibrations which induce shear stresses that are destructive to TEGs.
Figure 2.1 Wireless sensor powered by a TEG installed in an aircraft fuselage (Leeuwen,
2010).
7
(a) (b) (c)
Figure 2.2 Application of a TEG in an automobile (a) A thermoelectric generator in a
Chevy Suburban would provide up to a 5 percent improvement in fuel economy
(Williams, 2011), (b) fabricated TEG, (c) TEG mounted on exhaust system (Meisner,
2011).
Aircraft wing vibrations at large amplitudes for an extended period of time could
cause the wing to experience fatigue stress cycles that could in turn induce shear stresses
(Adams, 2010). Rigid thermoelectric devices placed on such a wing will fail for they do
not tolerate shear stresses. Figure 2.3 shows typical fatigue stress cycles that can affect
aircraft wing (Custom Thermoelectric, 2011; Alibaba, 2011).
Figure 2.3 Typical fatigue stress cycles. (a) Reversed stress; (b) repeated stress; (c)
irregular or random stress cycle (Custom Thermoelectric, 2011; Alibaba, 2011)
8
Figure 2.3a illustrates a completely reversed cycle of stress of sinusoidal form.
Figure 2.3b illustrates a repeated stress cycle in which the maximum stress σmax (Rmax)
and minimum stress σmin (Rmin) are not equal. Figure 2.3c illustrates a complicated
stress cycle which might be encountered in a part such as an aircraft wing which is
subjected to periodic unpredictable overloads due to gusts. Rigid thermoelectric devices
will not withstand such stress fluctuations (Liard Technologies, 2011). Stresses in figure
2.3 are also applicable to vehicle exhaust systems and braking systems in general. The
exhaust system vibrations, in particular, get amplified by unevenness of the road as the
vehicle rides over them, hence increasing the likelihood of a TEG failure.
2.2 Bismuth Telluride Limitations
Bismuth Telluride (Bi2Te3) is a material proven for use in the development of
TEG elements (Rowe, 2006; Tong, et al., 2010; Custom Theremoelectric, 2011). As a
semiconductor it can be manipulated relatively easily to acquire properties of a conductor
or an insulator. It is used widely in thermoelectric generators at temperatures about
450K. Its attractive properties include electrical conductivity of 1.1x105 S.m/m2 with
very low lattice thermal conductivity of 1.20 W/mK and electrical resistivity of 10µΩ m.
Aforementioned properties make Bismuth telluride suitable for attaining a high figure-of-
merit that is critical to TEG operations (Rowe, 2006). The figure of merit (Z) is the ratio
of the electrical power factor (σα 2) and the thermal conductivity (k), where σ is electrical
conductivity, and α is Seebeck coefficient. Figure 2.4 shows figure of merit of materials
used in classic thermoelectric devices. Materials which possess a Z > 0.5 x 10-3 are
usually regarded as thermoelectric materials. Further, their crystalline structure lends to
9
an anisotropic nature, with a density of 7.8587 g/cm3, a modulus of elasticity that ranges
from 45 GPa to 62.8 GPa, and Ultimate Tensile Strength (UTS) of 7.4 GPa (Tong et al.,
2010). As Bi2Te3’s tensile strength is lower than its compressive strength, it will show
brittle behavior (Moaveni, 2008; Beer et al., 2006; Rowe, 2006; McCarty, 2008). For this
reason, TEGs applications on non-planar surfaces, and those that experience tensile or
shear stresses have not feasible for practice (Custom Thermoelectric - PG, 2011).
Figure 2.4 Performance of Thermoelectric materials at various Temperatures (Ferrotec –
TR, 2011)
2.3 Flexible Thermoelectric Generators
Literature shows that nearly 60% of the world’s useful energy is wasted as heat
(Huang, 2009; Callier, n.d). It therefore makes sense to invest in ways to recover this
energy. In previous and current research on TEGs optimization (Hannan, et al., 2014; and
Quan et al., 2013), work has been done on ways to develop flexibility in energy
harvesting. Further, attempts to develop flexible TEGs have been made in the recent past.
Four types of flexible TEG designs have been developed. These have employed flexible
foil structures, wavy-slit technology, carbon nanotubes (CNT), and graphene nanoribbons
10
(GNRs). In their research, Qu, Plotner, and Fischer (2001) determined that flexible foil
substrate technology relies on embedding thermo-elements in epoxy. This design is
constrained by epoxy thickness. Foil substrates are typically made of flexible epoxy film
categorized as thin or thick, with an average 50 µm for thin and about 190 µm for thick.
Additionally, thermocouple strips capable of generating voltage have been embedded in
the epoxy film. Glatz, Muntwyler, and Hierold (2006) argued that because of their limited
thickness, thin film deposited materials have to be laid out laterally rather than vertically,
inducing thermal losses through the supporting material and limiting the integration
density. They further observed that placing a thermocouple onto a thin membrane
reduces thermal losses but does not allow for effective thermal contacting the cold and
hot side via the top and bottom surface of the thin device. He and his team therefore
suggested and developed a thermoelectric wafer in a 190 µm thick flexible polymer mold
formed by photolithographic patterning. Their preliminary efforts, together with Saqr and
Musa (2009) led to a proposal of a model of vertical micro thermoelectric generators. A
TEG of this design tends to have low power capacity due to its micro epoxy thickness
size.
Shiozaki, et al. (2004) proposed a flexible thermopile generator with slits (FTGS)
to permit application of TEGs to non-planar surfaces. Devices of this nature have
thermocouples placed on a polyimide sheet. Each thermocouple is placed at 45o angle
vertically, effectively separating p from n thermocouples. The cold junctions are formed
by bending the thermopile sheet to a wavy form. The design by Shiozaki, Toriyama,
Sugiyama, et al. (2004) forms the wavy and slit flexible thermopile generator. Using an
approach similar to Shiozaki’s, Lon E. Bell registered patent #6,700,052 B2, in March 2,
11
2004, in which he claimed “a flexible thermoelectric comprising: a plurality of
thermoelectric elements; and first and second substrates sandwiching the plurality of
thermoelectric elements and having electrical conductors that interconnect ones of the
plurality of thermoelectric elements, wherein at least one of the first and second
substrates is constructed of a substantially rigid material, said substrates configured to
flex in at least one direction” (see figure 2.5). One of the challenges facing the wavy-slit
design is the reconfiguration of thermal and electrical continuity from floating elements.
This design adds weight because of rerouting continuity components.
Figure 2.5 Bell’s thermoelectric device (U.S. Patent, 2004)
Carbon Nanotubes (CNTs) pose a means for developing composite thermoelectric
devices. CNTs are mechanically strong and light weight. However, their high thermal
and electrical conductivity pose a challenge to their integration into thermoelectric
devices. Koplow et al. (2008) observed that for highly efficient devices, efficient
12
generators should consist of materials with high Seebeck coefficients to provide
significant voltages, low electrical resistivity to minimize internal losses, and low thermal
conductivity to minimize heat losses. The solution to CNT use in TEGs lies in doping
them in order to effect desired properties. Dragoman et al. (2007) observed that the
Seebeck coefficient, α, is strongly dependent on the CNT conductance, G, e.g., the
transmission coefficient carrier through the CNT. The mobility of CNT is μ=(G)lfp/Ne,
where lfp is the mean free path of the carriers and Ne is the charge density. Therefore, the
Seebeck coefficient and the mobility are related through conductance G. Dragoman et al.
(2007) established that in CNTs, the mobility decreases with temperature, while α
increases rapidly at low temperatures and increases slowly in the 200 – 300K range.
In Graphene Nanoribbons (GNR) technology, thin strips are increasingly being
explored for use in TEGs. Their high electrical and thermal conductance places them in a
close category with CNTs. However, a unique difference exists such that thermal
conductivity is significantly decreased under tensile strain, but is insensitive to
compressive and torsional strains (Wei, Xu, Wang, et al., 2011).
2.4 TEGs In Aerospace
Thermal management has increasingly been identified as a significant challenge
for both military and civilian aircraft, according to Wissler, 2009. Recently, the Defense
Advanced Research Projects Agency (DARPA) identified thermal management as a main
obstacle to further enhancement of Department of Defense capacity (Cohen, 2012; &
Keller 2013). Martinez (2014) indicates commercial aircraft have also experienced
increased heat generation through advancement in avionics, passenger capacity, and
13
passenger service expectations. Although TEGs could be considered as part of thermal
management problems, the aircraft environment is unfortunately highly problematic for
their application. The brittle nature of TEGs makes them highly susceptible to
mechanical failure in the environment within aircraft where they might be employed,
such as in the engine, aircraft skin, and landing gear. Further, the rigid structures of these
devices restrict their use to flat surfaces. Were non-planar thermoelectric devices capable
of being developed, their suitability for aircraft applications would be enhanced
considerably. To this end, this study investigates a method of developing more compliant
thermoelectric configurations tailored to the unique structural and thermal loadings they
might be subjected to in an aircraft environment and be applied in locations where a non-
planar configuration is necessary.
Thermoelectric generators could be used to reduce aircraft weight associated with
thermal management by reducing the amount of heat that needs to be managed as a result
of conversion of thermal to electrical energy; improve system efficiency when paired
with solar cells to produce more power; reduce costs such as fuel by engine waste heat
harvesting; and reduce carbon emissions as suggested by Aljazeera and agencies (2015);
Huang (2009), and Callier (2010). Further, aircraft maintenance, the third highest aircraft
expense after labor and fuel, can be significantly reduced if TEGs are used to power
aircraft health monitoring sensors (Samson, Otterpohl, Kluge, et al., 2010). In addition,
Huang (2009) and Fleming, Ng, and Ghamaty (2004) suggest that the micro air-vehicle
(MAV) challenge of developing a light weight propulsion system could be enhanced by
converting waste heat into useful electricity through the use of a TEG.
