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DESIGN OF EXPERIMENTS FOR THERMOELECTRIC
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DESIGN OF EXPERIMENTS FOR THERMOELECTRIC MATERIALS
By

Kimberly A. Sarbo

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department‘of Mechanical Engineering

2006

ABSTRACT
DESIGN OF EXPERIMENTS FOR THERMOELECTRIC MATERIALS
By

Kimberly A. Sarbo

Direct conversion of heat energy to electricity has many applications. One
application of this technology is utilizing the heat from a nuclear reactor in a submarine
to generate electricity. Thermoelectric modules can also be used to convert the wasted
heat energy from an internal combustion engine directly to electric machines in a hybrid
vehicle. The goal of the following research is to use statistical methods to find the
optimal thermoelectric material in order to maximize the efficiency of the conversion of
rejected heat energy to useful work.

When a temperature gradient is applied across a thermoelectric module, an
electric current is produced. However, in order to make these modules practical, more
efficient materials must be developed. This presentation outlines how a D-Optimal, four-
component mixture experiment based on a silver, lead, antimony, and tellurium system
was used for this optimization.

For the ranges tested, resistance, ZT, and electrical conductivity were maximized
when lead levels were high and tellurium levels were low. However, an inverse
relationship appeared between silver and antimony levels. ZT increased when silver was
higher and antimony was lower; the opposite trends appeared in the electrical

conductivity and resistance data.

Acknowledgements

I would like to thank the many people that helped me over the course of this
research project. First, I would like to thank Dr. Harold Schock for giving me the
opportunity to work at the Automotive Research Experiment Station. Secondly, I owe a
great deal to Dr. John Cornell and Dr. Wendell Smith for supplying me with an endless
amount of statistical knowledge and expertise. Both of these men provided consulting on
this project without compensation; their only goal was to help a student solve a difficult
problem. I must also recognize the help of Dr. Mercouri Kanatzidis and his research
team. In addition, I appreciate the helpfulness of Dr. Tim Hogan and his research group,
including Adam Downey and Jarrod Short. I would also like to thank Ed Timm for
helping on this project, including making all the thermoelectric material. Thanks to Dr.
Giles Brereton and Dr. Tim Hogan for serving on my advisory committee. I must also
recognize and thank Pat Schock and Mulyanto Poort for editing this thesis and assisting
with various aspects of this work. I would like to thank all those at the Automotive
Research Experiment Station including J ia Ma, Melissa Jopke, Tom Stueken, Andreas
Panayi, Boon Keat Chui, Josh Bedford, and Mayank Mittal. I would also like express my
gratitude for the support of Andy F edewa throughout this entire endeavor. Finally, I
would like to thank my parents, Rod and Jeanette Sarbo, for their support and for making

my collegiate experience possible.

iii

TABLE OF CONTENTS

List of Tables ..................................................................................................................... vi
List of Figures ................................................................................................................... vii
List of Symbols and Abbreviations ..................................................................................... x
CHAPTER 1
Introduction .......................................................................................................................... 1
1 .1 Motivation ............................................................................................................ 1
1.1.1 Motivation for Thermoelectric Materials .................................................... 1
1.1.2 Motivation for Design of Experiments ........................................................ 2
1.2 Previous Work ..................................................................................................... 2
1.3 Introduction to Thermoelectric Materials ............................................................ 4
1.4 Introduction to Design of Experiments ................................................................ 8
CHAPTER 2
Experimental Equipment and Procedure ........................................................................... 16
2.1 Mixture Selection and Experimental Setup ....................................................... 16
2.1.1 Design-Expert® Software .......................................................................... 16
2.2 Material Preparation .......................................................................................... 19
2.2.1 Vacuum Sealing Line and Pump ............................................................... 19
2.2.2 Furnace and Temperature Control System ................................................ 20
2.2.3 Leematic 2000 Surface Grinder and Slicer ................................................ 22
2.3 Data Collection .................................................................................................. 24
2.3.] F our-Point Probe ........................................................................................ 24
2.3.2 ZT Machine ............................................................................................... 25
CHAPTER 3
Results and Discussion ...................................................................................................... 30
3.1 Resistance .......................................................................................................... 31
3.2 ZT ...................................................................................................................... 37
3 .3 Electrical Conductivity ...................................................................................... 4 1
CHAPTER 4
Conclusions ....................................................................................................................... 47
CHAPTER 5
Recommendations .............................................................................................................. 50
Appendices ........................................................................................................................ 51
Appendix A
Additional Plot ............................................................................................................... 52

iv

Appendix B
ZT and Electrical Conductivity Data ............................................................................. 53

References .......................................................................................................................... 57

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

List of Tables

The number of terms for some Scheffe’ polynomials for q components .......... 13
Molar composition of design points. ............................................................... 18
Mass composition of design points .................................................................. 19
ANOVA for quadratic model of transformed resistance response. ................. 35

R2 values and adequate precision for the transformed resistance response fit.

............................................................................................ 35
ANOVA for quadratic model of ln(ZT) response. .......................................... 39
R2 values and adequate precision for ln(ZT) response fit ................................ 39
ANOVA for quadratic model of electrical conductivity data. ......................... 43
R2 values and adequate precision for electrical conductivity response. .......... 43
Optimized component levels for tested properties. ......................................... 49

vi

Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.

Figure 6.

Figure 7.
Figure 8.

Figure 9.

Figure 10.
Figure 11.
Figure 12.
Figure 13.
Figure 14.
Figure 15.
Figure 16.
Figure 17.
Figure 18.
Figure 19.
Figure 20.

Figure 21.

List of Figures

Temperature and voltage gradients across a thermoelectric leg. ....................... 5
Example resistivity measurement on a thermoelectric leg. ............................... 6
A thermoelectric couple ..................................................................................... 7
Module configuration exploded view (left) and assembled (right). .................. 7
A proposed thermoelectric generator design consisting of six modules. .......... 8

Comparison of factorial (2 factors, 2 levels) and mixture designs for 2
components. ....................................................................................................... 9

Examples of two-, three-, and four-component mixture spaces, respectively. 10

A Constrained-region, four-component mixture space. .................................. 11
Mixture space with axes and point location for (A, B, C) = (1/6, 1/2, 1/3) ..... 12
Sample 2-dimensional contour and 3-dimensional surface plots. ................... 15
Vacuum sealing system. .................................................................................. 20
Furnace closed (left) and open (right). ............................................................ 21
Computer and ATS Controller. ....................................................................... 21
Furnace temperature as a function of time. ..................................................... 22
Leematic 2000 Surface Grinder and Slicer. .................................................... 23
Coin and leg orientation in a cast thermoelectric ingot ................................... 24
Four-point probe .............................................................................................. 25
Electrical diagram of the ZT Machine. ........................................................... 26
The ZT Machine (pictured without the outer vacuum chamber). ................... 27
A close-up of mounted legs on the ZT Machine. ............................................ 27
Example plot of resistance vs. frequency for a thermoelectric leg. ................ 28

vii

Figure 22.
Figure 23.

