Thermal Modeling of Shape Memory Alloy Wire

Thermal Modeling of Shape Memory Alloy Wire
Thermal Modeling of Shape Memory
Alloy Wire Actuators for Automotive
Applications
by
Huilong Ma
A thesis
presented to the University of Waterloo
in fulfillment of the
thesis requirement for the degree of
Master of Applied Science
in
Mechanical Engineering
Waterloo, Ontario, Canada, 2010
© Huilong Ma 2010
Author’s declaration
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,
including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.
ii
Abstract
Shape Memory Alloy is an amazing material, which can “remember” and return to its
original shape when heated due to its temperature dependent phase transformation. Shape
Memory Alloy wire has significant potential for application in the automobile industry
due to its high ratio of energy / weight and silent actuation. However, a dependable
method to measure the operating temperature of SMA wire and a reliable heat transfer
model to characterize the dynamics of the SMA wire limit its widespread use in the
automobile industry. This thesis presents a detailed description of the work performed to
develop a reliable method for determining surface temperature of current carrying SMA
wires and the development of a heat transfer correlation for natural convection cooling of
heated SMA wires. The major findings of the research are as follows:
When a spot welded thermocouple measures the temperature of a current carrying
SMA wire, there is a “spurious voltage” ∆V added to the thermo electro-motive force
(EMF) of the thermocouple as a result of a voltage drop across the two points of contact
that the thermocouple wires make with the SMA wire. This leads to an erroneous
temperature reading that can be higher or lower than the actual temperature depending on
the direction of current flow. When the carrying current is reversed in direction, the
“spurious voltage” becomes –∆V allowing a correct temperature reading to be obtained
by averaging the readings based on opposed current flow.
A two-step spot welding procedure for attaching thermocouples to SMA wire can
eliminate the influence of the “spurious voltage” in the temperature reading. By spot
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welding the thermocouple wires onto the SMA wire one by one, the thermocouple lead
offset is eliminated and the thermocouple provides an accurate point source reading.
Infrared thermal imaging can be a good supplement in the experiment to monitor errors
in temperature readings from thermocouples. Due to the curvature of the SMA wire, the
temperatures of the locations on the SMA wire that are the closest to the infrared camera
represent the temperature of the SMA wire. So a line analysis across the SMA wire on
the software “ThermaCAM” is required to determine the temperature of the SMA wire by
infrared thermal imaging and the highest temperature on the line is the temperature of the
SMA wire.
A new natural convective heat transfer correlation comprising the inclination angle φ is
developed based on experimental results, which can be used to predict the temperature of
a SMA wire given its diameter and inclination angle.
The comparisons show that the new correlation agrees with existing correlations in a
vertical orientation and for small Rayleigh numbers (0.001 < RaD < 0.05) in the
horizontal orientation. The correlation developed in this work for horizontal orientation
tends to overestimate values of Nusselt numbers as predicted in other correlations when
the Rayleigh number is high (0.05 < RaD < 0.6). It is speculated that this overestimation
can be attributed to a temperature distortion associated with thermocouple measurement
at or near ambient pressure conditions.
iv
Acknowledgements
I would like to express my sincerely appreciation to Professor Richard Culham (MME)
and Professor Robert Gorbet (ECE) for giving me the opportunity to work on this project
and valuable directions in my work. I also want to thank Professor Pete Teertstra (MME)
for sharing his extensive experience in laboratory techniques, which is a very important
part of my work. I would also like to thank all those who provided training, advice and
guidance throughout the course of my program.
v
Dedication
To all of my family members, for their warm support.
To everybody I worked with in this project, from whom I learnt.
vi
Table of Contents
Author’s declaration............................................................................................................ ii
Abstract .............................................................................................................................. iii
Acknowledgements..............................................................................................................v
Dedication .......................................................................................................................... vi
Table of Contents.............................................................................................................. vii
List of Figures ................................................................................................................... xii
List of Tables ................................................................................................................... xiv
Nomenclature.....................................................................................................................xv
Chapter 1 Introduction .........................................................................................................1
1.1 Background ................................................................................................................1
1.2 Motivation for the Current Work ...............................................................................3
1.3 Problem Statement .....................................................................................................4
1.4 Overview ....................................................................................................................5
Chapter 2 Literature Review................................................................................................8
2.1 Review of Temperature Measurement .......................................................................8
2.1.1 Invasive Temperature Measurement Methods ....................................................9
2.1.1.1 Thermometers.........................................................................................................10
2.1.1.2 Thermocouples .......................................................................................................11
2.1.1.3 Resistance Temperature Detectors .........................................................................17
2.1.1.4 Thermistors.............................................................................................................17
2.1.2 Semi-Invasive Methods .....................................................................................18
2.1.2.1 Heat-Sensitive Crystalline Solids and Paints..........................................................19
2.1.2.2 Pyrometric Cones ...................................................................................................19
2.1.2.3 Temperature Phosphors and Temperature-Sensitive Paints ...................................19
2.1.2.4 Thermochromic Liquid Crystals.............................................................................20
2.1.3 Non-Invasive Methods ......................................................................................21
2.2 Review of the Existing Heat Transfer Correlations .................................................25
2.2.1 Horizontal ..........................................................................................................26
vii
2.2.2 Vertical ..............................................................................................................28
2.2.3 Inclined Wire .....................................................................................................30
2.3 Summary of Literature Review ................................................................................32
Chapter 3 Infrared Thermometry in SMA Wire Application ............................................34
3.1 Principles of Infrared Thermometry.........................................................................34
3.1.1 Fundamentals of Thermal Radiation .................................................................34
3.1.2 Lambert’s Cosine Law.......................................................................................35
3.1.3 Planck’s Law .....................................................................................................36
3.1.4 Stefan-Boltzmann’s Law ...................................................................................38
3.1.5 Emissivity ..........................................................................................................39
3.2 Design Element ........................................................................................................39
3.2.1 Detectors............................................................................................................40
3.2.2 Thermal Imaging Scanner .................................................................................41
3.3 SMA Temperature Determinations Through Thermo Imaging ...............................42
3.3.1 Emissivity of SMA Wire ...................................................................................42
3.3.2 Determining Temperature of SMA Wire...........................................................45
3.3.3 Verifying of IR Camera.....................................................................................49
3.4 Summary of Infrared Thermal Imaging in SMA Wire Application.........................53
Chapter 4 Improved Spot Welded Thermocouples in SMA Wire Application .................54
4.1 Current’s Influence on Spot Welded Thermocouple................................................56
4.2 Current Reversal and Averaging Method.................................................................60
4.3 Pulse Shut-Off Measurement Method......................................................................61
4.4 Two-Step Spot Welded Thermocouple ....................................................................63
4.5 Temperature Distribution Along SMA Wire ...........................................................65
4.6 Summary of Improved Two-Step Spot Welded Thermocouple...............................69
Chapter 5 Experimental Methods ......................................................................................71
5.1 Background ..............................................................................................................71
5.1.1 Pressure Variation Method ................................................................................73
5.1.2 Data Collection ..................................................................................................76
viii
5.1.2.1 The Rayleigh Number (Ra) ....................................................................................76
5.1.2.2 The Nusselt Number (Nu) ......................................................................................77
5.1.2.3 The Inclination Angle.............................................................................................80
5.2 SMA Wire Test Fixture............................................................................................80
5.2.1 Level Temperature Distribution on the SMA Wire...........................................81
5.2.2 Sample SMA Wire.............................................................................................82
5.2.2.1 Diameter of the Sample SMA Wire .......................................................................83
5.2.2.2 Length of the Sample SMA Wire ...........................................................................83
5.2.2.3 Oxide of the Sample SMA Wire.............................................................................83
5.2.2.4 Ring Terminal.........................................................................................................84
5.2.3 SMA Wire Setup on the Fixture ........................................................................85
5.2.3.1 E-type 40 AWG Thermocouple..............................................................................86
5.2.3.2 Voltage Measurement Leads ..................................................................................87
5.2.3.3 Power Leads ...........................................................................................................87
5.2.3.4 Placement of TCs, Volt Measurement Leads, Power Leads and Ring Terminal ...88
5.2.3.5 String ......................................................................................................................88
5.2.3.6 Fixture ....................................................................................................................89
5.2.3.7 Mechanical Load ....................................................................................................89
5.3 Experimental Setup Diagrams..................................................................................90
5.4 Equipment ................................................................................................................95
5.4.1 Inclination Angle Control..................................................................................95
5.4.2 Pressure Control and Monitor............................................................................95
5.4.2.1 NRC 3117 Vacuum Station ....................................................................................95
5.4.2.2 CeramiCel VCMT-13 Vacuum Gauge ...................................................................96
5.4.3 Power Supply and Current Shunt ......................................................................96
5.4.3.1 Power Supply..........................................................................................................97
5.4.3.2 Current Shunt..........................................................................................................97
5.4.4 Data Acquisition ................................................................................................98
5.4.4.1 Keithley 2700 .........................................................................................................98
5.4.4.2 Keithley 7700 Module Card ...................................................................................99
5.4.4.3 ExceLINX.............................................................................................................100
ix
5.4.5 Welder and Microscope...................................................................................101
5.4.5.1 Capacitor-Discharge Spot-Welder........................................................................101
5.4.5.2 Microscope ...........................................................................................................102
5.4.5.3 Thermocouple Welder ..........................................................................................102
5.5 Experimental Procedure .........................................................................................103
5.6 Uncertainty Error Analysis.....................................................................................107
5.7 Summary of Experimental Methods ......................................................................107
Chapter 6 Results and Discussions ..................................................................................108
6.1 Experimental Data Reduction ................................................................................108
6.2 Convective Heat Transfer Modeling ......................................................................119
6.2.1 Development of Convective Heat Transfer Correlation..................................119
6.2.2 Verification of Convective Heat Transfer Correlation ....................................120
6.2.2.1 Comparisons in Vertical and Horizontal ..............................................................120
6.2.2.2 Temperature Distortion on SMA Wire.................................................................122
6.2.2.3 IR Picture Verifies the Temperature Distortion on SMA Wire ............................124
6.3 SMA Wire Temperature Prediction Through New Correlation.............................129
6.4 Summary of Results and Discussion......................................................................133
Chapter 7 Conclusion and Recommendations .................................................................136
7.1 Conclusion..............................................................................................................136
7.1.1 The Carrying Current’s Influence on Temperature Readings .........................136
7.1.2 Two-Step Spot Welding Thermocouple ..........................................................137
7.1.3 Infrared Thermal Imaging is a Good Supplement ...........................................137
7.1.4 Temperature Distortion Error in New Correlation’s Verification ...................138
7.1.5 SMA Wire Temperature Prediction Through New Correlation ......................139
7.2 Recommendations ..................................................................................................139
7.2.1 Improve Two-Step Spot Welding....................................................................139
7.2.2 Improve Thermocouple Method to Minimize Temperature Distortion...........140
7.2.3 Develop a Correlation of Temperature and Resistance ...................................140
Appendix A Ring Terminal Brings SMA Wire Temperature Down...............................141
Appendix B Uncertainty Analysis ...................................................................................143
x
B.1 Uncertainty Analysis Method................................................................................143
B.2 Uncertainty in Measured Values ...........................................................................144
B.2.1 Temperature T.................................................................................................145
B.2.2 Voltage V ........................................................................................................145
B.2.3 Current I ..........................................................................................................146
B.2.4 Pressure P .......................................................................................................146
B.2.5 Dimension .......................................................................................................147
B.2 The Inclination Angle φ .....................................................................................147
B.3 Uncertainty in Calculated Values ......................................................................147
B.4 Uncertainty of Experimental Result...................................................................149
Appendix C Conduction Wire Loss Via TC and Voltage Measurement Leads ..............150
References........................................................................................................................155
xi
List of Figures
Figure 1-1 Microscopic Diagram of the Shape Memory Effect ......................................... 2
Figure 2-1 Infrared Thermal Imaging of SMA Wire by FLIR ThermaCam S60 ............. 23
Figure 2-2 Inclination Angle φ ......................................................................................... 25
Figure 2-3 Correlations of NuD vs RaD for Horizontal Slender Cylinders........................ 28
Figure 2-4 Correlations of NuD vs RaD for Vertical Slender Cylinders............................ 30
Figure 2-5 The Effect of Inclination on NuD for a Slender Cylinder................................ 32
Figure 3-1 Electromagnetic Radiation Covers a Wide Range of Wavelengths................ 35
Figure 3-2 The Angular Distribution of Blackbody Intensity and Emissive Power......... 36
Figure 3-3 The Spectral Emissive Power of Blackbody With Wavelength...................... 38
Figure 3-4 Thermopile Sensor in Infrared Camera........................................................... 40
Figure 3-5 Two-Dimensional Scanning for a Small Detector Array................................ 41
Figure 3-6 Gier Dunkle Db100 Infrared Reflectometer ................................................... 43
Figure 3-7 Obtain Emissivity of SMA Wire Through Painting........................................ 44
Figure 3-8 Different Points on SMA Wire Observed by a Detector................................. 45
Figure 3-9 Temperature Bias on a Line Analysis Along the Wire ................................... 46
Figure 3-10 Line Analysis Performed Across the Wire.................................................... 47
Figure 3-11 A Thermistor is in Contact With the SMA Wire .......................................... 48
Figure 3-12 Gier Dunkel Emissivity Sample is Tested by IR Camera Imaging............... 50
Figure 3-13 IR Temperature Reading is Much Higher Than Room Temperature ........... 50
Figure 3-14 IR Camera Verified, Thermistor Bead Temperature from Two Sources...... 52
Figure 4-1 Thermocouple Reading is Affected by Spatial Offset Across TC Leads........ 55
Figure 4-2 SMA Wire with Thermocouple Attachment Points ........................................ 57
Figure 4-3 Comparison of Glue Spot TC and Spot Welded TC ....................................... 57
Figure 4-4 Temperature Measurement of Spot Welded TC and Glue Spot TC ............... 58
Figure 4-5 Spot Welded TC.............................................................................................. 59
Figure 4-6 Pulse Shut-Off Method ................................................................................... 62
Figure 4-7 Two-Step Spot Welded Thermocouple........................................................... 65
Figure 4-8 Five Thermocouples, Voltage Measurement Leads and Power Leads Layout66
Figure 4-9 No Current Influence on Two-Step Spot Welded Thermocouple Reading .... 67
Figure 4-10 The Temperature Reading From TC1 is Much Lower Than Other TCs....... 68
xii
Figure 5-1 Thermal Analysis on the SMA Wire Test Section at Steady-State................. 79
Figure 5-2 Expected Uniform Temperature Distribution on the SMA Wire.................... 82
Figure 5-3 Ring Terminal (DYNALLOY, INC)............................................................... 84
Figure 5-4 Sample SMA Wire Setup on the Fixture......................................................... 85
Figure 5-5 Test Fixture ..................................................................................................... 90
Figure 5-6 Experimental Setup Diagram .......................................................................... 92
Figure 5-7 Experimental Close Up Diagram of Bell Jar................................................... 93
Figure 5-8 SMA Wire Inclined From Horizontal at an Angle φ....................................... 94
Figure 5-9 Power Supply and Current Shunt.................................................................... 96
Figure 5-10 Keithley 2700 ................................................................................................ 98
Figure 5-11 Keithley 7700 Module................................................................................... 99
Figure 5-12 ExceLINX Channel Confiugration............................................................... 100
Figure 5-13 Capacitor-Discharge Spot-Welder .............................................................. 101
Figure 5-14 The Smallest Thermocouple Bead Made on THERM-X (Model 258B) .... 103
Figure 6-1 NuD vs RaD at Inclination Angles From 0 To 900 (RaD = 0.001 – 0.6) ......... 118
Figure 6-2 Natural Convective Flow Near Horizontal Cylinder and Vertical Cylinder. 124
Figure 6-3 IR Picture Shows Temperature Distortion at Thermocouple Attachment .... 125
Figure 6-4 Comparison To the Existing Correlations in Horizontal Orientation ........... 127
Figure 6-5 Comparison to the Existing Correlations in Vertical Orientation................. 128
Figure 6-6 Flowchart to Determine SMA Wire Temperature ........................................ 130
Figure 6-7 TSMA vs Carrying Current I (P = 1 atm)......................................................... 135
Figure A-1 Thermal Gradient After a Ring Terminal is Crimped on SMA Wire .......... 142
Figure C-1 SMA Wire Setup Diagram ........................................................................... 152
Figure C-2 Comparison of Power Input of 1TC and 2TCs in Vacuum .......................... 153
Figure C-3 Comparison of Power Input of 1TC and 2TCs at Pressures......................... 154
xiii
List of Tables
Table 4-1 Temperature Readings and Corresponding Voltages for TC1 ......................... 60
Table 6-1 Raw Data and the Results NuD, RaD at Inclination Angle φ = 00 ................... 111
Table 6-2 Raw Data and the Results NuD, RaD at Inclination Angle φ = 150 ................. 112
Table 6-3 Raw Data and the Results NuD, RaD at Inclination Angle φ = 300 ................. 113
Table 6-4 Raw Data and the Results NuD, RaD at Inclination Angle φ = 450 ................. 114
Table 6-5 Raw Data and the Results NuD, RaD at Inclination Angle φ = 600 ................. 115
Table 6-6 Raw Data and the Results NuD, RaD at Inclination Angle φ = 750 ................. 116
Table 6-7 Raw Data and the Results NuD, RaD at Inclination Angle φ = 900 ................. 117
Table 6-8 Correlation NuD = C RaDn at Each Inclination Angle φ.................................. 119
Table 6-9 The Values of k, µ and Pr at Different Temperature (25 – 70 0C) ................. 132
Table 6-10 TSMA vs Carrying Current I (P = 1 atm) ........................................................ 135
xiv
Nomenclature
A
area, m2
As
surface area, m2
Cp
constant pressure specific heat, kJ/(kg · K)
D
characteristic length, m
d
diameter of SMA wire, m
e
emissive power (usually with a subscript), W/m2, without a superscript e is usually
the exponential function
g
gravitational acceleration, m/s2
GrD
Grashof number
h
convective heat transfer coefficient, W/(m2 · K)
i
radiation intensity, W/(m2·sr)
iλ
spectral radiation intensity, W/(m2·µm·sr)
I
electric current, A
k
thermal conductivity, W/(m ·K)
L
length, m
l
length, m
M
molar mass, kg/kmol
NuD
Nusselt number
P
pressure, Pa
Pr
Prandtl number
q
energy flux, energy per unit area and per unit time, W/m2
qo
radiosity, W/m
qi
incident heat flux, W/m
Q
heat transfer rate, W
Qconv
heat transfer rate through convection, W
2
2
Qcond heat transfer rate through conduction, W
Qrad
heat transfer rate through radiation, W
R
gas constant, kJ/(kg·K)
RaD
Rayleigh number
Ru
universal gas constant, kJ/(kmol·K)
xv
T
temperature, K
Tm
mean temperature, K
Ts
surface temperature, K
Tss
steady-state temperature, K
T∞
ambient temperature, K
V
electric voltage, V
Z
compressibility factor
Greek Letters
β
volumetric expansion, 1/K
ε
emissivity
θ
zenithal angle, rad or degree
λ
wavelength, µm
µ
dynamic viscosity, kg/(m·s) or N·s/m2
ν
kinematic viscosity, = µ/ρ, m2/s
ρ
density kg/m3
ρ
liner resistance of wire, Ω/m
ρ
reflectivity
σ
Stefan-Boltzmann constant, 5.67 x 10-8 W/(m2·K4)
φ
azimuthal angle, rad or degree
φ
inclination angle, rad or degree
xvi
Chapter 1
Introduction
1.1 Background
Shape Memory Alloys (SMAs) are amazing and unique materials. After being
deformed, they can “remember” and recover their original shape when the temperature is
increased. According to Otsuka and Wayman (1998) [1], Arne Olander was the first who
observed the pseudo elastic behavior of the Au-Cd alloy in 1938. Kurdjumov and
Khandros [2] conducted experiments on CuZn and CuAl alloys and introduced the
concept of thermo elastic martensitic transformation to explain the reversible
transformation of martensite. The real breakthrough in the research and the application of
shape memory alloys is the discovery of NiTi by Buehler et al. [3] in 1963. The term
“NiTiNOL” was named after that in memory of their works in Naval Ordnance
Laboratory (NOL). Since that time, a series of new shape memory alloys were
investigated and adopted in commercial applications. In 1971, Cryofit, the first SMA
material, was used to join titanium tubing on the U.S. Navy / Grumman F-14A [4], [5].
The NiTiNb systems were widely used in nuclear reactors to repair battle damage [6].
High Temperature SMA (HTSMA), such as TiPd, TiPt and TiAu, which possess
transformation temperatures greater than 100 0C, were developed in earlier 1970’s [7]. In
1999, Miyazaki et al. [8] improved NiTiCu alloys. The improved fatigue life and low
cost of this new material made it suitable for a wide variety of engineering applications.
1
There are other types of SMA besides NiTi, such as: CuZnAl, CuAlNi, CuAlMn,
CoNiAl, NiMnGa, etc.
Figure 1-1 Microscopic Diagram of the Shape Memory Effect
Two major characteristics of shape memory alloys are shape memory and pseudoelasticity. Studies of micro structure shows: SMA has two different temperaturedependent crystal structures (or phases). One is martensite which exists at low
temperature; the other is austenite that exists at higher temperature. The austenite is
usually called the parent phase. It is the two different phases and a solid to solid phase
transition that gives rise to these special characteristics of SMA. As shown in Figure
1-1[9], at low temperature, SMA exists in the twinned martensite form. When a load is
applied to the SMA, it undergoes a macroscopic shape change and its micro structure
changes to the deformed martensite in which its molecules are no longer twinned. When
being heated, a reverse phase transformation (from detwinned martensite to austenite)
occurs which leads to a complete shape recovery. Cooling of the SMA causes the
2
formation of twinned martensite again with no associated shape change is observed. This
process is called shape memory effect (SME).
For many years, shape memory alloys have been a good choice for engineers and other
designers who apply the shape memory effects to convert thermal energy into mechanical
work. They have a variety of applications in aerospace, medicine and transportation. In
aerospace, they have been used in the Smart Wing program and the Smart Aircraft and
Marine Propulsion System demonstration (SAMPSON) [10], Rotorcraft [11] and the lowshock release mechanism in satellites [12]. In the field of medicine, SMAs can be used in
Orthodontic applications, Cardiovascular applications, Orthopedic applications and
Surgical Instrument applications. In transportation, SMAs have been used for
applications ranging from impact absorption to sensing and actuation, such as:
deployment of a protective panel; actuating blinds that cover the fog lamp to prevent
damage; and SMA spring-actuated valves to adjust the oil level in the gearbox [13].
1.2 Motivation for the Current Work
Shape memory alloy (SMA) actuators can be an ideal substitution for more traditional
actuators (e.g., magnetic solenoid) due to their unique “shape memory” property. Low
cost, light weight, scalable and having a high power/weight ratio, these actuators are
promising in a variety of mechanical systems, particularly in the automotive industry.
The grill of the louver in front of a car, for instance, can be opened and closed using the
shape memory alloy thus enabling you to control the aerodynamics of a car. Springs
made from shape memory alloys can be used as sensor-actuators in pressure control
valves or oil cooler bypass valves in automatic transmissions. Shape memory alloys can
3
even be designed to capture heat energy from engine exhaust gases via an electric
generator and transfer that energy to recharge batteries for hybrids or next generation
electric vehicles. With more and more scientists and engineers investigating this material,
shape memory alloy components are becoming increasingly popular for automotive
applications.
1.3 Problem Statement
SMA actuators rely on a reversible, thermally-driven phase transformation which
occurs as the alloy temperature is cycled between approximately 30 0C and 100 0C. The
difference in mechanical properties of the two material phases can be used to do
mechanical work. Notably, the NiTi SMA commonly used for actuation has a relatively
high electrical resistivity, enabling the design of electro-mechanical actuators using
SMA. An actuator design typically comprises a biased wire made of SMA, which
contracts in the presence of an electrical current, and expands as it cools when the current
is removed. The actuator behavior can be roughly divided into the thermo-electric
heating response which converts electrical to thermal energy and generates the phase
transformation, and the thermo-mechanical phase transformation which causes the
motion.
A good numerical thermo-electrical model of SMA wire is very important to protect
this material from overheating, while achieving maximum performance. The thermoelectrical process starts from Joule heating when a current passes the SMA wire,
undergoes heat exchange with environment and eventually obtains heat equilibrium. This
process may be incorporated with a phase transformation when the SMA wire
4
temperature goes above the transition temperature TAs at which martensite changes to
austenite. In the whole process, the temperature of the SMA wire increases from the room
temperature along a power curve, probably hesitates when phase transformation occurs,
and in the end reaches a balanced temperature. In order to accurately model the thermoelectrical behavior of SMA wire, it is necessary to develop a reliable technique to
measure the temperature of SMA wire and to understand clearly the mechanisms
whereby the SMA wire exchanges heat with the environment.
