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Optimum thermal performance of microchannel heat sink by adjusting channel width and height
Available online at www.sciencedirect.com
International Communications in Heat and Mass Transfer 35 (2008) 577 – 582 www.elsevier.com/locate/ichmt
Optimum thermal performance of microchannel heat sink by adjusting channel width and height ☆ Hong-Sen Kou ⁎, Ji-Jen Lee, Chih-Wei Chen Department of Mechanical Engineering, Tatung University, Taipei, Taiwan, ROC Available online 31 December 2007
Abstract A three-dimensional numerical model of the microchannel heat sink is presented to study the effects of heat transfer characteristics due to various channel heights and widths. Based on the theory of a fully developed flow, the pressure drop in the microchannel is derived under the requirement of the flow power for a single channel. The effects of two design variables representing the channel width and height on the thermal resistance are investigated. In addition, the constraint of the same flow cross section is carried out to find the optimum dimension. Finally, the minimum thermal resistance and optimal channel width with various flow powers and channel heights are obtained by using the simulated annealing method. © 2007 Elsevier Ltd. All rights reserved. Keywords: Microchannel; Heat sink; Thermal resistance; Simulated annealing method
1. Introduction The research of the microchannel heat sink started in the early 1980s because it has the advantages of high heat flux, low cost, small space, etc. By matching with the semi-conductor manufacturing process, the microchannel heat sink has become one of the focal points of current micro cooling technology. The microchannel heat sink was first introduced by Tuckerman and Pease in 1981 [1], who discussed the heat transfer characteristics by theories and the experimental measurement. In the theoretical study, the fully developed flow was considered and the capacity of dissipating heat flux could be promoted to 1000 W/cm2. In the experimental measurement, the cooling water could dissipate heat flux about 790 W/cm2 by forcing a coolant through the microchannel etched onto a silicon wafer. The pressure drop and thermal performance of the microchannel heat sink were investigated both experimentally and numerically by Qu and Mudawar [2,3]. The finite difference method and the SIMPLE algorithm were used to solve the conventional Navier–Stokes and energy equations. The numerical results ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (H.-S. Kou).
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.12.002
showed good agreement with the corresponding experimental data obtained from oxygen-free copper and fitted with a polycarbonate plastic cover plate. Besides, the temperature at the heated base surface of the heat sink reduced with the increase in the thermal conductivity of the solid substrate, especially near the channel outlet. The frictional factor was predicted by Toh et al. [4], who solved the steady, laminar flow and the heat transfer equations using the finite-volume method. The results showed that the frictional loss decreased with the increase of the heat input at a lower Reynolds number because the viscosity decreased as the temperature of water increased. Gamrat et al. [5] presented both three- and two-dimensional numerical analysis of convective heat transfer in a microchannel heat sink. The thermal entrance effect and conduction/convection coupling effect were included. The numerical analysis did not reveal any significant scale effect on heat transfer in the microchannel heat sink down to the smallest channel spacing size considered (0.1 mm). Lee and Garimella [6] proposed a generalized correlation for predicting Nesselt number along the axial distance, which was useful for the design and optimization of a microchannel heat sink. In recent optimum publications on the microchannel heat sink, Knight et al. [7] presented a fin model to redesign the previous investigators' microchannel heat sink and reduced thermal
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H.-S. Kou et al. / International Communications in Heat and Mass Transfer 35 (2008) 577–582
Nomenclature Cp F k p q Q R Rt T U x,y,z
specific heat, J/kg°C area of channel, μm2 thermal conductivity, W/m°C pressure, N/m2 heat flux, W/cm2 volume flow rate of single channel, m3/s thermal resistance, °C/W/cm2 total thermal resistance, °C/W temperature, °C velocity component in x direction, m/s cartesian coordinates
Greek symbols μ dynamic viscosity, kg/ms ρ density, kg/m3 Ω flow power, W
Subscripts inlet inlet of channel oulet outlet of channel l liquid m mean value max maximum value w wall s solid resistance from 10 to 35%. Wen and Choo [8] set up a thermal resistance model to study an optimum thermal design of the heat sink under three types of flow constraints: the constant coolant volume flow rate, constant pressure drop, and constant pumping power. The lowest total thermal resistance was 0.054 °C/W/cm2 when the channel aspect ratio was 15, the fin width to the channel width was 0.65, the channel width was 50 μm, and the pumping power was fixed at 7.5 W. Ryu et al. [9] used an ADI-type finitevolume method to solve traditional transport equations and then incorporated an optimization scheme of directional search to minimize the thermal resistance subject to a specified pumping power, where the channel aspect ratio was fixed. The optimum dimensions and corresponding thermal resistance had a powerlaw relation with respect to the pumping power. Kim [10] adopted
Fig. 2. Schematic drawing of a half channel and a half fin.
