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3D numerical optimization of a heat sink base for electronics cooling
International Communications in Heat and Mass Transfer 39 (2012) 204–208
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
3D numerical optimization of a heat sink base for electronics cooling☆ Ji Li ⁎, Zhong-shan Shi Laboratory of Electronics Thermal Management, College of Physics, Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China
a r t i c l e
i n f o
Available online 16 December 2011 Keywords: Base Electronics cooling Heat sink Numerical simulation Optimization
a b s t r a c t In this paper, the possible optimal thickness of a heat sink base has been explored numerically with different convective heat transfer boundary conditions in a dimensionless three dimensional heat transfer model. From the numerical results, relations among different heat transfer mechanisms (natural or forced, air or liquid), different area ratios of a heat sink to a heating source, and the lowest thermal resistance have been obtained and discussed. Also a simple correlation for these three parameters from data fitting is given for guiding a heat sink design. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction
analysis in a heat sink base can be simplified to a problem as shown in Fig. 1(b), since the heat enters the bottom surface of the heat sink base and leaves the top surface of the base over which a uniform heat transfer coefficient, or an external resistance, is prescribed. In the following, this argument will be tested. A numerical simulation result for a natural convection situation by a widely adopted commercial software (FLUENT) is given in Fig. 2. The temperature distribution in the base of the heat sink is compared between the full simulation and the result just for the base with an equivalent uniform heat transfer coefficient which is determined with the following definition,
It is well known that the performance reliability and life expectancy of electronic equipment are inversely related to the component's temperature. Conventionally, most optimization works on heat sinks assume that the heat is evenly distributed over the entire base area of the heat sink, and therefore, do not account for the additional temperature rise caused by a smaller heat source (the contact area is smaller than the base area) [1,2]. Several studies have been carried out to determine this spreading resistance (or called constriction resistance) for a system similar to Fig. 1(a) through numerical or analytical solutions [3-5]. However, an optimal result for the base thickness was not obtained in those works for a given boundary condition. In order to cohere with real situations and to pursue the minimum value of thermal resistance of a heat sink with different heat source contact areas, heat transfer boundary conditions and heating flux, in this investigation, a three-dimensional heat transfer models were developed. Through detailed calculations of dimensionless temperature with variations of base thickness and contact area ratios under different heat transfer boundary conditions, the optimal base thickness can be obtained for a certain heat sink configuration. Furthermore, a rigorous experimental test was carried out to verify the reliability of the numerical predictions.
the equivalent heat transfer coefficient; A Here, his base is the cross section area of the top surface of the base; h is the real convection heat transfer coefficient; Atotal is the total area of the fins adding the top surface area of the baseAbase. From Fig. 2(a) and (b), if a proper heat transfer coefficient is selected (through a detailed integration of h ⋅ dA along the fins), the difference between the full simulation for the heat sink and the simulation for the base only is very small. This comparison verifies the previous argument as used historically for the calculation of the spreading resistances [3-5].
2. Arguments on base's role in heat sinks' performance
3. Heat transfer modeling for a base optimization
For a problem like in Fig. 1(a), a heating source is mounted on a heat sink. Lee et al. [4] argued that the problem of a thermal resistance
From Fourier Law q=−λ ∂T/∂x, introducing the following dimensionless temperature
h⋅A base ¼ h⋅Atotal
θðx; y; zÞ ¼ ☆ Communicated by P. Cheng and W.Q. Tao. ⁎ Corresponding author. E-mail address: [email protected] (J. Li). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.12.001
λ ðT ðx; y; zÞ−T 0 Þ qw b
ð1Þ
ð2Þ
and the following dimensionless length, X =x/b, Y=y/b and Z=z/b, the dimensionless description of the problem can be obtained as, here if we
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
a)
b) Domain of the numerical simulation Fig. 1. Construction of the object in interest.
a) A full simulation with commercial software
b) A simplified simulation only for base with commercial software Fig. 2. Temperature distribution comparisons in the base of the heat sink.