14
Table 2.1 shows the extreme thermal environments present in various aircraft
systems. Heat generation from aircraft avionics, the more electrical aircraft (MEA), and
the landing gear, provides opportunities for energy harvesting using TEGs, as the
operating temperature of these systems falls within the acceptable range for TEGs
(Hufford, 2014). In avionics applications, the Institute for Interconnecting and Packaging
Electronic Circuits (IPC) suggests some limiting cases for different thermal parameters,
which should be maintained in commercial aircraft. For example, the recommended
worst case thermal conditions for commercial aircraft are a minimum of -55oC and a
maximum of 95oC. However, the actual thermal profile experienced by avionics systems
tends to go beyond these limits. The temperature extremes are much lower than the range
specified, with a high temperature of about 55oC. According to Das (1999) the
temperature difference in cycles and the number of cycles are much higher than the limits
set by IPC. Choi et al. (2011) and FerroTec (n.d.) reveal four additional factors that relate
to failure rate in thermal cycling include (1) the total number of cycles, (2) the total
temperature excursion over the cycle, (3) the upper temperature limit of the cycle, and (4)
the rate of temperature change.
Table 2.1 Maximum temperatures for different aircraft systems (The YF-12A) (Jenkins
and Quinn 1996).
Source Operating Temperature (oF[
oC])
Avionics front of aircraft
Mid fuselage
Engine front
450 [232]
200 – 350 [93-177]
450 – 550 [232-288] Engine Mid
Engine fins/wings
Brakes
600 [316]
450 [232]
750 - 1022[400-550]
Increased electrical power demands in MEA, has established a need for more
battery power. A response to such power demand has led to the creation of more
15
powerful batteries such as the lithium-ion cell LVP65-8-402 battery used in the Boeing
787 fleet. According to the National Transportation Safety Board (2013), a recent testing
showed that the heat generated inside the battery during the heaviest current loading
condition of a full auxiliary power unit (APU) start could expose a cell to temperatures
exceeding the maximum approved operating temperature of the battery (158ºF [70oC])
without detection by the battery’s monitoring system.
Brake temperatures on aircraft landing gears can reach much higher temperatures.
For example, a Boeing 767 fitted with carbon brakes realizes a maximum temperature of
427oC while the MD-11 is rated at 550oC as reported in Boeing (1990). High
temperatures of this nature used with TEGs can offer significant auxiliary power for
aircraft inspections and as supplement power on the ground or in flight. Although the
TEGs based upon Bismuth Telluride are limited to a maximum temperature of 250oC,
other TEG materials could work at these temperatures. Thermal Electric Corporation
(n.d.) provides an example of, Hybrid BiTe – PbTe can operate for temperatures up to
360oC, SnSe – PbSnTe up to 600oC, Calcium Manganese (CMO) up to 800oC, and CMO
cascade with Bi2Te3 stacked up to 600oC.
Although the aerospace industry has made inroads using TEGs as its “space
battery” for deep space applications (Das, 1999), the aircraft sector has lagged behind.
The main challenges the aircraft sector faces in the use of TEGs are vibrations and non-
planar surfaces. According to Adams (2010), aircraft vibrations at large amplitudes for an
extended period of time cause fatigue stress cycles that could in turn induce shear
stresses. Such vibrations are often experienced in wings and landing gears, which are
both prime candidates for use of TEG technology. Rigid thermoelectric devices placed on
16
such a wing or landing gear will fail due to their inability to withstand such shear stresses
(Laird Technologies, n.d.). In addition, the landing gear incorporates many non-planar
components which pose a challenge to clamping rigid TEGs for energy harvesting.
For purposes of reporting type of vibrations on aircraft, Carbaugh, Carriker,
Huber, et al. (n.d.) suggest two categories of which one is high frequency tactile vibration
typically, more than 25 Hz, and the other is low frequency, typically less than 20 Hz.
High frequency vibrations could be associated with sound that related to a small-mass
component acting on the frame, examples being a loose door, access panel or fairing. On
the other hand, low frequency vibrations relate to large-mass components acting on the
airframe, examples being the rudder, horizontal stabilizer, or elevator. Table 2.2 describes
the types of vibratory loads present in various aircraft and flight conditions.
17
Table 2.2 Vibratory environment in aircraft: Sample - Boeing series (Carbaugh, et al.,
n.d)
Airplane Flight Condition Symptom
737-300/-
400/-500
All phases of flight
Takeoff and approach
Climb and level flight
High-freq. vib. and noise; vary with speed Vibration and noise in wing
Low-freq. vib. in flight deck
747
Takeoff
Climb 17000–31000ft
Cruise
Vibration in the nose area
Vibration
Flight deck rumble
777 Taxi and takeoff
Climb
Loud grinding noise & vibration at door 2 area
floor
Strong vibration felt through floor near seat
row 19
DC-10
All phases of flight
Climb
Climb
High-level vibration and vibration near wing;
varies with airspeed
Buzz in floor and sidewall on left side of airplane forward of wing
Low frequency vibration in cabin adjacent to LE of wing
MD All phases of flight
Takeoff
Cruise
Vibration in forward gallery
Cabin vibration and associated with whining
noise
Cabin vibration
2.5 Potential Benefits
Aircraft maintenance is complex and costly. The use of TEGs in monitoring
applications could reduce cost by reducing the need of physical inspection. For example,
monitoring the external skin of the aircraft is important for the early detection of cracks
due to wear, or damage caused by bird collisions. Autonomously-powered sensors can be
installed in difficult-to-access locations and require no maintenance. TEGs can be used in
series to power not only the sensors, but also the electronics required to transmit the
18
sensor readings to a central location (Wright, 2010). The need is not only for predictive
maintenance, but to provide auxiliary power for inspection of the aircraft while on the
ground and supplemental power while in flight. In extreme design, a solar powered
aircraft combined with TEG technology could fly longer missions without the need of
fossil fuel propulsion [Impulse]. Unmanned aircraft vehicles would be prime targets for
such applications.
2.6 Other TEG Applications
The unique operation of TEGs is helping them play an increasing role in the
conversion of waste heat into electricity for temperatures of up to 230oC. But their
potential for future use is large. TEGs are unique in that they have no moving parts, are
reliable, are silent, and can be operated unattended in hostile, inaccessible environments
(Roeser, 1940; Telkes, 1947). These characteristics make TEGs the power provider of
choice for deep space expeditions and harsh climate operations using radioactive isotopes
as heat sources (Rowe, 1983; Macklin & Moseley, 1990). Increasingly, because of their
ruggedness, portability, and ready ability to produce electricity, TEGs are being adapted
to military and civilian application to power systems in aircraft, cars, and trains (Bell,
2008). However, Bi2Te3 TEGs are limited by their brittleness. This characteristic poses a
unique challenge for employing these devices on non-planar surfaces where they can
experience shear loadings or in vibratory environments (McCarty, 2008; Choi et al.
2011).
19
2.7 Prior TEG Research
Most of the early research on TEGs focused on use of metals and metal alloys
(Telkes, 1947; Macklin & Moseley, 1990; Telkes, 1954; Agabaev, 1979). In the recent
past and currently, research on material characterization, performance optimization, and
energy conversion has been conducted widely (Rowe, 2006; Rowe, 2012; Goldsmid,
2010; Sutton, 1966). Initiatives are being explored to improve the figure of merit which
is at the heart of TEG operations in power generation (Kim et al., 2015; Poudeu et al.,
2006; Tuley & Simpson, 2017). TEG configurations to improve efficiency in power
generation continue to be of interest (Crane & Bell, 2006; DiSalvo, 1999; Hashim et al.,
2016). A TEG research aspect that has not received much attention is the influence of
geometric configuration in power generation (Sahin & Yilbas, 2013). Past work on the
TEG geometric aspect has been examined (Thacher, 1982; LeBlanc, 2014; James, 1962)
without considerations of its effects on power generation and efficiency, and even
without consideration of current flow (Sahin & Yilbas, 2013). Creative ways have been
developed to use TEGs on non-planar surfaces. Some report use of flexible foil substrate
technology (Qu et al., 2001; Saqr & Musa, 2009; Glatz et al.); others have used flexible
thermopile generator with slits (Shiozaki et al.); and others a plurality of thermoelectric
elements (U.S Patent, 2004). In addition, new technologies such as carbon nanotubes
(Rowe, 2012; Rowe 2006); graphene nanoribbons (Goldsmid, 2010) and nano-films
(Arora et al., 2017) are being examined as alternatives to the brittle bismuth telluride.
In spite of its brittleness, the allure of using bismuth telluride and its alloys in
energy harvesting is very attractive due to their high energy conversion efficiency at
ambient temperature for achieving power generation without requiring any moving parts
20
or cooling systems in electronic devices (Mehta et al., 2012). Mativo and Hallinan (2017)
effort focused on tailoring the design of a printable TEG for specific structural and
thermal loads. However, this effort didn’t account for power generation.
2.8 Multifunctional Optimization Process
Design for a flexible TEG has competing objectives. A multifunctional
optimization is a tool used to evaluate competing objectives in a design space for the
purpose of identifying critical level of each one towards accomplishing a specific goal
successfully. Four multifunctional optimization processes are reviewed in this section
(Vincent, 1983; Athan & Papalamros, 1996; Pakala, 1994).
The Pareto method seeks to reduce one criterion without increasing any others.
Under this assumption, the set of solutions being considered can be reduced to an
attainable set termed the Pareto set which consists of Pareto optimal points. In this
method, a Pareto point in the design space is considered optimal if no feasible point exists
that would reduce one criterion without increasing the value of one or more criteria. The
designer can then select the optimal solution that meets subjective trade-off preferences
(Athan & Papalamros, 1996; Vincent & Grantham, 1981). Its analysis technique
prioritizes problem-solving work so that the first piece of work done resolves the greatest
number of problems.
Goal Programming Method: Ignizio, a renowned scholar in applying goal
programming to engineering problem solving, stated that there is not now, and probably
never shall be, one single, “best” approach to all types of multi-objective mathematical
programming problems. However, he suggested that, a general goal programming
21
method could be considered. His approach involves an initial prioritization of objective
criteria and constraints by the designer. Goals are selected for each criterion and
constraint, and “slack” variables are introduced to measure deviations from these goals at
different design solutions. Goal values are approached in their order of priority and
deviations from both above and below the goals are minimized (Ignizio, 1976)]. The
result is a compromised decision. The concept of Pareto optimality is not relevant to this
approach (Papalambros & Wilde, 2000).
Game theory has been used in multi-criteria optimization formulations (Vincent,
1983; Vincent & Grantham, 1981). If there is a natural hierarchy to the design criteria,
Stackelberg game models can be used to represent a concurrent design process (Pakala,
1994). Some game theory strategies will result in points that are not Pareto points,
because they make different assumptions about a preference structure. For example, a
rivalry strategy giving highest priority to preventing a competitor’s success would likely
result in a non-Pareto point.