This work is divided into two parts. The first part focuses on the efforts to
experimentally investigate appropriate temperature measurement techniques, and the
development of reliable thermocouple-based methods. The measurements are developed
for the validation of thermo-electric models for SMA wire heating in the laboratory, so a
reasonable level of accuracy and precision is desirable. In addition, the eventual
approach may also be used in direct temperature measurement in an end application, for
direct temperature control of the SMA actuator. In the second part, the existing natural
convective heat transfer correlations of different orientations were studied and analyzed.
Due to differences in experimental conditions, these existing correlations are not
applicable to the design condition stipulated for this work; however they provide useful
information for the development of new heat transfer models.
1.4 Overview
In chapter 2, different available methods of temperature measurement will be reviewed.
The studies of temperature measurement in different categories from researchers will also
be analyzed, that include: invasive, non-invasive and semi-invasive methods. The
5
knowledge about temperature measurement will be used to develop experimental
methods to measure the temperature of SMA wire. After this, the existing heat transfer
correlations for natural convection from slender cylinders will be reviewed. Methods for
determining the applicable Rayleigh number (Ra) and Nusselt number (Nu) range, the
fluid type, and the diameter of the cylinder will be examined. The review helps to
determine if these correlations can be applied in this work directly or they can only be the
reference to a new correlation.
In chapter 3, infrared thermal imaging will be studied. The basic principle of thermal
radiation and the structure of the IR camera will be reviewed. The problems that occur
when using an IR camera to find out the temperature of SMA wire will be investigated,
such as: the emissivity of SMA wire; the correct location to determine the temperature of
SMA wire; and the verification of the IR camera.
In chapter 4, the two step spot welded thermocouple will be introduced. This includes:
the problems encountered when deploying spot welded thermocouple to measure a
current carrying wire; the theory of the effective voltage of a thermocouple; and a series
of experiments that result in the improved two step spot welded thermocouple.
Chapter 5 will discuss the experimental method to examine the effect of other
parameters on convection and to determine a new heat transfer correlation, which
includes: the overview of the experiment, the SMA wire setup on fixture, the experiment
setup diagrams, the experimental equipments, the experiment procedures and the
uncertainty errors in the experiment.
Chapter 6 presents the results and discussion of the experiments and consists of three
sections. In the experimental data reduction section: the experimental data are processed
6
and presented in tables and figures. Based on these figures, a new convective heat
transfer correlation is developed in the second section. A comparison of the new
correlation to existing correlations in the horizontal and vertical orientations are
performed and discussed. In the last section, three examples are used to demonstrate how
to predict the temperature of SMA wire through the new correlation.
Chapter 7 gives the conclusions and recommendations.
7
Chapter 2
Literature Review
Shape Memory Alloy materials are becoming more prevalent in many engineering
applications such as switches and actuators as a low cost alternative to more elaborate
and complex systems. As new materials are developed with unique thermophysical
properties, it is imperative that measurement techniques be developed that allow for
accurate and reliable assessment of these properties. In certain instances new
measurement procedures may need to be prescribed where there has been little activity in
the past, however, in many instances a review of the open literature will provide valuable
insight into procedures that can potentially be adapted to temperature measurement in
current carrying SMA wires. The following review is a non-exhaustive examination of
measurement techniques commonly used to determine surface temperatures of objects
with relatively small characteristic length scales, such as wires and thin films. In addition,
the literature review will examine empirically derived heat transfer correlations that have
been developed for natural convection heat transfer from similar objects such as wires
and circular cylinders.
2.1 Review of Temperature Measurement
The topic of temperature and temperature measurement has garnered much attention
over the past several centuries. Although fundamentally simple in its concept,
temperature measurement can be challenging, especially when trying to isolate localized
temperature fields over small domains, such as wires and ribbons with characteristic
8
lengths in the sub-millimeter range. For these micro-scale applications, conventional
methods of temperature measurement often do not provide satisfactory results, primarily
due to effects of scale such as lead losses and contact resistance in contact methods, and
domain resolution in non-contact methods.
Typically temperature cannot be measured directly relying instead on measuring the
effects of some other physical phenomena and then relating this to temperature. Even our
most fundamental methods of temperature measurement such as a thermometer
(volumetric expansion of a fluid), a thermocouple (Seebeck voltage induced in a couple
of dissimilar metals) or resistance temperature detector (resistance change) depend on
indirect methods of temperature detection. By the nature of contact between the medium
of interest and the device, Childs [14] classified the various temperature measurement
techniques into three categories: invasive; semi-invasive and non-invasive.
The purpose of this work is to determine SMA temperatures either directly or indirectly
as the wire is transitioned through the austenitic/martensitic transformation regimes.
Measurement constraints include accurate measurement of surface temperature at
multiple points on SMA wires in the range of 0.3 – 1.0 mm diameter, with a current flow
of up to 1.5 A and a temperature ceiling of approximately 150 0C. The following review
describes several potential measurement techniques and critically examines their
applicability to the measurement constraints listed above.
2.1.1 Invasive Temperature Measurement Methods
As the name implies, invasive techniques include a wide variety of measurement
techniques where the detector or the transducer is in direct contact with the medium of
9
interest thereby “invading” the measurement field. In most situations, the act of
measurement influences the measurement parameter of interest. Examples of invasive
methods include: liquid-in-glass thermometers, bimetallic strips, thermocouples,
resistance temperature detectors and gas thermometry.
2.1.1.1 Thermometers
Standard tube or capillary thermometers use a calibrated relationship between
volumetric expansion of a fluid and heat input to determine temperature over a nominal
range of interest. Typical fluids used in capillary tube thermometers include mercury or
organic liquids such as alcohol.
Bimetallic thermometers rely on the fact that materials expand at different rates as they
are heated. By attaching two dissimilar metals together to form a composite strip, a strain
is introduced into the strip upon heating which can be calibrated to provide a measure of
temperature. While these devices are not as accurate as thermocouples or RTDs and they
do not readily lend themselves to temperature recording, they have found wide spread use
in devices used to control temperature levels [15].
Gas thermometry provides an accurate means of measuring temperature but tends to be
a specialist activity and is usually confined to standards laboratories and cryogenic
applications. The use of gas thermometry in applications requiring a high measure of
certainty has been detailed by several other researchers including, Steur [16], Steur and
Pavese [17], and Edsinger and Schooley [18]. Gas thermometers are generally not
commercially available.
10
Unfortunately, thermometers do not generate data that are easily recorded or
transmitted and they cannot easily be used to make spot or point measurements,
effectively restricting their use for measuring temperatures in SMA wires and ribbons.
2.1.1.2 Thermocouples
Thermocouples are a common choice for temperature measurement because of their
self-energization, low cost, robust nature and wide temperature range. Measurement
accuracy is typically not as good as RTDs or thermistors but “as received” accuracies of
0.75 – 2.0 0C are typical and accuracies of 0.25 – 0.5 0C can be achieved with minor
calibration corrections.
Like other solid bodies and surfaces, SMA wires and ribbons face the same problems
when measuring their temperatures as Michalski et al. [19] described in their Theory of
the Contact Method. They pointed out that temperature measurement of a solid surface
has a series of possible errors that could contribute to deviations between a measured
temperature and that of the undisturbed object. These errors include: temperature
distortion error of the solid surface when introducing a probe; contact thermal resistance
error between the solid surface and the probe; and conduction thermal resistance error in
the probe. From the reviews below, we can conclude that choosing probes as thin as
possible and making good thermal contacts between the solid surface and the probes can
minimize all these systematic errors in temperature measurement of SMA wires and
ribbons.
In respect to these three main errors: temperature distortion error, contact thermal
resistance error, influence of current on thermocouple readings and system error of
11
thermocouple thermometry, numerous research initiatives have been conducted and the
following methods have been proposed to eliminate or control these effects.
Temperature Distortion Error
Robertson [20] improved a surface temperature probe by utilizing a heater and a
differential temperature measurement to control the heater so as to minimize the
temperature gradient along the probe and temperature distortion on the surface as well.
Nakabeppu and Suzuki [21] developed an active method for temperature measurement
with sub-micron resolution and a thermal feedback system. Similar to Robertson, this
method can keep a sensor at the same temperature as the target by using a differential
thermocouple and an electrical heater.
Ishihara et al. [22] investigated the measurement of surface temperature of burning
solid using micro thermocouples. They found that the decrease in indicated surface
temperature (temperature distortion) due to heat conduction through the leads was as
large as 100 K, depending on the wire diameter, burning rate and lead wire angle.
Shaukatullah and Claassen [23] reviewed the effect of thermocouple size and
attachment method on the measurement of surface temperature. They concluded that
thermocouple wires with small diameter cross sections and low thermal conductivities are
better for minimizing errors due to heat losses through the wires.
From the above works, we can see that several procedures can be used to minimize the
temperature distortion error, such as: maintaining the probe temperature as close as
possible to the surface temperature, placing lead wires as close as possible to the target
12
surface, and choosing small size thermocouples and low thermal conductivity
thermocouple wire.
Contact Thermal Resistance Error
Renken [24] positioned a thermocouple adjacent the inner perimeter of a cavity and
traversed the length of the cavity of the substrate, thereby enhancing the heat transfer
efficiency from the substrate to the thermocouple.
Leath [25] recorded and controlled the temperature of a heater wire by pressing the
thermocouple elements on the wire with a spring. They believed the thermal contact was
improved in this way.
Shaukatullah and Claassen [23] concluded that using tapes and non-thermally
conductive epoxy to attach thermocouples results in larger errors.
Anon [26] investigated thermocouple attachment methods in semiconductor
manufacturing and chose cementing the thermocouple bead in a tapered hole in the wafer
with improved reliability.
Sobolik et al. [27] used Unsteady Surface Element (USE) methods to model a
thermocouple wire attached to a thin disk. They also assessed the effect of errors and
thermal factors with mathematical techniques.
Dunstan et al. [28] measured temperature variations in NiTiNOL working elements
during cycling using a copper constantan thermocouple made from stripped 0.33 mm
diameter wires spot-welded onto one of the elements. They found the effectiveness of the
operation of the engine was strongly dependent upon the diameter of the NiTiNOL
working elements and the performance was excellent with 1.23 mm diameter. By
13
embedding a thermocouple in a hole spark-machined through 0.75 mm of the wire
diameter at the point of maximum curvature and comparing the centre and corresponding
surface temperature profiles, they also suggested that the surface temperatures, which are
easier to measure, are adequate for analyses of engine characteristics.
Thermally conductive epoxy can be used in our experiment; presses can also be applied
on the sensors against the target surface to make sure they are closely contacted. Spot
welding methods have higher thermal conduction than other methods if ignoring the
influence of current on thermocouple readings. Though some results or conclusions may
not be used in this study, all of these experiments provided valuable information on how
to improve thermal contact between the thermocouples and the target surfaces.
Influence of Current on Thermocouple Readings
Volkov [29] proposed that thermocouple readings can be affected by current in such a
way as to add an effective voltage ∆V to the thermo electro-motive force (EMF) of the
thermocouple if it has direct contact with the current carrying surface. He studied this
influence in an experiment in which 10 thermocouples were attached to a 0.7 mm
diameter 650 mm long steel wire with three different attachment methods: direct
attachment of the thermocouple by spot welding, indirect attachment using a Teflon tube
or a film of mica. He concluded that the influence of current on thermocouple readings
was a function of the voltage gradient along the current carrying surface. Without
properly handling this error, it is impossible to obtain a true temperature of the SMA
wire.
14
System Errors of Thermocouple Thermometry
Zanstra [30] developed a junction welding technique that enabled the thermocouples to
present a very uniform time response. If rapidly fluctuating temperatures are to be
compared, sensors should have small and identical time responses. Thermocouples offer
the advantage of minute size, small heat capacity and negligible heat flow to or from the
measuring spot. However, characteristics of soldered and welded thermocouple junctions
vary considerably. A technique is described for welding uniformly-sized, fine
thermocouples, which are producible even by unskilled workers.
Wang et al. [31] presented a computerized calibration laboratory with automatic data
acquisition capabilities and software for thermocouples and RTDs. With this system, for
J- type thermocouples on three consecutive lengths, the reproducibility was ± 0.1 0F up to
500 0F and ± 0.7 0F at 1000 0F. For 100 ohm RTDs the reproducibility was 0.01 ohm ( ±
0.025 0C) at 0 0C and ± 0.04 ohm ( ± 0.1 0C) at 200 0C.
Maeno et al. [32] developed a simple differential thermometer using a thermocouple
with a SQUID (superconducting quantum interference device) detector, which
demonstrated a response time of 15 ms at 1 K and a temperature sensitivity of 10 −7 K with
a 10 Hz filter.
Sawada and Nishiwaki [33] conducted research on the accuracy and response of a
thermocouple to transient temperature changes in an attached metal and demonstrated
that the accuracy of temperature measurement can be estimated using a Fourier number
15
and can be improved by selecting suitable combinations of the two kinds of thermocouple
wire and their diameters.
Suyama et al. [34] evaluated the calibration accuracy of R-type thermocouples through
comparison with a platinum resistance thermometer. The mean deviation of each
thermocouple is found to be about − 0.15 0C ~ + 0.10 0C with a standard deviation of less
than ± 0.05 0C in the temperature range of 50 0C ~ 950 0C.
Ancsin [35] studied the stability, reproducibility and accuracy of thermocouples and
found that noble metal thermocouples can yield a thermoelectric temperature scale
accurate within ± 0.1 0C from room temperature to 962 0C if they are calibrated after they
have been heat treated into a reasonably stable thermoelectric state.
Cengel et al. [36] increased accuracy and resolution of thermocouples by using
operational amplifiers to amplify the voltage signal before sending them to the computer.
The collection of technical papers cited above related to time response, stability,
reproducibility and accuracy of thermocouples provide good suggestions on using
thermocouples and eliminating systematic errors.
Another research worthy of mentioning is Rego et al. [37], where he conducted in-situ
temperature measurements of optical fibers subjected to electric arc discharges. In this
work, temperature profiles in capillary and optical fibers were estimated by solving the
classical heat transfer equations subject to data obtained from thermocouple
measurements from fibers of various diameters. The coupling of empirical and analytical
data provides an effective mechanism for predicting temperature distribution in fibers and
wires of small diameter.
16
2.1.1.3 Resistance Temperature Detectors
RTDs (Resistance Temperature Detectors) have an advantage of high accuracy and
good repeatability; however their relatively large size restricts their use in temperature
measurement of wires and ribbons. The achievable uncertainty for a commercially
available PRT (Platinum Resistance Thermometer) is generally of order of ± 0.01 0C to ±
0.2 0C over the range of 0 0C to 300 0C [38]. But in order to avoid the instability caused
by mechanical shock and strain due to thermal expansion, the PRTs are inevitably
constructed too bulky to be used for spot temperature measurement [39].
2.1.1.4 Thermistors
Thermistors have very high sensitivities with values of the order of 50 mV/0C, which is
more than one hundred times that of PRTs and one thousand times that of thermocouples.
Thermistors can have either a negative temperature coefficient (NTC), where the
resistance reduces with temperature, or a positive temperature coefficient (PTC)
depending on the type of materials used. NTC thermistors are the most common and can
operate at temperatures between – 200 0C and 1000 0C. PTC thermistors are not normally
used in temperature measurement.
Commercial NTC thermistors can be broadly classified into two groups depending on
how the leads are connected to the thermistor body: bead thermistors and metalized
surface contacts. Types of bead thermistors include bare beads, glass coated beads, glass
probes, glass rods and bead-in-glass enclosures. Metalized surfaced contact thermistor
consists of a metalized body fired onto a ceramic thermistor material. A wide range of
17
metalized surface contact thermistors are available including disks, chips or wafers,
surface mounts, flakes, rods, and washers.
The high sensitivity and fast response of thermistors make them ideally suited to
precision temperature measurement where resolutions better than 5 µK can be
obtained[40]. A thermistor could be the preferred choice over other thermometers for
measuring temperatures of SMA wires and ribbons as long as sufficiently small bead
thermistors can be found. Quality Thermistor Inc. can supply MINI BEAD 0.038" MAX
OD thermistor with a tolerance ± 0.2 0C (0 0C to 70 0C) [41]. Honeywell can supply
small thermistors with bead diameters of 0.014" [42]. Some specifications of products
need to be verified and efforts should be made to assess the availability of smaller
thermistors.
2.1.2 Semi-Invasive Methods
Semi-invasive temperature measurement techniques are classified here as those
involving some form of treatment of the surface of interest such as the application of a
temperature-sensitive paint and remote observation of the temperature dependent
properties of the surface application.
A range of techniques have been developed including thermo-chromic liquid crystals,
heat-sensitive crystalline solids and paints, thermographic phosphors and pyrometric
cones.
18
2.1.2.1 Heat-Sensitive Crystalline Solids and Paints
Most temperature sensitive crystalline solids and paints melt on reaching a certain
temperature and are used in the form of labels or pellets to indicate whether a particular
process temperature has been attained. On heating these products above some critical
temperature the indicator material melts, fuses or changes composition providing that a
process temperature has been reached or exceeded.
2.1.2.2 Pyrometric Cones
Pyrometric cones and bars have been developed for the ceramics and glass industries to
provide an indication of both firing time and temperature.
Heat-sensitive crystalline solids and paints and pyrometric cones cannot be used to
measure the temperature of SMA wires or ribbons because they are developed to indicate
a specific temperature and the evidential transformation in cones and paints is not
reversible. In addition, pyrometric cones are too bulky to be considered in wire or ribbon
applications.
2.1.2.3 Temperature Phosphors and Temperature-Sensitive Paints
Temperature phosphors and temperature-sensitive paints are materials that can be
excited by the absorption of energy and subsequently emit light in a fashion inversely
proportional to their temperature.
The thermographic phosphor thermometry technique can offer sensitivities of 0.05 0C
and an uncertainty of 0.1 % – 5 % of the Celsius temperature reading [43]. Applications
have included temperature measurements of flat plates in supersonic flows [44], wind
19
tunnel models [45], [46], turbine blades and components [47], [48], curved surfaces [49],
surface temperature fields [50], color TV screens [51], and textiles during microwave
drying [52].
2.1.2.4 Thermochromic Liquid Crystals
Thermochromic liquid crystals exhibit brilliant changes in color over discrete
temperature bands. With a change in temperature, thermochromic liquid crystal mixtures
will turn from colorless, black against a black background, to red at a particular
temperature and then pass through the visible spectrum (red to orange to yellow to green
to blue to violet) before turning colorless again.
Thermochromic liquid crystals have been widely used in the characterization of heat
transfer on applications such as: turbine blades [53], jet engine nozzles and vehicle
interiors [54]. Simonich and Moffat [55] report a calibration uncertainty of ± 0.25 0C
using mercury vapor lamp illumination. Ireland and Jones [56] demonstrated that liquid
crystal formulations typically have a response time constant of 0.003 s.
Temperature phosphors and temperature-sensitive paints and thermochromic liquid
crystals present high sensitivities and low uncertainties. They can be tried in our work to
measure the temperature of SMA wires and ribbons, although the effect of illumination
conditions and the process of calibrating the acquired image to temperature will be
important to understand.
20
2.1.3 Non-Invasive Methods
Non-invasive techniques are those where the medium of interest is observed remotely.
Infrared thermometry is the most widely used form of non-invasive temperature
measurement. A number of other non-invasive measurement techniques have also been
developed in addition to the infrared methods which are particularly useful for
monitoring gas or plasma temperatures. These non-invasive techniques are based on
observing the variation of refractive index, absorption or emission spectroscopy,
scattering, fluorescence and acoustics.
Infrared thermometers can be broadly classified into the following categories [57]:
•
spectral band thermometers
•
total radiation thermometers
•
ratio thermometers
•
multi-waveband thermometers
•
special-purpose thermometers and methods
•
thermal imagers
Spectral band thermometers are the most common form of infrared temperature
measurement. They measure radiant energy across a relatively narrow band of
wavelengths somewhere within the range 0.5 µm to 25 µm. In this spectral region, short
wavelengths have the advantage that the rate of change of radiant energy with
temperature is high (up to 2 % – 3 % 0C) and they are highly sensitive.
21
Total radiation thermometers measure the total radiance of a surface. As a broadband
of wavelengths will contain a large amount of energy at any temperature these are
normally used for general purpose or low-temperature application temperature
measurements where the energy emitted is relatively low.
Ratio thermometers, which are also known as dual-wavelength or two-color
thermometers, measure the radiance at two wavelengths and determine the ratio. The
advantage of ratio thermometers is that it is not necessary to know the emissivity of the
target. However they are less sensitive than spectral band thermometers.
Multi-waveband thermometers allow measurement of surface temperature when the
emissivity is not constant at different wavelengths. Special-purpose infrared
thermometers and methods include auxiliary reflector methods, hot surface methods,
polar radiometer methods, reflectance methods, temperature-invariant methods and fiberoptic thermometers. These techniques have been mostly developed in an attempt to
provide a measurement of temperature that is independent of emissivity.
Thermal imagers can be used to measure the temperature distribution over a target area.
The use of a thermal image to provide quantitative information for the temperature
distribution, particularly of a surface comprising different material, has to be carefully
managed. Without correction for local emissivity values, the thermal imager will assume
a default value and apply this to the whole image.
22
ε = 0.97
Wire with
oxide
Painted
Wire
Figure 2-1 Infrared Thermal Imaging of SMA Wire by FLIR ThermaCam S60
In Figure 2-1, the emissivity setting on the IR camera is set to 0.97 and the
temperatures at point 1 and point 2 are 46.3 0C and 37.1 0C. As you will read in Section
3.3.1, point 1 and point 2 should have the same temperature, but with different
emissivity. So when they are checked by an IR camera with a fixed ε = 0.97, the
temperature difference can be 9.2 0C.
Generally, if a good contact thermometer can be used for the application it is certainly
capable of higher accuracy than a radiation thermometer in the same situation [58]. When
choosing a radiation thermometer, the specifications need also to be considered include:
temperature range, accuracy, operating wavelength, field of view, response time, mode of
readout, application/manufacturer and calibration, besides emissivity.
The FLIR ThermaCAM S60 equipped with a zoom lens provides all the necessary
features for measuring surface temperatures of SMA wires and ribbons, as required in our
application. This thermal camera has a 320 x 240 detector, standard temperature range 0
− 500 0C and an accuracy of 2 % (typically 2 0C) [59]. When equipped with a FLIR 50
23
Micron Macro Lens, which has a close focus of 14.3 x 10.8 mm, [60] the FLIR
ThermaCAM S60 has a resolution of 0.045 mm/pixel.
Other research initiatives on radiation thermometry discussing the measurement of
temperature of small-scale objects are also reviewed.
The Exergen Company [61] used an infrared thermocouple, which emulates a K-type
thermocouple, to measure the temperature of a bare cable in preparation for subsequent
treatment of the cable with a plastic coating. The problem with this method is the
relatively large spot size of the sensor compared with our wire diameter.
Borca-Tascius and Chen [62] employed photo-thermal radiometry to measure the
temperature of fine wires with an accuracy of 1 % to 2 %, which is of the same order of
magnitude as the accuracy of contact methods.
Moron et al. [63] obtained local temperature values by measuring the intensity of
infrared emission of the sample. It is a good reference for constructing a relation between
temperatures and other parameters such as current, power, etc.
Shimizu et al. [64] combined thermo reflectance and RTDs in their work to measure
the temperature of micro-scale metal thin films. This method has a 0.2 0C temperature
resolution and a 1 µm spatial resolution at a visibility of 100 %.
Iadicola et al. [65][66][67] combined an infrared imaging camera and K-type
thermocouples to monitor the change of temperature along an SMA wire (using a small
quantity of a thermally conductive paste). In order to ensure accuracy, the thermocouples
were calibrated to ASTM certified thermometers at two calibration temperatures. The
thermal imaging camera calibrates the emissivity of the specimen by comparing to the
24
thermocouple readings. In this way, they believed the infrared images provide an accurate
measurement to within ± 1 0C in absolute temperature and within about ± 0.2 0C in
differential temperature.
2.2 Review of the Existing Heat Transfer Correlations
Numerous works have been performed to investigate natural convection heat transfer
from slender cylinders in different orientations including: horizontal, vertical and various
angles in between, as shown in Figure 2-2. Many empirical correlation equations have
been developed based on experimental data, theoretical deduction or numerical
calculation. Studying and understanding these correlations is necessary in this work.
Some of these correlations can be used to model heat transfer from current carrying SMA
wire; while others that may not be directly applicable to SMA wire can be used as
reference material in the development of new correlations.
Figure 2-2 Inclination Angle φ
25
2.2.1 Horizontal
Jaluria [68] derived the average Nusselt number NuD by using an integral method,
which can be applied to a range: 10-10 < RaD < 107, in Equation 2-1.
Nu D =
2
π
π /2
∫
0
⎛x⎞ ⎛x⎞
Nu D ⎜ ⎟d ⎜ ⎟ = F (Pr)( Ra D )1 / 4
⎝D⎠ ⎝D⎠
(2-1)
where: F(Pr) = F(0.7) = 0.436.
