Comprehensive Optimization for Thermoelectric Refrigeration Devices
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A Thesis presented to the Graduate School Faculty University of Missouri – Columbia
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In Partial Fulfillment of the Requirements for a Master of Science - Mechanical Engineering
__________________________ By Robert A. Taylor Dr. Gary Solbrekken December 2005
Thesis Supervisor
Acknowledgements This research would never have been completed without help and guidance from the people who deserve to be mentioned here. I truly appreciate the patience and advice of Dr. Gary Solbrekken. His knowledge of the topic and his willingness to answer (many) questions was indispensable. My gratitude is also owed to Kasey Scheel for building an experimental set-up and doing the ‘grunt’ work of taking thermal resistance data. The MAE department secretaries were also instrumental in coordinating all the paperwork and ancillary tasks required in any research project. Thanks are also in order to family and friends who supported my efforts. They were a major driving force to my progress in these endeavors. Thanks to everyone for all the encouragement…
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Comprehensive Optimization for Thermoelectric Refrigeration Devices Robert A. Taylor Dr. Gary Solbrekken
Thesis Supervisor
Abstract R. E. Smalley, 1996 recipient of the Nobel Prize in chemistry, stated that energy is the number one problem facing humanity for the next 50 years [Smalley, 1996]. If this projection comes to fruition, as it most probably will, proper implementation of technologies that generate and convert energy will be of immense importance. A large market is currently in place for which thermoelectric (TE) technology can provide diverse energy solutions. This market should continue to grow as improvements are made to TE materials. In the last 10-15 years, researchers have developed TE materials that promise to double the current performance of currently available materials. The semi-conductor industry and an enormous amount of study are fueling this improvement. The current study is directed to develop and analyze system level optimization schemes that make the best use of those new materials, in addition to currently available materials. To fully realize the benefits of TE refrigeration, system level optimization is critical. This study takes an in-depth look at how the electric current and TE geometry can be optimized. In both cases, it is possible to optimize the overall system to maximize the coefficient of performance or to minimize the heat source temperature. A comparison between the two optimization techniques demonstrates conditions under which one approach would be chosen over the other. An interesting finding from the comparison is that there is an electric current and TE geometry that will provide the minimum heat source temperature AND the maximum COP. One may consider this point to be a true optimum that has not been previously published to the knowledge of the author. The models used to study the optimization strategies were validated experimentally. The validation measurements were conducted using a test bed built by an undergraduate researcher. The measurements reinforced the expected trends from the optimization
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study and corroborated the point where the COP is maximized at the same current and geometry that the heat source temperature is minimized. Finally, one of the trends observed with the optimization study is that when the heat flow from the source increases, the TE geometry optimization process suggests that a thinner TE element is needed. However it is known that micro-scale/interfacial effects will become dominant as the geometry shrinks.
A survey of micro-scale thermal and
electrical effects is briefly reviewed. The survey suggests that micro-scale effects will need to be accounted for when the TE geometry shrinks below 10 µm.
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Table of Contents Acknowledgements ........................................................................................................... ii Abstract............................................................................................................................. iii List of Tables.................................................................................................................... vii List of Illustrations......................................................................................................... viii Nomenclature .................................................................................................................. xii 1. Introduction...............................................................................................................- 1 2. Background ...............................................................................................................- 5 i. Thermoelectric Cooling/Heating...........................................................................- 5 ii. Generation of Electricity......................................................................................- 9 3. Recent Research ......................................................................................................- 15 i. Improving TE Material Performance ..................................................................- 15 ii. Using Current Materials.....................................................................................- 22 iii. Inexpensive Designs .........................................................................................- 23 4. Basic Thermoelectric Phenomena .........................................................................- 23 i. Seebeck Effect.....................................................................................................- 24 ii. Peltier Effect .....................................................................................................- 26 iii. Thomson Effect.................................................................................................- 27 5. Thermoelectric Cooling Optimization ..................................................................- 29 i. Baseline Model....................................................................................................- 29 ii.TE Model ............................................................................................................- 30 iii. Maximum Coefficient of Performance .............................................................- 34 vi. Operating Current Comparison .........................................................................- 40 -
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vii. Module Optimization .......................................................................................- 47 viii. Optimization Conclusions...............................................................................- 57 6. Experimental Testing ..............................................................................................- 58 i. Heat Sink Analysis ..............................................................................................- 60 ii. Apparatus Set-Up ...............................................................................................- 60 iii. Heat Sink Characterization ...............................................................................- 63 iv. TE Experimentation...........................................................................................- 65 v. Results ................................................................................................................- 67 vi. Error Analysis....................................................................................................- 76 7. Micro-Scale Effects .................................................................................................- 80 i. Range of Applicability ........................................................................................- 81 ii. Phonon Radiative Transfer.................................................................................- 82 iii. Boundary Resistances .......................................................................................- 86 a. Acoustic Mismatch Model………………………………………………....- 86 b. Diffuse Mismatch Model………………………………………………..…- 89 c. Electrical Contact Resistance……………………………………………...- 93 8. Conclusions..............................................................................................................- 95 9. Future Work ............................................................................................................- 96 10. References..............................................................................................................- 97 11. Appendix ..............................................................................................................- 101 -
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List of Tables Table 5.1. Optimization potential for the various parameters.......................................- 33 Table 5.2. Optimization approach comparison ............................................................- 40 Table 5.3. Decision process for using figure V.13 ........................................................- 50 Table A.1 Type E thermocouple data [Omega, 2005].................................................- 106 Table A.2. Uncertainty in measurements ....................................................................- 107 Table A.3. Specifications for commercially available modules.................................- 107 Table A.4. Experimental test for module 81036 ........................................................- 107 Table A.5. Experimental test for module 81460 ........................................................- 108 Table A.6. Experimental test for module 81085 ........................................................- 108 Table A.7. Experimental test for module 81026 ........................................................- 108 Table A.8. Polynomials for computing temperature dependent properties [Rowe, 2003].. 108 Table A.9. Reproducability study results, day 1. .......................................................- 109 Table A.10. Reproducability study results, day 2. .....................................................- 109 Table A.11. Reproducability study results, day 3.......................................................- 109 Table A.12. The values of specific heat and phonon velocity used in the EPRT.......- 110 -
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List of Illustrations Fig. 1.1 . Top: Number of transistors in Intel chips, Bottom: Power density comparison for the same components (units in W/cm2) [SIA, 2004] ..........................................- 2 Fig. 1.2. Components of a TE module: Left: metallized connection bars on a ceramic, Right: P-N thermoelectric elements ready to be connected in series [Melcor, 2003]- 3 Fig. 1.3. Components of a TE module: Left: Ferrotec Single-stage TE module, Right: Two-stage TE cooling. .............................................................................................- 4 Fig. 2.1. TE COP as it varies with ∆T. (Vapor compression [Ellsworth, 2001]) ..........- 7 Fig. 2.2. A comparison of TE to conventional electricity generation technology .......- 10 Fig. 2.3. The Seebeck coefficient along with thermal and electrical conductivity as functions of free carrier concentration (S is used as the Seebeck coefficient, α, and β is used for the thermal conductivity, k, in the figure). ...........................................- 11 Fig. 2.4. Maximum TE thermal efficiency versus ∆T (Th-Tc) with Z = .0033 (an approximation of currently available Bismuth Telluride materials) ......................- 13 Fig. 2.5. A Hi-Z employee installing thermoelectric modules in a class 8 diesel truck exhaust system [Hi-Z, 2003] ..................................................................................- 14 Fig. 3.1. Left: A sample quantum dot structure, Right: A Si/SiGe superlatice structure [Shakouri, 2003].....................................................................................................- 18 Fig. 3.2. Absolute cooling of a p-BiTe/SbTe superlattice as compared to the bulk material [Venkatasubramanian, 2001]..................................................................................- 19 Fig. 3.3. Top curves: Figure of merit for Bismuth Telluride materials. Bottom points: approximate measured figure of merit for single-walled carbon nano-tubes [Shi, 2003].......................................................................................................................- 20 Fig. 3.4. Improvement in figure of merit over the last few decades [Darpa, 2002].....- 22 Fig. 4.1. Cross-section of a TE generator/thermocouple showing the p-n junction .....- 25 Fig. 4.2. A simple Peltier cooling/heating design. .......................................................- 26 Fig. 4.3. A typical TE module assembly [Melcor, 2003] ..............................................- 27 Fig. 4.4. Thomson heat addition to a thermocouple......................................................- 28 -
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Fig. 5.1. The baseline model diagram...........................................................................- 30 Fig. 5.2. Baseline model thermal resistance network ...................................................- 30 Fig. 5.3. Sketch of a TE Refrigeration System. ............................................................- 31 Fig. 5.4. Thermal Resistance Network for TE Refrigeration........................................- 31 Fig. 5.5. An iteration technique for finding optimizing the COP.................................- 37 Fig. 5.6. The iteration scheme for getting a minimum junction temperature. .............- 39 Fig. 5.7. COP as a Function on Geometry for Both Methods.......................................- 41 Fig. 5.8. COP and junction temperature as a function of current (Tj,min: Q = 21 W, ψha = 0.6 K/W, N*γ = .213 m, ∆T = 0-75 K; COPopt: ∆T = 25 K, Tc = 292.7 K, N*γ = .213 m, Q = 1-43 W) ......................................................................................................- 42 Fig. 5.9. COP and junction temperature as a function of current for both methods (Tj,min: Q = 21 W, ψha = 0.6 K/W, N*γ = .213 m, ∆T = 0-75 K; ; COPopt ∆T = 51.157 K, Tc = 292.7 K, N*γ = .213 m, Q = -2-30 W) ...................................................................- 44 Fig. 5.10. COP and junction temperature as a function of current for both methods (Tj,min: Q = 21 W, ψha = 0.6 K/W, N*γ = 0.213 m, ∆T = 0-75; COPopt: ∆T = 75 K, Tc = 292.7 K, N*γ = 0.213 m, Q = -5-21 W) ...........................................................................- 45 Fig. 5.11. Junction temperature alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 29.84 K, 21.54 K, 13.36 K, Tc = 328.88 K, 340.1 K, 350.03 K; For both: N*γ = 0.883, 1.369 , 2.417 m – all respectively) ...........................................................................................................- 46 Fig. 5.12. Junction temperature alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 29.84 K, 21.54 K, 13.36 K, Tc = 328.88 K, 340.1 K, 350.03 K; For both: N*γ = 0.883, 1.369 , 2.417 m – all respectively) ...........................................................................................................- 47 Fig. 5.13. Q = 75 W, I = ITjmin, ψha = 0.4 K/W, γ = Independent Variable....................- 48 Fig. 5.14. ∆T = 20 K, I = Iopt, Tc = 340 K, γ = independent variable...........................- 49 Fig. 5.15. Current optimization for N=71, ψha = 0.4 K/W, ICOPopt=ITj,min, γ = independent variable ...................................................................................................................- 50 Fig. 5.16. Junction temperature plotted against Nγ; Qc = 75 W ψha = 0.2 K/W..........- 51 Fig. 5.17. Junction temperature as a function of geometry for both approaches..........- 52 -
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Fig. 5.18. Junction Temperature versus Qc. (COP and Nγ, for TE cooling, are also shown at each point, ψha = 0.2 K/W) .................................................................................- 53 Fig. 5.19. Optimization for Q = variable, Tj,min = 85oC, I=ITj,min, and γ= γmin ...............- 54 Fig. 5.20. The COP and junction temperature as it varies with geometry (Tj,min: Q = 100 W, ψha =0.4 K/W I= Imin COPopt: Tc = 340 K, ∆T = 21.6 K, I = 21.245 Amps; For both: N=71 and γ = independent variable) .............................................................- 55 Fig. 5.21. Junction temperature alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 28.1 K, 21.6 K, 13.4 K, Tc = 329 K, 340 K, 350.03 K; For both: I = 19.66 Amps, 21.245 Amps, 22.91 Amps; Optimum N*γ = 0.998, 1.367 , 2.4 m – all respectively).......................................- 56 Fig. 5.22. COP alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 28.1 K, 21.6 K, 13.4 K, Tc = 329 K, 340 K, 350.03 K; For both: I = 19.66 Amps, 21.245 Amps, 22.91 Amps; Optimum N*γ = 0.998, 1.367 , 2.4 m – all respectively) ..................................................................- 57 Fig.6.1. The airflow test chamber purchased from Airflow Measurement Systems, Inc..... 61 Fig. 6.2. Schematic of the airflow test chamber ...........................................................- 62 Fig. 6.3. The plexi-glass wind tunnel set-up................................................................- 63 Fig. 6.4. The baseline model diagram...........................................................................- 64 Fig. 6.5. Measured thermal resistance as a function of pressure drop ..........................- 65 Fig. 6.6. The thermoelectric module test-bed. ..............................................................- 66 Fig. 6.7. Plot of the material property variation as a function of temperature [Rowe, 2003] ................................................................................................................................- 68 Fig. 6.8. Junction temperature versus current for the 81026 module. ..........................- 69 Fig. 6.9. Junction temperature versus current for the 81460 module ...........................- 70 Fig. 6.10. The variation in minimum junction temperature for the different modules .- 71 Fig. 6.11. Junction temperature versus current for the 81036 module..........................- 72 Fig. 6.12. Junction temperature versus current for the 81085 module .........................- 72 Fig. 6.13. COP versus current for the 81026 module ...................................................- 73 -
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Fig. 6.14. COP versus current for the 81460 module. ..................................................- 74 Fig. 6.15. The change in the COPopt point as geometry changes ..................................- 75 Fig. 6.16. COP versus current for the 81036 module. ..................................................- 75 Fig. 6.17. COP versus current for the 81085 module. ..................................................- 76 Fig. 6.18. Testing plan for a reproducibility study........................................................- 77 Fig. 6.19. Junction temperature reproducibility results for the different days..............- 78 Fig. 6.20. Thermal resistance reproducibility study for the different days...................- 78 Fig. 6.21. The variation in junction temperature as it changes measurement trials......- 79 Fig. 6.22. The variation in thermal resistance as it changes measurement trials ..........- 80 Fig. 7.1. EPRT prediction of non-dimensional heat flux vs. acoustical thickness.......- 84 Fig. 7.2. EPRT prediction of thermal conductivity for BiTe as compared to constant bulk conductivity............................................................................................................- 85 Fig. 7.3. Specular phonon boundary scattering.............................................................- 87 Fig. 7.4. Diffusive boundary scattering ........................................................................- 89 Fig. 7.5. Boundary resistance ratio between the diffuse mismatch and the acoustic mismatch models [Swartz and Pohl, 1989]............................................................- 91 Fig. 7.6. The ratio of effective thermal conductivity to the bulk thermal conductivity for Bismuth Telluride. ..................................................................................................- 92 Fig. 7.7. Ratio of bulk resistivity to the effective resistivity.........................................- 94 Fig. A.1. Reliability data for a selected thermoelectric module [Ferrotec, 2005].......- 101 Fig. A.2. Change in electrical resistivity with doping concentration [Heremans, 2003]..... 102 Fig. A.3. Topographical map of a mountainous region...............................................- 103 Fig. A.4. Q = 75 W, W = 40 W, = 0.4 K/W, I = Independent Variable........................- 104 Fig. A.5. ∆T = 30 K, I = 16.9 A, γ = 0.006 m, ψha = independent variable.................- 105 -
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Nomenclature COP
Coefficient of Performance
I
Electrical current (Amps)
K
Thermal Conductance (W/°C)
L
TE element length (thickness) (m)
L’
Fin length (m)
N
Number of TEC thermocouples (#)
Q
Heat load (W)
Q’
Flow Rate (CFM)
R
Electrical resistance (Ohms)
T
Temperature (°C)
W
TE Input power (W)
Z
TE Figure of Merit (1/K)
Z’
Acoustic Impedance (kg/m2-s)
Greek Letters α
Seebeck coefficient (V/K)
α’
Transmission probability (dimensionless)
∆
Change in value
k
Thermal conductivity (W/mK)
kb
Boltzmann’s Constant (J/K-molecule)
Ψ
Thermal resistance (°C/W)
γ
TE element geometry metric (m)
xii
ρ
Electrical resistively (Ohm-cm)
Subscripts c
TEC cold side
h
TEC hot side
ha
TE hot side to ambient
hs
TE hot side to heat sink
j
Junction
jc
Junction to case
max
Based on the maximum of given quantity
min
Based on the minimum of given quantity
n
N-type semi-conducting material
p
P-type semi-conducting material
sa
Sink to Ambient
TE
Thermoelectric
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1. Introduction Improvements in manufacturing methods, driven by the electronics industry, have made TE devices effective in numerous applications [Peltier Device Info Directory, 2005]. Their compact size and light weight make TE modules especially well-suited for portable and dimensionally constrained applications. Since TE devices are very sensitive to boundary and operating conditions, proper choice of materials, geometry, and operating conditions play a critical role in creating the optimum TE technology for a specific need. Blind application of thermoelectric technology will most likely be more costly and less effective than an optimized approach. Thus, it is important to study these devices to derive their maximum performance. Since the bulk of the study is related to finding optimum cooling solutions, it is worth while to note the need for such designs. Currently the electronics industry is a large possible market for TE refrigeration applications. Figure 1.1 shows the increasing trend in the number of transistors and the power density in microprocessors.