205
206
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
a) h =50 W/m2K (ξ =0.6or Bih =3.18 x 10-3)
b) h =1500 W/m2K (ξ =0.4or Bih =6.36 x10-2)
c) h =22000 W/m2K (ξ =0.17or Bih =0.396) Fig. 3. Representative temperature distributions for different heat transfer boundary conditions when α = 2.
assume constant thermal properties for simplicity (which means the base material is all same),Governing equation 2
2
ð6Þ
∂θ ¼0 ∂X
ð7Þ
α ∂θ Bih ;− θ ¼ 2 ξ ∂X
ð8Þ
∂θ ¼0 ∂Y
ð9Þ
2
∂ θ ∂ θ ∂ θ þ þ ¼0 ∂X 2 ∂Y 2 ∂Z 2
ð3Þ
Boundary conditions Z ¼ 0; if 0 ≤ X ≤
else
∂θ Bih θ ¼ ξ ∂Z
Z ¼ ξ;−
1 1 ∂θ and 0 ≤ Y ≤ ; − ¼1 2 2 ∂Z
1 α 1 α ∂θ Bih b x ≤ and b y ≤ ; − θ ¼ 2 2 2 2 ξ ∂Z
X ¼ 0;−
X¼ ð4Þ
ð5Þ
and Y ¼ 0;−
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
(a) h =50 W/m2K
207
the heating source. Bih = hδ/λ is the Biot number based on the air nat ural convection heat transfer; Bih ¼ hδ=λ is the Biot number based on the equivalent convection heat transfer. In this research, a value of b = 0.025 m (e.g., a CPU chip) is implicated everywhere for related parameters calculation. Here aluminum is the material of interest due to its massive adoption in heat sink manufacturing, and λ = 237 W/mK is chosen in the numerical model. The convergent criterion was set as ∑∑∑jðθði; j; kÞ−θ0 ði; j; kÞÞ=θ0 ði; j; kÞj≤10
−3
ð11Þ
From a grid sensitivity analysis, a 100(x) × 100(y) × 12(z) mesh was adopted. The code validation was verified by a simple comparison with a 1D analytical solution of heat conduction problem. 4. Results and discussions With the arguments in the previous sections, we select three different equivalent heat transfer coefficients at Z =ξ to represent three common heat dissipation mechanisms in the real applications: (1) h = 50 W/m2K for air natural convection heat transfer; (2) h=1500 W/m2K? for air 2 forced convection heat transfer; (3) h =22,000 W/m K? for liquid forced convection heat transfer. Fig. 3a, b, and c shows the local cross-sectional temperature distribu = 50 (ξ = 0.6 orBih = 3.18 × 10 − 3), tion in the Y–Z plane at X = 0 for h = 6.36 × 10 − 2), h = 22,000 (ξ = 0.17 or h = 1500 (ξ = 0.4 or Bih Bih = 0.396) respectively. Fig. 4a, b, and c illustrates a comparison of the dimensionless maximum temperature at the central point at the base bottom surface with variations of relative base thickness for different Biot numberBih. The results demonstrate that there is no lowest point for equivalent = 50 W/m 2K. However, there exists the heat transfer coefficient h lowest point when the equivalent heat transfer coefficient is large enough. From Fig. 4b (h = 1500 W/m 2K), for α = a/b = 2, the lowest point of the temperature curves occurs at ξ = δ/b = 0.4 with a dimensionless temperature of 1.954; for α = a/b = 2.8, the lowest point occurs at ξ = δ/b = 0.6 with a dimensionless temperature of 1.243; for α = a/ b = 4, the lowest point occurs at ξ = δ/b = 0.8 with a dimensionless = 22,000 W/m2K), for α = a/ temperature of 0.8886. From Fig. 4c ( h b = 2, the lowest point occurs at ξ = δ/b = 0.17 with a dimensionless temperature of 0.438; for α = a/b = 2.8, the lowest point occurs at ξ = δ/b = 0.21 with a dimensionless temperature of 0.436; for α = a/ b = 4, the lowest point occurs at ξ = δ/b = 0.21 with a dimensionless temperature of 0.443. In this investigation, in order to test the accuracy of the predictions from this non-dimensional base optimization analysis, a rigorous experimental method was designed. Fig. 5 gives the experimental results compared with the numerical simulations. Here we compared the overall thermal resistance which is defined as,
(b) h =1500 W/m2K
(c) h =22000 W/m2K
RT ¼
Fig. 4. Comparison of maximum dimensionless temperature in the base for an equivalent heat transfer coefficient.