The simplest approach, as a first step, is to select from the set of objective
functions that one can be considered the most important criterion for the particular design
application. The other objectives are then treated as constraints by restricting the
functions within acceptable limits. One can explore the implied trade-offs of the original
multiple objectives by examining the change of the optimum design as a result of changes
in the imposed acceptable limits, in a form of sensitivity analysis or parametric study
(Papalambros & Wilde, 2000).
22
2.9 Summary
Flexible TEGs will open more waste heat harvest applications to both planar and
non-planar surfaces using the same type device therefore reducing or eliminating the need
to custom make for surface variations or making special adapters to accommodate
mounting of rigid devices to uneven surfaces. Demand for such TEGs will increase and
bulk production will reduce device cost.
23
CHAPTER III
METHODOLOGY
3.1 Goals
The broad goal of this research is to develop a general framework for
optimization of a printed TEG that can be used to design a TEG for applications where
the TEG is subject to both thermal and structural loads. Using this general framework, the
following specific goals are set; namely to optimize a TEG for: (a) structural loadings
alone; (b) thermal loadings alone; (c) for combined structural and thermal loadings; and
(d) for optimal TE power while also insuring some specified structural integrity.
Fundamentally, the last goal represents the priority for this research. Goals (a) – (c) are
simply posed in order to help develop confidence in the final TEG model.
This research considers the possibility of employing TEGs in environments where
vibratory or shear loadings are present. TEGs in general cannot sustain such loadings;
typically, they are designed only to support compressive loadings. Thus, in a sense, this
research aims to develop a flexible TEG material through the control of its topography.
This means that we are looking to create TEG topographies which are not completely
solid; instead we are looking to create controlled porosity in the TEG legs in order to
accommodate expected shear structural loadings.
24
3.2 General Framework
To achieve the aforementioned goals, a general optimization framework is
established based upon the possibility that a TEG could be printed, e.g., TE material
could be deposited on a surface in such a way as to support transverse structural loadings.
The framework posed ultimately relies upon the establishment of a unit cell model of a
potentially complex TEG system.
The approach posed follows the topology optimization approach originally
developed by Sigmund and Maute (2013). In this framework, a composite structure can
be optimized to meet the applied loadings.
Applied here, this methodology makes use of a solid isotropic material with
penalization (SIMP) to create an optimal composite and compliant TEG tailored to the
specific mechanical and thermal loads present in aircraft it may be subject to.
To this end, the following subsection describes the design domain established,
discusses the tool used in the numerical experiment, defines a unit cell model of the more
complicated physical systems, and discusses the practical implications of this approach to
the development of a compliant TEG device.
3.2.1 Design Domain
In our numerical experiment platform, the design domain is a space that allows
the development of topologies that have been subjected to loads and constraints. The
SIMP approach considers a design domain that is discretized into finite elements. Each
element is assigned a density amount, ω, which is treated as the design variable. More or
less, this approach permits control of the placement of material within these elements.
25
Also, the SIMP approach presumes that a constant load applied to a material with a high
stiffness (K) will result in a minimum displacement (min(u)) which can be written as
min(fu). The applied load and displacement is spread across the material and is
accurately represented as min(∫(𝑓𝑢) 𝑑Ω) where 𝑑Ω is the material domain. For a generic
collection of finite elements such as illustrated in figure 3.1, the discretized design
domain experiences small displacements (u) to the right due to an imposed shear force
(f). Considered here is a 2D design domain; thus, a unit thickness depth is assumed.
Figure 3.1 Discretized form elements (e) in a design domain
The design domain considered is space for building a TEG unit cell which
comprises two legs and a street (void) between the two legs. Ultimately the optimization
will control the placement of solid material and voids within the TEG legs. The street and
periphery to the legs is assumed to be effectively thermally insulated. Two fixed solid
regions in this design are the top and bottom covers of the TEG unit cell. The solid leg
region is variable and can be changed for optimizing power generation while supporting
both structural and thermal loads. Figure 3.2 shows the general TEG unit cell with
support and loads.
26
y Top cover Conductive metal
Leg: S1(x,y) Leg: S2(x,y)
Street (insulated)
Fixed x
Bottom cover
Figure 3.2 Illustration of a TEG unit cell
In this unit cell illustration, S1 and S2 are the unknown leg shape, size, with
topology of thermoelectric p-type and n-type legs (e.g., contiguous p- and n-type
materials) separated by an insulated space between them known as the street. The S1 and
S2 legs carry both structural and thermal loads. In this figure the legs are shown as
rectangular elements, however, here they are intended only to represent the geometrical
domain where a composite TEG system can be developed. The tool used to create the
domain space and ultimately unit cell is discussed followed by the creation of a unit cell.
3.2.2 Defining A Unit Cell For TEG Optimization
Before defining the unit cell considered for this the TEG application, it is first
important to describe the tool that will be employed to distribute material throughout the
domain in order to provide both the thermoelectric function and support loads (shear and
normal).
27
3.2.2.1 Tool Description
The “tool” used to layout the design domain and create and study the TEG unit
cell is a finite element based tool developed within MATLAB. This tool was originally
developed for simple well-posed topology optimization problems (Bendsoe and Sigmund,
2004). An example of a simple problem would be to develop and optimize a topology for
a column that would support a certain structural load, given specific constraints. Our
case, however, considers an ill-posed problem that optimizes the TEG legs to support
both structural and thermal loadings while maintaining capacity for generating power.
The tool is therefore modified to create a design domain that is used to seek an optimized
complex TEG unit cell. The known quantities for the TEG are structural and thermal
loads, support conditions, the area of the structure to be constructed, and restrictions such
as the location and size of prescribed holes (void) and solid areas. The physical size and
the shape and the connectivity of the structure are unknown.
The MATLAB code developed is divided into four main sections. First, the main
section of the program defines the inputs and starts the distribution of the material evenly
in the design domain. The numerical experiment begins with the initialization of design
variables and physical variables. Here, the design variable is the volume fraction limit of
the material; e.g., we will be developing optimal topologies for specified overall TE
material volume fractions. The physical densities are initially assigned a constant
uniform value, but are iteratively updated to yield an optimal ability to support the loads.
Further, in a discretized design domain like ours, the stiffness is determined by using a
power-law interpolation function between void and solid (Bendsoe, 1989). The power-
28
law penalizes intermediate density values, driving them towards solid or void, hence
producing the final output of the program.
The second code section is the Optimality Criteria (OC) which updates the design
variables depending on the used and remaining material. The OC is formulated on the
grounds that if constraint 0 ≤ x ≤ 1 is active, then convergence is achieved with a positive
move-limit of 0.2 and a damping coefficient of 0.3, for compliant designs (Liu and Tovar,
2014; Vijayan and Karthikeyan, 2013; Bendsoe, 1995). Material spread entirely on an
element is considered to have a value of 1. A 0.2 positive move-limit is the minimum
material on an element that has the ability to attract any other material from neighboring
elements. The damping coefficient of 0.5 or higher is associated with minimum
compliance problems and less than that value is associated with compliant designs. These
values can be changed to speed convergence. The iterative process continues with
termination occurring when a maximum number of iterations is reached or where a
tolerance of the material distribution is relatively small, for example if available material
is equal to 0.01(Liu and Tovar, 2014).
The third section of the code is a mesh-independency filter whose function is to
avoid numerical instabilities. Liu and Tovar (2014) discuss the function of “a basic filter
density” as
𝑖 = ∑ 𝑯𝑖𝑗𝑣𝑗𝑥𝑗𝑗𝜖𝑁𝑖
∑ 𝑯𝑖𝑗𝑣𝑗𝑗𝜖𝑁𝑖
…………………………………. (3.1)
where Ni is the neighborhood of an element xi with volume vi, and Hij is a weight factor.
The neighborhood is defined as
𝑁𝑖 = 𝑑𝑖𝑠𝑡(𝑖, 𝑗) ≤ 𝑅 ………………………………. (3.2)
29
where the operator dist(i, j) is the distance between the center of element i and the center
of element j, and R is the size of the neighborhood or filter size.” The weight factor is
defined as the distance between the neighborhood elements, such as
𝑯𝑖𝑗 = 𝑅 − 𝑑𝑖𝑠𝑡(𝑖, 𝑗) ………………………………… (3.3)
where j 𝜖 Ni. The filtered density 𝑖 defines a modified physical density that is now
incorporated in the topology optimization formulation (Liu and Tovar, 2014, p. 1177).
In the MATLAB code, the element sensitivity is modified during every iteration
using the following
𝜕𝑐(𝐱)
𝜕𝑥𝑖=
1
max (𝛾, 𝑥𝑖)∑ 𝐻𝑖𝑗𝑗𝜖𝑁𝑖
∑ 𝐻𝑖𝑗𝑥𝑗
𝑗𝜖𝑁𝑖
𝜕𝑐(𝐱)
𝜕𝑥𝑗……………(3.4)
where x is current design, and γ (= 10-3) is a small number in order to avoid division by
zero (Liu and Tovar, p. 1186). This process ensures that elements gain or lose material
depending on the critical amount available and reaction to loads, based on set filter
settings.
Finally, the fourth section of the code is the finite element code where a global
stiffness matrix is formed by a loop over all elements and executed. For the structural
evaluation, an 8 x 8 matrix is formed following a 4 node bi-linear element. Since heat
conduction is considered to move in mono-linear in an element, a 4 x 4 matrix is formed.
Supports and loads are added as well as the young’s modulus and Poisson’s ratio. The
complexity of combining the structural and thermal matrices is discussed in section 3.5.
The material redistribution algorithm, driven by a sensitivity filter, determines
changes in material distribution element to element. This material distribution is
dependent on what is being optimized and, of course, the boundary conditions applied to
30
the design domain. An ideal solution which optimizes for structural loads alone, must
render a full dense material, ω = 1. For thermal optimization alone, the ω = 1 solution for
an entire design domain will be optimal in order to minimize the temperature drop within
a material. However, different non-trivial solutions will result if the material cannot fill
an entire design domain.