Figure 24.

Figure 25.

Figure 26.

Figure 27.

Figure 28.
Figure 29.
Figure 30.
Figure 32.
Figure 33.
Figure 34.
Figure 35.
Figure 36.
Figure 37.
Figure 38.

Figure 39.

Figure 40.

Figure 41.

Correlation plot between tellurium and lead. .................................................. 31
Resistance data plotted by design point. ......................................................... 32
Internally studentized residuals vs. predicted resistance without inverse square
root transformation. ......................................................................................... 32
Internally studentized residuals vs. predicted resistance with inverse square
root transformation. ......................................................................................... 33
Predicted vs. Actual plot for resistance without transformation or ignored
points. .............................................................................................................. 33
Predicted vs. Actual plot for resistance with point 14 ignored and the inverse
square root transformation applied. ................................................................. 34
Contour plot of resistance at tellurium equal to 37.945 grams. ...................... 36
Contour plot of resistance at tellurium equal to 39.42 grams. ........................ 36
Contour plot of resistance at tellurium equal to 40.894 grams. ...................... 37
Contour plot of ZT at tellurium equal to 37.945 grams. ................................. 40
Contour plot of ZT at tellurium equal to 39.42 grams. ................................... 40
Contour plot of ZT at tellurium equal to 40.894 grams. ................................. 41
Electrical conductivity data plotted by design point. ...................................... 42

Contour plot of electrical conductivity at tellurium equal to 37.945 grams. ..44
Contour plot of electrical conductivity at tellurium equal to 39.42 grams. ....44
Contour plot of electrical conductivity at tellurium equal to 40.894 grams. ..45

Contour plot of electrical conductivity at tellurium equal to 40.894 grams
where all points were included in the analysis. ............................................... 46

Three-dimensional view of resistance response surface at tellurium equal to
39.42 grams ...................................................................................................... 47

Three-dimensional view of ZT response surface at tellurium equal to 39.42
grams ................................................................................................................ 48

viii

Figure 42.

Figure 43.
Figure 44.
Figure 45.
Figure 46.

Figure 47.

Three-dimensional view of electrical conductivity response surface at

tellurium equal to 39.42 grams. ....................................................................... 48
ZT vs. Electrical Conductivity for all design points. ...................................... 52
Data collected using ZT Machine (part 1). ..................................................... 53
Data collected using ZT Machine (part 2). ..................................................... 54
Data collected using ZT Machine (part 3). ..................................................... 55
Data collected using ZT Machine (part 4). ..................................................... 56

Images in this thesis are presented in color.

ix

AC

cm

DC

mol

List of Symbols and Abbreviations

cross-sectional area
alternating current
centimeters

direct current

grams

current

length

Kelvin

mass

moles

number of mixture components
resistance

spacing

Siemens

temperature

atomic mass (grams)
Seebeck coefficient
electrical resistivity
electrical conductivity

thermal conductivity

CHAPTER 1
Introduction
1.1 Motivation
1.1.1 Motivation for Thermoelectric Materials
This thesis is based on the thermoelectric effect discovered by Seebeck. Put
Simply, the system in this project provides a temperature gradient from which an electric
current will be created and used for other applications. One application of this
technology is utilizing the heat from a nuclear reactor in a submarine to generate
electricity. Thermoelectric modules can also be used to convert the wasted heat energy
from an internal combustion engine directly to electric machines in a hybrid vehicle.
It has been shown that from the energy in a given gallon of fuel, as much as 35% of the
energy is expelled as exhaust gases, and as much as 25% of the energy is released from
other various heat transfer losses. Thus, an engine is only using a fraction of the energy
provided by the fuel to propel the vehicle. The goal of the following research is to use
statistical methods to find the optimal thermoelectric material in order to utilize as much
as possible of the unused heat energy as possible.
Thermoelectric devices have many advantageous qualities over alternative

devices, such as:

o Comparatively lightweight, and small in Size
0 Heat pumping process is reversible
0 There are no moving parts

0 Very reliable (over 100,000 hours of operation)

0 Very fine temperature control with the correct electronic system

o No consumables

0 Generators can take on many different configurations for different
applications

0 Thermoelectric devices can be designed to be small without a change in
efficiency

The main disadvantage of thermoelectric systems is the low efficiency. The average
generators on the market have a conversion efiiciency of approximately 5%. However,
even a small increase in efficiency for an engine system would result in increased engine
life due to a decrease in engine stress.
1.1.2 Motivation for Design of Experiments

Design of experiments provides an organized, statistically sound method of
determining and executing experiments. There are many varieties of experiments, all
with different applications, strengths, and weaknesses. The main motives for using

design of experiments are the following:

0 Designed experiments are efficient and systematic
- The response can be analyzed with respect to one or more factors
0 Error, variance, and other statistical parameters can be interpreted

1.2 Previous Work

The experiments outlined in this paper were chosen based on the Ag-Pb-Sb-Te
system (or ‘LAST system,’ standing for Lead-Antimony-Silver-Tellurium), as published
by Hsu, et a1. [5]. The samples Anglgstezo and AngmeTelz were reported to have
high figures-of-merit. Although the results have been repeated within 20% by other

research groups [22], replication of the high ZT materials has been challenging, possibly

caused by the strong stoichiometric variations in the materials under the initial fabrication
techniques [6]. It is believed that a sweet spot exists in this system that may deviate
slightly from the above stoichiometries due to stratification of the components. It has
also been reported in the literature that LAST samples, when exposed to air, can react
with oxygen thus causing compositions different to those originally formulated [22].

Kosuga et a1. [8] also published data on the LAST system and reached a
maximum ZT of 0.33 at 673 K for composition Ag0,7PblngTe20. The compositions
AnglngTezo, AggnglngTezo and AgqubmeTezo exhibited the highest ZT at the
lowest level of silver tested, though the method of preparation differs greatly from that
outlined in this text. The results from the following mixture experiment suggest that
higher levels of Silver are needed relative to the ranges tested, though the highest level of
Silver tested for the mixture experiment is still lower than the lowest level tested by
Kosuga et al. The results from the above papers, combined with subsequent experiments
using the LAST system, were used to determine the component ranges for the following
mixture experiment.

The motivation behind using a mixture experiment for the analysis of the LAST
system is supported by several sources. Most notably, the article “Designing Mixture
Experiments” by Agreda and Agreda [1] states, “experiments involving mole or mass
fractions lend themselves naturally to the mixture design approach, as the mass or mole
fractions add to l by definition.” In addition, authors Cornell [2] and Smith [14], who
have published on the subject of mixture experiments, provided consulting on this project

in addition to the many examples in their respective publications.