Tsubouchi and Masuda [69] developed a correlation of NuD - RaD (10-6 < RaD < 10) as
shown in Equation 2-2, based on experiments, in which oil was used as the fluid and the
cylinder diameter range was 0.015 – 0.063 mm.
Nu D = 0.36 + 0.56 RaD0.25
(2-2)
As a result of the survey of a large number of experimental investigations, Morgan [70]
suggested a set of correlations for calculating heat transfer from long horizontal cylinders
as in Equation 2-3.
Nu D = 0.675RaD0.058 , RaD = 10−10 − 10−2
Nu D = 1.02 RaD0.148 , RaD = 10−2 − 102
(2-3)
Nu D = 0.85RaD0.188 , RaD = 102 − 104
Churchill and Chu [71] used a theoretical deduction method and gave a correlation for
RaD > 10-6, as in Equation 2-4, which can be used for both laminar and turbulent flow
regimes: But they didn’t specify the applicable range of cylinder diameter.
26
⎧
⎪
⎪⎪
Nu D = ⎨0.6 +
⎪
⎪
⎪⎩
⎫
⎪
1/ 6
⎪⎪
0.387 RaD
8 / 27 ⎬
⎡ ⎛ 0.559 ⎞9 / 16 ⎤ ⎪
⎟ ⎥ ⎪
⎢1 + ⎜
⎢⎣ ⎝ Pr ⎠ ⎥⎦ ⎪⎭
2
(2-4)
Raithby and Hollands [72] performed analysis on the thin-layer of laminar and
turbulent flows at a constant temperature of the surface of the cylinder and obtained the
following correlation in the Rayleigh number range 10-2 < RaD < 1012:
Nu D3.337 = [2 / ln(1 + π / 1.2943 / 4 Cl Ra1D/ 4 )]3.337 + (0.72Ct Ra1D/ 3 )3.337
(2-5)
where, Cl = 0.50[1 + (0.49 / Pr)9 / 16 ]−4 / 9 , Ct = [0.14 Pr 0.084 ,0.15]∗
The operator “[,]*” means that [A, B]* = B if A ≤ B and [A, B]* = A if A > B. Here Pr
= 0.7, so Ct = [0.14 ⋅ 0.70.084 ,0.15]∗ = [0.1359,0.15] = 0.15 .
A graphical representation of these correlations is shown in Figure 2-3 when. Pr = 0.7.
Due to the different methods used and also due to the different situations in each case, the
correlations are not exactly the same.
27
Nu D − Ra D (Horizontal)
1
0.9
0.8
Jaluria [68]
NuD
0.7
0.6
Tsubouchi & Masuda [69]
0.5
Morgan [70]
0.4
Chruchill & Chu [71]
0.3
Raithby & Hollands [72]
0.2
0.1
0
0.001
0.01
0.1
1
Ra D
Figure 2-3 Correlations of NuD vs RaD for Horizontal Slender Cylinders
2.2.2 Vertical
In the vertical orientation, Elenbass [73] started with the correlation for a horizontal
cylinder, he derived a theoretical equation, as in Equation 2-6, for average laminar
natural convection from a vertical cylinder by analogizing a vertical cylinder to a vertical
plate, without specifying the applicable Rayleigh number range.
⎛
2 ⎞
D⎞
⎛
⎟⎟ = 0.6⎜ RaD ⎟
Nu D exp⎜⎜ −
L⎠
⎝
⎝ Nu D ⎠
0.25
(2-6)
Yang [74] suggested a general empirical equation with constant wall temperature that
applies to all Prandtl numbers and Rayleigh numbers. But in the laminar region, it has big
28
differences when compared to the experimental data. The length / diameter ratios (L / D)
of test cylinders in the experiments are also smaller than the SMA wire in this work.
0.25
D⎞
⎛
0.670⎜ RaD ⎟
L⎠
⎝
Nu D = 0.36 +
9 / 16 4 / 9
⎡ ⎛ 0.492 ⎞ ⎤
⎟ ⎥
⎢1 + ⎜
⎣⎢ ⎝ Pr ⎠ ⎦⎥
(2-7)
Koch [75] proposed a correlation based on experimental data for diameters from 14
mm to 100 mm, the length / diameter ratio from 20 – 152, and RaD (L / D) from 109 to
1011.
L⎞
⎛
Nu D = 0.00562⎜ RaD ⎟
D⎠
⎝
0.312
(2-8)
Mueller [76] also gave a similar formula but for lower RaD ranging from 10-6 to 10-2.
Nu D = 1.0( RaD )0.11
(2-9)
Zitsev et al. [77] investigated the heat transfer in a gas of a very thin vertical wire (D =
10 – 100 µm, RaD (D /L) = 10-11 − 10-4.5) through which an electric current flows and
obtained the Equation 2-10 with accuracy within 2 %:
⎤
⎡
⎥
⎢
2
4.47
⎥
= ln ⎢1 +
0.26
⎢ ⎛
Nu D
D⎞ ⎥
⎢ ⎜ RaD ⎟ ⎥
L⎠ ⎦
⎣ ⎝
(2-10)
Equation 2-6 to Equation 2-10 are plotted when D = 0.5 mm, L = 250 mm, Pr = 0.7 as
shown in Figure 2-4.
29
Nu D − Ra D (Vertical)
1
0.9
NuD
0.8
0.7
Elenbass [73]
0.6
Yang [74]
Koch [75]
0.5
Mueller [76]
0.4
Zitsev & Sokovishin [77]
0.3
Present Correlation
0.2
0.1
0
0.001
0.01
0.1
1
Ra D
Figure 2-4 Correlations of NuD vs RaD for Vertical Slender Cylinders
Generally, Figure 2-3 and Figure 2-4 have the same trend when RaD increases, but NuD
is higher when the wire is horizontal compared to a vertical orientation. This also leads to
another conclusion where NuD changes when the wire is inclined at an angle from
horizontal.
2.2.3 Inclined Wire
Oosthuizen [78] studied flow over an inclined circular cylinder of lengths between
152.4 and 304.8 mm and outside diameter between 19.1 and 25.4 mm. The results showed
that the heat transfer coefficient decreases with increasing angle of inclination φ (wire
and horizontal).
30
In Oosthuizen’s result, the Grashof number for an inclined cylinder is determined by
the Equation 2-11:
GrD =
gβ (Ts − T∞ ) D 3 cos ϕ
(2-11)
ν2
In air when Pr = 0.7 the following dimensionless dependence can be used to describe
the correlation of NuD and RaD:
Nu D
⎛ Ra D
⎞
cos ϕ ⎟
⎜
⎝ 0.7
⎠
1/ 4
= 0.42
1/ 4
= 0.55,
Nu D
⎛ Ra D
⎞
cos ϕ ⎟
⎜
⎝ 0.7
⎠
Nu D
⎛ Ra D
⎞
cos ϕ ⎟
⎜
⎝ 0 .7
⎠
where, L* =
1/ 4
⎡ ⎛ 1.31
= 0.42 ⎢1 + ⎜⎜ 0.25
⎢⎣ ⎝ L*
⎞
⎟⎟
⎠
8
⎤
⎥
⎥⎦
when L* > 10
(2-12)
when L* < 1
(2-13)
0.125
when 1 < L* < 10
(2-14)
L
, and the RaD is from 2.8 × 104 to 6.3 × 104.
D ⋅ tg (ϕ )
In this case, as L = 250 mm, D = 0.5 mm and Pr = 0.7, L* > 10 when 0 < φ < 88.90. So it
is reasonable to assume that Equation 2-12 applies to any angle φ. Equation 2-12 is
plotted in Figure 2-5.
31
Nu D vs Angle φ
0.45
0.4
0.35
NuD
0.3
0.25
Ra=0.001
Ra=0.6
0.2
0.15
0.1
0.05
0
0
20
40
60
80
100
Angle φ (degree)
Figure 2-5 The Effect of Inclination on NuD for a Slender Cylinder
2.3 Summary of Literature Review
The Temperature Measurement
Thermocouples are more suitable in this work than other invasive methods such as:
thermometers, thermistors or RTDs. A thermometer is large in size and is not capable of
providing a recordable readout. A RTD is more accurate than a thermocouple, but it is
usually much larger than a thermocouple and can not give an accurate temperature
reading of a thin SMA wire or ribbon. A thermistor is also more accurate than a
thermocouple, but the smallest available commercial thermistor is still five times bigger
in diameter than the bead of a thermocouple. Though not as accurate as a RTD or a
32
thermistor, a thermocouple has an accuracy of 1 0C with an internal CJC (or an accuracy
of 0.1 0C with ice point CJC and annual calibration). In addition, thermocouples are
cheap, easy to fabricate in the lab and can be spot welded onto the SMA wire directly.
The main drawback of a thermocouple is the temperature distortion that may be involved.
Infrared thermal imaging can not be the primary method to measure the temperature of
the SMA wire in this work due to its low accuracy of 2 0C and the inability to provide a
recordable readout for modeling. (by the current FLIR ThermaCAM S60). But infrared
thermal imaging can be a good supplement to thermocouples because it does not need to
contact object to give a temperature reading. This can be of great help when monitoring if
there is any temperature distortions on the SMA wire, or any carrying current’s influence
on temperature readings.
The Existing Heat Transfer Correlations
There are very few correlations in the current literature that account for how the
Nusselt number changes with the Rayleigh number when a wire (or a slender cylinder) is
inclined from the horizontal. In order to predict the temperature of SMA wire and to
characterize SMA wire, it is required to develop a new natural convection heat transfer
correlation which comprises an inclination angle φ from the horizontal. Some correlations
in the horizontal and vertical orientations were developed through different methods such
as: theoretical deduction and numerical analysis, or based on the experimental data that
are quite different from this work, but they can be good references to verify the new
developed correlation as they are applicable over a similar Rayleigh number range with
the new correlation.
33
Chapter 3
Infrared Thermometry in SMA Wire Application
3.1 Principles of Infrared Thermometry
Infrared thermometry is the most widely used form of non-invasive temperature
measurement. Heat transfer can occur by means of three fundamental mechanisms:
conduction, convection and radiation. The last form can be stated more fully as the
transfer of heat energy by means of electromagnetic radiation, which is known as thermal
radiation. The emission of energy in the form of electromagnetic radiation can be
exploited to undertake a measurement of temperature. A typical infrared measurement
system might comprise the source, the medium through which the radiant energy is
transmitted, an optical system to gather the electromagnetic radiation, a transducer to
convert the radiation into a signal related to temperature and amplification and interface
circuitry to control, display and record the measurement.
3.1.1 Fundamentals of Thermal Radiation
Electromagnetic radiation consists of interacting self-sustaining electric and magnetic
fields that propagate through vacuum at a speed of 299,792,458 m/s. It does not
necessarily need a medium for transmission and can travel through vacuum or through a
body of fluid.
34
The known electromagnetic spectrum comprises electromagnetic waves with
wavelengths as illustrated in Figure 3-1 [79] The classifications are for convenience and
the band between 0.1 µm and 100 µm is known as thermal radiation and that between 0.78
µm and 1000 µm as infrared radiation.
Figure 3-1 Electromagnetic Radiation Covers a Wide Range of Wavelengths
3.1.2 Lambert’s Cosine Law
Lambert’s Cosine Law can be expressed as:
eλb (λ , θ , ϕ ) = iλb (λ ) cos θ = eλb (λ , θ )
(3-1)
where: eλb (λ ,θ , ϕ ) is the energy emitted by a black surface per unit time within a unit
small wavelength interval centred around the wavelength λ, per unit elemental surface
area, and into a unit elemental solid angle centred around the direction (θ, φ).
35
Figure 3-2 The Angular Distribution of Blackbody Intensity and Emissive Power
As shown in Figure 3-2 [80] Lambert's cosine law says that the total radiant power
observed from a "Lambertian" surface is directly proportional to the cosine of the angle θ
between the observer's line of sight and the surface normal. The law is also known as the
cosine emission law or Lambert's emission law. Flat diffusing surfaces are said to be
ideal diffusers or Lambertian if the geometrical distribution of radiation from the surfaces
obeys Lambert’s Law. Lambert’s Law has important consequences in the measurement of
radiation. The cosine factor needs to be considered when making meaningful
measurements of radiation with angular distribution.
3.1.3 Planck’s Law
Any object will emit energy due to its temperature. The term “thermal radiation” is
used to describe the energy that is emitted in the form of electromagnetic radiation at the
surface of a body that has been thermally excited.
36
The thermal radiation emitted by a surface is not equally distributed over all
wavelengths. The wavelength dependency of any radiative quantity or surface property is
referred to as spectral dependency.
For a blackbody the spectral distribution of hemispherical emissive power and radiant
intensity in vacuum are given as a function of wavelength and the blackbody’s absolute
temperature:
eλb ( λ , T ) = π ⋅ i ' λb ( λ , T ) =
2π ⋅ C1
λ (e C2 / λT − 1)
(3-2)
5
This is known as Planck’s spectral distribution of emissive power. C1 and C2 are
constants. As shown in Figure 3-3, [81] the hemispherical spectral emissive power is
given as a function of wavelength for several different values of the absolute temperature.
It is evident that the energy emitted for all wavelengths increases as the temperature
increases. The curves show that this is also true for the energy at each individual
wavelength. Another characteristic is that the peak spectral emissive power shifts toward
a smaller wavelength as the temperature is increased.
37
Figure 3-3 The Spectral Emissive Power of Blackbody With Wavelength
3.1.4 Stefan-Boltzmann’s Law
The hemispherical total emissive power of a blackbody radiating into a vacuum can be
expressed as:
∞
∞
0
0
eb = ∫ eλb (λ )dλ = ∫ π ⋅ i ' λb (λ )dλ = σ ⋅ T 4
(3-3)
This is the Stefan-Boltzmann’s law, and σ is the Stefan-Boltzmann’s constant. This
equation enables a calculation to be made to determine the amount of radiation emitted in
all directions over all wavelengths simply from knowledge of the temperature.
38
3.1.5 Emissivity
The magnitude of radiation from a real surface is a function of both the temperature of
the surface and the surface properties. The surface property limiting the quantity of
radiation is called emissivity, ε. Emissivity is defined on a scale from 0 to 1 and is the
ratio of the electromagnetic flux that is emitted from a surface to the flux that would be
emitted from an ideal blackbody at the same temperature.
The hemispherical total emissivity is defined by the ratio of the total emissive power of
a grey body to that of a blackbody at the same temperature:
(3-4)
e(T )
ε (T ) =
eb (T )
where: ε(T) is the total emissivity, e(T) is the emissive power for a grey body (W/m2).
The emissive power of a grey body at a given temperature can be readily calculated by
a modification to the Stefan-Boltzmann’s law:
∞
∞
0
0
e = ε ∫ eλb (λ )dλ = ε ∫ π ⋅ i 'λb (λ )dλ = ε ⋅ σ ⋅ T 4
(3-5)
3.2 Design Element
IR thermometers come in a wide variety of configurations pertaining to optics,
electronics, technology, size, and protective enclosures. All, however, have a common
chain of IR energy in and an electronic signal out. This basic chain consists of collecting
optics, lenses, and/or fiber optics, spectral filtering, and a detector as the front end.
39
3.2.1 Detectors
An essential component of an infrared thermometer is the transducer used to convert
the absorbed electromagnetic radiation into a signal that can be measured. These devices
are called detectors or thermal detectors and function by converting the absorbed
radiation into heat energy causing the detector temperature to rise accompanied by the
output.
Thermal detectors are widely used in infrared thermometry. They convert the absorbed
electromagnetic radiation into heat energy causing the detector temperature to rise. This
can be sensed by its effects on certain physical properties, such as: electrical resistance
used by a bolometer; thermo electro-motive force (EMF) used by thermocouple and
thermopile detectors.
Figure 3-4 Thermopile Sensor in Infrared Camera
40
As shown in Figure 3-4 [82], a differential temperature between the hot and cold
junctions is generated by using a geometry which conducts heat from the hot and cold
junction at different rates or by using materials with different thermal conductivity.
3.2.2 Thermal Imaging Scanner
Thermal imaging involves determining the spatial distribution of thermal energy
emitted from the surface of an object. This information can be manipulated to provide
qualitative and quantitative data of the distribution of temperature on a surface. In their
usual form they comprise an optical system, a detector, processing electronics and a
display.
Figure 3-5 Two-Dimensional Scanning for a Small Detector Array
As shown in Figure 3-5 [83], a single detector with some form of scanner can be used
to transmit the radiation signal from specific regions of the optical system to enable a
two-dimensional image of the temperature distribution to be built up.
41
3.3 SMA Temperature Determinations Through Thermo Imaging
As mentioned before, there are errors in temperature measurement of SMA wires when
using contact methods such as: thermocouples and RTDs. One of these errors is aroused
by the current that passes through the SMA wire. The other error is caused by the
temperature local distortion when the sensor is introduced. With these errors, it is
difficult to know the real temperature readings of SMA wire, and also it is hard to model
the electric-thermo characteristic of SMA wire.
Infrared temperature measurement is a non-contact, non-obtrusive method used to
obtain temperature readings, thereby eliminating many of the problems associated with
methods such as thermocouples, thermistors and RTDs
3.3.1 Emissivity of SMA Wire
In order to obtain an accurate measure of temperature using infrared techniques, it is
imperative to obtain an accurate measure of the surface emissivity of the SMA wire. A
typical method to obtain emissivity is to compare the infrared reading to the
thermocouple reading. Through adjusting the emissivity setting on the IR camera, the
infrared reading can be brought into agreement with the thermocouple reading. Then the
emissivity setting on IR camera is the emissivity of the object. But in this application, the
thermocouple readings contain large errors and this is why the infrared method is
introduced. So other ways must be found to obtain the emissivity of SMA wire.
The emissivity of an object can be measured using a Gier Dunkle DB100 Infrared
Reflectometer. (Figure 3-6) Due to: ε(T) + ρ(T) = 1 ( for diffuse-gray surface), so the
42
reflectometer measures the reflectivity of the object and the emissivity of the object can
be calculated. In this way, the emissivity of SMA (Flexinol, dia = 500 µm, spool# = 23185B, As= 90 0C) is obtained as: ε = 0.37. Given the curvature of the SMA wire, it can
be very difficult to obtain consistent readings over the perimeter of a wire due to
reflection and angle of incidence effects.
wavelength range: 4 – 40 µm
sample size: dia 20 mm min (flat)
accuracy: ± 0.02
Figure 3-6 Gier Dunkle Db100 Infrared Reflectometer
As shown in Figure 3-7, another method used in this work is to paint a section of SMA
wire with black paint that has a uniform emissivity of 0.97. The emissivity of a SMA
wire can be determined by using an indirect method as follows:
•
take IR images of painted and non-painted sections of the SMA wire with the
emissivity of the camera set to 0.97 (the known value of the painted section)
•
record the temperature of the painted section of the SMA wire
43
•
adjust the emissivity setting of the camera until the temperature of the
unpainted section is equivalent to the previously recorded temperature for the
painted section
ε = 0.97
Wire with
oxide
ε = 0.62
Wire with
oxide
Painted
Wire
Painted
Wire
Painted wire vs wire with oxide,
Painted wire vs wire with oxide,
Front view, I = 0.75A
Front view, I = 0.75A
Figure 3-7 Obtain Emissivity of SMA Wire Through Painting
The last emissivity setting is the SMA wire’s emissivity. The distance between point 1
and point 2 in Figure 3-7 is 4.5 mm. When at steady state condition, point 1 and point 2
have the same temperature. Due to the difference of the emissivity, the two points give
different temperatures when the emissivity on the IR camera is set to a fixed value. The
temperature of the SMA wire can be determined through the known emissivity of the
paint and the emissivity of the oxidized wire can then be found through the temperature
of the SMA wire.
44
3.3.2 Determining Temperature of SMA Wire
The infrared temperature readings are dependant on the location of the points that are
chosen for analysis. This can be easily understood through the Lambert Cosine Law.
(Figure 3-2) Lambert's Cosine Law says that the total radiant power eλb (λ ,θ , ϕ )
observed from a "Lambertian" (or diffusing) surface is directly proportional to the cosine
of the angle θ between the observer's line of sight and the surface normal. As shown in
Figure 3-8, the surface normal of the top point A overlaps the line from the detector to
the point A, so the total radiant power received by the detector is eλb (λ ,θ , ϕ ) . For the
point B, due to an angle θ between its surface normal and the line from the detector to the
point B, the total radiant power received by the detector is eλb (λ ,θ , ϕ ) ⋅ cos θ . The
detector receives a higher radiant power from the top point A than point B and gives a
different temperature reading: TA > TB.
Figure 3-8 Different Points on SMA Wire Observed by a Detector
45
Figure 3-9 Temperature Bias on a Line Analysis Along the Wire
This location error can be seen in Figure 3-9. In this figure, a SMA wire is placed
horizontally. A line analysis is conducted along the wire and centred on the wire. It is
assumed that the temperature along this line should be constant due to the thermal
equilibrium established in the SMA wire. But the IR camera readings indicate a
temperature difference of 6 0C between points on the line. The possible reason for this is
that the SMA wire is not straight itself. So when a line analysis is conducted along the
wire, some points are at different elevations in relation to the centre line. This causes a
temperature difference among the measured values.
A practical solution to this error is conducting a line analysis across the SMA wire
rather than along the wire. Each cross section is scanned for temperature from top to
bottom, with the maximum temperature over the cross section representing the centre line
temperature. When heat equilibrium is reached, each top point temperature should be the
same as shown in Figure 3-10. Another example is shown in Figure 3-11, in which, a
thermistor is laid on the SMA wire. It clearly shows that the introduced thermistor
46
disturbs the temperature distribution on the SMA wire and causes a local temperature
distortion on SMA wire.
Each temperature distribution line with a different color in the lower picture
corresponds to a line across the SMA wire in the top picture
Figure 3-10 Line Analysis Performed Across the Wire
47
Thermistor
Each temperature distribution line with a different color in the lower picture
corresponds to a line across the SMA wire in the top picture
Figure 3-11 A Thermistor is in Contact With the SMA Wire
48
3.3.3 Verifying of IR Camera
Calibration of the IR camera is an important step to insure the accuracy of IR
temperature readings. Several samples (at least two samples) with different emissivity
should be used to calibrate the IR camera. Typically, a thermocouple is attached to the
sample and the IR camera is then used to measure the point where the TC is attached.
When the temperature of that point on the sample is known, the emissivity setting of the
camera can be adjusted until a temperature match is achieved. If the IR temperature
equals the known temperature, the IR camera can be verified. Otherwise, the IR camera
needs to be calibrated.
There can be a problem with low emissivity, shiny metal surfaces. As mentioned
above: the Gier Dunkle DB100 Infrared Reflectometer can be used to measure the
emissivity of object. It has an emissivity sample which has a gold-plated side with a
standard emissivity of 0.04 and a black side with a standard emissivity of 0.90. At room
temperature, the sample’s temperature should equal the room temperature. When both
sides of the sample are photographed by the IR camera with the emissivity setting equal
to their standard emissivity values respectively, as shown in Figure 3-12, if the IR
temperature readings equal room temperature, the IR camera can be verified. The black
side is less affected by surrounding objects with higher temperature (e.g. IR camera
itself) and calibration can be done perfectly, due to its high emissivity and low
reflectivity. The radiant energy that the detector receives from the black side is mainly its
emissive power. But when measuring the gold-plated side, IR temperature reading is
around 30 0C, much higher than room temperature as shown in Figure 3-13.
49
Sample with
Standard
Emissivity
Gier Dunkle Calibrated Standard
Reflectivity
Emissivity
Gold
0.96
0.04
Black
0.10
0.90
Figure 3-12 Gier Dunkel Emissivity Sample is Tested by IR Camera Imaging
Gold side ε = 0.04, Tamb = 21.5 0C
Black side ε = 0.90, Tamb = 24.3 0C
Figure 3-13 IR Temperature Reading is Much Higher Than Room Temperature
50
This can be explained by radiosity in radiation heat transfer. The term radiosity is
defined as: the rate at which radiation energy leaves a surface by combined emission and
reflection of radiation.
qo = ε ⋅ σ ⋅ T 4 + ρ ⋅ qi = ε ⋅ σ ⋅ T 4 + (1 − ε ) ⋅ qi
(3-6)
where qo is radiosity of the surface. qi is the incident heat flux that arrives at the surface.
Surfaces emit radiation as well as reflecting it, and thus the radiation leaving a surface
consists of emitted and reflected components. The IR camera detects the total radiation
energy streaming away from the gold plated surface, with no regard for its origin. The
gold surface has a very small emissivity of 0.04 and a large reflectivity 0.96. At room
temperature, the energy it reflects to the IR camera is much greater than that it emits.
Because it faces the IR camera normal, it reflects the radiation of the camera back to the
camera itself. So the IR temperature reading represents the temperature of camera. This
agrees with the fact that IR camera’s working temperature is around 30 – 35 0C.
In order to verify the IR camera, another sample with an emissivity that is lower than
0.9 should be used, besides the black side of the emissivity sample. A thermistor can be a
perfect choice because the thermistor has good accuracy ± 0.2 0C (0 0C to 70 0C) [41] and
the size of the thermistor bead is big enough for an IR camera to read temperature on a
thermal picture.