The
corresponding thermal solution must become more effective since the microprocessor temperature must be still kept at 85oC or lower in spite of the power density increase [SIA, 2004]. Although TE performance is limited, new materials and module designs (discussed in later chapters) have potential to excel under these conditions.
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1000000000
10000
100000000 10000000
# of Transistors
1000000 100000
Transistors Heat Flux
10000
100
1000
Heat Flux (W/cm2)
1000
10
100 10 1 1975
1980
1985
1990
1995
2000
2005
2010
1 2015
Year
Fig. 1.1. Top: Number of transistors in Intel chips, Bottom: Power density comparison for the same components (units in W/cm2) [SIA, 2004] Conventionally, cheap air-cooled heat sinks have been used to dissipate CPU heat. It has been forecast that air cooling may have a bleak future due to the increasing power densities. The Semiconductor Industry Association (SIA) Roadmap indicates that power density will climb from 0.6 W/mm2 to over 1W/mm2 for cost-performance chips in the next ten years [SIA, 2004]. To put this in perspective, a 140 mm2 CPU will have an increase in heat dissipation from 84 W to 140 W, over a 65% increase. The roadmap states that the cooling solution is unidentified within ten years [SIA, 2004]. This means that conventional air cooling will eventually become inadequate, or completely uneconomic. Cold plates, heat pipes, and vapor compression incorporate liquid carrying components with electronic devices. Liquid cooling systems must be designed with extreme care and high reliability, making them much more expensive than conventional
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air-cooling solutions. If any one component malfunctions in these active solutions the heat dissipating device will overheat quickly, possibly in a damaging way. Noise from and space for the working fluid system can also be limiting factors. Although the performance of these systems is quite good, in general, they are not very portable or lightweight. The modularity of TE technology allows it to meet a wide variety of size constraints. Modules can be wired in series or parallel, depending on heat dissipation and power requirements. In this way, each individual module can be optimized according to specific boundary conditions. Therefore, each module would nominally be used over a favorable temperature range [Angrist, 1982]. (The importance of temperature on material properties will be discussed in chapter 2). Figure 1.2 shows the elements that make up a TE module.
Fig. 1.2. Components of a TE module: Left: metallized connection bars on a ceramic, Right: P-N thermoelectric elements ready to be connected in series [Melcor, 2003] Modules can also be staged or cascaded to achieve higher efficiency or performance. Staging also allows for a larger temperature difference, in the case of cooling. Angrist states that it is possible to gain maximum performance over a large temperature range by using a number of different materials/modules [Angrist, 1982]. Figure 1.3 shows a comparison between a single-stage module and a two-stage module.
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Fig. 1.3. Components of a TE module: Left: Ferrotec Single-stage TE module, Right: Two-stage TE cooling. Another advantage of TE energy conversion is that it is done in the solid state. As such, the devices have no moving parts that can wear out. One company, Ferrotec, sells modules that last an average of 68,000 thermal cycles, or about 20,000 hours - a thermal cycle is defined as 2.5 minutes from 30°C to 100°C and back down in 2.5 minutes from 100°C to 30°C, where a 5% change in electrical resistance denotes failure (data is in the appendix) [Ferrotec, 2003]. Another benefit of TE devices is that they convert thermal energy directly into electricity, or visa-versa.
Direct conversion eliminates losses
associated with multiple energy conversion processes. Direct conversion also means there is no need for additional equipment or materials, making for a simplified device. TE technology can be used for electricity generation as well as cooling. This flexibility results in a huge variety of possible applications. Even though this study will focus on the cooling side of TE technology, the principles learned during the course of this research could extend to any application. I believe solid state solutions and TE solutions, in particular, have the potential to enhance or replace a wide variety of conventional systems. During the course of this study some applications for TE technology are introduced. This is followed by a brief review of the thermoelectric phenomenon: the Peltier, Seebeck, and Thomson effects. A review of recent research and design of
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thermoelectric technology is presented. The review will include the search for and design of new materials and devices. The review will also include current methods of TE cooling optimization. Next, opportunities for optimization of TE modules are systematically explored in a parametric study. The results from the parametric study are used to guide the optimization of the applied electric current and the TE element geometry. To validate the operational optimization techniques, experimental measurements were carried out on TE modules with different geometric configurations. Small-scale thermal and electric transport effects are considered to identify the limits of the current modeling. Finally, some conclusions are offered and possible future work in TE design is identified. The next few sections will give an overview of TE cooling, generation, and some possible applications for TE technology.
2. Background TE devices have many possible applications beyond cooling CPU chips. Among these applications are portable coolers, environmental control for optoelectronic equipment, and power generation in remote environments. Some consumer applications include a TE powered watch [Seiko, 2002], a TE temperature controlled vest, a Cannon digital camer [Peltier Directory, 2005], and a Colemann portable cooler [Colmann, 2002]. The rest of this chapter will provide an in-depth discussion of the various TE uses.
i. Thermoelectric Cooling/Heating As mentioned earlier, thermal management of electronic equipment is a potentially heavy user of TE cooling. The power density trends shown in figure 1.1 are
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driving the need for advanced cooling solutions. Since the SIA roadmap suggests that by the year 2015 there is currently no known thermal solution to meet industry performance needs, several researchers, such as Solbrekken, et al, and Phelan, et al, have proposed TE refrigeration as a possible solution [Solbrekken, 2004; Phelan, 2002]. A drawback of current thermoelectric materials is that they generally have a low coefficient of performance. The COP is defined as:
COP =
Qout Wnet ,in
(II.1)
Qout refers to the amount or rate of thermal energy removal from the refrigerated component. Wnet,in refers to the amount or rate of electrical energy that must be input to the TE module to drive the heat removal. It turns out that for any refrigeration system COP is a strong function of the extreme temperatures, Th and Tc. The total optimization of TE refrigeration system will be discussed in more detail in subsequent chapters, but for now let us simply compare the COP of TE cooling to other forms of refrigeration. It was shown by Bierschenk et al that COP values greater than one can be achieved for TE cooling if the operating parameters are selected carefully [Bierschenk, 2004]. For current bismuth-telluride TE materials, the maximum COP for a given ∆T (Th-Tc) is plotted in figure 2.1. For ∆T’s less than 30oC, COP values greater than unity can be achieved. In fact, as the temperature difference is reduced further, the optimum COP increases rapidly. Figure 2.1 also compares TE cooling with conventional vapor compression refrigeration. The upper curve, in figure 2.1, represents a ‘good’ cooling system. In this case ‘good’ is assumed to be 30% of the efficiency of a reversible Carnot cycle.
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That is, 0.3 * COPCarnot is plotted for
comparison’s sake. This curve can be thought of as Stirling cooling system, a system which tries to approximate the Carnot cycle. 10 9
Optimum COP
8 30% Carnot COP TEC COP Vapor Compression
7 6 5 4 3 2 1 0 0
10
20
30
40
50
60
TE Element Delta T (Th-Tc) Fig. 2.1. TE COP as it varies with ∆T. (Vapor compression [Ellsworth, 2001]) The figure also shows that the TE cooling COP is at least the same order of magnitude as vapor compression. The controllability, portability, modularity, and size benefits of TE devices allow them to overcome the drawback in the COP in certain applications. TE technology can also be applied to applications that require heating. A similar COP metric can be defined for heating efficiency. It is given by:
β=
Q pumped
(2.2)
Wnet ,in
In general an electrical resistance heater is limited to a COP of unity. That is, all of the electrical work input to the device will be converted to heat. A TE heat pump can have COP values well above unity, making them more effective heaters. The benefit must, of course, be balanced with the additional cost. For example, a TE cooling system (a -7-
commercially available TE module, a cheap heat sink, and a 12-volt fan) would cost $40$50 [Ferrotec, 2005]. A comparable thermofoil electric heater, capable of providing 100120 W, would cost approximately $30 [Minco, 2005]. TE heating is probably best used in concert with TE cooling for temperature control of small spaces, such as pictures 2 and 4 of Figure 2.1. The relatively small size of TE elements, and the fact that electricity controls the heating/cooling, gives a nearly instantaneous temperature response. Cryopreservation and storage of biological tissue are applications where precise temperature control and high cooling rates are necessary. A good example of tissue storage is installed on the International Space Station. A device called the Advanced Thermoelectric Refrigerator/Freezer/Incubator or ARCTIC stores samples before they are returned to Earth for further scientific analysis [PIMA Handbook, 2003]. In cryopreservation cells can be severely damaged if the cooling rate is not controlled precisely. For TE refrigeration, even with a current commercially available module (Ferrotec: 81085), the cooling rate can exceed 7.6oC/s (under no heat load) [Hanneken, 2005]. Thus, a TE cooling system like this can cool at any rate between 0oC/s and 7.6oC/s, which is an enormous range of cooling rates. Heating can be applied at an even larger range of rates, if necessary, during the thawing stages. In an exhibition at the University of Missouri – Columbia, our lab demonstrated the heating/cooling rate capabilities for an off-the-shelf thermoelectric module. A single 4cm X 4cm Ferrotec TE module (81036 ~ $25, similar to the one in fig. 1.3) was attached to an air-cooled heat sink (~5cm X 5cm) and connected to a power supply. A drop of colored water was placed on the TE module surface opposite of the heat sink. Within 8
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seconds of applying power the water completely froze (15-20 seconds for large drops). The power supply polarity was reversed, and in less than 3 seconds the small drop completely boiled off the surface (4-6 seconds for large drops).
ii. Generation of Electricity As mentioned in the introduction TE modules can be utilized to generate electricity. They are particularly suited to recover electricity from waste heat sources as they require a relatively small temperature difference to generate electricity. Usable power can be derived from a temperature difference of just a few degrees or a few hundreds of degrees centigrade. Waste heat can come from any source that is typically expelled into the atmosphere such as car exhaust [Hi-Z, 2004], electronic components [Solbrekken, 2004], or even geothermal energy [NREL, 2004]. The TE electricity generation process is only slightly different than TE cooling. The same features (no moving parts and direct energy conversion) that are attractive for TE cooling are also advantages for TE electricity generation. Conventional electricity generation uses fossil fuels to create heat, which is then used to evaporate a working fluid, which then turns a mechanical turbine, which then uses a generator to create electricity.
The long supply chain also requires a large amount of materials and
equipment. TE generation, on the other hand, converts a temperature gradient directly into electricity; thus, a heat source (with temperatures above OR below ambient) is the only input. TE generation is not currently suited to be a primary generator (due to its lower efficiency), but it can be employed to use waste heat. Figure 2.2 shows a summary of both situations.
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Fossil Fuels
Conventional Generation Mechanical Energy
Heat Energy
Boiler
Electricity
Turbine
TE Generation
Heat Source (e.g. Geothermal [4])
Electricity
Fig. 2.2. A comparison of TE to conventional electricity generation technology The generation efficiency depends on the temperature difference across the converter, but in most cases efficiency is not critical when waste heat is used. The efficiency of any generation device is defined as,
η th =
Wnet ,out
(2.3)
Qin
The term Wnet, out is simply defined as the electrical power output, while Qin is the amount of heat energy input into the system. A maximum thermal efficiency for TE generation can found in many solid state texts [Angrist, 1982].
η t (max) =
(m'opt −1)(∆T / Th )
(2.4)
m'opt +Tc / Th
The quantity m’opt is given as,
m'opt = (1 + ZTavgerage )1/ 2
(2.5)
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The maximum efficiency, eqn. 2.4, is found by taking the derivative of efficiency with respect to m’opt and setting it equal to zero (Note: m' opt ≥ 1 ). The parameter ZTavg is a non-dimensional quantity called the figure of merit, where Z is given as the following:
Z=
(α n + α p )2
(2.6)
[( ρ n k n )1 / 2 + ( ρ p k p )1 / 2 ] 2
Together the material properties (electrical resistivity, ρ, thermal conductivity, k, and the Seebeck coefficient, α) give a good indication of how well a thermoelectric module will perform. Thus, the maximum efficiency can be increased by increasing the figure of merit, as implied by eqn. 2.4. This is because the numerator is multiplied the m’opt whereas the denominator is only an additive function of m’opt. Figure 2.5 shows how the material properties vary with carrier concentration. It illustrates, qualitatively, that semiconductor materials maximize the figure of merit.
Fig. 2.3. The Seebeck coefficient along with thermal and electrical conductivity as functions of free carrier concentration (S is used as the Seebeck coefficient, α, and β is used for the thermal conductivity, k, in the figure).
- 11 -
In this figure a concentration of 1025 carriers/m3 gives the point where the power factor α2σ is maximized. The bottom graph shows how the thermal conductivity varies as a function of free carrier concentration. These two graphs show that the figure merit will be maximized for semiconductor type materials. (Figure A.2, in the appendix shows, in some real materials, a more detailed description of how electrical resistivity varies with respect to carrier concentration.) It turns out, for bulk materials, that the figure of merit is usually maximized in doped Bismuth Telluride compounds. Eqn. 2.4 also predicts that a larger temperature difference will also increase the efficiency. Figure 2.4 shows that the maximum efficiency is a linear function of the hot and cold temperature difference. The figure also shows that higher absolute hot side temperature also improves efficiency.
25
Th = 1000 K Th = 800 K Th = 600 K Th = 400 K
Efficiency [%]
20
15
10
5
0 0
50
100
150
200 Th-Tc [K]
- 12 -
250
300
350
400
Fig. 2.4. Maximum TE thermal efficiency versus ∆T (Th-Tc) with Z = .0033 (an approximation of currently available Bismuth Telluride materials) Like refrigeration, TE generators are especially useful in portable applications. An example of a low power consumption device that requires portable power is the Seiko Thermic® watch [Seiko, 2002]. It works by using the temperature gradient between the human body (37oF) and the ambient temperature (20oF, nominally). The watch does have its limitations. The thermoelectric module cannot charge the battery if the temperature reservoirs are too close together. That is, in a 37 degree or higher ambient, the module will not be able to generate any electricity. Also if the watch is taken off for an extended period of time it will eventually loose battery power. A demonstration of a TE generator in a harsh environment is provided by Hi-Z technologies where the waste heat from the engine of a class 8 diesel truck is used to generate electricity. The company installed 72 HZ-14 modules after the engine turbocharger on the exhaust pipe. The other side of the TE modules was cooled by circulating water from the radiator. Figure 4 shows a picture of this installation. The arrows point to the placement of the TE generators, the exhaust pipe, and the engine. It can be seen that thermoelectric modules require only a minimal amount of space to generate an appreciable amount of electricity. In the picture the bundle of wires includes thermocouple wires for monitoring the exhaust temperature. 1kW of electricity was generated, allowing the alternator to be removed. Hi-Z stated in a research paper that driveshaft power increased by up to five horsepower due to the elimination of the alternator from the engine driveshaft [Hi-Z, 2003]. The truck reduced its emissions by increasing fuel efficiency.
- 13 -
TE Module Placement
Exhaust System
Engine Block
Fig. 2.5. A Hi-Z employee installing thermoelectric modules in a class 8 diesel truck exhaust system [Hi-Z, 2003] Solbrekken et. al. demonstrated that it is possible to use a thermoelectric generator to recover waste heat from a microprocessor to power a cooling fan [Solbrekken, 2004]. The system requires a TE module/heat sink in conjunction with a shunt heat sink. That is, some of the CPU heat is used for electricity generation, while the rest is dissipated to the ambient through a low thermal resistance path to ensure the CPU is kept below 85oC. It was shown that the TE device could generate sufficient electricity to drive a cooling fan to keep a 30 W heat source below 85oC [Solbrekken, 2004]. Overall, when small size is a major concern in temperature control, thermoelectric systems are a feasible option. With the success that thermoelectric cooling/heating modules have enjoyed, and a renewed focus on energy management at the national level, it is important that engineers take a leading role in the design of this flexible technology.
