- Email: [email protected]
An impingement heat sink module design problem in determining simultaneously the optimal non-uniform fin widths and heights
International Journal of Heat and Mass Transfer 73 (2014) 627–633
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
An impingement heat sink module design problem in determining simultaneously the optimal non-uniform fin widths and heights Cheng-Hung Huang ⇑, Yu-Hsiang Chen Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 4 November 2013 Received in revised form 10 February 2014 Accepted 10 February 2014 Available online 15 March 2014 Keywords: Optimal heat sink design problem Impingement heat sink module Levenberg–Marquardt Method Non-uniform fin heights and widths
a b s t r a c t The Levenberg–Marquardt Method (LMM) is applied in the present study to determine the optimal fin widths and heights of an impingement cooling heat sink module using a general purpose commercial code (CFD-ACE+). In this optimal heat sink design problem, the non-uniform fin widths and heights are chosen as the design variables. The objective of the present study is to minimize the system thermal resistance (Rth) of the fin array and to obtain the optimal dimensions of heat sink. The results obtained by numerical experiments demonstrate that by utilizing the optimal heat sink and operating at the design condition Re = 15,000, Rth can be decreased by 3.10% and 1.20% when compared with the original and Yang and Peng’s (2008) [1] heat sinks, respectively. Nu and COE can be increased by 3.20% and 3.20%, respectively, when compared with the original heat sink, and these parameters can be increased by 1.22% and 1.18%, respectively, when compared with the optimal heat sink proposed by Yang and Peng (2008) [1]. Consequently, the thermal performances of optimal impingement heat sink can be improved. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The hydraulic and thermal characteristics of various impinging heat sinks have extensively studied, due to its favorable unit price, weight and reliability, by many researchers. For instance, Sansoucy et al. [2] experimentally studied the heat transfer from a parallel flat-plate heat sink under a turbulent air jet impingement. Brignoni and Garimella [3] examined experimentally the optimization of confined impinging air jets used in conjunction with a pin-fin heat sink. The ranges of enhancement factors for the heat sink were found to be between 2.8 and 9.7 relative to a bare surface. Li et al. [4] and Li and Chen [5] investigated the thermal performance of pin-fin and plate-fin heat sinks with confined impingement cooling by using infrared thermography. Yang and Peng [1] numerically investigated the thermal performance of a heat sink with a non-uniform fin heights design with impingement cooling. In their work, they intuitively and without any theoretical basis designed four groups of heat sink with non-uniform fin heights. Along with the original design, the Nusselt numbers and coefficients of performance for the four groups of impingement heat sinks were calculated and compared. Based on their research outcomes they concluded that group IV of type-k design had the best thermal performance when operating at ⇑ Corresponding author. Tel.: +886 627 47018; fax: +886 627 47019. E-mail address: [email protected] (C.-H. Huang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.026 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
Re = 15,000. The above study is not related to any design problem because no design algorithm was applied. An effective heat sink design algorithm improves the efficiency of heat sink modules and reduces the duration and cost of the design process. Many investigations have proposed high-performance heat removal characteristics. For instance, Park et al. [6] used numerical optimization to estimate the shape of pin-fins for a heat sink to improve the cooling efficiency. Iyengar and Bar-Cohen [7] utilized the least-energy optimization algorithm to design plate fin heat sinks in the forced convection problem. The Taguchi experimental design method was utilized by Sahin et al. [8] to examine the effects of changes in the design parameters on the heat transfer and pressure drop characteristics of a heat exchanger. Recently, Huang et al. [9] examined the impingement heat sink design problem studied by Yang and Peng [10] by utilizing the LMM. In reference [10], they investigated the thermal performance of a heat sink with a non-uniform fin width design with impingement cooling. The objective of the study by Huang et al. [9] is to further reduce the thermal resistance of their optimal heat sink by employing the optimal design algorithm to determine the optimal widths of fin in an impingement heat sink and to further improve the thermal performance of the heat sink module proposed by [10]. Results indicated that under the design operating condition Re = 5000, Rth can be decreased by 12.98% and 4.81% compared to the original and to Yang and Peng’s optimal heat sinks, respectively.
628
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
Nomenclature Ab Ah Bi b COE G Hi J k L L1, L2, L3
bottom surface, mm2 heating surface, mm2 design variables, mm the thickness of heat sink base, mm coefficient of enhancement fin pitch, mm fin height, mm functional defined by Eq. (7) thermal conductivity of fin, W/(m-K) the width of heat sink base, mm the width, depth and height of the computational domain, mm
The purpose of the present work is to examine the impingement heat sink design problem studied by Yang and Peng [1] again. The objective is to further reduce the thermal resistance and improve the thermal performance of their optimal heat sink by employing our optimal design algorithm to determine simultaneously the optimal fin widths and heights of an impingement heat sink.
