Optimized Thermoelectric Module-Heat Sink ... - Tufts Digital Library

Optimized Thermoelectric Module-Heat SinkAssemblies for Precision Temperature Control

Submitted byRui Zhang

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTERS OF SCIENCE IN MECHANICAL ENGINEERING

School of EngineeringTufts University

Medford, Massachusetts

Aug. 2011

Author: Certified By:Rui Zhang Associate Professor Marc HodesDepartment of Mechanical Engineering Department of Mechanical EngineeringTufts University Tufts University

Committee: Committee:Professor Vincent P. Manno Ross WilcoxonDepartment of Mechanical Engineering Rockwell Collins, Inc.Tufts University Cedar Rapids, IA

ABSTRACT

Optimized Thermoelectric Module-Heat Sink Assemblies for Precision Temperature Control

by

Rui Zhang

Chair: Marc Hodes

Robust precision temperature control of heat-dissipating photonics components is achieved

by mounting them on thermoelectric modules (TEMs) which are in turn mounted on heat

sinks. However, the power consumption of such TEMs is high. Indeed, it may exceed that

of the component. This problem is exacerbated when the ambient temperature surrounding

a TEM and/or component heat load that it accommodates vary. In the usual packaging

configuration a TEM is mounted on an air-cooled heat sink of specified thermal resistance.

However, heat sinks of negligible thermal resistance minimize TEM power for sufficiently

high ambient temperatures and/or heat loads. Conversely, a relatively high thermal resis-

tance heat sink minimizes TEM power for sufficiently low ambient temperature and heat

load. In the problem considered, total footprint of thermoelectric material in a TEM, ther-

moelectric material properties, component operating temperature, relevant component-side

thermal resistances and ambient temperature range are prescribed. Moreover, the minimum

and maximum rates of heat dissipation by the component are zero and a prescribed value,

respectively. Provided is an algorithm to compute the combination of the height of the pel-

lets in a TEM and the thermal resistance of the heat sink attached to it which minimizes the

maximum sum of the component and TEM powers for permissible operating conditions. It

ii

is further shown that the maximum value of this sum asymptotically decreases as the total

footprint of thermoelectric material in a TEM increases. Implementation of the algorithm

maximizes the fraction of the power budget in an optoelectronics circuit pack available for

other uses. Use of the algorithm is demonstrated through an example for a typical set of

conditions.

iii

Acknowledgements

This thesis would not have been possible without the guidance and the help of several

individuals who in one way or another contributed and extended their valuable assistance in

the development and completion of this research.

First and foremost, my utmost gratitude to my advisor, Dr. Marc Hodes, who has

supported me thoughout my thesis with his patience, encouragement and knowledge whilst

allowing me the room to work in my own way for the last two years. Without him, this

thesis would not have been completed or written. His ideas and passions in science inspire

and enrich my growth as a student. One could not wish for a better or friendlier advisor

and I am indebted to him more than he knows.

I gratefully acknowledge my thesis committee members, Professor Vincent Manno and

Ross Wilcoxon for their advice, insight, encouragement and assistance. Special thank to

Ross for offering me a visit of the thermal lab at Rockwell Collins which was incredibly

valuable to me.

Many thanks to Dave Brooks, a fellow graduate student, and Martin Cleary, a post-

doctoral fellow, for their generous help and collaboration in the completion of this thesis

and the design and construction of the thermocouple calibration apparatus. Thanks are also

extended to Drew Mills, a fellow graduate student, for his help on LabVIEW and Latex and

Gennady Ziskind, a visiting professor, for his help and insight in apparatus design.

Finally, I would like to thank the Wittich Energy Sustainability Research Initiation Fund

for supporting this research.

iv

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

CHAPTER

I. Introduction And Background . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Thermoelectric Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thermoelectric Modules . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

II. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Expressions for General Case . . . . . . . . . . . . . . . . . 92.1.2 Expressions for Simplified Case . . . . . . . . . . . . . . . . 11

2.2 Prescribed, Independent and Dependent Variables . . . . . . . . . . 112.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 The case when qcp = qcp,max . . . . . . . . . . . . . . . . . . 132.3.2 The case when qcp = 0 . . . . . . . . . . . . . . . . . . . . 182.3.3 Role of φAsub . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Dependence of TEM Operating Mode on T∞, H and Ru−∞ . . . . . 21

III. Implementation of Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 25

v

3.1 Fixed Pellet Height . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Arbitrary Pellet Height . . . . . . . . . . . . . . . . . . . . . . . . . 27

IV. Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 33

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vi

LIST OF FIGURES

Figure

1.1 Cutaway view of a single stage TEM [1]. . . . . . . . . . . . . . . . . . . . 3

2.1 Schematic of a TEM embedded in a thermal resistance network. . . . . . . 8

2.2 Ru−∞,max as a function of H at T∞,min and T∞,max. . . . . . . . . . . . . . 14

2.3 Wt as a function of H at T∞,min and T∞,max when Ru−∞ = 0 and Ru−∞,max. 14

2.4 Wt and Wt,ml as a function of H at T∞,min and T∞,max when (T∞,max − Tcp)is small when Ru−∞ = 0 and Ru−∞,max. . . . . . . . . . . . . . . . . . . . . 16

2.5 Wt and Wt,ml as a function of H at T∞,min and T∞,max when (T∞,max − Tcp)is large when Ru−∞ = 0 and Ru−∞,max. . . . . . . . . . . . . . . . . . . . . 17

2.6 Wt as a function of H at T∞,max when qcp = 0 and qcp,max when Ru−∞ = 0and Ru−∞,max. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Wt as a function of H at T∞,min when qcp = 0 and qcp,max when Ru−∞ = 0and Ru−∞,max. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 Wt,mg as a function (φAsub)1/2. . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9 Delineation of TEM operating modes as a function of H and Ru−∞ whenT∞ = T∞,min < Tcp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.10 Delineation of TEM operating modes as a function of H and Ru−∞ whenT∞ = T∞,min < Tcp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 WTEM as a function of Ru−∞ (Tcp = 55 C). . . . . . . . . . . . . . . . . . 27

3.2 qcp,TEM,max as a function of H at T∞,min and T∞,max for Ru−∞ = 0. . . . . 28

vii

3.3 Ru−∞,max as a function H at T∞,min and T∞,max. . . . . . . . . . . . . . . . 29

3.4 Wt and Wt,ml as a function of H at Tcp = 55 C when Ru−∞ = 0 and Ru−∞,max. 29

3.5 Wt as a function of Ru−∞ at Hmg at T∞,min and T∞,max. . . . . . . . . . . . 30

3.6 Wt,mg as a function of (φAsub)1/2. . . . . . . . . . . . . . . . . . . . . . . . 30

A.1 Mathematica Codes for Eqs. 2.12 - 2.15 . . . . . . . . . . . . . . . . . . . . 38

B.1 Calculation Part1 - Page 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40







B.8 Calculation Part1 - . . . . . . . . . . . . . . . . . . . . . . . . . . . 47











viii



C.1 Type-T thermocouple measurement circuit. . . . . . . . . . . . . . . . . . . 64

C.2 Tigtech 116 SRL Thermocouple welder. . . . . . . . . . . . . . . . . . . . . 66

C.3 Handheld thermal wire strippers. . . . . . . . . . . . . . . . . . . . . . . . 66

C.4 Magnifying lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

C.5 Schematic of thermocouple calibration rig. . . . . . . . . . . . . . . . . . . 69

C.6 Copper annulus assembly design for measurement junction . . . . . . . . . 71

C.7 Teflon cover drawings for reference junction (Dimensions are in inches). . . 72

C.8 Teflon cover drawings for measurement junction (Dimensions are in inches). 73

C.9 Copper enclosure drawing (Dimensions are in inches). . . . . . . . . . . . . 74

C.10 Copper bottom cover drawing (Dimensions are in inches). . . . . . . . . . . 75

C.11 Copper rod drawing for reference junction (Dimensions are in inches). . . . 76

C.12 Copper rod drawing for measurement junction (Dimensions are in inches). . 77

C.13 Fluke 5610 thermistor probe. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

C.14 Black stack overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

C.15 2182A Nanovoltmeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C.16 2182A Nanovoltmeter voltage specifications. . . . . . . . . . . . . . . . . . 80

C.17 7001 Switch system front panel. . . . . . . . . . . . . . . . . . . . . . . . . 81

C.18 7001 Switch system back panel. . . . . . . . . . . . . . . . . . . . . . . . . 81

C.19 7011-S Switch card with thermocouples attached. . . . . . . . . . . . . . . 81

C.20 LabView VI front panel for thermocouple calibration. . . . . . . . . . . . . 83

ix

C.21 LabView VI block diagram for thermocouple calibration. . . . . . . . . . . 84

C.22 Thermocouple Junctions Attached to the Copper Rod Example. . . . . . . 84

C.23 Calibration readouts of thermocouples and thermistors. . . . . . . . . . . . 86

C.24 Calibration results after reference junction compensation. . . . . . . . . . . 87

C.25 Experimental data, Least-squares Fit data and data from reference tables(ITS-90) for thermocouple 1, 11, 21 and 31 . . . . . . . . . . . . . . . . . . 88

C.26 Comparison of experimental data, least-squares fit data and that in referencerables for thermocouple 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89




C.30 Absolute errors versus temperatures in voltage for thermocouple 1. . . . . . 91

C.31 Absolute errors versus temperatures in temperature for thermocouple 1. . . 91

C.32 Calibration report of reference thermistor probe (Serial Number: B072717)page 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93






x




C.41 Calibration report of measurement thermistor probe (Serial Number: B072719)page 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102








C.49 Calibration report of Black Stack page 1. . . . . . . . . . . . . . . . . . . . 110



C.52 Mathematica Code for Least-squares Fit and Reference Tables page 1. . . . 113



xi

LIST OF APPENDICES

Appendix

A. Mathematica Codes for Eqs. 2.12 - 2.15 . . . . . . . . . . . . . . . . . . . . . 37

B. Mathematica Codes for Example Calculation . . . . . . . . . . . . . . . . . . 39

C. Thermocouple Welding and Calibration . . . . . . . . . . . . . . . . . . . . . 63

xii

Nomenclature

AP Pellet cross-sectional area [m2]

Asub Footprint of TEM substrates [m2]

k Thermal conductivity [W/(mK)]

K Thermal conductance [W/C]

H Pellet height [m]

H1 H corresponding to Region I-II transition [m]

H2 H corresponding to Region II-III transition [m]

Hl Lower limit of H [m]

Hu Upper limit of H [m]

I Current [A]

Ki−j Thermal conductance from node i to node j [W/C]

N Number of thermocouples

q Heat transfer rate [W]

R Ohmic resistance [Ω]

Rec−ρ Electrical contact resistivity [Ωm2]

Rec−R Electrical contact resistance [Ω]

Ri−j Thermal resistance from node i to node j [C/W]

T Temperature [K]

V Voltage [V]

W Electrical power [W]

Greek Symbols

α Seebeck coefficient [V/K]

xiii

φ Pellet packing density

Φ Current flux [A/m2]

ρ Electrical resistivity [Ωm]

Subscripts

∞ Ambient

c Controlled side

cp Control plane

cp-c Control plane to controlled side

cp-∞ Control plane to ambient

csi Controlled-side interface

max Maximum

mg Global minimum maximum quantity

min Minimum

ml Local minimum maximum quantity

n n-type

open Open circuit mode

p Peltier effect

p p-type or pellet

pe Quantity when efficiency maximized

p, n p-n (as in difference in Seebeck coefficients)

sh Short circuit mode

u Uncontrolled side

u-∞ Uncontrolled side to ambient

TEM Thermoelectric module

t Total power consumption

xiv

CHAPTER I

Introduction And Background

1.1 Thermoelectric Effects

Thermal energy may be reversibly converted into electrical energy and vice versa in

electrically conducting materials by three thermoelectric effects [2]. The Seebeck effect occurs

when a conductor subjected to a temperature gradient generates an electric potential gradient

under open circuit conditions. It is described by

dV = αdT (1.1)

where V is voltage, α is the Seebeck coefficient of a conductor (or semiconductor) and T

is temperature. The Seebeck coefficient may be positive or negative and depends upon the

scattering properties of a conductor.

When current, I, flows through the interface between conductors, heat is absorbed or

rejected by the Peltier effect due to varying levels of electrical energy associated with current

flow in different conductors. The rate of heat absorption, qp, is

qp = I(αB − αA)T (1.2)

when the direction of current (i.e., flow of positive charge carriers) is from conductor A to

1

conductor B.

When current flows through a conducting material in the presence of a temperature gra-

dient, heat is also absorbed or evolved due to the Thomson (bulk) effect [3]. The irreversible

effects of bulk and interfacial Ohmic heating and heat conduction must also be considered

in the analysis of thermoelectric circuits.

1.2 Thermoelectric Modules

Thermoelectric modules (TEMs) are solid-state devices well suited to the precision tem-

perature control of photonics components through Peltier cooling and heating. They possess

no moving parts; therefore, they are reliable and free of maintenance. Additionally, TEMs

are lightweight, silent and moderately priced. A shortcoming of them, however, is their

low thermodynamic efficiency compared to alternative technologies. This is largely a conse-

quence of large currents being driven through Bi2Te3-based semiconducting materials with

high electrical resistivity and modest thermal conductivity. Moreover, TEMs operate at

high currents and low voltages and accompanying DC to DC power conversion losses can be

significant.

A cutaway view of a single-stage TEM is shown in Fig. 1.1. Adjacent pairs of negatively-

doped (n-type) and positively-doped (p-type) semiconductor pellets are interconnected elec-

trically in series and thermally in parallel and are attached to ceramic substrates. Each pair

of n- and p-type pellets is referred to as a thermocouple and there are N thermocouples

in a TEM. The substrates to which a photonics component and heat sink are attached to

are referred to as the controlled-side and uncontrolled-side substrates, respectively [3]. The

interfaces between the semiconducting pellets and conductors forming the interconnects on

the respective sides of the TEM are referred to as the controlled-side interface (csi) and

uncontrolled-side interface (usi). The Peltier effect essentially occurs at these interfaces. A

bipolar DC power supply is connected to the external leads attached to the uncontrolled side

of a TEM operating in precision temperature control mode as shown in Fig. 1.1. Current di-

2

Figure 1.1: Cutaway view of a single stage TEM [1].

rection is defined as positive when positive charge carriers flow from n-type to p-type pellets

at the csi. Moreover, because αn and αp are negative and positive quantities, respectively,

heat is absorbed by the Peltier effect at the csi and is released by it at the usi for positive

currents. This is referred to as Peltier cooling mode. Analogously, when current is negative,

the Peltier effect causes heat to be absorbed at the usi and released at the csi. This is re-

ferred to as Peltier heating mode. Generation mode, i.e., when a TEM supplies power to an

electric load for the purpose of generating power is not of interest in this analysis. Rather,

it is shown that it may be exploited to eliminate TEM power consumption altogether in

certain situations.

1.3 Problem Statement

Especially when the maximum ambient temperature surrounding a TEM and/or heat

load imposed on it are large, it may consume as much or more power than the photonics

component mounted to it. The usual configuration where an essentially constant thermal

resistance air-cooled heat sink is attached to the uncontrolled-side substrate of a TEM ex-

3

acerbates this problem. Indeed, a heat sink of negligible thermal resistance minimizes TEM

power for sufficiently high ambient temperature and/or heat load, but a relatively high

thermal resistance one minimizes it for sufficiently low ambient temperature and heat load.