- 14 -
3. Recent Research TE technology has a lot of room for improvement. Currently, many researchers are trying to expand the market of applicability for TE devices. Researchers are extremely varied in their approach to this challenge. The intent of this section is to review concepts for improving TE technology. There are three basic categories to which these approaches can be lumped. First, since TE performance is fundamentally limited by the materials used, the most notable approach is to improve TE material properties. The second approach, and the focus of this study, is to achieve the best possible performance for a given set of materials. In other words, the second approach is optimizing the system to minimize losses. The third category involves finding cheap materials or manufacturing processes to reduce the cost of TE devices. Less expensive module that are less effective could still prove advantageous compared to a better, but more expensive, design. i. Improving TE Material Performance TE performance can be enhanced through designing TE materials that maximize the figure of merit. This method has shown noteworthy progress in recently published papers. A major limitation to advance seems to come in translating lab discoveries to real-world devices. This is due to various system level losses and, in some cases, microscale effects. That being noted, there are many promising techniques that could bring about drastic improvements in coming years.
- 15 -
So, what can be done to increase the figure of merit? As shown in eqn. 2.6, if thermal conductivity is somehow decreased then the figure of merit (and TE performance accordingly) will be improved. Thus, many new materials are being created to find ways to reduce the thermal conductivity. Looking again at figure 2.3, we can see that in semiconductors a major portion of the thermal conductivity is due to the lattice (a.k.a. phonon) contribution. It is possible to scatter phonons (lattice vibrations) at ‘barriers’ within the material. Barriers are engineered boundaries to phonon propagation, which cause thermal conductivity to diminish, resulting in an increased overall figure of merit. The following equations demonstrate how the entire material can be advantageously engineered. We start with a form of the figure of merit derived from the Boltzmann transport equation [Venkatasubramanian, 2001],
ε −µ ZT =
kT Le + L ph
(3.1)
The term, ε-µ, is the energy difference between the Fermi level and the conduction band for a given material. The two dimensionless Lorenz numbers in the denominator are given as [Venkatasubramanian, 2001],
Le = kρ
q2 k 2T
L ph = k ρ
(3.2)
q2 k 2T
(3.3)
- 16 -
In these equations k, q, ρ, k, and T represent the Boltzmann constant, the carrier density, electrical resistivity, thermal conductivity, and temperature, respectively. Breaking the equations up in this manner shows that there are many parameters to engineer. The Fermi level can be changed by controlled doping. The thermal conductivity can be reduced by creating barriers to phonon and electron flow. Unfortunately reducing the electron thermal conductivity also increases the electrical resistivity (by the Weidmeir-Franz relationship). Therefore, it is desirable to reduce the phonon thermal conductivity while changing the electrical resistivity as little as possible. The temperature can also be controlled to improve material performance. Two types of engineered materials, quantum dots and superlattices, have shown promise in reducing the material thermal conductivity without significantly reducing the power factor (α2σ). The term quantum dot implies that electrons and/or phonons are confined in all three dimensions (a 0-D structure). Hence, when a phonon hits a quantum dot it scatters in all three dimensions. A superlattice is composed of many thin layers of semi-conducting material. Each new layer boundary confines energy carriers (a 2-D structure). In this situation the material will scatter phonons in one dimension. Figure III.1 gives examples of both of these structures.
- 17 -
Fig. 3.1. Left: A sample quantum dot structure, Right: A Si/SiGe superlatice structure [Shakouri, 2003]. Superlattice materials are usually carefully constructed with molecular beam epitaxy (MBE) [Fan, 2001]. Quantum dot structures are usually self-assembled. In selfassembly, conditions such as the substrate material, time, and temperature must be precisely controlled to grow uniform dots. Unfortunately, devices created using both methods are somewhat expensive to fabricate and require a substantial amount of time to procure enough material. Researchers have also had trouble reliably making these materials [Moyzhes, 1998]. One researcher, Venkatasubramanian, demonstrated a significant increase of the figure of merit for the superlattice material mentioned above [Venkatasubramanian, 2001]. A non-dimensional figure of merit, ZT, of approximately 2.4 at 300 K was achieved. Another researcher, Harman, has noted a ZT of ~2 for 100µm n-PbSeTe/PbTe quantum dot structure. Both of these are significant improvements over the ZT~ 0.8 for current BiTe bulk materials. These material gains make a large difference in the cooling
- 18 -
performance. Figure 3.2 shows how the cooling performance of these improved ZT materials compares with bulk ZT values.
Fig. 3.2. Absolute cooling of a p-BiTe/SbTe superlattice as compared to the bulk material [Venkatasubramanian, 2001] Another idea to improve materials employs single-walled carbon nano-tubes. In theory, nano-tubes provide a similar benefit to superlatices. They are small enough geometrically, as 1-D structures, to affect the bulk thermal conductivity. So far carbon nano-tubes have not presented an improvement in the figure of merit, but this is mostly due to low electrical conductivity and high thermal conductivity [Shi, 2003]. They are still very heavily researched, and are predicted to have much better material properties in the future. Figure 3.3 shows some data for the figure of merit in single walled carbon nano-tubes (SWCN) – the tubes are of 10 nm to 148 nm in diameter. Figure III.3 also
- 19 -
shows how this compares to conventional bulk materials over the same temperature range.
1.E+01 1.E+00 10
110
210
310
410
1.E-01
ZT
1.E-02 1.E-03 1.E-04 1.E-05
Temperature Dependent Properties, BiTe Constant Properties, BiTe
1.E-06
SWCN (approx.) 1.E-07 Temperature [K]
Fig. 3.3. Top curves: Figure of merit for Bismuth Telluride materials. Bottom points: approximate measured figure of merit for single-walled carbon nano-tubes [Shi, 2003] A comparison between carbon nano-tubes and conventional materials quickly shows that they have a long way to go before nano-tubes will be comparable in performance to current materials. Nano-tube researchers have great expectations, but currently they are many orders of magnitude away from improving on current materials. A different technique seeks to take advantage of the Thomson effect (discussed in more detail in the next chapter). In most cases the Thomson effect is small compared to the other TE effects and is considered negligible. It is, however, theoretically possible to
- 20 -
employ a non-negligible Thomson effect to enhance the performance of TE devices. A paper by Huang et. al suggests that cooling efficiency can be improved by cleaver usage of the Thomson effect [Huang, 2005] (this effect will be covered in detail in the next section). The authors of the study show that when the Thomson coefficient is relatively large, cooling performance can be optimized. The extra cooling is a result of the fact that the Thomson effect conducts heat out the sides of the elements. Under favorable conditions the effect can increase a given non-dimensional temperature gradient by 5060%, relative to a negligible Thomson effect [Huang, 2005]. In order to accomplish this, the material would have to be designed to maximize the Thomson coefficient. The system would also have to be designed to carry the dissipated heat away from the element sides. It may be possible to utilize this effect, but it has not yet been proven beyond a theoretical stage. Overall, it can be seen that there are opportunities to engineer materials that increase the figure of merit. Again, any improvement in the figure of merit will provide a benefit in the overall TE system performance, illustrated by eqn. 2.4. The last ten years have shown significant progress due to the hard work of many researchers. Figure 3.4 shows the significant improvement (at room temperatures) that has been realized in recent years. Each scalable improvement in future material design is sure to cause considerable gain in the relative attractiveness of TE technology.
- 21 -
3.5 3
Figure of Merit, ZT
2.5 2 1.5 1 0.5 0 1940
1950
1960
1970
1980
1990
2000
2010
Year
Fig. 3.4. Improvement in figure of merit over the last few decades [Darpa, 2002] ii. Using Current Materials Recently, there have been multiple studies that explore the use of TE refrigeration in electronic applications. The main thrust of these studies is to design TE systems in order to eliminate or reduce losses in efficiency and performance. Simons, et al completed a case study using conventional off-the-shelf TE modules applied to a server application [Simons, 2000]. Their conclusion was that current TE materials cannot provide large enough COP’s to be competitive with conventional vapor compression refrigerators. A similar finding was expressed by Phelan, et al [Phelan, 2002]. A study by Solbrekken, et al presented an operational envelope over which TE refrigeration provides a performance advantage over an air-cooled heat sink [Solbrekken, 2005]. That study also presented a strategy for determining the operating current such that the junction temperature is minimized in the presence of a finite thermal resistance heat sink.
- 22 -
As mentioned earlier, a study by Bierschenk suggested that current materials can operate with COPs well above unity. The remainder study is dedicated to exploring optimum cooling condition as well as comparing the different operational envelopes. iii. Inexpensive Designs Another main research track is to make TE materials cheaper. Polymers have made many products economically feasible in the last century. Work is being conducted to try to apply this paradigm to electronic equipment. Organic materials, such as polyethylene oxide, can be doped in order to make them electrically approximate to semiconductors [Reeves, 1982; Shakouri, 1999; Heremans, 2003; Martin, 2003]. These materials are by their very nature less efficient than Bi2Te3, but the material costs are dramatically lower. The main challenge in this area comes in trying to attain low electrical resistance along with and high Seebeck coefficients. Electrons must be forced to ‘hop’ between polymer chains in order to carry the heat energy away from the device. The converse is true when generating electricity from these materials: the generated current is lower due to the fact that charge carriers must ‘hop’ across polymer chains to flow through the load resistance. If feasible materials become available, organic thermoelectric devices could become very inexpensive in the future.
4. Basic Thermoelectric Phenomena When two dissimilar materials - one of a lower energy state (usually a positively doped material) and one of a higher energy state (usually a negatively doped material) - 23 -
come in contact, a thermocouple is formed. A thermoelectric module is simply an array of thermocouples. These couples are connected electrically in series and thermally in parallel (see figure 4.1). Conventional thermoelectric modules operate based on the three thermoelectric effects: the Seebeck effect, the Peltier effect, and the Thomson effect. Each of these will now be discussed in more detail. i. Seebeck Effect The Seebeck effect was discovered by an Estonian physicist, Thomas Johann Seebeck, in 1821. Seebeck found that a compass needle would be deflected when a closed loop was formed of two dissimilar metals with a temperature difference between the junctions. The deflection is a response to the magnetic field created when current flows through the circuit [Angrist, 1982]. The best way to describe this effect involves an open circuit. If the circuit is broken, and a voltmeter is placed in the circuit, a voltage proportional to the temperature difference between the junctions can be measured. Figure 4.1 shows this situation.
- 24 -
Heat Flow
Hot Junction
Electrode
N-type material
P-type material
Electrode
Electrode
Cold Junction
DVM
Fig. 4.1. Cross-section of a TE generator/thermocouple showing the p-n junction The open circuit voltage in this type of device is given as [Angrist, 1982] T2
V = ∫ (α B (T ) − α A (T ))dT
(4.1)
T1
Assuming constant Seebeck coefficients, eqn. 4.1 can be written as [Angrist, 1982],
V = (α B − α A )(T2 − T1 )
(4.2)
Equation 4.2 shows that an increase in the difference between Seebeck coefficients or a larger temperature difference will increase the open circuit voltage. It should be stated that without current flow there are no losses due to Ohm’s law. If it is also assumed that there is no voltage drop in the electrodes, the total module voltage is N times the voltage created in one couple. Therefore, eqn. 4.2 can be changed to:
V = N (α B − α A )(T2 − T1 )
(4.3)
- 25 -
ii. Peltier Effect When electric current is passed between two electrically dissimilar materials heat is absorbed or liberated at the junction. The direction of the heat flow depends on the direction of the applied electric current and the relative Seebeck coefficient of the two materials. This effect is the one utilized in TE heat pumps and coolers. This effect was revealed 13 years after the Thomas Seebeck made his discovery, in 1834, by Jean Peltier. Figure 4.2 shows an analogous (to figure 4.1) schematic of this effect. Heat Absorbed
Cold Junction
Electrode
N-type material
P-type material
Electrode
Electrode
Heat Rejected
Hot Junction
Power Source
Fig. 4.2. A simple Peltier cooling/heating design. The amount of Peltier heat absorbed by the cold junction (the cooling power) is given by [Angrist, 1982], Q& = (Π B − Π A ) I
(4.4)
The terms, ΠA(B), are the Peltier coefficients, in units of Watts per Amp (W/A). The perceptive reader will notice that the two previous effects are closely tied together. In fact, the coefficients can be related in the following equation:
- 26 -
Π = αT
(4.5)
Like the Seebeck effect, each couple will absorb and reject a certain amount of energy, and thus, N couples will absorb N times the amount of heat of a single couple, assuming they are all at the same temperature. Figure 4.3 shows a module with multiple junctions connected electrically in series, and thermally in parallel.
Fig. 4.3. A typical TE module assembly [Melcor, 2003] The total heat absorbed or liberated at either the hot side or cold side of the module is given as:
Q& = N (Π B − Π A ) I
(4.6)
iii. Thomson Effect The Thomson effect predicted in 1855 by Sir William Thomson (a.k.a. Lord Kelvin), suggested that a material carrying current along a temperature gradient should absorb or dissipate heat. The Thomson coefficient is defined as positive if heat must be added to keep the temperature constant. It is dependent on the absolute temperature and
- 27 -
the change in the Seebeck coefficient with respect to temperature. Thomson heating is dependent on the temperature change and the current along as well as the Thomson coefficient, as given by the sign convention. The sign convention is best described in conjunction with a sketch, such as figure 4.4. If the current is flowing in the positive xdirection, as it is in the p-type material, then the Thomson coefficient is negative. If it is the opposite situation, as in the n-type material, the Thomson coefficient is positive.
Current Flow P-type material Qthomson
Th
Tc
N-type material X=L X=0
X
Hot Side
Cold Side Fig. 4.4. Thomson heat addition to a thermocouple The coefficient is proportional to the change in the Seebeck coefficient per change in temperature. Thus, the equation is given as [Angrist, 1982] (under constant pressure),
τ ' = −T
dα dT
(4.7)
To get a further understanding of the Thomson effect on a material, we need to look at heat rejection/absorption. When current flows through a material with a constant Seebeck coefficient ( τ '= 0 ), energy is lost solely through Joule heating. When the
- 28 -
Seebeck coefficient is a function of temperature ( τ '≠ 0 ), the Thomson effect becomes important, given as [Angrist, 1982],
Qs = { I 2R − 1 Iτ2 ' ∆3 T Joule Heating
(4.8)
T hom son Heating
The subscript, s, denotes that this heat is transferred from the sides of the material. The resistance term, R, is also be slightly dependent on temperature. Now that we have a basic understanding of how TE devices work, we can look at how these devices can be optimized. The remainder of this study will focus on applying the cooling phenomenon discovered by Jean Peltier. The next couple of chapters, in particular, will investigate optimum refrigeration conditions for general and commercially available TE modules.
5. Thermoelectric Cooling Optimization In order to discuss the intricate issues of TE cooling optimization, we must include the TE module in a system. A TE cooling system simply places a TE module in between a heat dissipation device and an air-cooled heat sink. To illustrate this change it is helpful to first introduce a baseline cooling system. i. Baseline Model The baseline model assumed for this study consists of an air-cooled heat sink attached directly to a heat source (or CPU) with a thermal interface material placed between the heat sink and heat source. Figure 5.1 shows this configuration.
- 29 -
Ta
Heat Sink
Heat Source
Tj
Interface Material
Fig. 5.1. The baseline model diagram A thermal resistance network can be generated for the model. Figure 5.2 shows the simple one dimensional resistance network. Note that the heat flow, Qc, is the same through both the interface material and the heat sink. Tj
ψhs
Th
ψsa Ta
Qc
Fig. 5.2. Baseline model thermal resistance network
ii.TE Model A sketch of a TE cooling system is shown in figure 5.3. The system simply adds a TE module to the baseline system, which is placed between the heat sink and heat source. A thermal interface material (grease) is assumed to be placed in each of the interfaces. The thermal modeling of the TE refrigerated system is aided through the drawing of a resistance network as shown in figure 5.4.
- 30 -
Ta
Heat Sink
Th Interface Material
TE Module Tc
Tj
Heat Source
Fig. 5.3. Sketch of a TE Refrigeration System. Tj ψjc
Tc
Th ΨTE
Ψhs
ΨTsaa
TE Unit Qc
W
Qc + W
Fig. 5.4. Thermal Resistance Network for TE Refrigeration In figure 5.4, Ψjc and Ψhs are the interface resistances, and Ψsa is the thermal resistance of the heat sink. Analysis of the TE system demands a more rigorous model than the baseline design. As implied by figure 5.4, the operation of the TE module requires external work. That work is needed to drive heat from cold to hot. The input work is converted to heat in the TE module and must be dissipated by the heat sink (as illustrated in figure 5.4 by the additional heat flow term in the heat sink). This additional heat load limits the application of TE refrigeration relative to the baseline configuration, and must be accounted for in the system level modeling.