2. The direct problem Fig. 1 shows a three-dimensional model of impingement heat sink module and will be considered in the present study to illustrate the methodology in designing the optimal fin shape by
Nu P q Rth T1 V Wi
Nusselt number number of design variables applied heat flux, W/m2 thermal resistance, °C/W ambient temperature, °C specified fin array volume, mm3 fin widths, mm
Greek symbols X total computational domain
using the LMM to minimize the thermal resistance of the heat sink module. Let X represents the computation domain and {X} = {X1 [ X2}, where X1 represents the heat sink fin array region and X2 represents the air flow region. All outer boundary fin surfaces are subjected to Robin-type boundary conditions with heat transfer coefficient h and ambient temperature T1. A heat flux q is imposed on heating surface Ah of the bottom surface of heat sink module Ab, while the remainder of the bottom surface of the fin array remains insulated. Fig. 1(a) shows the geometry of the computational domain of the impingement heat sink module, and Fig. 1(b) illustrates its bottom and heating surfaces. The three-dimensional motion of the fluid is an incompressible and unsteady flow and is calculated by using the three-dimensional Reynolds-Averaged Navier–Stokes (RANS) equations. In addition, the fluid is in a state of turbulence and thus the standard k e equation was selected for turbulence modeling, and the standard wall function was used near the wall. The velocity-inlet and pressure-outlet are chosen as the fluid boundary conditions in this study. The solution for the above three-dimensional impingement heat sink module is obtained using CFD-ACE+ [11] since Huang et al. [9] have examined its validity. The direct problem considered here is the determination of the velocity and temperature distributions for the air and heat sink when all the boundary conditions and the heat flux on Ah are known. 3. The design problem: obtain minimum thermal resistance The objective of this study is to utilize the Levenberg–Marquardt Method (LMM) design algorithm to re-examine the same impingement heat sink design problem that was recently studied by Yang and Peng [1]. The goal is to further reduce the thermal resistance of their optimal heat sink by determining the optimal fin heights and widths of the impingement heat sink. The design variables for a 6 by 6 squared fins heat sink module is illustrated in Fig. 2. The fin width and height can vary from fin to fin, and the fin pitch and volume are fixed. The fin array volume V can be calculated using the following equation:
V ¼ ½W 21 H1 þ W 2 H2 ðW 2 þ 2W 1 Þ þ W 3 H3 ðW 3 þ 2W 1 þ 2W 2 Þ 4 þ ðL2 bÞ mm3 L ¼ 2 W 1 þ 2 W 2 þ 2 W 3 þ 5 G mm Fig. 1. The (a) geometry of the computational domain of the impingement cooling heat sink module and (b) the bottom and heating surfaces of the fin array.
ð1Þ ð2Þ
where L and b are the width and thickness of the heat sink base, respectively. W1 to W3 and H1 to H3 are the fin widths and fin
629
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
determined from the solution of the direct problem given previously by using an updated optimal fin shape. P represents the total number of design variables. For the impingement heat sink module considered here we have Bi = {W1, W2, H1, H2} and P = 4. 4. The Levenberg–Marquardt Method (LMM) for minimization In mathematics and computing, the Levenberg–Marquardt Method (LMM) [9,12], also known as the damped least-squares (DLS) method, provides a numerical solution to the problem of minimizing a function, generally nonlinear, over a space of parameters of the function. These minimization problems arise especially in least squares curve fitting and nonlinear programming. Eq. (7) is minimized with respect to the estimated parameters Bi to obtain the following expression:
@J½X1 ðBi Þ @Rth ðBi Þ Rth ¼ 0; ¼ @Bi @Bi Fig. 2. The design variables of an impingement heat sink.
height, respectively, and G is the fin pitch. Here, L, b and G are known parameters, while W1, W2, H1 and H2 are the design variables for the heat sink modules considered in this study. W3 can be determined using Eq. (2) with a desired fin array the width of heat sink base L, and H3 can be determined using Eq. (1) with a desired fin array volume V. The shape of an impingement heat sink module can be constructed with known or estimated values for the above variables. For the impingement heat sink design problem considered here, a fixed fin array volume V is given while the design variables W1, W2, H1 and H2 are unknown. In addition, the thermal resistance of the heat sink is required to be minimized to increase the thermal performance and efficiency of the module. Based on the heating condition and the geometry of the heat sink considered here, the position of the highest temperature should be located at the center of the heating surface Ah. The thermal resistance of the present heat sink module is given below:
Rth ðBi Þ ¼
T ave ðBi Þ T 1 Q
ð3Þ
where Tave is the average temperature of the base plate of the heat sink module and is a function of the design variables Bi. The optimal design problem is to determine the optimal shape for the heat sink module. The Nusselt number and coefficient of enhancement (COE) for the present heat sink are defined as follows:
h¼
Q
ð4Þ
Ah ðT ave T 1 Þ
Nu ¼
hd ka
COE ¼
ð5Þ
Nunew Nuold
ð6Þ
where ka is the thermal conductivity of air and d is the diameter of the nozzle. The solution of the present impingement cooling heat sink module design problem is to be obtained by minimizing the following functional:
J½X1 ðBi Þ ¼ Rth ðBi Þ2 ¼ AT A;
i ¼ 1 to P
ð7Þ
where X1 indicates the domain of the heat sink module and is a function of the design variables Bi. Rth represents the estimated or computed thermal resistance of the fin array. This quantity can be