3.2.2.2 Physical Model
A unit cell physical model is posed. This model is comprised of two legs, a street,
heat spreader, and a heat sink (figure 3.3). In practice, one or many more unit cells would
be present. The unit cell shown in this figure considers mechanical boundary conditions
that include the applied compressive loads from the top, and shear loads from the side on
the upper end of the left leg, and a fixed (zero deflection) or anchored surface, at the
bottom surface. The upper surface is assumed capable of deflection. Additionally, the
geometrical domain is assumed fixed. The thermal loading applied on each TEG leg is
subjected to a distributed heating on the top surface. The sides are assumed insulated.
The temperature of the bottom surface is set to a constant value, controlled by the heat
sink to which heat is being rejected.
31
Figure 3.3 Illustrated dimensions of a TEG unit cell subjected structural and thermal
loads
A Marlow TG12-6 TEG is used as the baseline dimension for the design and
study of the unit cell. A purposeful choice to use Marlow TG12-6 TEG was made
because: a) of access to industrial design and manufacturing information was available
through industry connection; b) of being a single stage TEG that reduces the design
complication; c) its power generation had been studied before and published and could be
compared for accuracy (Kanimba & Tian, 2016); and d) its temperature range would
allow its application in aircraft avionics and inspection aspects. These reasons were
considered as a baseline model was sought. Therefore, a unit cell model of a
dimensionally proportional design domain of 60 units in the x- direction and 20 units in
the y-direction, a 3:1 ratio for breadth to height, was created (figure 3.3).
Dimensions of the unit cell model are as follows: the width of each TEG leg is
0.055” which is also the measure of the street width. The total width is therefore 0.165”
(4.2 mm). The TEG leg height is 0.055” (1.4 mm), a 3:1 ratio for breadth to height. The
cross-sectional area of each leg space is 0.055” x 0.055” which is 0.003 in2 (1.95 mm2).
32
The maximum uniformly distributed compression load for the Marlow TG 12-6
generator is set to 200 psi (Marlow Industries, 2015). While there is no shear load
recommendation for Marlow TG 12-6 generator, the model uses unitless displacements
for both shear and adjoint of 1 and -1 respectively. The scale for the displacement ranged
from maximum (1) to minimum (0), with the negative sign indicating an opposing
displacement. These values ensure a balanced lateral loading.
A spring-like support is used to depict the lateral shear and adjoint displacements.
Rather than applying a point load or force, a displacement boundary element is used to
enforce the displacement. The boundary elements are quite useful for the sub-structuring,
a process in which a portion of a model is analyzed with a finer mesh without re-
computing the response of the entire model. The spring is aligned with the global axes
(Maoveni, 2008).
The configuration shown in figure 3.3 could represent an aircraft skin, with heat
input from the fuel (used for heat exchange to the external environment) or a number of
other combined thermal and structurally loaded environments. In this unit cell, S1 and S2
are the unknown leg shape, size, with topology of thermoelectric p-type and n-type legs
(e.g., contiguous p- and n-type materials) separated by an insulated space between them
known as the street. The S1 and S2 legs carry both structural and thermal loads. In this
figure the legs are shown as rectangular elements, however, here they are intended only
to represent the geometrical domain where a composite TEG system can be developed.
The street serves as insulation space between the legs and is configured as fixed void
space in the unit cell design model. To counter the applied shear load indicated in figure
3.3, an adjoint load is introduced to the right of the model.
33
What follows are specific applications of the general framework to structural
loadings alone, thermal loadings alone, and combined structure and thermal loadings.
3.3 Optimizing For Structure Alone
3.3.1 Exploring New Structure
The exploration for a new compliant structure involves the removal of material to
determine if it can support structural loads while maximizing power generation. The
effort in the study is geared toward optimizing the holes shown in figure 3.4 model, to
find the best void structure, and develop optimal configurations for different loads for any
environment.
Figure 3.4 Exploring new structure
3.3.2 Process
Figure 3.5 shows a flowchart that describes the process to create a unit cell and
test it for structural loading. The flowchart is a representation of the standard tool which
is a MATLAB code developed by Sigmund and Bendsoe1999. It starts with inputs such
as the perimeter area to create the design space. Boundary conditions and loads are
prescribed. Input of the design variable is loaded. An even distribution of the design
34
variable is made across the design space. The program is run as described in the general
framework section 3.2.2.1, and as depicted in figure 3.5.
No
Yes
Figure 3.5 Standard topology optimization flow chart
Finite Element Analysis – resulting
displacements and Strains
Sensitivity Analysis – design variable
through equilibrium equations and
finding derivatives of displacements
with respect to design variables
Update material distribution
Is material
distributed to achieve
min. obj. function?
Initialize design space, e.g. volume fraction –
homogeneous distribution of material
Material distribution
Displacements
Outputs
Filtering – moves material to
elements in areas where critical mass
is formed according to set filter size
Stop
Resulting density
distribution plotted
35
As shown in the flowchart, also discussed by Liu & Tovar, 2014, the iterative
process continues with termination occurring when a maximum number of iterations is
reached or where the change in material distribution from iteration to iteration is within a
specified tolerance, for example if available material is equal to or less than 0.01.
Convergence takes place because the design variable is less than the convergence criteria.
The finite element code makes use of the sparse option in MATLAB. The top left
and top right node element numbers are used to insert the element stiffness matrix at the
right places in the global matrix. Both nodes and elements are numbered column-wise from
left to right. Each node has a designation of two degrees of freedom which are translational
for horizontal and vertical of an element node (figure 3.5). Fixed and free degrees of
freedom are easily entered in the code. Youngs modulus and Poisson’s ratio can be entered
in the code as well.
Step one in the process first requires specification of the applied loads, support
conditions, volume domain of structure to be constructed, and geometry of the void and
solid areas. In developing a design domain, a volume fraction (φ) is defined as a ratio of
the object volume versus the available space volume in the material domain (Ω).
3.3.3 Finite Elements
The design domain is discretized into elements. A single element is shown (figure
3.6). Each element has four nodes each with two degrees of freedom (dof); totaling to 8
dof. Ke is the element stiffness matrix with a dimension of 8 x 8.
36
Figure 3.6 Element showing 8 dof
A 2D-quad element illustrated in figure 3.5 has four nodes. Each node has the potential to
translate in the x direction (horizontal) and in the y direction (vertical), when a force is
applied to the element. The force applied is proportional to element stiffness and resulting
displacement. The relationship is captured in equation 3.5.
0 = Ku-f 3.5
where applied force f, is equal to the product of element stiffness K and its displacement
u. The stiffness matrix K depends on the vector ρ of the element-wise constant material
densities in each element, e = 1… N. Therefore, K can be written as
𝑲 = ∑ 𝜌𝑒𝑝𝑲𝑒
𝑁
𝑒=1
………………………… …………………………………3.6
where Ke is the global element stiffness and p is power which penalizes intermediate
densities. A p of 3 or greater is preferred (Bendsoe & Sigmund, 2004; Liu & Tovar,
2014) because a high p level makes it uneconomical to have intermediate densities in the
optimal design. Further, the p of 3 or greater will result in obtaining a true “0-1” design
for our active volume constraint (Bendsoe & Sigmund, 2004).
37
3.3.4 Structural Model
Table 3.1 lists the design parameters considered for the structural model. These
parameters include: design domain, geometry, fixed void region, fixed solid region,
variable solid region, fixed nodes, free nodes, uniformly distributed compression loads,
shear displacement, adjoint displacement, and mechanical material properties.
Table 3.1: Structural design parameters
Design parameter Quantity
Design domain 60 units in x direction 20 units in y direction Fixed void region 20 in x direction and 16 in y direction
Leg height 16
Leg cross-sectional area space 400 Bismuth Telluride Density 7.8587 g cm-3
*Bismuth Telluride Young’s Modulus 8.1 – 50 GPa
Bismuth Telluride Ultimate Tensile Strength
Bismuth Telluride density Bismuth Telluride melting point
7.4 GPa
7.37 g/cm3 585oC
Poisson’s Ratio 0.23
* Santamaria, Alkorta, and Gil Sevillano, 2013 realized a plastic value for Bismuth
Telluride under high pressure is assumed to be 3 GPa.
The imposed boundary conditions are represented as follows:
• Fixed end: x0 = 0; xf =0; where x0 is initial position of x, and xf is final position of
x.
• Free end: xL’ = 0; xL
’’ = δ; where xL’ is the initial position at the top of the
element, and xL’’ is the final position at the top of the element, and δ is the lateral
displacement.
Also, the length of the legs is fixed, but the width of the legs can vary subject to the
ability to sustain the imposed structural loads.
A matrix determined analytically using symbolic manipulation software by
Bendsoe and Sigmund, 2004 is modified and used in this numerical experiment. Earlier,
38
in figure 3.6 an illustration of elements’ degrees of freedom was given. It follows that, the
matrix terms k(1), k(2), …, k(8) represent node stiffness. The kij conditions apply for
each node, such that a force in the “i” direction to have a deformation of 1 in the “j”
direction only. Table 1 provides physical properties of Young’s modulus and Poisson’s
ratio used in the matrix computation. A Young’s modulus (E) of 8.1 and Poisson’s ratio
(ν) of 0.23 are used in the matrices.
𝑘 = [0.5 − 𝑣
6,0.125 + 𝑣
8,0.25 − 𝑣
12,−0.125 + 3𝑣
8,−0.25 + 𝑣
12 ,−0.125 − 𝑣
8,𝑣
6,0.125 − 3𝑣
8]
𝑘𝑒 =E
1 − 𝑣2∗
[
𝑘(1)𝑘(2)𝑘(3)𝑘(4)𝑘(5)𝑘(6)𝑘(7)𝑘(8)
𝑘(2)𝑘(1)𝑘(8)𝑘(7)𝑘(6)𝑘(5)𝑘(4)𝐾(3)
𝑘(3)𝑘(8)𝑘(1)𝑘(6)𝑘(7)𝑘(4)𝐾(5) 𝑘(2)
𝑘(4)𝑘(7)𝑘(6)𝑘(1)𝑘(8)𝐾(3) 𝑘(2) 𝑘(5)
𝑘(5)𝑘(6)𝑘(7)𝑘(8)𝐾(1) 𝑘(2) 𝑘(3)𝑘(4)
𝑘(6)𝑘(5)𝑘(4)𝐾(3) 𝑘(2) 𝑘(1)𝑘(8) 𝑘(7)
𝑘(7)𝑘(4)𝐾(5) 𝑘(2) 𝑘(3)𝑘(8) 𝑘(1) 𝑘(6)
𝑘(8)𝐾(3) 𝑘(2) 𝑘(5)𝑘(4) 𝑘(7) 𝑘(6) 𝑘(1)]
where ke is the global element stiffness matrix.
