## A three-dimensional heat sink module design problem

## with experimental veriﬁcation

### Cheng-Hung Huang

a,⇑_{, Jon-Jer Lu}

a_{, Herchang Ay}

b
a
Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC b

Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan, ROC

### a r t i c l e

### i n f o

Article history: Received 7 October 2010

Received in revised form 11 November 2010 Accepted 11 November 2010

Available online 18 December 2010 Keywords:

Levenberg-Marquardt Method Heat sink module design Optimum design

### a b s t r a c t

A three-dimensional heat sink module design problem is examined in this work to estimate the optimum design variables using the Levenberg–Marquardt Method (LMM) and a general purpose commercial code CFD-ACE+. Three different types of heat sinks are designed based on the original ﬁn arrays with a ﬁxed volume. The objective of this study is to minimize the maximum temperature in the ﬁn array and to determine the best shape of heat sink. Results obtained by using the LMM to solve this 3-D heat sink module design problem are ﬁrstly justiﬁed based on the numerical experiments and it is concluded that for all three cases, the optimum ﬁn height H tends to become higher and optimum ﬁn thickness W tends to become thinner than the original ﬁn array, as a result both the ﬁn pitch D and heat sink base thickness U are increased. The maximum temperature for the designed ﬁn array can be decreased drastically by uti-lizing the present ﬁn design algorithm. Finally, temperature distributions for the optimal heat sink mod-ules are measured using thermal camera and compared with the numerical solutions to justify the validity of the present design.

Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, the tendency to design electronic products becomes lighter, thinner, shorter, and smaller. Due to the fact that shrinking in the dimension of these electronic products will result in drastic increase in the heat generation rate when comparing with previous products. For this reason, an efﬁcient cooling system to remove the high heat generation, and consequently maintain the stability and reliability of the products, have received much attention.

The heat sink module is the most common heat exchanger for CPUs and has been extensively used in order to provide cooling function for electronic components. The conventional heat sink module utilized the forced convection cooling technique; dissipate heat from CPUs to the ambient air. The combination of the fan and heat sink design usually involved in this forced convection cooling technique.

The forced convection cooling technique becomes one of the most commonly used devices to cool CPUs since it has the advan-tageous of simple maintenance process, more reliability and lower manufacturing cost. It has been seen by many researchers that a heat sink with good geometrical design will provide better cooling performance and higher efﬁciency. It implies that the optimization process must be an effective tool for the heat sink design problem.

If an efﬁcient heat sink design algorithm is provided, it will greatly improve the reliability and prolong the life span of the CPUs. Many investigations of the optimum design parameters and the selection of heat sink module have been proposed in order to offer a high-performance heat removal characteristic. For instance, Kraus and Bar-Cohen[1]presented the fundamental the-ories for heat transfer and hydrodynamics characteristics of heat sinks. Shih and Liu[2]and Furukawa and Yang[3]presented an ap-proach to design the plate-ﬁn heat sinks by minimizing the entro-py generation rate in order to reach the most efﬁcient heat transfer. Leon et al.[4]and Small et al.[5]used computational ﬂuid dynamics (CFD) to study ﬂow and heat transfer behaviors for stag-gered heat sinks in detail. Iyengar and Bar-Cohen[6]utilized the least-energy optimization algorithm to design the plate ﬁn heat sinks in the forced convection problem. Yang and Peng[7,8] inves-tigated numerically the thermal performances of the heat sink with un-uniform ﬁn width and ﬁn height designs with an impingement cooling. Zhou et al.[9]considered a multi-parameter constrained optimization procedure to design the plate ﬁnned heat sinks by minimizing their rates of entropy generation. Park and Moon[10]

utilized the progressive quadratic response surface model to esti-mate the optimum ﬁn design variables for a plate-ﬁn type heat sink. Srisomporn and Bureerat [11] considered a geometrical design problem for the plate-ﬁn heat sinks by using hybridization of the multiobjective evolutionary algorithms (MOEAs) and a response surface method (RSM). Shan et al.[12] established the

0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.11.044

⇑Corresponding author.

E-mail address:chhuang@mail.ncku.edu.tw(C.-H. Huang).

Contents lists available atScienceDirect

## International Journal of Heat and Mass Transfer

direct link between the pressure drop of heat sinks and system operating curve for the selected fan to optimize a parallel plate impingement heat sink. Park et al.[13]applied the numerical opti-mization to determine the shape of pin-ﬁns for a heat sink to im-prove the cooling efﬁciency. Sahin et al. [14] used the Taguchi experimental design method to examine the effects of design parameters on the heat transfer and pressure drop characteristic of a heat exchanger.