Y¼
α ∂θ Bih ;− θ ¼ 2 ξ ∂Y
ð10Þ
Here, α = a/b is the relative length of the base to the length of the heating source; ξ = δ/b is the relative base thickness to the length of
T case −T 0 Qw
ð12Þ
Here, RTis the thermal resistance of heat sink; Tcase is the maximum temperature in heat sink; T0 is the ambient temperature. For = 50), the optimal base thickness does the natural convection ( h not exist both from the simulations and the experiments. For the forced convection (h = 1500), the optimal base thickness is around ξ = δ/b = 0.6 from the simulations and 0.8 from the experiments. This discrepancy may be resulted from the difference in the value of the equivalent heat transfer coefficient and the bulk material thermal properties between the numerical simulations and the real situations. Generally, the prediction from the present optimization model is acceptable. Table 1 summarizes the optimization results from the present mode, similar to Lee and other pioneers [3-5], here Bib ¼ hb=λ. For
208
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
Fig. 5. Comparison between the numerical predictions and the experimental results when α = 2.8.
Table 1 Optimal dimensionless base thickness ξ for different heat transfer boundary conditions with different length ratios of heat sink over the heating element. Bib
5.29 × 10− 3
5.29 × 10− 2
0.16
2.33
2 2.8 4
– – –
0.6 0.7 0.9
0.4 0.6 0.8
0.17 0.21 0.21
α
guidance purposes in the design of heat sinks, a fitting correlation was developed here based upon the numerical results, ξ¼c
α 0:47 Bi1:76 b
ð13Þ 2
logðcÞ ¼ −0:45 þ 1:83 logðBib Þ−3:98½ logðBib Þ −3:21½ logðBib Þ
3
ð14Þ
The above correlation should be used within 5 × 10 − 2≤ Bib ≤3 with a rough error of 10–20%. 5. Conclusive remarks The spreading resistance in a heat sink base with different contact areas and different heat transfer boundary conditions has been
investigated numerically with a nondimensionalized 3D heat transfer model and the optimal dimensionless base thickness were obtained. Through the numerical calculations, the detailed temperature distribution in the heat sink base was revealed more clearly than the previous analytical solutions in historic literatures, and the mechanism for which optimal base exists can be understood further. Acknowledgements This work was supported by the National Science and Technology Ministry (2009GB104001) and supported by the National Natural Science Foundation of China (Project No. 51176202). References [1] S. Lee, Optimum design and selection of heat sinks, IEEE Transactions on Components, Packaging, and Manufacturing Technology, Part A 18 (4) (1995) 812–817. [2] A. Bar-Cohen, M. Ivengar, Design and optimization of air-cooled heat sinks for sustainable development, IEEE Transactions on Components and Packaging Technologies 25 (4) (2002) 584–591. [3] S. Lee, S. Song, V. Au, K.P. Moran, Constriction/spreading Resistance Model for Electronics Packaging, ASME/JSME Thermal Engineering Conference 4 (1995) 199–206. [4] D.P. Kennedy, Spreading resistance in cylindrical semiconductor devices, Journal of Applied Physics 31 (1960) 1490–1497. [5] M.M. Yovanovich, S.S. Burde, J.C. Thompson, Thermal constriction resistance of arbitrary planar contacts with constant flux, AIAA progress in astronautics and aeronautics, Thermophysics of Spacecraft and Outer Planet Entry Probe 56 (1977) 127–139.