the fin model [7], the porous model [11], and the numerical optimization method to explore the thermal resistance. The optimized results were compared among these three approaches with the pumping power at 2.56 W. Similar to the above studies [2–6,9,10], the present study investigates the heat transfer characteristics of the microchannel heat sink based on the fully developed flow. The commercial software “CFD-ACE+” is used as the computational code. In order to clarify the thermal resistance, the unit volume including a half channel and a half fin is considered for the computational domain shown in Fig. 1 to find its optimal channel width at different channel heights with the fixed flow power of the single channel. Furthermore, the constraint of the same flow cross section is carried out to find the optimal dimensions. By using the simulated annealing method, the minimum thermal resistance and the optimal channel widths can be obtained at various flow powers and channel heights. 2. Numerical analysis 2.1. The physical model and computational domain The microchannel heat sink comprising of an adiabatic cover plate and a silicon substrate with many microchannels fabricated on the other side. The coolant flows though these channels and takes away the heat generated by IC chips. Based on geometric symmetry, a half of the microchannel along with the fin is chosen for the computational domain as illustrated in Fig. 1. 2.2. Governing equations and boundary conditions
Fig. 1. Schematic of a microchannel heat sink and computational domain (dotted line).
In order to simplify the Navier–Stokes and energy equations, some assumptions like steady, incompressible, fully developed flows are considered. However, the buoyancy force and radiative heat transfer are neglected. Then, the transport equations can be rewritten as For the cooling fluid region, the momentum equation is written as
H.-S. Kou et al. / International Communications in Heat and Mass Transfer 35 (2008) 577–582
Momentum equation (fluid region): 1 ∂p ∂2 U ∂2 U ¼ 2 þ 2 A ∂x ∂y ∂z
ð1Þ
For a rectangular duct on the coordinate system shown in Fig. 2, the fully developed velocity profile of Eq. (1) is derived as 2 3 m1 cosh mpy X l 2 2 zl ð1Þ mpz 4 4zl dp 15 U ð y; zÞ ¼ 3 cos p A dx m¼1;3;5 zl m3 cosh mpyl 2zl
ð2Þ Then, the energy equation of the cooling fluid region is written as 2 3 m1 X cosh mpy l 2 2 zl 4zl dp ð1Þ mpz 4 ∂T 15 cos mpyl zl ∂x p3 A dx m¼1;3;5 m3 cosh 2zl 2 ∂ T ∂2 T ∂2 T ¼a þ þ 2 ∂x2 ∂y2 ∂z ð3Þ As to the energy equation of the solid region, it is written as 0¼
∂2 T ∂2 T ∂ 2 T þ þ 2 ∂x2 ∂y2 ∂z
Table 2 Comparison of streamwise velocity between numerical solution and fully developed flow model, where zl = 56 (μm), zs = 44 (μm), yl = 320 (μm), and Δp = 103.42 (kPa) y (μm)
z (μm)
U (m/s) Theory (Eq. (2))
Present study (numerical solution) x = 0 (cm)
x = 0.5 (cm)
x = 1 (cm)
0 0 0 80 120 140
0 10 15 0 0 0
4.053 3.528 2.883 4.007 3.609 2.695
4.052 3.527 2.882 4.005 3.601 2.683
4.052 3.527 2.882 4.005 3.601 2.683
4.052 3.527 2.882 4.005 3.601 2.683
silicon region, implying that the normal derivative of the temperature gradient is taken as zero. Also, boundary conditions at the interface of the solid fin and the liquid flow should be without contact resistance. After integrating velocity in the x plane, the mean velocity can be obtained as
ð5Þ
1 1 1 ∂T ¼0 ð2Þoutlet: x ¼ x0 ; yl b y b yl ; 0 b z b zl ; p ¼ poutlet ; 2 2 2 ∂x
In the present study, a uniform heat flux is added on the bottom of silicon substrate. 1 1 ∂T ð3Þheated wall: y ¼ yl y0 ; 0 b x b x0 ; 0 b z b z0 ; ks ¼ qw 2 2 ∂y
ð7Þ As to the other boundary conditions of the computational domain, the symmetry boundary conditions are imposed on the Table 1 Comparison of the present results to experimental data [1] and numerical solution [9] Δp (kPa)
56 55 50