So a thermistor senses the temperature through its bead and the
temperature can be read by a data logger (such as a Keithley 2700). The temperature of
its bead can also be measured through the IR camera in the same time. Adjust the
emissivity setting of the camera until the IR temperature reading equals the reading from
the data logger and the emissivity of the thermistor bead is obtained. Keep this emissivity
51
setting on the IR camera and heat the thermistor bead with a heat gun. When the
temperature readings from the data logger and the IR camera agree over a range of
temperatures, it shows that the IR camera can give correct temperature readings when
given the new emissivity of the thermistor. Thus the IR camera can be verified because it
is correct at two different emissivity samples: the black side of the emissivity sample and
the thermistor. As shown in Figure 3-14, a TPC thermistor (NI24MA0502G) was used in
the experiment. The temperature readings from the two sources agree at all temperature,
so the IR camera can be verified.
Temperature From Thermistor vs IR Camera
65
55
0
Temperature ( C )
60
50
45
40
Temperature from Thermistor
35
Temperature from IR Camera
30
0
1
2
3
4
5
6
7
Test Number
Figure 3-14 IR Camera Verified, Thermistor Bead Temperature from Two Sources
52
3.4 Summary of Infrared Thermal Imaging in SMA Wire Application
Infrared thermal imaging is useful in this work, but it can not be used independently. It
has a relatively poor accuracy of 2 0C and the readout can not be easily recorded for
modeling heat transfer (given the existing equipment: FLIR ThermaCAM S60).
Infrared thermal imaging belongs in the category of non-invasive temperature
measurement methods. When the surface emissivity is known, an IR camera can give the
temperature of the SMA wire through the analysis software without touching the SMA
wire. This provides advantages over the invasive temperature measurement methods such
as: thermocouples, thermistors, etc. As described in Section 2.1.1, an invasive probe
senses the temperature of an object through direct contact with the object; thereby
transferring heat between the object and the probe. Once the sensor establishes thermal
equilibrium with the object, the temperature of the probe accurately reflects the
temperature of the target object. Given the heat transfer between the object and the probe,
the temperature of the object will be affected by the measurement probe. So there is a
temperature distortion on the target object when a probe is introduced and this distortion
can be significant at times. Infrared thermal imaging, on the other hand, provides the
temperature of an object through detecting the radiant power from the object and
interpreting this power strength to corresponding temperature value. An IR camera
doesn’t need to contact an object to give its temperature, so it can be used to monitor
temperature distortion when attaching a thermocouple or a thermistor to the SMA wire.
53
Chapter 4
Improved Spot Welded Thermocouples in SMA Wire Application
Thermocouples are a common choice for temperature measurement because of their
self-energization, low cost, robust nature and wide temperature range. With minor
calibration corrections, thermocouples can have accuracies of 0.25 – 0.5 0C. Michalski et
al. [19] conducted research on measuring the temperature of solid bodies with a
thermocouple and pointed out that temperature measurement of a solid surface has
possible errors including: temperature distortion error of the solid surface when
introducing a thermocouple, and contact thermal resistance error between the solid
surface and the thermocouple.
Volkov [29] observed that thermocouple readings can be affected by current in such a
way as to add a “spurious voltage” ∆V to the thermo electro-motive force (EMF) of the
thermocouple if it has direct contact with the current carrying surface. He concluded that
the influence of current on thermocouple readings was a function of the voltage gradient
along the current carrying surface. In Figure 4-1, when the pre-formed thermocouple
bead is spot welded on the SMA wire, the thermocouple legs contact the SMA wire
directly and there is a spatial offset ∆X between two legs along the axial direction of the
SMA wire. When a current passes the SMA wire, there is a voltage ∆V = I * R between
two legs, which is proportional to ∆X. R is the resistance between two thermocouple legs.
Furthermore, both Dutton [84] and Mulford [85] propose a method using two
thermocouple junctions of opposing polarity, affixed next to each other on a currentcarrying conductor, to measure and compensate for the current-induced error.
54
Figure 4-1 Thermocouple Reading is Affected by Spatial Offset Across TC Leads
Kuribayashi [86] used a thermocouple twist junction attached in close proximity to the
SMA wire to avoid measurement errors due to the current going through the SMA wire.
He also used isolation transformers to electrically separate the sensor circuit from the
SMA wire heating circuit.
There is little further evidence of consideration of this problem in the literature,
perhaps due to the particular nature of the SMA wire. For temperature measurement of
larger-diameter conductors, electrically-insulated thermocouples are recommended[29].
However, the insulation increases the volume of the bead relative to the thin wire under
test, introducing further uncertainty in the measurement.
As described in Section 2.3, thermocouple is better over other methods to measure the
temperature of SMA wire. In this chapter, experiments are performed to investigate how
to eliminate the carrying current’s influence when using thermocouples to measure the
temperature of SMA wire. This includes: different methods to attach thermocouples to
SMA wire; improving the spot welding attachment; analysis and calculation of the
55
“spurious voltage” ∆V, etc. The errors are also investigated according to the theory of
temperature measurement of a solid surface with a thermocouple in this chapter and
solutions are proposed to minimize these errors.
4.1 Current’s Influence on Spot Welded Thermocouple
In an early experiment comparing thermocouple attachment methods, two 500 mm long
samples of 500 µm diameter SMA wire (Flexinol, from Dynalloy) were prepared by
affixing seven T-type 36 AWG thermocouples along the wire length at regular 50mm
separations between 100 mm and 400mm along the wire. 36 AWG copper and constantan
wires are used to make the thermocouples. Thermocouple beads are prepared using a
Therm-X Model 258B welder. On one wire, the pre-formed thermocouple beads were
spot-welded using a capacity welder after first removing the oxide layer from the SMA
using an abrasive paper, the power setting on the capacity welder was 1 W. While on the
other wire, they were affixed with adhesive gel (QuickTite, from Loctite) and cured in
room temperature for 1 hour. Examples are shown in Figure 4-2.
A current of 250 mA was applied to each of the wires, first in one direction then in the
opposite, and the steady-state readings from all thermocouples recorded. The results are
shown in Figure 4-4. For the glued thermocouples, there are measurement differences
(less than 0.9 0C, the biggest one is between TC6 and TC7) along the length of the wire
which can be easily ascribed to thermocouple accuracy and variability in the gluing
process as evidenced by comparing Figure 4-3 (a) and Figure 4-3 (b). The difference in
readings when current is reversed is 0.2 0C. The spot-welded thermocouples, on the other
hand, show wider variability between thermocouples. TC4 is 28.2 0C, TC5 is 22.8 0C, and
56
Figure 4-2 SMA Wire with Thermocouple Attachment Points
(a), TC 2 glue spot
(b), TC 4 glue spot
(c), TC 7 spot welded
Figure 4-3 Comparison of Glue Spot TC and Spot Welded TC
TC4 is 5.4 0C higher than TC5. In addition, the current direction has a large influence on
readings, producing mirror-image curves which appear symmetric about approximately
26 0C. As shown in Figure 4-4, when the current is reversed, TC4 becomes 23.6 0C, TC5
is 29.1 0C, and TC4 is 5.5 0C higher than TC5. The average reading from the glued
thermocouples is also about one degree lower than the average from the spot-welded
57
thermocouples, which is not unexpected given the greater thermal resistance of the
adhesive.
Reverse Current Method (Current = 250 mA )
30
Temperature (0C )
29
28
TC GLUE SPOT
27
TC GLUE SPOT
(Reverse Current)
TC WELD SPOT
26
TC WELD SPOT
(Reverse Current)
25
Tamb
24
23
22
1
2
3
4
5
6
7
TC Number
Figure 4-4 Temperature Measurement of Spot Welded TC and Glue Spot TC
Looking closely at the spot-welded thermocouples in Figure 4-3(c), Figure 4-5 (a),
(b), and (c), (the close up pictures at the top left corners) it can be seen that the spatial
offsets between each of the thermocouple leads on the SMA wire are not exactly the
same.
In the absence of current in the SMA wire, and assuming a constant wire
temperature, this offset should have no influence on the thermocouple thermo electromotive force (EMF). However, we hypothesize that the wire current induces an ohmic
drop proportional to the lateral separation of the leads. This additional voltage corrupts
the thermoelectric voltage generated by the thermal field, leading to an erroneous reading
58
which deviates from the average reading of 26 0C. The fact that the error-induced offset
reverses its sign when the current is reversed supports this hypothesis.
(a), TC 1 spot welded
(b), TC 3 spot welded
(c), TC 4 spot welded
Figure 4-5 Spot Welded TC
To further verify this idea, we can calculate the lateral offset suggested by the observed
errors, by comparing the measured thermocouple voltages with the expected thermo
electro-motive force (EMF) for a T-type thermocouple at 26 0C. The equation is
V26C − Vmeasured = I ⋅ ρ ⋅ ∆x
(4-1)
Where V is the T-type thermocouple thermo electro-motive force (EMF), I is the wire
current (250 mA), ρ is the linear resistance of the wire (6.3 Ω/m [87]), and ∆Χ is the
thermocouple lead separation. T-type thermocouple thermo electro-motive force (EMF)
for TC 1 readings from Figure 4-4 were computed from thermocouple coefficients in the
NIST ITS-90 database [88], and are given in Table 4-1.
The lateral offset between the leads in TC1 can then be estimated using Equation 4-1,
giving a separation of 21 µm. Similarly, the offsets of TC3, TC4 and TC7 can be
estimated at 17 µm, 61 µm and 9 µm respectively. It can be seen from this experiment
and analysis that even very small offsets can introduce significant measurement error,
59
particularly at relatively high currents and temperatures close to room temperature. Since
currents of 250 mA or higher, and room temperature operation are very normal conditions
for an SMA actuator, it is important to develop a method to account for these currentinduced measurement errors.
Table 4-1 Temperature Readings and Corresponding Voltages for TC1
Temp (0C)
V (mV)
Comment
26.0
1.033
average reading
26.8
1.066
reverse current
25.2
1.000
forward current
4.2 Current Reversal and Averaging Method
Since the error component of the measured thermocouple voltage is due to an ohmic
drop, its sign is dependent on the direction of current flow, as seen in Figure 4-4. One
approach to compensate for this effect which has been previously proposed is to use a
“compensated thermocouple” comprised of three thermocouple leads [85]. Two leads of
a similar metal are affixed to the surface under measurement, and the third lead, of a
dissimilar metal, is affixed in the centre. This creates two thermocouple junctions which
will have opposite current-induced errors, and the average voltage can be read to get the
“true” temperature. This approach still relies on the precise relative positioning of the
leads on the surface, but residual errors due to lateral spacing differences would be
smaller than the original current-induced errors.
60
Often, the amplitude or pulse-width modulation (PWM) duty cycle of the current in an
SMA wire is controlled by a computer or embedded microcontroller, in order to control
temperature and hence actuator motion. The greater the amplitude or duty cycle, the
more power is delivered to the wire and the faster the heating. In this case, another
approach using a single traditional thermocouple can be used to reduce or eliminate
current-induced measurement errors. Since current direction does not affect heating, the
control electronics can be used to alternate current direction while achieving the same
overall desired control. If temperature measurements are synchronized with the current
reversals, the average thermocouple thermo electro-motive force (EMF) can be used to
eliminate the effects of the ohmic drop on the temperature reading.
4.3 Pulse Shut-Off Measurement Method
The pulse shut-off method takes advantage of the fact that the electrical time constant
is much smaller than the thermal time constant of the system: the problematic ohmic drop
disappears as soon as the current is removed, while the wire takes longer to cool. This is
illustrated in Figure 4-6, which shows the response of a K-type 40 AWG thermocouple
spot-welded to a 500 µm diameter Flexinol wire. Thermocouple readings were recorded
during two trials, in which 750 mA was run through the wire, in opposite directions in
each trial. At steady state with current flowing, the measured thermocouple temperatures
are - 4 0C with current in one direction, and 78 0C after the polarity is switched. When
the current is shut off, the thermocouple reading in both cases jumps almost immediately
to 37 0C, the average of the two. The jump is followed by an identical slow decrease in
measured temperature as the wire cools to ambient temperature.
61
Pulse Shut Off Method
0
Temperature ( C )
90
80
Temp
70
60
Temp-Rev
50
40
30
20
10
0
-10
-20
1
201
401
601
801
1001
1201
Time (sec )
Figure 4-6 Pulse Shut-Off Method
By synchronizing the temperature measurement with the removal of the current in the
wire, a single spot-welded thermocouple can be used to get an accurate reading, avoiding
errors induced due to ohmic drop.
This is particularly suited to SMA actuator
applications where heating is controlled using a PWM current signal, as the measurement
can be synchronized with the off-cycle of the control signal.
To further confirm the validity of the averaged and pulse shut-off measurements, we
can use a first-order heating model accounting for Joule-heating and convection, to
approximate the expected steady-state temperature of the wire in our experiments[89]:
Tss = T∞ +
ρ
I2
h ⋅π ⋅ d
(4-2)
62
where Tss is the steady-state temperature, T∞ is the ambient temperature, I is the wire
current (250 mA), ρ is the linear resistance of the wire (6.3 Ω/m, [87]), d is the wire
diameter (500 µm), and h is the convection coefficient (75 W/m2·K, [89]). (4-2 confirms
that at a measured ambient temperature of 23 0C, the steady state temperatures for our
wire at 250 mA and 750 mA are approximately 25 0C and 38 0C respectively. Within
thermocouple measurement error, these are the two average temperatures observed in
Figure 4-4 and Figure 4-6.
4.4 Two-Step Spot Welded Thermocouple
Different from the two methods above, the zero-offset spot weld method focuses on the
joint point where the thermocouple is attached on the SMA wire and eliminates the ohmic
drop through improved spot welding techniques. The conventional way to attach a
thermocouple on an SMA wire is to make the thermocouple bead first and then spot weld
the bead on the SMA wire. Experimental results show that the readings of a regular spot
welded thermocouple contain current effects that make temperature readings higher or
lower than real temperatures. The picture of a regular spot welded thermocouple (Figure
4-5) shows that two thermocouple leads contact the SMA wire at two distinct points
despite the formation of a single thermocouple bead prior to spot welding. Inevitably,
there is voltage drop across the thermocouple bead at this kind of junction when a current
passes through the SMA wire. Another potential method of attachment is to spot weld the
two thermocouple leads directly onto the SMA wire. However, experiments show that
two distinct points of contact result even when using this method.
63
In addition, the
problem with using this procedure is the control and placement of the very thin
thermocouple leads under the microscope during the spot welding process.
A two-step spot weld thermocouple provides a good solution to ensure that the two
thermocouple leads are attached to the SMA wire at a single point in a small junction.
Figure 4-7 shows the procedure for the two-step spot welding of thermocouples. The
goal of this procedure is to ensure that the two thermocouple wires can be welded exactly
at the same point on the SMA wire, which negates the current effect and provides good
thermal contact. The attachment procedure is performed under a 3X microscope. The
bead that is formed from one thermocouple wire is welded onto the SMA wire prior to
the attachment of the second thermocouple wire, providing minimal separation of the
wires thus minimizing current effects.
The two-step spot welding of thermocouples needs to be investigated using the average
reverse current method or the pulse shut off method to see if there is any influence of the
passing on the temperature readout. The verification procedure involves applying a
current on the SMA wire and reading the temperature T1, then reversing the polarity of
the same current and reading another temperature T2. If T1 = T2, this verifies a good spot
welded thermocouple. If T1 ≠ T2, this indicates a poor spot welded thermocouple. Then
the two step welding procedure needs to be repeated until T1 = T2. If T1 ≠ T2, the average
of T1 and T2 represents the real temperature of SMA wire.
64
Make a bead from one thermocouple wire first; spot weld the bead on the SMA wire.
Attach the other thermocouple wire on the bead; the current influence can be eliminated
Figure 4-7 Two-Step Spot Welded Thermocouple
4.5 Temperature Distribution Along SMA Wire
Examining the temperature distribution along an SMA wire is an important step in
modeling the heat transfer and characterization of the thermal properties of this material.
According to thermodynamic theory, when a current passes through the SMA wire and
reaches a thermal equilibrium, the temperature along the SMA wire should be the same.
This assumption is verified in the experimental heat transfer coefficient measurement, as
shown in Figure 4-8.
In Figure 4-8, a 50 cm long, 0.5mm diameter SMA wire was supported on a fixture.
Two ring terminals were crimped onto the SMA wire at points 12.5 cm away from each
end. The two ring terminals were then tied and suspended onto the fixture. Five E-type 40
65
AWG thermocouples were attached on the centre section of the SMA wire with an offset
of 5 cm from one another using the two-step spot welding method.
Figure 4-8 Five Thermocouples, Voltage Measurement Leads and Power Leads
Layout
When each thermocouple was welded onto the SMA wire, a pulse shut off method was
used to examine the influence of current influence on the thermocouples. Their
temperature and time profiles are shown in Figure 4-9. It could be noted that all the
thermocouples attached on the SMA wire showed no current influence on their readouts.
The complete setup together with all thermocouples was placed in the vacuum of 10-8
torr. Different currents were fed to the SMA wire and the temperature of the 5
thermocouples were recorded and plotted as in Figure 4-10
66
TC 2 (3A, 1000Hz)
120
120
100
100
Temp (0C)
Temp (0C)
TC 1 (3A, 1000Hz)
80
60
40
20
80
60
40
20
0
0
1
1937 3873 5809 7745 9681 11617 13553 15489 17425 19361
1
Time (0.001s)
TC 3 (3A, 1000Hz)
TC 4 (3A, 1000Hz)
120
140
120
100
100
80
60
40
Temp (0C)
Temp (0C)
752 1503 2254 3005 3756 4507 5258 6009 6760 7511 8262 9013
Time (0.001s)
80
60
40
20
20
0
0
1
1
1032 2063 3094 4125 5156 6187 7218 8249 9280 10311
1388 2775 4162 5549 6936 8323 9710 11097 12484 13871
Time (0.001s)
Time (0.001s)
TC 5 (3A, 1000Hz)
120
Temp (0C)
100
80
60
40
20
0
1
736 1471 2206 2941 3676 4411 5146 5881 6616 7351 8086 8821
Time (0.001s)
Figure 4-9 No Current Influence on Two-Step Spot Welded Thermocouple Reading
In Figure 4-10, when the current increased, the temperature readings of the 5 points
also increased. When the current was reversed during the experiment, the temperature
values didn’t change. This further proved evidence that there was no current effect on
thermocouple readings. TC2, 3 and 4 agreed at all current levels, TC5 was a little lower
than TC2, 3 and 4. TC1 was the lowest value and TC1 became much lower than others at
larger currents.
67
-8
Temp Distribution of SMA Wire in Vacuum (10 torr )
160
120
5.437 mA
0
Temperature ( C )
140
100
11.777 mA
15.086 mA
20.213 mA
25.084 mA
80
60
40
20
0
1
2
3
4
5
Thermocouple Number
Figure 4-10 The Temperature Reading From TC1 is Much Lower Than Other TCs
The two-step spot welding method was used to attach thermocouples onto the SMA
wire, the pulse shut off method and the average reverse current method were both used to
verify thermocouples before and during the experiment to show that there was no current
influence on the temperature readouts. There were systematic errors in the thermocouple
circuit, such as: cold reference junction, low accuracy of measurement meter, but none of
these errors were significant enough to produce the temperature differences observed
between TC5 and the other TCs. A possible reason is that TC1 has a poorer spot weld
than other TCs. If the spot weld is poor, the thermal resistance between the thermocouple
and the SMA wire will be higher, so it will give a lower temperature reading at the same
condition. This can be verified by checking temperature profiles in Figure 4-9. It can be
seen that TC1 cools at a higher rate than other TCs when the power is turned off.
68
4.6 Summary of Improved Two-Step Spot Welded Thermocouple
It can be concluded that a two step spot weld thermocouple is the best practice in this
work to measure the temperature of SMA wire.
A glue-spot attachment is not suitable to measure the fine SMA wire due to its higher
thermal contact resistance than metal. Glue-spot attachment also means an extra material
will have to be introduced and more power will be consumed when heat equilibrium is
reached between the thermocouple bead and the SMA wire. This results in a higher
temperature distortion on the SMA wire and a bigger error in measurement. In addition,
glue has a lower melt point, which makes the thermocouple bead detach from the SMA
wire when temperature increases to a degree.
A pre-formed thermocouple bead is not an ideal option, either. At first, due to the
technical constraint, the bead can not be made small enough. The smallest bead that can
be made with the current equipment is a diameter of 0.25 mm. Similar to the glue spot
procedure, spot welding a bead on an SMA wire also introduces extra material to the
attachment and brings a higher error to the measurement. Another vital drawback with
the pre-formed bead-spot welding is that this method makes it hard to control where two
thermocouple legs contact the SMA wire. Improper arrangement of two thermocouple
legs on the SMA wire results in the spatial offset between the two legs, which leads to
errors in temperature measurement.
The two-step spot welding procedure successfully controls the placement of the two
thermocouple legs. Two legs can be precisely spot welded onto an SMA wire one after
another, so the spatial offset will not exist and the spurious voltage will not occur.
69
Because the two legs are spot welded onto the SMA wire one by one during the spot
welding, there is no bead and the error related to temperature distortion is minimized.
70
Chapter 5
Experimental Methods
A series of experiments were performed to collect the data necessary to model
convective heat transfer of a heated SMA wire. As described in the literature review, no
existing heat transfer correlations can be directly applied to SMA wires with confidence
since these correlations were developed under different conditions. The primary goal of
the experiment conducted in this work is to obtain data that can be used to produce a
natural convective heat transfer model that relates Nusselt number to Rayleigh number
for SMA wires of sub millimeter diameter.
5.1 Background
An accurate heat transfer model is of great importance when modeling SMA actuator
dynamics. SMA wires are ideal as actuators in automobile applications because of the
unique temperature dependent phase transformations. They can be deformed in the low
temperature martensite (M) phase, but return to their original un-deformed configuration
when heated to a higher temperature austenite (A) phase. This phase transformation is
accompanied by a large force during the re-arrangement of crystalline structure and the
two phases have very different thermal, mechanical and electrical properties. The SMA
constitutive and phase kinetic behaviors are directly controlled by the heat transfer rate to
and from the wires and as a result a detailed heat transfer model is needed to monitor and
predict the thermophysical behavior of SMA wires.
71
The SMA wire actuator is usually driven by a current. When a current is passed
through a SMA wire, the SMA wire is heated via the Joule effect. The input energy is
then dissipated to the environment through convective, radiative and conductive heat
transfer. The convective heat transfer is a complicated mechanism which is dependent on
a series of conditions, such as: the geometry, orientation and the fluid properties. The
convective heat transfer coefficient h can be nondimensionlized with the Nusselt number
(Nu) and the parameters that reflect these conditions are expressed as: Rayleigh number
(Ra).
The Nusselt number (Nu) is the ratio of the convective heat transfer associated with
movement of the surrounding fluid through a thin boundary layer to the conductive heat
transfer through the same fluid layer when the fluid is motionless (Nu= Qconv / Qcond). The
Nusselt number (Nu) is a measure of the heat transfer enhancement associated with
convection as compared to conduction in the fluid. The Rayleigh number is also a
dimensionless number associated with the transition from laminar to turbulent flow for
natural convection heat transfer within a fluid. The Rayleigh number is defined as the
product of the Grashof number (Gr), which describes the relationship between buoyancy
and viscosity within a fluid, and the Prandtl number (Pr), which describes the
relationship between momentum diffusivity and thermal diffusivity. A simple empirical
correlation for the average Nusselt number (Nu) in natural convection can be expressed
as: Nu = C ⋅ Ra n . The constant C and n depend on the geometry of the surface and the
flow regime, which is characterized by the Rayleigh number (Ra).
A correlation can be determined based on prescribed experimental conditions, where
test conditions such as wire temperature, geometry and ambient properties can be used to
72
obtain a range of Rayleigh numbers. A 500 µm diameter SMA wire is used as a sample to
model convective heat transfer. The convective heat transfer coefficient can not be
measured directly; it can only be calculated by the parameters that will be collected in the
experiment. This section describes the pressure variation method and the type of data that
is necessary to compute Nu and Ra for heat transfer modeling.
5.1.1 Pressure Variation Method
The goal of this experiment was to accurately measure the natural convective heat
transfer coefficient from a heated SMA wire over a wide range of Rayleigh numbers
(Ra). The Rayleigh number is the product of the Grashof number (GrL) and the Prandtl
number (Pr) and can be written as:
RaL = GrL ⋅ Pr =
gβ (Ts − T∞ ) L3c
ν
2
(5-1)
Pr [90]
where:
RaL is the Rayleigh number
Pr is the Prandtl number
g is the gravitational acceleration (m/s2)
β is the coefficient of volume expansion (1/K)
Ts is the temperature of the surface (K)
T∞ is the temperature of the fluid sufficiently far from the surface (1/K)
Lc is the characteristic length of the geometry (m)
73
ν is the kinematic viscosity of the fluid (m2/s)
g is a constant. Pr is independent of pressure and changes in a very small range from
0.6935 to 0.7539 for air temperature changes from -150 0C to 2000 0C, so it can be
treated as constant as well. Assuming that the air in the test environment is an ideal gas,
β=
(5-2)
1
T∞
T∞ is usually 296 K (23 0C), so β can also be treated as a constant. The Rayleigh
number (RaL) can thereby be expressed as a function of Ts, Lc and ν.