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
3D numerical optimization of a heat sink base for electronics cooling☆ Ji Li ⁎, Zhong-shan Shi Laboratory of Electronics Thermal Management, College of Physics, Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China
a r t i c l e
i n f o
Available online 16 December 2011 Keywords: Base Electronics cooling Heat sink Numerical simulation Optimization
a b s t r a c t In this paper, the possible optimal thickness of a heat sink base has been explored numerically with different convective heat transfer boundary conditions in a dimensionless three dimensional heat transfer model. From the numerical results, relations among different heat transfer mechanisms (natural or forced, air or liquid), different area ratios of a heat sink to a heating source, and the lowest thermal resistance have been obtained and discussed. Also a simple correlation for these three parameters from data fitting is given for guiding a heat sink design. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction
analysis in a heat sink base can be simplified to a problem as shown in Fig. 1(b), since the heat enters the bottom surface of the heat sink base and leaves the top surface of the base over which a uniform heat transfer coefficient, or an external resistance, is prescribed. In the following, this argument will be tested. A numerical simulation result for a natural convection situation by a widely adopted commercial software (FLUENT) is given in Fig. 2. The temperature distribution in the base of the heat sink is compared between the full simulation and the result just for the base with an equivalent uniform heat transfer coefficient which is determined with the following definition,
It is well known that the performance reliability and life expectancy of electronic equipment are inversely related to the component's temperature. Conventionally, most optimization works on heat sinks assume that the heat is evenly distributed over the entire base area of the heat sink, and therefore, do not account for the additional temperature rise caused by a smaller heat source (the contact area is smaller than the base area) [1,2]. Several studies have been carried out to determine this spreading resistance (or called constriction resistance) for a system similar to Fig. 1(a) through numerical or analytical solutions [3-5]. However, an optimal result for the base thickness was not obtained in those works for a given boundary condition. In order to cohere with real situations and to pursue the minimum value of thermal resistance of a heat sink with different heat source contact areas, heat transfer boundary conditions and heating flux, in this investigation, a three-dimensional heat transfer models were developed. Through detailed calculations of dimensionless temperature with variations of base thickness and contact area ratios under different heat transfer boundary conditions, the optimal base thickness can be obtained for a certain heat sink configuration. Furthermore, a rigorous experimental test was carried out to verify the reliability of the numerical predictions.
the equivalent heat transfer coefficient; A Here, his base is the cross section area of the top surface of the base; h is the real convection heat transfer coefficient; Atotal is the total area of the fins adding the top surface area of the baseAbase. From Fig. 2(a) and (b), if a proper heat transfer coefficient is selected (through a detailed integration of h ⋅ dA along the fins), the difference between the full simulation for the heat sink and the simulation for the base only is very small. This comparison verifies the previous argument as used historically for the calculation of the spreading resistances [3-5].
2. Arguments on base's role in heat sinks' performance
3. Heat transfer modeling for a base optimization
For a problem like in Fig. 1(a), a heating source is mounted on a heat sink. Lee et al. [4] argued that the problem of a thermal resistance
From Fourier Law q=−λ ∂T/∂x, introducing the following dimensionless temperature
h⋅A base ¼ h⋅Atotal
θðx; y; zÞ ¼ ☆ Communicated by P. Cheng and W.Q. Tao. ⁎ Corresponding author. E-mail address: [email protected] (J. Li). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.12.001
λ ðT ðx; y; zÞ−T 0 Þ qw b
ð1Þ
ð2Þ
and the following dimensionless length, X =x/b, Y=y/b and Z=z/b, the dimensionless description of the problem can be obtained as, here if we
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
a)
b) Domain of the numerical simulation Fig. 1. Construction of the object in interest.
a) A full simulation with commercial software
b) A simplified simulation only for base with commercial software Fig. 2. Temperature distribution comparisons in the base of the heat sink.