1.3 Introduction to Thermoelectric Materials

The technology behind thermoelectric materials is based on the Seebeck effect.
In the 19Lh century, Seebeck found that a metal bar with a temperature gradient across it
would also have a voltage across the bar. Seebeck also found that a compass needle
would be deflected when it was placed near a closed circuit of two different conductors
when one of the junctions in the circuit was heated. This was due to the fact that a
current flowed through the circuit, thus causing a magnetic field in the vicinity of the
circuit. Unfortunately, Seebeck incorrectly reasoned that the magnetic field was
responsible for the effect, and pursued this line of reasoning, rather than using his
findings to make thermocouples, and thus using the temperature difference to create
electricity. However, Seebeck experimented with many different materials, some of
which are now classified as semiconductors [7].

The inverse of Seebeck’s finding was discovered by Peltier about 13 years later.
Peltier found that a temperature gradient was created when an electric current was
supplied to a closed circuit composed of two different conductors. Peltier never
compared his findings to those of Seebeck. It was not until 1838 that Lenz explained the
Seebeck and Peltier effects. He showed that heat is transferred in a certain direction
depending on the direction of the current flow in a circuit composed of two conductors
[11].

The efficiency of thermoelectric materials is related to the parameter known as the

figure-of-merit, Z, in equation (1.1).

Z=—=—— (1.1)

The unit of Z is UK. Since Z varies with temperature, a nondimensional figure of merit,
ZT, is often used. The Seebeck coefficient, a, is a measure of the change in voltage over
the change in temperature across a thermoelectric leg, as seen in Figure 1 and equation

(1.2) below.

Thot +
AT AV
Tcold -

 

Figure 1. Temperature and voltage gradients across a thermoelectric leg.

dV

:5 (1.2)

a

Thermal conductivity, 9», depends weakly on the carrier concentration of the
thermoelectric material. Resistivity, p, is measured as seen in Figure 2, and is the inverse

of electrical conductivity, 6.

 

 

 

 

 

Figure 2. Example resistivity measurement on a thermoelectric leg.

VA 1
=—=— 1.3
p 11 a ( )

A thermoelectric module is composed of positively and negatively doped legs,
hot-Side material, cold-Side material, and the necessary electrical wiring. The device is
constructed by connecting the thermoelectric legs thermally in parallel and electrically in
series. This configuration allows for the best temperature gradient across the system
while providing for proper current flow. Figure 3 shows a “couple,” a unit consisting of
P- and N-type material connected by a conductive material, such as copper, to provide the

electrical connection.

 

Figure 3. A thermoelectric couple.
Many of these couples are connected and sandwiched between thermally conductive

plates to make modules similar to the one in Figure 4.

 

Figure 4. Module configuration exploded view (left) and assembled (right).

 

 

Figure 5. A proposed thermoelectric generator design consisting of six modules.
1.4 Introduction to Design of Experiments

There are many kinds of experimental designs including, but not limited to,
factorial, fractional factorial, nested, and response surface designs. The simplest type of
design is full factorial. This design consists of completing experiments for every
combination of factors. For instance, two factors being investigated at two levels would
result in 4 experimental runs. This method cannot be used for mixture experiments
because factorial designs do not take proportions of components into account. Using a
factorial design for a mixture experiment would result in unnecessary, repeated tests.
The figure below demonstrates the difference between factorial and mixture spaces.
Experimental runs are placed at the points of the factorial space, whereas runs, or design
points, are placed along the constraint line in the mixture space. Response surface
designs can be thought of as a hybrid of factorial and mixture designs. They can be

organized similarly to designs in the factorial family. a response surface is generated

8

Similar to those of mixture experiments, but the response does not depend on the ratios of

the components.

L c.

 

 

 

 

 

 

 

 

 

A A

Figure 6. Comparison of factorial (2 factors, 2 levels) and mixture designs for 2
components.

The statistical analyses in this text are based on a mixture design, which is under
the category of response surface designs. A D-Optimal mixture design was chosen since
the design space was complex and component ranges varied in size. A D-Optimal design
is also useful for reducing the error of the model coefficients. What differentiates

mixtures from other designs is the constraint below, which applies for q components.
‘1
Ex,- =1 (1.4)
i =1

This constraint defines the boundaries of the mixture space. Examples of simplexes for
two-, three-, and four-component mixtures are shown in Figure 7, where a two-
component mixture is a line, a three-component mixture is a plane, and a four-component

mixture is a tetrahedron.

X
X1 X1 1
X2
X
x; 3

Figure 7. Examples of two-, three-, and four-component mixture spaces, respectively.
In order to accommodate the four-component, three-dimensional space response
surface plot, triangular slices are plotted along one axis as seen in Figure 8. The vertices
of the largest tetrahedron represent pure blends, or the design points where only one
component is present. The smaller tetrahedron represents a constrained mixture space,

where only certain component ranges are being investigated.

 

 

Sb Composition

Figure 8. A Constrained-region, four-component mixture space.

Each triangular slice has three axes that extend from each vertex and bisect the
opposing side, where the maximums for each component are located at the vertices.
Areas outside the constraints are shaded gray. Points in the design space are located by
drawing a perpendicular line across each axis at each component level. An example of

locating point (A, B, C) = (1/6, 1/2, 1/3) is shown below.

 

Figure 9. Mixture space with axes and point location for (A, B, C) = (1/6, 1/2, 1/3).
Design points are chosen based on the number of components and the type of
curve fit needed, based on subject matter knowledge and preliminary experiments.
Examples of mixture polynomials (or curve fits) are linear, quadratic, special cubic,
cubic, special quadratic, and quadratic. Table 1 shows the number of terms needed for q
components for Scheffé polynomials, the standard polynomials used for most mixture

experiments [12] [2] [14].

Table l. The number of terms for some Scheffé polynomials for q components.

 

 

q Linear Quadratic 833:1? Cubic (821:3: Quartic
2 2 3 - 4 - 5

3 3 6 7 10 9 l 5

4 4 10 14 20 22 35

5 5 l 5 25 3 5 45 70

6 6 2 1 41 56 81 126

 

Table 1 demonstrates how the number of experiments increases with the number of
components and the complexity of the contours of the response surface. A linear curve
fit only requires design points at the vertices, or each of the pure blends. The quadratic
model adds mixtures of two blends, or points along the edges of the design space, and so
on for higher level polynomials.

In addition to the points in Table 1, design points are needed to test for lack of fit
and pure error. These points are used in the statistical analysis in order to determine the
correct response surface and how well the response surface fits the collected data.