- 31 -
It is of general interest to establish the amount of heat that can be cooled by the TE refrigerator. This is found through an energy balance on the cold junction of the TE module, and is given by: Q c = 2 N α IT c − =
2 N α IT c 1 424 3 Electron Heat Pumping
−
∆T 1 2 I R TE − ψ TE 2
(5.1)
1 2 2 Nρ I − 21 N γ k4 ∆3 T 42 γ 2 142 43 Conduction Joule Heat Heating Leak
Where:
γ=
(5.2)
Ae L
Note: ∆T = Th- Tc and the parameters, ψTE and RTE, are referred to as the module thermal and electrical resistance, respectively. These are defined as, RTE =
ψ TE =
2 Nρ
(5.3)
γ 1 2 Nkγ
(5.4)
The input work, or electric power, used by the TE must overcome the Seebeck voltage as well as the Joule heating.
W =
2142 N α I4 ∆3T Electron Heat Pumping
+
I 2 R TE 123 Joule
(5.5)
Heating
- 32 -
Using these equations along with knowledge of the interface and heat sink thermal resistance, the CPU junction temperature can be found from:
T j = Ta + (Qc + W )ψ ha − ∆T + Qcψ jc
(5.6)
The heat sink thermal resistance has been added to the interface thermal resistance to obtain Ψha. Since there are a lot of parameters that impact the performance and efficiency of the system, it is useful to look at each individually. Table 5.1 gives an indication as to whether each parameter can be optimized and the effect that it has on the performance and efficiency.
Positive correlation means that an increase in the parameter will cause
and increase in the given system characteristic, holding everything else constant. Negative correlation means that an increase in the parameter will cause a decrease in the given system characteristic, all other parameters held constant. Indeterminate correlation means that an increase will cause either a positive or a negative response to the characteristic, providing the opportunity for optimization.
For instance, if we can
decrease the thermal conductivity, holding everything else constant, the heat load would be increased allowing the junction temperature and the COP to rise. Table 5.1. Optimization potential for the various parameters Parameters Optimization? Correlation to Tj Heat Input, Qc Temperature Difference, ∆T Current, I Thermal Resistance, ψha Hot Side Temp., Th Cold Side Temp.,
Correlation to COP
No
Positive
Positive
No Yes
Negative Indeterminate
Negative Indeterminate
No
Positive
Indirectly negative
No
Depends on Tc and ∆T
Depends on Tc and ∆T
No
Depends on Th and ∆T
Depends on Th and ∆T
- 33 -
Tc Ambient Temp., Ta Total Geometry, N*γ Seebeck Coefficient, α Electrical Resistivity, ρ Thermal Conductivity, k
No
Positive
Negative
Yes
Indeterminate
No
Positive
Indeterminate Positive if Tc>∆T, Negative if Tc COPTj,min (Regime 1)
0 0
0.01
0.02
0.03 0.04 Gamma [m]
Fig. 5.7. COP as a Function on Geometry for Both Methods
- 41 -
0.05
0.06
The COP’s in figure 5.7 are surprisingly high. This is the result of using a low ∆T ( ∆T ≤ 30 K ). As one might expect, the COP optimization approach provides a higher COP then the Tj,min approach over most TE module geometries. There are points for each cooling load where the two current setting schemes have the exact same COP (and the exact same system temperatures). However, there are some cases at small values of γ where the Tj,min approach actually has a larger COP. A more thoughtful analysis is required to explain this unexpected result. Recall from above that the TE module geometry was calculated by assuming a temperature difference across the TE module for the COP optimization scheme while the ∆T for the Tj,min approach is allowed to float. This creates three performance regimes that can be examined – where ∆T for Tj,min is larger, equal to, and smaller than that for the COPopt ∆T. A graph of COP and Tj for the first regime where ∆T is larger in Tj,min is shown in Figure 5.8. 8
320
COP opt curve
315
COP for Tj min
6
Tj min Curve
310
5
Tj curve
305
Tj,min
4
300
Tj @ COPopt
3
295
2
290
COP @ Tj,min 1
Junction Temperature [K]
COP
7
285
COPopt
0
0
2
4
280
6
8
10
12
Current [A]
Fig. 5.8. COP and junction temperature as a function of current (Tj,min: Q = 21 W, ψha = 0.6 K/W, N*γ = .213 m, ∆T = 0-75 K; COPopt: ∆T = 25 K, Tc = 292.7 K, N*γ = .213 m, Q = 1-43 W) - 42 -
The upper parabolic curve (solid triangles) and the straight line (open squares) correspond to the junction temperature for the Tj,min and COP optimization approaches. The convex dotted parabolic (solid squares) curve and the smooth exponentially decaying (open triangles) curve give COP as a function of current, where the parabolic curve corresponds to the COP optimization approach. It is interesting to observe that for the Tj,min approach, the Tj curve indeed has a minimum value, but the COP curve has no bound. The opposite can be said for the optimum COP curves. For the condition illustrated in figure 5.8, where ∆T for the COP optimum approach is assumed to be 25 K, it can be seen that optimum current for the Tj,min approach will have a COP that is less than the optimum COP. Also, at the Tj,min current, the COP is lower than that for COPopt. In this case, the performance is as expected.
It should be restated that these two
comparisons are not really equal, but it is helpful to explore the location the optimum currents for different input conditions. The second regime occurs when the optimum current for Tj,min is the same as for COPopt. Figure 5.9 shows the COP and Tj for the two approaches are the same. At the common optimum operating current all parameters are exactly the same. In this special case, there is no difference in the system performance or efficiency between the two models, and this point could be considered to be a true optimum.
- 43 -
1 0.8 COP
310
COP opt COP for Tj min Tj for COP opt Tj Min.
305 300
COPopt, COP @ Tj,min
0.6 295 0.4 290
0.2 Tj,min, Tj @ COPopt
0 3
5
7
9 Current [A]
Junction Temperature [K]
1.2
285 11
13
Fig. 5.9. COP and junction temperature as a function of current for both methods (Tj,min: Q = 21 W, ψha = 0.6 K/W, N*γ = .213 m, ∆T = 0-75 K; ; COPopt ∆T = 51.157 K, Tc = 292.7 K, N*γ = .213 m, Q = -2-30 W) Finally, an example of the third regime where the Tj,min approach provides a COP that is larger than that given by the COPopt model is shown in figure V.8. In this plot, it is seen that the optimum Tj occurs at a current that is smaller than that for COPopt. This provides a ∆T that is smaller than that which is assumed for the COPopt approach. From the plot (and figure 2.1), it is seen that as ∆T is reduced, the COP increases. Hence the Tj,min approach has a larger COP due to the COPopt model forcing a ∆T across the TE module. In light of this result, one could conjecture that when using the COPopt model, it may be wise to iterate the value of ∆T to truly provide the maximum COP. It should also be pointed out that the model will sometimes predict unrealistic negative COP values. This can only happen if we are extracting work (generation) from the system. This situation would take us away from running in a refrigeration mode.
- 44 -
2
310
1.8 1.6 1.4
COP for Tjmin
Tj,min
COP
1.2
Tj min Curve COP @ Tj,min
1
Tj of COP opt
306 304 302 300
0.8
Tj @ COPopt
0.6
298
COPopt
0.4
296
Junction Temperature [K]
308
COP opt curve
294
0.2 0
292 5
6
7
8
9
10
11
12
13
14
Current [A]
Fig. 5.10. COP and junction temperature as a function of current for both methods (Tj,min: Q = 21 W, ψha = 0.6 K/W, N*γ = 0.213 m, ∆T = 0-75; COPopt: ∆T = 75 K, Tc = 292.7 K, N*γ = 0.213 m, Q = -5-21 W) Figures 5.8 and 5.10 demonstrate how use of one optimization method can be limiting. When the two approaches align, as in the optimum point of figure 5.9, they do so under a critical heat load and thermal resistance for the Tj,min model, and a certain ∆T and Tj for the COPopt model. This is as expected since both the current equations are being employed to find the optimum point, leaving two free variables for each approach. It is possible to always choose the currents so that they will align on an optimum point. Figures 5.11 and 5.12 show how these trends differ with different heat loads.
- 45 -
380 375 370
Tj [K]
365 360 355
Tj, 80 W Tj, 100 W Tj, 120 W Optimum Current 85oC
350 345 340 10
15
20
25
30
Current [Amps]
Fig. 5.11. Junction temperature alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 29.84 K, 21.54 K, 13.36 K, Tc = 328.88 K, 340.1 K, 350.03 K; For both: N*γ = 0.883, 1.369 , 2.417 m – all respectively) Figure 5.11 shows that, in general, current must go up as the amount of heat being pumped is increases. The junction temperature and the cold side of the TE module, Tc, are also increasing at each heat load. We can see that the junction temperature surpasses 85oC at approximately 21 Amps, corresponding to a heat load of about 97 Watts. Therefore, for this optimization technique, 97 Watts is the maximum amount of heat that can be dissipated using TE refrigeration for an electronic application. To keep the two approaches equal for these heat loads ∆T must be reduced and the geometry ratio must go up with heat dissipation rate.
Figure 5.12 shows the COP
relationships for the same conditions. Similar trends are observed, the higher COP values are related to the fact that ∆T decreases at higher heat inputs. This happens because the
- 46 -
temperature input, Tc, is relaxed at each new heat load. In other words, the Tc input is increased at each optimum point. This allows for the additional heat to be dissipated through the same heat sink and reduces the temperature gradient across the TE. 5 4 4 COP, 80 W COP, 100 W COP, 120 W Optimum Current
COP
3 3 2 2 1 1 0 10
12
14
16
18
20
22
24
26
28
Current [Amps]
Fig. 5.12. Junction temperature alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 29.84 K, 21.54 K, 13.36 K, Tc = 328.88 K, 340.1 K, 350.03 K; For both: N*γ = 0.883, 1.369 , 2.417 m – all respectively) For this type of approach we are really only choosing the heat inputs (80, 100, 120 W) and the thermal resistance (0.4 K/W). Thus at each critical point (the triangles), both optimum current equations are being employed, reducing the number of total free variables to two (those two being the thermal resistance and the heat load). If only one variable is a true constraint we can also employ an optimum geometry. The next section will provide an overview of how geometry can also be optimized. vii. Module Optimization
- 47 -
It was shown earlier that geometry is also a parameter that can optimized. It may also be desirable to optimize geometry and current for minimum junction temperature or maximum COP. First, we can try this very thing to minimize the junction temperature. Using the Tj,min current, plotted against geometry, we can see that there is minimum. In figure 5.13 each point uses an optimum current, but the geometry is simply linearly increased over a reasonable range. We can see that the minimum junction temperature can be found for one particular geometry and operating current. In the figure, this happens when the current is approximately 17 amps and γ ~ 0.0141 m.
0.005
0.007
0.009
Gamma [m] 0.011 0.013
0.015
0.017
0.019 354 352
10
COP Tj_min Current Tjmin
1
350 348 346
Tj [K]
COP, Current [Amps]
0.003 100
344 342
0
340
Fig 5.13. Q = 75 W, I = ITjmin, ψha = 0.4 K/W, γ = Independent Variable Along the same lines, we can try to get an optimum COPopt. This is done by plotting the optimum COP (using Iopt) for various geometries. Figure 5.14 shows the results of this relationship. For comparison 0.25% of the Iopt current is plotted in the figure. We can see that there is no optimum-optimum for COP. If the parameters Tc and ∆T are both held constant (as they are in the derivatives) any γ value will give the best efficiency. Under these conditions that COP is 2.35.
- 48 -
6.00
110
90
Iopt COPopt Q
4.00
70
3.00
50
2.00
30
1.00
10
0.00 0.000
-10 0.005
0.010
0.015
Heat Load [W]
COP, 0.25% Current [A]
5.00
0.020
Gamma [m]
Fig. 5.14. ∆T = 20 K, I = Iopt, Tc = 340 K, γ = independent variable One can also look at what happens to the various parameters with respect to geometry if the two optimum currents are set equal to each other. Figure 5.15 shows this relationship. Each point in figure 5.15 is operating at the critical current which will provide both optimum performance (Tj) and optimum efficiency (COP). There are two independent parameters left to define, the heat sink thermal resistance and the geometry. It is interesting to note that as we get to larger γ values, each parameter levels off, except COP. This indicates that a dramatic increase in geometry becomes necessary to increase the heat load. This leveling off can be thought of as the maximum theoretical optimum cooling allowed for this heat sink. The high COP values are again related to its rapid increase as ∆T decays.
- 49 -
10.0
160
COP Heat Load Current Tj
140 120
COP
8.0
100 80
6.0
60
4.0
40 2.0 0.0 0.000
20
0.020
0.040
0.060
0.080
Q [W], Current [A], Tj [K]
12.0
0 0.100
Gamma [m]
Fig. 5.15. Current optimization for N=71, ψha = 0.4 K/W, ICOPopt=ITj,min, γ = independent variable Figure 5.15 can be used in designing an optimum refrigeration system. Suppose the module and heat sink thermal resistance are defined as Nγ = 1.37 and ψha = 0.4 K/W, respectively. The heat load and current that would allow this module to run optimally would be 100 W and about 21.2 Amps, respectively. At this optimum condition the junction temperature and COP would be 360 K and 2.17, respectively. Thus, a designer would follow the steps of table 5.3. Table 5.3. Decision process for using figure 5.15 STEP ACTION 0 For ψha = 0.4 K/W, ICOP,opt=ITj,min 1 Pick one other constraint (γ, Q, I, Tj, COP) off figure 5.15 2 All other variables are dependent upon this choice It was mentioned previously that the grouping 2Nγ is an important definition of TE module geometry. One could question why the individual element aspect ratio, γ, and the number of thermocouples, N, should be combined into one parameter. This is due to the fact that the grouping Nγ can be mathematically isolated in the case of the optimum current. That is, Nγ becomes meaningful a parameter grouping for the optimum current.
- 50 -
To illustrate, a number of cases were run using the Tj,min model for which γ and N were varied over a large range of values. In each case, the product of N and γ was calculated and used to correlate the resulting junction temperature. Figure 5.16 shows that for the Tj,min approach, the minimum junction temperature for all values of γ corresponds to a single value of Nγ. 180
Junction Temperature [C]
160 γ = 0.01 γ = 0.005 γ = 0.001 γ = 0.0005 HS
140 120 100 80 60 40 20 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
N*gamma
Fig. 5.16. Junction temperature plotted against Nγ; Qc = 75 W ψha = 0.2 K/W Using this method of optimizing geometry, we can compare a TE refrigerated system to the baseline configuration, shown in figure 5.1, to learn under what conditions TE refrigeration has an advantage. Figure 5.17 shows the junction temperature over a range of γ values for three different heat loads. The straight middle line is the constant junction temperature used in a COP optimizing approach. The straight marked lines, in figure 5.17, correspond to the baseline design where an air-cooled heat sink is placed directly on top of a heat source. The parabolic curves, denoted by closed markers in
- 51 -
figure 5.17, give the minimum junction temperature approach. We can see that for 100 W (and below) there is geometric configuration in which the minimum junction temperature is below 85oC. The figure also shows that there only select geometry conditions where TE refrigeration provides an advantage over a heat sink alone. This window for TE cooling shrinks as the heat load is increased. That is, the choice of geometry is more closely constrained around the optimum geometry as the heat load is increased. However, it appears that there might always be an optimum geometry using the Tj,min current that can provide a performance advantage over a simple air-cooled heat sink. Tj min @ 80 W
110
Tj min @ 100 W
Junction Temperature [oC]
105
Tj min @ 120 W
100
Fixed at 85 oC Baseline @ 80 W
95
Baseline @ 100 W
90
Baseline @ 120 W
85 80 75 70 65 0
0.02
0.04
0.06 Gamma [m]
0.08
0.1
0.12
Fig. 5.17. Junction temperature as a function of geometry for both approaches. If the optimum Nγ is found in conjunction with Tj,min, it is possible to show that using a TE module will indeed ALWAYS have a lower Tj than when using an air-cooled heat sink. Figure 5.18 shows that for a given heat sink thermal resistance of 0.2 K/W, this is indeed true. The reason that Tj,min and an optimum Nγ will always have a lower Tj is because as Qc increases, the optimum Nγ must increase dramatically. One interpretation
- 52 -
of this phenomena, as shown on the secondary scale in figure 5.18, is that the thickness of the TE elements becomes vanishingly thin, which implies ∆T will go to 0. In this limiting case, the TE module ‘disappears’, and the configuration effectively becomes the baseline case of the heat sink only. From a TE module design perspective, however, these findings suggest that thinner TE elements will increase the envelope over which TE refrigeration will have an advantage over just a heat sink alone.
Of course
interfacial/micro-scale effects need to be considered in the analysis for elements on the order of 10 µm and smaller.