44 45 50
320 287 302
103.42 117.21 213.73
2
2
0
0
The volumetric flow rate of the coolant for the single channel is given as Q ¼ U m yl z l l 16yl z3l dp X 1 1 zl mpyl 1 ¼ 4 tanh 2 p A dx m¼1;3;5 m4 mp yl 2zl ð9Þ
ð6Þ
yl (μm)
4
ð8Þ
1 1 1 ð1Þ inlet: x ¼ 0; yl b y b yl ; 0 b z b zl ; P ¼ Pinlet ; 2 2 2 T ¼ Tinlet
zs (μm)
R 1yl R 1zl
U ð y; zÞdydz yl zl l 16z2l dp X 1 1 zl mpyl 1 ¼ 4 tanh 2 p A dx m¼1;3;5 m4 mp yl 2zl
Um ¼
ð4Þ
At the inlet and outlet sections of the channel flow, the boundary conditions of the flow are
zl (μm)
579
Rt (°C/W) Experimental data [1]
Numerical prediction [9]
Present study
0.11 0.113 0.09
0.111 0.117 0.092
0.11 0.113 0.09
The flow power of the coolant for the single channel is given as dp X¼Q xl dx l 16xl yl z3l dp 2 X 1 1 1 zl mpyl ¼ tanh dx m¼1;3;5 m4 2 mp yl p4 A 2zl ð10Þ Since a fully developed flow has the same pressure gradient along the flow direction, the pressure drop between the inlet and outlet sections of the channel is determined as dp pinlet poutlet ¼ xl dx ( )12 l 16yl z3l X 1 1 1 zl mpyl ¼ tanh p4 AXxl m¼1;3;5 m4 2 mp yl 2zl ð11Þ
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H.-S. Kou et al. / International Communications in Heat and Mass Transfer 35 (2008) 577–582
2.3. Numerical method Since the energy equations presented in the above section are complex, the commercial software “CFD-ACE+” is used throughout this study to solve these equations. In this code, the finite-volume method is adopted along with the SIMPLEC algorithm that serves to enhance the well-known SIMPLE algorithm. Besides, the user subroutines are added into CFDACE+ to input the pressure drop in the boundary conditions under various flow powers. After carrying out numerical computation, the thermal resistance is found to be R¼
Tmax Tinlet qw
ð12Þ
where Tmax is the maximum temperature observed in the heat sink. 2.4. Optimization technique In order to find the minimum value of the thermal resistance under several variables, the simulated annealing method is used here to be the optimization technique. The simulated annealing method is a recent technique for finding the global optimum of a given cost function in a large search space. The aim of the simulated annealing method is to find the minimization of an energy state. In this paper, the thermal resistance is used to be the energy state by searching covering designs. 3. Results and discussions In this paper, the dimensions of single channel along with one fin are fixed with x0 = 1 cm, y0 = 1100 μm, and z0 = 100 μm or 200 μm. The solid region of the heat sink is made of silicon with thermal conductivity of ks = 148 W/m°C, and the coolant
Fig. 3. The effects of the channel width on thermal resistance when Ω = 0.01 Watt and Ω = 0.001 Watt.
Fig. 4. The effects of the channel height on thermal resistance when Ω = 0.01 Watt and Ω = 0.001 Watt.
used is water, where ρ l = 1000 kg/m 3 , μ l = 0.001 kg/ms, Cpl = 4179 J/kg°C and kl = 0.613 W/m°C. Uniform heat flux of 100 W/cm2 is applied to the bottom surface of the heat sink. A distributed 16× 17 ×12 grid is used. Then, the problems are solved after the solution converges with the minimum reduction in normalized residuals for each variable at less than 10E-4. In order to validate this simulation model, Table 1 shows a comparison of the present results to the experimental data [1] and the numerical solution [9] based on the same dimensions with the heat sink 1 cm in depth and 1 cm in width. Under various pressure drops and channel dimensions, the results show closely agreement. In addition, the streamwise velocity distributions with different positions are listed here to demonstrate that the fully developed flow assumption is valid in this study. The numerical solution of a velocity profile at
Fig. 5. The effects of a constant channel area on thermal resistance.
