Evaluation of Spreading Thermal Resistance for Heat Generating Multi-electronic Components

EVALUATION OF SPREADING THERMAL RESISTANCE FOR HEAT GENERATING MULTI-ELECTRONIC COMPONENTS

Yun Ho Kim a, Seo Young Kim b, and Gwang Hoon Rhee a

a Department of Mechanical & Information Engineering, University of Seoul, Seoul 130-743, South Korea

b Thermal/Flow Control Research Center, Korea Institute of Science and Technology, Seoul 130-650, South Korea

Phone: +82-2-958-5673, Fax: +82-2-958-958-5689 Email: [email protected]

ABSTRACT

The Lee et al.'s equation has been widely used to predict spreading thermal resistance in a single, centered heat source and axi-symmetric condition. However, the eccentric and multiple heat sources are mounted on the base plate in many electronic applications. Thus, it is necessary to determine the spreading thermal resistance for multiple heat sources. For this purpose, we establish the correlation to predict spreading thermal resistance in this study. The correlation which transforms four heat sources to a single equivalent heat source is proposed and then the spreading thermal resistance can be obtained with the Lee et al.’s equation. When the four heat sources are mounted on a square base plate, the correlation is expressed as a function of the heat source size, the length of base plate, and the distance between heat sources. Compared to the results of three-dimensional numerical analysis, the spreading thermal resistance by the proposed correlation is in good agreement within 10 percent accuracy.

KEYWORDS: Spreading thermal resistance, Correlation, Geometric equivalence, Multi-heat sources, Numerical analysis

NOMENCLATURE A heat source size, m2 Ac contact area of non-circular plate, m2

Aeq equivalent size of a single heat source, 2

eqm , m2

As base pate area of non-circular plate, m2

a heat source radius, m b base plate radius, m d distance between heat sources, m h convection heat transfer coefficient, W/m2·K k thermal conductivity, W/m·K l length of base plate, m m length of heat source, m meq length of a single equivalent heat source, m Q heat flow rate, W R1D one dimensional thermal resistance, °C/W Rs spreading thermal resistance, °C/W Rt total thermal resistance, °C/W T temperature, °C t base plate thickness, m

Subscripts f external air ave average value b bottom surface of base plate c contact area max maximum value s top surface of base plate

INTRODUCTION

The estimation of spreading thermal resistance is very important to determine the cooling efficiency and to provide the cooling guideline in modern electronics. Kennedy [1] first performed the study on the spreading thermal resistance analytically. He built up the analytical foundation about the spreading thermal resistance for axi-symmetric problems with a uniform heat-flux source on a finite cylinder. The general solution for the spreading thermal resistance on a cylindrical and a single centered source with a film coefficient at the lower surface was presented by Yovanovich [2] and Negus et al. [3]. However, Yovanovich’s analytical solution was difficult to use because of the expression based on Fourier series expansions. To overcome this problem, Lee [4] and Song et al. [5] presented the approximate equation of the spreading thermal resistance based on the Yovanovich’s analytical solutions.

The spreading thermal resistance has proven to be sensitive to geometric configurations. Many studies have been carried out to determine the spreading thermal resistance for various geometrics [6-8]. Muzychka et al. [9, 10] presented the analytical solutions to determine the spreading thermal resistance for compound annular sectors and orthotropic systems. The edge boundary condition was adiabatic for the general analytical solutions as yet, but Yovanovich [11] presented the analytical solutions adopting the cooling boundary conditions at edges. Muzychka et al. [12, 13] presented a simple geometric transformation for predicting the spreading thermal resistance in isotropic and compound rectangular flux plates using the solution for an isotropic or compound circular flux plates.

To date, the study on the spreading thermal resistance was mainly limited to a single heat source. However, the eccentric and multiple heat sources are generally mounted on the base plate in many electronics. Recently, Muzychka et al. [14] obtained an analytical solution to predict the temperature

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distribution on the top surface of the base plate having multiple, arbitrarily located heat sources. However, it is difficult to determine the spreading thermal resistance, since no unique value of the spreading thermal resistance is definable, when more than one heat source is present.

In this paper, therefore, we present a correlation that transforms multiple heat sources to a single equivalent heat source maintaining the same thermal resistance. This method will provide a convenient tool for estimating spreading thermal resistance in the electronic applications with multiple heat sources.