Once data are collected at locations within the design space, a polynomial is used
to fit a response surface. The most common polynomials used are linear, quadratic,
special cubic, and cubic Scheffé polynomials. For a four-component mixture composed
of A, B, C, and D, the equations for the predicted response, 37, are shown for linear (1.5),
quadratic (1.6), and special cubic (1.7) curve fits, where the ci’s represent the regression
coefficients.

y=qA+QB+QC+QD (L3

13

j)=CIA+CzB+C3C+C4D+C5AB+...+610CD (16)
j} = CIA + CzB+C3C+C4D+CsAB+m+610CD+C1lABC+...+ 014BCD (1.7)

The linear Scheffé model coefficients are estimates of the response at each of the
vertices. A quadratic model can be thought of as an augmentation of the linear model,
where combinations of components are being investigated. A similar augmentation is
noted in the special cubic model. The effect of a component, or a combination of
components, is only seen in the slope of the corresponding response surface.

A statistical software package is used to fit all predicted responses as well as the
corresponding p-values for each response. P-values determine the significance of a
model, or a term in a model. Graphically, a p-value for this analysis is the area under the
F-curve, to the right of the calculated F-value. For most statistical applications, a model
or term with a p-value of 0.05 or below is considered statistically significant. Models and
terms with p-values above 0.10 are considered statistically insignificant. Lack of fit tests
and model summary statistics are used to determine the model that best represents the
collected data. Once the best model is chosen, an analysis of variance (ANOVA) is
completed for the selected model and other statistical measures are calculated. After the
statistical analysis is complete, the response surfaces are plotted in 2-dimensional contour

views or 3-dimensional surface views similar to those below.

14

 

Figure 10. Sample 2-dimensional contour and 3-dimensional surface plots.

CHAPTER 2
Experimental Equipment and Procedure

2.1 Mixture Selection and Experimental Setup

Sample ranges were chosen based on the previous experiments made by the
Kanatzidis research group and previous mixtures prepared at the Automotive Research
Experiment Station by Tim. In order to satisfy the mixture experiment constraint that
all samples had to equal a constant amount (Equation 1.1), all chemical compounds were
converted into 100-gram samples. This was done since the samples did not add to a
constant number of total moles. Conversions were made by choosing a reference
component, silver in this case, and scaling the number of grams for all components as

seen in Equations (2.1) through (2.4).

 

 

 

 

 

 

 

 

 

 

m -[ m 1 *mol *u -m *(E—) *(m_ol) (21)
Ag u*mol Ag Ag Ag Ag u Ag mol Ag '
I
mpb=( m J *molpb*upb=mAg* “Pb * mon (2.2)
u*m01 Ag ”Ag molAg
me=[ m ) *mole*qu=mAg* "Sb * mOISb (2.3)
u*mol Ag uAg mol/4g
l
mTe=[ m ] *molTe*uTe=mAg* ”Te * more (2.4)
u*mol Ag uAg mol/4g

2.1 .1 Design-Expert® Software

The Design-Expert® software, developed by Stat-Ease®, was used to perform all
statistical analyses. The design of experiments program has the ability to set up an
experiment, analyze data using analysis of variance as well as other tools, and display

data using a variety of methods. The program offers many experimental designs

l6

including: factorial, fractional factorial, Taguchi, Placket-Burman, response surface, and
mixture designs. However, only the mixture design was needed for the purpose of this
project.

Once gram ranges are found using equations (2.1) through (2.4), high and low
gram amounts are entered into Design-Expert®. It is necessary to enter the type of
design, a D-Optimal Mixture; the highest polynomial order that can be accommodated,
Special cubic; and any other adjustments that are needed, such as additional design points
into the program. For the purpose of this experiment the defaults of 5 replicates and 5
check blends were used for a total of 24 experimental runs. The program then generates
a list of gram amounts for each of the runs, or design points, that need to be prepared and
tested. Once the experimental results are entered into the software, the statistical analysis
is completed, a model is chosen by the user, and color-coded contour plots are generated
in order to view the response surface. Tables 2 and 3 display the list of experimental runs
in moles and grams, respectively. Replicate design points are italicized. An ingot is

made for each of the design points below.

17

Table 2. Molar composition of design points.

 

 

[Lesign Point Type Molar Composition
omt Ag Pb Sb Te
1 Vertex 0.628 18 1.005 21.254
2 Plane Centroid 0.406 18 1.433 19.970
3 Triple Blend 0.610 18 1.273 19.532
4 Vertex 0.596 18 0.955 18.739
5 Vertex 0.191 18 0.955 19.051
6 Vertex 0.201 18 1.466 21.274
7 Vertex 0.629 18 1.466 20.913
8 Triple Blend 0.480 18 1.447 20.380
9 Plane Center 0.437 18 0.972 19.728
10 Center Edge 0.195 18 0.974 20.004
1 1 Center Edge 0.195 18 0. 974 20. 004
12 Vertex 0.191 18 0. 955 19. 051
13 Vertex 0. 628 18 1. 005 21.254
14 Vertex 0. 596 18 0. 955 18. 739
15 Plane Centroid 0.431 18 1.201 18.797
16 Vertex 0.605 18 1.410 19.008
17 Plane Centroid 0.196 18 1.256 19.993
18 Axial Check Blend 0.303 18 1.320 20.627
19 Vertex 0.198 18 0.993 20.995
20 Triple Blend 0.336 18 0.984 20.411
21 Axial Check Blend 0.296 18 1.297 19.374
22 Vertex 0.201 18 1.466 21.274
23 Plane Centroid 0.617 18 1.180 20.216
24 Vertex 0.191 18 1.394 18.785

 

18

Table 3. Mass composition of design points.

 

 

[ifsign Point Type Grams
ornt Ag Pb Sb Te
1 Vertex 1.021 56.239 1.846 40.894
2 Plane Centroid 0.674 57.413 2.686 39.227
3 Triple Blend 1.021 57.889 2.406 38.684
4 Vertex 1.021 59.188 1.846 37.945
5 Vertex 0.327 59.225 1.846 38.602
6 Vertex 0.327 56.132 2.686 40.855
7 Vertex 1.021 56.132 2.686 40.162
8 Triple Blend 0.789 56.870 2.686 39.654
9 Plane Center 0.736 58.162 1.846 39.256
10 Center Edge 0.327 58.079 1.846 39.748
11 Center Edge 0.327 58. 079 1.846 39. 748
12 Vertex 0. 327 59. 225 1.846 38. 602
13 Vertex 1.021 56. 239 1.846 40. 894
14 Vertex 1. 021 59.188 1. 846 3 7. 945
15 Plane Centroid 0.736 59.005 2.313 37.945
16 Vertex 1.021 58.348 2.686 37.945
17 Plane Centroid 0.327 57.781 2.369 39.523
18 Axial Check Blend 0.499 56.897 2.452 40.152
19 Vertex 0.327 56.933 1.846 40.894
20 Triple Blend 0.558 57.466 1.846 40.130
21 Axial Check Blend 0.499 58.352 2.471 38.678
22 Vertex 0.32 7 56. 132 2. 686 40. 855
23 Plane Centroid 1.021 57.208 2.204 39.568
24 Vertex 0.327 59.042 2.686 37.945

 

2.2 Material Preparation
2.2.1 Vacuum Sealing Line and Pump
A BOC Edwards pump system is used to achieve the desired vacuum in the quartz

ampoule during the sealing process. The system consists of a rotary pump, diffusion

19

pump, electrical controller, and active gauge display. The pump system is attached to a

series of glass tubes and valves to control the airflow.