H.-S. Kou et al. / International Communications in Heat and Mass Transfer 35 (2008) 577–582 Table 3 The optimal channel widths and thermal resistances as determined by various flow powers and channel heights Input
Optimal Output
Geometric configuration (μm) yo = 1100, zo = 100, zl = 10–90
yo = 1100, zo = 200, zl = 20–180
yl 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400
Flow power (Watt)
Channel width (μm)
Thermal resistance (°C/W/cm2)
Mean velocity (m/s)
Ω 0.0001
zl 71.212 73.184 75.573 78.791 60.536 61.769 63.578 66.268 49.546 49.434 49.950 51.318 38.460 37.567 37.030 37.230 129.242 132.404 136.135 142.262 99.468 101.042 102.873 106.233 70.427 70.579 70.761 71.665 48.170 47.605 46.931 46.503
R 0.232 0.248 0.273 0.316 0.127 0.139 0.156 0.182 0.078 0.090 0.104 0.123 0.055 0.067 0.079 0.096 0.229 0.255 0.295 0.368 0.139 0.158 0.187 0.236 0.093 0.109 0.130 0.165 0.068 0.082 0.099 0.124
Um 0.220 0.249 0.291 0.360 0.640 0.723 0.847 1.051 1.830 2.045 2.376 2.941 5.090 5.631 6.464 7.943 0.295 0.332 0.383 0.461 0.821 0.923 1.068 1.298 2.185 2.446 2.823 3.446 5.704 6.346 7.282 8.864
0.001
0.01
0.1
0.0001
0.001
0.01
0.1
different positions is given in Table 2, which shows satisfactory agreement with a fully developed flow model, where zl = 56 μm, yl = 320 μm, and Δp = 103.42 kPa. Under the fully developed flow model, therefore, the flow power obtained from Eq. (10) can be used as an input variable to calculate the pressure drop of Eq. (11) and then to proceed the present numerical calculation. The effects of the channel width on the thermal resistance are shown in Fig. 3 when the flow power of a single channel is fixed at Ω = 0.01 Watt or Ω = 0.001 Watt. It can be observed that an optimum width of the channel exists under constant flow power. The thermal resistance, R, decreases as the flow power or flow rate increases. When the flow power is larger than Ω = 0.01 Watt and the channel width is between 20 μm and 80 μm, the thermal resistance will be lower than 0.15 °C/Wcm2. When the flow power is fixed at Ω = 0.001 Watt, the effect due to the channel width will be more significant especially for the shorter channel width. Because the interface area between the coolant fluid and the heat sink is not large enough to dissipate heat flux quickly, the shorter channel width has higher thermal resistance.
581
Besides, the influence exerted by the channel height is shown in Fig. 3. Fig. 4 shows the effects of the channel height on thermal resistance when Ω = 0.01 Watt and Ω = 0.001 Watt. Since the height of the sink is fixed in this study, increasing the height of the channel induces the decrease of the substrate thickness. It is found that thermal resistance monotonically decreases with the decrease of the substrate thickness. The channel height above 1070 μm is not explored here out of consideration of the grid's error. The trend in the change of thermal resistance due to the channel height is the same as flow power. When yl b 1070 μm, a lower channel height always has higher thermal resistance. The effects of thermal resistance in various channel flow area are also considered here. The results are shown in Fig. 5, where the parameter, F = yl × zl, is defined as the cross section area of a flow channel. Both a larger flow area and flow power can obtain lower thermal resistance. Finally, minimum thermal resistance and the optimum channel width with various flow powers and channel heights are obtained by using the simulated annealing method. In the process of the optimization procedure, the channel width is searched from 10 μm to 90 μm. Table 3 shows that flow power plays the most important role in determining thermal resistance. The optimum channel width increases as the channel height decreases when the flow power is fixed at 0.0001 W and 0.001 W. However, the optimum channel width is not significantly influenced with the increase of the channel height when the flow power is fixed at 0.01 W and 0.1 W. A similar trend can be observed in the other optimum case, where the size of a single channel is fixed by x0 = 1 cm, y0 = 1100 μm, z0 = 200 μm and the optimal channel width ranges from 20 μm to 180 μm. It is noted that the whole thermal resistance of a heat sink can be determined by multiplying the thermal resistance of a single channel to the heat sink width and depth. For practical application, hence, engineers can refer to this Table in designing the optimal microchannel heat sink. 4. Conclusions In this study, the minimum thermal resistance of a heat sink including a microchannel with a fixed dimension is investigated to search for its optimum channel width. The validity of a fully developed flow model in a rectangular channel has been proved which can be used to determine the flow power and pressure drop of a microchannel heat sink. The results show that a larger flow area, larger flow power, and shorter substrate thickness can obtain lower thermal resistance. Meanwhile, the optimum width and height of a channel is available from this numerical procedure if the heat sink has the same flow area. After carrying out the optimum method, the optimal channel width is obtained with different channel heights. The optimal channel width is not significantly influenced by the decrease in the channel height when the flow power is fixed at 0.01 Watt and 0.1 Watt. On the contrary, the optimum channel width strongly depends on the channel height when the flow power is fixed at 0.001 Watt and 0.0001 Watt.
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References [1] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE Electronic Device Letters EDL-2 (1981) 126–129. [2] W. Qu, I. Mudawar, Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink, International Journal of Heat and Mass Transfer 45 (2002) 2549–2565. [3] W. Qu, I. Mudawar, Analysis of three-dimensional heat transfer in microchannel heat sinks, International Journal of Heat and Mass Transfer 45 (2002) 3973–3985. [4] K.C. Toh, X.Y. Chen, J.C. Chai, Numerical computation of fluid flow and heat transfer in microchannels, International Journal of Heat and Mass Transfer 45 (2002) 5133–5141. [5] G. Gamrat, M. Favre-Marinet, D. Asendrych, Conduction and entrance effects on laminar liquid flow and heat transfer in rectangular microchannels, International Journal of Heat and Mass Transfer 48 (2005) 2943–2954.