i ¼ 1 to P
ð8Þ
Eq. (8) is linearized by expanding Rth(Bi) in a Taylor series and retaining the first-order terms. Then, a damping parameter ln is added to the resulting expression to improve the convergence, leading to the Levenberg–Marquardt Method [12], which is given by:
ðF þ ln IÞDB ¼ D
ð9Þ
F ¼ WT W
ð10Þ
D ¼ WT A
ð11Þ
DB ¼ Bnþ1 Bn
ð12Þ
Here, the superscripts n and T represent the iteration index and the transpose matrix, respectively, I is the identity matrix, and W denotes the Jacobian matrix, which is defined as:
W¼
@Rth
ð13Þ
@BT
Eq. (9) is now written as follows, which is suitable for iterative calculation: 1
Bnþ1 ¼ Bn þ ðWT W þ ln IÞ WT Rth
ð14Þ n
The algorithm for choosing this damping value l is described in detail by Marquardt [12]. 5. Results and discussion The LMM is applied in the present study to determine simultaneously the optimal fin widths and heights of an impingement cooling heat sink module. To illustrate the reliability of the LMM in predicting the optimal fin widths and heights of the heat sink with a specified fin array volume by minimizing the thermal resistance of the heat sink, the following optimal heat sink design problem will be considered. Aluminum alloy will be considered for the heat sink module material, the thermal conductivity is taken as k = 168 W/(m-K), and the input heating power is chosen as Q = 100 W. Here, L = 80 mm, L4 = 40 mm, b = 8 mm, G = 6.4 mm and V = 143,360 mm3 are used for all the cases considered in this work. The applied boundary heat flux generated by electric power is calculated as q = Q/Ah=100/0.0016 = 62500 W/m2 and is utilized in all the cases considered in this work. The dimension for the computational domain is taken as L1 L2 L3 = 210 210 112 mm. A regular heat sink module with a constant fin width and height was used as the original design by Yang and Peng [1]. The dimensions and shape of the original heat sink module are given
630
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
Table 1 The dimensions and computational results for the heat sink module. Original W1 (mm) W2 (mm) W3 (mm) H1 (mm) H2 (mm) H3 (mm)
Yang & Peng [1]
8.0 8.0 8.0 40.0 40.0 40.0
8.0 8.0 8.0 50.0 45.0 35.0
150.7 106.9 89.8 80.6 73.8
155.6 106.1 88.5 79.4 73.1
Optimal 9.25 8.11 6.63 40.26 38.84 40.83
Tave (°C)
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
Rth (°C/W)
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
1.287 0.849 0.678 0.586 0.518
1.336 0.841 0.665 0.574 0.511
1.281 0.826 0.657 0.569 0.505
Nu
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
147.719 223.927 280.404 324.427 367.016
142.301 226.057 285.886 331.209 372.043
148.411 230.162 289.367 334.120 376.464
COE
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
0.963 1.010 1.020 1.021 1.014
1.005 1.028 1.032 1.030 1.026
1 1 1 1 1
150.1 104.6 87.7 78.9 72.5
in Table 1 and Fig. 3(a), respectively. By using the following working conditions: Q = 100 W, Re = 15,000 and T1 = 295 K, the temperature distribution of the original heat sink can be obtained using CFD-ACE+ code with a total of 878,230 numerical grids and is shown in Fig. 4(a). The number of grid is determined after grid independent test. The thermal resistance is calculated as Rth = 0.678 °C/W.
The dimensions and shape of the proposed optimal fin module by Yang and Peng [1], i.e., the Type k model, are given in Table 1 and Fig. 3(b), respectively. When applying the same working conditions, the CFD-ACE+ package can be used to compute the temperature distributions for the Type k fin array. The results of this computation are illustrated in Fig. 4(b). The thermal resistance is calculated as Rth = 0.665 °C/W, implying a 1.92% decrease in thermal resistance for the Type k fin array. It is interesting to examine whether the use of an optimal design algorithm can further improve the thermal performance of the heat sink by further reducing its thermal resistance. After performing the optimal design computation the optimal non-uniform fin widths and heights can be obtained by using the present algorithm to minimize thermal resistance Rth. The estimated design variables W1, W2, H1 and H2 and the calculated variable W3 and H3 are obtained and are shown in Table 1, the optimal heat sink is shown in Fig. 3(c) and the temperature distributions of optimal heat sink working at Re = 15,000 is shown in Fig. 4(c). The thermal resistance for this optimal heat sink is calculated to be Rth = 0.657 °C/W, i.e., there is a 3.10% decrease in thermal resistance for this optimal fin array compare to the original heat sink. The designed optimal heat sink has better thermal performance than both original and Yang and Peng’s heat sinks. Next, we would like to change the impinging velocity and compute the thermal performances of these three heat sinks. When impinging velocity changes to Re = 5000, the corresponding computed temperature distribution for original, Yang and Peng’s and optimal heat sinks are given in Fig. 5(a–c), respectively. The thermal resistances for the original, Yang and Peng’s [1] and optimal heat sinks are calculated to be Rth = 1.287, 1.336 and 1.281 °C/W, respectively, and this implies that there is a 0.47% decrease in thermal resistance for this optimal fin array while there is a 3.81% increase in thermal resistance for Yang and Peng’s fin array. This indicates that the designed optimal heat sink still has better thermal performance than the original heat sink, but Yang and Peng’s [1] heat sink is not a good design when operating at Re = 5000 since
Fig. 3. The (a) original heat sink and (b) Yang and Peng’s heat sink, (c) optimal heat sink.
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
631
Fig. 4. The temperature distribution for (a) the original, (b) Yang and Peng’s and (c) optimal heat sink modules at an impingement velocity Re = 1,5000.