Optimized TEM-heat sink assemblies reduce the severity of this problem.

The primary purpose of this thesis is provision of an algorithm to compute the combina-

tion of the height of the pellets in a TEM and the thermal resistance of a heat sink attached

to it which minimizes the maximum sum of the component and TEM powers for permissible

operating conditions. In the problem considered, total footprint of thermoelectric material

in a TEM, thermoelectric material properties, component operating temperature, relevant

component-side thermal resistances and ambient temperature range are prescribed. The

minimum ambient temperature is assumed to be less than or equal to the component oper-

ating temperature. Moreover, the minimum and maximum rates of heat dissipation by the

component are zero and a prescribed value, respectively.1 (When a fraction of the power

supplied to a photonics component is converted to light rather than heat this should be

considered.) Further insight into the design problem is provided by considering the impact

of increasing the total footprint of thermoelectric material on decreasing the minimum max-

imum sum of component and TEM powers. Finally, the TEM operating modes relevant to

the optimization are discussed in some depth.

1.4 Previous Work

Hodes [4] presented a means to compute the optimal value of heat sink thermal resistance

for the problem at hand when all other parameters are prescribed. Elsewhere, the premise

that the ability to adjust heat sink thermal resistance is beneficial because as the ambient

temperature changes different values of heat sink thermal resistance minimize TEM power

has been addressed. This has been discussed in the context of TEM-finned variable conduc-

1The latter two assumptions imply TEM operation in refrigeration mode is necessary at some, if not all,permissible operating conditions.

4

tance heat pipe assemblies [5] and moving shrouds which slide along a heat sink to vary the

number of fins exposed to air [6]. Moreover, Wilcoxon et al. [7] proposed utilizing a liquid

metal substrate to separate the uncontrolled side of a TEM from a low thermal resistance

heat sink. Then the dominant thermal resistance between the uncontrolled-side interface

and ambient is the caloric thermal resistance of the liquid metal, which may be controlled

by varying its mass flow rate. Another approach is to implement a variable speed fan to

vary the thermal resistance of a conventional air-cooled heat sink. However, this is typically

not viable as the airflow supplied to optoelectronic circuit packs is provided by shelf-level

fans running at a fixed speed. Finally, it is noted that the height of the pellets which mini-

mizes the power of a TEM operating in refrigeration mode is relevant and calculable for an

arbitrary heat sink thermal resistance [8]. However, this is not necessarily the optimal pellet

height for the problem at hand as discussed in the analysis section below.

1.5 Outline of Thesis

The governing equations for the optimization problem are presented next. Then, the

algorithm is developed and, where appropriate, TEM operating modes are discussed. Lastly,

application of the algorithm is illustrated by an example for a typical set of operating con-

ditions. Further power savings achievable by increasing the total footprint of thermoelectric

material in a TEM are quantified in the example.

5

CHAPTER II

Methodology

2.1 Governing Equations

Assuming one-dimensional flow of charge and heat a schematic of a photonics compo-

nent mounted to a TEM embedded in a thermal resistance network is shown in Figure 2.1.

The control plane (denoted by the subscript cp) represents the component, which is mod-

eled as a planar heat source that dissipates heat at the rate of qcp (0 ≤ qcp ≤ qcp,max)

and must be maintained at a temperature of Tcp. Ambient, controlled-side interface and

uncontrolled-side interface temperatures are denoted by T∞, Tc and Tu, respectively. Mini-

mum and maximum ambient temperatures are denoted by T∞,min and T∞,max, respectively,

and T∞,min ≤ T∞ ≤ T∞,max. The thermal resistances between the ambient and control plane,

control plane and controlled-side interface, and uncontrolled-side interface and ambient are

denoted by Rcp−∞, Rcp−c and Ru−∞, respectively. Constriction and spreading resistances in

the controlled- and uncontrolled-side substrates, respectively, are accounted for in Rcp−c and

Ru−∞. The heat sink resistance is included in Ru−∞.

Surface energy balances at the control plane, csi and usi yield, respectively [8]

qcp = Kcp−c (Tcp − Tc) +Kcp−∞ (Tcp − T∞) (2.1)

Kcp−c (Tcp − Tc) =φAsub2Ap

[Iαp,nTc −K (Tu − Tc)−

I2 (R +Rec−R)

2

](2.2)

Ku−∞ (Tu − T∞) =φAsub2Ap

[Iαp,nTu −K (Tu − Tc) +

I2 (R +Rec−R)

2

], (2.3)

7

Figure 2.1: Schematic of a TEM embedded in a thermal resistance network.

where αp,n = αp−αn and thermal conductances (Ki−j) are the inverse of thermal resistances.

The pellet packing density(φ), bulk electrical resistance of a thermocouple (R), thermal

conductance of a thermocouple (K) and electrical contact resistance associated with the

four pellet-interconnect interfaces in a thermocouple (Rec−R) are, respectively

φ =2NApAsub

(2.4)

R =2ρH

Ap(2.5)

K =2kApH

(2.6)

Rec−R =4Rec−R

Ap, (2.7)

where ρ,H, k and AP are the electrical resistivity, height, thermal conductivity and cross-

sectional area of a pellet, respectively, and Rec−ρ is the electrical contact resistivity at the

8

interface between a pellet and an interconnect. Finally, Asub is the footprint of a TEM such

that φAsub is the total footprint of thermoelectric material.

Based on the definitions of φ, R, K and Rec−ρ and , the surface energy balances at the

csi and usi become, respectively,

Kcp−c (Tcp − Tc) = φAsub

[Φαp,nTc

2− k (Tu − Tc)

H− Φ2

(ρH

2+Rec−ρ

)](2.8)

Ku−∞ (Tu − T∞) = φAsub

[Φαp,nTu

2− k (Tu − Tc)

H+ Φ2

(ρH

2+Rec−ρ

)](2.9)

where the flux of current through the pellets (Φ) is

Φ =I

Ap. (2.10)

The electrical power consumed by a TEM, WTEM , is [8]

WTEM =φAsubΦ

2[αp,n (Tu − Tc) + Φ (2ρH + 4Rec−ρ)] . (2.11)

WTEM is positive when DC power is supplied to a TEM and negative when it operates in

generation mode.

Equations 2.8, 2.9 and 2.11 show that the current flux through the pellets in a TEM

is the relevant variable rather than the individual values of I and AP . Based on Eq. 2.1,

Eqs. 2.8 - 2.9 and Eq. 2.11, expressions for qcp and WTEM for a general case and a simplified

case are provided as follows. The Mathematica codes for these expressions are provided in

Appendix A.

2.1.1 Expressions for General Case

Tc and Tu are computed as a function of Φ, H and Ku−∞ from the csi and usi surface

energy balances (Eqs 2.8 and 2.9). Then, as per Eqs 2.1 and 2.11, expressions for qcp and

9

WTEM , respectively, which are independent of Tc and Tu are

qcp = Kcp−∞ (−T∞ + Tcp)−AsubKcp−cφ

[HΦ2 (2Rec−ρ +Hρ)

(2Ku−∞ − Asubαp,nφΦ)

+ 4k(Ku−∞T∞ + Asub (2Rec−ρ +Hρ)φΦ2

)− (4kKu−∞ +Hαp,nΦ (2Ku−∞ − Asubαp,nφΦ))Tcp] /

[4Asubk (Kcp−c +Ku−∞)φ

+ H (2Ku−∞ − Asubαp,nφΦ) (2Kcp−c + Asubαp,nφΦ)] (2.12)

WTEM = AsubφΦ [8Asubk (Kcp−c +Ku−∞)Rec−ρφΦ

+H2ρΦ (4Kcp−cKu−∞ − AsubKcp−cαp,nφΦ + AsubKu−∞αp,nφΦ)

+H(AsubKu−∞φΦ

(T∞α

2p,n + 4kρ+ 2Rec−ραp,nΦ

)+ 2Kcp−c(Ku−∞(T∞αp,n + 4Rec−ρΦ)

+ AsubφΦ (2kρ−Rec−ραp,nΦ)))

− HKcp−cαp,n (2Ku−∞ − Asubαp,nφΦ)Tcp] /

[4Asubk (Kcp−c +Ku−∞)φ

+ H (2Ku−∞ − Asubαp,nφΦ) (2Kcp−c + Asubαp,nφΦ)] . (2.13)

10

2.1.2 Expressions for Simplified Case

Assuming that Ku−∞ → ∞ implies that Tu = T∞. Then the surface energy balance at

the usi is unnecessary and qcp and WTEM are, respectively1

qcp = −2T∞ [HKcp−cKcp−∞ + Asubk (Kcp−c +Kcp−∞)φ]

+ AsubHKcp−∞T∞αp,nφΦ

+ AsubHKcp−cφΦ2 (2Rec−ρ +Hρ)

− [2HKcp−cKcp−∞ + 2Asubk (Kcp−c +Kcp−∞)φ

+ AsubH (Kcp−c +Kcp−∞)αp,nφΦ] /

(2HKcp−c + 2Asubkφ+ AsubHαp,nφΦ) (2.14)

WTEM = AsubφΦ [2HKcp−cT∞αp,n + (4HKcp−c (2Rec−ρ +Hρ)

+ Asub(8KRec−ρ +HT∞α

2p,n + 4Hkρ

)φ)

Φ

+ AsubHαp,n (2Rec−ρ + hρ)φΦ2 − 2HKcp−cαp,nTcp]/

(4HKcp−c + 4Asubkφ+ 2AsubHαp,nφΦ) (2.15)

2.2 Prescribed, Independent and Dependent Variables

The relevant properties of the pellets, i.e., ρ, k, αn (a negative quantity), αp (a positive

quantity assumed equal in magnitude to αn) and the electrical contact resistivity at the

interconnects in a TEM, Rec−ρ, are prescribed constants. Tcp, Kcp−∞, Kcp−c, and φAsub

are also prescribed constants.2 A range of ambient temperatures, subject to the constraint

that T∞,min ≤ T∞ ≤ T∞,max is specified. This constraint, a typical, albeit not universal,

condition for photonics components utilized in telecommunications circuit packs, implies

1These expressions are necessary to determine the permissible range of H and H2 and Hpe in the nextsection.

2The benefit of increasing φAsub is considered below.

11

that the optimal value of Ru−∞ = 0 when T∞ = T∞,max. It is noted that (T∞,max−Tcp) may

not exceed a maximum value as discussed below. A procedure to compute the minimum

value of Tcp and thus the maximum value of (T∞,max − Tcp) when all variables aside from

H are prescribed is given by Hodes [8]. (See Section V.A.2 of this reference). It should be

invoked for the limiting case when Ru−∞ = 0 and qcp = qcp,max. If the result is that T∞,max is

too large, the limiting case when Ru−∞ = 0 and qcp = 0 may be checked as this corresponds

to a TEM of infinite footprint. If the result is favorable then the required TEM footprint

should be computed. Otherwise, the problem at hand may not be accommodated by the

prescribed thermoelectric material properties and/or component-side thermal resistances.

Finally, qcp ranges from zero to a prescribed maximum value, qcp,max. However, total power

consumption Wt, that is (WTEM + qcp), decreases when qcp drops below qcp,max. Therefore,

only qcp,max need be considered in the algorithm as discussed in the next section.

Pellet height (H) and the thermal resistance between the uncontrolled-side interface and

ambient (Ru−∞) are the independent variables optimized to minimize the maximum total

power consumption (Wt). It is noted that qcp,max is a prescribed constant and minimizing

Wt is equivalent to minimizing WTEM .3 For the sake of generality we assume that H and

Ru−∞ may assume arbitrary values larger than 0. (The optimal H does not approach 0 in

practice. However, unless H is constrained to be a sub-optimal value, the optimal value of

Ru−∞ equals 0 and it should be made as low as possible.)

The current flux through a TEM (Φ), the controlled-side interface temperature (Tc) and

the uncontrolled-side interface temperature (Tu) are dependent variables that assume the

necessary values to maintain Tcp over the range of ambient temperatures and heat loads for

permissible combinations of H and Ru−∞. WTEM , itself a dependent variable, follows from

Eq. 2.11 once the other dependent variables have been computed. Recall, the objective here

is minimization of the maximum value of WTEM (or, equivalently, Wt) over all permissible

combinations of qcp and T∞. The corresponding value of WTEM or Wt is subsequently denoted

3The ordinate of (most of) the log-log plots below is Wt rather than WTEM . Otherwise, negative valuesof WTEM arising when generation mode is encountered may not be displayed.

12

by the subscript “ml” or “mg” as are the corresponding values of H and Ru−∞,ml or those of

Hmg and Ru−∞,mg. The subscripts “ml” and “mg” denote local minimums (at a prescribed

H) and global minimums (for all H), respectively. Permissible combinations of H and Ru−∞

are determined by the existence of an H and Ru−∞-dependent maximum (and positive) value

of the temperature difference between the ambient and the control plane as described above.

2.3 Analysis

2.3.1 The case when qcp = qcp,max

The range of permissible pellet heights (Hl ≤ H ≤ Hu) in a TEM is determined by

setting qcp = qcp,max and T∞ = T∞,max. At the minimum and maximum pellet heights, a

TEM operates in Peltier cooling mode as a refrigerator; therefore, the corresponding usi-to-

ambient thermal resistance equals zero. (Smaller values of Hl and larger values of Hu are

permitted when T∞ < T∞,max and/or qcp < qcp,max).

Hl and Hu are determined following the (numerical) procedure given by Hodes [8]. First,

differentiating the expression for qcp when Ru−∞ = 0 (Eq. 2.14) with respect to Φ and setting

the result equal to zero determines a numerical function for the current flux (Φmax (H))

which maximizes qcp as a function of H. The corresponding values of qcp are denoted by

qcp,TEM,max(H). Next, setting qcp,TEM,max (H) = qcp,max, Hl and Hu may be computed. The

maximum value of Ru−∞, for Hl ≤ H ≤ Hu is denoted by Ru−∞,max (H) and, moreover,

it is finite for Hl < H < Hu. It is determined by differentiating qcp with respect to Φ

and defining a numerical function Φmax (Ru−∞, H) as the corresponding root. Then, by

inserting this result into the expression for qcp and setting qcp = qcp,max, Ru−∞,max (H) may

be computed. Representative results are shown in Fig. 2.2. for conditions corresponding to

T∞,min and T∞,max. Smaller values of Ru−∞,max (H), are permissible when T∞ = T∞,max and

thus dictate the maximum value of Ru−∞,max for a given H.

When T∞ = T∞,max, qcp = qcp,max and Hl (T∞,max) ≤ H ≤ Hu (T∞,max), Wt corresponding

13

T∞, max

, qcp, max

T∞, min

, qcp, max

Impermissible Region

H

Hl(T

∞, max) H

u(T

∞, max)

0

Hu(T

∞, min)

Ru

-∞,

max

Figure 2.2: Ru−∞,max as a function of H at T∞,min and T∞,max.