[6] P.S. Lee, S.V. Garimella, Thermally developing flow and heat transfer in rectangular microchannels of different aspect ratios, International Journal of Heat and Mass Transfer 49 (2006) 3060–3067. [7] R.W. Knight, D.J. Hall, J.S. Goodling, R.C. Jaeger, Heat sink optimization with application to microchannels, IEEE Transactions on Components, Hybrids, and Manufacturing Technology 15 (5) (1992) 832–842. [8] Z. Wen, K.F. Choo, The optimum thermal design of microchannel heat sinks, IEEE/CPMT Electronic Packaging Technology Conference (1997) 123–129. [9] J.H. Ryu, D.H. Choi, S.J. Kim, Numerical optimization of the thermal performance of a microchannel heat sink, International Journal of Heat and Mass Transfer 45 (2002) 2823–2827. [10] S.J. Kim, Method for thermal optimization of microchannel heat sinks, Heat Transfer Engineering 25 (1) (2004) 37–49. [11] C.L. Tien, S.M. Kuo, Analysis of forced convection in microstructures for electronic system cooling, Proc. Int. Symp. Cooling Technology for Electronic Equipment, Honolulu, HI (1987) 217–226.
International Communications in Heat and Mass Transfer 35 (2008) 577 – 582 www.elsevier.com/locate/ichmt
Optimum thermal performance of microchannel heat sink by adjusting channel width and height ☆ Hong-Sen Kou ⁎, Ji-Jen Lee, Chih-Wei Chen Department of Mechanical Engineering, Tatung University, Taipei, Taiwan, ROC Available online 31 December 2007
Abstract A three-dimensional numerical model of the microchannel heat sink is presented to study the effects of heat transfer characteristics due to various channel heights and widths. Based on the theory of a fully developed flow, the pressure drop in the microchannel is derived under the requirement of the flow power for a single channel. The effects of two design variables representing the channel width and height on the thermal resistance are investigated. In addition, the constraint of the same flow cross section is carried out to find the optimum dimension. Finally, the minimum thermal resistance and optimal channel width with various flow powers and channel heights are obtained by using the simulated annealing method. © 2007 Elsevier Ltd. All rights reserved. Keywords: Microchannel; Heat sink; Thermal resistance; Simulated annealing method
1. Introduction The research of the microchannel heat sink started in the early 1980s because it has the advantages of high heat flux, low cost, small space, etc. By matching with the semi-conductor manufacturing process, the microchannel heat sink has become one of the focal points of current micro cooling technology. The microchannel heat sink was first introduced by Tuckerman and Pease in 1981 [1], who discussed the heat transfer characteristics by theories and the experimental measurement. In the theoretical study, the fully developed flow was considered and the capacity of dissipating heat flux could be promoted to 1000 W/cm2. In the experimental measurement, the cooling water could dissipate heat flux about 790 W/cm2 by forcing a coolant through the microchannel etched onto a silicon wafer. The pressure drop and thermal performance of the microchannel heat sink were investigated both experimentally and numerically by Qu and Mudawar [2,3]. The finite difference method and the SIMPLE algorithm were used to solve the conventional Navier–Stokes and energy equations. The numerical results ☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (H.-S. Kou).
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.12.002
showed good agreement with the corresponding experimental data obtained from oxygen-free copper and fitted with a polycarbonate plastic cover plate. Besides, the temperature at the heated base surface of the heat sink reduced with the increase in the thermal conductivity of the solid substrate, especially near the channel outlet. The frictional factor was predicted by Toh et al. [4], who solved the steady, laminar flow and the heat transfer equations using the finite-volume method. The results showed that the frictional loss decreased with the increase of the heat input at a lower Reynolds number because the viscosity decreased as the temperature of water increased. Gamrat et al. [5] presented both three- and two-dimensional numerical analysis of convective heat transfer in a microchannel heat sink. The thermal entrance effect and conduction/convection coupling effect were included. The numerical analysis did not reveal any significant scale effect on heat transfer in the microchannel heat sink down to the smallest channel spacing size considered (0.1 mm). Lee and Garimella [6] proposed a generalized correlation for predicting Nesselt number along the axial distance, which was useful for the design and optimization of a microchannel heat sink. In recent optimum publications on the microchannel heat sink, Knight et al. [7] presented a fin model to redesign the previous investigators' microchannel heat sink and reduced thermal
578
H.-S. Kou et al. / International Communications in Heat and Mass Transfer 35 (2008) 577–582
Nomenclature Cp F k p q Q R Rt T U x,y,z
specific heat, J/kg°C area of channel, μm2 thermal conductivity, W/m°C pressure, N/m2 heat flux, W/cm2 volume flow rate of single channel, m3/s thermal resistance, °C/W/cm2 total thermal resistance, °C/W temperature, °C velocity component in x direction, m/s cartesian coordinates
Greek symbols μ dynamic viscosity, kg/ms ρ density, kg/m3 Ω flow power, W
Subscripts inlet inlet of channel oulet outlet of channel l liquid m mean value max maximum value w wall s solid resistance from 10 to 35%. Wen and Choo [8] set up a thermal resistance model to study an optimum thermal design of the heat sink under three types of flow constraints: the constant coolant volume flow rate, constant pressure drop, and constant pumping power. The lowest total thermal resistance was 0.054 °C/W/cm2 when the channel aspect ratio was 15, the fin width to the channel width was 0.65, the channel width was 50 μm, and the pumping power was fixed at 7.5 W. Ryu et al. [9] used an ADI-type finitevolume method to solve traditional transport equations and then incorporated an optimization scheme of directional search to minimize the thermal resistance subject to a specified pumping power, where the channel aspect ratio was fixed. The optimum dimensions and corresponding thermal resistance had a powerlaw relation with respect to the pumping power. Kim [10] adopted
Fig. 2. Schematic drawing of a half channel and a half fin.