205
206
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
a) h =50 W/m2K (ξ =0.6or Bih =3.18 x 10-3)
b) h =1500 W/m2K (ξ =0.4or Bih =6.36 x10-2)
c) h =22000 W/m2K (ξ =0.17or Bih =0.396) Fig. 3. Representative temperature distributions for different heat transfer boundary conditions when α = 2.
assume constant thermal properties for simplicity (which means the base material is all same),Governing equation 2
2
ð6Þ
∂θ ¼0 ∂X
ð7Þ
α ∂θ Bih ;− θ ¼ 2 ξ ∂X
ð8Þ
∂θ ¼0 ∂Y
ð9Þ
2
∂ θ ∂ θ ∂ θ þ þ ¼0 ∂X 2 ∂Y 2 ∂Z 2
ð3Þ
Boundary conditions Z ¼ 0; if 0 ≤ X ≤
else
∂θ Bih θ ¼ ξ ∂Z
Z ¼ ξ;−
1 1 ∂θ and 0 ≤ Y ≤ ; − ¼1 2 2 ∂Z
1 α 1 α ∂θ Bih b x ≤ and b y ≤ ; − θ ¼ 2 2 2 2 ξ ∂Z
X ¼ 0;−
X¼ ð4Þ
ð5Þ
and Y ¼ 0;−
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
(a) h =50 W/m2K
207
the heating source. Bih = hδ/λ is the Biot number based on the air nat ural convection heat transfer; Bih ¼ hδ=λ is the Biot number based on the equivalent convection heat transfer. In this research, a value of b = 0.025 m (e.g., a CPU chip) is implicated everywhere for related parameters calculation. Here aluminum is the material of interest due to its massive adoption in heat sink manufacturing, and λ = 237 W/mK is chosen in the numerical model. The convergent criterion was set as ∑∑∑jðθði; j; kÞ−θ0 ði; j; kÞÞ=θ0 ði; j; kÞj≤10
−3
ð11Þ
From a grid sensitivity analysis, a 100(x) × 100(y) × 12(z) mesh was adopted. The code validation was verified by a simple comparison with a 1D analytical solution of heat conduction problem. 4. Results and discussions With the arguments in the previous sections, we select three different equivalent heat transfer coefficients at Z =ξ to represent three common heat dissipation mechanisms in the real applications: (1) h = 50 W/m2K for air natural convection heat transfer; (2) h=1500 W/m2K? for air 2 forced convection heat transfer; (3) h =22,000 W/m K? for liquid forced convection heat transfer. Fig. 3a, b, and c shows the local cross-sectional temperature distribu = 50 (ξ = 0.6 orBih = 3.18 × 10 − 3), tion in the Y–Z plane at X = 0 for h = 6.36 × 10 − 2), h = 22,000 (ξ = 0.17 or h = 1500 (ξ = 0.4 or Bih Bih = 0.396) respectively. Fig. 4a, b, and c illustrates a comparison of the dimensionless maximum temperature at the central point at the base bottom surface with variations of relative base thickness for different Biot numberBih. The results demonstrate that there is no lowest point for equivalent = 50 W/m 2K. However, there exists the heat transfer coefficient h lowest point when the equivalent heat transfer coefficient is large enough. From Fig. 4b (h = 1500 W/m 2K), for α = a/b = 2, the lowest point of the temperature curves occurs at ξ = δ/b = 0.4 with a dimensionless temperature of 1.954; for α = a/b = 2.8, the lowest point occurs at ξ = δ/b = 0.6 with a dimensionless temperature of 1.243; for α = a/ b = 4, the lowest point occurs at ξ = δ/b = 0.8 with a dimensionless = 22,000 W/m2K), for α = a/ temperature of 0.8886. From Fig. 4c ( h b = 2, the lowest point occurs at ξ = δ/b = 0.17 with a dimensionless temperature of 0.438; for α = a/b = 2.8, the lowest point occurs at ξ = δ/b = 0.21 with a dimensionless temperature of 0.436; for α = a/ b = 4, the lowest point occurs at ξ = δ/b = 0.21 with a dimensionless temperature of 0.443. In this investigation, in order to test the accuracy of the predictions from this non-dimensional base optimization analysis, a rigorous experimental method was designed. Fig. 5 gives the experimental results compared with the numerical simulations. Here we compared the overall thermal resistance which is defined as,
(b) h =1500 W/m2K
(c) h =22000 W/m2K
RT ¼
Fig. 4. Comparison of maximum dimensionless temperature in the base for an equivalent heat transfer coefficient.