T∞, max

, qcp, max

, Ru-∞, max

(T∞, max

)

T∞, min

, qcp, max

, Ru-∞, max

(T∞, max

)

T∞, max

, qcp, max

, Ru-∞

= 0

T∞, min

, qcp, max

, Ru-∞

= 0

Wt

H2

Hl(T

∞, max) H

u(T

∞, max)H

pe

H

H1

Hu(T

∞, min)

qcp,max

Region I Region II Region III

Figure 2.3: Wt as a function of H at T∞,min and T∞,max when Ru−∞ = 0 and Ru−∞,max.

14

to Ru−∞ = Ru−∞,max (T∞,max) and Ru−∞ = 0 as a function of H is given by the solid and

dashed blue curves, respectively, in Fig. 2.3. Clearly, Wt is minimized when Ru−∞ = 0.

Hpe, the pellet height which minimizes WTEM in Peltier cooling mode when Ru−∞ = 0 is

denoted by the solid orange circle. A (numerical) procedure to compute it (for arbitrary

values of Ru−∞) is available [8]. The corresponding curves when T∞ = T∞,min are shown in

red. When T∞ = T∞,min and T∞,min < Tcp, the height of the pellets may be made arbitrarily

small when Ru−∞ = 0 as per the dashed red curve. However, when T∞ = T∞,min and Ru−∞

= 0, Hu (T∞,min) may not be made arbitrarily large as for a sufficiently high H, a TEM

operates in Peltier cooling mode.

H1 and H2, along with Hl (T∞,max) and Hu (T∞,max), divide the permissible range of H

into three different regions as per Figs. 2.4 and 2.5.4 The inset plots in Figs. 2.4 and 2.5 are

of Wt versus Ru−∞ for a representative H within the various regions delineated. The inset

plots for Region I are semilog plots due to the large disparity between Ru−∞,max(T∞,max) and

Ru−∞,max(T∞,min), but those for Region II and III are linear. Moreover, the blue and red

curves on each inset plot span the permissible range of Ru−∞ from 0 to Ru−∞,max(T∞,max)

and from 0 to Ru−∞,max(T∞,min).

Region I applies for Hl (T∞,max) ≤ H < H1. Within it, either Wt is higher for T∞,min

for all permissible values of Ru−∞ as per Fig. 2.4 (see inset plot) or Wtot is higher for

T∞,max for all permissible values of Ru−∞ as per Fig. 2.5 (see inset plot). The former case

is relevant when (T∞,max − Tcp) is sufficiently small. Then, H1 is computed by utilizing

Eq. 2.13 to set WTEM (H,Φ (H,Ru−∞,max (H)) , Ru−∞,max (H)) when T∞ = T∞,max equal to

that when T∞ = T∞,min. In this case a numerical function for Φ (H,Ru−∞,max (H)) follows

from Eq. 2.12. In the latter case H1 is computed by setting WTEM (H,Φ (H)) for Ru−∞ = 0

when T∞ = T∞,max equal to that when T∞ = T∞,min utilizing Eq. 2.15. In this case Φ (H)

is a numerical function that follows from Eq. 2.14 when qcp = qcp,max. (Since Eq. 2.14 is

quadratic in current, the smaller of the two roots is the one of interest as it corresponds to

4Fig. 2.4 is identical to Fig. 2.3, but contains more information.

15

a lower TEM power.)

T∞, max

, qcp, max

, Ru-∞, max

(T∞, max

)

T∞, min

, qcp, max

, Ru-∞, max

(T∞, max

)

T∞, max

, qcp, max

, Ru-∞

= 0

T∞, min

, qcp, max

, Ru-∞

= 0

Wt, ml

Wt

H2

Hl(T

∞, max) H

u(T

∞, max)H

pe

H

H1

Hu(T

∞, min)

qcp,max

T∞,max

T∞,min

Ru-∞

Ru-∞

Wt

Wt


Region I

Region II

qcp,max

T∞,max

T∞,min

Ru-∞

Wt

Region III

qcp,max

T∞,max

T∞,min

Figure 2.4: Wt and Wt,ml as a function of H at T∞,min and T∞,max when (T∞,max − Tcp) issmall when Ru−∞ = 0 and Ru−∞,max.

Region II applies for H1 < H < H2. The distinguishing feature of Region II is that

for sufficiently low Ru−∞, Wt is larger for when T∞ = T∞,min, but the converse is true for

sufficiently high Ru−∞ as per the inset plots in Figs. 2.4 and 2.5. H2 is computed by setting

WTEM (H,Φ (H) , T∞,min)=WTEM(H,Φ (H) , T∞,max) when Ru−∞ = 0 as per Eq. 2.15 and

selecting the larger root.5

Region III applies for H2 < H ≤ Hu (T∞,max). Within it, Wt is largest when T∞ = T∞,max

for all permissible values of Ru−∞ as per Figs. 2.4 and 2.5 (see inset plots). As H increases

beyond H2, less power is required at T∞,min and for sufficiently high H a TEM must operate

in generation mode. In practice this may be accomplished by applying a variable (electrical)

load resistance across the external leads of a TEM or with the necessary configuration of a

DC power supply. Beyond generation mode, Peltier cooling is required at T∞,min.

5When (T∞,max − Tcp) is sufficiently large, H1 and H2 are the smaller and larger roots, respectively, for

which WTEM (H,Φ (H) , T∞,min) = WTEM (H,Φ (H) , T∞,max) when Ru−∞ = 0 as per Fig. 2.5.

16

T∞, max

, qcp, max

, Ru-∞, max

(T∞, max

)

T∞, min

, qcp, max

, Ru-∞, max

(T∞, max

)

T∞, max

, qcp, max

, Ru-∞

= 0

T∞, min

, qcp, max

, Ru-∞

= 0

Wt, ml

H2

Hl(T

∞, max) H

u(T

∞, max)H

peH

u(T

∞, min)

Wt

qcp, max

H1

H

T∞,max

T∞,min T

∞,max

T∞,min

Ru-∞

Ru-∞

Wt

Wt

qcp,max

qcp,max

Wt

Ru-∞

Region I

Region I

Region II Region III

T∞,max

T∞,min

Region II

Region III

Figure 2.5: Wt and Wt,ml as a function of H at T∞,min and T∞,max when (T∞,max − Tcp) islarge when Ru−∞ = 0 and Ru−∞,max.

The minimum value of the maximum Wt as a function of H (Wt,ml) is indicated by the

(dashed) light blue curves in Figs. 2.4 and 2.5. In general, for a given H, Ru−∞ is set equal to

the value which minimizes the maximum value of Wt to plot this curve. The characteristics

of the inset plot of Wt versus Ru−∞ in Fig. 2.4 for Region I apply for sufficiently small

(T∞,max − Tcp). First, Wt is larger when T∞ = T∞,min than that when T∞ = T∞,max for all

permissible values of Ru−∞. Secondly, Wt when T∞ = T∞,min decreases monotonically with

Ru−∞. Therefore, the largest permissible value of Ru−∞, i.e., Ru−∞,max (H), equals Ru−∞,ml

and corresponds to Wt,mg, the local minimum of the maximum value of Wt. Analogously, the

characteristics of the inset plot of Wt versusRu−∞ in Fig. 2.5 for Region I apply for sufficiently

large (T∞,max − Tcp). First, Wt is larger when T∞ = T∞,max than when T∞ = T∞,min for

all permissible values of Ru−∞. Secondly, Wt increases monotonically with Ru−∞; therefore,

Ru−∞,ml = 0.

When H lies within Region II, i.e., H1 < H < H2, regardless of the value of (T∞,max−Tcp),

the optimal value of Ru−∞ (i.e., Ru−∞,ml) lies between 0 and Ru−∞,max (H). This may be

17

discerned from the inset plots in Figs. 2.4 and 2.5. They show that such an intermediate

value of Ru−∞ for which WTEM (H,Φ (H) , T∞,min) = WTEM (H,Φ (H) , T∞,max) corresponds

to Wt,ml. To compute Ru−∞,ml at a prescribed H, first qcp is set equal to qcp,max and Eq. 2.12 is

used to compute numerical functions for Φ (Ru−∞) when T∞ equals T∞,min and T∞,max. Then,

Eq. 2.15 is used to set WTEM (Φ (Ru−∞) , Ru−∞, T∞,min) = WTEM (Φ (Ru−∞) , Ru−∞, T∞,max)

and Ru−∞,ml is numerically computed. Finally, when H lies within Region III, i.e., H2 <

H ≤ Hu (T∞,max), Wtot is largest when T∞ = T∞,max; therefore Ru−∞,ml is 0.

Ideally, the designer of TEM-heat sink assemblies has the ability to select H in addition

to Ru−∞. Then, a global optimum must be found and the corresponding values of Ru−∞

and Wt are denoted by Ru−∞,mg and Wt,mg, respectively. When (T∞,max−Tcp) is sufficiently

small, Hmg = H2 and Ru−∞,mg = 0 represent this optimum as per Fig. 2.4. Otherwise, Ru−∞

remains 0, but Hmg = Hpe represent this optimum as per Fig. 2.5. In general, Hpe can be

smaller, equal to or larger than H2, but, unless they are equal the larger of the two along

with Ru−∞ = 0 represent the global optimum and this is the key result of the present work.

Since Hmg may equal H2 or Hpe, determining the optimum H is more difficult than when a

TEM operates exclusively in refrigeration mode and it equals Hpe a priori [8].

2.3.2 The case when qcp = 0

Total power consumption decreases when qcp decreases as described above. Comparisons

of Wt versus H for qcp = 0 and qcp = qcp,max are provided in Figs. 2.6 - 2.7 to illustrate this

point.

When T∞ = T∞,max, a TEM operates in refrigeration mode regardless of the value of qcp

as per Fig. 2.6. Therefore, reducing qcp reduces WTEM and thus Wtot for all H and Ru−∞.

The plot of Wt versus H for Hl ≤ H ≤ Hu illustrates this point. It is noted that the range

of H for qcp = 0 is larger than that for qcp = qcp,max.

The corresponding plot when T∞ = T∞,min rather than T∞ = T∞,max is shown in Fig. 2.7.

Wt is lower when qcp = 0 than when qcp = qcp,max for both Ru−∞= 0 and Ru−∞ = Ru−∞,max

18

for all H. Moreover, when qcp = qcp,max (or any finite qcp) a TEM may operate in power

generation mode for certain combinations of H and Ru−∞ as the value of Wt is below that

of qcp,max as per the inset plot.

T∞,max

, qcp,max

, Ru-∞,max

(T∞,max

)

T∞,max

, qcp,max

, Ru-∞

= 0

T∞,max

, qcp

= 0 , Ru-∞,max

(T∞,max

)

T∞,max

, qcp

= 0 , Ru-∞

= 0

Hl(q

cp,max) H

u(q

cp,max)H

l(q

cp= 0) H

u(q

cp= 0)

Wt

H

qcp,max

Figure 2.6: Wt as a function of H at T∞,max when qcp = 0 and qcp,max when Ru−∞ = 0 andRu−∞,max.

T∞, min

, qcp, max

, Ru-∞, max

(T∞, max

)

T∞, min

, qcp, max

, Ru-∞

= 0

T∞, min

, qcp

= 0 , Ru-∞, max

(T∞, max

)

T∞, min

, qcp

= 0 , Ru-∞

= 0

Hl(q

cp,max) H

u(q

cp,max)H

l(q

cp= 0) H

u(q

cp= 0)

Wt

qcp,max

H

Figure 2.7: Wt as a function of H at T∞,min when qcp = 0 and qcp,max when Ru−∞ = 0 andRu−∞,max.

19

2.3.3 Role of φAsub

Assuming cross sections of pellets in a TEM are square, the global minimum maximum

total power (Wt,mg) is plotted versus the side length of a square TEM occupied by its pellets,

(φAsub)1/2, in Fig. 2.8. It decreases with (φAsub)

1/2 because the heat flux imposed upon a

TEM decreases with this parameter and Wt at the optimum H corresponds to refrigeration

mode at T∞,max and qcp,max. Indeed, it is well known that decreasing the heat flux imposed

on a TEM increases its coefficient of performance [8, 9]. The benefit of increasing φAsub

becomes negligible once heat flux has been sufficiently reduced as per Fig. 2.8.

Global Minimum Maximum Total Power

ΦAsub

1/2

Wt,

mg

Figure 2.8: Wt,mg as a function (φAsub)1/2.

2.4 Algorithm

The optimization algorithm consists of the following steps as described in the previous

section.

1. Prescription of Constants. Prescribe ρ, k, αn, αp, Rec−ρ,Tcp, T∞,min, T∞,max, qcp,max,

Kcp−∞, Kcp−c and φAsub.

20

2. Determine the permissible Range of H. Use the smallest range as determined by T∞,max

and qcp,max.

3. Compute Ru−∞,max (H). Use the smallest value at a given H as determined by T∞,max

and qcp,max.

4. Plot Wt versus H as in Figs. 2.4 and 2.5. Delineate Regions I-III in the permissible

range of H and whether H2, if it exists, or Hpe corresponds to the global optimal total

power.

5. Determining the Optimal Combination of H and Ru−∞. The larger of H2 or Hpe is the

global optimal (Hmg) along with Ru−∞ = 0 based on the plot in Step 4. If Step 4 is

skipped, both H2 and Hpe must be computed and compared to determine which one

corresponds to the global optimal total power (Wt,mg). However, it is suggested that

Wt versus H be plotted as per Step 4 rather than proceeding directly to Step 5. This is

because although Hpe or H2 has been shown to be the optimum value of H in all cases

considered in this thesis, proof that this is universally true for arbitrary combinations

of the prescribed constants is not obvious.

6. Plot Wt,mg versus (φAsub)1/2. If the value of (φAsub)

1/2 used in Steps 1-5 is below that

corresponding to saturation, increase it to one nearer saturation to the extent that

packaging constraints permit.

2.5 Dependence of TEM Operating Mode on T∞, H and Ru−∞

Dependent upon the values of T∞, H and Ru−∞, a TEM operates in one of 5 different

modes. Peltier cooling, Peltier heating and generation modes were defined above. Open

circuit and short circuit modes correspond to disconnected and shorted external leads of

a TEM.6 The uncontrolled-side to ambient thermal resistances corresponding to them are

6See Hodes [3] for further discussion on TEM operating modes, albeit using not an entirely consistent setof definitions.

21

denoted by Ru−∞,open (H) and Ru−∞,sh (H), respectively.

The case discussed here assumes T∞,min ≤ Tcp ≤ T∞,max. In Fig. 2.9, Ru−∞,open (H),

Ru−∞,sh (H) and Ru−∞,max are plotted versus H for T∞,min. When T∞ = T∞,min and for

sufficiently low H, as Ru−∞ is increased from 0 to Ru−∞,max (H) the operating modes tra-

versed are Peltier heating, open circuit, generation, short circuit and Peltier cooling modes,

successively.