Fig. 5. The temperature distribution for (a) the original, (b) Yang and Peng’s and (c) optimal heat sink modules at an impingement velocity Re = 5000.
the thermal resistance becomes larger than that for the original one. When considering Re = 10,000, 20,000 and 25,000, the computed thermal performances of thermal resistance (Rth), Nusselt number (Nu) and coefficient of enhancement (COE) can be ob-
tained from the numerical temperature data. The calculated values of Rth, Nu and COE are summarized in Table 1. The comparisons of the thermal performances of the Rth, Nu and COE for three heat sinks are shown in Fig. 6(a–c), respectively. For instance, at the design operating condition Re = 15,000, Rth of the
632
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
1.4
(a)
(b) 400
Original Yang and Peng [1] Optimal
1.2
Original Yang and Peng [1] Optimal
350
300
Nu
O
Rth, C/W
1 250
0.8 200
0.6 150
0.4
100
4000
8000
12000
16000 Re
(c)
20000
24000
28000
4000
8000
12000
16000 Re
20000
24000
28000
1.04
COE
1.02
1
Original Yang and Peng [1] Optimal
0.98
0.96 4000
8000
12000
16000 Re
20000
24000
28000
Fig. 6. The comparisons of (a) thermal resistance, (b) Nusselt number and (c) coefficient of enhancement among three types of heat sinks at different Reynolds numbers.
optimal heat sink can be decreased by 3.10% and 1.20% when compared with the original and Yang and Peng’s [1] heat sinks, respectively. Nu and COE of the optimal heat sink can both be increased by 3.20%, when compared with the original heat sink, and these parameters can be increased by1.22% and 1.18%, respectively, when compared with the optimal heat sink proposed by Yang and Peng [1]. Moreover, the thermal performance of the optimal heat sink is always better than that for the original and Yang and Peng’s heat sinks under all working conditions considered here, but the thermal performance of the Yang and Peng’s heat sink is worse than that for the original heat sink. Therefore, it can be concluded that the present design algorithm is valid and effective for obtaining the optimal non-uniform fin widths and heights.
6. Conclusions A three-dimensional optimal fin design problem in determining the optimal fin widths and heights for impingement heat sink module is successfully examined using the Levenberg–Marquardt Method (LMM) and the CFD-ACE+ commercial package. The minimization of the thermal resistance of the fin array is utilized in the
optimal design process. The numerical experiment results show that the heat sink module with optimal non-uniform fin widths and heights had the best thermal performances among three types of heat sinks with fixed fin volumes. At the design operating condition Re = 15,000, Rth can be decreased by 3.10% and 1.20%, compared to the original heat sink and Yang and Peng’s heat sink, respectively. This improvement is not significant in number but is significant in reality applications since any improvement of the performance for heat sink under the same working conditions and heat sink volume is always preferable and desirable. Meanwhile, when working at the non-designed conditions, the thermal performance of the optimal heat sink is always the best among three heat sinks, however, the thermal performance of the Yang and Peng’s heat sink is worse than that for the original heat sink at Re = 5000. This shows that the present optimal design algorithm can be applied successfully in the fin design problem to obtain the optimal fin shape.
Acknowledgment This work was supported in part through the National Science Council, ROC, Grant number NSC-100-2221-E-006-011-MY3.
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
References [1] Y.T. Yang, H.S. Peng, Numerical study of pin-fin heat sink with un-uniform fin height designs, Int. J. Heat Mass Transfer 51 (2008) 4788–4796. [2] E. Sansoucy, P.H. Oosthuizen, G.R. Ahmed, An experimental study of the enhancement of air-cooling limits for telecom/datacom heat sink applications using an impinging air jet, J. Electron. Packag. 128 (2006) 166–171. [3] L.A. Brignoni, S.V. Garimella, Experimental optimization of confined air jet impingement on a pin fin heat sink, IEEE Trans. Compon. Packag. Technol. 22 (3) (1999) 399–404. [4] H.Y. Li, S.M. Chao, G.L. Tsai, Thermal performance measurement of heat sinks with confined impinging jet by infrared thermography, Int. J. Heat Mass Transfer 48 (2005) 5386–5394. [5] H.Y. Li, K.Y. Chen, Thermal performance of plate-fin heat sinks under confined impinging jet conditions, Int. J. Heat Mass Transfer 50 (2007) 1963–1970.