 

Figure 11. Vacuum sealing system.

The mixture components are weighed and put into a 22-mm diameter glass tube.
A vacuum of 1x10'4 millibar is created using the BOC Edwards pump, and the quartz
tube is sealed off using a propane-oxygen torch.
2.2.2 Furnace and Temperature Control System

The furnace is a three-zone, series 3210 split-tube built by Applied Test Systems,
Inc. The outer shell is stainless steel and uses Kanthal® A1 resistive heating elements.
The inner diameter is two inches, and the outer diameter is ten inches. The furnace is

19.25 inches in length, where 15 of those inches are heated. Each of the three five-inch-

20

long zones has a thermocouple port. and the fumace has one overall control thermocouple

port. The furnace is rated to have a maximum temperature of 1 100°C.

 

Figure 12. Furnace closed (left) and open (right).

The temperature control system, also a product of Applied Test Systems, Inc.,
regulates the power given to the heating elements. The system uses Barber Colman 2404
controllers to attain the specified temperature. The control system uses iTools, a software
program developed by Eurotherrn. Inc. which allows the user to program a heating and

cooling profile for the fumace.

 

Figure 13. Computer and ATS Controller.

21

The sealed tube from the step above is placed in the ATS furnace where the
mixture is heated, rocked, and cooled. The temperature is increased at 50°C/hr up to
1020°C where the liquid mixture is rocked for 8 hours. The mixture is then cooled to
850°C at 85°C/hr, then to 750°C at 5°C/hr, and finally, from 750°C to 50°C at 30°C/hr as

seen in Figure 14.

 

 

—Zone 1
—Zone 2

1000 / \\ — Zone 3 f
800 _

600 /

/ \

200 / \ I
/ . , . . . \

o . . . . .
0:00 12:00 0:00 12:00 0:00 12:00
Day 1 | Day 2 1 Day 3

 

 

 

 

 

Temperature (°C)

 

 

 

Figure 14. Furnace temperature as a function of time.
2.2.3 Leematic 2000 Surface Grinder and Slicer

The Leematic surface grinder/slicer from K.O. Lee Company utilizes a high-speed
abrasive disk for making precision cuts. For this application, a diamond cutting wheel is
rotated at 5000 rpm to cut the thermoelectric material. The process is controlled via a
computer which actuates the three axis traverse. This allows the entire process to be

automated. Multiple cutting passes are made at 5 cm/sec, each at a slightly greater depth

22

than the previous. Such a high-speed cutting system makes it possible to make highly

accurate cuts on the relatively brittle thermoelectric materials.

.0“

OI “hi
:I Q

 

Figure 15. Leematic 2000 Surface Grinder and Slicer.

After the ingots have been cooled, they are glued onto an aluminum bar for
cutting. Each ingot is sliced into four coins 7 mm thick. Each coin is then individually
sliced into seven 5 mm by 5 mm by 7 mm legs (Figure 16). Legs are labeled, and

position within the ingot is noted for future studies.

23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 fl 5] .
§/./1“"'“\lc
% .g' i if
m: 6 g 7 ..
/ l l
[ [F | 1 i ‘ j
I 1' ' I II
er I I? i. ‘1 .
\\ 1 I 2 3 //
g in / l
I \NLIMTJ :
.i L L n

 

 

 

 

 

Figure 16. Coin and leg orientation in a cast thermoelectric ingot.
2.3 Data Collection
2.3.1 Four-Point Probe

Each leg was first screened using the four-point probe. Intact legs were tested on
each of the four longer sides, unless a side was too damaged due to holes or cracks in the
material. The probes are rested on the material for a nearly-immediate resistance reading.
Holes and other imperfections can cause severe fluctuations in the resistance
measurement, so some measurements were averaged or ignored, depending on the
disparity between measurements. The variability between resistances of legs was a
function of ingot composition or position of the leg within the ingot.

A known direct current is passed through the outer probes using a Keithley 2400
sourcemeter, and a voltage drop is measured between the inner two probes using a
Keithley 2182 nanovoltmeter. Ohm’s law can be used to calculate the resistance, R, of

the sample (2.5).

R = (2.5)

NIV.‘

24

The resistance is then used to find the resistivity, p, of the material using Equation (2.6).
p = —— = — (2.6)

It should be noted that for samples of finite size, a correction factor should be used to
obtain an accurate resistivity, where the dimensions of the sample are used to find the
appropriate correction factor from a chart. However, since all thermoelectric legs were
fabricated to be the same size, the measured resistance is proportional to the corrected
resistivity. Therefore, the resistance given by the four-point probe was used to make

relative comparisons between samples.

DC Power Supply l Ammeter
Voltmeter

 

I

 

 

 

 

 

U U U
/(—)/(—-)/(——)/
s s 5
Sample A
K 1 >

 

 

 

 

 

Figure 17. F our-point probe.
2.3.2 ZT Machine

The ZT Machine, developed by Adam Downey of the Hogan research group [4],
is capable of measuring electrical conductivity and ZT. The apparatus is able to measure

sixteen samples at a time over a period of approximately 2 ‘/2 days. Three to ten legs

25

were chosen from each ingot for testing based on the lowest number of cracks and holes
in the material. Flux is applied to the end of each leg; and Cerromatrix Bi 48%-Pb
28.5%-Sn 14.5%-Sb 9% solder is applied to the end of the leg and two pieces of thin
copper foil. The foil and leg ends are heated to form bonds between the legs and foil
pieces. The current-carrying wires are bonded to the copper foil and attached to pin and
socket circuit boards for mounting to the device, so that the legs are electrically in series.
A voltage difference is measured across each leg by two 0.003 inch copper wires spark-
welded to one side as seen in Figure 18. Samples are then suspended by the current—
carrying wires in approximately a 10-utorr vacuum, created using a rough pump and

turbo pump, which keeps each sample in thermal isolation.

 

 

 

. /' ' ’
Copper for] Sample
Woods metal solder

Figure 18. Electrical diagram of the ZT Machine.

26

 

 

Figure 20. A close-up of mounted legs on the ZT Machine.