Peltier heating is required when H and Ru−∞ are sufficiently low in order to maintain Tcp

above T∞,min as per the region under the blue curve in Fig. 2.9. When Ru−∞ = Ru−∞,open (H),

the conduction resistance of the TEM is precisely that required to maintain the component

at its operating temperature. When Ru−∞,open < Ru−∞ < Ru−∞,sh, a TEM must operate

in generation mode. Essentially as Ru−∞ increases from Ru−∞,open (H) to Ru−∞,sh (H) the

requisite electrical resistance of a load across a TEM decreases from∞ to 0. It is noted that

Peltier cooling occurs at the csi when a TEM operates in generation mode and, moreover, the

current and thus the rate of Peltier cooling increases as Ru−∞ increases from Ru−∞,open (H)

to Ru−∞,sh (H). Therefore, larger values of Ru−∞ may be accommodated as load resistance

decreases. When Ru−∞,sh < Ru−∞ ≤ Ru−∞,max, a TEM must operate in cooling mode to

overcome the insulative nature of the attached heat sink. Finally, when Ru−∞ > Ru−∞,max,

operation is impermissible as the insulation of the attached heat sink surpasses the Peltier

cooling ability of a TEM.

In the case of T∞,max, Ru−∞,open (H) and Ru−∞,sh (H) do not exist, and Ru−∞,max is

plotted versus H in Fig. 2.10. For an arbitrary H, a TEM operates in Peltier cooling mode

at T∞,max as T∞,max > Tcp. Note that, when H is sufficiently high, only cooling mode is

permissible regardless of the value of T∞ as per Figs. 2.9 and 2.10.

22

Ru-∞,max

at T∞,min

Ru-∞,open

at T∞,min

Ru-∞,sh

at T∞,min

Peltier Cooling Mode

Generation Mode

Peltier Heating Mode

Ru

-∞

H


Open Circuit Mode

Short Circuit Mode

Figure 2.9: Delineation of TEM operating modes as a function of H and Ru−∞ when T∞ =T∞,min < Tcp.

Ru-∞,max

at T∞,max

Peltier Cooling Mode


Ru

-∞

H0

Hl

Hu

Figure 2.10: Delineation of TEM operating modes as a function of H and Ru−∞ when T∞= T∞,min < Tcp.

23

CHAPTER III

Implementation of Algorithm

Examples are provided to illustrate the use of the optimization algorithm and its implica-

tions and the corresponding Mathematica codes are provided in Appendix B. Thermoelectric

material properties used are those typical of Bi2Te3-based TEMs operating near 300 K such

that the Seebeck coefficient of a thermocouple (αp,n) is 4.0 × 10−4 V/K, pellet thermal con-

ductivity (k) is 1.5 W/(mK) and pellet electrical resistivity (ρ) is 1.25 × 10−5 Ωm [10]. The

electrical contact resistivity at the interconnects (Rec−ρ) is 1 × 10−9 Ωm2, representative of a

typical TEM [11]. The footprint of the TEM (Asub) is 30 mm × 30 mm. The corresponding

total footprint of thermoelectric material in the TEM (φAsub) is 278.3 mm2.1 The maximum

rate of heat dissipation of the component mounted to the control plane is qcp,max = 10 W

and its operating temperature is Tcp = 55 C. The prescribed ambient temperature range

is T∞,min = -5 C ≤ T∞ ≤ T∞,max = 65 C. The conductance between the control plane

and csi (Kcp−c) is set equal to 17.17 W/C. This is based on a typical thermal contact resis-

tance between a component and TEM substrate with three-dimensional conduction through

an alumina substrate [8]. The conductance between the control plane and controlled-side

ambient (Kcp−∞) is set equal to 0.00015 W/C as the control plane is assumed to be well

insulated from the ambient. A list of the prescribed variables is provided in Table 3.1. In the

first set of calculations pellet height is assumed to equal 1 mm, a pellet height in Region II,

and usi-to-ambient thermal resistance is optimized. Then pellet height and usi-to-ambient

1The number of the thermocouples (N) affects the operating voltage and spreading/constriction resis-tances in the substrates; however, none of these paremeters are required to perform the optimization.

25

thermal resistance are simultaneously optimized. Finally, the effect of increasing φAsub is

considered.

Table 3.1: List of prescribed variables.

Prescribed Variables Prescribed Values

αp,n 4.0 × 10−4 V/K

k 1.5 W/(mK)

ρ 1.25 × 10−5 Ωm

Rec−ρ 1 × 10−9 Ωm2

Asub 30 mm × 30 mm

φAsub 278.3 mm2

qcp,max 10 W

Tcp 55 C

T∞,min -5 C

T∞,max 65 C

Kcp−c 17.17 W/C

Kcp−∞ 0.00015 W/C

3.1 Fixed Pellet Height

When H = 1 mm, TEM power is plotted versus the usi-to-ambient thermal resistance

(Ru−∞) when T∞ equals T∞,min and T∞,max, in Fig. 3.1.2 WTEM increases with Ru−∞ when

T∞ = T∞,max as the TEM operates in Peltier cooling mode. Ru−∞,max (T∞,max) = 1.07

C/W and Ru−∞ = 0 C/W minimizes WTEM when T∞ = T∞,max. When T∞ = T∞,min, the

TEM maintains the control plane temperature substantially (i.e., 60 C) above the ambient

temperature. Then Ru−∞,max (T∞,min) = 5.44 C/W as, the lower the ambient temperature,

the higher the value of Ru−∞,max. The permissible range of Ru−∞ is 0 C/W to 1.07 C/W,

which corresponds to Ru−∞,max when T∞ = 65 C. It is noted that when T∞ = -5 C, a

2TEM power (WTEM ) rather than total power (Wt) is plotted to clearly show the range of Ru−∞ forwhich the TEM operates in generation mode.

26

TEM operates in Peltier heating mode when 0 C/W ≤ Ru−∞ <3.65 C/W, in open circuit

mode at Ru−∞ = 3.65 C/W, in generation mode when 3.65 C/W < Ru−∞ < 4.68 C/W,

in short circuit mode at Ru−∞ = 4.68 C/W and in Peltier cooling mode when 4.68 C/W

< Ru−∞ ≤ 5.36 C/W. The minimum maximum value of WTEM occurs when Ru−∞ = 0.17

C/W as per Fig. 3.1 and corresponds to 4.25 W.

-5

0

5

10

15

20

25

0 1 2 3 4 5 6

T∞,max

= 65 °C

T∞,min

= -5 °C

WT

EM

[W]

Ru-∞

[°C/W]

-0.1

-0.05

0

0.05

0.1

3.62 3.99 4.35 4.71

Ru-∞,max

(65 °C) Ru-∞,max

(- 5°C)

Ru-∞,open

Ru-∞,sh

Figure 3.1: WTEM as a function of Ru−∞ (Tcp = 55 C).

3.2 Arbitrary Pellet Height

When H is not prescribed, Step 2 of the algorithm is followed to compute the permissible

range of H. The maximum heat rate that may be dissipated from the control plane is plotted

versus TEM pellet height when Ru−∞ = 0 and T∞ = T∞,max (blue curve) and T∞,min (red

curve) in Fig. 3.2. The horizontal dashed line corresponding to qcp,max = 10 W intersects the

Peltier cooling mode curve at Hl (T∞,max) and Hu (T∞,max). When H is within this range

(0.017 mm ≤ H ≤ 4.158 mm), the TEM may maintain Tcp = 55 C when qcp ≤ 10 W. Values

of H outside of this range are impermissible. It is noted that the permissible range of H

determined confirms that the initial pellet height of 1 mm used above is a valid one.

27

0

200

400

600

800

10-2

10-1

100

101

T∞,max

= 65°C

T∞,min

= -5°C

qcp

= 10W

qcp

,TE

M,m

ax

[W]

H [mm]

Figure 3.2: qcp,TEM,max as a function of H at T∞,min and T∞,max for Ru−∞ = 0.

Step 3 of the algorithm requires computation of Ru−∞,max as a function of H. It is plotted

versus pellet height for T∞,max = 65 C, and T∞,min = -5 C in Fig. 3.3. Recall that Ru−∞,max

is determined by its value at T∞,max.

Based on Step 4 of the algorithm, total power at the optimum Ru−∞ (H) is plotted versus

H in Fig. 3.4 to illustrate the minimum maximum power (Wt,m) over the permissible range

of H. In Region I, where 0.017 mm ≤ H < 0.175 mm, the TEM requires more power

at T∞,min = -5 C than at T∞,max = 65 C; therefore, setting Ru−∞ = Ru−∞,max (T∞,max)

minimizes maximum Wt. In Region II, where 0.175 mm ≤ H ≤ 1.121 mm, a finite Ru−∞ for

which WTEM(H, Φ(H), 65 C) = WTEM(H, Φ(H), -5 C) minimizes maximum total power

consumption. Finally, in Region III, where 1.121 mm < H ≤ 4.158 mm, Wt is higher at

T∞,max = 65 C than T∞,min; therefore, setting Ru−∞ = 0 C/W minimizes the maximum

Wt.

Wt initially decreases with H regardless of the values of T∞ and Ru−∞ as per Fig. 3.4

due to the adverse effects of heat conduction at sufficiently small H. It is noted that the

electrical contact resistance may dominate bulk resistance at sufficiently lowH and increasing

H essentially decreases heat conduction without affecting total ohmic resistance. Beyond

28

0

1

2

3

4

5

6

7

10-3

10-2

10-1

100

101

T∞,max

= 65 °C

T∞,min

= - 5 °C

Ru

-∞,m

ax

[°C

/W]

H [mm]

Figure 3.3: Ru−∞,max as a function H at T∞,min and T∞,max.

10

100

1000

T∞,max

= 65°C , qcp

= 10W , Ru-∞,max

(65°C)

T∞,min

= - 5°C , qcp

= 10W , Ru-∞,max

(65°C)

T∞,max

= 65°C , qcp

= 10W , Ru-∞

= 0

T∞,min

= - 5°C , qcp

= 10W , Ru-∞

= 0

Wt,ml

H [mm]

Wt[W

]

1.1210.017 0.6590.175 7.1604.158


Figure 3.4: Wt and Wt,ml as a function of H at Tcp = 55 C when Ru−∞ = 0 and Ru−∞,max.

Hpe (0.659 mm) Wt increases with H when qcp = qcp,max (10 W) and T∞ = T∞,max(65 C).

However, increasing H from Hpe to H2 (1.121 mm) is of benefit because of the accompanying

reduction in Wt when T∞ = T∞,min despite the increase in Wt when T∞ = T∞,max. The global

minimum maximum total power (Wt,mg = 13.75 W) occurs at H = 1.121 mm and Ru−∞ =

0 C/W. Typical TEM heights are between 1 mm and 3 mm and when Ru−∞ is optimized

29

the Wt,ml in this range varies from 14.25 W to 18.73 W. Wt,ml is reduced by 26% (WTEM,ml

is reduced by 57%) when H is 1.121 mm rather than 3 mm, for example. Finally, Wtot is

plotted versus Ru−∞ in Fig. 3.5 at H = 1.121 mm and it verifies the optimal value of Ru−∞

equals zero as expected.

5

10

15

20

25

30

35

0 1 2 3 4 5 6

T∞,max

= 65°C

T∞,min

= - 5°C

Wt[W

]

Ru-∞

[°C/W]

13.77

13.8

13.83

0 0.0075 0.015

9.92

10

10.1

3.3 4.4

Figure 3.5: Wt as a function of Ru−∞ at Hmg at T∞,min and T∞,max.

Global Minimum Maximum Total Power

13

14

15

16

17

18

19

10 100

ΦAsub

1/2[mm]

Wt,

mg

Figure 3.6: Wt,mg as a function of (φAsub)1/2.

30

Wt,mg versus (φAsub)1/2 is plotted in Fig. 3.6. Wt,mg may be reduced to 13.3 W for

sufficiently high (φAsub)1/2. In practice, the footprint of a TEM is constrained to about 50

mm x 50 mm, but arbitrarily large footprint arrays of them may be used. However, the “real

estate” in optoelectronics circuit packs is limited and thus φAsub is too.

31

CHAPTER IV

Conclusion and Future Work

An algorithm to find the unique value of pellet height and uncontrolled-side interface to

ambient thermal resistance to minimize maximum total power utilized for precision tempera-

ture control of photonics components has been provided. This algorithm is confined to steady

state conditions and assumes that the total power consumption during transients does not

exceed the maximum one predicted by this algorithm because of the thermal capacitance of

the system. However, further investigation on transient phenomena is required to verify this.

Prescribed paremeters are total footprint of thermoelectric material, thermoelectric material

properties, heat load, component operating temperature, relevant component-side thermal

resistances and ambient temperature range. “Blindly” choosing an off-the-shelf TEM for a

particular precision temperature control problem may henceforth be avoided. The algorithm

has been invoked to show through example calculations the benefit of the optimization.

It was shown that considerable power savings can be achieved, a significant result for op-

toelectronics circuit packs in telecommunications hardware with specified power budgets.

Moreover, additional power savings can be accomplished by varying the total footprint of

thermoelectric material. The physics of the algorithm have been interpreted in the context

of TEM operating modes.

Although the results presented here have demonstrated that implementation of the algo-

rithm provided in this thesis maximizes the fraction of the power budget in an optoelectronics

circuit pack available for other uses, experimental verification of the algorithm is suggested.

Because precise temperature measurement is very important in such experiments, the first

33

step in the future work is calibrating the thermocouples. An apparatus and a procedure to

calibrate thermocouples with a high accuracy is provided in Appendix C.

34

BIBLIOGRAPHY

[1] M. Hodes. Thermoelectric Modules: Principles and Research. Notes from short coursepresented at ITHERM 2010, Las Vegas, NV.

[2] C. L. Foiles. Thermoelectric effects. New York: McGraw-Hill, 2nd edition, 2005.

[3] M. Hodes. On one-dimensional anaysis of thermoelectric modules(TEMs). IEEE Tran.Compon. Packag. TEchnol., 28(2):218–229, 2005.

[4] M. Hodes. Precision temperature control of an optical router. New York: McGraw-Hill,2005.

[5] C. Melnick, M. Hodes, G. Ziskind, M. Cleary, and V. P. Manno. Thermoelectric module-variable conductance heat pipe assemblies for reduced power temperature control. IEEETran. Compon. Packag. TEchnol., 2010. to appear.

[6] Temperature Control of Thermooptic Devices, 2007.

[7] R. Wilcoxon, N. Lower, and D. Dlouhy. A compliant thermal spreader with internalliquid metal cooling channels. In in Proc. 25th IEEE SEMI-THERM Symposium, SantaClara, CA, February 2010.

[8] M. Hodes. Optimal design of thermoelectric refrigerators embedded in a thermal resis-tance network. IEEE Tran. Compon. Packag. Technol., 2011. to appear.

[9] M. Hodes. Optimal pellet geometries for thermoelectric refrigeration. IEEE Tran.Compon. Packag. Technol., 30(1), 2007.

[10] Melcor thermal solutions, 2002.

[11] L. W. da Silva and M. Kaviany. Micro-thermoelectric cooler: Interfacial effects onthermal and electrical transport. Int. J. Heat Mass Tran., 47, 2004.

[12] National Institute of Standards, Technology (US), C. Croarkin, P. Tobias, C. Zey, andInternational SEMATECH. Engineering statistics handbook. The Institute, 2001.

[13] American Society for Testing and Materials. Annual book of astm standards. AmericanSociety for Testing and Materials, 1992.

[14] JV Nicholas and D.R. White. Traceable temperatures: an introduction to temperaturemeasurement and calibration. John Wiley & Sons Inc, 2001.

35

[15] Hart Scientific. A fluke company catalog., 2010.