3.3.5 Objective Function
A representative static load is applied in the design tool to study and seek for
structural optimization. No vibratory loads are applied in the design tool. The process to
achieve this desired outcome is to reduce structural material carefully without
compromise of structural fidelity. The systematic material reduction is conducted by
manipulation of the volume fraction for the TEG leg. Therefore, an objective function for
maximization of the TEG leg displacement has the form in equation 3.7.
39
max (u)…………………………..………………………… 3.7
subject to the following constraints:
0 ≤ 𝜌 < 1 is material distribution (design variable)
between 0 and 100%
𝒖∗ ≥ 0.029
r = 0 = ku-f
∑𝑣𝑒𝜌𝑒 = 𝑉 ∗ 𝜑, 0 < 𝜌𝑚𝑖𝑛 ≤ 𝜌𝑒
𝑁
𝑒=1
≤ 1, 𝑒 = 1,… ,𝑁
where ρ is the element-wise constant material densities in element, e = 1, …, N. The u* is
the calculated minimum displacement, equation 3.8, that is able to supports the loads.
The stiffness matrix K depends on the ρ. r is the residual in obtaining the structural
equilibrium. For topology optimization the equilibrium r = 0 is found using an iterative
procedure. u and f are displacement and load vectors, respectively. The shear and adjoint
loads are unit valued displacements at the boundary. The displacements can be applied in
lieu of a force on a node (Porter, M. A., 1994).
The preferred displacement u* should be ≥ 0.029. This calculated value of 0.029
is considered minimum displacement from a 100% volume fraction of a unit cell. The u*
value is obtained using the following equation:
𝒖∗ =𝑃𝐿3
3𝐸𝐼 ………………………………………….. 3.8
where P is a point load, L is the length, E is the modulus of elasticity, and I is the inertia.
The goal is to find a maximum displacement of TEG legs that will insure that the
mechanical loads are sustained (i.e., that there is sufficient compliance to tolerate the
loads) – ultimately while maintaining the ability to generate maximal power.
40
3.3.6 Exploring Acceptable Topologies
The interest in this study is to create compliant TEGs. Therefore, the topology
optimization process controls the placement of material in the two legs of the unit cell. A
successful optimized structural model requires that developed TEG should be able to
accommodate translation induced by the shear load. Each leg is expected to support
translation. A leg with one point connected to the TEGs top cover would allow a minimal
lateral displacement before it results to undesired potential rotation. Two points
connecting the TEG leg to the top cover would allow a higher lateral translation with less
induced friction that three or more points would cause. According to Alread & Leaslie,
2007, there are three basic ways that structural members can be connected. Pin
connection allow members to rotate but not translate, moment connections allow neither
translation nor rotation relative to one another, and rollers that allow only translation. Our
design follows the later.
Table 3.2 shows the optimal TEG structures obtained for variable porosity. In the
table, the topologies shown in columns 2 – 4 are created by performing two changes, first
by varying TEG leg width size, an action done by varying width of the street, and second
by changing the volume fraction or the amount of the TEG material in the legs.
For our case, a baseline model B is a topology whose street width size is equal to
the width size of each leg as shown in column 3. Topologies with a double street width
size 2B, halves the leg width size, are shown in column 2, while those with half the
baseline street width size ½B are presented in column 4.
The volume fraction is varied from 100% to 60%. This effort is done to examine
topologies whether topologies created with a lower volume fraction of TEG material can
41
achieve greater compliance (u) while supporting the structural loads. The volume fraction
is presented in column 1, while the displacement is shown in column 5.
Results of the effort presented in Table 3.2 are categorized in two, i.e., topologies
that are suitable for a vibratory environment and those that aren’t. A suitable topology is
a leg with two points that connect to the top plate. The two points indicate ability for the
leg to translate without causing a moment. From the table, and enlarged topologies in
figure 3.7 a-c, it is observed that the baseline models presented in column 3 with a
volume fraction around 80% have suitable topologies. Other leg topologies have one
point touching the top cover which would restrict desired translation and possibly initiate
rotation.
42
Table 3.2: Optimal TEG considering only structural loadings alone for varied leg size
Volume
Fraction
2 x Baseline Street
(2B)
Baseline Street
(B)
½ Baseline Street
(½B)
Displacement
100
2B (u=0.1259)
B (u=0.0504)
½ B (u=0.0286)
90
2B (u=0.1259)
B (u=0.0555)
½ B (u=0.0376)
80
2B (u = 0.1260)
B (u=0.0636) ½ B (u= 0.0494)
70
2B (u=0.1630)
B (u=0.0784) ½ B (u=0.0688)
60
2B (u=0.1902) B (u=0.1031)
½ B (u =0.0962)
Figures 3.7 a and b are enlarged topologies illustrating legs with one, or two points
touching the top cover.
Figure 3.7 a: Suitable leg with two distinct points [1 & 2] touching the top cover
Figure 3.7 b: Unsuitable leg with one point touching the top cover
43
Table 3.3 shows the optimal topology as a function of the porosity of the TEG for
fixed leg width. Here, the porosity ranges from 0 to 0.8. Each of the shown cases
represents the maximum transverse translation at a given porosity. The associated shear
translation respectively ranges from 0.0504 to 0.6079. As the porosity increases, the
shear translation increases. Thus, more porosity increases the compliance to shear
loadings. The largest displacements occurred between volume fractions between 0% and
30%. This displacement is however not considered in this research due to minimal
material to even converge and form TEG legs.
Of particular interest is the leg pattern at the top. Topologies with a volume
fraction about 80% have two points on each leg. Two points are important because they
indicate the ability to accommodate lateral movement at the top of the TEG unit cell
which would allow use in a vibratory environment. While Table 3.2 depicts exploration
of topologies and displacements of three different leg sizes with variable volume fraction,
Table 3.3 presents further exploration of the optimized leg size that emerged from Table
3.2, in pursuit of an optimal displacement as it explores from maximum to minimum
volume fractions for the TEG leg.
44
Table 3.3 Percent material, displacements, and topology
Volume
fraction
Displacements (u) Structural topology
100 0.0504
90 0.0555
80 0.0636
70 0.0784
60 0.1031
50 0.1290
40 0.1420
30 0.1778
20 0.2340
10 0.6079
0 0.6079
Figure 3.8 presents the transverse displacement as a function of volume fraction.
It is clear that when the volume fraction exceeds 40%, the slope of the displacement
45
reduces markedly. Thus, in a sense, the structural integrity stabilizes rapidly from this
point on. In contrast, with a volume fraction less than 40%, the displacement slope
increases rapidly. This may not be bad. If the environment calls for more compliance a
higher void percentage is required.
Figure 3.8 Plot of shear displacement versus volume fraction for structurally optimized
TEG
Table 3.4 shows the optimal structurally optimized topologies for volume fraction
near 80%. The volume fraction presented in column 1 range from 76% to 84%, while the
corresponding topologies are shown in column 2, and the displacements in column 3. It
can be observed that, topologies of volume fraction of 80% and above have legs with the
same form. The leg topology at the volume fraction of 78% displays a slightly different
formation for each leg, which could be a concern for even translation. The topology with
a 76% volume fraction has its legs connected to one point at the top cover. From the
topologies presented and the displacements shown indicate the volume fraction of 80% as
46
the candidate for the vibratory environment because it has the highest displacement, and
maintains two distinct points for each leg connecting to the TEGs top cover.
Table 3.4: Structurally optimized topologies for volume fraction near 80%
Volume
fraction (%)
Topologies Displacement (u)
84
0. 0604
82
0.0620
80
0.0636
78
0.0662
76
0.0688
3.4 Optimizing For Thermal Only
3.4.1 Thermal Model
Optimal topologies for thermal loadings alone based upon the logic shown in the
flowchart given by figure 3.2 are developed for variable volume fractions. A one-
dimensional steady-state heat conduction model is assumed for the unit cell.
47
The heat transfer rate is governed by Fourier’s Law (equation 3.9)
𝑞𝑥 = −𝑘𝐴𝜕𝑇
𝜕𝑥………………………………………….. (3.9)
where qx is the x- component of the heat transfer rate, k is thermal conductivity of the
material, A is the area normal to heat flow, and 𝜕𝑇
𝜕𝑥 is the temperature gradient. When the
length, 𝑙, of the member or element is considered, equation 3.9 can be re-written as
(equation 3.10)
𝑞 = 𝑘𝐴𝑑𝑇
𝑙…..……………………………………. (3.10)
Since we have a constant thermal conductivity, k, Laplace’s equation governs the change
in temperature in the x-direction.
𝑑2𝑇
𝑑𝑥2 = 0 ………………………………………… (3.11)
Integrating equation 3.11 twice results to equation 3.12.
T(x) = c1x + c2 ………………………………….. (3.12)
To solve for c1 and c2, we assume two boundary equations as follows: at x = x0, T=T0,
and at x = xL, T = TL. The solution to equations (3.11) and (3.12) is as follows in
equations 3.13a and 3.13b.
𝑐1 = −𝑇0−𝑇𝐿
𝑥𝐿−𝑥0 ……………………………….……… (3.13a)
𝑐2 = 𝑇0 + [𝑇0−𝑇𝐿
𝑥𝐿−𝑥0] 𝑥𝐿 ……………………………… (3.13b)
Similar to the structural model, the thermal model has a conductance matrix [K], the
temperature matrix T, and the heat source matrix q. These matrices are created and
assembled for each leg and computed to find thermal flow depicted as q = KT. Bendsoe
and Sigmund, 2004, presented matrix that was determined analytically using a symbolic
manipulation software. This 4 x 4 matrix used in the thermal model.
48
𝑘𝑒 =
[ 2
3,−1
6,−1
3,−1
6−1
6,2
3,−1
6,−1
3−1
3,−1
6,2
3,−1
6−1
6,−1
3,−1
6,2
3]
where ke is thermal conductance.