Y¼
α ∂θ Bih ;− θ ¼ 2 ξ ∂Y
ð10Þ
Here, α = a/b is the relative length of the base to the length of the heating source; ξ = δ/b is the relative base thickness to the length of
T case −T 0 Qw
ð12Þ
Here, RTis the thermal resistance of heat sink; Tcase is the maximum temperature in heat sink; T0 is the ambient temperature. For = 50), the optimal base thickness does the natural convection ( h not exist both from the simulations and the experiments. For the forced convection (h = 1500), the optimal base thickness is around ξ = δ/b = 0.6 from the simulations and 0.8 from the experiments. This discrepancy may be resulted from the difference in the value of the equivalent heat transfer coefficient and the bulk material thermal properties between the numerical simulations and the real situations. Generally, the prediction from the present optimization model is acceptable. Table 1 summarizes the optimization results from the present mode, similar to Lee and other pioneers [3-5], here Bib ¼ hb=λ. For
208
J. Li, Z. Shi / International Communications in Heat and Mass Transfer 39 (2012) 204–208
Fig. 5. Comparison between the numerical predictions and the experimental results when α = 2.8.
Table 1 Optimal dimensionless base thickness ξ for different heat transfer boundary conditions with different length ratios of heat sink over the heating element. Bib
5.29 × 10− 3
5.29 × 10− 2
0.16
2.33
2 2.8 4
– – –
0.6 0.7 0.9
0.4 0.6 0.8
0.17 0.21 0.21
α
guidance purposes in the design of heat sinks, a fitting correlation was developed here based upon the numerical results, ξ¼c
α 0:47 Bi1:76 b
ð13Þ 2
logðcÞ ¼ −0:45 þ 1:83 logðBib Þ−3:98½ logðBib Þ −3:21½ logðBib Þ
3
ð14Þ
The above correlation should be used within 5 × 10 − 2≤ Bib ≤3 with a rough error of 10–20%. 5. Conclusive remarks The spreading resistance in a heat sink base with different contact areas and different heat transfer boundary conditions has been
investigated numerically with a nondimensionalized 3D heat transfer model and the optimal dimensionless base thickness were obtained. Through the numerical calculations, the detailed temperature distribution in the heat sink base was revealed more clearly than the previous analytical solutions in historic literatures, and the mechanism for which optimal base exists can be understood further. Acknowledgements This work was supported by the National Science and Technology Ministry (2009GB104001) and supported by the National Natural Science Foundation of China (Project No. 51176202). References [1] S. Lee, Optimum design and selection of heat sinks, IEEE Transactions on Components, Packaging, and Manufacturing Technology, Part A 18 (4) (1995) 812–817. [2] A. Bar-Cohen, M. Ivengar, Design and optimization of air-cooled heat sinks for sustainable development, IEEE Transactions on Components and Packaging Technologies 25 (4) (2002) 584–591. [3] S. Lee, S. Song, V. Au, K.P. Moran, Constriction/spreading Resistance Model for Electronics Packaging, ASME/JSME Thermal Engineering Conference 4 (1995) 199–206. [4] D.P. Kennedy, Spreading resistance in cylindrical semiconductor devices, Journal of Applied Physics 31 (1960) 1490–1497. [5] M.M. Yovanovich, S.S. Burde, J.C. Thompson, Thermal constriction resistance of arbitrary planar contacts with constant flux, AIAA progress in astronautics and aeronautics, Thermophysics of Spacecraft and Outer Planet Entry Probe 56 (1977) 127–139.