160
1000000 TE HS N*gamma COP
120
100000 10000
Tj [C]
100
1000
80
100
60
10
40
COP, N*gamma [m]
140
1
20 0
0.1 0
100
200
300
Q [W] Fig. 5.18. Junction Temperature versus Qc. (COP and Nγ, for TE cooling, are also shown at each point, ψha = 0.2 K/W) A relationship can be drawn using the minimum junction temperature as the constant free variable. This is plotted against the input heat in figure 5.19. The geometry
- 53 -
minimizes junction temperature as does the current in the figure. The thermal resistance and geometry float in order to allow this optimum junction temperature to be set to 85oC. Heat Load [W] 50
100
150
200
250
Gamma [m], Thermal Resistance [K/W]
10
1
Gamma HA COP Current
300
350
400 1000
100
0.1
10
0.01
1
0.001
COP, ITj,min [A]
0
0.1
Fig. 5.19. Optimization for Q = variable, Tj,min = 85oC, I=ITj,min, and γ= γmin The conclusions that can be drawn from figure 5.19 are numerous. First, it can be seen that to be able to dissipate more than 150 W (@ 85oC) we need a heat sink thermal resistance of less than 0.2 K/W. This is beyond the capacity of most current commercial heat sinks. We can also see that although the current starts out low, 3.6 Amps @ 20 W, it must become very large, 154 Amps, to pump the required heat, 350 W out of the system. The geometry also grows to a relatively large value at 350 W of heat dissipation. An interesting result of this analysis is that the points of optimum geometry and current which minimize junction temperature also align with the optimum COP. This is an artifact of choosing the optimum junction temperature as a constraint in the COPopt model. Since there is only one point where the geometry and current are optimized, as in
- 54 -
figure 5.9, the COPopt current must identically match with the Tj,min approach. This is also explained by the fact that there is no optimum geometry when the optimum current is picked, as shown in figure 5.14. Now that optimum geometry has been compared with the optimum current and a baseline heat sink system, we should compare the two optimum geometries together. A similar comparison to the current approaches can be applied here. Figure 5.20 shows that we can find define modules where the two optimums overlap. In the figure, the curves look almost identical to the current comparison. 3.0
366
COPopt COPtj Tjmin TjCop
COP max, COP of Tj min
2.6 2.4 COP
2.2 2.0
364 363 362
Tj min, Tj of COP max
1.8
365
Tj [K]
2.8
361
1.6
360
1.4
359
1.2 1.0 0.01
358 0.015
0.02
0.025
0.03
0.035
Gamma [m]
Fig. 5.20. The COP and junction temperature as it varies with geometry (Tj,min: Q = 100 W, ψha =0.4 K/W I= ITj,min COPopt: Tc = 340 K, ∆T = 21.6 K, I = 21.245 Amps; For both: N=71 and γ = independent variable) We can see that here again (for the given conditions) there is one point where the current for the two optimizing techniques line up. This is very useful when one knows two constraints (in this case the heat load and the heat sink thermal resistance) and has the freedom to employ a module with any element geometry ratio. This method of matching the two optimum geometries conditions can be extended to various heat loads
- 55 -
as was accomplished with the current comparison. Figures 5.21 and 5.22 show similar results to the results found in the previous operating current comparison. 380 375 370
Tj [K]
365
Tj, 80 W Tj, 100 W Tj, 120 W Optimum Gamma 85oC
360 355 350 345 340 0.005
0.015
0.025 0.035 Gamma [m]
0.045
0.055
Fig. 5.21. Junction temperature alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 28.1 K, 21.6 K, 13.4 K, Tc = 329 K, 340 K, 350.03 K; For both: I = 19.66 Amps, 21.245 Amps, 22.91 Amps; Optimum Nγ = 0.998, 1.367 , 2.4 m – all respectively) Here again we can see that as the heat load is increased the junction temperature and the geometry must also be increased. When geometry is increased, the modules must be made thinner, as was also shown in figure 5.18. The cold side temperature of the TE device, Tc, also increases with increased heat loading. The values of ∆T that allow the two approaches to be equal must go down with increased heat loads. Figure 5.22 illustrates that the COP increases with increasing heat loads. The reduced value of ∆T drives this trend. The plots of figures 5.22 and 5.23 are only slightly different conditions than the ones of figure 5.11 and 5.12.
- 56 -
4.5 4.0 3.5
COP
3.0 2.5
COP, 80 W COP, 100 W COP, 120 W Optimum Gamma
2.0 1.5 1.0 0.5 0.0 0.005
0.015
0.025
0.035
0.045
0.055
Gamma [m]
Fig. 5.22. COP alignment as a function of current for both methods (Tj,min: Q = 80,100,120 W, ψha = 0.4 K/W; COPopt: ∆T = 28.1 K, 21.6 K, 13.4 K, Tc = 329 K, 340 K, 350.03 K; For both: I = 19.66 Amps, 21.245 Amps, 22.91 Amps; Optimum N*γ = 0.998, 1.367 , 2.4 m – all respectively) viii. Optimization Conclusions The above discussion suggests that, any arbitrary application of a single optimizing approach will not necessarily provide the global optimum.
In the past,
researchers have not completely optimized their models, by arbitrarily setting more than one constraint. For instance, if we wish to have a junction temperature less than 85oC and pump at least 100 W, it is not good practice to simply set these values. It may be possible to actually do significantly better. A different sized module or a different operating current could possibly pump more heat or produce a lower junction temperature. A module sized to do this may be of equal or lesser cost. In general, the designer should try to use as many of the four optimization equations as possible.
- 57 -
The comprehensive optimization covered in this study included many ideas. The application of TE refrigeration for multiple general heat loads.
The analysis was
extended to allow for the optimization of the TE module geometry as well as the usual operating current optimization approaches. Performance and COP were compared with respect to the two techniques for both current and geometry.
A significant finding
of this study is that TE refrigeration can ALWAYS provide a performance benefit relative to an air-cooled heat sink alone if current and geometry are optimized. This performance benefit will be very small at high heat loads (less than 1K). All the aforementioned optimization models assume that some factors can be neglected.
It is taken for granted that the heat flows through the elements in one
dimensional. Also, the air gaps between elements are not included in the models. Thus, some amount of heat could be transferred out of the system by natural convection or radiation, hurting performance. It was also assumed that the material properties are constant.
In reality, the properties depend on temperature.
The next section uses
experiments, on off-the-shelf modules, to investigate how well these models predict real behavior.
6. Experimental Testing The test bed was built by Kasey Scheel as part of his undergraduate honors thesis [Scheel, 2005].
A test bed was constructed to simulate both the baseline and TE
configurations given in figures 5.1 and 5.3. The experimental testing can be separated into two stages. The first stage simulates the baseline design to calibrate the heat sink thermal resistance as a function of the air velocity. The second stage approximates the
- 58 -
TE configuration model to validate the two current optimization approaches. The second stage also seeks to validate the existence of a common optimum point. Experiments in the second stage can be grouped in two categories. The first category is intended to validate the Tj,min optimized current for a given thermoelectric module, as discussed in chapter 5. The parameter of interest is the temperature to which the heat source temperature will be exposed. This is calculated through the following equation: T j = Tc + Qc *ψ jc
(6.1)
The thermoelectric cold side, Tc, is measured with a thermocouple. The input heat, Qc, is held constant and calculated by readings from the power supply.
The
interfacial thermal resistance is also set to a constant desired value. The input current to the thermoelectric module is varied around the predicted optimum to demonstrate the ability of the analytic model to predict junction temperature and COP. Tests in the second category are conducted to validate the COPopt model with commercially available TE modules. To do this, the testing parameter is the coefficient of performance given by the following: COP =
Qc Win
(6.2)
To find this quantity, the heat input of a thermo-foil heater, Qc, and the power input into the thermoelectric, Win, is measured. In these tests the performance, Tj, is held constant at varying currents. In order to do this the heat input and thermal resistance must be changed to new values (predicted by the models) at each respective current. These inputs ensure that the junction temperature and ∆T are held constant for each input current.
- 59 -
i. Heat Sink Analysis One of the most difficult parameters to control in validating the models is the heat sink thermal resistance. To determine this parameter Kasey Scheel, an undergraduate student at the University of Missouri, built and calibrated a flow bench. A flow bench controls air flow through a duct in which a heat sink can be placed. From a simple energy balance of the flow bench, the thermal resistance model can be expressed as [Scheel, 2005],
ψ sa =
Ts − Ta = Qc
1
(6.3)
⎡ ⎛ − hPL' ⎞⎤ ⎟⎥ m& c p ⎢1 − exp⎜ ⎜ ⎟ ⎢⎣ ⎝ m& c p ⎠⎥⎦
The right hand side of the equation includes the mass flow rate, m& , the specific heat of air, cp, the heat transfer coefficient, h, and fin geometry, PL’. Therefore, the thermal resistance is simply a function of flow rate and geometry. Since the geometry is fixed for a given heat sink, the only parameter that needs to be varied to characterize thermal resistance is the flow rate. It can also be seen through the middle part of eqn. 6.3 that heat sink base temperature, Ts, ambient temperature, Ta, and heat input, Qc, measurements will be enough to calculate the magnitude of thermal resistance.
ii. Apparatus Set-Up The equipment needed for experimentation include two power supplies, four different off-the-shelf thermoelectric modules (Ferrotec modules: 81036, 81460, 81085, 81026 – module specifications are in the appendix), type-E thermocouples, a digital multi-meter (HP, 3478 A), a thermofoil heater (Minco, 412130), insulation (Thermafiber,
- 60 -
092403W), a slug of aluminum, a heat sink, and a flow bench (Airflow Measurement Syastems, 2516 - including a variable fan and standard ASME nozzles), pressure taps, and a heat sink duct. The flow bench, purchased from Airflow Systems - shown in figure 6.1, is utilized to provide a controlled, repeatable heat sink thermal resistance. Airflow through the bench is driven by an AC fan that is controlled with a variac. The flow rate is calculated from a pressure drop measurement across a set of ASME nozzles as specified in the AMCA210-99 standard.
Fig.6.1. The airflow test chamber purchased from Airflow Measurement Systems, Inc. Figure 6.2 shows a schematic of the internal components of the flow bench. It should be mentioned that the flow direction is from right-to-left and that the nozzle placement is reversed for the experiments.
- 61 -
Blower
Airflow
Airflow
Duct Pressure Tap
Nozzle Plate
Pressure Tap
Fig. 6.2. Schematic of the airflow test chamber The nozzle plate inside the chamber contains three different standard nozzles. The nozzle actually used in the tests was chosen based on a differential pressure between 25 and 1000 Pa. The lower limit of 25 Pa ensures that the flow through the nozzle will be completely turbulent. The higher limit is chosen so that compressibility effects of the airflow are negligible. The nozzle used has an outlet area of ~1.75 in2. During testing, a pressure drop across the nozzle plate is measured. This measurement can be used to calculate volumetric flow rate, in CFM. Q' = 1096Y
∆P
ρ
Σ(CA)
(6.4)
The constant, 1096, is a conversion factor. That is, when A, ∆P, and ρ are in units of ft2, inches of water, and lbm/ft3 the 1096 gives Q’ in CFM.
The quantity Y is the
compressibility factor and C is a discharge coefficient. Due to our choice of nozzle, the compressibility factor should be unity throughout the tests. The discharge coefficient is found through a correlation to the Reynolds number [Scheel, 2005]. The heat sink is placed in a wind tunnel constructed from four pieces of plexiglass, and is shown in figure 6.3. The plexi-glass sides are 0.5 inches thick with the
- 62 -
bottom pieces 0.25 inches thick. The duct length is 36 inches long, 4 inches wide and 0.983 inches high. The height was chosen to ensure that the heat sink is closely ducted. The duct is bolted onto the front end of the flow bench, also shown in figure 6.3.
Heat Sink Plexi-Glass Duct Flow Bench
Heater/TE Assembly
Fig. 6.3. The plexi-glass wind tunnel set-up To accurately measure the pressure drop over the standard nozzle, the test bed uses a highly accurate differential capacitance manometer from MKS Instruments (Model: 698A11TRB). The resolution of this instrument is 10-4 percent of full scale and an accuracy of 0.08 percent of the reading. Since the full scale is 10 Torr, the resolution is 10-5 Torr. The accuracy part means that if we are measuring small pressure differences, say 0.2 Torr (approximately the lowest achievable by the fan), the uncertainty would be 1.6 X 10-4 Torr. For larger measurements, e.g. 10 Torr, the uncertainty would be 0.008 Torr. iii. Heat Sink Characterization
- 63 -
To determine the thermal resistance of the heat sink, Kasey Scheel measured three quantities: the ambient air temperature, Ta, the heat sink base temperature, Ts, and the amount of heat, Qc, added to the system at various flow rates. A thin-film thermofoil heater, capable of supplying 100 W (at less than 150oC), was used to apply heat to the system. The heater was placed, with thermal grease, against the bottom of the heat sink. Figure 6.4 is shows this set-up along with the placement of the base temperature measurement. Each of the measured quantities is shown in the figure. Ta
Tj
Heat Sink
Ts
Heat Source, Qc Interface Material
Fig. 6.4. The baseline model diagram A curve of the heat sink thermal resistance versus pressure drop can be approximated. Figure 6.5 shows the results of the measured thermal resistance values, taken by Kasey Scheel. The data is compared with manufacturer data. We can see that the data agrees quite well.
- 64 -
Thermal Resistance
1.2
Measured Mfg. Results
1 0.8 0.6 0.4 0.2 0 150
250
350
450
550
650
Flowrate (LFM)
Fig. 6.5. Measured thermal resistance as a function of pressure drop It should be noted that Kasey Scheel observed a long term drift in the control of the fan motor. Fortunately, the thermal resistance measured at a given instant in time appears to be consistent with measurements taken during stabilized experiments and manufacturers data [Scheel, 2005]. iv. TE Experimentation The second phase of testing is conducted to confirm the theoretical current optimization modeling. A picture of this set-up is given as figure 6.6. When data is actually being taken, the heat sink base will be level with the bottom of the wind tunnel (hence, higher than what is shown in the picture).
- 65 -
Wind Tunnel Heat Sink
Heater Placement Insulation
TE Module
Fig. 6.6. The thermoelectric module test-bed. These experiments seek to validate the curves in figure 5.9. To explore the two optimization techniques we can pull various operating points off the model for these specific devices (this data is given in the Appendix, as Tables A.7-A.10). For the COPopt model each data point requires an accurate heat input, current input, and heat sink thermal resistance. All three of these inputs change as the current is increased in order to maintain a constant junction temperature. In the Tjmin model, accurate inputs are still required, but current is the only thing that changes along the curves. After the system reached steady state for each data point, 20-30 measurements are collected. These measurements are then averaged and recorded. Reproducibility tests were then done on the entire system.
The purpose of
conducting a these tests was to find the amount of variability in the system. Four different modules were used in the experiments. This meant that the entire system had to
- 66 -
be taken apart and put back together with the new module. Each of the three parameters that vary during testing, Qc, Win, flow rate, along with assembly were tested under different conditions, which included re-assembly. Thirty measurements were taken at each of the 24 possible experimental permutations.
v. Results When the data was first compared to the models, the agreement was not very close. It seemed as the average TE temperature deviated farther from ambient the data less closely matched the model. The main reason for this is that the models assumed constant material properties. This assumption, in general, is inaccurate. D. M. Rowe reports empirical relationships between the TE material properties and temperature [Rowe, 2003]. The polynomials for computing these temperature dependent properties are given in the Appendix as Table A.11. Figure 6.7 shows the variable properties as functions of temperature.
- 67 -
1.E+00 100
200
300
400
1.E-01
α_vary ρ_vary k_vary
1.E-02
6.5 5.5 4.5
1.E-03 3.5 2.5
1.E-04
1.5
Thermal Conductivity [W/m-K]
Seebeck Coefficient [V/K], Electrical Resistivity [Ohm-m]
0
8.5 500 7.5
1.E-05 0.5 1.E-06
-0.5 Temperature [K]
Fig. 6.7. Plot of the material property variation as a function of temperature [Rowe, 2003] Once these properties were input in the models the data matched much more closely. It was also assumed that there is some small heat loss, 1.0-1.5 W, which was incorporated into the modeling. This value was chosen due to a first order analysis of heat transfer by conduction out the sides by conduction. With an approximation of error propagation (again, the derivation is in the appendix) we can see that the data match very well with the mathematical predictions.
Figures 6.8-6.12 compare the analytic
predictions of Tj with the experimental data. The ‘_M’ and ‘_EX’ designation in the legend denote the model and the experimental data, respectively.