633
[6] K. Park, D.H. Choi, K.S. Lee, Numerical shape optimization for high performance of a heat sink with pin-fins, Numer. Heat Transfer, Part A 46 (2004) 909–927. [7] M. Iyengar, A. Bar-Cohen, Least-energy optimization of forced convection plate fin heat sinks, IEEE Trans. Compon. Packag. Technol. 26 (2003) 62–70. [8] B. Sahin, K. Yakut, I. Kotcioglu, C. Celik, Optimum design parameters of a heat exchanger, Appl. Energy 82 (2005) 90–106. [9] C.H. Huang, Y.H. Chen, H.Y. Li, An impingement heat sink module design problem in determining optimal non-uniform fin widths, Int. J. Heat Mass Transfer 67 (2013) 992–1006. [10] Y.T. Yang, H.S. Peng, Numerical study of the heat sink with un-uniform fin width designs, Int. J. Heat Mass Transfer 52 (2009) 3473–3480. [11] CFD-ACE+ user’s manual, ESI-CFD Inc., 2005. [12] D.M. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11 (1963) 431–441.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
An impingement heat sink module design problem in determining simultaneously the optimal non-uniform fin widths and heights Cheng-Hung Huang ⇑, Yu-Hsiang Chen Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 4 November 2013 Received in revised form 10 February 2014 Accepted 10 February 2014 Available online 15 March 2014 Keywords: Optimal heat sink design problem Impingement heat sink module Levenberg–Marquardt Method Non-uniform fin heights and widths
a b s t r a c t The Levenberg–Marquardt Method (LMM) is applied in the present study to determine the optimal fin widths and heights of an impingement cooling heat sink module using a general purpose commercial code (CFD-ACE+). In this optimal heat sink design problem, the non-uniform fin widths and heights are chosen as the design variables. The objective of the present study is to minimize the system thermal resistance (Rth) of the fin array and to obtain the optimal dimensions of heat sink. The results obtained by numerical experiments demonstrate that by utilizing the optimal heat sink and operating at the design condition Re = 15,000, Rth can be decreased by 3.10% and 1.20% when compared with the original and Yang and Peng’s (2008) [1] heat sinks, respectively. Nu and COE can be increased by 3.20% and 3.20%, respectively, when compared with the original heat sink, and these parameters can be increased by 1.22% and 1.18%, respectively, when compared with the optimal heat sink proposed by Yang and Peng (2008) [1]. Consequently, the thermal performances of optimal impingement heat sink can be improved. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The hydraulic and thermal characteristics of various impinging heat sinks have extensively studied, due to its favorable unit price, weight and reliability, by many researchers. For instance, Sansoucy et al. [2] experimentally studied the heat transfer from a parallel flat-plate heat sink under a turbulent air jet impingement. Brignoni and Garimella [3] examined experimentally the optimization of confined impinging air jets used in conjunction with a pin-fin heat sink. The ranges of enhancement factors for the heat sink were found to be between 2.8 and 9.7 relative to a bare surface. Li et al. [4] and Li and Chen [5] investigated the thermal performance of pin-fin and plate-fin heat sinks with confined impingement cooling by using infrared thermography. Yang and Peng [1] numerically investigated the thermal performance of a heat sink with a non-uniform fin heights design with impingement cooling. In their work, they intuitively and without any theoretical basis designed four groups of heat sink with non-uniform fin heights. Along with the original design, the Nusselt numbers and coefficients of performance for the four groups of impingement heat sinks were calculated and compared. Based on their research outcomes they concluded that group IV of type-k design had the best thermal performance when operating at ⇑ Corresponding author. Tel.: +886 627 47018; fax: +886 627 47019. E-mail address: [email protected] (C.-H. Huang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.026 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
Re = 15,000. The above study is not related to any design problem because no design algorithm was applied. An effective heat sink design algorithm improves the efficiency of heat sink modules and reduces the duration and cost of the design process. Many investigations have proposed high-performance heat removal characteristics. For instance, Park et al. [6] used numerical optimization to estimate the shape of pin-fins for a heat sink to improve the cooling efficiency. Iyengar and Bar-Cohen [7] utilized the least-energy optimization algorithm to design plate fin heat sinks in the forced convection problem. The Taguchi experimental design method was utilized by Sahin et al. [8] to examine the effects of changes in the design parameters on the heat transfer and pressure drop characteristics of a heat exchanger. Recently, Huang et al. [9] examined the impingement heat sink design problem studied by Yang and Peng [10] by utilizing the LMM. In reference [10], they investigated the thermal performance of a heat sink with a non-uniform fin width design with impingement cooling. The objective of the study by Huang et al. [9] is to further reduce the thermal resistance of their optimal heat sink by employing the optimal design algorithm to determine the optimal widths of fin in an impingement heat sink and to further improve the thermal performance of the heat sink module proposed by [10]. Results indicated that under the design operating condition Re = 5000, Rth can be decreased by 12.98% and 4.81% compared to the original and to Yang and Peng’s optimal heat sinks, respectively.
628
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
Nomenclature Ab Ah Bi b COE G Hi J k L L1, L2, L3
bottom surface, mm2 heating surface, mm2 design variables, mm the thickness of heat sink base, mm coefficient of enhancement fin pitch, mm fin height, mm functional defined by Eq. (7) thermal conductivity of fin, W/(m-K) the width of heat sink base, mm the width, depth and height of the computational domain, mm
The purpose of the present work is to examine the impingement heat sink design problem studied by Yang and Peng [1] again. The objective is to further reduce the thermal resistance and improve the thermal performance of their optimal heat sink by employing our optimal design algorithm to determine simultaneously the optimal fin widths and heights of an impingement heat sink.
