BOOKCOMP, Inc. — John Wiley & Sons / Page 1016 / 2nd Proofs / Heat Transfer Handbook / Bejan 1016 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1016], (70) Lines: 1825 to 1867 ——— 4.11815pt PgVar ——— Short Page * PgEnds: Eject [1016], (70) where θ is a configuration factor which must be determined empirically. However, to obtain engineering estimates of the thermal performance, Bar-Cohen et al. (1987) have proposed a value of 31.5 for θ. The upper bound on the performance maps is therefore marked with both the CHF limit and the condensation limit. 13.8 THERMOELECTRIC COOLERS The Peltier effect is the basis for the thermal electric cooler (TEC), which is a solid- state heat pump. If a potential is placed across two junctions, heat will be absorbed into one junction and expelled from the other in proportion to the current. Most material combinations exhibit the Peltier effect to some degree. However, it is most obvious across a p-n junction as shown in Fig. 13.34. As electrons are transported from the p-side of a junction to the n-side, they are elevated to a higher-energy state and thus absorb heat, resulting in cooling the surrounding area. When they are transported from the p-side to the n-side, they release heat. The materials that have been used to make TEC include bismuth telluride (Bi 2 Te 3 ), lead telluride (PbTe), and silicon germanium (SiGe). To obtain optimum parameters, these semiconductors are doped during fabrication. Bi 2 Te 3 has the best performance at temperatures of interest for electronic components and is most commonly used. A TEC device is constructed by placing from one to several hundred thermocouples electrically in series and thermally in parallel between two pieces of metallized, thermally conductive ceramic acting as an electrical insulator. For continuous cooling at the low-temperature side of the TEC, the heat absorbed at the cold side, as well as the heat generated by the flow of electricity, must be removed from the hot side by one of the thermal transport mechanisms described previously. Solution of the governing equations for a thermoelectric couple yields a relation for the maximum temperature differential obtainable with such a device, as ∆T max = α 2 T 2 c 2KR (13.93) K = k a A a L a + k b A b L b (13.94) R = ρ a L a A a + ρ b A b A b (13.95) It may thus be observed that the maximum temperature differential can be enhanced by minimizing the product of the thermal conductance K and electrical, resistance R. A TEC device is frequently rated by a figure of merit, as given by FOM = α 2 s ρ TE k TE (13.96) BOOKCOMP, Inc. — John Wiley & Sons / Page 1017 / 2nd Proofs / Heat Transfer Handbook / Bejan THERMOELECTRIC COOLERS 1017 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1017], (71) Lines: 1867 to 1899 ——— -1.24886pt PgVar ——— Short Page PgEnds: T E X [1017], (71) Figure 13.34 Thermoelectric cooler. where α 2 s is the Seebeck coefficient and ρ TE is the resistivity of the TEC element. Values of this figure of merit are typically in the range 0.002 to 0.005 K −1 ,but extensive research is under way to improve this FOM by as much as an order of magnitude in the next few years. The coefficient of performance of a TEC is defined as the ratio of heat pumped to the input power: COP = heat pumped input power = q c P in (13.97) The optimum COP is given by COP opt = T avg ∆T B − 1 B + 1 − 0.5 (13.98) where B = 1 +(FOM · T avg ) (13.99) A TEC may be selected from the performance and COP curves for a given set of design criteria. It is essential that the TEC and heat sink, as well as the power supply used to operate the TEC, be selected together. For increased cooling capacity, TECs may be operated in parallel. However, for lower chip temperatures it may be necessary to cascade several TEC devices or to operate them in series. The biggest limitation in the use of a TEC for commercial electronics is the very low FOM. Recent years have witnessed extensive research in the area of TECs from a material point of view to increase the FOM. Thin-film supperlatice and quantum-well structures have shown evidence of providing high FOM compared to the state-of-the- art TEC. BOOKCOMP, Inc. — John Wiley & Sons / Page 1018 / 2nd Proofs / Heat Transfer Handbook / Bejan 1018 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1018], (72) Lines: 1899 to 1975 ——— 0.20338pt PgVar ——— Normal Page PgEnds: T E X [1018], (72) 13.9 CHIP TEMPERATURE MEASUREMENT As mentioned previously, the total thermal resistance of a package is usually the sum of θ jc (junction to case) and θ ca (case to ambient). The object of most experimental programs is to determine these resistances. This knowledge then allows the junction temperature to be predicted as a function of power and cooling medium temperature and flow rate. θ jc and θ ca are typically measured in a lab environment with the help of test chips popularly known as test vehicles. There are basically two measurement methods commonly used to determine the temperature of a test vehicle die. The first makes use of atemperature-sensitive parameter on the testvehicle todetermine the die temperature, such as one or more dedicated diodes of resistors. Calibration of these temperature sensors can be very time consuming. Furthermore, as a test vehicle can have more than one temperature sensor and there could also be part-to-part variability in the test vehicle coming out of the factory, time to calibrate each temperature sensor could be enormous. Solbrekken and Chiu (1998) introduced a simplified calibration procedure that utilizes single-resistance measurements either at room temperature or at the anticipated test temperature. In their proposed method 30 temperature sensors are selected randomly and are calibrated at three set-point temperatures. Then the average intercept of resistance versus temperature for the 30 temperature sensors is calculated. Later, for the calibration of other test vehicles for the same generation of products, the die is calibrated at a single temperature and the average intercept calculated for the 30 samples is used in conjunction with the single-point calibration for determining the temperature of the test vehicle. A second category of chip tem- perature measurement techniques includes liquid crystals, thermographic phosphors, laser scanners, infrared camera, and so on. Each of these methods presents a unique set of concerns, ranging from temperature resolution to cost of implementation. 