- 68 -
Tj of COPopt_M Tj of COP_EX Tjmin_M Tjmin_EX
335
Tj [K]
325 315 305 295 285 275 1.0
2.0
3.0
4.0
5.0
6.0
7.0
Current [A] Fig. 6.8. Junction temperature versus current for the 81026 module. We can see in figure 6.8, and the rest of these experiments, that it is rather problematic to set the junction temperature at a constant value when trying to replicate the COPopt curves. For the COPopt model the parameters, current, heat load, and thermal resistance must be changed to get each new data point.
With this fan/heat sink
combination it is difficult to get the required thermal resistance at each point. The higher currents on figure 6.8 require a thermal resistance of below 0.4 K/W for the COPopt model. A thermal resistance of around 0.45 K/W seems to be the limit for this set-up. This is simply not achievable for the heat sink/maximum flow rate combination used in these tests. Since the thermal resistance required for the Tjmin experiments are consistent and achievable, the results are much more accurate. Similar results were found for the next module tested (Ferrotec – 81460), given in figure 6.9.
- 69 -
310.0
Tj of COPopt_M Tj of COP_EX Tjmin_M Tjmin_EX
Tj [K]
305.0 300.0 295.0 290.0 285.0 3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0 11.0
Current [A] Fig. 6.9. Junction temperature versus current for the 81460 module Recall that there is one optimum point in which both methods align for a given module and a given heat sink, if we intend to set the two optimum currents equal. Therefore, each module should be run at the one optimum point. Recall that as the module geometry gets larger (that is, γ ↑ ) the heat load that can be pumped also increases, shown in figure 6.10. If we look at the optimum points in order from the smallest total module footprint area (N*γ) to the largest, we can be seen that lower junction temperatures are associated with smaller modules. That is, Tj,min of the 81026 module (N*γ = 0.168) is less than the Tj,min of the 81460 module (N*γ = 0.188) by few degrees. This is due to the fact that the heat load for the 81026 module is less than the optimal heat load for the 81460 module. This trend continues for the other modules. Figure 6.10 shows this curve compared with the theoretical optimum points.
- 70 -
330 320 Opt. Tj_M 310
Opt. Tj_EX
Tj [K]
300
81085
81036 81460
290 81026
280 270 260 250 240 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
N*Gamma [m]
Fig. 6.10. The variation in minimum junction temperature for the different modules Figure 6.10 shows that there is pretty good agreement with the model as geometry increases. This gives some validation of figure 5.21 in the last chapter. It should be noted that in figure 6.10 the heat load is rising at each point, making the junction temperature increase accordingly. The change is only 16 K from the smallest to the largest module, but it could be very significant if the junction temperature is close to becoming to hot. In figure 6.11 we can see that the junction temperature data of the COPopt approach matches well within the uncertainty. The better results are due to the fact that each data point was held much longer in order to wait for stabilization in the fan. The range of thermal resistances chosen was also easier to achieve.
- 71 -
318
Tj of COPopt_M Tj of COP_EX Tjmin_M Tjmin_EX
313
Tj [K]
308 303 298 293 288 3.0
4.0
5.0
6.0
7.0
8.0
9.0 10.0 11.0 12.0 13.0
Current [A] Fig. 6.11. Junction temperature versus current for the 81036 module.
Tj of COPopt_M Tj of COP_EX Tjmin_M Tjmin_EX
350.0 340.0
Tj [K]
330.0 320.0 310.0 300.0 290.0 1.0
2.0
3.0
4.0 5.0 Current [A]
6.0
7.0
8.0
Fig. 6.12. Junction temperature versus current for the 81085 module All of these curves demonstrate the fact that there is an optimum point, and that with careful choice of operating conditions a real TE system can employ this point. Of
- 72 -
course the experimental data does not lie exactly on the predicted values, but most points are within the estimated measurement uncertainty. The 81085 module is the furthest off the model, with an average deviation of 5 K, due to the fact that the heat loading is the highest. A higher heat load means more losses through the insulation and lower thermal resistances, which are hard to achieve. COP was also calculated during the course of these experiments. Figures 6.136.17 show how the COP prediction curves (COP vs. I) match with the laboratory tests. Since both input conditions that determine COP, the heat load and current, were set to the desired values, it is much easier to get the COP models to line up with experimental data. The temperatures were dependent on these inputs and therefore deviate further.
3.0
COPopt_M COPopt_EX COP of Tjmin_M COP of Tjmin_EX
2.5 2.0 COP
1.5 1.0 0.5 0.0 -0.5 1.0 -1.0
2.0
3.0
4.0 Current [A]
Fig. 6.13. COP versus current for the 81026 module
- 73 -
5.0
6.0
7.0
One very notable feature of these results is that the COP is greater than 2.5 at 1.5 Amps. This proves the previous claims that COP can achieve values well above unity if the ∆T is small enough.
3.0 COPopt_M COPopt_EX COP of Tjmin_M COP of Tjmin_EX
2.5
COP
2.0 1.5 1.0 0.5 0.0 -0.5
2.0
4.0
6.0 8.0 Current [A]
10.0
Fig. 6.14. COP versus current for the 81460 module. Similarly, for these experiments we can see that as the module size gets larger the COP rises. Figure 6.15 shows how this optimum point changes as compared to the geometry of the different modules. The optimum modeling points are also shown in figure 6.15, for comparison. We can see that the experiments match the model trend as module size (and heat load) is increased.
- 74 -
1.2
1 Opt. COP_M Opt. COP_EX
COP
0.8
0.6
81085
81036
0.4
81460 81026
0.2
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
N*Gamma [m]
Fig. 6.15. The change in the COPopt point as geometry changes
3.0
COPopt_M COPopt_EX COP of Tjmin_M COP of Tjmin_EX
2.5 2.0 COP
1.5 1.0 0.5 0.0 -0.5 3.0 -1.0
5.0
7.0
9.0
Current [A]
Fig. 6.16. COP versus current for the 81036 module.
- 75 -
11.0
13.0
COPopt_M COPopt_EX COP of Tjmin_M COP of Tjmin_EX
3.0 2.5 2.0 COP
1.5 1.0 0.5 0.0 -0.5 1.0 -1.0
3.0
5.0
7.0
9.0
Current [A]
Fig. 6.17. COP versus current for the 81085 module. In these results we can again see that the experimental trends follow the models. In summary, the two optimization methods for current have been shown to accurately approximate a real TE cooling system. The experiments show that there is one, and only one, solution (for a given heat sink and geometry) where these methods line. The only deviation from the simple model is that temperature dependent material properties were included in the model and a small, constant amount of heat was assumed to be lost. The experiments show that it is especially important to use temperature dependent material properties.
vi. Error Analysis A reproducibility study was done over a three day period to explore the system deviations. Each day the apparatus was taken completely apart and put back together.
- 76 -
Each of the input parameters was varied during the experiments each day. Figure 6.18 shows a tree of how the reproducibility trials were conducted.
DAY 1,2,3 Qc = 24 W
Ψha = 0.6K/W Ψha = 1 K/W
Qc = 24 W
Ψha = 0.6K/W Ψha = 1 K/W
I = 3.5A I = 5A I = 3.5A I = 5A I = 3.5A I = 5A I = 3.5A I = 5A
Fig. 6.18. Testing plan for a reproducibility study Figure 6.19 shows how junction temperature varied over the three days for the same set of conditions. Figure 6.20, similarly, indicates how the thermal resistance varied between days.
- 77 -
340
Junction Temperature [K]
330 320
DAY 1 DAY 2 DAY 3
310 300 290 280 270 0.5
1
1.5
2
2.5
3
3.5
DAY
Fig. 6.19. Junction temperature reproducibility results for the different days 1 0.95
Thermal Resistance [K/W]
0.9 0.85 0.8
DAY 1 DAY 2 DAY 3
0.75 0.7 0.65 0.6 0.55 0.5 0.5
1
1.5
2
2.5
3
DAY
Fig. 6.20. Thermal resistance reproducibility study for the different days
- 78 -
3.5
It can be seen that the thermal resistance and junction temperature are relatively stable between days.
There is, of course, some variation from the mean for both
parameters, but it is always under 10%, and is under 5% in most cases. Figures 6.21 and 6.22 present this same data on a per measurement basis. That is, each grouping of data points represents the same measurement trial. We can see that the points lie almost on top of each other. The raw data of these tests is given in the appendix. 340
Junction Temperature [K]
330
DAY 1 DAY 2 DAY 3
320 310 300 290 280 270 0
2
4
6
8
Measurement
Fig. 6.21. The variation in junction temperature as it changes measurement trials
- 79 -
10
1
Thermal resistance, HA [K/W]
0.95 0.9 0.85
Day 1 Day 2 Day 3
0.8 0.75 0.7 0.65 0.6 0.55 0.5 0
2
4
6
8
10
Measurement
Fig. 6.22. The variation in thermal resistance as it changes measurement trials
7. Micro-Scale Effects It has been shown during the course of this study that thinning the elements will allow higher heat loads to be dissipated. Recent research has also shown that the last 1015 years have brought dramatic changes in the sciences. A major motivating factor for these changes is the ability to fabricate devices on the micro and nano-scale. Micro-scale devices do not always obey the macro-scale models. This is especially true in the area of heat transfer. The familiar Fourier mode of conduction becomes invalid when the size of the device is close to the mean free path of the energy carriers [Tzou, 1997]. In the micro-scale, using the bulk thermal conductivity (listed in standard tables) can cause very grievous errors in heat transfer calculations.
- 80 -
An effective thermal conductivity can be
found through micro-scale modeling and subsequently used in the TE modeling equations. Electrical properties are also affected in a micro-scale regime, but at smaller scales. The electrical resistivity (also listed in standard tables) is the main parameter of interest in this study. If has any significant perturbation from bulk values, it would also need to be included in the TE models. The next few sections will discuss when these effects become important. They will also propose and discuss some of the applicable small scale models. In light of such changes we will also take a brief look at the response in figure of merit. i. Range of Applicability Thermoelectric materials, as mentioned earlier, are generally made from doped semi-conducting materials, such as Bismuth Telluride. These materials have relatively few free electrons. Therefore, heat is carried almost solely by phonons and transferred by phonon interactions. Recall that if these phonons are confined or bound in some way heat transfer is diminished. For a device to be considered micro-scale its thickness need only be of the same order of magnitude as the average phonon mean free path. For the most abundant semi-conductor, intrinsic silicon, the mean free path is approximately 300 nm [Swartz and Pohl, 1989].
If this value is used as an estimate for TE materials, the
scale under which micro-scale effects should be checked for a TE device is 3 µm. Throughout the course of this study it was shown that if we wish to operate at optimum conditions - geometry and current - that we will have a minimal element side length of 500 µm, for the 20 W heat load shown in figure 5.19. This number was found by assuming one element side length equals the element height. Mathematically, that is:
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x1 x 2 = γ = x 2 if x1 = L L
(7.1)
It is therefore not necessary to incorporate the effects in the modeling, when using the aforementioned optimization techniques. If performance alone is considered, at high heat loads, we may need the elements to thin close to the micro-scale, as was shown in figure 5.18. Fast transient response can also be a domain where micro-scale effects become prevalent. Since this study only considers TE cooling for steady state conditions, transient effects can be neglected. Although these effects are not expected to be an issue in bulk devices, new TE materials (low dimensional structures) definitely fall in a microscale regime.
ii. Phonon Radiative Transfer Due to the fact that phonon interactions are the dominant mode of heat transfer, we can choose a relatively simple model to describe their behavior. We can assume a one-dimensional phonon radiative transfer model (PRT) model. It is named radiative transfer because the final equation ends up looking very similar to a conventional radiation. This model is much easier to use than the full Boltzmann transport equation or molecular dynamics models. The PRT model was developed by Majamdar in 1993 and is derived from the Boltzman transport equation. It is given as [Majamdar, 1993], o
f − fw ∂f w ∂f ⎛ ∂f ⎞ ≅ w + vx w = ⎜ w ⎟ τ ∂t ∂x ⎝ ∂t ⎠ scattering
(7.2)
The subscripts (x and w) represent the direction of phonon motion and the frequency of phonon vibration, respectively. The equation basically states that a perturbation from the
- 82 -
equilibrium state f wo , results from phonon scattering for the entire relaxation time. After some complicated integrations and algebraic manipulations, the equation for phonon radiative transfer can be simplified to [Majumdar, 1993],
q =
σ (T 1 4 − T 04
)
(7.3)
3 ⎛ L⎞ ⎜ ⎟ +1 4 ⎝ l ⎠
The parameters of this equation include the Stefan-Boltzmann constant, σ, the temperature on either side of the material in question, T1 and T0, and the length of the material divided by the mean free path,
L (the acoustic thickness). Eqn. 7.3 is, at this l
point, very similar to a standard heat transfer approach to radiation. This equation can be reduced to a non-dimensional heat flux by dividing the left hand side of by the numerator on the right hand side.
This equation can be considered an effective thermal
conductance, and is given as [Majumdar, 1993]: q 1 = 4 3⎛L⎞ σ T − T0 ⎜ ⎟ +1 4⎝ l ⎠
(
4 1
(7.4)
)
The heat flux varies from a value of unity at zero thickness a value of zero for an infinitely thick material.
Figure 7.1 shows how this relationship with the acoustic
thickness. It was shown by Majumdar that this equation matches almost identically to a theoretical prediction of the modified diffusion equation. This approximation is used because the diffusion method is much more time consuming and complicated.
- 83 -
1
Non-Dimensional Heat Flux
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10
15
20
25
30
35
40
Acoustical Thickness (L/l)
Fig. 7.1. EPRT prediction of non-dimensional heat flux vs. acoustical thickness This equation can also be manipulated to give an effective thermal conductivity. Note: It
(
)(
)
is assumed here that T12 + T02 T1 + T0 = 4T 3 ⎤ ⎡ ⎥ ⎢ 4σT L 1 l ⎥ = Cv ⎢ = 3⎛L⎞ 4 ⎛ l ⎞⎥ 3 ⎢ ⎜ ⎟ +1 ⎢1 + 3 ⎜ L ⎟ ⎥ 4⎝ l ⎠ ⎝ ⎠⎦ ⎣ 3
k eff
(7.5)
The new equation can be plotted against the acoustic thickness. The values for specific heat, C, can be found in property tables for silicon. The values for phonon velocity, v, are found through the following equation [Majumdar, 1993]: v=
16σT 3 C
(7.6)
- 84 -
The values used in this study are summarized in table A.12, found in the appendix. We can see that these values are strong functions of temperature. Figure 7.12 shows how the effective thermal conductivity varies with acoustic thickness at various temperatures. It should be noticed in the above figure that the bulk thermal conductivity of 1.5 W/m-K is approached by the model at 300 K.
14 200 K Bulk, STP 300 K 400 K 500 K 600 K
Thermal Conductivity (W/mK)
12 10 8 6 4 2 0 0
10
20
30
40
50
60
Acoustic Thickness (L/l)
Fig. 7.2. EPRT prediction of thermal conductivity for BiTe as compared to constant bulk conductivity If the TE devices used in this system were at or below an acoustic thickness of approximately 10-15, we would have to incorporate a reduced effective thermal conductivity. Looking again at the figure of merit of eqn. 2.6, we can see that this would actually be beneficial. For bismuth telluride, if the effective thermal conductivity were to be reduced by 10%, it would cause an 11.1% increase in figure of merit, and if the
- 85 -
effective thermal conductivity were reduced by 25%, it would cause a 33% increase in figure of merit!
iii. Boundary Resistances The thermal boundary resistance models offer a different description of the thermal character of small scale devices. The two models that will be discussed here are the acoustic mismatch model and the diffuse mismatch model.
A comprehensive
examination of these models with application to solid-solid interface was presented by Swartz and Pohl in 1989. a. Acoustic Mismatch Model The acoustic mismatch model is basically analogous to Snell’s law of reflection. If, like photons, phonons are incident on the interface at a certain angle, a similar process is thought to occur. If the incident angle of phonon propagation is less than a critical angle from the normal, the phonon will transmit through the second material at an angle related to Snell’s law. This relationship is given as: sin θ 1 sin θ 2 = c1 c2
(7.7)
The c parameters in this equation represent the phonon wave speed in each material, while the incident and refracted angles given by θ. If the incident angle is greater than the critical angle the phonon will reflect at an angle that is opposite the incident angle. The two conditions are shown in figure 7.3.