RaL = f (Ts , Lc ,ν )
(5-3)
In this case, the temperature of the surface Ts is TSMA; the characteristic length Lc is the
diameter of SMA wire D; ν is the kinematic viscosity of air.
Changing TSMA can vary RaL, but TSMA can only change RaL over a small range. If T∞=
296 K, air pressure is 1 atm, (β = 1/296 1/K; Pr=0.7282; ν=1.562 × 10-5 m2/s), for a 0.5
mm diameter SMA wire, a change in TSMA from room temperature 296 K (23 0C) to 673 K
(400 0C) results in a change in RaL from 0.3 to 6.97. Since the temperature of the SMA
wire should not be higher than 200 0C, using temperature to obtain a range of Rayleigh
number is very limiting.
Changing the diameter of SMA wire is not effective either to adjust RaL because there
are limited diameters of SMA wire to choose from. The diameter of SMA wire varies
from 0.025 mm to 0.5 mm and when the diameter is small it becomes very difficult to
measure the temperature of the SMA wire.
74
However, varying the ambient fluid properties can be a very effective method for
obtaining a wide range of Rayleigh number. The kinematic viscosity is given:
ν=
µ
ρ
(5-4)
where:
µ is the dynamic viscosity of air (kg/(m.s))
ρ is the density of air (kg/m3)
µ is independent of air pressure and alters only in a small range from 1.849 × 10-5 to
2.181 × 10-5 kg/(m.s) when Tm changes from 296 K (23 0C) to 373 K (100 0C), so it can be
treated as a constant. ρ varies significantly from 1.184 kg/m3 to 1.72 × 10-3 kg/m3 when
the air pressure varies from 1 atm to a near vacuum when air pressure is 1 torr. When
TSMA = 373 K (100 0C), T∞= 296 K (23 0C), (β = 1/296 1/K; Pr = 0.7202; µ =2.008 × 10-5
kg/(m.s)), for a 0.5 mm diameter SMA wire, if ρ varies from 1.184 kg/m3 to 1.72 × 10-3
kg/m3 , RaL varies from 6.4 to 7.86 × 10-6 according to Equation 5-1.
The air in the test environment is assumed to be an ideal gas, so:
ρ=
(5-5)
P
RTm Z
where;
P is the pressure of air (Pa);
R is the gas constant of air (kJ/(kg.K));
Tm is the average temperature of air and SMA wire (K) (Tm = (TSMA + T∞) / 2),
75
Z is the compressibility factor of air.
If Tm, R and Z are assumed to be constant, Equation 5-5 clearly shows a direct
relationship between density and pressure, therefore an experiment in which the ambient
pressure is reduced will result in a lowing of air density. P is easy to control with a
vacuum station using a bell jar and a mechanical / diffusion vacuum pump to evacuate
the air pressure from 1 atm to 0.001 torr. The pressure variation method is an ideal choice
in this experiment to alter RaL over a wider range.
5.1.2 Data Collection
The following section provides detailed descriptions on collecting data that are
necessary to determine an empirical correlation of Nusselt number versus Rayleigh
number for natural convection heat transfer from a heated SMA wire.
5.1.2.1 The Rayleigh Number (Ra)
A modified version of the Rayleigh number can be obtained by substituting Equations
5-2, 5-4 and 5-5 into Equation 5-1, where the characteristic length Lc and the
temperature of the surface Ts are replaced by the diameter of the SMA wire D and TSMA.
RaD =
(5-6)
g (TSMA − T∞ ) D 3 P 2
Pr
µ 2T∞ R 2 Z 2Tm2
g, Pr, µ, R, Z are constants. g = 9.8 m/s2, R = 0.287 kJ/(kg·K). Z is the compressibility
factor of air, where air is assumed to be ideal gas and Z = 1. Pr and µ can be obtained
from standard air property tables when the mean temperature Tm is determined. Tm is the
average of the ambient temperature and the temperature of the SMA wire: Tm = (TSMA +
76
T∞) / 2 (K). The remaining parameters TSMA, T∞, P and D need to be measured in the
experiment to calculate RaD.
5.1.2.2 The Nusselt Number (Nu)
The Nusselt number (Nu) is the non-dimensional convective heat transfer coefficient
which can be expressed as:
Nu L =
(5-7)
hLc
k
where:
h is convection heat transfer coefficient (W/(m2 K))
Lc is the characteristic length (m)
k is the thermal conductivity of the fluid (W(/m K))
The convective heat transfer coefficient h can be defined as: the rate of heat transfer
between a solid surface and a fluid per unit surface area per unit temperature difference.
The convective heat transfer coefficient is then expressed as:
h=
(5-8)
Qconv
As ⋅ (Ts − T∞ )
where:
h is the convective heat transfer coefficient (W/m2 ·K)
Qconv is the convective heat transfer rate (W),
As is the convective heat transfer surface area (m2)
Ts is the temperature of the surface (K)
77
T∞ is the temperature of the fluid sufficiently far away from the surface (K)
In this case, the characteristic length Lc is the diameter of the SMA wire D. Substituting
Equation 5-8 into 5-7, the Nusselt number becomes:
Nu D =
(5-9)
hD
Qconv D
=
k
kAs (Ts − T∞ )
k is constant which can be obtained from standard air property tables. D and l can be
measured, therefore As can be calculated, As = π·D·l (mm2).
The convective heat transfer rate Qconv can be determined by performing a heat balance
on the test section of SMA wire when at a thermal equilibrium, as shown in Figure 5-1.
Usually the SMA wire is heated by passing a current through the wire. When the
temperature of the SMA wire increases, heat transfer occurs between the SMA wire and
the environment through conduction, convection and radiation due to the temperature
difference between the SMA wire and the environment. At a steady state, the power input
to the SMA wire should equal the heat dissipation from the SMA wire via conduction,
convection and radiation as shown in Figure 5-1, where VI = Qconv + Qcond + Qrad. If VI,
Qcond and Qrad are known, Qconv can be easily determined.
78
Figure 5-1 Thermal Analysis on the SMA Wire Test Section at Steady-State
Qcond is the heat dissipation from the SMA wire through the attached thermocouple
leads, the voltage measurement leads, the power leads, the string and the ring terminals.
As described in the Appendix A and C, the conduction losses through above paths can be
minimized and neglected when the SMA wire is properly setup on the fixture.
Qrad is the heat exchange between the SMA wire and the environment through
electromagnetic waves as a result of the temperature difference between TSMA and the
surrounding environment at T∞.
4
Qrad = ε ⋅ σ ⋅ As ⋅ (TSMA
− T∞4 )
(5-10)
where:
ε is the emissivity of the SMA wire;
σ is the Stefan-Boltzmann’s constant (σ = 5.67×10-8 W/(m2·K4)
As is the surface area of the test section of SMA wire (m2)
TSMA, is the temperature of the SMA wire (K)
79
T∞ is the temperature of the surrounds (K)
Since radiative heat transfer is independent of the transfer medium, Qrad for the SMA
will be the same in a vacuum as in ambient air at 1 atm if T∞ is preserved. At thermal
equilibrium in a vacuum, the power input to the SMA wire equals Qrad only. For the same
section of SMA wire, Qrad at a prescribed temperature and air pressure is equivalent to
the power input at the same temperature in a vacuum:
(5-11)
Qrad = VIvacuum
With the power input VI and the radiation Qrad determined, and the conduction losses
considered negligible, the convection can be calculated as: Qconv = VI – Qrad, or
(5-12)
Qconv = VI – VIvacuum
5.1.2.3 The Inclination Angle
When an SMA wire is inclined at an angle φ, as shown in Figure 5-8, the convective
heat transfer is affected as a result of an axial component of air flow which decreases the
convection. To develop a correlation that contains the inclination angle φ, the SMA wire
was inclined from the horizontal at angles: 150, 300, 450, 600, 750, and 900 (vertical) by
elevating one side of the test fixture as described in the Section 5.4.1.
5.2 SMA Wire Test Fixture
As described previously, a sample SMA wire was used to collect empirical data
necessary to model natural convection heat transfer. The sample SMA wire was heated
by an electric current; its temperature and power input were then measured.
80
5.2.1 Level Temperature Distribution on the SMA Wire
The test section of the SMA wire should be maintained at a uniform temperature during
the test procedure; otherwise there can be large errors when calculating the Nusselt
number (Nu) and the Rayleigh number (Ra). First, to collect data and to heat up the SMA
wire, thermocouples, voltage measurement leads and power leads have to be attached to
the SMA wire. These extraneous connections can lead to temperature distortions in the
test section of SMA wire, and introduce errors in the results. Similarly, when the ring
terminals are crimped onto the sample SMA wire to suspend the SMA wire to the fixture,
they also introduce temperature distortion to the SMA wire due to their larger surface
area which results in enhanced convection to the air.
The expected uniform temperature distribution over the test section of the SMA wire is
shown in Figure 5-2. The factors that distort the uniform temperature distribution on the
SMA wire are: the size of the SMA wire; the attached thermocouple, voltage
measurement leads, power leads and ring terminals, as described in the next Sections
5.2.2. Choosing a larger diameter sample SMA wire and properly arranging the locations
of the thermocouple, voltage measurement leads, power leads and ring terminals on the
sample SMA wire can minimize the conduction heat losses and keep a uniform
temperature distribution on the sample SMA wire.
81
Figure 5-2 Expected Uniform Temperature Distribution on the SMA Wire
5.2.2 Sample SMA Wire
The SMA wire is a commercial product from DYNALLOY, INC: FLEXINOL
Actuator Wire. It is a new roll of SMA wire with a total length of 20 meters. The
specifications of the SMA wire are: diameter: 500 µm (0.020”), As Temp: 90 0C, Spool #:
9CF9513K. The selection of the specific diameter, length, preparation method and setup
is based on experimental constraints, the temperature distortion on the sample SMA wire
and the attachment accessories.
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5.2.2.1 Diameter of the Sample SMA Wire
There are several diameters of SMA wire that can be chosen: 500 µm, 380 µm and 250
µm, etc., but only 500 µm is used in this experiment due to the need to spot weld
thermocouples to the SMA wire. As described in the previous chapter, the temperature
measurement of the SMA wire is performed using a thermocouple which is spot welded
onto a section of SMA wire using a two-step spot welding procedure. At present, the twostep spot welding method works well only on a 500 µm diameter SMA wire. Another
reason to use 500 µm diameter SMA wires is that it helps to decrease the temperature
distortion over the SMA wire.
5.2.2.2 Length of the Sample SMA Wire
The maximum length of the SMA wire that can be used is 25 cm given the need to
work within the bell jar of the vacuum chamber. The experiment is operated within a
vacuum chamber with a 18” diameter (45.7 cm) Pyrex bell jar to allow the pressure
variation method to be employed. As discussed in the following sections, the sample
SMA wire is suspended on a supporting fixture which is placed on a metallic stand inside
the bell jar. There should be a distance of 10 cm between the bell jar and the fixture to
allow the bell jar to move down freely to cover the baseplate; therefore the length of the
sample SMA wire must be less than 25 cm.
5.2.2.3 Oxide of the Sample SMA Wire
The SMA oxide should be removed before attaching thermocouples, voltage
measurement leads and power leads so that good electrical contact can be achieved. Paper
83
sheet (3M Wetordry431Q, in 180 0C weight) was used to remove the oxide as described
in Section 5.5.
5.2.2.4 Ring Terminal
(b) After crimping
(a) Before crimping
Figure 5-3 Ring Terminal (DYNALLOY, INC)
Ring terminals from Dynalloy, Inc are attached to the SMA wire to provide a means to
connect to the fixture. The detailed specifications can be downloaded from the website
(http://www.dynalloy.com/CrimpPdfs/6983.pdf). The ring terminals provide a good
method to hold the sample SMA wire tightly when they are properly crimped onto the
sample SMA wire. A crimping tool is required to obtain a good connection with the SMA
wire. In this work, a crimping tool SPC CCT-8424-01 is used to crimp the ring terminals
onto the sample SMA wire. Before crimping, a piece of metal is curved to form a groove
on the lower part of the ring terminal, where the end of the SMA wire can be placed, as in
Figure 5-3 (a). When the sample SMA wire end is in the groove, the crimping tool
applies a force on the groove and the two pieces of metal attach firmly onto the SMA
wire, as in Figure 5-3 (b). The ring terminal can bear up to 4.5 kg of force.
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5.2.3 SMA Wire Setup on the Fixture
As described previously, it is very important to keep a uniform temperature distribution
on the SMA wire during the experiment, so the SMA wire should be as big in diameter as
possible, while the thermocouple leads, voltage measurement leads; power leads should
be as thin as possible to minimize conduction losses. The placement of the
thermocouples, the voltage measurement leads, the power leads and the ring terminals
should also be properly arranged to keep a uniform temperature distribution on the SMA
wire. The SMA wire setup on the test fixture is given in Figure 5-4.
Figure 5-4 Sample SMA Wire Setup on the Fixture
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In Figure 5-4, the test section of the SMA wire is 50 cm in length. Two ring terminals
are crimped onto the SMA wire 12.5 cm from each end. One ring terminal is suspended
to the left side of the fixture on the screw by a piece of string while the other ring
terminal is connected to a mechanical load with another piece of string across a pulley.
An E-type 40 AWG thermocouple is attached to the centre of the 50 cm sample SMA
wire using a two-step spot welding procedure which is described in the Section 4.4. Two
constantan 40 AWG thermocouple wires are spot welded on the SMA wire 2.5 cm closer
to the centre of the sample SMA wire. Two 30 AWG copper wires are connected to the
two ends of the SMA wire using two alligator clips. The two ends of the SMA wire are
then suspended in a horizontal orientation above the fixture.
5.2.3.1 E-type 40 AWG Thermocouple
When thermocouples are attached to an SMA wire, it can lead to conductive heat
transfer along the thermocouple wire, thus making the temperature of the SMA wire
lower at the attachment point than other locations. In order to minimize the temperature
drop of the SMA wire at the attachment point, thermocouples should be as thin in
diameter as possible and its thermal conductivity should be as low as possible in order to
minimize the temperature distortion that the thermocouple may bring to the SMA wire.
40 AWG thermocouple wire is one of the thinnest thermocouples available. E-type
thermocouples contain constantan and chromel thermocouple wires. Constantan’s
thermal conductivity is 21.8 W/(m·K), chromel’s thermal conductivity is 19.2 W/(m·K),
which are much lower than copper’s thermal conductivity: 391 W/(m·K) that is found in
T-type thermocouples.
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A E-type 40 AWG thermocouple was attached to the centre of the SMA wire test
section. It is assumed that the temperature reading at this point represents the temperature
along the entire test section of the SMA wire. The thermocouple was spot welded onto
the SMA wire using the two-step spot welding method in order to obtain a joint that
provides good thermal contact and eliminates the influence of the current that passes
through the SMA wire.
5.2.3.2 Voltage Measurement Leads
Similar to the selection of the thermocouples, two 40 AWG, constantan wires were
used as voltage measurement leads in the experiment due to their small diameter and low
thermal conductivity. The voltage measurement leads were spot welded onto the SMA
wire to provide good electrical connections. The location of the voltage measurement
leads is discussed in Section 5.2.3.4.
5.2.3.3 Power Leads
Power leads also contribute to temperature distortion in the SMA wire; therefore power
leads can not be too thick in diameter. On the other hand, power leads should be large
enough to carry the required current up to 2 A. As a balance of these two factors, 30
AWG copper wires were used as power leads in the experiment. Copper has high thermal
conductivity (391 W/(m·K)), and acts as a very good conductor. The power leads were
connected to the two ends of the SMA wire using two alligator clips. The placement of
power lead is discussed in the following section.
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5.2.3.4 Placement of TCs, Volt Measurement Leads, Power Leads and Ring
Terminal
In order to keep a uniform temperature over the test section of the SMA wire, the
locations of the thermocouple leads, voltage measurement leads and power leads must be
given proper consideration.
A 2.5 cm distance between the ring terminal and the voltage measurement leads on the
SMA wire is necessary to help reduce the temperature drop that the ring terminal brings
to the test section of the SMA wire. If the SMA wire test section is too close to the ring
terminal, there is large temperature difference over the SMA wire and the SMA wire that
is farther away from the ring terminal will have a higher temperature than the SMA wire
adjacent to the ring terminal. This will lead to uncertainties when modeling the heat
transfer from the SMA wire. The infrared thermal imaging in Appendix A shows that
when the SMA wire is 1.5 cm away from the ring terminal, it is not significantly affected
by the ring terminal.
A 12.5 cm offset is required to reduce the influence from the power leads for the same
reason. The power leads are 30 AWG copper wires, which have a large thermal
conductivity (391 W/(m·K)). When the power leads are attached too close to the SMA test
section, the measured temperature of the SMA wire appears to be lower than expected, as
a result of a conduction loss along the highly conductive power leads.
5.2.3.5 String
The strings in Figure 5-4 are all cotton string, which connect the sample SMA wire to
the fixture, the load, and the level stand above the fixture. Cotton string is flexible and it
88
is easy to tie cotton string onto the ring terminal or the fixture. Cotton string is also strong
enough to bear the force when a load is applied to the SMA wire. There is no conduction
loss along the cotton string when it connects the sample SMA wire to the fixture.
Two cotton strings are used to tie each end of the SMA wire to the level stand above
the fixture so that they do not contact with the metallic fixture. If the SMA wire touches
the fixture, a conductive loss occurs which makes the temperature of the SMA wire at
these locations lower than other locations on the SMA wire.
5.2.3.6 Fixture
The test fixture was made in the machine shop of the Faculty of Engineering at the
University of Waterloo, according to the fixture schematic shown in Figure 5-5. On the
left side, the SMA wire is suspended to a screw by a piece of cotton string; on the right
side, the SMA wire is pulled downward by a mechanical load through another piece of
string along a pulley. The pulley is designed to allow the SMA wire to contract when
heated and relax when cooled without exerting any unwanted forces.
5.2.3.7 Mechanical Load
The load is a block with a weight of 300 g, which pulls the SMA wire to keep it in
tension. Because the SMA wire contracts and extends when turning on /off the power
input, it is necessary to use a load to keep the SMA in tension during the experiment,
which conform with the assumption of a straight thin circular cylinder for the heat
transfer model.
89
Figure 5-5 Test Fixture
5.3 Experimental Setup Diagrams
The experimental setup diagram is shown in Figure 5-6. There are four parts in the
experimental setup: the inclination angle control; the pressure control and monitor; the
power supply circuit and the data acquisition system. A close up diagram of the bell jar
vacuum chamber is shown in Figure 5-7, which gives more details on the SMA setup on
the fixture in the bell jar.
Inclination Angle Control
As shown in Figure 5-7, the inclination angle control is realized by an orientation
adjustment string in the bell jar. The orientation adjustment string is tied to the left side of
the fixture at the location A, at the other end; it connects to the level stand at the location
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B. By adjusting the string length between A and B, the SMA wire is inclined from
horizontal at a desired angle, as shown in Figure 5-8.
The Pressure Control and Monitor
The pressure control is performed by using an air release valve, the vacuum pump
valve, vacuum pump and vacuum gauge that are connected to the platform by metal
tubing. A Pyrex bell jar is used to cover the test fixture and establish a controlled ambient
environment around the fixture. The air pressure is measured using a Varian CeramiCel
vacuum gauge which was installed on the baseplate of the NRC 3117 vacuum station.
The Varian CeramiCel vacuum gauge transfers the pressure signals to a Keithley
2700/7700 for recording and analyzing.
The Power Supply Circuit
The power supply circuit consists of the DC power supply, the current shunt and the
SMA wire, which are connected in a series arrangement.
The Data Acquisition
As shown in Figure 5-6, data acquisition was performed using a Keithley 2700/7700,
with measurements for: Tamb, TSMA, VSMA, Vshunt and Vpressure. The Keithley 2700/7700 is
connected to a computer and is operated under the control of the manufacturer’s
software: Keithley ExceLINX.
91
Inclination Angle Control
Data Acquisition System
Power Supply System
Figure 5-6 Experimental Setup Diagram
92
Pressure Control & Monitor
Figure 5-7 Experimental Close Up Diagram of Bell Jar
93
Figure 5-8 SMA Wire Inclined From Horizontal at an Angle φ
94
5.4 Equipment
5.4.1 Inclination Angle Control
The orientation adjust control consists of a cotton string, which connects the fixture at
the location A and the stand at the location B, as shown in Figure 5-7. When the SMA
wire needs to be inclined from horizontal at an angle φ during the experiment, for
example: 150, 300, 450, 600, 750 and 900 (vertical), the fixture can be raised at the left side
to make the SMA wire inclined at a prescribed angle. A protractor with ± 0.50 accuracy is
used to verify that the SMA wire is inclined by the prescribed angle.
5.4.2 Pressure Control and Monitor
5.4.2.1 NRC 3117 Vacuum Station
Pressure variation is controlled using a NRC 3117 vacuum station, which includes a
18” dia x 30” tall Pyrex bell jar with implosion cage on a 20” dia stainless steel baseplate;
electromechanical hoists installed on the baseplate; a pumping system that connects the
baseplate to a 17 CFM mechanical roughing pump, a NRC VHS 6 diffusion pump rated
at 2400 L / s, a NRC 316 series LN 2 trap and electropneumatic valves. The fixture and
the sample SMA wire are placed on the stand which is covered by the bell jar.
The mechanical roughing pump can achieve a vacuum from 1 atm (760 torr) to around
0.001 torr, which meets the air pressure requirement for modeling natural convective heat
transfer. But the diffusion pump can make a higher vacuum from 10-7 torr to 10-3 torr,
95
which is necessary to estimate the conduction heat loss at a prescribed temperature,
which is described in the Appendix A.
5.4.2.2 CeramiCel VCMT-13 Vacuum Gauge
The air pressure in the bell jar is measured using a CeramiCel / Varian Capacitance
Diaphragm Gauge VCMT-13. The CeramiCel VCMT-13 is a highly accurate, compact
and temperature-compensated vacuum pressure transducer that can operate at ambient
temperature. The CeramiCel VCMT-13 features a ceramic capacitive sensor element
which transfers pressure signals of air in the bell jar to direct voltage signals that can be
measured and recorded by the Keithley 2700/7700. The CeramiCel VCMT-13 is powered
by 24 VDC and outputs 0 – 10 VDC corresponding to the air pressure in the bell jar from
0 torr to 1000 torr, allowing P to be calculated as
P = Vpressure×100×133.3 Pa
(5-13)
The CeramiCel VCMT-13 has a resolution of 0.0015 % at full scale.
5.4.3 Power Supply and Current Shunt
Figure 5-9 Power Supply and Current Shunt
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5.4.3.1 Power Supply
A BK PRECISION 1760A was used as the power supply to heat up the sample. The
BK PRECISION 1760A has a constant current power supply mode, which is very helpful
to monitor the magnitude of power input. The BK PRECISION 1760A can fix the
voltage to a value and adjust the current that passes through the circuit, so the magnitude
of the current reflects the magnitude of the power input during the experiment. The BK
PRECISION 1760A can accurately adjust the current with an increment of 1 mA, this
feature is very useful when adjusting power input to bring the SMA wire to a prescribed
temperature in vacuum.
5.4.3.2 Current Shunt
The current shunt provides a fixed and stable resistance, so when the voltage across the
shunt is measured, the current passing through the shunt can be accurately calculated by
the formula: I = V / R. The shunt was connected with the SMA wire in a series circuit, so
the current passing through the SMA wire could be determined by measuring Vshunt and
calculating I in the circuit. In this experiment, the shunt has resistance 0.05 Ω, and current
can be determined as:
(5-14)
ISMA = Vshunt / 0.05
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5.4.4 Data Acquisition
5.4.4.1 Keithley 2700
Figure 5-10 Keithley 2700
The Keithley 2700 as shown in Figure 5-10 was used in this experiment for it’s auto
range feature and it’s manual operation feature. It can change its range automatically
according to the magnitude of the measuring value. This feature provides great
convenience when measuring VSMA, Vshunt and Vpressure because they are quite different in
magnitude and require different measurement ranges. Another important feature of the
Keithley 2700 is that it can operate manually. If you want to verify an unexpected value,
you measure a value without considering any software or system factors.
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5.4.4.2 Keithley 7700 Module Card
Figure 5-11 Keithley 7700 Module
The Keithley 7700 card is a 20-Channel, Multiplexer Module, as in Figure 5-11. Up to
20 channels different signals can be connected to the 7700 Module directly and these
signals can be recorded by the computer for further analysis when the Keithley 2700 is
operating. The Keithley 7700 module has an on board cold junction compensate feature
for temperature measurement.