From references mentioned above, the optimum design prob-lems for an efﬁcient heat sink module become a primary challenge in the electronic industry. In order to obtain an optimum design for heat sink modules, the proper types for heat sink modules and a suitable optimization algorithm should be chosen before proceed-ing to the design problem. Among numerous existproceed-ing designs of heat sink modules, the design of heat sink with the plate ﬁn array is widely utilized in the cooling enhancement of current electronic equipment. Therefore this type of heat sink module with modiﬁca-tions will be considered in this work. Besides, the present study will also focus on the thermal performance of the fan–sink assem-bly subjected to forced air cooling.

The Levenberg–Marquardt Method (LMM)[15]has proven to be a powerful algorithm in inverse design calculation for engineering applications. This inverse design method had been applied to pre-dict the form of a ship’s hull in accordance with the desired hull pressure distribution by Huang et al. [16]. Subsequently, Chen and Huang[17]applied it to predict an unknown hull form based on the preferable wake distribution in the propeller disk plane. Chen et al.[18]further applied it to the aspect of optimal design for a bulbous bow. Huang and Lin[19]applied LMM in the theoret-ical and experimental Studies to estimate the optimum shape for gas channel for a serpentine PEMFC. The LMM will be adopted in the present study as an optimization algorithm.

This work addresses the development of an efﬁcient method for parameter estimation in estimating the design variables for heat sink modules that satisﬁes the constraint of minimizing the maxi-mum surface temperature. However, without experimental veriﬁca-tion it is difﬁcult to show that the present design algorithm can be utilized in reality. For this reason in the present study the estimated optimal heat sinks will be fabricated and they will be used in exper-iment to measure the temperatures by using infrared thermal scan-ner. Finally these temperatures will be compared with the calculated temperatures to show the accuracy of our computations.

2. The direct problem

The following three-dimensional heat sink module is consid-ered to illustrate the methodology for developing expressions for

use in determining the design variables for heat sink module in the present inverse design problem by using LMM and CFD-ACE+

[20].

It is assumed thatXrepresents the domain of computation and fXg ¼ fX1[X2g, whereX1indicates the domain of ﬁn array and

X2represents the air ﬂow region. The boundary conditions on all

the outer boundary surfaces are subjected to the Robin boundary conditions with heat transfer coefﬁcient h and ambient tempera-ture T1. A heat ﬂux q is imposed at the heating surface Shwhile

the rest of bottom surface Sb of ﬁn array remains insulated. Fig. 1(a) shows the geometry of the computational domain of heat sink module andFig. 1(b) indicates the bottom and heating sur-faces of the ﬁn array.

The mathematical formulation of this 3-D heat conduction problem for the ﬁn domainX1is given by:

Nomenclature

Bj design variables

D ﬁn pitch (calculated variable)

D1 the width of row passage (calculated variable) H ﬁn height (design variable)

J functional deﬁned by Eq.(10)

k thermal conductivity of ﬁn

L1, L2 width and depth of the computational domain N number of ﬁn plate in each row

q applied heat ﬂux Sb bottom surface Sh heating surface Tf calculated ﬁn temperature T1 ambient temperature ui ﬂow velocity

U ﬁn base thickness (calculated variable) V speciﬁed ﬁn array volume

W ﬁn thickness (design variable) W1 ﬁn width (design variable) Y desired maximum temperature Greek symbols

W Jacobian matrix deﬁned by Eq.(16)

X total computational domain

### e

convergence criterion### l

n_{damping parameter}

Fig. 1. The (a) geometry of the computational domain of heat sink module and (b) bottom and heating surfaces of the ﬁn array.

kðX1Þ @2TfðX1Þ @x2 þ @2TfðX1Þ @y2 þ @2TfðX1Þ @z2 " # ¼ 0; in

### X

1 ð1aÞ k@Tf@z ¼ q; on the heating surface Sh ð1bÞ

@Tf

@z ¼ 0; on the surface fSb Shg ð1cÞ

where Tfindicates the ﬁn temperature distribution and k is the

ther-mal conductivity of ﬁn.