the fin model [7], the porous model [11], and the numerical optimization method to explore the thermal resistance. The optimized results were compared among these three approaches with the pumping power at 2.56 W. Similar to the above studies [2–6,9,10], the present study investigates the heat transfer characteristics of the microchannel heat sink based on the fully developed flow. The commercial software “CFD-ACE+” is used as the computational code. In order to clarify the thermal resistance, the unit volume including a half channel and a half fin is considered for the computational domain shown in Fig. 1 to find its optimal channel width at different channel heights with the fixed flow power of the single channel. Furthermore, the constraint of the same flow cross section is carried out to find the optimal dimensions. By using the simulated annealing method, the minimum thermal resistance and the optimal channel widths can be obtained at various flow powers and channel heights. 2. Numerical analysis 2.1. The physical model and computational domain The microchannel heat sink comprising of an adiabatic cover plate and a silicon substrate with many microchannels fabricated on the other side. The coolant flows though these channels and takes away the heat generated by IC chips. Based on geometric symmetry, a half of the microchannel along with the fin is chosen for the computational domain as illustrated in Fig. 1. 2.2. Governing equations and boundary conditions
Fig. 1. Schematic of a microchannel heat sink and computational domain (dotted line).
In order to simplify the Navier–Stokes and energy equations, some assumptions like steady, incompressible, fully developed flows are considered. However, the buoyancy force and radiative heat transfer are neglected. Then, the transport equations can be rewritten as For the cooling fluid region, the momentum equation is written as
H.-S. Kou et al. / International Communications in Heat and Mass Transfer 35 (2008) 577–582
Momentum equation (fluid region): 1 ∂p ∂2 U ∂2 U ¼ 2 þ 2 A ∂x ∂y ∂z
ð1Þ
For a rectangular duct on the coordinate system shown in Fig. 2, the fully developed velocity profile of Eq. (1) is derived as 2 3 m1 cosh mpy X l 2 2 zl ð1Þ mpz 4 4zl dp 15 U ð y; zÞ ¼ 3 cos p A dx m¼1;3;5 zl m3 cosh mpyl 2zl
ð2Þ Then, the energy equation of the cooling fluid region is written as 2 3 m1 X cosh mpy l 2 2 zl 4zl dp ð1Þ mpz 4 ∂T 15 cos mpyl zl ∂x p3 A dx m¼1;3;5 m3 cosh 2zl 2 ∂ T ∂2 T ∂2 T ¼a þ þ 2 ∂x2 ∂y2 ∂z ð3Þ As to the energy equation of the solid region, it is written as 0¼
∂2 T ∂2 T ∂ 2 T þ þ 2 ∂x2 ∂y2 ∂z
Table 2 Comparison of streamwise velocity between numerical solution and fully developed flow model, where zl = 56 (μm), zs = 44 (μm), yl = 320 (μm), and Δp = 103.42 (kPa) y (μm)
z (μm)
U (m/s) Theory (Eq. (2))
Present study (numerical solution) x = 0 (cm)
x = 0.5 (cm)
x = 1 (cm)
0 0 0 80 120 140
0 10 15 0 0 0
4.053 3.528 2.883 4.007 3.609 2.695
4.052 3.527 2.882 4.005 3.601 2.683
4.052 3.527 2.882 4.005 3.601 2.683
4.052 3.527 2.882 4.005 3.601 2.683
silicon region, implying that the normal derivative of the temperature gradient is taken as zero. Also, boundary conditions at the interface of the solid fin and the liquid flow should be without contact resistance. After integrating velocity in the x plane, the mean velocity can be obtained as
ð5Þ
1 1 1 ∂T ¼0 ð2Þoutlet: x ¼ x0 ; yl b y b yl ; 0 b z b zl ; p ¼ poutlet ; 2 2 2 ∂x
In the present study, a uniform heat flux is added on the bottom of silicon substrate. 1 1 ∂T ð3Þheated wall: y ¼ yl y0 ; 0 b x b x0 ; 0 b z b z0 ; ks ¼ qw 2 2 ∂y
ð7Þ As to the other boundary conditions of the computational domain, the symmetry boundary conditions are imposed on the Table 1 Comparison of the present results to experimental data [1] and numerical solution [9] Δp (kPa)