27

Four cycles of a square wave, low frequency current of 1 mHz to 100 mHz are fed
through each leg, followed by 10 cycles of high frequency current from 1 Hz to 11 Hz.
The high frequency current, or alternating current, prevents the development of a
temperature gradient across the thermoelectric leg. The low frequency current, or direct
current, allows a temperature gradient to form. Resistance measurements are taken at
both current frequencies and are used in calculating the figure of merit. An example plot
of resistance vs. frequency is shown below in Figure 21. The ratio of the low frequency,
or RDC, measurements on the left of the plot over the high frequency, RAC, measurements

on the right Side of the plot, minus 1, is used to determine ZT.

 

 

l T T l I
c - -
V ZT : 0.78
C9
0 — -
C
I“
fl
.2 - -
If:
a: _ ‘ .
- —— . o 0 o -'
I I I I I

 

 

 

0.0001 0.001 0.01 0.1 1 10 100
Frequency (Hz)

Figure 21. Example plot of resistance vs. frequency for a thermoelectric leg.

Once measurements are completed, Equation (2.7) is used to calculate ZT.

28

ZT = 529—1 (2.7)
RAC

Electrical conductivity is calculated using AC resistance, cross-sectional area, and length
between the voltage probes according to Equation (2.8) below.

1
0':—

s
._ 2.8
RAC A ( )

29

CHAPTER 3
Results and Discussion

The following chapter outlines the results for the resistance, ZT, and electrical
conductivity measurements. Unfortunately, after the experimental data had been
collected, Smith discovered there were collinearities in the design. This is usually due
either to small ranges of components, a large disparity between the proportions of the
ranges, or a combination of both. Figure 22 shows the correlation of -0.9 between lead
and tellurium. The negative correlation means that when the coefficient of lead is
increased, the coefficient of tellurium is decreased. This means that the coefficients in
the Scheffé polynomial are not correct. However, even though numerical values on the
response surface plots cannot be trusted, the general trends predicted by the software are
still useful. The collinearity is a property of the mixture design and applies to all of the

resistance, ZT, and electrical conductivity response surfaces below.

30

 

40.89“ I13 I
40.15-— I I I
g I ' I I
3 39.41—
a - -
I.—
38.68- ' I .
37.94 I II

 

 

 

l l l l l
56.13 56.90 57.67 58.45 59.22
Lead

Figure 22. Correlation plot between tellurium and lead.
3.1 Resistance

Resistance data for each ingot is plotted below in Figure 23. In order to eliminate
heteroscedasticity of the residuals, an inverse square root transformation was used.
Transformations are used to create homogeneity of variance and normality, and the
inverse square root transformation was applied upon the recommendation of the Design-
Expert® program. Heteroscedasticity of the residuals, or the undesirable cone shape, can
be seen in Figure 24 below. Figure 25 shows the desired random scatter of the residuals
after the transformation has been applied. Design point 14 was ignored for the statistical
analysis. Figures 26 and 27 demonstrate these two changes. Figure 26 displays the
predicted vs. actual plot for the resistance without any transformation or ignoring any
data points. Figure 27 contains the predicted vs. actual plot after both changes have been

applied.

31

0.015

Reslstance (Ohms)
O
O
N

.o
O
_s

5..

    

13 5 7 911131517192123
DeaIgnPoInt

Figure 23. Resistance data plotted by design point.

 

 

 

 

 

 

3.36— I
E
«I
:I
E
g 1.77—
m I
'0 I
o I
”E 0.18— = I I = I
-8 "' 'I I
3 I ' I
(I) If
é‘ -1.41-
I
§ I
‘_E
-3.00

 

I
0.000 0.003 0.006 0.009 0.012
Predicted

Figure 24. Internally studentized residuals vs. predicted resistance without inverse
square root transformation.

32

 

 

 

3.00
g
(D
3
.9
(I)
(to 1.50‘ I I.
'U
‘5' 0.00 I :
'U I
3 I I
a I ' . I
>
fi -1.50—
E
0 I
"E I
-3.00

 

 

 

 

I I l l I
10.37 14.60 18.83 23.05 27.28
Predicted

Figure 25. Internally studentized residuals vs. predicted resistance with inverse square
root transformation.

 

 

0.031 ~
0023—
'0
2
% 0.016~
a
O.
0.0085
0.000—

 

 

 

I

l l l I
0.001 0.008 0.015 0.023 0.030
Actual

Figure 26. Predicted vs. Actual plot for resistance without transformation or ignored
points.

33

 

34.00-

 

27.00~
'0
2
43’ 20.00—
a
D.

13.00—

6.00

 

 

 

I

l l I l
6.87 13.41 19.94 26.47 33.00
Actual

Figure 27. Predicted vs. Actual plot for resistance with point 14 ignored and the inverse
square root transformation applied.

34

Table 4. ANOVA for quadratic model of transformed resistance response.

 

 

Source Sum of Squares DF 3:32:16 F-Value 1:31:21;

Model 673.39 9 74.82 1.64 0.2012
Linear Mixture 181.42 60.47 1.33 0.3076
AB 36.39 1 36.39 0.80 0.3875
AC 2.46 1 2.46 0.054 0.8198
AD 52.73 1 52.73 1.16 0.3014
BC 12.78 1 12.78 0.28 0.6051
BD 146.44 1 146.44 3.22 0.0962
CD 1.92 1 1.92 0.042 0.8404

Residual 591.76 13 45.52

Lack of Fit 499.82 9 55.54 2.42 0.2052

Pure Error 91.94 4 22.98

Cor Total 1265.15 22

 

Table 5. R2 values and adequate precision for the transformed resistance response fit.

 

R2 Rzadjg Rzp,¢d_ Adeq. Precision

 

0.53 0.21 -0.82 3.80

 

The R2 values are low, but this may be caused by the collinearities in the design.
The adequate precision, a signal-to-noise ratio, should be at a minimum of 4 for a
reasonable fit. Due to the large number of imperfections in the material, it is not
surprising the adequate precision values were low for all properties tested.

The response surface plots for resistance are shown below. The transformation
has been applied, but plots are shown in the original scale. Three slices have been taken
from the three-dimensional design space along the tellurium axis, where tellurium is

equal to 37.945 g, 39.42 g, and 40.894 g. Note that the pink dots represent design points.

35

 

.88 0.33 5.60
Pb C: Sb

Figure 28. Contour plot of resistance at tellurium equal to 37.945 grams.

2338

0.0017

  

ME ‘
58.41 0.33 4.12
B: Pb C: Sb

Figure 29. Contour plot of resistance at tellurium equal to 39.42 grams.
36

 

56.93 0.33 2.65
B: Pb C: Sb

Figure 30. Contour plot of resistance at tellurium equal to 40.894 grams.
3.2 ZT

The scatter plot containing the ZT data is below. Points are color-coded by coin.
The ZT values for the statistical analysis were obtained by averaging the measured values
for each leg to obtain one value per design point for entry into the Design-Expert®
software. A natural log transformation of the ZT response was modeled due to the

heteroscedasticity of the residuals. No design points were removed for this analysis.