For the air ﬂow region,X2, a fan is located right at the top surface

of the heat sink module to drive the air and the ﬂow is assumed to be a three-dimensional steady and incompressible ﬂow, in addition, the thermophysical properties of the ﬂuid are assumed to be constant. Both the buoyancy and radiation heat transfer effects are neglect. The three-dimensional governing equations of mass, momentum, energy in the steady turbulent main ﬂow using the standard k–

### e

model, turbulent kinetic energy and turbulent energy dissipation rate are shown in Eqs.(2)–(6), respectively[7,8]:

@qui @xi ¼ 0 ð2Þ

### q

uj @ui @xj ¼ @p @xi þ @ @xj### l

t @ ui @xj þ@uj @xi ð3Þ### q

uj @Ta @xj ¼ @ @xj### l

1### r

1þ### l

t### r

t_{@T}a @xj " # ð4Þ

### q

uj @k @xj ¼ @ @xj### l

t### r

k @k @xj þ### l

t @ ui @xj þ@uj @xi_{@ }

_{u}i @xj

### qe

ð5Þ### q

_{u}

_{j}@e @xj ¼ @ @xj

### l

t### r

e @e @xj þ C1### l

t### e

k @ ui @xj þ@uj @xi_{@}

_{u}i @xj C2

### q

### e

2 k ð6ÞSince the Navier–Stokes equations are solved inside the domain, no-slip boundary condition is applied to all the walls in the do-main. Therefore, at all of the surfaces ui¼ 0.

The solution for the above 3-D ﬂuid-heat conjugate problem in the irregular solid and air ﬂow domains fX1[X2g is solved using

CFD-ACE+. The direct problem considered here is concerned with the determination of the velocity and temperature distributions for the heat sink when all the boundary conditions and heat ﬂux on Share known.

3. The heat sink design problem

Three commonly seen heat sink modules are used in the present study, they are (1) two-row plate ﬁn type (Type A),(2)three-row plate ﬁn type (Type B) and(3)four-row plate ﬁn type (Type C), and are illustrated inFig. 2(a)–(c), respectively. The ﬁn array vol-ume V can be calculated by using the following equations:

V ¼ ½ðW H W1 N 2Þ þ ðL1 L2 UÞ mm3_{;}

for Type A heat sink ð7Þ

V ¼ ½ðW H W1 N 3Þ þ ðL1 L2 UÞ mm3_{;}

for Type B heat sink ð8Þ

V ¼ ½ðW H W1 N 4Þ þ ðL1 L2 UÞ mm3_{;}

for Type C heat sink ð9Þ

where N indicates number of ﬁn plate in each row, L1 and L2 are the width and depth of heat sink, respectively; W and H are the ﬁn thickness and ﬁn height, respectively, W1 indicates the ﬁn width.

D is the ﬁn pitch and D1 represents the width of row passage. Here H, W and W1 are the design variables for the heat sink modules con-sidered here. Heat sink base thickness U can be determined by V and design variables while D and D1 can be calculated by using W and W1. Here U, D and D1 represent the calculated variables. When V, H, W and W1 are given, the shapes for heat sinks for Type A, B and C heat sinks can be constructed.

For the heat sink design problem, a ﬁxed ﬁn array volume is given while the design variables H, W and W1 are regarded as being un-known; in addition, the highest temperature on heating surface Sh

of heat sink are required to become as low as possible to increase the efﬁciency of heat sinks. According to the heating condition and geometry of the heat sink considered here, the position of the highest temperature is located at the center of heating surface Sh.

Let the desired temperatures located at the center of Shbe

de-noted by Y, the inverse design problem can then be stated as fol-lows: utilizing the above mentioned desired temperature Y, design the optimal shapes for Type A, B and C heat sinks.

The solution of the present heat sink design problem is to be ob-tained in such a way that the following functional is minimized:

J½X1ðBjÞ ¼ ½Tf;centerðBjÞ Y

2

¼ ATA; j ¼ 1 to P ð10Þ

Here Tf,centerrepresents the estimated or computed temperatures at

the center of Shand it is the maximum temperature in the ﬁn array.

This quantity is determined from the solution of the direct problem

given previously by using an original ﬁn shape. B indicates the design variable and P represents the total number of design variable. For all types of heat sinks considered here, we have Bj= {H, W, W1} and P = 3.

4. The Levenberg–Marquardt Method (IMM) for minimization

Eq.(10)is minimized with respect to the estimated parameters Bjto obtain: @J½X1ðBjÞ @Bj ¼ @Tf;center @Bj ½Tf;center Y ¼ 0; j ¼ 1 to P ð11Þ

Eq.(11) is linearized by expanding Tf,center(Bj) in Taylor series

and retaining the ﬁrst order terms. Then a damping parameter

### l

nis added to the resulting expression to improve convergence, lead-ing to the Levenberg–Marquardt Method[15], given by:

ðF þ

### l

n_{IÞDB ¼ D}

_{ð12Þ}F ¼ WT W# ð13Þ D ¼ WT

_{A}ð14Þ

### DB ¼ B

nþ1 Bn ð15ÞHere, the superscripts n and T represent the iteration index and transpose matrix, respectively, I is the identity matrix, andW de-notes the Jacobian matrix, deﬁned as:

W¼@Tf;center

@BT ð16Þ

The Jacobian matrix deﬁned by Eq.(16)is determined by perturbing the unknown parameters Bjone at a time and computing the

result-ing change in temperatures on Sbfrom the solution of the direct

problem, Eqs.(1)–(6).