2. The direct problem Fig. 1 shows a three-dimensional model of impingement heat sink module and will be considered in the present study to illustrate the methodology in designing the optimal fin shape by
Nu P q Rth T1 V Wi
Nusselt number number of design variables applied heat flux, W/m2 thermal resistance, °C/W ambient temperature, °C specified fin array volume, mm3 fin widths, mm
Greek symbols X total computational domain
using the LMM to minimize the thermal resistance of the heat sink module. Let X represents the computation domain and {X} = {X1 [ X2}, where X1 represents the heat sink fin array region and X2 represents the air flow region. All outer boundary fin surfaces are subjected to Robin-type boundary conditions with heat transfer coefficient h and ambient temperature T1. A heat flux q is imposed on heating surface Ah of the bottom surface of heat sink module Ab, while the remainder of the bottom surface of the fin array remains insulated. Fig. 1(a) shows the geometry of the computational domain of the impingement heat sink module, and Fig. 1(b) illustrates its bottom and heating surfaces. The three-dimensional motion of the fluid is an incompressible and unsteady flow and is calculated by using the three-dimensional Reynolds-Averaged Navier–Stokes (RANS) equations. In addition, the fluid is in a state of turbulence and thus the standard k e equation was selected for turbulence modeling, and the standard wall function was used near the wall. The velocity-inlet and pressure-outlet are chosen as the fluid boundary conditions in this study. The solution for the above three-dimensional impingement heat sink module is obtained using CFD-ACE+ [11] since Huang et al. [9] have examined its validity. The direct problem considered here is the determination of the velocity and temperature distributions for the air and heat sink when all the boundary conditions and the heat flux on Ah are known. 3. The design problem: obtain minimum thermal resistance The objective of this study is to utilize the Levenberg–Marquardt Method (LMM) design algorithm to re-examine the same impingement heat sink design problem that was recently studied by Yang and Peng [1]. The goal is to further reduce the thermal resistance of their optimal heat sink by determining the optimal fin heights and widths of the impingement heat sink. The design variables for a 6 by 6 squared fins heat sink module is illustrated in Fig. 2. The fin width and height can vary from fin to fin, and the fin pitch and volume are fixed. The fin array volume V can be calculated using the following equation:
V ¼ ½W 21 H1 þ W 2 H2 ðW 2 þ 2W 1 Þ þ W 3 H3 ðW 3 þ 2W 1 þ 2W 2 Þ 4 þ ðL2 bÞ mm3 L ¼ 2 W 1 þ 2 W 2 þ 2 W 3 þ 5 G mm Fig. 1. The (a) geometry of the computational domain of the impingement cooling heat sink module and (b) the bottom and heating surfaces of the fin array.
ð1Þ ð2Þ
where L and b are the width and thickness of the heat sink base, respectively. W1 to W3 and H1 to H3 are the fin widths and fin
629
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
determined from the solution of the direct problem given previously by using an updated optimal fin shape. P represents the total number of design variables. For the impingement heat sink module considered here we have Bi = {W1, W2, H1, H2} and P = 4. 4. The Levenberg–Marquardt Method (LMM) for minimization In mathematics and computing, the Levenberg–Marquardt Method (LMM) [9,12], also known as the damped least-squares (DLS) method, provides a numerical solution to the problem of minimizing a function, generally nonlinear, over a space of parameters of the function. These minimization problems arise especially in least squares curve fitting and nonlinear programming. Eq. (7) is minimized with respect to the estimated parameters Bi to obtain the following expression:
@J½X1 ðBi Þ @Rth ðBi Þ Rth ¼ 0; ¼ @Bi @Bi Fig. 2. The design variables of an impingement heat sink.
height, respectively, and G is the fin pitch. Here, L, b and G are known parameters, while W1, W2, H1 and H2 are the design variables for the heat sink modules considered in this study. W3 can be determined using Eq. (2) with a desired fin array the width of heat sink base L, and H3 can be determined using Eq. (1) with a desired fin array volume V. The shape of an impingement heat sink module can be constructed with known or estimated values for the above variables. For the impingement heat sink design problem considered here, a fixed fin array volume V is given while the design variables W1, W2, H1 and H2 are unknown. In addition, the thermal resistance of the heat sink is required to be minimized to increase the thermal performance and efficiency of the module. Based on the heating condition and the geometry of the heat sink considered here, the position of the highest temperature should be located at the center of the heating surface Ah. The thermal resistance of the present heat sink module is given below:
Rth ðBi Þ ¼
T ave ðBi Þ T 1 Q
ð3Þ
where Tave is the average temperature of the base plate of the heat sink module and is a function of the design variables Bi. The optimal design problem is to determine the optimal shape for the heat sink module. The Nusselt number and coefficient of enhancement (COE) for the present heat sink are defined as follows:
h¼
Q
ð4Þ
Ah ðT ave T 1 Þ
Nu ¼
hd ka
COE ¼
ð5Þ
Nunew Nuold
ð6Þ
where ka is the thermal conductivity of air and d is the diameter of the nozzle. The solution of the present impingement cooling heat sink module design problem is to be obtained by minimizing the following functional:
J½X1 ðBi Þ ¼ Rth ðBi Þ2 ¼ AT A;
i ¼ 1 to P
ð7Þ
where X1 indicates the domain of the heat sink module and is a function of the design variables Bi. Rth represents the estimated or computed thermal resistance of the fin array. This quantity can be