37

0.35 -'

 

 

 

 

 

A ..
0.30 - ‘ t
2 ’ . : I B
0.25- ‘ . ; . . 2 ‘ _ ‘ C
II; 0.20 T . : , ' ’ ; ‘ 2 g ' D
0.15 d E z ‘ e 3 1 : : . . . : - g : :
0.10 d ‘ _ z t i . Q I i 2 . .
0.05-: . ‘. H I .;. .35;
0.00 "

 

a 4 e a 10 11214 16 10 20°20 24
Design Point
Figure 31. ZT data for each leg measured coded by coin.

The AN OVA for the quadratic Scheffé fit of ln(ZT) is in Table 6. Some may
argue that a linear fit is more reasonable, but the extreme collinearity of the data causes
exaggerated standard errors of the component estimates, especially those of terms
consisting of Silver and antimony. As a result, all model terms were included in order to
avoid Type 11 error, or the error in concluding that a coefficient estimate is insignificant
when the estimate is Significant. AS a result values in the AN OVA are not accurate; the

tables below have been included for completeness.

38

Table 6. ANOVA for quadratic model of ln(ZT) response.

 

 

Source Sum of Squares DF SEC/[[32:16 F-Value 1:313:12
Model 1.88 9 0.21 1.10 0.4196
Linear Mixture 1.05 3 0.35 1.85 0.1844
AB 0.077 1 0.077 0.41 0.5344
AC 0.11 1 0.11 0.58 0.4588
AD 0.034 1 0.034 0.18 0.6798
BC 7.167E-004 1 7.167E-004 3 .77 8E-003 0.9519
BD 0.069 1 0.069 0.37 0.5549
CD 2.450E-004 1 2.450E-004 l .291 E-003 0.9718
Residual 2.66 14 0.19
Lack of Fit 1.91 9 0.21 1.42 0.3645
Pure Error 0.75 5 0.15
Corrected Total 4.54 23

 

Table 7. R2 values and adequate precision for ln(ZT) response fit.

 

R2 Rzadj. Rzpm Adeq. Precision

 

0.41 0.04 -1.10 4.57

 

The contour plots for ZT are shown below. Responses are shown in the original

scale, though the transformation has been applied.

39

 

59.88 0.33 5.60
B: Pb C: Sb

Figure 32. Contour plot of ZT at tellurium equal to 37.945 grams.

A: Ag
2.60

 

58.41 0.33 4.12
B: Pb C: Sb

Figure 33. Contour plot of ZT at tellurium equal to 39.42 grams.

40

 

56.93 0.33 2.65
B: Pb C: Sb

Figure 34. Contour plot of ZT at tellurium equal to 40.894 grams.
3.3 Electrical Conductivity

All electrical conductivity data points were averaged for each ingot. Due to the
large difference in scale, point 12 was ignored for the statistical analysis of the electrical

conductivity data. No transformation was needed for this response.

41

3500

3000

2500

2000

1500

1000

Electrical Conductivity (Slcm)

500

 

13 5'7 911131517192123
DesignPoint

Figure 35. Electrical conductivity data plotted by design point.
The AN OVA and R2 tables are provided below. Once again, inflated standard

errors have led to insignificant p-values where significant p-values may exist.

42

Table 8. ANOVA for quadratic model of electrical conductivity data.

 

 

Source Sum of Squares DF 81:61:; F-Value 131' 3: 1:;

Model 3.470E+006 9 3.856E+005 3.46 0.021 1
Linear Mixture 2.328E+006 3 7.758E+005 6.97 0.0049
AB 2261.67 1 2261.67 0.020 0.8888
AC 1.858E+005 1 1.858E+005 1.67 0.2189
AD 1058.54 1 1058.54 9.511E-003 0.9238
BC 1 .826E+005 1 1.826E+005 1.64 0.2226
BD 4.937E+005 l 4.937E+005 4.44 0.0552
CD 1.948E+005 1 1.948E+005 1.75 0.2087

Residual 1.447E+006 13 1.1 13E+005

Lack of Fit 1.264E+006 1.404E+005 3.07 0.1459

Pure Error 1.829E+005 4 45716.96

Corrected Total 4.917E+006 22

 

Table 9. R2 values and adequate precision for electrical conductivity response.

 

R2 Rzadj Rzprcd Adeq. Precision

 

0.706 0.502 -0.02 7.39

 

43

 

59.88 0.33 5.60
B: Pb C: Sb

Figure 36. Contour plot of electrical conductivity at tellurium equal to 37.945 grams.

5:.

-

6:9

58.41 0.33 4.12
B: Pb C: Sb

 

Figure 37. Contour plot of electrical conductivity at tellurium equal to 39.42 grams.

 

56.93 0.33 2.65
B: Pb C: Sb

Figure 38. Contour plot of electrical conductivity at tellurium equal to 40.894 grams.
For comparison, Figure 39 shows the electrical conductivity response surface for
tellurium equal to 40.894 grams when point 12 is included in the analysis. Though the
predicted values on the contours cannot be trusted, it is interesting to see how one
aberrant data point can affect the response surface. It is usually recommended that data
points not be removed unless it is believed there is something wrong with the sample or
measurement. Design point 12 was removed from the above analysis due to the known
material properties of the legs. Holes and cracks were visible on many of the legs,

suggesting that microcracks may also exist.

45

 

56.93 0.33 2.65
B: Pb C: Sb

Figure 39. Contour plot of electrical conductivity at tellurium equal to 40.894 grams
where all points were included in the analysis.

46

CHAPTER 4
Conclusions
Resistance, ZT, and electrical conductivity data have been analyzed for 24 cast

ingots. The goal of the mixture experiment was to find a material with minimum
electrical resistance, maximum ZT, and maximum electrical conductivity. Of these,
maximum ZT is the most important. Unfortunately, poor material properties and
collinearities have led to high variability and error. However, response surface plots can
be used to determine general trends for future work. Figures 40 through 42 contain three-
dimensional views of the fitted response surfaces for each property at different areas of

interest.

0.0030 1

0.0025 ‘

  
   
 

0.0020 1

Resistance

0.0015:

 

0.0010 1

 

Ag(2.603) ”“56 132)
, 3001.122)

Sb(1.846)

Pb(58.407)

Figure 40. Three-dimensional view of resistance response surface at tellurium equal to
39.42 grams.