[16] Hart Scientific. Black Stack 1560 Thermometer Readout User’s Guide., 2010.

[17] Keithley Instruments Inc. Models 2182 and 2182A Nanovoltmeter User’s Manual., 2004.

[18] Keithley Instruments Inc. Model 7001 Switch System Instruction Manual Rev. H, 1991.

[19] Keithley Instruments Inc. Models 7011-S and 7011-C Instruction Manual, 1991.

[20] G.W. Burns, MG Scroger, GF Strouse, MC Croarkin, and WF Guthrie. Temperature-electromotive force reference functions and tables for the letter-designated thermocoupletypes based on the its-90. NASA STI/Recon Technical Report N, 93:31214, 1993.

36

APPENDIX A

Mathematica Codes for Eqs. 2.12 - 2.15

Mathematica Codes for Eqs. 2.12 - 2.15

37

Exit

<< Notation`

Symbolize@TuD; Symbolize@TcD; Symbolize@Rec-RD; Symbolize@Ku-¥,uD; Symbolize@Rec-ΡD; SymbolizeAqcpE;Symbolize@APD; Symbolize@AsubD; Symbolize@T¥D; SymbolizeAKcp-cE; SymbolizeAKcp-¥,cE;

R = 2 * Ρ * H AP; K = 2 * k * AP H; Rec-R = 4 * Rec-Ρ AP; M = yy H2 * APL;

csi = Kcp-c ITcp - TcM M Ix * Α * Tc - K HTu - TcL - x2 * R 2 - x2 * Rec-R 2M;usi = Ku HTu - TambL M Ix * Α * Tu - K HTu - TcL + x2 * R 2 + x2 * Rec-R 2M;W = M * Ix * Α * HTu - TcL + x2 * HR + Rec-RLM;

H*Current Flux but other rate form*L

csif = FullSimplify@csi . x ® AP FDusif = Simplify@usi . x ® AP FDwf = Simplify@W . x ® AP FD

2 Kcp-c Tc +2 k HTc - TuL yy

H+ Tc yy Α F yy H2 Rec-Ρ + H ΡL F

2+ 2 Kcp-c Tcp

2 Ku HTamb - TuL +yy H2 k HTc - TuL + H F HTu Α + 2 Rec-Ρ F + H Ρ FLL

H 0

1

2yy F H-Tc Α + Tu Α + 4 Rec-Ρ F + 2 H Ρ FL

aa = Solve@8csif, usif<, 8Tc, Tu<D@@1DD; Tc = FullSimplify@Tc . aaD; Tu = FullSimplify@Tu . aaD;

Tc

Iyy IH H2 Rec-Ρ + H ΡL F2 H2 Ku - yy Α FL + 4 k IKu Tamb + yy H2 Rec-Ρ + H ΡL F

2MM +

2 Kcp-c H2 H Ku + 2 k yy - H yy Α FL TcpM I4 k IKu + Kcp-cM yy + H H2 Ku - yy Α FL I2 Kcp-c + yy Α FMM

Tu

I4 Ku Tamb IH Kcp-c + k yyM + 2 H Ku Tamb yy Α F + 2 yy IH Kcp-c + 2 k yyM H2 Rec-Ρ + H ΡL F2

+

H yy2 Α H2 Rec-Ρ + H ΡL F3

+ 4 k Kcp-c yy TcpM I4 k IKu + Kcp-cM yy + H H2 Ku - yy Α FL I2 Kcp-c + yy Α FMM

IKcp-c ITcp - TcM + Kcp-¥,c ITcp - TambMM

Kcp-¥,c I-Tamb + TcpM +

Kcp-c ITcp - Iyy IH H2 Rec-Ρ + H ΡL F2 H2 Ku - yy Α FL + 4 k IKu Tamb + yy H2 Rec-Ρ + H ΡL F

2MM + 2 Kcp-c

H2 H Ku + 2 k yy - H yy Α FL TcpM I4 k IKu + Kcp-cM yy + H H2 Ku - yy Α FL I2 Kcp-c + yy Α FMMM

H*General Case*L

qcp@F_, H_D = FullSimplify@%D

Kcp-¥,c I-Tamb + TcpM - IKcp-c yy IH H2 Rec-Ρ + H ΡL F2 H2 Ku - yy Α FL + 4 k IKu Tamb + yy H2 Rec-Ρ + H ΡL F

2M +

H-4 k Ku + H Α F H-2 Ku + yy Α FLL TcpMM I4 k IKu + Kcp-cM yy + H H2 Ku - yy Α FL I2 Kcp-c + yy Α FMM

Work@F_, H_D = SimplifyAM * Ix * Α * HTu - TcL + x2 * HR + Rec-RLM . x ® AP FE

Iyy F I8 k IKu + Kcp-cM Rec-Ρ yy F + H2 Ρ F I4 Ku Kcp-c + Ku yy Α F - Kcp-c yy Α FM +

H I-2 Kcp-c yy F H-2 k Ρ + Rec-Ρ Α FL + Ku I2 Kcp-c HTamb Α + 4 Rec-Ρ FL + yy F ITamb Α2

+ 4 k Ρ + 2 Rec-Ρ Α FMMM +

H Kcp-c Α H-2 Ku + yy Α FL TcpMM I4 k IKu + Kcp-cM yy + H H2 Ku - yy Α FL I2 Kcp-c + yy Α FMM

H*Simplified Case*L

qscp@F_, H_D = FullSimplifyALimitAqcp@F, HD, Ku ® ¥EE

-I2 Tamb IH Kcp-c Kcp-¥,c + k IKcp-c + Kcp-¥,cM yyM + H Kcp-¥,c Tamb yy Α F + H Kcp-c yy H2 Rec-Ρ + H ΡL F2

-

I2 H Kcp-c Kcp-¥,c + 2 k IKcp-c + Kcp-¥,cM yy + H IKcp-c + Kcp-¥,cM yy Α FM TcpM I2 H Kcp-c + 2 k yy + H yy Α FM

Works@F_, H_D = FullSimplify@Limit@Work@F, HD, Ku ® ¥DD

Iyy F I2 H Kcp-c Tamb Α + I8 H Kcp-c Rec-Ρ + 8 k Rec-Ρ yy + H Tamb yy Α2

+ 4 H IH Kcp-c + k yyM ΡM F +

H yy Α H2 Rec-Ρ + H ΡL F2

- 2 H Kcp-c Α TcpMM I4 H Kcp-c + 4 k yy + 2 H yy Α FM

Figure A.1: Mathematica Codes for Eqs. 2.12 - 2.15

38

APPENDIX B

Mathematica Codes for Example Calculation

Calculation Part 1

39

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Î½»® ã ðòðíçå Î¹®» ã îòëè ö ïðóëå

Î»½ó® ã ï ö ïðóçå

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îð

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½½ ØÁÎ»¿´ æã ¾¾ ï ö¯½°ô³¿¨ö

Figure B.1: Calculation Part1 - Page 1

40

Ð´±¬ ½½ Ø ô Øô ðòððððïô ðòððï

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41

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43

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ðòððððíëêïîëô íîòèéêêèçèðô ïòëïçîì l ïðéô ïéðòêëèô óîòïëîëê l ïðéô íéîòëëî

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ðòððððêîèèìëô ïðòíîéíèèïîô çòëèðç l ïðêô èïòëìíëô óïòîíðêè l ïðéô ïìéòêéé

ðòððððéðìëèîô èòëêíëéðééïô èòéçççí l ïðêô éïòçëìô óïòïðïðí l ïðéô ïîìòðïï

ðòððððéèçììîô éòïèïìîîéíêô èòïðîçë l ïðêô êíòççêéô óçòèëððì l ïðêô ïðìòìèë

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ðòððððççïðëíô ëòîðêíéèéèíô êòçîïìë l ïðêô ëïòèííëô óéòèéçïë l ïðêô éìòèêéê

ðòðððïïïðìïô ìòìçìéëèìéëô êòìïççì l ïðêô ìéòïèçèô óéòðìíðç l ïðêô êíòêìîè

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ðòðððïîììïëô íòçïíîçíîììô ëòçêèîë l ïðêô ìíòîèïîô óêòîçîïé l ïðêô ëìòîííè

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´´î ã Ô·¬ ðòðððïçîðíðëô ðòðððîïðéïèìô ðòðððîíïîîëðô ðòðððîëíéîéîô

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ðòðððììîçëêðô ðòðððìèêðêííô ðòðððëíííêëéô ðòðððëèëîéïìô ðòðððêìîîîèëô ðòðððéðìéîèëô

ðòðððééííïðèô ðòðððèìèëêéìô ðòðððçíïïìééô ðòððïðîïéêìêô ðòððïïîïîððð å

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ðòðððîëíéîéô ïòçíììêìíèëô íòçìéêé l ïðêô îçòîðèëô óíòðîéëç l ïðêô îðòëìëì

ðòðððîéèìïçô ïòéçêêçéîèçô íòéëèðç l ïðêô îèòîðïíô óîòéíèìê l ïðêô ïèòïïðï

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ðòðððíêéèéïô ïòìéðéëìëçëô íòîëçîï l ïðêô îëòèìéçô óîòððìðç l ïðêô ïîòíïïè

ðòðððìðíêéîô ïòíèëíðééðêô íòïïíïë l ïðêô îëòîìêèô óïòéçéèï l ïðêô ïðòéèí

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ðòðððìðíêéîô ïòíèëíðééðêô íòïïíïë l ïðêô îëòîìêèô óïòéçéèï l ïðêô ïðòéèí

ðòðððììîçëêô ïòíðçïçèïêíô îòçéëçï l ïðêô îìòéîïçô óïòêðèîë l ïðêô çòìïéðì

ðòðððìèêðêíô ïòîìïíçîíçîô îòèìêèë l ïðêô îìòîêëèô óïòìíìðé l ïðêô èòïçëìê

ðòðððëíííêêô ïòïèïðîêçðêô îòéîëíè l ïðêô îíòèéîïô óïòîéìðî l ïðêô éòïðîîî

ðòðððëèëîéïô ïòïîéíèìééíô îòêïðçé l ïðêô îíòëíëîô óïòïîé l ïðêô êòïîíëí

ðòðððêìîîîçô ïòðéçèéêíéëô îòëðíïî l ïðêô îíòîëðíô óççï çììòô ëòîìéìí

ðòðððéðìéîçô ïòðíèðîìèïîô îòìðïìï l ïðêô îíòðïíô óèêé çðéòô ìòìêíêï

ðòðððééííïïô ïòððïëððíðëô îòíðëìé l ïðêô îîòèîðêô óéëì ðïëòô íòéêíîî

ðòðððèìèëêéô ðòçêçççêðèîîô îòîïìèé l ïðêô îîòêêèëô óêìç ììïòô íòïíèíè

ðòðððçíïïìèô ðòçìííêçéèðëô îòïîçí l ïðêô îîòëëìîô óëëí ììîòô îòëèîìí

ðòððïðîïéêô ðòçîïëéîçéìèô îòðìèìî l ïðêô îîòìéìêô óìêë íîëòô îòðèçëè

ðòððïïîïîô ðòçðìêèîêïïïô ïòçéïçë l ïðêô îîòìîéíô óíèì ìëêòô ïòêëìéè

´´í ã Ô·¬ ðòððïïëèëìëêô ðòððïïçéïíëïô ðòððïîíéðïððô ðòððïîéèîïíðô

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ðòððîèççéðêçô ðòððîççêîçîðô ðòððíðçêðçìîô ðòððíïççîîðéô ðòððííðëéèîîô ðòððíìïëèçíïô

ðòððíëîçêéïéô ðòððíêìéîìððô ðòððíéêèéîììô ðòððíèçìîëëíô ðòððìðîíçêéìô ðòððìïëèðððð å

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¬¬î æã Ú·²¼Î±±¬ ¯ Úô Õ«ô îêèòïëô Ø ïð ô

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ðòððïïëèëëô ðòèçççíïêìëíô ïòçìëçê l ïðêô îîòìïéêô óíëé ëìîòô ïòëïììï

ðòððïïçéïìô ðòèçëèííîéìèô ïòçîðìè l ïðêô îîòìïïêô óííï ìíîòô ïòíèðëë

ðòððïîíéðïô ðòèçîìðêððêèô ïòèçëìç l ïðêô îîòìðçô óíðê ïðîòô ïòîëíðì

ðòððïîéèîïô ðòèèçêéïçìíìô ïòèéðçé l ïðêô îîòìðçéô óîèï ëíðòô ïòïíïéî

ðòððïíîðéçô ðòèèéêëéîïèîô ïòèìêçî l ïðêô îîòìïíêô óîëé êçíòô ïòðïêìê

ðòððïíêìéèô ðòèèêíçîëìççô ïòèîííí l ïðêô îîòìîðëô óîíì ëéðòô ðòçðéïïï

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Ú·²¼Î±±¬ ïð ¯³¿¨ Õ«ô Ì¿³¾ô Ø ô Õ«ô ïòî ô Ó»¬¸±¼ r þÒ»©¬±²þô ß½½«®¿½§Ù±¿´ r îô


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ðòððïéïêêîô ðòçðîìîèéíîêô ïòêêççé l ïðêô îîòëìïèô óçð çèçòíô ðòîçéîíï

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ðòððïçëéðïô ðòçíèðïîðéêçô ïòëçðêè l ïðêô îîòêëíìô óîï êçëòïô ðòðêîîéìë