- 86 -
Critical Angle
θ1
θ1’
θ1’
c1 > c2
Material 1 Material 2
θ2 Normal
Fig. 7.3. Specular phonon boundary scattering It is therefore assumed in this model that no scattering occurs at the interface. This approach best applies to low temperature interactions, where the phonon wavelength is much longer than other interface lengths [Silva, 2003]. That is, surface roughness and defects would not be large enough to cause unpredicted scattering. In the TE study temperatures most often stay at or above ambient. That being said, this model should be discussed, because it ends up being a decent approximation for solid-solid interfaces [Swartz and Pohl, 1989]. To describe this model we must first define a quantity called the energy transmission probability, which is given as [Swartz and Pohl, 1989]:
α '1→2 =
4Z1 Z 2 (Z1 + Z 2 ) 2
(7.8)
The subscripts represent the material, and this equation is merely a function of the acoustic impedances [Swartz and Pohl, 1989]:
- 87 -
Z 'i = ρ i ci
(7.9)
The variable, ρ, is density, not be confused with electrical resistivity. For simplicity,
α '1→2 is assumed constant with respect to the incident angle. This is essentially assuming that each material has isotropic properties. Using this information, we can define the equation for boundary resistance as [Swartz and Pohl]: −1
R Am
⎡ ⎡ ⎤⎤ = ⎢2.04 × 1010 ⎢∑ c 1−, 2j Γ1, j ⎥ ⎥ T −3 ⎢⎣ ⎣ j ⎦ ⎥⎦
(7.10)
It should be noted that there are three phonon wave speeds, denoted by the subscripts, j: two transverse (assumed to be the same) and one longitudinal. The term Γ1,j, is simply equal to
1 α 1→2 under these assumptions. The constant before the summation is found by 2
multiplying out fundamental constants and the final units of R Bd are K/W-cm2. To change this into something that can be used in the TE modeling, we need to relate the boundary resistance to heat transfer. Taking the simple one dimensional case, we can assume that heat flow is restricted by a boundary resistance on each side of the TE element in addition to the conventional bulk resistance (due to the element itself). Since all of the heat in this situation (two solids in contact) is transferred by conduction, it can be reasonably assumed that: Rbulk + 2 R Am =
∆x ∆T = k eff . A Q
(7.11)
The term ∆x is the length of the material of interest and the area, A, is the contact area between the two materials. We can now rearrange the equation to get a function for effective thermal conductivity.
- 88 -
k eff . =
(2 R Am
∆x + Rbulk )A
(7.12)
Note: The bulk resistance term is found by the following:
Rbulk =
∆x
(7.13)
k bulk A
b. Diffuse Mismatch Model The diffuse mismatch model is very different in concept from the acoustic mismatch model.
For this model the interface is characterized by fully diffusive
scattering. Figure 7.4 shows a visual aid of this process. Incident Phonon
N1 > N2
Diffuse Scattering
Material 1 Material 2
Fig. 7.4. Diffusive boundary scattering The nature of scattering on each side of the interface depends on each materials density of phonons. In solid materials this quantity (the density of phonons - analogous to the density of states) describes the continuous range of energies for the phonon modes.
- 89 -
Phelan showed that this model agrees fairly well with experimental data with high applied temperatures [Phelan, 1998]. Therefore, this method would be the one of choice if small scale thermal resistance could not be neglected in the TE device modeling. Following a similar development as was done for the acoustic mismatch model, the transmission probabilities is given as [Swartz and Pohl, 1989]:
α i (w) =
∑c j
3−1, j
∑c
i, j
N 3−1, j (w, T )
(7.14)
N i , j (w, T )
i, j
Here again it should be pointed out that the first subscript, j, refers to the different phonon waves: two transverse and one longitudinal. The parameter N is called the density of phonons and is dependent on temperature and frequency.
If we assume a similar
isotropic approximation, while still allowing the different phonon speeds, we can get a similar term for boundary resistance, which is given as [Swartz and Pohl, 1989]: −1
R Dm
⎡ ⎡ ⎤⎡ ⎤⎤ −2 −2 ⎢ ⎢∑ c 1, j ⎥ ⎢∑ c3−1, j ⎥ ⎥ j ⎦⎣ j ⎦ ⎥ T −3 = ⎢⎢1.02 × 1010 ⎣ −2 ⎥ c 1, j ∑ ⎥ ⎢ i, j ⎥⎦ ⎢⎣
(7.15)
The constant found before the summations is a simplification of the fundamental constants and the units for R Bd are, again, K/W-cm2. It can be seen that the acoustic and diffuse mismatch models differ by a factor of two in the preceding constant, and have slightly different summation terms. Assuming everything is kept constant between the models, we can get a ration of effective thermal conductivities [Swartz and Pohl, 1989]:
- 90 -
R Dm R Am
⎤ ⎤⎡ ⎡ −2 −2 ⎢∑ c 1, j Γi , j ⎥ ⎢∑ c i , j ⎥ j ⎦ ⎦⎣ j =2⎣ ⎤ ⎤⎡ ⎡ −2 −2 ⎢∑ c 1, j ⎥ ⎢∑ c3−1, j ⎥ ⎦ ⎦⎣ j ⎣ j
(7.16)
Swartz and Pohl showed that, at least in the solid-solid interface, this difference was not very significant.
Figure 7.5 gives a non-dimensional representation of the
boundary resistance ratio. If also shows that the models will predict the same result within approximately 40%.
Fig. 7.5. Boundary resistance ratio between the diffuse mismatch and the acoustic mismatch models [Swartz and Pohl, 1989] We again would like to cast the diffuse mismatch model prediction into something that can be simply plugged into the TE simulations. This can be done using the same logic as before; hence, the effective thermal conductivity can be written as:
- 91 -
k eff . =
∆x (2 R Dm + Rbulk )A
(7.17)
Examining both eqn. 7.13 and eqn. 7.15, we can see that as the heat transfer length becomes long, the boundary resistance becomes of less importance. This happens because the boundary resistance is constant while the bulk resistance keeps increasing with length. The bulk resistance basically dominates and forces the effective thermal conductivity to the bulk value at large values of ∆x. We can plot the thermal conductivity ratio (effective: bulk) as a function of heat transfer length.
Figure 7.6 shows this
situation. 1.2
k_eff/k_bulk
1 0.8 0.6 0.4 0.2 0 1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 L [m]
Fig. 7.6. The ratio of effective thermal conductivity to the bulk thermal conductivity for Bismuth Telluride. From this analysis it can be seen that 95% of the bulk value of thermal conductivity is reached when the heat transfer length becomes about 10 µm. This size
- 92 -
will still be much smaller than the smallest element reached using the optimization methods of this study. If the higher heat loads ever do approach this range the effective thermal conductivity can simply be plugged into the model, as mentioned before. c. Electrical Contact Resistance If we take another look at the figure of merit, we see that an increase in electric resistivity will have adverse effects. For Bismuth Telluride, an increase of 10% in the resistivity will cause a 9.1% reduction in the figure of merit. If the resistivity is increased by 30%, the figure of merit is decreased by about 23%. This decrease would certainly shift the optimum points that were found in the last chapter. Joule heating would increase and the efficiency/performance would subsequently go down. The electrical contact resistance will now be discussed to see what, if any, deviation from bulk can be expected. Before a semiconductor and a metal are in contact their electronic energy levels are different. This difference sets up a potential barrier which can be described as electronic contact resistance. The contact resistance can be described in a very similar way to the thermal boundary resistance. We can define an effective electrical resistivity, given as:
ρ eff . =
(Rbulk
+ 2 Rbd )A L
(7.18)
The well-known bulk resistance is given as: Rbulk =
ρ bulk L
(7.19)
A
Figure 7.7 describes how the ratio of bulk resistivity to effective resistivity changes with the contact length, L. The contact resistance values are given in terms of Rc = Rbd A . The value Rc = 2.6 × 10 −12 Ω − m 2 chosen to plot was a value measured by Silva and Kaviany
- 93 -
[Silva, 2003]. Arbitrary resistance values were chosen on either side to show what would happen if a larger or smaller electrical resistance were actually used. We can see that even in a worst case scenario, Rc = 5 × 10 −12 Ω − m 2 (~ double the real theoretical value), that the resistivity will have reached 95% of its bulk value by 20 µm. With the other contact
resistances
of
figure
7.7,
95%
of
bulk
is
reached
by
10
µm
(assuming Rc = 2.6 × 10 −12 Ω − m 2 ) and ~3 µm for the best contact. 1.2 Rc =5*10^-12 Ωm2 Rc = 2.6*10^-12 Ωm2 Rc = 10^-12 Ωm2
1.0
rho_bulk/rho_eff
0.8
0.6
0.4
0.2
1.E-11
1.E-09
1.E-07
1.E-05
1.E-03
0.0 1.E-01
L [m]
Fig. 7.7. Ratio of bulk resistivity to the effective resistivity In conclusion, here again we are not in a regime where the contact resistance becomes important. If we were to operate in that regime the effective electrical resistivity would just need to be substituted into the TE equations.
- 94 -
8. Conclusions Several things can be concluded about the optimization of TE devices from this study. It is evident that a large market is in place for which TE technology could provide solutions. With the help of primary research into the improvement of TE materials, we can expect this market to grow significantly in coming years. System design of thermoelectric modules for refrigeration applications has been slightly overlooked due to the fact that material design seems much more pressing. Noting this fact, this study took an in-depth look into system optimization. It was shown that current as well as geometry can (and must) be optimized in order to achieve a design which is truly optimized. Depending on the constraints, it was shown that there are four possible equations with which to optimize the entire system. The geometry equations, in particular, were findings of this study. It was shown that all four of these equations are important and as many as possible should be used in system design. It was shown that the two optimum currents could be set equal to each other getting a system which has an optimum COP as well as a minimal junction temperature. This is also true of the geometry: module geometry can be defined which would allow the system to be optimized for both COP and Tj. It was then shown through experimentation that these two optimum currents could indeed be set equal to each other. This validated the existence of an optimum point for the given geometry and heat sink thermal resistance. Through the use of multiple modules, the general trends of geometry optimization were also validated.
- 95 -
The
experimental data agreed inside the measurement uncertainty (found in the appendix), as long as the thermal resistance could be achieved. Lastly, thermal and electrical effects in the micro-scale were then discussed. It was found that even though these effects need not be considered under the conditions of this study that they could be considerable at less than 10 µm. While a reduction in thermal conductivity would help the system, an increased resistivity would adversely affect the design. The results of this research also demonstrate the intricate nature in which phenomena are intertwined in TE design. Each parameter must be carefully considered before an optimum solution can arise. Since TE devices are very sensitive to boundary conditions, a slight error could mean significant deviation from that optimum.
9. Future Work There are plenty of directions that future work could take. First and foremost, it would be interesting to more comprehensively validate the total optimum points. This would require a much wider selection of TE modules to test optimum geometry conditions. Second, a study could be conducted to determine if shrinking the geometry could provide a reduced thermal conductivity without affecting the electrical properties. If this could be done the optimum points would be shifted beneficially. It would also be of interest to this researcher to examine the generation side of TE devices. It may be possible to arrive at similar optimum conditions. In closing, there is plenty of room for intelligent design with TE technology. The number of ways in which TE devices could be applied to solve real world problems is practically unlimited.
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10. References Angrist, S. W., Direct Energy Conversion, 4th ed., Allyn and Bacon Inc., Boston, 1982. Bierschenk, Jim and Johnson, Dwight “Extending the Limits of Air Cooling with Thermoelectrically enhanced Heat Sinks,” Marlow Industries, Inc. Dallas, TX, 2004. Brooks, R.D. and Mattes, H. G., “Spreading Resistance Between Constant Potential Surfaces” The Bell System Technical Journal, pp.775-784, March 1971 Buchner, Bernard, “Thermoelectric Generation,” Institute for Solid State Research, Available [online] at http://www.ifw-dresden.de/iff/index.htm Cheeke, David and Ettinger, Harry, “Microscopic Calculation of the Kapitza Resistance between Solids and Liquid Helium,” Physical Review Letters, Volume 37, Number 24, pp. 1625-1628, 13 December, 1976. Cheeke, David, Ettinger, Harry, and Hebral, B. “Analysis of Heat Transfer Between Solids at Low Temperatures” Canadian Journal of Physics, Volume 54, pp.1749-1763, May 1976. Chen, Gang, “Diffusion-Transmission Interface Condition for Electron and Phonon Transport” Applied Physics Letters, Volume 82, Number 6, pp. 991-993, 10 February 2003. Electricool “Ready to Use Thermoelectric Products,” Available [online] at http://www.electracool.com/products.html, 2003. Elsworth, Micheal J., “The Challenge of Operating at Ultra-Low Temperatures.” Electronics Cooling, 2001. Fan, X., Croke, C., J.E. Bowers, A. Shakouri, et al., “SiGeC/Sisuperlattice micro cooler,” Applied Physics Lett. Volume 78, Number 11, 2001. Ferrotec, “Reliability of Thermoelectric Cooling Modules,” Available [online] at http://www.ferrotec.com/usa/thermoelectric/ref/3ref10.htm, 2003 Hanneken, Mike, “Thermoelectric Cooling for Cryopreservation,” University of Missouri Undergraduate Thesis [Internal Document], 12 May 2005. Harman, T.C., Taylor, P.J., Walsh, M.P, and LaForge, B.E., “Quantum Dot Superlattice Thermoelectric Materials and Devices” Science, Vol. 297, pp. 2229-2232, September 2002.
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Harman T. et al., “Improvement in ZT using Quantum Dot Sturctures,” MIT Lincoln Lab, Science, Volume 27, Sept. 2002. Heremans, P. and Arkhipov, V.I, Emelianova, E.V. and Adriaenssens, G.J., and Bassler, H. “Charge Carrier Mobility in Doped Semiconducting Polymers” Applied Physics Letter, Vol. 82, Number 19, pp. 3245-3247, May 2003. Hi-Z “Hi-Z Technology,” Available [online] at http://www.hi-z.com , 2003. Huang, Mei-Jiau, Yen, Ruey-Hor, and Wang, An-Bang, “The influence of the Thomson effect on the performance of a thermoelectric cooler.” Heat and Mass Transfer, Vol. 48, pp. 413-418, 2005. Incropera FP, De Witt DP, 1990, Fundamentals of Heat and Mass Transfer, Wiley and Sons, New York, NY. Li, Deyu, Huxtable, Scott T., Habramson, Alexis R., Majumdar, Arun, “Thermal Transport in Nanostructured Solid-State Cooling Devices” Journal of Heat Transfer, Volume 127, pp. 108-114, January 2005. Li, Deyu, Huxtable, Scott, Abramson, Alexis, and Majumdar, Arun, “Thermal Transport in Nanostructured Solid-State Cooling Devices” Transactions of the ASME, Vol. 127, pp. 108-114, January 2005. Majumdar, A., “Microscale Heat Cunduction in Dielectric Thin Films” Journal of Heat Transfer, Volume 115 pp. 7-16, February 1993. Martin, Bjorn, Wagner, Achim, and Kliem, Herbert, “A Thermoelectric Voltage Effect in Polyethylene Oxide” Journal of Physics D: Applied Physics, Vol. 36, pp. 343347, 2003. Melcor “Melcor. Your Total Thermoelectric Cooling Solution,” Available [online] at http://www.thermoelectrics.com/introduction.html, 2003. Minco, Personal Correspondence (phone conversation - for pricing information), October, 2005. Moyzhes, B. and Nemchinsky, V., “Thermoelectric Figure of Merit of MetalSemiconductor Barrier Structure Based on Energy Relaxation Length” Applied Physics Letters, Vol. 73, Number 13, pp.1895-1897, September 1998. Munkinmaentie et. Al. “TE Energy” Available [online] at www.teenergy.fi, May 2005. NREL, “Geothermal Energy,” Available [online] at http://geothermal.inel.gov/maps/index.shtml, 2005.
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Omega, “Technical Reference Guide - Thermocouples” Available [online] at http://www.omega.com/temperature/Z/zsection.asp, May 2005 Pearse, A. G. E. “Rapid Freeze-Drying of Biological Tissues with a Thermoelectric Unit,” Journal of Science Instruments Vol. 40 pp. 176,177. Peltier Device Information Directory, “Peltier Device Manufacturers,” Available [online] at http://www.peltier-info.com/manufacturers.html. Phelan, P.E., V.A. Chiria, and T-Y T Lee, “Current and Future Miniature Refrigeration Cooling Technologies for High Power Microelectronics,” IEEE Transactions on Components and Packaging Technologies, Vol. 25, No. 3, 2002. Phelan, P.E. “Applications of diffuse mismatch theory to the prediction of thermal boundary resistance in thin-film high Tc superconductors,” ASME Journal of Heat Transfer, Volume 120, pp. 37-43, 1998 PIMA ISS Acceleration Handbook, “Advanced Thermoelectric Refrigerator/Freezer/Incubator (ARCTIC)” Available [online] at http://pims.grc.nasa,gov, May 12, 2003. Reeves, Geoffrey and Harrison, Barry H. “Contact Resistance of Polysilicon/Silicon Interconnections” Electronics Letters, Volume 18, Number 25, pp.1083-1086, 9 December 1982. Reeves, Geoffrey and Harrison, Barry H. “Determination of Contact Parameters of Interconnecting Layers in VLSI Circuits” IEEE Transaction on Electron Devices, Volume ED-33, Number 3, pp. 328-334, 3 March 1986. Scheel, Kasey, “Heat Sink Characterization,” University Undergraduate Thesis [Internal Document], September 2005.
of
Missouri
Scott, David B., Hunter, William R., and Shichijo, Hisashi, “A Transmission Line Model for Silicided Diffusions: Impact on the Performance of VLSI Circuits” IEEE Transaction on Electron Devices, Volum ED-29, Number 4, pp. 651-660, April 1982. Seiko Instruments Inc. “Endless Evolution” Available [online] at http://www.sii.co.jp/info/eg/thermic_main.html, 2002. Semiconductor Industry Association, International Technology Roadmap for Semiconductors: 2004 Update. Shakouri, Ali and Li, Suquan, “Thermoelectric Power Factor for Electrically Conductive Polymers” Proceeding of the International Conference on Thermoelectrics, September 1999.