i ¼ 1 to P
ð8Þ
Eq. (8) is linearized by expanding Rth(Bi) in a Taylor series and retaining the first-order terms. Then, a damping parameter ln is added to the resulting expression to improve the convergence, leading to the Levenberg–Marquardt Method [12], which is given by:
ðF þ ln IÞDB ¼ D
ð9Þ
F ¼ WT W
ð10Þ
D ¼ WT A
ð11Þ
DB ¼ Bnþ1 Bn
ð12Þ
Here, the superscripts n and T represent the iteration index and the transpose matrix, respectively, I is the identity matrix, and W denotes the Jacobian matrix, which is defined as:
W¼
@Rth
ð13Þ
@BT
Eq. (9) is now written as follows, which is suitable for iterative calculation: 1
Bnþ1 ¼ Bn þ ðWT W þ ln IÞ WT Rth
ð14Þ n
The algorithm for choosing this damping value l is described in detail by Marquardt [12]. 5. Results and discussion The LMM is applied in the present study to determine simultaneously the optimal fin widths and heights of an impingement cooling heat sink module. To illustrate the reliability of the LMM in predicting the optimal fin widths and heights of the heat sink with a specified fin array volume by minimizing the thermal resistance of the heat sink, the following optimal heat sink design problem will be considered. Aluminum alloy will be considered for the heat sink module material, the thermal conductivity is taken as k = 168 W/(m-K), and the input heating power is chosen as Q = 100 W. Here, L = 80 mm, L4 = 40 mm, b = 8 mm, G = 6.4 mm and V = 143,360 mm3 are used for all the cases considered in this work. The applied boundary heat flux generated by electric power is calculated as q = Q/Ah=100/0.0016 = 62500 W/m2 and is utilized in all the cases considered in this work. The dimension for the computational domain is taken as L1 L2 L3 = 210 210 112 mm. A regular heat sink module with a constant fin width and height was used as the original design by Yang and Peng [1]. The dimensions and shape of the original heat sink module are given
630
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
Table 1 The dimensions and computational results for the heat sink module. Original W1 (mm) W2 (mm) W3 (mm) H1 (mm) H2 (mm) H3 (mm)
Yang & Peng [1]
8.0 8.0 8.0 40.0 40.0 40.0
8.0 8.0 8.0 50.0 45.0 35.0
150.7 106.9 89.8 80.6 73.8
155.6 106.1 88.5 79.4 73.1
Optimal 9.25 8.11 6.63 40.26 38.84 40.83
Tave (°C)
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
Rth (°C/W)
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
1.287 0.849 0.678 0.586 0.518
1.336 0.841 0.665 0.574 0.511
1.281 0.826 0.657 0.569 0.505
Nu
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
147.719 223.927 280.404 324.427 367.016
142.301 226.057 285.886 331.209 372.043
148.411 230.162 289.367 334.120 376.464
COE
Re = 5000 Re = 10,000 Re = 15,000 Re = 20,000 Re = 25,000
0.963 1.010 1.020 1.021 1.014
1.005 1.028 1.032 1.030 1.026
1 1 1 1 1
150.1 104.6 87.7 78.9 72.5
in Table 1 and Fig. 3(a), respectively. By using the following working conditions: Q = 100 W, Re = 15,000 and T1 = 295 K, the temperature distribution of the original heat sink can be obtained using CFD-ACE+ code with a total of 878,230 numerical grids and is shown in Fig. 4(a). The number of grid is determined after grid independent test. The thermal resistance is calculated as Rth = 0.678 °C/W.
The dimensions and shape of the proposed optimal fin module by Yang and Peng [1], i.e., the Type k model, are given in Table 1 and Fig. 3(b), respectively. When applying the same working conditions, the CFD-ACE+ package can be used to compute the temperature distributions for the Type k fin array. The results of this computation are illustrated in Fig. 4(b). The thermal resistance is calculated as Rth = 0.665 °C/W, implying a 1.92% decrease in thermal resistance for the Type k fin array. It is interesting to examine whether the use of an optimal design algorithm can further improve the thermal performance of the heat sink by further reducing its thermal resistance. After performing the optimal design computation the optimal non-uniform fin widths and heights can be obtained by using the present algorithm to minimize thermal resistance Rth. The estimated design variables W1, W2, H1 and H2 and the calculated variable W3 and H3 are obtained and are shown in Table 1, the optimal heat sink is shown in Fig. 3(c) and the temperature distributions of optimal heat sink working at Re = 15,000 is shown in Fig. 4(c). The thermal resistance for this optimal heat sink is calculated to be Rth = 0.657 °C/W, i.e., there is a 3.10% decrease in thermal resistance for this optimal fin array compare to the original heat sink. The designed optimal heat sink has better thermal performance than both original and Yang and Peng’s heat sinks. Next, we would like to change the impinging velocity and compute the thermal performances of these three heat sinks. When impinging velocity changes to Re = 5000, the corresponding computed temperature distribution for original, Yang and Peng’s and optimal heat sinks are given in Fig. 5(a–c), respectively. The thermal resistances for the original, Yang and Peng’s [1] and optimal heat sinks are calculated to be Rth = 1.287, 1.336 and 1.281 °C/W, respectively, and this implies that there is a 0.47% decrease in thermal resistance for this optimal fin array while there is a 3.81% increase in thermal resistance for Yang and Peng’s fin array. This indicates that the designed optimal heat sink still has better thermal performance than the original heat sink, but Yang and Peng’s [1] heat sink is not a good design when operating at Re = 5000 since
Fig. 3. The (a) original heat sink and (b) Yang and Peng’s heat sink, (c) optimal heat sink.
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
631
Fig. 4. The temperature distribution for (a) the original, (b) Yang and Peng’s and (c) optimal heat sink modules at an impingement velocity Re = 1,5000.