The heat conductance in a global form can be written as KthT = Fth. T is the
temperature, and Fth is the thermal load vector which characterizes the heat flux at the
boundary. Kth is the conductivity matrix and takes the form
𝑲𝑡ℎ = ∑𝑲𝑡ℎ𝑒 𝝆
𝑁
𝑒=1
……………………………………………3.14
where N is the number of elements, and ρ is the design variable. The elemental thermal
conductivity Keth is given by:
𝑄
𝑡= 𝑘𝐴
∆𝑇
𝑑 ……………………………………………… 3.15
where Q/t is the rate of thermal energy transferred through the material. The thermal
conductivity is represented by k. The cross-sectional area is represented by A. The
difference in temperature is represented by ΔT. The thickness or length through which
heat transfers is represented by d.
The thermal design parameters shown in table 3.5 include geometrical
characteristics, material properties, and heat source and heat sink temperatures. A
Marlow T12_6 TEG geometry serves as the baseline.
49
Table 3.5 Thermal design parameters
Design parameter Quantity
Design domain 60 units in x direction 20 units in y direction Fixed void region 20 in x direction and 16 in y direction
Leg height 16
Leg cross-sectional area space 400 Bismuth Telluride Seebeck coefficient, αp, αn 0.000215, -0.000215 v K-1
Bismuth Telluride Thermal conductivity, kp, kn 1.47 Wm-1K-1
Temperature – heat source 110 < 𝑇 ≤ 230 Temperature – heat sink 50oC
Ultimately, the maximum heat flow is dependent on the structure. Rowe, 2006,
states that the ideal geometry for thermo-elements used in TEGs should be long and thin
(p. 1-11).
3.4.2 Objective Function
In this model, the TEG legs are assumed to be made of homogenous material
(figure 3.9). The leg volume has a uniform cross sectional area and perfect insulation is
assumed for all sides.
Qin
y Thermal loading
Leg: S1(x,y) Leg: S2(x,y)
Street (insulated)
Fixed x
Qout_1 Heat Sink Qout_2
Figure 3.9 TEG unit cell subject to thermal loading
50
The performance desired outcome is to maximize the temperature gradient by
maximize heat flow through the TEG. The objective function for the quest is shown in
equation 3.16.
max (q) … .. ..……………………………………………… 3.16
subject to the following constraints:
0 ≤ 𝜌 < 1 is material distribution (design variable)
between 0 and 100%
Tmin ≤ T ≤ Tmax
∑𝑣𝑒𝜌𝑒 = 𝑉 ∗ 𝜑, 0 < 𝜌𝑚𝑖𝑛 ≤ 𝜌𝑒
𝑁
𝑒=1
≤ 1, 𝑒 = 1,… ,𝑁
where the ρ is the element wise constant material densities in element, e = 1, …, N.
Maximizing the heat flow on the TEG legs by varying volume fraction φ, subject
to the equilibrium constant [KthT - Fth = 0] and material volume constraint
∑ 𝑣𝑒𝜌𝑒 ≤ 𝑉, 0 < 𝜌𝑚𝑖𝑛 ≤ 𝜌𝑒 𝑁𝑒=1 ≤ 1, when e = 1,…,N are used in the analysis. The
conductivity matrix takes the form in equation 3.17:
𝑲𝑡ℎ = ∑𝑲𝑡ℎ𝑒 𝝆
𝑁
𝑒=1
……………………………………………3.17
where N is the number of elements.
3.4.3 Cases Examined
Optimal topologies were developed for varying volume fraction, as shown in
figure 3.10 below. Figure 3.10 a-c show the optimal topologies for respectively100%,
51
80%, and 60% volume fractions. Common to each of the configurations is a constant
cross-sectional area. The heat is simply being uniformly distributed to the upper surface.
(a) (b) (c)
Figures 3.10: Optimal topologies for volume fractions of a). 100%, b). 80%, and c) 60%
The heat conductance patterns mirror Bejan’s Constructal Law (Bejan 2015;
Bejan & Lorente 2008a; Bejan, Lorente, & Lee 2008b, and Bejan 2005). These
researchers argued that for a flow system to persist in time it must evolve in such a way
that it provides easier access to its currents. Tree like flow patterns in figures 3.9b and
3.9c support Bejan’s Constructal Law. Further, in his observation, Bejan suggested that
nature is the law of configuration generation, or the law of design. Where others saw
disorder, Bejan found a pattern. “One of the basic aims of design is to get the most
amount of work done with the least amount of effort. What engineers try to accomplish
through their plans, nature achieves on its own. Zane (2007) wrote about going with the
flow and gave examples such as animate (people, trees) and inanimate (rivers, mud
cracks) phenomenon, if given freedom, will organize themselves into patterns or shapes
that help them get from here to there in an easier manner”.
3.5 Multi-functional Optimization for Combined Structural and Thermal Loads
In this section, a general multi-dimensional optimization approach is considered,
whereby an optimal topology can be developed to maximize for displacement and heat
flux. The weighting of structural and thermal densities is represented by equation 3.18.
52
𝑐 = 𝑤𝑠𝑢 + (1 − 𝑤𝑠)𝑞 ……………………………………. (3.18)
Above, c is the objective function, ws is the 8 by 8 structural matrix, u is the
displacement, and q is the heat flux. Effectively, these weightings define which model is
more important.
Equation 3.18 can also be expressed as:
max(𝑢, 𝑞) = 𝑘1𝑢 + 𝑘2𝑞 ………………………………….. (3.19)
where k1 is the structural matrix, and k2 is the thermal matrix. Both equations 3.18 and
3.19 are used to find the optimal densities (e.g., localized void fraction) for the given
weightings of the thermal and structural models.
Results are shown in Table 3.6 for a volume fraction of 0.8 only. No other results
were obtained as this approach is generally not useful. The reality is that the goal always
will be to generate the most power (maximize heat flux) while insuring that the structural
loads are accommodated. Thus mechanical loads represent a constraint; not something to
be optimized.
Table 3.6 Search for best combination of heat flux and displacement
Weighting solid (ws) Heat flux (q) Displacement (u)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
.302
.272
.242
.211
.181
.151
.121
.091
.061
.031
0 .045
.091
.136
.182
.227
.272
.312
.364
.410
As shown in the table, a weighting of about ws of 0.4 yielded the best combination for
heat flux and displacement.
53
Shape optimal solutions were developed for TEG legs for variable volume
fraction of material. With boundary conditions in place, models were developed to study
how thermal flow effects and patterns were affected by thermal conductivity and varying
volume fractions. This exercise was intentional because we wanted to investigate thermal
flow for an optimal structural model. A heat source of a constant temperature 230oC was
applied and a heat sink temperature of 50oC was set.
Table 3.7 shows different patterns for structural only and thermal only loadings.
Shown are shape optimal solutions for material volume fractions of 100%, 80%, and 60%
for separate thermal and structural loading. In all thermal models, a uniform heat load is
distributed across the upper face of the top plate. At the bottom face, the material is
contiguous. The topologies created match those obtained in the previous sections for
respectively structural and thermal optimization alone.
Table 3.7 Topology differences of structural only and thermal only loadings
Percentage
volume
fraction
Structural only topology under
compression and shear loads
Thermal only topology
100
80
60
54
3.6 Optimization of TEG Power for Combined Structural and Thermal Loads
Power generation in a TEG is a function of the leg geometry, figure of merit (ZT),
thermal conductivity, and electrical resistivity. In traditional thermoelectric designs, TEG
legs have constant cross-sectional which lend them to be applied into thermoelectric
equations easily. Because we are optimizing for power (thermal loads) a constant cross-
sectional area is assumed to simplify analysis. In reality, because the heat flow through a
TEG with distance from the source declines slightly, one could relax this assumption.
However, maximal TEG power conversion efficiencies based upon the 2nd Law of
Thermodynamics are no greater than 0.1, the assumption for constant cross-section is
reasonable.
3.6.1 TEG Heat Conversion to Power
Three phenomena control the power generation within a TEG; namely the
Seebeck effect, Joule effect, and Thomson effect.
The Seebeck effect describes how a temperature difference between two
dissimilar conductors produces electromotive force between them. The p-type leg
contains positive charge carrier (holes) while the n-type leg is a negative charge carrier
(electrons). If the hot and cold junctions are maintained at different temperatures Th and
Tc and Th > Tc, an open circuit electromotive force, V, is developed at the resistance load,
RL. This action is represented in equation 3.20:
V=α(Th – Tc) or α = V/ΔT ……………………………………(3.20)
where V is electromotive force or voltage, α is Seebeck coefficient, Th is temperature at
the heat source, and Tc is temperature at the sink.
55
The electromotive force is a catalyst for the thermally excited electrons to move.
Although the temperature distribution along the legs is the same, the heat flux is higher in
the n-type leg. Different charge carriers in the legs cause unique rate of vibrations
resulting to current flow.
Joule heating occurs when current flows through a material offering resistance to
the flow. The amount of heat produced is represented as shown in equation 3.21.
𝑄𝑗 = 𝐼2𝑅 ……………………………………………….. (3.21)
where Qj is joule heating, I is current, and R is electrical resistance.
The Thomson effect relates to the rate of generation of reversible heat resulting
from the flow of current along an individual conductor along which there is a temperature
difference (Rowe, 2006; Saqr & Musa, 2009). The Thomson effect is defined by equation
3.22.
𝑄𝑡 = 𝛽𝐼∆𝑇 …………………………………………… (3.22)
where β is the Thomson coefficient, Qt is the rate of reversible heat absorption, I is the
current flow, and ΔT is the temperature difference between the ends of the conductor.
This effect is normally minimal.
The performance of the thermoelectric generator is greatly influenced by the
figure of merit (Z) depicted in equation 3.23 below.
𝑍 =𝛼2𝜎
𝑘 ………………………………………………. (3.23)
Here α2σ is the electrical power factor and k is the thermal conductivity. When
multiplied with temperature, Z becomes unitless and is represented as ZT, a
dimensionless figure of merit. A higher ZT is associated with higher TEG power
generation. Further, in order to obtain maximum figure of merit (Kanimba & Tian, 2016),
56
the geometry and material properties for the TEG should satisfy equation 3.24 (Rowe,
2006; Kanimba & Tian, 2016; Chen et al., 2012).