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Shi, Li, Li, Deyu, Yu, Choongho, Jang Wanyoung and Kim, Dohyung, “Measuring Thermal and Thermoelectric Properties of One-Dimensional Nanostructures Using a Microfabricated Device” Journal of Heat Transfer, Vol. 125, pp. 881-888, October 2003. Silva, Luciana W., and Kaviany, Massoud, “Micro-Thermoelectric Cooler: Interfacial Effects on Thermal and Electrical Transport” International Journal of Heat and Mass Transfer, Volume 47, pp. 2417-2435, 2003. Simons, R. E., and R. C. Chu. “Applications of Thermoelectric Cooling to Electronic Equipment: A Review and Analysis,” Sixteenth IEE Semi-Therm, March 2123, 2000. Smalley, R.E., Nobel Laureate 1996, “Our Energy Challenge,” Columbia University, NYC, September, 2003. Solbrekken, G. L., K. Yazawa, and A. Bar-Cohen, “Thermal Management of Protable Electronic Equipment using Thermoelectric Energy Conversion,” Itherm04, 2004. Solbrekken, G. L., K. Yazawa, and A. Bar-Cohen, 2003, “Chip Level Refrigeration Of Portable Electronic Equipment Using Thermoelectric Devices,” Proceedings of InterPack 2003, Maui, HA, Jul 6-11, Paper IPACK2003-35305. Solbrekken, Gary, Zhang, Yan, Bar-Cohen, Avram, and Shakouri, Ali, “Use of Superlattice Thermoionic Emission for ‘Hot Spot’ Reduction in a Convectively-Cooled Chip” InterPack, 2002. Stratton, R., “Diffusion of Hot and Cold Electrons in Semiconductor Barriers” Physical Review, Volume 126, Number 6, pp. 2002-2014, 15 June 1962. Swartz, E.T. and Pohl, R.O, “Thermal Boundary Resistance” Reviews of Modern Physics, Volume 61, Number 3, pp. 605-669, July 1989. Tzou, D. Y., Macro-to Microscole Heat Transfer: The Lagging Behavior, pp. 160, Taylor and Francis, 1997 Venkatasubramania, Rama, Siivola, Edward, Colpitts, Thomas and O’Quinn, Brooks, “Thin-Film Thermoelectric Devices with High Room-Temperature Figures of Merit” Nature, Vol. 413 pp. 597-602, October 2001. Venkatasubramanian et al., “High ZT High ZT p-BiTe/SbTe BiTe/SbTe Superlattice” RTI, Nature, Volume11, Oct. 2001.
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11. Appendix i. Chapter 2 Thermoelectric technology has proven to be very reliable. The modules have no moving parts and can withstand many thermal cycles, as was mentioned in chapter 2. Figure A.1 shows the data for one such reliability test. We can see that the resistance change is well below 2% all the way until 28,000 thermal cycles are reached. The number of cycles before failure in this module was actually below the average.
Fig. A.1. Reliability data for a selected thermoelectric module [Ferrotec, 2005] ii. Chapter 3 Chapter 3 stated that the amount of carriers in a material could significantly change the material properties. This is what led to the choice of semi-conductors for TE materials. Even in the sub-set of semi-conducting material there is a lot of variation in material properties. Figure A.2 demonstrates this by showing how doping concentration can change the electrical resistivity.
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Fig. A.2. Change in electrical resistivity with doping concentration [Heremans, 2003]. During the course of optimization it was stated that there are critical values of thermal resistance and heat input that would cause the optimum geometry to be undefined. These values can be found by the following relationships:
kρI ⎞ ⎛ Qcrit . = 2 NI ⎜ αTc − ⎟ α ⎠ ⎝
(A.1)
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ψ ha ,crit
Q 2N = Q ⎞ ⎛ (2 Nα )⎜αITc − ⎟ − 2 NkρI 2 2N ⎠ ⎝
αITc −
(A.2)
iii. Chapter 5 Chapter 5 stated that optimization through one technique will not necessarily give a true optimum. Figure A.3 shows, by means of a topographical map, how a mistake in optimization can easily be made. If we arbitrarily choose a longitudinal coordinate and follow it down, the chances are slim that we will achieve a true maximum.
12,000 ft 9,000 ft
Max ?
14,000 ft
Fig. A.3. Topographical map of a mountainous region. In this situation it would seem that top right peak is the maximum elevation on the map, at 12,000 feet. Of course, the lower right peak is 2,000 feet higher, but the chances are close to zero that it will be pin-pointed in a one-dimensional optimization method.
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The optimization process a designer must choose is intrinsically dependent upon the constraints. For instance, it may be necessary to use a particular COP in a system. Assuming this, let us choose a constant heat input of 75 W a constant work input of 40 W. Figure A.4 shows how the junction temperature and element geometry would respond to different current inputs. Current [A] 9.000 10.000
14.000
19.000
24.000
29.000 362.000 360.000 358.000
Gamma [m], COP
1.000
356.000 COP Gamma Tj
0.100
354.000 352.000 350.000 348.000 346.000
0.010
344.000 342.000 0.001
340.000
Fig. A.4. Q = 75 W, W = 40 W, = 0.4 K/W, I = Independent Variable Although there is a minimum junction temperature as a function of current, a constant work is very impractical for any real device. For this to be possible, the geometry must increase at each current, as shown in the figure. Following the same brute force method many different things can be attempted. As another example, let us see if there is an optimum as thermal resistance, ψha, is changed. Figure A.5 shows how COP and junction temperature change with respect to thermal resistance. It can be seen that there is no optimum in either COP or junction temperature, both parameters simply
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climb, exponentially, with an increasing heat sink resistance. COP climbs because the work is artificially held constant (due to a constant module and current input). 3.5
900
3.0
Q Tj COP
700 600
2.5
500
2.0
400
1.5
COP
Heat Load [W], Junction Temp. [K]
800
300
1.0
200 0.5
100 0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Thermal Resistance [K/W]
Fig. A.5. ∆T = 30 K, I = 16.9 A, γ = 0.006 m, ψha = independent variable . There are many more relations of this form, but they are rarely useful and most do not provide any sort of optimum, as indicated by the table. For the rest of this study we narrow the focus to optimizing under combinations of the four optimum equations (eqns. 5.8, 5.11, 5.12, and 5.13).
Under this assumption, the search for comprehensive
optimums (given constraints) is greatly simplified. Let us start out with the better known current optimization schemes.
iv. Chapter 6 Chapter 6 requires a lot of temperature measurement during the course of the experiments. The majority of the temperature readings in this study were taken with a - 105 -
type E thermocouple, purchased from Omega. The polynomial for converting the voltage reading into temperature data is given in table A.1.
Table A.1 Type E thermocouple data [Omega, 2005] Type E Polynomial 0.104967248 17189.45282 -282639.085 12695339.5 -448703084.6 11086600000 -1.76807E+11 1.71842E+12 -9.19278E+12 2.06132E+13
a0 a1 a2 a3 a4 a5 a6 a7 a8 a9
To estimate the measurement error of the experiments in chapter 6, one can do uncertainty propagation. This is done by taking partial derivatives of the quantity of interest. The following equations were found for each of the parameters of interest, junction temperature, coefficient of performance, and heat sink thermal resistance in order.
uTj = [uTc ] 2 + [u vQ I Qψ jc ] 2 + [u I VQψ jc ] 2 ⎛ IQ u COP = [u vQ ⎜⎜ ⎝ VTE I TE
u HA
⎛ V ⎞2 ⎟⎟] + [u IQ ⎜⎜ Q ⎝ VTE I TE ⎠
(A.3)
⎞2 ⎛ −V I ⎟⎟] + [uVTE ⎜⎜ 2 Q Q ⎠ ⎝ VTE I TE
⎛ ⎛ ⎞2 ⎛ − (Th − Ta ) I Q −1 ⎜ [u ] 2 + [u ⎜ ⎟] + [uVQ ⎜ Ta ⎜ ⎟ ⎜V I +V I ⎜ Th Q Q ⎝ VTE I TE + VQ I Q ⎠ ⎝ TE TE =⎜ ⎜ ⎛ ⎞ ⎛ − (Th − Ta ) I TE ⎞ 2 ⎟] + [u ITE ⎜ − (Th − Ta )VTE ⎟] 2 ⎜ [uVTE ⎜⎜ ⎜ ⎟ ⎟ ⎜ ⎝ VTE I TE + VQ I Q ⎠ ⎝ VTE I TE + VQ I Q ⎠ ⎝
⎛ −V I ⎞2 ⎟⎟] + [u ITE ⎜ 2TE TE ⎜ I I ⎠ ⎝ Q Q
⎞2 ⎟] ⎟ ⎠
(A.4)
1
⎞2 ⎛ − (Th − Ta )VQ ⎟] + [u IQ ⎜ ⎟ ⎜V I +V I Q Q ⎠ ⎝ TE TE
⎞2 ⎟] ⎟ ⎠
⎞2 +⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(A.5)
The subscripts Tx, Q, and TE are readings for the given temperature, heat load, and work input into the thermoelectric. Table A.2 gives the approximate uncertainties for each of these readings.
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Table A.2. Uncertainty in measurements Error Device
Uncertainties
E-type T/C [K] Q volt meter reading [V]
1 0.001
Analog Q current reading [A] TE Voltage Reading [V] TE Current Reading [A]
0.05 0.003 0.015
Once the uncertainty has been calculated we can run the experiments of chapter 6. The models dictate the actual operating conditions for the tests. These were found by first plugging in the real module parameters into the models as given by table A.3. Table A.3. Specifications for commercially available modules Modules 9504/071/150B (81036) 9504/071/120B (81460) 9500/127/085B (81085) 9500/127/060B (81026)
Couples
Element Area
Height
γ
N*γ
71.000
0.000
0.002
0.003
0.218
71.000
0.000
0.002
0.003
0.188
127.000
0.000
0.001
0.002
0.238
127.000
0.000
0.002
0.001
0.168
A map for the experiments can now be defined by the optimum models. That is, a couple of (hopefully) achievable points can be taken off each model for experimentation. Tables A.4-A.7 give these desired values for their respective modules. Table A.4. Experimental test for module 81036
Module: 81036 Tj,min HA (K/W) 0.6 0.6 0.6 0.6 0.6 0.6 0.6
Desired Work (W) 11.72 18.5 27 52.72 62.7125 87.98 138.48
Settings Qc (W) 21.7 21.7 21.7 21.7 21.7 21.7 21.7
I (Amps) 4.1 5 6 8 8.65 10 12
COPopt Work (W) 38.927 54 62.7125 67.5 83.4
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Desired HA (K/W) 1 0.7 0.6 0.55 0.45
Settings Qc (W) 12.7 19 21.7 23 27
I (Amps) 6.7 8 8.65 9 10
Table A.5. Experimental test for module 81460
Module: 81460 Tj,min HA (K/W) 0.6 0.6 0.6 0.6 0.6
Desired Work (W) 13.52 32.22 45.5 50.6954 102.6
Settings
COPopt
Qc (W) 17.9 17.9 17.9 17.9 17.9
I (Amps) 4 6 7 7.93 10
HA (K/W) 1.2 1 0.6 0.54 0.4
Desired Work (W) 31.192 35.04 50.6954 67.62 92.565
Settings Qc (W) 7.25 10 17.9 19.6 24.2
I (Amps) 5.6 6 7.93 8.4 9.9
Table A.6. Experimental test for module 81085
Module: 81085 Tj,min HA (K/W)
Desired Work (W)
Settings
3.825 18.3 35.96 59.25 102.69
24 24 24 24 24
0.6 0.6 0.6 0.6 0.6
I (Amps)
Qc (W)
1.5 3 4 5 6.3
COPopt HA (K/W) 1.29 0.96 0.6 0.4
Desired Work (W) 32.305 40.2 59.25 94.248
Settings Qc (W) 13.5 18 26 34.7
I (Amps) 3.5 4 5 6.3
Table A.7. Experimental test for module 81026
Module: 81026 Tj,min HA (K/W) 0.6 0.6 0.6 0.6 0.6
Desired Work (W) 5.67 25.05 50.63 77.8 123
Settings Qc (W) 15.5 15.5 15.5 15.5 15.5
I (Amps) 1.5 3 4.15 5 5.9
COPopt HA (K/W) 1.15 0.83 0.6 0.5 0.4
Desired Work (W) 28.08 36.855 50.63 60.2875 71.55
Settings Qc (W) 7 11 15.5 18.1 20.75
I (Amps) 3 3.5 4.15 4.55 5.2
It was also mentioned in chapter 6 that material dependent properties were later included in the models to match with experiments. This data was given in polynomial form by Rowe. Table A.8 shows the polynomial coefficients. Table A.8. Polynomials for computing temperature dependent properties [Rowe, 2003] Coefficient
Seebeck
Thermal Conductivity
Electrical Resistivity
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0 1 2 3 4 Multiplying Factor
1648.16 -16.2202 0.0644158 -0.0001068 6.201E-08
109.7932 -1.077964 0.005036171 -1.13525E-05 1.02126E-08
6.630679 -0.06096743 0.000216728 -2.94957E-07 1.35E-10
0.000001
0.1
0.00001
A reproducibility study was also run during the course of the experiments in chapter 6. The data collected from the various studies is listed in tables A.9-A.11. For each test thirty measurements were taken. The average and standard deviations of this data are listed here. Table A.9. Reproducability study results, day 1. DAY 1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
Avg. HA
0.601351
0.588014
0.939756
0.907949
0.609216
0.586902
0.91772
0.900215
SD HA
0.002364
0.002386
0.002073
0.002552
0.003348
0.001904
0.002243
0.002321
Avg. Tj
305.0574
305.0516
319.6716
331.9844
282.1237
282.3098
290.8729
302.5615
SD Tj
0.006407
0.005094
0.004932
0.012956
0.006281
0.006357
0.007015
0.004063
2
2
2
2
Measurement #
Table A.10. Reproducability study results, day 2. DAY 2 Measurement #
2
2
2
1
2
3
4
5
6
7
8
Avg. HA
0.600528
0.582687
0.915577
0.900031
0.598458
0.582163
0.936795
0.913142
SD HA
0.009736
0.00306
0.001728
0.000724
0.003735
0.002611
0.002291
0.001191
Avg. Tj
305.2655
304.686
318.6743
331.4732
282.6441
282.93
292.0598
303.576
0.00349
0.008026
0.086578
0.088373
0.004511
0.007895
0.038918
0.025682
SD Tj
Table A.11. Reproducability study results, day 3. DAY Measurement # Avg. HA
3
3
3
3
3
3
3
3
1
2
3
4
5
6
7
8
0.602516
0.576816
0.919588
0.912568
0.600132
0.580251
0.908651
0.898948
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SD HA
0.000651
0.000373
0.000441
0.000417
0.000642
0.000347
0.000571
0.000477
Avg. Tj
305.5557
303.8543
318.6331
333.2986
282.0694
282.2992
291.1829
303.3841
SD Tj
0.011944
0.027314
0.045159
0.022591
0.003603
0.006848
0.021817
0.047185
v. Chapter 7 Chapter 7 stated that the specific heat is a function of temperature.
It was
calculated in order to get the effective thermal conductivity for the EPRT. Table A.9 shows the values used in this calculation. Table A.12. The values of specific heat and phonon velocity used in the EPRT T (K) 200 300 400 500 600
C (J/KgK) 557 710 788 831 860
v (m/s) 13760.524 13760.524 13760.524 13760.524 13760.524
StephenBoltz 40.37853687 40.37853687 40.37853687 40.37853687 40.37853687
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