Fig. 5. The temperature distribution for (a) the original, (b) Yang and Peng’s and (c) optimal heat sink modules at an impingement velocity Re = 5000.
the thermal resistance becomes larger than that for the original one. When considering Re = 10,000, 20,000 and 25,000, the computed thermal performances of thermal resistance (Rth), Nusselt number (Nu) and coefficient of enhancement (COE) can be ob-
tained from the numerical temperature data. The calculated values of Rth, Nu and COE are summarized in Table 1. The comparisons of the thermal performances of the Rth, Nu and COE for three heat sinks are shown in Fig. 6(a–c), respectively. For instance, at the design operating condition Re = 15,000, Rth of the
632
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
1.4
(a)
(b) 400
Original Yang and Peng [1] Optimal
1.2
Original Yang and Peng [1] Optimal
350
300
Nu
O
Rth, C/W
1 250
0.8 200
0.6 150
0.4
100
4000
8000
12000
16000 Re
(c)
20000
24000
28000
4000
8000
12000
16000 Re
20000
24000
28000
1.04
COE
1.02
1
Original Yang and Peng [1] Optimal
0.98
0.96 4000
8000
12000
16000 Re
20000
24000
28000
Fig. 6. The comparisons of (a) thermal resistance, (b) Nusselt number and (c) coefficient of enhancement among three types of heat sinks at different Reynolds numbers.
optimal heat sink can be decreased by 3.10% and 1.20% when compared with the original and Yang and Peng’s [1] heat sinks, respectively. Nu and COE of the optimal heat sink can both be increased by 3.20%, when compared with the original heat sink, and these parameters can be increased by1.22% and 1.18%, respectively, when compared with the optimal heat sink proposed by Yang and Peng [1]. Moreover, the thermal performance of the optimal heat sink is always better than that for the original and Yang and Peng’s heat sinks under all working conditions considered here, but the thermal performance of the Yang and Peng’s heat sink is worse than that for the original heat sink. Therefore, it can be concluded that the present design algorithm is valid and effective for obtaining the optimal non-uniform fin widths and heights.
6. Conclusions A three-dimensional optimal fin design problem in determining the optimal fin widths and heights for impingement heat sink module is successfully examined using the Levenberg–Marquardt Method (LMM) and the CFD-ACE+ commercial package. The minimization of the thermal resistance of the fin array is utilized in the
optimal design process. The numerical experiment results show that the heat sink module with optimal non-uniform fin widths and heights had the best thermal performances among three types of heat sinks with fixed fin volumes. At the design operating condition Re = 15,000, Rth can be decreased by 3.10% and 1.20%, compared to the original heat sink and Yang and Peng’s heat sink, respectively. This improvement is not significant in number but is significant in reality applications since any improvement of the performance for heat sink under the same working conditions and heat sink volume is always preferable and desirable. Meanwhile, when working at the non-designed conditions, the thermal performance of the optimal heat sink is always the best among three heat sinks, however, the thermal performance of the Yang and Peng’s heat sink is worse than that for the original heat sink at Re = 5000. This shows that the present optimal design algorithm can be applied successfully in the fin design problem to obtain the optimal fin shape.
Acknowledgment This work was supported in part through the National Science Council, ROC, Grant number NSC-100-2221-E-006-011-MY3.
C.-H. Huang, Y.-H. Chen / International Journal of Heat and Mass Transfer 73 (2014) 627–633
References [1] Y.T. Yang, H.S. Peng, Numerical study of pin-fin heat sink with un-uniform fin height designs, Int. J. Heat Mass Transfer 51 (2008) 4788–4796. [2] E. Sansoucy, P.H. Oosthuizen, G.R. Ahmed, An experimental study of the enhancement of air-cooling limits for telecom/datacom heat sink applications using an impinging air jet, J. Electron. Packag. 128 (2006) 166–171. [3] L.A. Brignoni, S.V. Garimella, Experimental optimization of confined air jet impingement on a pin fin heat sink, IEEE Trans. Compon. Packag. Technol. 22 (3) (1999) 399–404. [4] H.Y. Li, S.M. Chao, G.L. Tsai, Thermal performance measurement of heat sinks with confined impinging jet by infrared thermography, Int. J. Heat Mass Transfer 48 (2005) 5386–5394. [5] H.Y. Li, K.Y. Chen, Thermal performance of plate-fin heat sinks under confined impinging jet conditions, Int. J. Heat Mass Transfer 50 (2007) 1963–1970.
633
[6] K. Park, D.H. Choi, K.S. Lee, Numerical shape optimization for high performance of a heat sink with pin-fins, Numer. Heat Transfer, Part A 46 (2004) 909–927. [7] M. Iyengar, A. Bar-Cohen, Least-energy optimization of forced convection plate fin heat sinks, IEEE Trans. Compon. Packag. Technol. 26 (2003) 62–70. [8] B. Sahin, K. Yakut, I. Kotcioglu, C. Celik, Optimum design parameters of a heat exchanger, Appl. Energy 82 (2005) 90–106. [9] C.H. Huang, Y.H. Chen, H.Y. Li, An impingement heat sink module design problem in determining optimal non-uniform fin widths, Int. J. Heat Mass Transfer 67 (2013) 992–1006. [10] Y.T. Yang, H.S. Peng, Numerical study of the heat sink with un-uniform fin width designs, Int. J. Heat Mass Transfer 52 (2009) 3473–3480. [11] CFD-ACE+ user’s manual, ESI-CFD Inc., 2005. [12] D.M. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11 (1963) 431–441.