Eq.(12)is now written in a form suitable for iterative calcula-tion as:

Bnþ1_{¼ B}n

þ ðWTWþ

### l

n_{IÞ}1

_{W}T

ðTf;center YÞ ð17Þ

The algorithm for choosing this damping value

### l

nis described in detail by Marquardt[15], so it is not repeated here.

The bridge between CFD-ACE+ and LMM is the INPUT/OUTPUT ﬁles. These ﬁles should be arranged such that their format can be recognized by CFD-ACE+ and LMM. A sequence of forward prob-lems is solved by CFD-ACE+ in an effort to update the design vari-ables for heat sink by minimizing a residual measuring the difference between estimated and desired temperatures located on Sbunder the present algorithm.

5. Computational procedure

The iterative computational procedure for the solution of this heat sink design problem using the Levenberg–Marquardt Method can be summarized as follows:

Step 1. Choose the original design variables for B at the zeroth iteration to start the computations.

Step 2. Solve the direct problem given by Eqs.(1)–(6)to obtain computed temperatures Tf,centeron Sh.

Step 3. Construct the Jacobian matrix in accordance with Eq.

(16).

Step 4. Update B from Eq.(17).

Step 5. Check the stopping criterion

### e

; if not satisﬁed go to Step 2 and iterate.calculated temperatures forFig. 12(a) are 316.25 K and 316.82 K, respectively, which implies only 0.181% error, the maximum error occurred at the 18th ﬁn and the error is calculated as 0.61%. For

Fig. 12(b), the averaged measured and calculated temperatures are obtained as 318.36 K and 318.85 K, respectively, which implies only 0.154% error. The maximum error occurred at the 24th ﬁn and the error is calculated as 0.43%.

For Type C ﬁn, the comparisons of measured and calculated temperatures at higher and lower measured positions are shown inFig. 14(a) and (b), respectively. The averaged measured and cal-culated temperatures forFig. 13(a) are obtained as 316.201 K and 316.892 K, respectively, which implies only 0.219% error, the max-imum error occurred at the 35th ﬁn and the error is calculated as 0.643%. ForFig. 13(b), the averaged measured and calculated tem-peratures are obtained as 318.077 K and 318.936 K, respectively, which implies only 0.27% error. The maximum error occurred at the 35th ﬁn and the error is calculated as 0.431%.

Based on the above results it can be concluded that the accuracy of the numerical solutions are guaranteed and this makes the opti-mal design of ﬁn array in this work valid in reality.

8. Conclusions

The Levenberg–Marquardt Method (LMM) combined with CFD-ACE+ code was successfully applied for the solution of the three-dimensional inverse design problem to estimate the optimal design variables for heat sink modules. Three types of heat sinks were con-sidered in the optimal design process and the objective of minimiz-ing the maximum temperature in the ﬁn array can always be achieved. Results based on the numerical experiments show that the optimum ﬁn height H and ﬁn thickness W tend to become higher thinner than the original ﬁn array, respectively, as a result both the ﬁn pitch D and heat sink base thickness U are both increased. The temperature distributions for optimum heat sink modules are mea-sured using thermal camera and compared with the numerical solu-tions. Results show that the measured temperatures match quite well with the calculated temperatures and this is a good reference to justify the validity of the present design algorithm.

Acknowledgment

This work was supported in part through the National Science Council, ROC, Grant number: NSC-99-2221-E-006-238-MY3. References

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[20] CFD-RC user’s manual, ESI-CFD Inc., 2005.

0 20 40 60 x (mm) 316 320 324 328 Temperature, T(x,0,12.5) (K)

Optimal Fin (Experimental) Optimal Fin (Numerical)

0 20 40 60 x (mm) 316 320 324 328 Temperature, T(x,0,1) (K)

Optimal Fin (Experimental) Optimal Fin (Numerical)

### (a)

### (b)

Fig. 14. The numerical and experimental temperatures for Type C optimal ﬁn at (a) z = 12.5 mm and (b) z = 1.0 mm.