𝐴𝑝2 𝐿𝑛
2
𝐴𝑛2 𝐿𝑝
2 = 𝑘𝑛𝜌𝑝
𝑘𝑝𝜌𝑛 ………………………………………….. (3.24)
where Ap is cross sectional area positive leg, An is cross sectional area negative leg, Ln is
length negative leg, Lp is length positive leg, kn is thermal conductivity negative leg, kp is
thermal conductivity positive leg, ρp is electrical resistivity positive leg, and ρn is
electrical resistivity negative leg.
The continuity equation for a the steady-state current flow in a thermoelectric
material can be represented as
∇ . 𝑗 = 0 ……………………………………………… (3.25)
where ∇ is the differential operator with respect to length and 𝑗 is current density. The
current density 𝑗 and temperature gradient ∇ 𝑇 affect the electric field . Differentiating
equation 3.23 with respect to length results to an electric field shown in equation 3.26
(Lee, 2013).
= 𝑗 𝜌 + 𝛼∇ 𝑇 ………………………………………… (3.26)
where ρ is electrical resistivity and α is Seebeck coefficient.
The heat flux is represented as (equation 3.27)
𝑞 = 𝛼𝑇𝑗 − 𝑘∇ 𝑇 ……………………………………… (3.27)
The heat diffusion for a steady state is given as (equation 3.28)
−∇ . 𝑞 + 𝑞 = 0 ……………………………………….. (3.28)
where 𝑞 is the heat generated by volume and is function of (equation 3.29)
. 𝑗 = 𝐽2𝜌 + 𝑗 . 𝛼∇ 𝑇 ……………………………………………. (3.29)
57
Substituting equations 3.27 and 3.29 into equation 3.28, results in equation 3.30
∇ . (𝑘∇ 𝑇) + 𝐽2𝜌 − 𝑇𝑑𝛼
𝑑𝑇𝑗 . ∇ 𝑇 = 0 …………………………………… (3.30)
where ∇ . (𝑘∇ 𝑇) is thermal conduction, 𝐽2𝜌 is Joule heating, and 𝑇𝑑𝛼
𝑑𝑇 is the Thomson
coefficient (Lee, 2013).
3.6.2 Examining One Dimension Heat Transfer
The governing equations given by equations 3.25 – 3.30 simplify significantly
with a one-D heat flow assumption, which is appropriate given the assumption of
constant cross-sectional area and nearly constant heat flow in the y-direction. Thus, the
energy balance given by figure 3.11 can represent the heat and power flow through a
thermoelectric leg with x=L being the hot side and x=0 depicting the cold side. The
schematic below represents the global energy balance on a leg.
Insulated 𝜇𝑒𝐽𝐴(𝑇𝑥 − 𝑇𝑥+∆𝑥) + (𝑒𝐽)2 (𝐴𝐿
𝜎
∆𝑥
𝐿)
𝑄|𝑘,𝑥 = −𝑘𝐴(𝑥)𝑑𝑇
𝑑𝑥|𝑥 𝑄|𝑘,𝑥+∆𝑥 = −𝑘𝐴(𝑥+∆𝑥)
𝑑𝑇
𝑑𝑥|𝑥+∆𝑥
x=L Insulated x=0
Figure 3.11: Illustration of equations used to evaluate heat conduction through a TEG leg
A differential energy balance inclusive of the Thomson and Joule heating terms is
thus given by equation 3.31 (Lee, 2013):
−𝑄|𝑘,𝑥 + 𝑄|𝑘,𝑥+∆𝑥 + 𝑄𝜇 + 𝑄𝑗 = 0 ………………………….... (3.31)
Further, using Fourier’s Law and the definitions for Joule and Thomson heating given
respectively by equations 3.21 and 3.22, equation 3.31 can be expressed as
𝑑
𝑑𝑥(𝑇 + ∆𝑇)𝑘𝐴 −
𝑑𝑇
𝑑𝑥𝑘𝐴 + 𝜇𝑒𝐽𝐴∆𝑇 + (𝑒𝐽)2 𝐴𝐿
𝜎
∆𝑥
𝐿= 0 ……… (3.32)
58
and re-written as
𝑘𝐴𝑑𝑇
𝑑𝑥|𝑥+∆𝑥 − 𝑘𝐴
𝑑𝑇
𝑑𝑥|𝑥 + 𝜇𝑒𝐽𝐴∆𝑇 + (𝑒𝐽)2 𝐴𝐿
𝜎
∆𝑥
𝐿= 0 ……… (3.33)
where T, k, A, μ, eJ, σ, and L are respectively temperature, thermal conductivity, cross
section area, Thomson coefficient, electrical current density, electrical conductivity, and
length of a TEG leg.
The boundary conditions given by equations 3.34 and 3.35 are applied to equation
3.33,
Tp (x=0) = Tn (x=0) = Tc ……………………………………… (3.34)
Tp (x=Lp) = Tn (x=Ln) = Th …………………………………… (3.35)
Here the subscripts c, and h represent cold side (heat sink), and hot side (heat source),
respectively; and then taking the limit as ∆𝑥 goes to 0, the following differential
equations (3.36 & 3.37) result for the p- and n-type legs.
p-type leg
𝑘𝑝𝐴𝑝𝑑2𝑇𝑝
𝑑𝑥2 + 𝜇𝑝𝐼𝑑𝑇𝑝
𝑑𝑥+
𝐼2
𝜎𝑝𝐴𝑝 ……………………….……… (3.36)
n-type leg
𝑘𝑛𝐴𝑛𝑑2𝑇𝑛
𝑑𝑥2 − 𝜇𝑛𝐼𝑑𝑇𝑛
𝑑𝑥+
𝐼2
𝜎𝑛𝐴𝑛 ……………………………… (3.37)
where I = eJA is the electrical current.
The heat flow through each of the legs is given respectively by equations 3.36 and
3.37 for respectively the hot and cold sides of the TE.
𝑄ℎ = 𝑁(𝛼𝐼𝑇ℎ + 𝑘𝑝𝐴𝑝𝑑𝑇𝑝
𝑑𝑥|𝑥=𝐿𝑝
+ 𝑘𝑛𝐴𝑛𝑑𝑇𝑛
𝑑𝑥|𝑥=𝐿𝑛
…… (3.38)
𝑄𝑐 = 𝑁(𝛼𝐼𝑇𝑐 + 𝑘𝑝𝐴𝑝𝑑𝑇𝑝
𝑑𝑥|𝑥=0 + 𝑘𝑛𝐴𝑛
𝑑𝑇𝑛
𝑑𝑥|𝑥=0 ……… (3.39)
59
where α is the Seebeck coefficient and represents αph – αnh for x = L, and αpc – αnc for
x=0. These equations can be further simplified by using the following relationship for
thermal and electrical conductance given by equations 3.40 and 3.41.
𝐾 =𝑘𝑝𝐴𝑝
𝐿𝑝+
𝑘𝑛𝐴𝑛
𝐿𝑛 ……………………………………….. (3.40)
𝑅 =𝐿𝑝
𝜎𝑝𝐴𝑝+
𝐿𝑛
𝜎𝑛𝐴𝑛 ……………………………………….. (3.41)
Using equations 3.34, 3.35, 3.38 – 3.41, we can generate the equations 3.42 – 3.43
𝑄ℎ = 𝑁(𝛼ℎ𝐼𝑇ℎ + 𝐾∆𝑇 − 1
2𝐼2𝑅 −
1
2𝜇𝐼∆𝑇) …….…… (3.42)
𝑄𝑐 = 𝑁(𝛼𝑐𝐼𝑇𝑐 + 𝐾∆𝑇 + 1
2𝐼2𝑅 +
1
2𝜇𝐼∆𝑇) ……..…… (3.43)
The equation to determine power is given as
𝑃 = 𝑄ℎ − 𝑄𝑐 = 𝑁(𝛼𝐼∆𝑇 − 𝐼2𝑅)… ………………. (3.44)
The equation used to calculate the thermal power efficiency of the TEG is
𝜂 =𝑃
𝑄ℎ ……………………………..……….. (3.45)
Power output in a thermoelectric generator is a function of electrical current. In this case,
the interest is in flexibility of the TEG leg, and the power that could be generated in that
state. In addition, although shown in equations 3.42 and 3.45, Thomson effect is
neglected in this thermoelectric computation as it has been identified as having minimal
influence (Rowe, 2006).
The integrated model is a function of separately run structural model matrices and
thermal/power model matrices with their results normalized, weighed and combined to
obtain optimized outcomes (Mativo & Hallinan, 2017). The reason to run the two models
separately is because of their uneven degrees of freedom (dof) that originate from
differences in their discrete elements in the reference domain. The structural element
60
node has the potential to translate (x, y, z) and the ability to rotate (x, y, z). This operation
results to 6 dofs. The thermal element node can only translate (x,y,z) and hence has 3
dofs. All the boundary conditions stated for both structural and thermal models apply in
the integrated model.
3.6.3 Integrated Model
The combined model is used to examine what heat flow can be allowed subject to
the constraint of meeting applied loads. Design parameters for both structure and thermal
are presented in Table 3.8.
Table 3.8: Design parameters for integrated model
Design parameter Quantity
Design domain 60 units in x direction 20 units in y direction Fixed void region 20 in x direction and 16 in y direction
Leg height 16
Leg cross-sectional area space 400 Bismuth Telluride Density 7.8587 g cm-3
Bismuth Telluride Young’s Modulus 8.1 – 50 GPa
Bismuth Telluride Ultimate Tensile Strength
Bismuth Telluride density Bismuth Telluride melting point
Bismuth Telluride Seebeck coefficient, αp, αn
Bismuth Telluride Thermal Conductivity, kp, kn Temperature – heat source
Temperature – heat sink
7.4 GPa
7.37 g/cm3 585oC
0.000215, -0.000215 v K-1
1.47 Wm-1K-1 110oC < T ≤ 230oC
50oC
Poisson’s Ratio 0.23
The objective function for the combined structural and thermal load is shown as
maximizing displacement and maximizing heat flow as represented by:
max (u, q) ..……………………………………………… 3.46
subject to the following constraints:
0 ≤ 𝜌 < 1 is material distribution (design variable)
between 0 and 100%
61
Tmin ≤ T ≤ Tmax
𝒖∗ ≥ 0.029
r = 0 = ku-f
∑ 𝑣𝑒𝜌𝑒 = 𝑉 ∗ 𝜑, 0 < 𝜌𝑚𝑖𝑛 ≤ 𝜌𝑒
𝑁
𝑒=1
≤ 1, 𝑒 = 1,… ,𝑁
where the ρ is the element wise constant material densities in element, e = 1, …, N. Tmin is
the temperature at the heat sink, while Tmax is the temperature at the heat source. The u*
is the minimum calculated displacement. The stiffness matrix K depends on the ρ. r is
the residual in obtaining the structural equilibrium. For topology optimization the
equilibrium r = 0 is found using an iterative procedure. u and f are displacement and load
vectors, respectively.