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5.4.4.3 ExceLINX
Figure 5-12 ExceLINX Channel Confiugration
Figure 5-12 is a screenshot of ExceLINX. ExceLINX is a Microsoft Excel add-in
software. It is flexible and easy-to-use software with no programming required. Channel
configuring and scan list setting can be completed in Microsoft Excel. Data are acquired
from the Keithley 2700/7700 and are written into Microsoft Excel directly for further
analysis.
100
5.4.5 Welder and Microscope
The two-step spot welding procedure was performed on the capacitor-discharge spotwelder; using a microscope to monitor the operation due to the small size of the
thermocouple and the SMA wire.
5.4.5.1 Capacitor-Discharge Spot-Welder
The capacitor-discharge spot-welder consists of a power control unit, an electrode
holder and a foot step. Figure 5-13 shows the power control unit and the electrode
holder.
Force Adjustment Knob
Electrode
Power Control Unit
Figure 5-13 Capacitor-Discharge Spot-Welder
The power control unit supplies power to the electrodes, an analog meter that displays
power output, an adjust knob to control the power output and a power switch to turn on /
off the power unit. The power output needs to be set to 0.85 W due to the small size of the
101
thermocouple wire and the SMA wire. Larger power output can damage the
thermocouple wire and the SMA wire instead of joining them together.
The electrode holder holds two electrodes, one is at the top location which can move
downward allowing two electrodes to touch, and the other one is fixed at the bottom
location. There is a force adjusting knob which controls the magnitude of the force
applied on the top electrode when it presses onto the bottom electrode. In this work, the
force is set to the minimum, again due to the small size of the thermocouple wire and the
SMA wire. Large force settings can damage the wires during the spot welding.
The foot step controls the movement of the top electrode.
5.4.5.2 Microscope
The microscope is a 3X optical microscope. The location of the microscope needs to be
properly adjusted so that the object lens focuses on the top of the bottom electrode and
the spot welding operation can be monitored clearly through the microscope.
5.4.5.3 Thermocouple Welder
The ambient thermocouple fixed to the stand to measure the air temperature Tamb in the
bell jar, as shown in Figure 5-7. This thermocouple bead can be made on the
thermocouple welder: THERM-X (Model 258B). The detailed information about the
welder can be downloaded from the website: http://www.therm-x.com/assets/5/258b.pdf .
The minimum bead size that can be made on this welder is 0.25 mm dia, as shown in
Figure 5-14. Though the beads are very small in size, they are still big when attaching
them onto the SMA wire to measure TSMA. Spot welding these beads onto the SMA wire
102
can cause a temperature jump/drop when measuring the temperature of a current carrying
SMA wire. But for the temperature measurement of air in the bell jar, there is no problem
with carrying current, so the thermocouple welder can be used to make these
thermocouples.
Figure 5-14 The Smallest Thermocouple Bead Made on THERM-X (Model 258B)
5.5 Experimental Procedure
The experiment consisted of three phases: general setup preparation; determination of
Qrad and convection heat transfer modeling. In the first phase, a sample SMA wire was
prepared and suspended onto the fixture according to Figure 5-4; with the equipment
wiring shown in Figure 5-6. In the second phase, the air pressure was decreased to 10-8
torr by the mechanical pump and the diffusion pump to determine the Qrad of the test
103
section of the SMA wire at 100 0C. In the third phase, the test section of the SMA wire
was heated to 100 0C again to collect data for heat transfer modeling.
The experimental steps are as follow:
General Setup Preparation
1. Cut a piece of Flexinol SMA wire with a diameter 0.5 mm (0.02”), length is 50
cm. Record its product’s specification including: the spool number, As and D.
2. Sand two ends of the SMA wire using Paper Sheet (3M Wetordry431Q, in
180C weight).
3. Crimp two Dynalloy ring terminals using the crimping tool (SPC CTT-842401) at the locations that are 12.5 cm away from the two ends. Tie the ring
terminals onto the fixture with cotton string as shown in Figure 5-4.
4. Sand the centre point of the SMA wire as in step 2 for attaching thermocouple.
5. Choose E-type 40 AWG thermocouple 40 cm. Strip the ends of the two
thermocouple wires and spot weld onto the SMA wire using the two step spot
welding technique described in the Section 4.4.
6. Test the thermocouple using the pulse shut off method and the reverse current
method that are described in the Section 4.2 and 4.3, to see if there is current
influence on the temperature readout. If there is a current influence, re-weld the
thermocouple on SMA wire according to the spot welding procedure in step 5.
7. Choose two constantan 40 AWG thermocouple wires 40 cm, strip the ends and
spot weld onto the SMA wire at the points 2.5 cm away from the ring terminal
104
toward the centre. These two thermocouple wires are used as the voltage
measurement leads.
8. Make an E-type 40 AWG, thermocouple on the thermocouple welder: THERMX (Model 258B).
9. Place the fixture on the stand and wire the thermocouples, the voltage
measurement leads, the power leads as in Figure 5-6.
10. Apply Dow Corning high vacuum grease on the edge of the bell jar and cover
the bell jar on the baseplate. Check the state of all the valves status on the
vacuum station NRC 3117, make sure that the air release valve is opened and
the vacuum pump valve is closed.
Determination of Qrad
11. Turn on the mechanical pump. After 5 minutes, when the vacuum pump is
running normally, close the air release valve and open the vacuum pump valve
12. Keep evacuating, when the pressure is lower than 0.02 torr, proceed to the next
step.
13. Turn on the diffusion pump.
14. After half an hour, run ExceLINX, which is pre-configured as in Figure 5-12 at
a scan speed of 5 s. Observe the temperature readout on ExceLINX.
15. Turn on the power supply, set to the constant voltage mode. Adjust the current
output slowly to bring the SMA wire temperature to 100 0C. When the
temperature readout on ExceLINX repeats 10 times, it shows the temperature is
105
stable and the sample SMA wire is in steady state. Turn off the power supply
and ExceLINX.
16. Turn off the diffusion pump.
17. After 1 hour, turn off the mechanical pump. Open the air release valve to bring
the pressure to 760 torr.
18. Close ExceLINX.
Convection Heat Transfer Modeling
19. Run ExceLINX, which is pre-configured as in Figure 5-12 at a scan speed of 5s.
Observe the temperature readout on ExceLINX
20. Turn on the power supply, set to the constant voltage mode. Adjust the current
output slowly to bring the SMA wire temperature to 100 0C. When the
temperature readout on ExceLINX repeats 10 times, it shows the temperature is
stable and the sample SMA wire is in steady state. Turn off the power, the SMA
wire temperature drops to the room temperature.
21. Turn on the mechanical vacuum pump to bring the pressure to 600 torr. Turn
on the power and adjust the current to bring the SMA wire temperature to
1000C. When the temperature readout on ExceLINX repeats 10 times, turn off
the power, the SMA wire temperature drops to the room temperature.
22. Change pressure to 500 torr, 400 torr, 300 torr, 200 torr, 100 torr, 66 torr, 33
torr and 1 torr, repeat the step 21.
23. Close ExceLINX.
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24. Open the bell jar, change the inclination angle from 00 (horizontal) to 150, 300,
450, 600, 750, 900 (vertical), then repeat above step from 19 to 23.
5.6 Uncertainty Error Analysis
A full uncertainty analysis was carried out to estimate the uncertainty errors of each
measurement and calculation results. Analysis detail is presented in Appendix B. The
uncertainty of Qrad is 0.142 %. The average uncertainties for the Rayleigh numbers (RaD)
and the Nusselt numbers (NuD) are 11.2 % and 4.65 %, respectively. The uncertainties for
each individual RaD and NuD are presented in Table 6-1 to 6-7.
5.7 Summary of Experimental Methods
The detailed experimental methods have been discussed step by step: from the
background of the experiment to the setup of sample SMA wire on the test fixture; from
the experimental setup diagrams to the descriptions of the equipment which are required
to collect data. In the end, the experimental procedure and the uncertainty error analysis
method were provided. The experiment tests were performed following the descriptions
in this chapter and the experimental results are present and discussed in the next chapter.
107
Chapter 6
Results and Discussions
In the Section 6.1, the raw data are processed and presented; the computational results
are tabulated and plotted for modeling. A new heat transfer correlation is developed from
the results of Nu and Ra in the Section 6.2. The comparisons between the new correlation
and the existing correlations are presented and discussed. Section 6.3 gives three
examples of SMA wire in different diameters and inclination angles to explain how to use
the new correlation to predict the temperature of the SMA wire when current I is
provided.
6.1 Experimental Data Reduction
Following the steps described in the Section 5.5, a set of data was collected during the
experiment. Using the appropriate equations, the Rayleigh numbers (RaD) and the Nusselt
numbers (NuD) at each inclination angle were calculated. The data and the results are
presented in this section.
g, Pr, µ, R, Z are constants. g = 9.8 m/s2, R = 0.287 kJ/(kg·K). Z is the compressibility
factor of air. Since air is assumed to be an ideal gas, Z = 1.
The experiment was designed to heat the SMA wire to TSMA = 373 K (100 0C), under
which NuD and RaD were determined and a heat transfer correlation was developed. The
ambient temperature T∞ = 296 K (23 0C), the mean temperature is Tm = 334 K (61.5 0C).
The constants can be determined at Tm = 334 K: µ = 2.008 ×10-5 kg/(m.s); Pr =0.7202.
108
The diameter and the length of the SMA wire test section were measured as: D = 0.49
mm; l = 191 mm.
Air pressure can be calculated from Vpressure using Equation 5-13, as described in the
Section 5.4.2.2 (P = Vpressure×100×133.3 Pa).
The current of the SMA wire is calculated using Equation 5-14 (ISMA = Vshunt / 0.05 A),
as described in Section 5.4.3.2
A preliminary test was performed in vacuum to determine the power input to the SMA
wire test section VIvacuum =0.0924 ± 0.00013 W, when TSMA = 373 K. From Equation 511, Qrad = 0.0924 ± 0.00013 W. Its uncertainty is calculated by Equation B-19 in
Appendix B
The Rayleigh number (RaD) is calculated using Equation 5-6, its uncertainty is
calculated by Equation B-22 in Appendix B, the Nusselt number (NuD) is calculated by
Equation 5-9, its uncertainty is calculated by Equation B-20 in Appendix B
The calculation results along with the raw data TSMA, T∞, Vshunt, VSMA, and Vpressure, are
tabulated in Table 6-1 to 6-7. NuD vs RaD is plotted in Figure 6-1.
The NuD-RaD curves in Figure 6-1 are generally smooth except for two instances in the
range 0.4 < RaD < 0.6 when the inclination angles are 150 and 600. These inconsistencies
are a result of instability in the temperature field leading to fluctuations in the natural
convection flow, primarily for an ambient air pressure between 600 and 760 torr. As a
result, the SMA wire temperature must be closely monitored and the power adjusted to
maintain a temperature of approximately 100 0C. The two instances in Figure 6-1 are a
result of not responding quickly enough to a sudden change in wire temperature.
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Experimental data reduction has been performed. The raw data and the computational
results of NuD and RaD, along with the uncertainty of each NuD and RaD, have been
tabulated for each inclination angle. NuD vs RaD for all inclination angles have been
plotted and are ready for development of new heat transfer correlation in the next section.
110
Table 6-1 Raw Data and the Results NuD, RaD at Inclination Angle φ = 00
TSMA (K)
T∞ (K)
Vshunt (V)
VSMA (V)
Vpressure (V)
NuD
RaD
δNuD/NuD
δRaD/RaD
373.64
295.49
0.0719
1.0461
7.6000
1.0703
0.6069
4.784 %
11.531 %
373.05
295.49
0.0699
1.0171
6.2186
1.0166
0.4035
4.784 %
11.531 %
373.02
295.60
0.0684
0.9950
5.2037
0.9724
0.2815
4.765 %
11.483 %
373.22
295.64
0.0666
0.9684
4.1407
0.9162
0.1783
4.757 %
11.465 %
373.16
295.70
0.0643
0.9345
3.1119
0.8505
0.1004
4.745 %
11.434 %
373.10
295.77
0.0613
0.8920
2.0746
0.7705
0.0445
4.733 %
11.405 %
373.15
295.86
0.0570
0.8278
1.0416
0.6552
0.0112
4.717 %
11.363 %
373.26
295.90
0.0548
0.7959
0.6850
0.5997
0.0048
4.711 %
11.349 %
373.12
295.96
0.0518
0.7525
0.3434
0.5303
0.0012
4.700 %
11.321 %
111
Table 6-2 Raw Data and the Results NuD, RaD at Inclination Angle φ = 150
TSMA (K)
T∞ (K)
Vshunt (V)
VSMA (V)
Vpressure (V)
NuD
RaD
δNuD/NuD
δRaD/RaD
373.25
295.64
0.0712
1.0386
7.6000
1.0581
0.6035
4.758 %
11.465 %
372.41
295.60
0.0690
1.0065
6.1761
1.0016
0.3948
4.766 %
11.486 %
373.17
295.70
0.0674
0.9817
5.1369
0.9426
0.2742
4.746 %
11.436 %
373.02
295.73
0.0656
0.9558
4.1146
0.8929
0.1753
4.740 %
11.422 %
373.06
295.77
0.0634
0.9239
3.0854
0.8303
0.0984
4.734 %
11.405 %
373.23
295.84
0.0608
0.8851
2.0561
0.7558
0.0437
4.722 %
11.375 %
372.98
295.94
0.0567
0.8256
1.0279
0.6501
0.0109
4.703 %
11.329 %
372.99
295.97
0.0544
0.7919
0.6796
0.5928
0.0048
4.698 %
11.316 %
372.98
296.00
0.0514
0.7482
0.3398
0.5224
0.0012
4.693 %
11.305 %
112
Table 6-3 Raw Data and the Results NuD, RaD at Inclination Angle φ = 300
TSMA (K)
T∞ (K)
Vshunt (V)
VSMA (V)
Vpressure (V)
NuD
RaD
δNuD/NuD
δRaD/RaD
373.59
296.37
0.0701
1.0202
7.6000
1.0315
0.5919
4.629 %
11.144 %
373.08
296.43
0.0683
0.9939
6.1760
0.9839
0.3879
4.619 %
11.119 %
372.98
296.51
0.0667
0.9711
5.1481
0.9389
0.2685
4.607 %
11.089 %
373.11
296.55
0.0650
0.9468
4.1187
0.8881
0.1719
4.600 %
11.072 %
373.06
296.61
0.0629
0.9154
3.0891
0.8273
0.0965
4.589 %
11.045 %
373.18
296.68
0.0602
0.8755
2.0599
0.7504
0.0429
4.578 %
11.017 %
373.04
296.76
0.0560
0.8149
1.0384
0.6428
0.0109
4.565 %
10.984 %
373.01
296.78
0.0538
0.7833
0.6838
0.5887
0.0047
4.562 %
10.978 %
373.18
296.81
0.0508
0.7394
0.3403
0.5159
0.0012
4.556 %
10.964 %
113
Table 6-4 Raw Data and the Results NuD, RaD at Inclination Angle φ = 450
TSMA (K)
T∞ (K)
Vshunt (V)
VSMA (V)
Vpressure (V)
NuD
RaD
δNuD/NuD
δRaD/RaD
373.24
295.33
0.0697
1.0158
7.6000
0.9609
0.5947
4.674 %
11.256 %
373.45
295.30
0.0681
0.9918
6.1983
0.8891
0.3989
4.671 %
11.249 %
373.08
295.31
0.0662
0.9640
5.1667
0.8689
0.2758
4.671 %
11.249 %
373.38
295.30
0.0647
0.9418
4.1391
0.8195
0.1762
4.660 %
11.223 %
373.01
295.34
0.0625
0.9094
3.0924
0.7778
0.0991
4.654 %
11.208 %
373.18
295.43
0.0599
0.8717
2.0656
0.7099
0.0438
4.645 %
11.184 %
373.19
295.49
0.0558
0.8124
1.0364
0.6089
0.0109
4.636 %
11.161 %
373.15
295.53
0.0536
0.7802
0.6812
0.5549
0.0047
4.626 %
11.136 %
373.07
295.61
0.0502
0.7296
0.3394
0.4846
0.0012
4.614 %
11.107 %
114
Table 6-5 Raw Data and the Results NuD, RaD at Inclination Angle φ = 600
TSMA (K)
T∞ (K)
Vshunt (V)
VSMA (V)
Vpressure (V)
NuD
RaD
δNuD/NuD
δRaD/RaD
372.97
296.11
0.0678
0.9865
7.6000
0.9609
0.5947
4.674 %
11.256 %
373.56
296.12
0.0656
0.9544
6.2100
0.8891
0.3989
4.671 %
11.249 %
373.22
296.12
0.0647
0.9421
5.1758
0.8689
0.2758
4.671 %
11.249 %
373.17
296.19
0.0630
0.9160
4.1419
0.8195
0.1762
4.660 %
11.223 %
373.50
296.22
0.0616
0.8958
3.1038
0.7778
0.0991
4.654 %
11.208 %
373.20
296.28
0.0589
0.8571
2.0682
0.7099
0.0438
4.645 %
11.184 %
373.00
296.33
0.0549
0.7982
1.0359
0.6089
0.0109
4.636 %
11.161 %
372.96
296.39
0.0526
0.7653
0.6830
0.5549
0.0047
4.626 %
11.136 %
373.28
296.46
0.0496
0.7220
0.3425
0.4846
0.0012
4.614 %
11.107 %
115
Table 6-6 Raw Data and the Results NuD, RaD at Inclination Angle φ = 750
TSMA (K)
T∞ (K)
Vshunt (V)
VSMA (V)
Vpressure (V)
NuD
RaD
δNuD/NuD
δRaD/RaD
373.25
296.73
0.0648
0.9444
7.6000
0.8768
0.5900
4.569 %
10.996 %
373.05
296.71
0.0633
0.9213
6.2100
0.8339
0.3929
4.573 %
11.006 %
373.23
296.76
0.0621
0.9036
5.1732
0.7995
0.2723
4.565 %
10.984 %
373.14
296.82
0.0606
0.8822
4.1423
0.7616
0.1738
4.554 %
10.959 %
373.29
296.86
0.0587
0.8538
3.1028
0.7081
0.0975
4.548 %
10.942 %
373.09
296.90
0.0563
0.8186
2.0706
0.6480
0.0433
4.542 %
10.929 %
373.03
296.97
0.0528
0.7680
1.0368
0.5634
0.0108
4.530 %
10.898 %
373.08
297.02
0.0509
0.7403
0.6823
0.5188
0.0047
4.522 %
10.878 %
373.09
297.06
0.0485
0.7054
0.3421
0.4647
0.0012
4.517 %
10.865 %
116
Table 6-7 Raw Data and the Results NuD, RaD at Inclination Angle φ = 900
TSMA (K)
T∞ (K)
Vshunt (V)
VSMA (V)
Vpressure (V)
NuD
RaD
δNuD/NuD
δRaD/RaD
372.94
297.05
0.0612
0.8898
7.6000
0.7814
0.5811
4.518 %
10.868 %
373.06
297.06
0.0603
0.8762
6.2063
0.7547
0.3878
4.516 %
10.863 %
373.34
297.10
0.0593
0.8617
5.1730
0.7255
0.2699
4.508 %
10.844 %
373.23
297.18
0.0579
0.8419
4.1392
0.6910
0.1724
4.497 %
10.817 %
373.03
297.25
0.0565
0.8214
3.1084
0.6573
0.0968
4.486 %
10.789 %
373.27
297.28
0.0548
0.7966
2.0738
0.6126
0.0431
4.481 %
10.777 %
373.32
297.32
0.0519
0.7536
1.0362
0.5407
0.0108
4.475 %
10.762 %
373.14
297.36
0.0502
0.7288
0.6850
0.5026
0.0047
4.469 %
10.747 %
373.21
297.40
0.0481
0.6984
0.3419
0.4556
0.0012
4.462 %
10.729 %
117
Nu D − Ra D (500 µ Dia Wire)
1.1
horizontal
15deg
1
30deg
45deg
0.9
60deg
75deg
vertical
NuD
0.8
0.7
0.6
0.5
0.4
0.001
0.01
0.1
Ra D
Figure 6-1 NuD vs RaD at Inclination Angles From 0 To 900 (RaD = 0.001 – 0.6)
118
1
6.2 Convective Heat Transfer Modeling
6.2.1 Development of Convective Heat Transfer Correlation
The NuD can be correlated with RaD in a power equation form: Nu D = CRa Dn by using
the “add trend line” feature in MS Excel, at each inclination angle.
Table 6-8 Correlation NuD = C RaDn at Each Inclination Angle φ
Inclination Angle
Nu D = CRa Dn
R-squared value
0
Nu D = 1.1194 RaD0.1153
0.9965
15
Nu D = 1.0969 RaD0.1137
0.9959
30
Nu D = 1.0854 RaD0.113
0.998
45
Nu D = 1.049 RaD0.1144
0.9984
60
Nu D = 0.9971RaD0.1085
0.9982
75
Nu D = 0.9122 RaD0.1036
0.9961
90
Nu D = 0.8128 RaD0.0882
0.9976
(degree)
From Table 6-8, the constant coefficient C at each inclination angle can be further
correlated with the inclination angle in a two order polynomial, by using the “add trend
line” feature in MS Excel again:
C = −0.0033 ⋅ ϕ + 1.158
(6-1)
R = 0.8994
2
119
Similarly, the constant exponent n can be expressed relative to the inclination angle φ
as:
n = −0.0003 ⋅ ϕ + 0.1195
(6-2)
R = 0.7124
2
A correlation that comprises the inclination angle φ can be expressed as:
Nu D = (−0.0033 ⋅ ϕ + 1.158) ⋅ Ra D( −0.0003⋅ϕ +0.1195)
(6-3)
6.2.2 Verification of Convective Heat Transfer Correlation
The verification of the new convective heat transfer correlation is performed by
comparing to existing correlations at various angles between horizontal and vertical. Due
to very limited data available for inclined orientations and the only reviewed correlation
for inclined wires from Oosthuizen [78] showing a much lower value compared to the
new correlation, the verification focuses on the comparison to existing correlations in
horizontal and vertical orientations. The new correlation is plotted in Figure 6-4 in the
horizontal orientation and Figure 6-5 in the vertical orientation at a Rayleigh number
range: 0.001 < RaD < 0.6, with the respective existing correlations as reviewed in the
Section 2.2.
6.2.2.1 Comparisons in Vertical and Horizontal
Figure 6-5 shows that the new correlation has the same trend as other correlations.
However, there are minor differences between the new correlation and the other existing
correlations. These discrepancies are the result of the different methods and conditions, or
even the errors when the existing correlations were developed. Considering the new
120
correlation is positioned in the middle among the existing correlations: lower than
Elenbass [73] and Mueller [76]; but higher than Zitsev & Sokovishin [77] and Yang [74],
the new correlation should be acceptable in the vertical orientation.
Figure 6-4 shows that the discrepancies among the existing correlations are relatively
smaller than that in Figure 6-5, and the trends of the existing correlations are very
similar. The new correlation agrees with the existing correlations when the Rayleigh
number (RaD) is low (0.001 < RaD < 0.05), but when the Rayleigh number is high, the
new correlation is higher than the existing correlations. It is quite unusual that these four
existing correlations have very similar trends and small discrepancies: ∆NuD between
maximum and minimum is 0.1 when RaD=0.001; ∆NuD = 0.2 at RaD = 0.6, because they
were developed for different conditions with completely different methods. Tsubouchi
and Masuda [69] used oil as the fluid to developed their correlation, Morgan [70]
suggested his correlation after analyzing numerous of correlations developed from others;
though Churchill and Chu [71],
Raithby and Hollands [72] all developed their
correlations using a theoretical method, their respective theoretical methods were
completely different. The difference between the current correlation and the existing
correlations is insignificant when Rayleigh number is low, but with Rayleigh number
increasing, this difference becomes prominent, ∆NuD between the current correlation and
Raithby, Hollands [72] is 0.2 at RaD=0.6, which is 15.5 % higher than Raithby and
Hollands’ data. There are many possible reasons for these differences with increasing
Rayleigh number including: air pressure, or temperature distortion on the SMA wire
which is the result of the thermocouple attachment.
121
6.2.2.2 Temperature Distortion on SMA Wire
As it has been stated in the Section 2.1.2, there can be a temperature distortion error
when measuring the temperature of a solid surface using a thermocouple. Before a
thermocouple is attached to the SMA wire, the SMA wire is in thermal equilibrium in
which the heat generated due to the Joule effect is dissipated to the environment through
convection and radiation. There is no conduction along the SMA wire since the
temperature of the SMA wire is constant along its length. When a thermocouple is spot
welded to the SMA wire, conduction occurs along the thermocouple lead drawing heat
away from the SMA wire in addition to convection. As a result, the temperature at the
attachment point starts to drop and conduction along the SMA wire occurs in the
direction towards to the attachment point due to the temperature gradient that now exists
along the SMA wire. When a new thermal equilibrium is reached at the attachment point,
its temperature stabilizes at a lower value than the previous temperature on the SMA
wire. The difference between the previous temperature and the new balanced temperature
at the attachment point is referred to as temperature distortion. The thermocouple
measures the temperature at the attachment, but the SMA wire is actually at a higher
temperature leading to a higher convection heat transfer rate.