56 55 50
44 45 50
320 287 302
103.42 117.21 213.73
2
2
0
0
The volumetric flow rate of the coolant for the single channel is given as Q ¼ U m yl z l l 16yl z3l dp X 1 1 zl mpyl 1 ¼ 4 tanh 2 p A dx m¼1;3;5 m4 mp yl 2zl ð9Þ
ð6Þ
yl (μm)
4
ð8Þ
1 1 1 ð1Þ inlet: x ¼ 0; yl b y b yl ; 0 b z b zl ; P ¼ Pinlet ; 2 2 2 T ¼ Tinlet
zs (μm)
R 1yl R 1zl
U ð y; zÞdydz yl zl l 16z2l dp X 1 1 zl mpyl 1 ¼ 4 tanh 2 p A dx m¼1;3;5 m4 mp yl 2zl
Um ¼
ð4Þ
At the inlet and outlet sections of the channel flow, the boundary conditions of the flow are
zl (μm)
579
Rt (°C/W) Experimental data [1]
Numerical prediction [9]
Present study
0.11 0.113 0.09
0.111 0.117 0.092
0.11 0.113 0.09
The flow power of the coolant for the single channel is given as dp X¼Q xl dx l 16xl yl z3l dp 2 X 1 1 1 zl mpyl ¼ tanh dx m¼1;3;5 m4 2 mp yl p4 A 2zl ð10Þ Since a fully developed flow has the same pressure gradient along the flow direction, the pressure drop between the inlet and outlet sections of the channel is determined as dp pinlet poutlet ¼ xl dx ( )12 l 16yl z3l X 1 1 1 zl mpyl ¼ tanh p4 AXxl m¼1;3;5 m4 2 mp yl 2zl ð11Þ
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2.3. Numerical method Since the energy equations presented in the above section are complex, the commercial software “CFD-ACE+” is used throughout this study to solve these equations. In this code, the finite-volume method is adopted along with the SIMPLEC algorithm that serves to enhance the well-known SIMPLE algorithm. Besides, the user subroutines are added into CFDACE+ to input the pressure drop in the boundary conditions under various flow powers. After carrying out numerical computation, the thermal resistance is found to be R¼
Tmax Tinlet qw
ð12Þ
where Tmax is the maximum temperature observed in the heat sink. 2.4. Optimization technique In order to find the minimum value of the thermal resistance under several variables, the simulated annealing method is used here to be the optimization technique. The simulated annealing method is a recent technique for finding the global optimum of a given cost function in a large search space. The aim of the simulated annealing method is to find the minimization of an energy state. In this paper, the thermal resistance is used to be the energy state by searching covering designs. 3. Results and discussions In this paper, the dimensions of single channel along with one fin are fixed with x0 = 1 cm, y0 = 1100 μm, and z0 = 100 μm or 200 μm. The solid region of the heat sink is made of silicon with thermal conductivity of ks = 148 W/m°C, and the coolant
Fig. 3. The effects of the channel width on thermal resistance when Ω = 0.01 Watt and Ω = 0.001 Watt.
Fig. 4. The effects of the channel height on thermal resistance when Ω = 0.01 Watt and Ω = 0.001 Watt.
used is water, where ρ l = 1000 kg/m 3 , μ l = 0.001 kg/ms, Cpl = 4179 J/kg°C and kl = 0.613 W/m°C. Uniform heat flux of 100 W/cm2 is applied to the bottom surface of the heat sink. A distributed 16× 17 ×12 grid is used. Then, the problems are solved after the solution converges with the minimum reduction in normalized residuals for each variable at less than 10E-4. In order to validate this simulation model, Table 1 shows a comparison of the present results to the experimental data [1] and the numerical solution [9] based on the same dimensions with the heat sink 1 cm in depth and 1 cm in width. Under various pressure drops and channel dimensions, the results show closely agreement. In addition, the streamwise velocity distributions with different positions are listed here to demonstrate that the fully developed flow assumption is valid in this study. The numerical solution of a velocity profile at
Fig. 5. The effects of a constant channel area on thermal resistance.