ðòððîðîîîô ðòçëïîðïéëïêô ïòëéïêç l ïðêô îîòêèìô óëêëìòèçô ðòðïëêííë

ðòððîðèçëëô ðòçêêëíêìîïçô ïòëëíðî l ïðêô îîòéïëíô ççðïòèîô óðòðîêîçèï

ðòððîïëçïëô ðòçèìíðêçïèëô ïòëíìêê l ïðêô îîòéìéô îì çèçòïô óðòðêíëéèî

ðòððîîíïðéô ïòððìèêêëíèô ïòëïêëç l ïðêô îîòééèêô íç êîðòêô óðòðçêîêïî

ðòððîíðëíçô ïòðîèêìèíëçô ïòìçèè l ïðêô îîòèïðïô ëí èðçòíô óðòïîìíçè

ðòððîíèîïèô ïòðëêïèèêëêô ïòìèïîç l ïðêô îîòèìðçô êé ëêèòô óðòïìèðíè

ðòððîìêïëîô ïòðèèïëçðìðô ïòìêìðí l ïðêô îîòèéðèô èð çðèòèô óðòïêéîîé

ðòððîëìíëïô ïòïîëìïïëëéô ïòììéðî l ïðêô îîòèççíô çí èìíòìô óðòïèîððç

ðòððîêîèîíô ïòïêçðìíðìçô ïòìíðîë l ïðêô îîòçîêíô ïðê íèíòô óðòïçîìîè

ðòððîéïëéèô ïòîîðìèçïîìô ïòìïíéï l ïðêô îîòçëïïô ïïè ëíçòô óðòïçèëîí

ðòððîèðêîìô ïòîèïêêëíèêô ïòíçéíè l ïðêô îîòçéíîô ïíð íîïòô óðòîððííè

ðòððîèççéïô ïòíëëïèëïçëô ïòíèïîë l ïðêô îîòççîëô ïìï éíçòô óðòïçéçïì

ðòððîççêîçô ïòìììéðèêéíô ïòíêëíï l ïðêô îíòððèïô ïëî èðíòô óðòïçïîçí

ðòððíðçêðçô ïòëëëëîíéêíô ïòíìçëê l ïðêô îíòðïçêô ïêí ëîïòô óðòïèðëïç

ðòððíïççîîô ïòêçëëêíîïêô ïòíííçé l ïðêô îíòðîêéô ïéí çðîòô óðòïêëêíé

ðòððííðëéèô ïòèééîçîëêìô ïòíïèëí l ïðêô îíòðîèëô ïèí çëëòô óðòïìêêçç

ðòððíìïëèçô îòïîïìèëííðô ïòíðíîí l ïðêô îíòðîììô ïçí êèëòô óðòïîíéëé

ðòððíëîçêéô îòìêëëîëëíçô ïòîèèðê l ïðêô îíòðïìô îðí ïðïòô óðòðçêèêçë

ðòððíêìéîìô îòçèìïëïêèìô ïòîéíðï l ïðêô îîòççêêô îïî îïðòô óðòðêêïðîë

ðòððíéêèéîô íòèëïêìêèîéô ïòîëèðê l ïðêô îîòçéïìô îîï ðïêòô óðòðíïëîèí

ðòððíèçìîêô ëòëçðîéîìèïô ïòîìíî l ïðêô îîòçíéçô îîç ëîêòô ðòððêééïëé

ðòððìðîíçéô ïðòèðìîëèëëô ïòîîèìï l ïðêô îîòèçëîô îíé éììòô ðòðìèéðëì

ðòððìïëèô íðèîòíèîèîëô ïòîïíêç l ïðêô îîòèìîéô îìë êéëòô ðòðçìïêçë



48

Calculation Part 2

49

Û¨·¬

ää Ò±¬¿¬·±²À



¯ ÚÁáÒ«³»®·½Ïô Õ«Á áÒ«³»®·½Ïô Ì¿³¾Á áÒ«³»®·½Ïô ØÁ áÒ«³»®·½Ï ã

Õ½°óô½ óÌ¿³¾ õ Ì½° ó Õ½°ó½ §§ Ø î Î»½ó® õ Ø ® Úî î Õ« ó §§ ¿ Ú õ ì µ Õ« Ì¿³¾ õ §§ î Î»½ó® õ Ø ® Úî õ

óì µ Õ« õ Ø ¿ Ú óî Õ« õ §§ ¿ Ú Ì½° ì µ Õ« õ Õ½°ó½ §§ õ Ø î Õ« ó §§ ¿ Ú î Õ½°ó½ õ §§ ¿ Ú å

É±®µ ÚÁáÒ«³»®·½Ïô Õ«Á áÒ«³»®·½Ïô Ì¿³¾Á áÒ«³»®·½Ïô ØÁ áÒ«³»®·½Ï ã

§§ Ú è µ Õ« õ Õ½°ó½ Î»½ó® §§ Ú õ Øî ® Ú ì Õ« Õ½°ó½ õ Õ« §§ ¿ Ú ó Õ½°ó½ §§ ¿ Ú õ

Ø óî Õ½°ó½ §§ Ú óî µ ® õ Î»½ó® ¿ Ú õ Õ« î Õ½°ó½ Ì¿³¾ ¿ õ ì Î»½ó® Ú õ §§ Ú Ì¿³¾ ¿î õ ì µ ® õ î Î»½ó® ¿ Ú õ

Ø Õ½°ó½ ¿ óî Õ« õ §§ ¿ Ú Ì½° ì µ Õ« õ Õ½°ó½ §§ õ Ø î Õ« ó §§ ¿ Ú î Õ½°ó½ õ §§ ¿ Ú å









ß° ã ïòçê ö ïðóêå


Ò¬»³ ã éïå



¬ ã ðòððëå µ½ ã ðòðíå Õ½°óô½ ã ðòðððïëå §§ ã îéèòí ö ïðóêå

Ð´±¬ ¯ Úô ííèòïëô ðòððï ô Úô ïðð ðððô ïð ððð ððð


óîð

óïð

ïð

îð

íð

öÚ×ÒÜ ÌØÛ ÎßÒÙÛ ÑÚ Ø ÚÑÎ Ìô³¿¨ö

¾¾ æã ÒÓ¿¨·³·¦» ¯ Úô ííèòïëô Ø ô Úô Ó»¬¸±¼ r þÒ»´¼»®Ó»¿¼þô

ß½½«®¿½§Ù±¿´ r êô Ð®»½··±²Ù±¿´ r ô É±®µ·²¹Ð®»½··±² r è

½½ ØÁÎ»¿´ æã ¾¾ ï ö¯½°ô³¿¨ö


50



óìð

óîð

îð

ìð

êð

èð

ïðð

Ð®·²¬ þÍ¬»° ïæ Ú×ÒÜ ÌØÛ ÎßÒÙÛ ÑÚ Øþ

Í¬»° ïæ Ú×ÒÜ ÌØÛ ÎßÒÙÛ ÑÚ Ø

ö¯½°ãïðÉ Ú·²¼ Ø´ ¿²¼ Ø«ö


Ø r ðòððìïëèìê

Ú·²¼Î±±¬ ½½ Ø ïðô Øô ðòððððï öØ´ ¿¬ ¯½° ã ïðÉö

Ø r ðòððððïêççìé

ö¯½°ã ðÉ Ú·²¼ Ø´ ¿²¼ Ø«ö

Ú·²¼Î±±¬ ½½ Ø ðô Øô ðòðððî öØ« ¿¬ ¯½° ã ðÉö

Ø r îçòïèîé

Ú·²¼Î±±¬ ½½ Ø ðô Øô ðòððððï öØ´ ¿¬ ¯½° ã ðÉö

Ø r ðòððððïëîëèë

öÚ×ÒÜ ÌØÛ ÎßÒÙÛ ÑÚ Ø ÚÑÎ Ìô³·²ö

¾¾ æã ÒÓ¿¨·³·¦» ¯ Úô îêèòïëô Ø ô Úô Ó»¬¸±¼ r þÒ»´¼»®Ó»¿¼þô

ß½½«®¿½§Ù±¿´ r êô Ð®»½··±²Ù±¿´ r ô É±®µ·²¹Ð®»½··±² r è

½½ ØÁÎ»¿´ æã ¾¾ ï ö¯½°ô³¿¨ö



ïëð

îðð

îëð

íðð

íëð

ìðð

ö¯½°ãïðÉ Ú·²¼ Ø´ ¿²¼ Ø«ö


Ø r ðòððéïêðíî

2 New calculation_optimal_TEM Power between H1 and H2.nb


51

Ð®·²¬ þÍ¬»° îæ Ú×ÒÜ Øî ¿²¼ Ø°»þ

Í¬»° îæ Ú×ÒÜ Øî ¿²¼ Ø°»

¬¬ï æã Ú·²¼Î±±¬ ¯ Úô ííèòïëô Ø ïð ô Úô î ððð ðððô ïð ðððô ëð ððð ððð ô



¬¬î æã Ú·²¼Î±±¬ ¯ Úô îêèòïëô Ø ïð ô Úô óï ððð ðððô óçðð ððð ðððô ð ô


Ú²ë æã

Ú ò

¬¬îå

Éêë ØÁÎ»¿´ æã É±®µ Úêëô ííèòïëô Ø å

ÉÒë ØÁÎ»¿´ æã É±®µ Ú²ëô îêèòïëô Ø å

Ð´±¬ É±®µ Úêëô ííèòïëô Ø ô É±®µ Ú²ëô îêèòïëô Ø ô Øô ðòðððïô ðòððî

ðòððïð ðòððïë ðòððîð

ë

ïð

ïë

ººï ã Ú·²¼Î±±¬ Éêë Ø ÉÒë Ø ô Øô ðòððïëèô ðòððïô ðòððî ô

Ó¿¨×¬»®¿¬·±² r ïððô ß½½«®¿½§Ù±¿´ r ïðô Ð®»½··±²Ù±¿´ r

Ø r ðòððïïîïî

¬¬ ã ÒÓ·²·³·¦» Éêë Ø ô ðòðððï ä Ø ä ðòððï ô Ø

íòíçêîéô Ø r ðòðððêëèéïë

¬¬ïð æã Ú·²¼Î±±¬ ¯ Úô ííèòïëô Ø ð ô Úô ïðð ðððô ïðððô ëðð ððð ô


Úêëð æã Ú ò ¬¬ïðå

¬¬îð æã Ú·²¼Î±±¬ ¯ Úô îêèòïëô Ø ð ô Úô óï ððð ðððô óçðð ððð ðððô ð ô


Ú²ëð æã

Ú ò

¬¬îðå

Éêëð ØÁÎ»¿´ æã É±®µ Úêëðô ííèòïëô Ø å

ÉÒëð ØÁÎ»¿´ æã É±®µ Ú²ëðô îêèòïëô Ø å

Ð´±¬ ¯ Úô ííèòïëô ðòðððï ô Úô ïð ðððô ïð ðððô ë ððð ððð

ï l ïðê î l ïðê í l ïðê ì l ïðê ë l ïðê

óíð

óîð

óïð

ïð

îð

íð

New calculation_optimal_TEM Power between H1 and H2.nb 3


52

Ð´±¬ É±®µ Úêëðô ííèòïëô Ø ô É±®µ Ú²ëðô îêèòïëô Ø ô Øô ðòððïêô ðòðï

ðòððì ðòððê ðòððè ðòðïð

ï

î

í

ì

ë

ê

Ð®·²¬ þØî ¿²¼ Ø°» ¼±» ²±¬ »¨·¬þ

Øî ¿²¼ Ø°» ¼±» ²±¬ »¨·¬

Ð®·²¬ þÍ¬»° îæ Ú×ÒÜ ÌØÛ ÎßÒÙÛ ÑÚ Î«óô« ßÒÜ ÌÛÓ ÐÑÉÛÎ ßÌ Î«óô³¿¨þ

Í¬»° îæ Ú×ÒÜ ÌØÛ ÎßÒÙÛ ÑÚ Î«óô« ßÒÜ ÌÛÓ ÐÑÉÛÎ ßÌ Î«óô³¿¨

¼¼ Õ«ÁÎ»¿´ô Ì¿³¾ÁÎ»¿´ô ØÁÎ»¿´ æã ÒÓ¿¨·³·¦» ¯ Úô Õ«ô Ì¿³¾ô Ø ô Úô

Ó»¬¸±¼ r þÒ»´¼»®Ó»¿¼þô ß½½«®¿½§Ù±¿´ r êô Ð®»½··±²Ù±¿´ r ô É±®µ·²¹Ð®»½··±² r è

ºº Õ«ÁÎ»¿´ô Ì¿³¾ÁÎ»¿´ô ØÁÎ»¿´ æã ¼¼ Õ«ô Ì¿³¾ô Ø îô ïô î ö¯½°ô³¿¨ö

¯³¿¨ Õ«ÁÎ»¿´ô Ì¿³¾ÁÎ»¿´ô ØÁÎ»¿´ æã ¯ ºº Õ«ô Ì¿³¾ô Ø ô Õ«ô Ì¿³¾ô Ø å

»» Ì¿³¾ÁÎ»¿´ô ØÁÎ»¿´ æã

Ú·²¼Î±±¬ ïð ¯³¿¨ Õ«ô Ì¿³¾ô Ø ô Õ«ô îë ô Ó»¬¸±¼ r þÒ»©¬±²þô ß½½«®¿½§Ù±¿´ r ïô





¬¬î æã Ú·²¼Î±±¬ ¯ Úô Õ«ô îêèòïëô Ø ïð ô Úô óìð ððð ðððô óëð ððð ðððô ð ô


Ú²ë æã

Ú ò

¬¬îå



´´ ã Ô·¬ ðòððððïèððððô ðòððððîðïêéçô ðòððððîîëçêçô ðòððððîëíïèëô

ðòððððîèíêéèô ðòððððíïéèììô ðòððððíëêïîëô ðòððððíççðïêô ðòððððììéðéíô ðòððððëððçïèô

ðòððððëêïîìçô ðòððððêîèèìëô ðòððððéðìëèîô ðòððððéèçììîô ðòððððèèìëîïô ðòððððççïðëíô