The temperature distortion becomes larger when the air pressure is higher. In vacuum,
there is no convection and the thermocouple wires dissipate heat through radiation when
the heat flows into the thermocouple wires from the SMA wire at the attachment point.
When there is a positive air pressure, the thermocouple wires have higher rates of heat
dissipation because of convection in addition to radiation. As a result, the conduction
leaving the attachment point increases and the temperature at the attachment point
122
becomes lower and the temperature distortion becomes larger. The convection increases
with the air pressure, so the temperature distortion also increases with the air pressure.
This explains that the difference between the new correlation and the existing correlation
is larger when the Rayleigh number (RaD) is larger.
When the SMA wire is in vertical, the temperature distortion is not as large as that
when the SMA wire is horizontal because the air adjacent to the SMA wire in vertical
orientation is hotter which makes the attachment point temperature higher. When the
SMA wire is placed horizontally, the air adjacent is heated up and becomes lighter. It
then rises to form a plume at the top and then leaves the SMA wire. The cooler air fills
the position where the heated air leaves as shown in Figure 6-2 (a) & (b). [91] When the
SMA wire is placed in a vertical orientation, as shown in Figure 6-2 (c) [92], the adjacent
air is heated by the SMA wire and rises to leave its original position. But the air that fills
in is from below, which has previously been heated up by the SMA wire, therefore is
hotter than in the horizontal orientation. The relative hotter air that rises from below can
provide extra energy that balances the conduction via the thermocouple wires. The
conduction along the SMA wire is not as large as in the horizontal orientation, and the
temperature gradient along the SMA wire is not as large as in horizontal SMA wire as
well. This is the reason why the temperature distortion in the vertical SMA wire is
smaller than in the horizontal SMA wire. This explains why there is not an obvious
difference between the new correlation and the existing correlations when the SMA wire
is in the vertical orientation.
123
(a)
(b)
(c)
Figure 6-2 Natural Convective Flow Near Horizontal Cylinder and Vertical
Cylinder
6.2.2.3 IR Picture Verifies the Temperature Distortion on SMA Wire
The temperature distortion error is hard to measure using any contact methods since
any contact temperature sensors bring in new distortions. Invasive infrared thermal
imaging can easily find the temperature distortion when using a thermocouple to measure
the temperature of the SMA wire. The infrared thermal photo in Figure 6-3 shows that: at
the location on the SMA wire where the thermocouple is attached, the color is yellow,
while at other locations on the SMA wire, the colors are white. The temperature
distribution along the SMA wire below the thermal photo, which is the result of the line
analysis on the thermal photo as stated in the Section 3.3.2, shows clearly that the
temperature where the thermocouple is attached onto the SMA wire is 7 0C lower than
other locations of the SMA wire.
124
0
102 C
100
98
96
94
0
50
100
150
200
250
300
92
mm
Figure 6-3 IR Picture Shows Temperature Distortion at Thermocouple Attachment
Due to the temperature distortion on the SMA wire, it is plausible to assume the actual
temperature of the SMA wire is 110 0C at 1 atm when the thermocouple has a reading
100 0C. Using Equations 5-6 and 5-9, when TSMA = 383K (110 0C), the recalculated
results are: RaD = 0.6608; NuD = 0.96, which are different from the results in Table 6-1.
Since the temperature distortion increases with the air pressure, it can be assumed that
there is a potential correlation based on the actual SMA wire temperature, as shown in
Figure 6-4. It is clear that if there is no temperature distortion on the SMA wire, the
potential correlation agrees well with the theoretical correlation from Raithby and
Hollands [72].
125
Temperature distortion error is an inherent error when measuring a solid surface using
contact temperature sensors, especially a thin wire. Infrared thermal imaging can measure
the temperature of a thin wire without contacting it, but there are some constraints when
using an infrared camera with close up lens, ie, an infrared camera may not be used in
vacuum. An improved thermocouple [21] used to measure the temperature of micro chip
maybe a good choice to eliminate temperature distortion to measure the temperature of
thin wires.
126
New Correlation In Horizontal Orientation
1.20
1.00
New correlation
Potential correlation
Tsubouchi & Masuda [69]
Morgan [70]
Chruchill & Chu [71]
Raithby & Hollands [72]
0
Assume T SMA = 383 K (110 C) ,
Nu D =0.96, Ra D =0.6608
NuD
0.80
0.60
0.40
0.20
0.00
0.0010
0.0100
0.1000
Ra D
Figure 6-4 Comparison To the Existing Correlations in Horizontal Orientation
127
1.0000
New Correlation in Vertical Orientation
1
0.9
0.8
0.7
NuD
0.6
0.5
0.4
0.3
New correlation
Elenbass [73]
Yang [74]
Mueller [76]
Zitsev & Sokovishin [77]
0.2
0.1
0
0.001
0.01
0.1
Ra D
Figure 6-5 Comparison to the Existing Correlations in Vertical Orientation
128
1
6.3 SMA Wire Temperature Prediction Through New Correlation
The SMA constitutive and phase kinetic behaviors are directly controlled by the heat
transfer rate to and from the wires, a detailed heat transfer model is needed to monitor and
predict the temperature and the behavior of the SMA wires. The new correlation is developed
based on experimental results; it then helps to predict SMA wire temperature when giving the
diameter and the inclination angle of the SMA wire.
Assuming a short SMA wire is a section of a sufficiently long SMA wire, when it is heated
by a current and reaches a steady state, there is no conduction along the axial direction of the
SMA wire due to the uniform temperature distribution along the SMA wire. The heat transfer
equation of this short section of SMA wire is in the form:
(
I 2 RSMA = hAs (T − T∞ ) + εσAs T 4 − T∞4
)
(6-4)
Following the flowchart in Figure 6-6, the temperature of the SMA wire can be
determined.
For example: a 0.5 mm diameter SMA wire is inclined 500 to the horizontal in 1 atm air
pressure. According to Equation 6-3, NuD-RaD becomes:
NuD = 1.0314·RaD0.1128
(6-5)
129
Figure 6-6 Flowchart to Determine SMA Wire Temperature
130
Substitute Equation 5-6 and 5-9 into Equation 6-5, Equation 6-5 becomes:
⎞
⎛
⎟
⎜
3 2
hD
⎟
⎜ g (TSMA − T∞ ) D P
= 1.0314⎜
⋅ Pr ⎟
2
k
⎟
⎜ µ 2T∞ R 2 Z 2 ⎛⎜ TSMA + T∞ ⎞⎟
⎟
⎜
2
⎝
⎠
⎠
⎝
(6-6)
0.1128
The heat transfer coefficient is:
⎞
⎛
⎟
⎜
3 2
1.0314 ⋅ k ⎜ g (TSMA − T∞ ) D P
⎟
h=
⋅ Pr ⎟
2
⎜
D
⎟⎟
⎜⎜ µ 2T∞ R 2 Z 2 ⎛⎜ TSMA + T∞ ⎞⎟
2
⎝
⎠
⎠
⎝
(6-7)
0.1128
Substitute Equation 6-7 into Equation 6-4, the heat transfer equation of the SMA wire
becomes:
I 2 RSMA
⎞
⎛
⎟
⎜
3 2
1.0314 ⋅ k ⎜ g (TSMA − T∞ ) D P
⎟
=
⋅ Pr ⎟
2
⎜
D
⎟⎟
⎜⎜ µ 2T∞ R 2 Z 2 ⎛⎜ TSMA + T∞ ⎞⎟
2
⎝
⎠
⎠
⎝
0.1128
4
⋅ πDl (TSMA − T∞ ) + εσπDl (TSMA
− T∞4 )
(6-8)
The resistance of a unit length of 0.5 mm diameter SMA wire is 4.3 Ω/m [87], from
Dynalloy, Inc. The length of this short section of SMA wire is assumed to be 1 cm. So its
resistance is RSMA = 0.043 Ω.
The values of k, µ and Pr [93] change with temperature, which are tabulated in Table
6-9.
131
Table 6-9 The Values of k, µ and Pr at Different Temperature (25 – 70 0C)
Tm (K)
µ kg/(m.s)
k W/(m.K);
Pr
298
1.849 x 10-05
0.02551
0.7296
303
1.872 x 10
-05
0.02588
0.7282
308
1.895 x 10
-05
0.02625
0.7268
313
1.918 x 10-05
0.02662
0.7255
318
1.941 x 10-05
0.02699
0.7241
323
1.963 x 10
-05
0.02735
0.7228
333
2.008 x 10
-05
0.02808
0.7202
343
2.052 x 10-05
0.02881
0.7177
Use Microsoft Excel “add trend line” feature, µ, k and Pr can be expressed as functions
of Tm in the forms:
(6-9)
µ = 0.9228 ⋅ (5 × 10−8 ⋅ Tm + 5 × 10−6 )
R 2 = 0.9999
(6-10)
k = 1.04 ⋅ (7 × 10 −5 ⋅ Tm + 0.0037)
R2 = 1
(6-11)
Pr = 5 × 10−7 ⋅ Tm2 − 0.0006 ⋅ Tm + 0.8645
R2 = 1
P = 1atm = 1.013 ×105 Pa. T∞ is assumed to be 296 K (23 0C). g, R and Z are constant,
g = 9.8 m/s2, R = 0.287 kJ/(kg·K), Z = 1. ε = 0.63, σ = 5.67×10-8 W/(m2·K4).
132
Substitute Equation 6-9, 6-10 and 6-11, g, R and Z into Equation 6-8, the results of
TSMA vs I can be tabulated as in Table 6-10 and plotted as in Figure 6-7:
Similarly, when a 0.381 mm diameter SMA wire is inclined from horizontal at 350, the
correlation becomes:
NuD = 1.0764·RaD0.1138
(6-12)
Its unit length resistance is 8.3 Ω/m. [87]
When a 0.254 mm diameter SMA wire is inclined from horizontal at 650, the
correlation becomes:
NuD = 0.9684·RaD0.1086
(6-13)
Its unit length resistance is 18.5 Ω/m. [87]
TSMA vs I in both cases are also tabulated in Table 6-10 and plotted as in Figure 6-7.
As shown in Figure 6-7, since a thinner diameter SMA wire has a larger electrical
resistance to heat the SMA wire to a same temperature, the thinner SMA wires need less
current.
6.4 Summary of Results and Discussion
The experimental results have been presented and discussed in this chapter. The raw
data was processed and presented in the form of tables and figures. A new heat transfer
correlation was developed based on the empirical data. The comparisons of the new
correlation to the existing correlations were discussed and the analysis showed the
133
discrepancies were attributed to the temperature distortion on the sample SMA wire. At
the end of this chapter, three examples were cited to describe the prediction of SMA wire
temperature by applying the new correlation.
134
Table 6-10 TSMA vs Carrying Current I (P = 1 atm)
I (A)
I (A)
I (A)
D = 0.5 mm
D = 0.381 mm
D = 0.254 mm
φ = 500
φ = 350
φ = 650
298
0.137
0.094
0.057
303
0.273
0.187
0.114
313
0.445
0.305
0.186
323
0.577
0.396
0.241
333
0.690
0.474
0.288
343
0.791
0.543
0.330
353
0.884
0.607
0.369
363
0.972
0.668
0.405
373
1.056
0.725
0.440
383
1.135
0.779
0.473
393
1.213
0.832
0.504
403
1.287
0.883
0.535
413
1.360
0.933
0.565
TSMA (K)
436
416
TSMA (K)
396
376
356
336
D=0.5mm, 50 deg inclined
D=0.381mm, 35 deg inclined
316
D=0.254mm, 65 deg inclined
296
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
I (A)
Figure 6-7 TSMA vs Carrying Current I (P = 1 atm)
135
1.600
Chapter 7
Conclusion and Recommendations
7.1 Conclusion
There are five major conclusions following the experiment and the discussion around
the temperature measurement of SMA wire and the natural convective heat transfer
model for SMA wire.
7.1.1 The Carrying Current’s Influence on Temperature Readings
The carrying current affects the temperature readings by adding a “spurious voltage”
∆V to the thermo electro-motive force (EMF) of the thermocouple which is spot welded
onto the SMA wire. The “spurious voltage” is a result of the ohmic drop between two
thermocouple legs due to the carry current which are actually at two different locations
on the SMA wire. The magnitude of ∆V is dependent on the spatial offset between the
two thermocouple legs, ∆x, and the carrying current I. When the direction of the carrying
current is reversed, ∆V becomes negative. The carrying current’s influence on
temperature reading can be eliminated by three methods: 1, the pulse shut off method; 2,
the reverse current and average method and 3, the improved two-step spot welding
thermocouple method.
136
7.1.2 Two-Step Spot Welding Thermocouple
The two-step spot welding method can spot weld the thinnest E-type 40 AWG
thermocouple wires onto the SMA wire with the smallest junction and effectively negate
the influence of the carrying current on the temperature readings, which has been
successfully applied in a pending patent [94]. Unlike the conventional method, in which a
thermocouple bead is made on a thermocouple welder, and then the bead is spot welded
onto the SMA wire using a capacitor current discharge welder, the two-step spot welding
thermocouple spot welds the thermocouple wires directly onto the SMA wire. With the
aid of a microscope, the two-step spot welding method takes good control of the thin
thermocouple wires and the SMA wire, precisely adds the second thermocouple wire on
the spot where the first thermocouple wire is welded on the SMA wire. As a result, the
junction size and the distance between the sense point of thermocouple and the SMA wire
are both the smallest that can be obtained, yet are big enough to offset the carrying
current’s influence on the thermocouple readings.
7.1.3 Infrared Thermal Imaging is a Good Supplement
Infrared thermal imaging is a good supplement to thermocouples in verifying the
occurrence of temperature distortion. Infrared thermal imaging cannot be the primary
method to determine the temperature of SMA wire and to model convection heat transfer
for SMA wires with confidence because of radiosity effect from the surroundings, the
SMA wire tends to move during the phase transformation which greatly changes the
137
temperature readings, and the IR software is unable to provide recordable data which are
required in modeling heat transfer (by the current IR camera: ThermaCAM S60). But as
an invasive method, infrared thermal imaging has an advantage which can give the
temperature of the SMA wire without having a sensor contact the SMA wire. An infrared
camera can take the thermal pictures of the SMA wire and the thermocouple in the twostep spot welding method. By comparing the temperature at the location where the
thermocouple is attached to the locations elsewhere on the SMA wire in the thermal
image, the infrared thermal imaging can verify if there is a temperature distortion on the
SMA wire and how big the distortion is.
7.1.4 Temperature Distortion Error in New Correlation’s Verification
The new correlation in this work was verified in the horizontal and vertical
orientations. The verification in the vertical orientation is acceptable as the new
correlation is positioned in the middle of the existing correlations: lower than Elenbass
[73] and Mueller [76]; but higher than Zitsev & Sokovishin [77] and Yang [74] In the
horizontal orientation, the new correlation agrees with the existing correlations from
Tsubouchi & Masuda [69], Morgan [70], Churchill & Chu [71] and Raithby & Hollands
[72] in the Rayleigh number range: 0.001 < RaD < 0.05; but is higher than these
correlations in the Rayleigh number range: 0.05 ≤ RaD < 0.6. This discrepancy is likely
due to the temperature distortion that the thermocouple brings to the SMA wire when the
air pressure increases and the temperature distortion becomes larger.
138
7.1.5 SMA Wire Temperature Prediction Through New Correlation
Based on the present experimental results, a new correlation can be determined as:
Nu D = (−0.0033 ⋅ ϕ + 1.158) ⋅ RaD( −0.0003⋅ϕ +0.1195) . For a given SMA wire with a diameter
D and an inclination angle φ, the new correlation can be used to determine the convective
heat transfer coefficient of the SMA wire h. When a current I passes the SMA wire and
reaches a steady state, the temperature of the SMA wire can be predicted through the heat
(
)
transfer equation: I 2 RSMA = hAs (T − T∞ ) + εσAs T 4 − T∞4 .
7.2 Recommendations
The recommendations to this work are in three aspects: to improve the two-step spot
welding thermocouple technique on 250 µm diameter SMA wire; to improve the
thermocouple measurement that can minimize the temperature distortion and to develop a
correlation of SMA wire based on electrical resistance instead of temperature.
7.2.1 Improve Two-Step Spot Welding
The two-step spot welding thermocouple technique works well on a 500 µm diameter
SMA wire, but it becomes harder when working on a 250 µm diameter SMA wire. The
difficulty on a 250 µm diameter SMA wire is the smaller size of the SMA wire makes it
harder to control the welding spot on the SMA wire and to find a proper power on the
welder.
139
7.2.2 Improve Thermocouple Method to Minimize Temperature Distortion
Temperature distortion leads to big errors in heat transfer model. The infrared thermal
imaging can identify the distortion, but it can not accurately model this temperature
distortion due to the inherent drawbacks as described in the Chapter 3. The temperature
distortion can be decreased by improving the thermocouple measurement as suggested by
Nakabeppu [21].
7.2.3 Develop a Correlation of Temperature and Resistance
The current work is based on temperature measurement to model heat transfer. Because
of the drawbacks with the temperature measurement, invasive or non-invasive, especially
working on thin wires, there are still some errors when measuring the temperature of
SMA wires for heat transfer modeling. If changes in electrical resistance as a result of
change in SMA temperature can be accurately measured, this maybe a more reliable
method of modeling heat transfer.
140
Appendix A
Ring Terminal Brings SMA Wire Temperature Down
When the ring terminal is crimped on the SMA wire, the SMA wire temperature can
decrease. This is demonstrated in the infrared picture depicted in Figure A-1. The ring
terminal is at the right side of IR picture. The color on the SMA wire close to the ring
terminal is deep yellow in the IR picture, which represents a lower temperature, while the
SMA wire on the other side is bright yellow in the IR picture which represents higher
temperature. Through the line analysis across the SMA wire on the research software, we
can find that the maximum temperature on each line decreases from 96 0C to 84.7 0C
when the line moves toward to ring terminal. This proves that there is a temperature
distribution on the SMA wire when the ring terminal is connected because the maximum
temperature on each line represents the SMA wire temperature at the point where the line
crosses.
The picture in Figure A-1 was taken on a 500 µm diameter SMA wire which was
suspended in air at 1 atm. The fact that the SMA wire close to the ring terminal has a
lower temperature shows that the ring terminal’s temperature is lower than the SMA
wire. This is easy to understand because the ring terminal itself does not generate heat. It
obtains heat from the SMA wire through conduction and then it dissipates the heat to the
environment through convection and radiation. Due to the larger surface of the ring
terminal, it has a higher heat dissipation rate than the SMA wire and hence acts as a heat
sink. Analysis of the IR picture shows that the voltage measurement lead should be
placed at least 15 mm away from the ring terminal in order to eliminate its influence.
141
1.5 cm
Figure A-1 Thermal Gradient After a Ring Terminal is Crimped on SMA Wire
142
Appendix B
Uncertainty Analysis
The uncertainty analysis based on the accuracy of all the equipment used in the test
apparatus is provided in this section.
B.1 Uncertainty Analysis Method
The uncertainty analysis method used in this work is adapted from the experiment
uncertainty analysis of Wheeler and Ganji [95]. In their analysis, they think if a result R
is the function of n measured variables x1, x2, x3, …., xn in an experiment as:
R = f ( x1 , x2 , x3 ,..., xn )
(B-1)
Then a small change in R, δR can be expressed by the variables x1, x2, x3, …., xn, in a
differential formula:
δR = δx1
n
∂R
∂R
∂R
∂R
= ∑ δxi
+ ⋅ ⋅ ⋅ ⋅ +δxn
+ δx1
∂xi
∂xn i =1
∂x1
∂x1
(B-2)
If R is a calculated result based on measured xi’s, the values of the δxi’s can be replaced
by the uncertainty in the variables, which can be denoted by wxi’s and δR can be replaced
by the uncertainty in the result, denoted by wR, as:
(B-4)
wR = δR
The maximum uncertainty in R can be estimated by forcing all terms on the right-hand
side of Equation B-2 to be positive as:
143
n
wR = ∑ wxi
i =1
(B-5)
∂R
∂xi
The uncertainty that is produced by Equation B-5 might be very high because the
variables wxi can be positive simultaneously. So a better estimate for the uncertainty is
given by:
2
⎡n ⎛
∂R ⎞ ⎤
⎟ ⎥
wR = ⎢∑ ⎜⎜ wxi
∂xi ⎟⎠ ⎥
⎢⎣ i =1 ⎝
⎦
(B-6)
1/ 2
If the result R is dependent only on a product of the measured values:
(B-7)
R = Cx1a x2b ⋅ ⋅ ⋅ xnN
The Equation B-6 can be expressed as:
2
2
2
⎛ wn ⎞ ⎤
wR ⎡⎛ w1 ⎞ ⎛ w2 ⎞
⎟ ⎥
= ⎢⎜ a ⎟ + ⎜ b ⎟ + ⋅ ⋅ ⋅ + ⎜⎜ N
R ⎢⎜⎝ x1 ⎟⎠ ⎜⎝ x2 ⎟⎠
xn ⎟⎠ ⎥
⎝
⎦
⎣
1/ 2
(B-8)
Equation B-8 is a better form since the fractional error in the result R is related directly
to the fractional errors in the individual measurements.
B.2 Uncertainty in Measured Values
In this experiment, there are six measured values, temperature T, voltage V, current I,
pressure P, dimension d and l. Each of them has an uncertainty in measurement.
144
B.2.1 Temperature T
The temperature is measured by E-type thermocouples and is read through the Keithley
2700 / 7700 with an internal junction on the module. According to the data from Keithley
Instrument, Inc, the uncertainty in the temperature reading is ± 1 0C, or
(B-9)
δT
⎛ 10 C ⎞
= ±⎜⎜ 0 ⎟⎟
T
⎝ T ( C) ⎠
B.2.2 Voltage V
The uncertainty of voltage measurement depends on the voltage reading and the
measurement that the range of measurement that the readings fall in. Referring the
Keithley 2700 / 7700 specification, the uncertainty of voltage V is as following:
δV
⎛
3.5 × 10 −7 ⎞
⎟ in the range of 100 mV
= ±⎜⎜ 3.0 × 10 − 5 +
V
V (V ) ⎟⎠
⎝
(B-10)
δV
⎛
7.0 × 10−6 ⎞
⎟ in the range of 1 V
= ±⎜⎜ 3.0 × 10 − 5 +
V
V (V ) ⎟⎠
⎝
(B-11)
δV
(B-12)
⎛
5.0 × 10−5 ⎞
⎟ in the range of 10 V
= ±⎜⎜ 3.0 × 10 − 5 +
V
V (V ) ⎟⎠
⎝
145
B.2.3 Current I
Current is obtained by measuring the voltage across the current shunt, so the
uncertainty of current depends on the voltage reading of shunt; the uncertainty of voltage
measurement and the uncertainty of shunt. The uncertainty of current is in the form as
following, which is the result from Teertstra [96]
(B-13)
1/ 2
2
⎡⎛
⎤
3.5 × 10− 6 ⎞
−5
⎟⎟ + 0.00142 ⎥
= ± ⎢⎜⎜ 3.0 × 10 +
I
Vshunt (V ) ⎠
⎢⎣⎝
⎥⎦
δI
B.2.4 Pressure P
The air pressure in the bell jar is measured using a Varian CeramiCel pressure gauge,
which senses the pressure and outputs a voltage signal. The uncertainty of the pressure
depends on the uncertainty of the Varian CeramiCel pressure gauge and the uncertainty
of voltage measurement. According to Teertstra [96], the uncertainty of pressure has the
following form:
2
2
⎡⎛
−5 ⎞ ⎤
⎞ ⎛
×
1
5
10
−
5
⎟ + ⎜ 3 × 10 +
⎟ ⎥
= ± ⎢⎜ 5
P
V pressure (V ) ⎟⎠ ⎥
⎢⎜⎝ 10 V pressure (V ) ⎟⎠ ⎜⎝
⎣
⎦
δP
146
1/ 2
(B-14)
B.2.5 Dimension
The dimensions in the experiment are measured by an electronic digital caliper with a
resolution of ± 0.005 mm. The uncertainties of the diameter, the length and the surface
area of SMA wire are in the following forms:
δd
d
δl
l
0.005
d (mm)
(B-15)
0.005
l (mm)
(B-16)
=±
=±
⎡⎛ 0.005 ⎞ 2 ⎛ 0.005 ⎞ 2 ⎤
⎟⎟ ⎥
⎟⎟ + ⎜⎜
= ± ⎢⎜⎜
A
⎢⎣⎝ d (mm) ⎠ ⎝ l (mm) ⎠ ⎥⎦
δA
(B-17)
1/ 2
B.2 The Inclination Angle φ
The inclination angle is measured using a plastic protractor which has an uncertainty of
± 0.50. The uncertainty of angle is:
(B-18)
δϕ
0 .5 0
=± 0
ϕ
ϕ( )
B.3 Uncertainty in Calculated Values
Qrad is the power input in vacuum: Qrad = VIvacuum, so the uncertainty of Qrad is:
147
δQrad
Qrad
⎡⎛ δV
= ⎢⎜⎜ SMA−vacuum
⎢⎣⎝ VSMA−vacuum
2
⎞ ⎛ δI SMA−vacuum
⎟⎟ + ⎜⎜
⎠ ⎝ I SMA−vacuum
⎞
⎟⎟
⎠
2
⎤
⎥
⎥⎦
(B-19)
1/ 2
When modeling heat transfer, the Nusselt number (NuD) and the Rayleigh number
(RaD) are calculated by Equation 5-6 and 5-9, so the uncertainty of NuD and RaD are
calculated by the following Equation B-20 and B-21.