13.10 SUMMARY The fundamental principles and concepts for thermal management of electronics were presented in this chapter. Simplified equations for first-order analysis of the temperature on electronic components were also introduced. A variety of cooling techniques, including heat sinks, jet impingement cooling, liquid immersion cooling, and thermoelectric cooling, were also discussed. NOMENCLATURE Roman Letter Symbols A area, m 2 a heat source area, m 2 B constant, dimensionless b heat source size, m PCB spacing, m BOOKCOMP, Inc. — John Wiley & Sons / Page 1019 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 1019 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1019], (73) Lines: 1975 to 1975 ——— 0.55142pt PgVar ——— Normal Page PgEnds: T E X [1019], (73) Bi Biot number, dimensionless [ = hL/k ] C constant in heat transfer coefficient correlations, dimensionless C p specific heat, J/hg ·K CHF critical heat flux, dimensionless COP coefficient of performance, dimensionless c speed of sound, m/s D diameter, m d diameter, m El Elenbaas number, dimensionless El modified Elenbaas number, dimensionless F ij radiation view factor from i to j, dimensionless FOM figure of merit, dimensionless f impingement area ratio, dimensionless g gravitational acceleration, m/s 2 H height, m Kennedy (1959) spreading resistance factor, dimensionless hardness, N/m 2 h heat transfer coefficient, W/m 2 · K h fv latent heat of vaporization, J/kg ·K ¯ h Planck’s constant, dimensionless K conductance, W/K k thermal conductivity, W/m · K k b Boltzmann constant, dimensionless L length, width, or thickness, m l mean free path, m m mass, kg ˙m mass flow rate, kg/s N number of atomic layers, dimensionless n number, dimensionless Nu Nusselt number, dimensionless [ = hL/k ] P pressure, N/m 2 plate to air parameter, dimensionless perimeter, m p volume fraction, dimensionless p c threshold volume fraction, dimensionless Pr Prandtl number, dimensionless = µC p /k Q total heat flow, W q heat flow, W q heat flux, W/m 2 R thermal resistance, °C/W or K/W r radius, m asymmetry parameter, dimensionless r T asymmetry parameter, dimensionless Ra Rayleigh number, dimensionless = (gβ∆TL 3 /ν 2 )Pr Re Reynolds number, dimensionless [ = ρVL/µ ] BOOKCOMP, Inc. — John Wiley & Sons / Page 1020 / 2nd Proofs / Heat Transfer Handbook / Bejan 1020 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1020], (74) Lines: 1975 to 2078 ——— 0.41089pt PgVar ——— Normal Page PgEnds: T E X [1020], (74) S shape factor, m 2 shape parameter, dimensionless T temperature, °C or K t time, s V velocity, m/s volume, m 3 W characteristic length (length or width), m x length or distance, m Y area-weighted interfacial gap, m z clear space between fins, m ∆k change in thermal conductivity, W/m · K ∆T temperature difference, °C or K ∆x distance between two points, m Greek Letter Symbols α thermal diffusivity, m 2 /s Seebeck coefficient, dimensionless β volumetric expansion coefficient, K −1 δ thickness or gap width, m penetration depth, m heater thickness, m ε emissivity, dimensionless ζ ratio of heat source area to substrate area, dimensionless η fin efficiency, dimensionless θ thermal resistance, °C/W asperity angle, deg κ conductance, W/K conduction factor, dimensionless µ dynamic viscosity, N · s/m 2 ν kinematic viscosity, m 2 /s π pi, dimensionless ρ density, kg/m 3 σ Stefan–Boltzmann constant, W/m 2 · K 4 surface tension, N/m surface roughness, m τ time constant, s φ Debye temperature, K ϕ particle volume fraction, dimensionless maximum packing fraction, dimensionless ϕ m maximum packing fraction, dimensionless χ ratio of logarithmic conductances, dimensionless Subscripts 0 initial (at time = 0) or based on channel inlet 1 item or entity 1 BOOKCOMP, Inc. — John Wiley & Sons / Page 1021 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 1021 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1021], (75) Lines: 2078 to 2078 ——— 0.00604pt PgVar ——— Normal Page PgEnds: T E X [1021], (75) 1D one-dimensional 2 item or entity 2 a ambient or entity a ad adiabatic amb ambient avg average b boundary or base or entity b bj bare junction base heat sink base area bulk bulk c case or cutout ca case to ambient cm control material co contact D diameter e encapsulant en equivalent normal eq equivalent in-plane ex external or excursion f fluid f g between saturated liquid and saturated vapor filler filler film film fin fin fl flow g interstitial fluid or gas (vapor) gap gap h hydraulic HS heat sink i counter ICP integrated circuit package in inlet j junction a counter jc junction to case jet jet jj mode of sound m continuous phase or matrix and maximum packing fraction max maximum mc between interface and matrix nc noncontacting nom nominal opt optimum out outlet p particle plane BOOKCOMP, Inc. — John Wiley & Sons / Page 1022 / 2nd Proofs / Heat Transfer Handbook / Bejan 1022 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1022], (76) Lines: 2078 to 2108 ——— -5.80669pt PgVar ——— Normal Page PgEnds: T E X [1022], (76) filler at constant pressure r radiation or radial direction real real s silicon mean condition sat saturation sk heat sink source source sp spreading ss steady state sub subcooled T total or overall TE thermoelectric device TIM thermal interface material v vapor w wall or wetted x layer designation z position along the z-coordinate direction Superscripts n normal direction p in-plane direction REFERENCES Asheghi, M., Touzelbaev, M. 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