H.-S. Kou et al. / International Communications in Heat and Mass Transfer 35 (2008) 577–582 Table 3 The optimal channel widths and thermal resistances as determined by various flow powers and channel heights Input
Optimal Output
Geometric configuration (μm) yo = 1100, zo = 100, zl = 10–90
yo = 1100, zo = 200, zl = 20–180
yl 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400 1000 800 600 400
Flow power (Watt)
Channel width (μm)
Thermal resistance (°C/W/cm2)
Mean velocity (m/s)
Ω 0.0001
zl 71.212 73.184 75.573 78.791 60.536 61.769 63.578 66.268 49.546 49.434 49.950 51.318 38.460 37.567 37.030 37.230 129.242 132.404 136.135 142.262 99.468 101.042 102.873 106.233 70.427 70.579 70.761 71.665 48.170 47.605 46.931 46.503
R 0.232 0.248 0.273 0.316 0.127 0.139 0.156 0.182 0.078 0.090 0.104 0.123 0.055 0.067 0.079 0.096 0.229 0.255 0.295 0.368 0.139 0.158 0.187 0.236 0.093 0.109 0.130 0.165 0.068 0.082 0.099 0.124
Um 0.220 0.249 0.291 0.360 0.640 0.723 0.847 1.051 1.830 2.045 2.376 2.941 5.090 5.631 6.464 7.943 0.295 0.332 0.383 0.461 0.821 0.923 1.068 1.298 2.185 2.446 2.823 3.446 5.704 6.346 7.282 8.864
0.001
0.01
0.1
0.0001
0.001
0.01
0.1
different positions is given in Table 2, which shows satisfactory agreement with a fully developed flow model, where zl = 56 μm, yl = 320 μm, and Δp = 103.42 kPa. Under the fully developed flow model, therefore, the flow power obtained from Eq. (10) can be used as an input variable to calculate the pressure drop of Eq. (11) and then to proceed the present numerical calculation. The effects of the channel width on the thermal resistance are shown in Fig. 3 when the flow power of a single channel is fixed at Ω = 0.01 Watt or Ω = 0.001 Watt. It can be observed that an optimum width of the channel exists under constant flow power. The thermal resistance, R, decreases as the flow power or flow rate increases. When the flow power is larger than Ω = 0.01 Watt and the channel width is between 20 μm and 80 μm, the thermal resistance will be lower than 0.15 °C/Wcm2. When the flow power is fixed at Ω = 0.001 Watt, the effect due to the channel width will be more significant especially for the shorter channel width. Because the interface area between the coolant fluid and the heat sink is not large enough to dissipate heat flux quickly, the shorter channel width has higher thermal resistance.
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Besides, the influence exerted by the channel height is shown in Fig. 3. Fig. 4 shows the effects of the channel height on thermal resistance when Ω = 0.01 Watt and Ω = 0.001 Watt. Since the height of the sink is fixed in this study, increasing the height of the channel induces the decrease of the substrate thickness. It is found that thermal resistance monotonically decreases with the decrease of the substrate thickness. The channel height above 1070 μm is not explored here out of consideration of the grid's error. The trend in the change of thermal resistance due to the channel height is the same as flow power. When yl b 1070 μm, a lower channel height always has higher thermal resistance. The effects of thermal resistance in various channel flow area are also considered here. The results are shown in Fig. 5, where the parameter, F = yl × zl, is defined as the cross section area of a flow channel. Both a larger flow area and flow power can obtain lower thermal resistance. Finally, minimum thermal resistance and the optimum channel width with various flow powers and channel heights are obtained by using the simulated annealing method. In the process of the optimization procedure, the channel width is searched from 10 μm to 90 μm. Table 3 shows that flow power plays the most important role in determining thermal resistance. The optimum channel width increases as the channel height decreases when the flow power is fixed at 0.0001 W and 0.001 W. However, the optimum channel width is not significantly influenced with the increase of the channel height when the flow power is fixed at 0.01 W and 0.1 W. A similar trend can be observed in the other optimum case, where the size of a single channel is fixed by x0 = 1 cm, y0 = 1100 μm, z0 = 200 μm and the optimal channel width ranges from 20 μm to 180 μm. It is noted that the whole thermal resistance of a heat sink can be determined by multiplying the thermal resistance of a single channel to the heat sink width and depth. For practical application, hence, engineers can refer to this Table in designing the optimal microchannel heat sink. 4. Conclusions In this study, the minimum thermal resistance of a heat sink including a microchannel with a fixed dimension is investigated to search for its optimum channel width. The validity of a fully developed flow model in a rectangular channel has been proved which can be used to determine the flow power and pressure drop of a microchannel heat sink. The results show that a larger flow area, larger flow power, and shorter substrate thickness can obtain lower thermal resistance. Meanwhile, the optimum width and height of a channel is available from this numerical procedure if the heat sink has the same flow area. After carrying out the optimum method, the optimal channel width is obtained with different channel heights. The optimal channel width is not significantly influenced by the decrease in the channel height when the flow power is fixed at 0.01 Watt and 0.1 Watt. On the contrary, the optimum channel width strongly depends on the channel height when the flow power is fixed at 0.001 Watt and 0.0001 Watt.
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