47

0.14

 

0.121

0.09 ‘

ZT

0.07 1

0.05 ‘

 

 

 

     
 

 

Ag(2.62)

  

 

Pb(56.13)

' "1r ’ Sb(4.14)

Sb(1.85) ' [AI-0(1) 33)

Pb(58.43)

Figure 41. Three-dimensional view of ZT response surface at tellurium equal to 39.42
grams.

 

1 900

1550‘

1200 *

 

850 '

Electrical Conductivity

500 ‘

 

Ag(2.62) . - SbI4-14)

 
 
  
 

Sb(1.85) Ag(0.33)

Pb (58.43)

Figure 42. Three-dimensional view of electrical conductivity response surface at
tellurium equal to 39.42 grams.

48

Table 10 summarizes the suggested levels of each component in order to optimize
each property. There is agreement between all responses for components in lead and
tellurium. However, an inverse relationship exists in silver between ZT and the other two
responses, and in antimony between ZT and conductivity.

Table 10. Optimized component levels for tested properties.

 

 

Ag Pb Sb Te
Resistance Low High Variable Low
ZT Max Max Min Min
Electrical Conductivity Min High Max Low

 

 

49

CHAPTER 5

Recommendations
One of the largest areas of needed improvement is in the reduction of
variability at every step of the processes outlined above. The first step is to
ensure the purity of the raw components as well as the accuracy of
measurement of these components.
One issue that could be addressed in future research is the chemical makeup
of the mixtures after undergoing the heating and cooling process. Members of
the Michigan State therrnoelectrics group have noticed traces of elements on
the sides of the quartz tubes in which the ingots were cast, suggesting that the
chemical composition prescribed was not the one tested. Methods of
assessing the composition of finished ingots could prove statistically useful.
I recommend that future mixture experiments be completed using ingots and
legs that have better material properties than those used for the above
experiment, since many imperfections were visible in the material. This may
be accomplished by hot pressing the current materials or by using other types
of thermoelectric material.
In order to reduce collinearity, the experimental design range sizes should be
proportional and larger. If this cannot be accommodated, a standard response

surface design should be used.

50

Appendices

51

Appendix A

Additional Plot

 

 

O

p

O

O

C

mm
PN
aw
NV
mp
mw
FF

FmI-OI‘

O

a

9

O

O

O

 

 

coon

ooov

F

€06. .28 Beam.

ooom

ooom

 

 

—
e
I I
a u
0 On
e
0.9.!
oele «I
I p.
O ..
4* 0
a I. eOAWI
6 Co
a a e s I
0 clc .-
.I...fd %
_. a 1
II II
‘0
.I C .04 I
IcI
O
n
I I
I
I
I I
I
he r

 

 

oogu

mogv

orgy

mFAU

ONAU

mNAV

omav

mmav

Figure 43. ZT vs. Electrical Conductivity for all design points.

52

Appendix B

ZT and Electrical Conductivity Data

Design Electrioar rfiyefaga .

Point Com Leg ZT Average ZT

[QM—14.34.444.54

wwwwoo

2
2
2
2
3
3

m>>UU
\IU‘I

mAAAe-AAAAAA 00
OD ww>>UU
AuMAAoa—aN—L A N

>00

 

Figure 44. Data collected using ZT Machine (part 1).

53

AVA-IQAN—IU'IUI:

-#
I
‘1

11
1
11
11
11 D

 

Figure 45. Data collected using ZT Machine (part 2).

54

Design
Point

Coin Leg ZT lAVerage

12

18
18
18 D

 

Figure 46. Data collected using ZT Machine (part 3).

55

.' , Average
Leg ZT AverageZT Electrleel Electrical

D
D
A
B
C
B
B
D
A
A
B

U

23
23
23
23
23
24
24

U DJW>>

OCD>>

24

24
24 D

 

Figure 47. Data collected using ZT Machine (part 4).

56

References

57

10.

ll.

l2.

13.

14.

References

Agreda, CL. and V.H. Agreda, Designing Mixture Experiments. ChemTech,
1989. 19(9): p. 3.

Cornell, J.A., Experiments with Mixtures. 2nd ed. Wiley Series in Probability and
Mathematical Statistics. 1990, New York: Wiley-Interscience.

Cornell, J .A., A Comparison Between Two Ten-point Designs for Studying
Three-component Mixture Systems. Journal of Quality Technology, 1986. 18(1):
p. 15.

Downey, A.D., Advancing Thermoelectrics Research with New Measurement
Systems and Thermoelectric Modeling for AC Electrical Measurements. 2006,
Michigan State University: East Lansing, Michigan.

Hsu, K.F., et al., Cubic AngmeTe2+mz Bulk Thermoelectric Materials with High
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Humphry, T.E., et al., Reversible Thermoelectric Nanomaterials. Physical Review
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Ioffe, A.F., Semiconductor Therrnoelements and Thermoelectric Cooling.
Infosearch Limited, 1957, London.

Kosuga, A., et al., Thermoelectric Properties of Ag.-belngTe20 (x=0, 0.1, 0.3).
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Mason, R.L., R.F. Gunst, and J .L. Hess, Statistical Design and Analysis of

Experiments : with Applications to Engineering and Science. 1989, New York:
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Rice, J .A., Mathematical Statistics and Data Analysis. 1988, Monterey, Calif:
Brooks/Cole Pub. Co.

Rowe, D.M., CRC Handbook of T hermoelectrics. 1995, Boca Raton, FL: CRC
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Scheffé, H., Experiments with Mixtures. J. Royal Statistical Soc. B, 1958. 20(2):
p.16

Schenck, H.V.N., Theories of Engineering Experimentation. 2d ed. 1968, New
York,: McGraw-Hill.

Smith, Wendell F., Experimental Design for Formulation. ASA-SIAM Series on
Statistics and Applied Probability, SIAM, Philadelphia, ASA, Alexandria, VA,
2005.

58

15.

16.

17.

18.
19.

20.

21.

22.

Smith, W.F., Data Set E-mail, K. Sarbo, Editor. 2006: East Lansing, Michigan.

Smith, W.F., Mixture Experiment E-mail, K. Sarbo, Editor. 2006: East Lansing,
Michigan.

Smith, W.F., Collinearity E-mail, K. Sarbo, Editor. 2006: East Lansing,
Michigan.

Smith, W.F., Data E—mail, K. Sarbo, Editor. 2006: East Lansing, Michigan.
Stat-Ease, Design-Expert. 2006, Stat-Ease, Inc.: Minneapolis.

Stat-Ease, Mixture Design for Optimal Formulations. 2006, Stat-Ease, Inc.:
Minneapolis.

Tellurex. An Introduction to Thermoelectrics. 2001 [cited; Available from:
http://www.tellurex.com/cthermo.html.

Wang, H., et al., High-perfonnance AgongIgflSbTezo Thermoelectric Bulk
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Applied Physics Letters, 2006. 88(9): p. 3.

59

   

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