The MATLAB tool used for structural model and thermal model was significantly
modified to integrate both models and extract power. An illustration flow chart, figure
3.12, depicts the process. The flow chart shows a tool that simultaneously computes a
TEG leg topology by distributing available material in the integrated design domain and
creating corresponding heat path resulting to a reconfigured leg and power generation,
respectively.
The MATLAB code for the combined model is divided into five sections. The
first main section of the program defines the inputs and starts the distribution of the
material evenly in the design domain. An initialization of design variables and physical
variables is done. In this case, the design variable is the volume fraction limit of the
material that is used to develop optimal topologies for special TE material volume
fractions. The physical densities are initially assigned a constant uniform value, but are
iteratively updated to form an optimal ability to support both structural and thermal loads.
62
Power-law is used to penalize intermediate density values, driving them towards solid or
void.
The second code section is the Optimality Criteria (OC) which updates the design
variables depending on the used and remaining material. The OC is formulated on the
conditions that if constraint 0 ≤ x ≤ 1 is active, then convergence is achieved with a
positive move-limit of 0.2 and a damping factor of 0.3, for compliant designs (Liu and
Tovar, 2014; Vijayan and Karthikeyan, 2013; Bendsoe, 1995). Material spread entirely
on an element is considered to have a value of 1, while an element without any material
has a value of 0. The iterative process continues with termination occurring when a
maximum number of iterations is reached or where a tolerance of the material
distribution is realtively small, for example if available material is equal to or less than
0.01 (Liu and Tovar, 2014).
The third section of the code is a mesh-independency filter whose function is to
avoid numerical instabilities. Details of the process are given in tool description section
3.2.2.1. The filtered density defines a modified physical density that is now incorporated
in the topology optimization formulation. During each iteration, the element sensitivity is
modified. This process ensures that elements gain or lose material depending on the
critical amount available and reaction loads, based on set filter settings.
The finite element code is the fourth section of the program. Here, two separate
global stiffness matrices are simultaneously created by a loop overall all elements and
executed. A 4 node bi-linear element enables formation of an 8 x 8 stiffness matrix for
structural loading while a 4 x 4 heat conduction matrix is formed for the mono-linear
thermal element. The two matrices evaluation simultaneously is the uniqueness of the
63
tool. Both matrices are weighted and normalized for optimized topologies before power
extraction.
Finally, the fifth section is power generation. It should be noted that while the
goal is to maximize power generation; the reality is that maximizing heat flux is the
equivalent of this.
64
Thermal Finite Element Analysis Structural
No
Yes
Figure 3.12 Detailed flowchart of integrated TEG model with power extraction
End
Is material distributed
to achieve min. obj.
function?
Extract power
Sensitivity Analysis – design variable through equilibrium equations and
finding derivatives of displacements with respect to design variables
Filtering – moves material to elements in areas where critical mass is formed
according to set filter size
Update material distribution
Initialize design space, e.g. volume fraction
– homogeneous distribution of material
Calculate temperature & flux
Get element ke
Assemble global K Assemble global Q
T= [K]-1 Q
Calculate displacement & forces
Get element ke Assemble global K
Assemble global F
u= [K]-1 F
65
3.6.4 Results
Numerical experiments were conducted to study the integrated structural and
thermal with power generation. Shear loading cases that were used in the structural only
models are also applied to the integrated models and the resulting adjoint displacements
calculated. The integrated models use optimized structural models and seek to maximize
heat flux. In order to achieve an even structural and thermal density in the design domain,
the baseline model at 100% volume fraction was used to determine the objective
functions for both the structural and thermal models. For the structural model, both the
mechanical loads and the constraints were applied resulting to an objective function of
57.43. For the thermal model, both the thermal loads for the heat source and heat sink;
and constraints of insulated sides were applied resulting to an objective function of 79.91.
Together their total was 137.34 which made the structural density to be at 41.8% and
thermal to be 58.2%. To normalize the density to a 50-50 ratio, the density for the
structure was multiplied by 1.2 while the thermal was multiplied by 0.86.
Table 3.9 shows the effects of weighting on topologies. Weighting is done
between 0 (Thermal only) and 1 (Structural only). It was observed that the higher the
structural weighting, the less the displacement and the increase in power generated.
66
Table 3.9 Effects of weighting (Ws) on the combined model
Percent
Structural
weighting
(Ws)
Percent
structural
weighting
(Ws)
0
0.2
0.4
0.6
0.8
1
Table 3.10 shows the optimal configurations for representative results for volume
fractions ranging from 100% to 40%. The volume fractions between 30% and 0% have
no material to create a leg to allow heat travel. As seen, the volume fraction
simultaneously subjected to structural and thermal loads results to the topologies
presented. Consistently, heat is traveling the near the street where the most material is
found. The overall constraint is structural because in the end it provides a path to heat
flow. The heat travels to the sink where it is readily distributed. The less the volume
fraction the more displacement is realized but less power generated. In order to sustain
shear load, less material is required, which means that the thermal resistance increases.
Thus, for a given temperature difference, there is reduced heat transfer, and thus power
generation potential.
67
Table 3.10 Displacement, power, and shape of an integrated TEG for the vibratory
environment
Volume
Fraction
Displace-
ment
Power
generation
at 110oC
Power
generatio
n at 170oC
Power
generatio
n at
230oC
Shape
100 0.0504 0.1163 0.4467 1.0499
90 0.0555 0.1078 0.4319 0.9725
80 0.0636 0.0992 0.3972 0.8937
70 0.07841 0.0900 0.3604 0.8115
60 0.1031 0.0799 0.3231 0.7264
50 0.1290 0.0704 0.2824 0.6374
40 0.1420 0.0591 0.2362 0.5313
Power generation is favored highly with models that have more material. The
highest power generation is shown to be from the model without a void while the least
power generation has the least material. Power generated from models with volume
fraction of 50% and less are not to be considered because they are not structurally
capable of supporting all loads. Comparisons of maximum power generation for the
baseline model, and the reconfigured integrated model are shown in table 3.11.
68
Table 3.11 Maximum power generation for the baseline and the integrated model
Experiment/
Temperature
Baseline Integrated
230C 1.0499 0.8937
170C 0.4667 0.3972 110C 0.1163 0.0992
The study was set to search for TEG geometry that could be used in a vibratory
environment. Based on the investigation, several models emerged with different
potentials. Tradeoffs are seen in table 3.11 and in figure 3.13. Figure 3.13 presents a
relationship between displacement and power generation. As can be seen, the less the
displacement, the higher the power generation. This is expected because there is more
material to carry heat.
Figure 3.13 Chart display power generation versus displacement
The less void in a model, the higher the power and less shear displacement. For a
balanced model that power generates and is structurally stable, a 80% volume fraction is
69
recommended for it can allow a displacement of 0.0636 and produce about 80% baseline
power.
70
CHAPTER IV
CONCLUSION AND DISCUSSION
4.1 Flexible TEGs Enhancement Of Energy Harvesting
This study successfully investigated the development of flexible TEGs that can
potentially be used in vibratory environment thus increasing the reach of locations that
can be accessed for energy harvesting. First, a numerical tool was created to perform
TEG unit cell numerical experiments. The creation of the tool required a creation of a
design domain were material distribution would be done to study effects of creating a
TEG unit cell with varying volume fraction. Structural boundary conditions of fixed and
free nodes, fixed solid region, and fixed void region, compression load, shear
displacement, and adjoint displacement were applied to the design domain. A Marlow TG
12-6 was used as baseline for this study. A baseline TEG unit cell model was successfully
created and tested. Once baseline parameters were established, numerical experiments for
structural only model were conducted with the aid of varying material distributed within
the design domain to establish compliant designs. Various structural topologies were
created and studied. The optimal structural only complaint topology was established.
Next, thermal boundary conditions were added to the structural model and tested
for heat flow patterns. It was shown that some of the complaint models had void areas
that prevented heat flow across the legs. The study helped in determining the model used
in the integrated model. The integrated model was a combination of structural and
thermal loading, and power extraction. A successful integrated model was established
and is presented in the dissertation. With a range of new integrated TEG legs that offer
71
opportunity for energy harvesting in vibratory environments, it is expected that more
waste heat, previously not accessible due to vibratory environments, can be reached and
converted to electricity.
4.2 Recommendations
A potential exists to add filler material in the TEG leg areas that have void spaces.
Filler materials such as conducting polymers are candidates because they offer a path for
heat flow while strengthening the TEG leg models. Whether the higher conductivity
could help relative to power production is uncertain.
Additionally, it has been demonstrated that it is feasible to build compliant TEGs;
however, there remains the technical obstacle associated with manufacturing the concept
developed. 3-D printing might be used to manufacture. This needs to be explored.
4.3 Summary
Topologies that allow sustainable displacements were determined. These
topologies have at least two leg contacts to the top cover allowing them to translate when
shear load is applied. Also, the TEG covers would not rotate because the shear load is
balanced or countered by the adjoint loads.
The size, shape and optimized TEG legs are very different from the baseline
design. These new legs are created to provide flexibility while carrying compression,
shear, and thermal loads. Traditional legs are optimized for compression and thermal
loads only. Though the new legs provide a certain amount of flexibility, they pose a new
challenge of narrow links shown in the topologies. Numerical analysis did not indicate
this as a problem however; a possibility of accumulation of heat in the narrow links could
72
be a concern. One suggestion is to identify polymers with similar thermal properties to
Bismuth Telluride and adding them to the spaces around the optimized legs. This
addition would do two things: 1) it will increase stiffness to the leg and reduce the
likelihood of breaking, and 2) it will allow even flow of heat from heat source to the sink.
73
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