ðòðððïïïðìïìô ðòðððïîììïëïô ðòðððïíçíççêô ðòðððïëêïèèèô ðòðððïéëðððð å

Ú±® · ã ïô · èô · õã ïô

Ø ã ´´ · å





53

ðòððððïèô çêíòðèêîëéêô ëðëòïíìô ïîêðòðê

ðòððððîðïêéçô îèðòïïççêçíô ìïêòèèîô ïðîðòçë

ðòððððîîëçêçô ïìêòïíèìðçëô íìëòíéô èîçòëïí

ðòððððîëíïèëô çðòçðèêðîîçô îèéòíëô êéëòçîì

ðòððððîèíêéèô êïòéíïéìîéïô îìðòîîëô ëëîòìíç

ðòððððíïéèììô ììòîîêîîèëéô îðïòèçô ìëîòçîç

ðòððððíëêïîëô íîòèéêêèçèðô ïéðòêëèô íéîòëëî

ðòððððíççðïêô îëòïîíëçðçïô ïìëòïêïô íðéòìêé

»» Ì¿³¾ÁÎ»¿´ô ØÁÎ»¿´ æã



¬¬ï æã Ú·²¼Î±±¬ ¯ Úô Õ«ô ííèòïëô Ø ïð ô Úô îî ððð ðððô ïð ðððô ëð ððð ððð ô



¬¬î æã Ú·²¼Î±±¬ ¯ Úô Õ«ô îêèòïëô Ø ïð ô Úô óèðð ðððô óëð ððð ðððô óïð ððð ô


Ú²ë æã

Ú ò

¬¬îå



Ú±® · ã çô · ïéô · õã ïô

Ø ã ´´ · å



ðòððððììéðéíô ïçòêîíìèéðëô ïòîëìèí l ïðéô ïîìòíðíô óïòéïççï l ïðéô îëìòêîë

ðòððððëððçïèô ïëòêðèïïíïèô ïòïììïé l ïðéô ïðéòîô óïòëíèðì l ïðéô îïïòêðï

ðòððððëêïîìçô ïîòêðçîïëèéô ïòðìëéî l ïðéô çíòïíéèô óïòíéëé l ïðéô ïéêòìêì

ðòððððêîèèìëô ïðòíîéíèèïîô çòëèðç l ïðêô èïòëìíëô óïòîíðêè l ïðéô ïìéòêéé

ðòððððéðìëèîô èòëêíëéðééïô èòéçççí l ïðêô éïòçëìô óïòïðïðí l ïðéô ïîìòðïï

ðòððððéèçììîô éòïèïìîîéíêô èòïðîçë l ïðêô êíòççêéô óçòèëððì l ïðêô ïðìòìèë

ðòððððèèìëîïô êòðèëîëðìïëô éòìéçè l ïðêô ëéòíéðêô óèòèïðèì l ïðêô èèòíïíê

ðòððððççïðëíô ëòîðêíéèéèíô êòçîïìë l ïðêô ëïòèííëô óéòèéçïë l ïðêô éìòèêéê


»» Ì¿³¾ÁÎ»¿´ô ØÁÎ»¿´ æã






¬¬î æã Ú·²¼Î±±¬ ¯ Úô Õ«ô îêèòïëô Ø ïð ô Úô óïðð ðððô óïð ððð ðððô óïð ððð ô


Ú²ë æã

Ú ò

¬¬îå





54

Ú±® · ã ïéô · îïô · õã ïô

Ø ã ´´ · å




ðòðððïîììïëô íòçïíîçíîììô ëòçêèîë l ïðêô ìíòîèïîô óêòîçîïé l ïðêô ëìòîííè

ðòðððïíçìô íòìíìïêëîéêô ëòëêðîë l ïðêô íçòçéççô óëòêïéïì l ïðêô ìêòíïíç

ðòðððïëêïèçô íòðíêîéíïîèô ëòïçðêî l ïðêô íéòïèîíô óëòððçç l ïðêô íçòêîðï

ðòðððïéëô îòéðíììïðïìô ìòèëìéì l ïðêô íìòèðìíô óìòìêíîé l ïðêô ííòçíçì

´´î ã Ô·¬ ðòðððïéëððððô ðòðððïçîðíðëô

ðòðððîïðéïèìô

ðòðððîíïîîëðô

ðòðððîëíéîéîô

ðòðððîéèìïçíô

ðòðððíðëëïìíô

ðòðððííëîìêîô

ðòðððíêéèéïìô

ðòðððìðíêéïéô

ðòðððììîçëêðô

ðòðððìèêðêííô

ðòðððëíííêëéô

ðòðððëèëîéïìô

ðòðððêìîîîèëô

ðòðððéðìéîèëô

ðòðððééííïðèô

ðòðððèìèëêéìô

ðòðððçíïïìééô

ðòððïðîïéêìêô

ðòððïïîïîððð å

»» Ì¿³¾ÁÎ»¿´ô ØÁÎ»¿´ æã

Ú·²¼Î±±¬ ïð ¯³¿¨ Õ«ô Ì¿³¾ô Ø ô Õ«ô î ô Ó»¬¸±¼ r þÒ»©¬±²þô ß½½«®¿½§Ù±¿´ r íô


¬¬ï æã Ú·²¼Î±±¬ ¯ Úô Õ«ô ííèòïëô Ø ïð ô Úô îðð ðððô ðô ë ððð ððð ô

ß½½«®¿½§Ù±¿´ r íô Ð®»½··±²Ù±¿´ r ô Ó¿¨×¬»®¿¬·±² r ïðð å


¬¬î æã Ú·²¼Î±±¬ ¯ Úô Õ«ô îêèòïëô Ø ïð ô

Úô óìçðô óè ððð ðððô ð ô ß½½«®¿½§Ù±¿´ r íô Ð®»½··±²Ù±¿´ r ô Ó¿¨×¬»®¿¬·±² r ïðð å

Ú²ë æã

Ú ò

¬¬îå



Ú±® · ã ïô · çô · õã ïô

Ø ã ´´î · å

Õ«óô« ã »» ííèòïëô Ø ïô î å

¦¦ ã Ï«·»¬ Ú·²¼Î±±¬ Éêë Õ«ô Ø ãã ÉÒë Õ«ô Ø ô Õ«ô íô Õ«óô«ô îð ô Ó»¬¸±¼ r þÒ»©¬±²þô

ß½½«®¿½§Ù±¿´ r íô Ð®»½··±²Ù±¿´ r ô É±®µ·²¹Ð®»½··±² r ïðô Ó¿¨×¬»®¿¬·±² r ïðð å

Õ« ã ¦¦ ïô î å

Ð®·²¬ Øô þô þô Õ«óô«ô þô þô Õ« öôþô þô

Úêë öôþô þôÉêë Õ«ôØ ôö þô þôÚ²ëö ô þô þô ÉÒë Õ«ô Ø å Õ« ãòå Ø ãò



55

ðòðððïéëô îòéðíììïðîïô îòéðíçðçïîïô ííòçìîî

ðòðððïçîðíïô îòìéïðèçîçîô îòìééçëïéèìô îçòçéðí

ðòðððîïðéïèô îòîêèíïðïëëô îòîèèîïêïéíô îêòëïê

ðòðððîíïîîëô îòðçðêëêèéêô îòïîçìçíèìîô îíòëðë

ðòðððîëíéîéô ïòçíììêìíèëô ïòççéêééëîïô îðòèéìë

ðòðððîéèìïçô ïòéçêêçéíìðô ïòèèçëçéêéïô ïèòëéïì

ðòðððíðëëïìô ïòêéìèíììèïô ïòèðîçððîçíô ïêòëëðì

ðòðððííëîìêô ïòëêêééîëêêô ïòéíëçèìðèëô ïìòééíï

ðòðððíêéèéïô ïòìéðéëìëçëô ïòêèéçççëðéô ïíòîðêç

¬¬ï æã Ú·²¼Î±±¬ ¯ Úô Õ«ô ííèòïëô Ø ïð ô Úô î ìîð ðððô ðô ì ððð ððð ô



¬¬î æã Ú·²¼Î±±¬ ¯ Úô Õ«ô îêèòïëô Ø ïð ô Úô óîð ðððô óïð ððð ðððô óð ô


Ú²ë æã

Ú ò

¬¬îå



Ú±® · ã ïðô · îïô · õã ïô

Ø ã ´´î · å

Õ«óô« ã »» ííèòïëô Ø ïô î å

¦¦ ã Ï«·»¬ Ú·²¼Î±±¬ Éêë Õ«ô Ø ãã ÉÒë Õ«ô Ø ô Õ«ô ëô Õ«óô«ô çðð ððð ððð ô Ó»¬¸±¼ r þÒ»©¬±²þô

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ðòðððìèêðêíô ïòîìïíçîíçîô ïòêêîìçéëçîô çòëïìèí

ðòðððëíííêêô ïòïèïðîéîðêô ïòéðïêèîêììô èòëëðèè

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ðòðððêìîîîçô ïòðéçèèïìêçô ïòèçììðêìèðô êòçîèðï

ðòðððéðìéîçô ïòðíèðìïîêìô îòðèëíëïéêìô êòîììéï

ðòðððééííïïô ïòððïëððíðëô îòíçêèîîèïìô ëòêííîè

ðòðððèìèëêéô ðòçêçççêðèîîô îòçììèìíðçëô ëòðèëîï

ðòðððçíïïìèô ðòçìííêçéèðëô ìòðéèëïíìçîô ìòëçíðè

ðòððïðîïéêô ðòçîïëéîçéìèô éòëìëðîêðéðô ìòïëðìî

ðòððïïîïîô ðòçðìêèîêïïïô ïèë èéíòèèìíô íòéëïëë



56

Calculation Part 3

57

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58

Ð´±¬ É±®µ Úêëô ííèòïëô Ø ô É±®µ Ú²ëô îêèòïëô Ø ô Øô ðòðððïô ðòðððí

ðòðððïë ðòðððîð ðòðððîë ðòðððíð

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Ú²ë æã

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Ï«·»¬ Ú±® · ã ïô · êô · õã ïô

§§ ã ´´ · ö ïðóêå

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2 WTEM verses phiAsub_list plot.nb


59

ðòðððïïðîêç ôèòéðïìê ôðòðððïíèçêê ôèòìêèçí

ðòðððïçìïéê ôêòïðìçç ôðòðððïçíììí ôêòïðìçë

ðòðððíîíéêì ôìòçïíèì ôðòðððîêééçé ôìòèìïîè

ðòðððëîëèðè ôìòîéííë ôðòðððíéíîçí ôìòðèíçì

ðòðððèìðèðé ôíòçðíïî ôðòðððëîéðïî ôíòêðëîï

ðòððïííïë ôíòêèðíí ôðòðððéëëéëì ôíòîçìéï

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»»î æã Ú·²¼Î±±¬ ¯ Úô îêèòïëô Ø ïð ô Úô óï ððð ðððô óçðð ððð ðððô ð


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ðòððëïíêðí ôíòìðîé ôðòððîììééé ôîòèêçéè

ðòððèðïêçì ôíòíêèïé ôðòððíéïîëî ôîòèïîëé

ðòðïîëððë ôíòíìêïì ôðòððëêéêíè ôîòééëíé

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60

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ðòêéïêìë ôíòíðéëê ôðòîçíèêì ôîòéðèéè

ïòðìëîè ôíòíðéí ôðòìëéîïé ôîòéðèíî

4 WTEM verses phiAsub_list plot.nb


61

APPENDIX C

Thermocouple Welding and Calibration

Temperature measurement is very important in experimental heat transfer as temper-

ature is a key variable in any heat transfer problem. Depending on the media, operating

range and application, temperature can be measured by many different methods. Thermo-

couples are the most widely used of all temperature sensors due to their reliability and low

cost. As high accuracy thermocouples are required in the future experimental work, numer-

ous thermocouples were made and calibrated against NIST-traceable thermistor probes in a

custom-built laboratory apparatus.

Introduction and Background

Theory of Thermocouples

Thermocouples consist essentially of two dissimilar conductors electrically joined together

at one end. The other ends of the conductors are connected directly to a temperature-

compensated voltmeter or to an intermediate reference junction at a precisely controlled

temperature. In any case, in order to use a thermocouple to measure temperature, one end

must be maintained at a known temperature, the so-called reference temperature, TR. The

other junction, which is placed in contact with the body or fluid of interest of unknown

temperature, is called the measurement junction, T j, as in Fig. C.1. The electromotive force

63

Figure C.1: Type-T thermocouple measurement circuit.

(emf) generated by a thermocouple is proportional, though not necessarily linearly, to the

temperature difference between the measurement junction and the reference junction. It is

extremely important to note that a thermocouple does not measure temperature directly.

Rather it measures the temperature difference between the two junctions.

Thermocouple Calibration

The materials of the conductors used for thermocouple determine the magnitude of the

emfs generated by the thermocouple. In other words, the response of a particular thermo-

couple depends on the precise composition of the metals used to construct it, meaning two

thermocouples of nominally the same material may respond differently (albeit slightly) un-

der identical measurement conditions. As a result, thermocouples need to be calibrated to

produce interpretable measurement information [12]. One may purchase calibrated thermo-

couples from manufacturers with a high accuracy calibration report; however this is not a

cost effective means of using thermocouples. ASTM Standard E230-87 in the 1992 Annual

Book of ASTM Standards [13] specifies that the initial calibration tolerances for “off-the-

shelf” type-T commercial thermocouples be ± 1 C or ± 0.75 % (whichever is greater)

between 0 C and 350 C, and ± 1 C or ± 1.5 % (whichever is greater) between -200 C

and 0 C. Nicholas et al. [14] specify the tolerances of type T thermocouples (Class 1) as

± 1.5 C or 0.4 % between -40 C and 350 C when the reference temperature is 0 C.

Sometimes a thermocouple is not as accurate as expected when one compares measurements

from different methods. The only way of ensuring the readings are accurate and trustwor-

64

thy is by regular calibration. Therefore, a thermocouple calibration rig was designed and

thermocouples and/or thermocouple probes (i.e., metal-sheathed thermocouples) were cali-

brated using precision thermistors (precisely calibrated by the manufacturer) and very high

accuracy electronics.

Thermocouple Welding

A thermocouple can be crudely made by simply tying the two ends of wires comprising

it together. For engineering applications, however, a welding machine or solder is typically

used to form the junction. A Tigtech 116 SRL Thermocouple Welder was used to form the

two junctions on thermocouple wires in this thesis, except in the case of the thermocouple

probes, where one of the junctions was formed by the manufacturer. The welding is done in

non-oxidizing environment (i.e., Argon gas) to avoid corrosion. During the welding process

an electric arc in an argon atmosphere generates sufficient heat to melt the wires thereby

fusing them together. To make sure that the junction is acceptable, one can inspect it

visually under a magnifying lens. Required materials and devices to form a junction are:

• Tigtech 116 SRL Thermocouple Welder (See Fig. C.2).

• Type T thermocouple wires (Omega Engineering Part Number: TT-T-36-100).

• Hot Wire stripper (Part Number: TC-1) (See Fig. C.3).

• Goggle for eye protection.

• Magnifying lens (See Fig. C.4).

The experimental procedure is as follows.

1. Wear safety goggle to protect eyes.

2. Expose approximately 1 cm of the thermocouple wires from one end using the handheld

thermal wire strippers.

65

Figure C.2: Tigtech 116 SRL Thermocouple welder.

Figure C.3: Handheld thermal wire strippers.

66

Figure C.4: Magnifying lens.

3. Regulate Argon supply source to a pressure between 2 and 4 psig at the welder inlet

and then turn on the power of the welder.

4. Open the spring-loaded wire holding device and place the two bare wires side by side

in the center of the jaws. Insert the wire holding device into the welding receptacle

and make sure it is centered and touching the metal back plate.

5. Push the “weld” button until the “arc” light comes on and then release the “weld”

button. The weld cycle is complete when the “arc” light goes out.

6. Examine the junction formed using the magnifying lens. If the weld does not appear

pristine, “current” and/or “time” control settings may need to be adjusted.

7. Repeat the welding process if necessary.

Thermocouple Calibration Experimental Rig Design

A new thermocouple calibration rig was designed, constructed and utilized. The focus of

this section is on the design of the thermocouple calibration rig and the relevant aspects of

various components.

67

Design Requirements

The first step in the design of the thermocouple calibration rig was to identify the key

design requirements that the system must meet. They were as follows.

• Measurement Junction Temperature. The measurement junction temperature should

be able to be stabilized for a certain length of time at any required temperature as per

control by a constant temperature bath or other sources. The measurement junction

should be isothermal and its temperature should not be affected by a temperature

gradient along the thermocouples wires.

• Reference Junction Temperature. The reference junction must be maintained at a

constant chosen temperature and it should be controlled to be a higher accuracy than

that expected from the thermocouple calibration.

• Readout Instruments. The resolution and accuracy of the voltmeters for reading the

voltages produced by thermocouples must produce a very small uncertainty in the

temperature measurement.

• Wiring Connection to Readout Instruments. Copper wires should be used between the

reference junction and readout instruments.

• Calibration Range and Interval. Calibration temperature range is from -10 C to 90 C.

Thermocouples need to be calibrated in sufficiently small temperature intervals such

that linear interpolation is valid between adjacent calibration points on the calibration

curve.

Experiment Rig Schematic

A schematic of the experimental rig is shown in Fig. C.5. The use of an ice bath (mixture

of melting shaved ice and water) to enclose the reference junction has the advantage of

simplicity in that its temperature is (i.e., 0 C) is precisely known. Ice baths are also cost

68

effective, albeit a bit inconvenient as ice needs to be periodically replenished. In this work, for

convenience, a copper block inserted in a circulating bath of a chiller rather than an ice bath

was utilized for the reference junctions. As described in more detail below, within the copper

block is a copper rod containing a precision thermistor probe and the reference junctions

are adhered to this copper rod as per Fig. C.5. The measurement junctions utilized the

same method. During the calibration, the reference junction was maintained at a constant

temperature, i.e., 0 C, controlled by a chiller and the measurement junction was set to one of

the required calibration temperature points in intervals of 5 C from -10 C to 90 C. Once the

measurement junction stabilized at each of the calibration temperature points, temperatures

of the two standard thermistor probes were readout by Black Stack thermometer and the

thermocouples were scanned by a Keithley 7001 switch system and connected to a Keithley

2182A Nanovoltmeter. Finally, all of the data were recorded and sent to a computer running

Labview for data acquisition. Specific details of the calibration system and its components

are provided in the next section.