2
2
2
⎡⎛ δV ⎞ 2 ⎛ δI ⎞ 2 ⎛ δV
⎞ ⎛ δI SMA − vacuum ⎞ ⎛ δd SMA ⎞ ⎤
SMA
SMA
SMA − vacuum
⎟⎟ + ⎜⎜
⎟⎟ + ⎜⎜
⎟⎟ + ⎜⎜
⎟⎟ + ⎜⎜
⎟⎟ ⎥
⎢⎜⎜
δNu D ⎢⎝ VSMA ⎠ ⎝ I SMA ⎠ ⎝ VSMA − vacuum ⎠ ⎝ I SMA − vacuum ⎠ ⎝ d SMA ⎠ ⎥
=
⎥
Nu D ⎢⎢ ⎛ δA ⎞ 2 ⎛ δT ⎞ 2 ⎛ δT ⎞ 2
⎥
SMA
SMA
∞
⎟⎟ + ⎜⎜
⎟⎟
⎢+ ⎜⎜ A ⎟⎟ + ⎜⎜ T
⎥
⎣ ⎝ SMA ⎠ ⎝ SMA ⎠ ⎝ T∞ ⎠
⎦
1/ 2
(B-20)
δRaD
RaD
⎡⎛ δP ⎞ 2 ⎛ δd ⎞ 2 ⎛ δT ⎞ 2 ⎛ δT ⎞ 2 ⎛ δT ⎞ 2 ⎛ δT ⎞ 2 ⎤
= ⎢⎜ 2 ⎟ + ⎜ 3 ⎟ + ⎜⎜ SMA ⎟⎟ + ⎜⎜ ∞ ⎟⎟ + ⎜⎜ 2 m ⎟⎟ + ⎜⎜ ∞ ⎟⎟ ⎥
⎢⎣⎝ P ⎠ ⎝ d ⎠ ⎝ TSMA ⎠ ⎝ T∞ ⎠ ⎝ Tm ⎠ ⎝ T∞ ⎠ ⎥⎦
1/ 2
(B-21)
⎡⎛ δT ⎞ 2 ⎛ δT ⎞ 2 ⎤
TSMA + T∞
δTm
= ± ⎢⎜⎜ SMA ⎟⎟ + ⎜⎜ ∞ ⎟⎟ ⎥
, then
Because Tm =
2
Tm
⎢⎣⎝ TSMA ⎠ ⎝ T∞ ⎠ ⎥⎦
1/ 2
, so Equation B-21 is
actually in the form of:
δRaD
RaD
2
⎡⎛ δP ⎞ 2 ⎛ δd ⎞ 2 ⎛ δT ⎞ 2
⎛ δT∞ ⎞ ⎤
SMA
⎟⎟ + 6⎜⎜
⎟⎟ ⎥
= ⎢⎜ 2 ⎟ + ⎜ 3 ⎟ + 5⎜⎜
⎢⎣⎝ P ⎠ ⎝ d ⎠
⎝ T∞ ⎠ ⎥⎦
⎝ TSMA ⎠
148
1/ 2
(B-22)
B.4 Uncertainty of Experimental Result
The uncertainties of the Nusselt number (NuD) and the Rayleigh number (RaD) can be
calculated by the Equation B-20 and B-22. The raw data, the results of NuD, RaD and
their uncertainties are tabulated in Table 6-1 to Table 6-7.
149
Appendix C
Conduction Wire Loss Via TC and Voltage Measurement Leads
Conduction wire loss through the thermocouple and the voltage measurement leads can
be estimated by adding a thermocouple onto the testing SMA wire and comparing the
power inputs in two cases at the same temperature.
Figure C-1 is the setup diagram to estimate the conduction wire losses via the
thermocouple and the voltage measurement leads. The SMA wire test section between
two voltage measurement leads in Figure C-1 has a unique temperature when in heat
equilibrium because it is 2.5 cm away from the terminal rings and it is 15 cm away from
the power leads.
The experiment to estimate conductive wire losses needs to run two times: one with
only one thermocouple TC1; the other one with two thermocouples TC1 and TC2. The
experiment steps are as follows:
1. Setup the experiment as in Figure C-1
2. Spot welded the thermocouple “TC1” at the centre of the testing SMA wire (no
TC2 at this time
3. Place the setup into the bell jar
4. Evacuate the bell jar by the mechanical pump and the diffusion pump until the
pressure is below 10-7 torr
5. Turn on the power to heat the SMA to 100 0C
150
6. After the TC1 is stable at 100 0C for 1 minute, increase the pressure in the bell
jar to 0.01 torr.
7. Adjust the current knob on the power supply to keep TC1 at 100 0C
8. After the TC1 is stable at 100 0C for 1 minute, increase the pressure in the bell
jar to the pressure 33 torr, 66 torr, 100 torr, 200 torr, …,760 torr as in step 6
9. At each pressure, adjust the current knob on the power supply to bring TC1 to
the temperature 100 0C, when the temperature 100 0C is stable for 1 minute, as
in step 7
10. Spot weld TC2 in the same manner as TC1 at the location as shown in Figure
C-1
11. repeat above steps from 3 to 9
12. compare the power inputs in the two cases
151
Figure C-1 SMA Wire Setup Diagram
The power inputs of the two cases in vacuum are compared in Figure C-2. A same
power input of 0.0924 W is required in both cases to keep the SMA wire at 100 0C. This
means that adding a second thermocouple doesn’t increase the power input in order to
keep the SMA wire test section at the same temperature 100 0C in vacuum. In other
words, the conductive wire loss through thermocouple is very small and can be neglected
in vacuum. Because the voltage measurement leads use the same size and type wire as
thermocouple, the wire loss through the voltage measurement leads can also be neglected.
152
Power Input Comparison of 1TC & 2TCs in Vacuum
0.115
Two thermocouples
Power Input (W )
0.11
One thermocouple
0.105
0.1
0.095
0.09
0.085
0.08
1
164 327 490
653 816 979 1142 1305 1468 1631 1794 1957
Time (5s )
Figure C-2 Comparison of Power Input of 1TC and 2TCs in Vacuum
The power inputs in both cases of one thermocouple and two thermocouples are plotted
in Figure C-3. It shows that the power input for one thermocouple agree with two
thermocouples at all pressures, so adding a thermocouple onto the testing SMA wire
doesn’t need to increase the power input to keep the SMA wire test section at the same
temperature 100 0C at all pressures.
This experiment concludes that the conductive wire losses via spot welded E-type 40
AWG thermocouple and constantan, 40 AWG voltage measurement leads are very small
and can be neglected at all pressures in the experiment.
153
Power Input of 1TC & 2TCs via Pressure
1.5
Power Input (W )
1.4
1.3
1.2
1.1
1
0.9
0.8
One thermocouple
0.7
Two thermocouples
0.6
0
200
400
600
Pressure (torr )
Figure C-3 Comparison of Power Input of 1TC and 2TCs at Pressures
154
800
References
[1] Otsuka, K., Wayman, CM., Shape Memory Materials, Cambridge, Cambridge University
Press, 1998
[2] Kurdjumov, G. V., Khandros, L. G., “First Reports Of The Thermoelastic Behaviour Of The
Martensitic Phase Of Au-Cd Alloys”, Doklady Akademii Nauk, SSSR 66, 1949, pp 211–213
[3] Buehler, W. J., Gilfrich, J. V., Wiley, R. C., “Effects of low-temperature phase changes on the
mechanical properties of alloys near composition TiNi”, Journal of Applied Physics, 34,
1963, p 1475
[4] Schetky, L., “Shape-Memory Alloys”, Scientific American, 241 pp 74-82
[5] Wayman, M., Harrison, J., “The Origins Of The Shape Memory Effect”, Journal of Minerals,
Metals, and Materials, 41 (99), 1989, pp 26–28
[6] Wu, M. H., Schetky, L. M., “Industrial Applications For Shape Memory Alloys”, Proceedings
of the International Conference on Shape Memory and Superelastic Technologies, Pacific
Grove, California, 2000
[7] Doonkersloot, H. C., Vucht, V., “Martensitic Transformations in Au-Ti, Pd-Ti and Pt-Ti
Alloys”, Journal of Less-Common Metals, 20, 1970, pp 83–91
[8] Miyazaki, S., Mizukoshi, K., Ueki, T., Sakuma, T., Liu, Y., “Fatigue Life of Ti-50”, Science
and Engineering, 1999, pp 658–663
[9] Oulu University - http://herkules.oulu.fi/isbn9514252217/html/x317.html
[10] Sanders, B., Crowe, R., Garcia, E., “Defense Advanced Research Projects Agency – Smart
Materials And Structures Demonstration Program Overview”, Journal of Intelligent
Material Systems and Structures, 15, 2004, pp 227–233
[11] Birman, V., “Review Of Mechanics Of Shape Memory Alloy Structures”, Applied
Mechanics Reviews, 50 (11), 1997, pp 629–645
[12] Johnson, A., “Non-explosive separation device”, U.S. Patent 5,119,555, June, 1992
[13] Otsuka, K., Kakeshita, T., “Science And Technology Of Shape-Memory Alloys”, New
developments, bulletin, February, 2002
[14] Childs, Peter R. N., Practical Temperature Measurement, Butterworth-Heinemann, Oxford,
London, 2001, p 7
[15] http://www.omega.com/prodinfo/temperaturemeasurement.html
155
[16] Steur, P.P.M., “The Interpolating Constant-volume Gas Thermometer And Thermal
Anchoring”, Metrologia, 36, 1999, pp 33-39
[17] Steur, P.P.M., Pavese, F., “He-3 Constant Volume Gas Thermometer As Interpolating
Instrument: Calculations Of The Accuracy Limit Versus Temperature Range And Design
Parameters”, Cryogenics, 29, 1989, pp 135-138
[18] Edsinger, R.E., Schooley, J.F., “Differences Between Thermodynamics Temperature And
(IPTS-68) In The Range 2300C to 6600C”, Metrologia, 26, 1989, pp 95-106
[19] Michalski, L., Eckersdorf, K., Kucharski, J., Temperature Measurement, John Wiley & Sons
Ltd, London, 1991, p 333
[20] Robertson, D., United States Patent, 3834237, 1972
[21] Nakabeppu, O., Suzuki, T., “Microscale Temperature Measurement By Scanning Thermal
Microscopy”, Journal of Thermal Analysis and Calorimetry, Vol. 69, 2002, pp 727–737
[22] Ishihara, A., Tang, K.C., Brewster, M.Q., “Temperature Measurement of a Burning Surface
by a Thermocouple II”, American Institute of Aeronautics and Astronautics, 2005-3576, 41st
AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Tucson, Arizona, 10 - 13
July, 2005,
[23] Shaukatullah, H., Claassen, A., “Effect of Thermocouple Wire Size and Attachment Method
on Measurement of Thermal Characteristics of Electronic Packages”, Nineteenth Annual
IEEE
Semiconductor
Thermal
Measurement
and
Management
Symposium,
Cat.
No.03CH37437, 2003, pp 97-105
[24] Renken, W., United States Patent, 5746513, 1997
[25] Leath, C.W., United States Patent, 3476910, 1968
[26] Anon, C., “Thermocouple Attachment Method”, IBM Technical Disclosure Bulletin, v 27, n
10A, March, 1985, p 5534
[27] Sobolik, K.B., Keltner, N.R., Beck, J.V., “Measurement Errors For Thermocouples Attached
to Thin Plates. Application to Heat Flux Measurement Devices”, American Society of
Mechanical Engineers, Heat Transfer Division, (Publication) HTD, v 112, 1989, p 15-22
[28] Dunstan, P. S., Kennon, N. F., Middleton, L. A., Dunne, D. P., “Thermal Characteristics of a
Nitinol Heat Engine”, Journal of Materials Science, 21, 1986, pp 1637-1641
[29] Volkov, S.D., “The Use of Welded Thermocouples on Current Carrying Surfaces”,
http://www.laboratory.ru/articl/tech/at150e.htm (accessed Nov 30, 2005)
156
[30] Zanstra, P.E., “Welding Uniform Sized Thermocouple Junctions From Thin Wires”, Journal
of Physics E (Scientific Instruments), v 9, n 7, July, 1976, pp 526-528
[31] Wang, T. P., Martincavage, J., Bediones, D., “Precision Calibration of Thermocouples &
RTD'S and a New Calibration Laboratory with Computerized Data Acquisition System”,
ISA, 1987, pp 107-128
[32] Maeno, Y., Haucke, H., Wheatley, J., “Simple Differential Thermometer Using a
Thermocouple with a SQUID Detector”, AIP Conference Proceedings, n 103, 1983, p 467
[33] Sawada, T., Nishiwaki, N., “Response of a Thermocouple to Transient Temperature Changes
in a Metal to Which it is Attached”, International Journal of Mechanical Sciences, v 33, n 7,
1991, pp 551-561
[34] Suyama, Y., Miyazato, M., Hamada, T., “An Evaluation Test on Calibration Accuracy of
Type R Thermocouples by Comparison with a Platinum Resistance Thermometer”,
Transactions of the Society of Instrument and Control Engineers, v 33, n 4, April, 1997, pp
302-304
[35] Ancsin, J., “A Study of Thermocouple Stability, Reproducibility and Accuracy (Pt vs. Pt-Rh
and Pt vs. Au)”, Metrologia, v 28, n 4, November. 1991, pp 339-347
[36] Cengel, Y. A., Zing, P. T. L., Kalinski, M. J., “Use of Operational Amplifiers in
Temperature Measurements with Thermocouples for Increased Accuracy and Resolution”
American Society of Mechanical Engineers, Fluids Engineering Division (Publication)
FED, v 44, 1986, pp 87-91
[37] Rego, G., Santos, L.M.N.B.F., Schroder, B., Marques, P.V.S., Santos, J.L., Salgado, H.M.,
“In Situ Temperature Measurement of An Optical Fiber Submitted To Electric Arc
Discharges”, Photonics Technology Letters, IEEE, Volume16, Issue 9, September, 2004, pp
2111-2113
[38] Childs, Peter R. N., Practical Temperature Measurement, Butterworth-Heinemann, Oxford,
London, 2001, p146
[39] Nicholas, J.V., White, D.R., Traceable Temperatures, 2nd Edition, John Wiley & Sons,
England, 2001, p 207
[40] Nicholas, J.V., White, D.R., Traceable Temperatures, 2nd Edition, John Wiley & Sons,
England, 2001, p 250
[41] http://www.thermistor.com/pdf/qtmb.pdf
157
[42] http://sensing.honeywell.com/index.cfm?ci_id=140301&la_id=1&pr_id=145382
[43] Allison, S.W., Gillies, G.T., “Remote Thermometry With Thermographic Phosphors:
Instrumentation and Applications”, Review of Scientific Instruments, 68(7), 1997, pp 26152650
[44] Bradley, L.C., Review of Scientific Instruments, 24, 1953, p 219
[45] Czysz, P., Dixin, W.P., “Thermographic Heat Transfer Measurement”, Instruments and
Control Systems, 41, 1968, pp 71-76
[46] Czysz, P., Dixin, W.P., “Quantitative Heat Transfer Measurement Using Thermographic
Phosphors”, SPIE Journal, 1969, pp 77-79
[47] Tobin, K.W., Allison, S.W., Cates, M.R., Capss, G.J., Beshears, D.L., Cyr, M., Boel, B.W.,
“High-Temperature Phosphor Thermometry Of Rotating Turbine Blades”, AIAA Journal,
28(8), 1990, pp 1485-1490
[48] Noel, B.W., Borella, H.M., Lewis, W., Turley, W.D., Beshears, D.L., Capps, G.J., Gates,
G.J., Cates, M.R., Muhs, J.D., Tobin, K.W., “Evaluating Thermographic Phosphors In An
Operating Turbine Engine”, Journal of Engineering for Gas Turbines and Power, 113, 1991,
pp 242-245
[49] Ervin, J., Murawski, C., Macarthur, C., Chyu, M., Bizzak, D., “Temperature Measurement of
A Curved Surface Using Thermographic Phosphors”, Experimental Thermal and Fluid
Science, 11(4), 1995, pp 387-394
[50] Edge, A.C., Laufer, G., Krauss, R.H., “Surface Temperature-field Imaging Withlaserinduced Thermographic Phosphorescence”, Appl. Optics, 39(4), 2000, pp 546-553
[51] Kusama, H., Jpn. Appl. Phys., 15, 1976, p 2349
[52] Dever, M., Bugos, A., Dyer, F., Cates, M., Tobin, K., Beshears, D., Capps, G.,
“Measurement Of The Surface Of Textiles During Microwave Drying Using A
Thermographic Phosphor”, Journal of Microwave Power and Electromagnetic Energy, 25,
1990, pp 230-235
[53] Cambell, R.P., Molezzi, M.J., “Application Of Advanced Liquid Crystal Video
Thermography To Turbine Cooling Passage Heat Transfer Measurement”, ASME Paper 96GT-225, 1996
[54] Lee, S.J., Yoon, J.H., “Temperature Field Measurement Of Heated Ventilation Flow In A
Vehicle Interior”, International Journal of Vehicle Design, 19, 1998, pp 228-243
158
[55] Simonich, J.C., Moffat R.J., “Liquid Crystal Visualization Of Surface Heat Transfer On A
Concavely Curved Turbulent Boundary Layer”, Journal of Engineering for Gas Turbines
and Power, 106, 1984, pp 619-627
[56] Ireland, P.T., Jones, T.V., “The Response Time Of A Surface Thermometer Employing
Encapsulated Thermochromic Liquid Crystals”, Journal of Physics E., 20, 1987, pp 11951199
[57] Childs, Peter R. N., Practical Temperature Measurement, Butterworth-Heinemann, Oxford,
London, 2001, p248
[58] Nicholas, J.V., White, D.R., Traceable Temperatures, 2nd Edition, John Wiley & Sons,
England, 2001, p 367
[59] http://www.flirthermography.com/media/S60_datasheet.pdf
[60]http://www.flirthermography.com/english/accessories/accessory/1231/accessory_category_id
/1035/
[61] http://www.exergen.com/industrl/spotlight/wiretemp/index.htm
[62] Borca-Tasciuc, T., Chen G., “Temperature Measurement Of Fine Wires By Photothermal
Radiometry”, Rev Sci. Instrum., Vol. 68, No. 11, November, 1997, pp 4080-4083
[63] Moron, C., Aroca, C., SAnchez, M.C., Lbpez, E., Sanchez, P., “Local Temperature In
Amorphous Ribbons During Current Annealing”, IEEE Transactions on Magnetics, Vol. 30,
No. I , January, 1994, pp 53-63
[64] Shimizu, Y., Ishii, J., Baba, T., “Reflectance Thermometry for Microscale Metal Thin
Films”, Japanese Journal of Applied Physics, Vol. 46, No. 5A, 2007, pp 3117–3119
[65] Iadicola, M. A., Shaw, J. A., “The Effect of Uniaxial Cyclic Deformation on the Evolution of
Phase Transformation Fronts in Pseudoelastic NiTi Wire”, Journal of Intelligent Material
Systems and Structures, Vol. 13, February/March, 2002, pp 143-155
[66] Iadicola, M. A., Shaw, J. A., “An Experimental Setup for Measuring Unstable Thermomechanical Behavior of Shape Memory Alloy Wire”, Journal of Intelligent Material
Systems and Structures, Vol. 13, February/March, 2002, pp 157-166
[67] Chang, B. C., Iadicola, M. A., Shaw, J. A., “Thermodynamics of Shape Memory Alloy Wire:
Modeling, Experiments, and Application”, Continuum Mech. Thermodyn., 18, 2006, pp 83–
118
159
[68] Jaluria, Y., Natural Convection Heat and Mass Transfer, Pergamon, Oxford, U.K., 1980, p
87
[69] Tsubouchi, T., Masuda, H., “Heat Transfer By Natural Convection From Horizontal
Cylinders at Low Rayleigh Numbers”, Sci. Rep. Res. Inst., Tohoku University, Ser. B, 19,
1967 – 1968, pp 205-219
[70] Morgan. V. T., Adv. Heat Transfer, 11, 1975, p 199
[71] Churchill. W., Chu, H. H. S., Int. J. Heat Mass Transfer, 19, 1976, p 1127
[72] Raithby, G. D., Hollands, K. G. T., Trans. ASME, Ser. C, J. Heat Mass Transfer, 98, 1976, p
72
[73] Elenbaas, W., “The Dissipation of Heat by Free Convection-Horizontal and Vertical
Cylinders”, Physica’s Grav., vol. 9, 1942, pp 665–672
[74] Yang, S.M., “General Correlating Equations for Free Convection Heat Transfer from a
Vertical Cylinder”, Proc. Int. Symposium on Heat Transfer, Tsinghua University, Peking,
1985, pp 153–159
[75] Koch, W., “Uber Die Warmeabgaba Geheizter Rohre Bei Veerschiedener Neigung Der
Rohrachse”, Gesundh Ing., Beih., Ser. I, No. 22, 1927, pp 1-29
[76] Mueller, A. C., “Heat Transfer from Wires to Air in Parallel Flow”, Trans. Amer. Inst.
Chem. Eng. 38, 1942, pp 613-627
[77] Zaitsev, V. A., Sokovishin, Yu. A., Heat and Mass Transfer-VI, vol. 1, part 3, Minsk, 1980,
pp 82–86
[78] Oosthuizen, P. H., J. Heat Transfer 98, 1976b, p 672
[79] Çengel, Y. A., Heat Transfer: A Practical Approach, McGraw-Hill, 2nd edition, p 563
[80] Siegel R., Howell, J.R., Thermal Radiation Heat Transfer, 4th edition, Taylor & Francis,
2002, p 16
[81] Çengel, Y. A., Heat Transfer: A Practical Approach, McGraw-Hill, 2nd edition, p 567
[82] Childs, Peter R. N., Practical Temperature Measurement, Butterworth-Heinemann, Oxford,
London, 2001, p255
[83] Childs, Peter R. N., Practical Temperature Measurement, Butterworth-Heinemann, Oxford,
London, 2001, p279
160
[84] Dutton, R. and Lee, E.C., “Surface Temperature Measurement of Current Carrying Objects”,
Journal, Instrument Society of America, vol 6 no 12, December, 1959
[85] Mulford, S.F., “Compensated Thermocouple” US Patent no. 2,612,779, October 7, 1952
[86] Kuribayashi, K., “Improvement of the Response of an SMA Actuator using a Temperature
Sensor”, Int. J. Robot., Res.10, 1991, pp 13–20
[87] http://www.dynalloy.com/TechData.html (accessed Aug 28 2009), Dynalloy, Inc. “Flexinol
Technical and Design Data”,
[88] Croarkin, M., Guthrie, W., Burns, G., Kaeser, M., and Strouse, G., “TemperatureElectromotive Force Reference Functions and Tables for the Letter-Designated
Thermocouple Types Based on the ITS-90”, Natl. Inst. Stand. Technol. Monograph 175;
1993, p 630 (available online at http://srdata.nist.gov/its90/main/, accessed Aug 28 2009)
[89] Gorbet, R.B., Wang, D.W.L, Morris, K.A., “Preisach Model Identification of a 2-Wire SMA
Actuator,” Proceedings of the 1998 ICRA, 1998, pp 2161-2167
[90] Çengel, Y. A., Heat Transfer: A Practical Approach, McGraw-Hill, 2nd edition, p 466
[91] Martynenko, O. G., Khramtsov, P. P., Free-Convective Heat Transfer, Springer-Verlag
Berlin Heidelberg, 2005, pp 229-230
[92] Martynenko, O. G., Khramtsov, P. P., Free-Convective Heat Transfer, Springer-Verlag
Berlin Heidelberg, 2005, p 244
[93] Çengel, Y. A., Heat Transfer: A Practical Approach, McGraw-Hill, 2nd edition, p 874
[94] Thermocouple Measurement in a Current Carrying Path, GM Ref. No. P005402, OLG Ref.
No. GM1834PUS
[95] Wheeler, A.J., Ganji, A.R., Introduction To Engineering Experimentation, Prentice Hall,
Inc., 1996, pp 159 - 161
[96] Teertstra, P., “Models and Experiments for Laminar Natural Convection from Heated Bodies
in Enclosures,” Department of Mechanical Engineering, University of Waterloo, Waterloo,
Ontario, Canada, 2003, pp 192 - 200
161
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