Figure C.5: Schematic of thermocouple calibration rig.

69

Measurement Junction and Reference Junction Design

The primary components in the calibration rig are the measurement junction and refer-

ence junction because both of them must be highly stable when the data are collected. As

mentioned in the previous section, key feature of the design is using a copper block immersed

in the circulating bath of a chiller as the reference junction rather an ice bath to provide

isothermal environment. The assembly of the copper block consists of a Teflon cover with an

O-ring providing a (moisture proof) seal from the top, a copper enclosure, a bottom cover

ensuring a good seal from the bottom and a copper rod into which a standard thermistor

is inserted to measure the temperatures of it as per Fig. C.6. The copper enclosure and

the bottom cover are compressed together with high pressure and also sealed by JB-Weld to

avoid any leak of the anti-freeze (50/50 Ethylene glycol and DI water) when the copper block

is immersed in the bath of the chiller during the calibration. Copper was selected to make the

block because of its acceptable thermal diffusivity and conductivity such that heat transfer

is rapid and insignificant temperature gradients are present at steady-state. However, a high

thermal mass or thermal capacity is required to “flatten out” the temperature fluctuations

of the bath in the chiller. Thus a large copper annulus (diameter is 3”, wall thickness 0.25”

and total height 5.1”) was fabricated and placed on standoffs inside the chiller. The copper

rod stands in the middle of the copper annulus and is somewhat insulated from the bottom

of it by a thermal pad (Part Number: TflexTM 2130 V0). Alloy 110 was chosen for the

copper block and the standoff is made of Aluminum. It is noted that the larger the thermal

mass is the more stable the environment it provides, however, the larger its thermal time

constant. Therefore, some compromises need to be made depending on the size of the bath,

the diameters of the probes and other requirements in the calibration. The mechanical draw-

ings for these parts are shown in Fig. C.7 - C.12. Note that the designs of the Teflon cover,

Fig. C.7, and copper rod, Fig. C.11, for the measurement junction are slightly different from

those for the reference junction, Fig. C.8 and C.12. This is because thermocouple probes

were calibrated after the calibrations of the thermocouples and required slight modifications.

70

Figure C.6: Copper annulus assembly design for measurement junction

In Fig. C.12, extra holes with different diameters were drilled for the thermocouple probe

calibration. Finally, the temperatures of these two junctions are controlled by two chillers,

NESLab Model RTE7, which can operate from -24 C to 150 C with an accuracy of ± 0.1

C.

71

Figure C.7: Teflon cover drawings for reference junction (Dimensions are in inches).

72

Figure C.8: Teflon cover drawings for measurement junction (Dimensions are in inches).

73

Figure C.9: Copper enclosure drawing (Dimensions are in inches).

74

Figure C.10: Copper bottom cover drawing (Dimensions are in inches).

75

Figure C.11: Copper rod drawing for reference junction (Dimensions are in inches).

76

Figure C.12: Copper rod drawing for measurement junction (Dimensions are in inches).

77

Standard Thermistor Probes and Readout Instruments

The two standard thermistor probes (Serial Number: B072717 and Serial Number:

B072719) selected for use in this calibration are stainless steel-sheathed thermistors man-

ufactured by Fluke Corporation (Model number 5610) as shown in Fig. C.13 [15]. They

are Secondary Reference Series Thermistors which are accurate to ± 0.01 C. The standard

thermistor probe for the reference junction (Serial Number: B072717) spans the temperature

range of 0 C to 100 C, and, the one for measurement junction (Serial Number: B072719)

was calibrated by Fluke Corporation to meet the special required temperature range of -10

C to 90 C in this calibration. (See the calibration reports of these thermistor probes in

the last section of this appendix.)

Figure C.13: Fluke 5610 thermistor probe.

The corresponding readout instrument for these two standard thermistor probes is the

Black Stack thermometer readout (Model 1560 Base Unit and Standards Thermistor Module

2563) also manufactured by Fluke Corporation as shown in Fig. C.14 [16]. The base module

consists of two parts: a display with the main processor and a power supply. It supplies

power, communication management and software coordination for all of the other modules

attached to it. It has the display, control buttons and an RS-232 port built-in. Standard

thermistor module 2536 is attached to the rear of the base unit. There are two input channels

and it displays temperature readouts directly after all of the calibration information of the

78

Figure C.14: Black stack overview.

thermistors are entered into it. The Black Stack readout has a temperature accuracy of ±

0.0013 C at 0 C and ± 0.0015 C at 75 C with a resolution of 0.0001 C. The report of

calibration is provided in the last section of this appendix. The combined accuracy of the

Black Stack in combination with the thermistor probes is ± 0.0113 C at 0 C and ± 0.0115

C at 75 C.1

The key readout instrument for the thermocouples selected is a nanovoltmeter (Model

2182A) made by Keithley as shown in Fig. C.15. It combines the accuracy of a digital

multimeter with low noise at high speeds for high-precision metrology applications. Its low

noise, high signal observation time, fast measurement rates, and 2 ppm (ppm = parts per

million) accuracy provide the most cost-effective meter available today. Voltage specifications

of it are shown in Fig. C.16. A range of 10 mV in Channel 1 with a resolution of 1 nV was

utilized. The accuracy for 2 years in voltage is ± 0.040 µV at 0 V and ± 0.192 µV at 3.134

mV and the corresponding accuracy in temperature is 0.0010 C at 0 C and 0.00432 C at

75 C. The combined accuracy of the Black Stack in combination with the thermistor probes

and the Keithley Naonovoltmeter is 0.0123 C at 0 C and 0.01582 C at 75 C. Because

40 thermocouples were calibrated at the same time, a switch system (Model 7011) with a

1The accuracy is conservatively estimated here and next in the total combination accuracy because it isalready high enough for the purpose of this thermocouple calibration.

79

switch card (Model 7011-S) was also selected from Keithley to scan each channel before

sending the signals to 2182A Nanovoltmeter. The 2182A Nanovoltmeter communicates with

the computer and the 7001 switch system through a GPIB cable and a trigger link cable was

also used between 2182A Nanovoltmeter and 7001 switch system for an easier scan control

[17]. Details about the Model 7001 and Model 7011-S are provided next.

Figure C.15: 2182A Nanovoltmeter.

Figure C.16: 2182A Nanovoltmeter voltage specifications.

The Model 7001 is a half-rack, high density, two-slot mainframe as shown in Figs. C.17 -

C.18. It simply communicates with GPIB and built-in scan control eliminates the need for

the computer to control every step of the test procedure. It is simple to control the number

of scans and the channels required to be scanned. Each slot of the 7001 can accommodate up

to 40 channels, so only one switch card of 40 channels is required in this calibration [18]. The

switch card selected, Model 7011-S, is a dual 1 x 10 4-pole multiplexer with a screw terminal

80

connector board which has four independent banks of 1 x 10 switching (Bank A, B, C and D)

as in Fig. C.19 [19]. The thermocouples are simply connected through screw terminals. Each

of the four multiplexer outputs on this card connects to the Model 7001 analog backplane

through removable jumpers. It automatically configures the 7001 mainframe.

Figure C.17: 7001 Switch system front panel.

Figure C.18: 7001 Switch system back panel.

Figure C.19: 7011-S Switch card with thermocouples attached.

81

Data Acquisition

LabView was utilized to record the measurements of all of the thermocouples and ther-

mistors. A LabView data acquisition VI was programmed for the Keithley Nanovoltmeter

and Black Stack to collect the voltage readouts of the thermocouples and the temperature

readouts of the standard thermistor probes as shown in Fig. C.20-C.21. Sequence structures

were utilized in writing the VI program to make sure that the 2182A Nanovoltmeter and

7001 switch system are both set up with required scan spacing, scan numbers and list of

channels to scan at the first step. Basically, the VI works as follows. First, both 2182A

Nanovoltmeter and 7001 switch are reset and specified with the scan spacing, scan list and

number of each scan. Then the 7001 switch system scans all of the 40 channels in sequence

with a spacing of 0.001 s and saves them in its buffer and the 2182A Nanovoltmeter displays

the corresponding voltages of the thermocouples. Next, the two standard thermistor probes

are scanned. Finally, all of the voltages and temperatures are sorted, plotted versus time

and saved to a separate data file (*.lvm) on the computer. It takes 24 s to completed one

scan and 100 samples need to be averaged. The detail calibration procedure is provided in

the next section.

Thermocouple Calibration Procedure

1. Use the Tigtech 116 SRL Thermocouple Welder and the above welding procedure to

make a thermocouple with two junctions as shown in Fig. C.2.

2. Attached all of the 40 measurement junctions to the measurement junction copper rod

and all of the reference junctions to the reference junction copper rod tightly using

aluminum tape as shown in Fig. C.22.

3. Assemble the copper blocks for the measurement junction and reference junction and

insert the two standard thermistor probes into the center holes of the copper rods.

82

Figure C.20: LabView VI front panel for thermocouple calibration.

Note that probe # B072717 is for the reference junction and probe # B072719 is for

the measurement junction.

4. Immerse the two copper blocks in the two NESLab RTE7 circulating chiller baths as

shown in Fig. C.5.

5. Connect the other two ends of the thermocouples into the 7011-S switch cards using

the screw terminals as shown in Fig. C.22 to measure the generated emf of the ther-

mocouples and connect the four leads of the thermistor probes to the terminals of the

Black Stack according to its manual.

6. Adjust the reference chiller temperature to get a constant reference temperature. Usu-

ally, by setting the chiller temperature at -0.7 C a reference temperature around 0.0018

C is obtained. Note that setting reference around 0 C was not ideal in this calibration

because the temperature range of the standard thermistor probe is from 0 to 100 C,

83

Figure C.21: LabView VI block diagram for thermocouple calibration.

Figure C.22: Thermocouple Junctions Attached to the Copper Rod Example.

so in the future a reference temperature should be chosen carefully depending on the

range of the standard thermocouples or thermistors.

7. Set the measurement temperature at one calibration temperature point, which is con-

trolled by adjusting the temperature of the measurement junctions-containing chiller.

84

Usually set the chiller temperature slightly lower than the required measurement tem-

perature when it is lower than the ambient temperature but slightly higher when it

is higher than the ambient temperature. For example, setting chiller temperature at

-5.5 C results in a measurement junction temperature of about -5.18 C, however, to

obtain a temperature of 45 C for the measurement junction, the chiller needs to be

set at about 45.3 C.

8. Run the LabView VI for the calibration and acquire the thermistor temperatures and

thermocouple voltages. Wait until the curves are flat, which implies that steady-state

conditions have been reached.

9. Record the voltages generated from the thermocouples which are in the a scale of

millivolts (i.e., -0.3 to 4 mV) and the temperatures of the standard thermistor probes

at the same time by clicking the “Record Data” button in the VI’s front panel. Wait

until 100 samples are obtained for each thermocouple and then click the “Stop” button

to stop recording, which usually takes 40 minutes. Voltages of 40 thermocouples and

temperatures of two thermistor probes will be saved in two difference data files (*.lvm)

on the computer.

10. Set the measurement temperature at the next calibration temperature point and repeat

steps 8 and 9. The calibration temperature points are obtained by increasing the

measurement junction from the lowest to the highest temperature, i.e., from -10 C to

90 C by a step of 5 C.

Data Analysis

Calibration results for thermocouple 1, 11, 21 and 31 were used as examples in the analysis

as per Fig. C.23. The first column is the reference junction temperature and the second

column is the measurement junction temperature. Reference temperature was controlled by

the chiller which was set at - 0.6 C and adjusted by ± 0.1 C depending on the ambient

85

temperature. Note that thermocouples were calibrated from -8.7 C to 87.6 C, because the

standard thermistors cover the range of 0 C and 100 C and setting the reference junction

temperature at the lowest or the highest temperature can not ensure the required accuracy.

After the cold junction compensations, the new data are provided in Fig. C.24. The Seebeck

coefficient of type T thermocouples at 0 C selected in the cold junction compensation is

38.748 µV/K [20].

Figure C.23: Calibration readouts of thermocouples and thermistors.

A “Least-squares Fit” was used to find a polynomial expression of the voltage of a

thermocouple as a function of the temperature using Mathematica which is provided at the

end of this appendix. Experiment data of thermocouples 1, 11, 21 and 31, Least-squares Fit

data and data from reference tables (ITS-90) [20] are provided in Fig. C.25. Comparisons of

these data are provided as in Fig. C.26 - C.29. It shows that the experimental data matches

the data in thermocouple reference tables for type-T thermocouple very well and the least-

squares fit method data also agrees very well with the experimental data. The absolute

error of experimental data comparing to the reference table data and that of least-squares

fit data corresponding to the experimental data are shown in Fig. C.30 - C.31. It is shown

86

Figure C.24: Calibration results after reference junction compensation.

that the experimental data has an absolute error within 0.00442 mV or 0.11087 C and the

least-squares fit method has an absolute error within 0.00151 mV or 0.03780 C.

87

Figure C.25: Experimental data, Least-squares Fit data and data from reference tables (ITS-90) for thermocouple 1, 11, 21 and 31

88

-1

0

1

2

3

4

-20 0 20 40 60 80 100

Experimental Data

Least-squares Fit Data

Reference Tables (ITS-90)

Vo

ltag

e[m

V]

Temperature [°C]

Thermocouple #1

Figure C.26: Comparison of experimental data, least-squares fit data and that in referencerables for thermocouple 1.

-1

0

1

2

3

4

-20 0 20 40 60 80 100

Thermocouple #11

Experimental Data



Vo

ltag

e[m

V]

Temperature [°C]


89

-1

0

1

2

3

4

-20 0 20 40 60 80 100

Thermocouple #21

Experimental Data



Vo

ltag

e[m

V]

Temperature [°C]


-1

0

1

2

3

4

-20 0 20 40 60 80 100

Thermocouple #31

Experimental Data



Vo

ltag

e[m

V]

Temperature [°C]


90

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

-20 0 20 40 60 80 100

|Experimental Data - Reference Table Data|

|Experimental Data - Least-squares Fit Data|

Ab

so

lute

Err

or

of

Vo

ltag

e[m

V]

Temperature [°C]

Figure C.30: Absolute errors versus temperatures in voltage for thermocouple 1.

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-20 0 20 40 60 80 100

|Experimental Data - Reference Table Data|

|Experimental Data - Least-squares Fit Data|

Ab

so

lute

Err

or

of

Tem

pera

ture

[°C

]

Temperature [°C]

Figure C.31: Absolute errors versus temperatures in temperature for thermocouple 1.

91

Calibration Reports and Mathematica Code for Data Analysis

92

Figure C.32: Calibration report of reference thermistor probe (Serial Number: B072717)page 1.

93


94


95


96


97


98


99


100


101

Figure C.41: Calibration report of measurement thermistor probe (Serial Number: B072719)page 1.

102


103


104


105


106


107


108


109

Figure C.49: Calibration report of Black Stack page 1.

110


111


112

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