THERMAL SPREADING RESISTNACE

When a heat source is mounted on a base plate or a heat sink as shown in Fig. 1, the total thermal resistance is expressed by

,, , , , b ave fc ave s ave s ave b avet

T TT T T TR

Q Q Q

−− −= + + (1)

where Q is the heat input. Tc,ave, Ts,ave and Tb,ave are the average temperatures at heat source area, top surface and bottom surface, respectively, and Tf is the ambient temperature. In the equation (1), the total thermal resistance consists of the spreading thermal resistance, the conduction thermal resistance and the convection thermal resistance. Here, the conduction and convection thermal resistances are given by

1

1D

b b

Rt

kA hA= + (2)

The spreading thermal resistance is defined by the

average temperature difference between the heat source and the top surface of the base plate [15]. In many applications, the spreading thermal resistance can be more crucial than the other thermal resistances as the thickness of base plate becomes thinner and the heat source area becomes smaller for the base plate area [5].

Fig. 1 Thermal resistances in electronics

VALIDATION OF NUMERICAL ANALYSIS The Lee et al’s approximate equation used to determine

the spreading thermal resistance for a single heat source can be obtained from the analytical solution for a cylindrical configuration as shown in Fig. 2. The heat source is located on the center with a uniform heat flux. The top and edge surfaces of the base plate are adiabatic. Natural convection boundary condition is applied at the bottom. The dimensionless approximate equation obtained from the analytical solution is given by

( )3 / 211

2 nεΨ = − Φ (3)

where

( )

( )

tanh 1,

1 tanh

Bi

Bi

n

n n

n

n

n

λ τλ π

πελ τ

λ

λ

+Φ = = +

+ (4)

where , / , / , Bi /s sk A R a b t b hb kε τΨ ≡ = = = , and nλ is the eigenvalue [16].

In this paper, the numerical analysis was performed and compare with the Lee et al’s approximate equation for a rectangular plate with a single heat source as shown in Fig. 3. In Fig. 3, l=220 mm, t=6 mm, k=50 W/m·K, Q=5 W, h=10 W/m2·K, and Tf=25°C. Since the heat source and the base plate are non-circular, the equivalent heat source and the base plate radius are obtained as follows [5]:

, c sa bA Aπ π

= = (5)

where Ac is the contact area and As is the surface area of the base plate for a square plate.

Fig. 2 Thermal modeling of the Lee et al.'s solution

259


Fig. 3 Thermal modeling for the numerical analysis

The spreading thermal resistances obtained by the

numerical analysis are presented in Fig. 4. It is found that the results of the present numerical analysis are in good agreement with the Lee et al.’s approximate equation according to the heat source size. However, when the heat source size is small for the base plate size, the present numerical analysis produces approximately 12% error at m=20 mm.

The Lee et al.’s approximate equation has limitation to determine the spreading thermal resistance for a single and centered heat source. Recently, Muzychka et al. [14] presented the analytical solution for temperature field in multiple heat sources as shown in Fig. 5. They suggested the analytical solution to predict temperature distribution on the top surface of the base plate using the superposition method of eccentric heat source solution. The analytical solution is given by [14]

1

N

j ifi

T T θ=

− = ∑ (6)

where

( )

( )

( ) ( )

,

01

,

1 1 1

, ,

1cos sin

22

1cos sin

22 4

1 1cos sin cos sin

2 2

m c j m ji i

i mm m j

n c j n ji i

n mnn m nn j

n c j n j m c j m j

m j n j

X cA A

c

Y dA A

d

Y d X c

c d

λ λθ

λ

δ δ

δ

δ δ λ λ

λ δ

∞

=

∞ ∞ ∞

= = =

= +

+ +

×

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑

∑ ∑∑ (7)

Fig. 4 Comparison of thermal resistances between the numerical analysis and the Lee et al.'s approximate equation

Fig. 5 Thermal modeling of Muzychka et al.'s solution with multiple heat sources

Equations (6) and (7) represent the sum of the effects of

all sources over the arbitrary locations [14]. Here, fT Tθ = − is the mean temperature excess, N is the number of the heat source, cj, dj are the size of the each heat source, Xc,j, Yc,j are the distance to each heat source, and λm=mπ/a, δn=nπ/b are the eigenvalues. ,, ,i i i

m n m nA A A are the Fourier coefficients,

respectively. 0

iA is the uniform flow solution for each heat

20 40 60 80 100 1200.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

Rs : Lee et al.'s approximate equation Rs : Present numerical analysis

- Definition of Mikic & Rohsenow [16]

m [mm]

Rs [°

C/W

]

260


source and it is given by [14]

0

1Q tA

ab k h= +⎛ ⎞

⎜ ⎟⎝ ⎠

(8)

Temperature distributions on the top surface of the base

plate for the Muzychka et al.’s analytical solution and the numerical analysis are displayed in Figs. 6 and 7. Here a=b=300 mm, c=d=25 mm, t=10 mm, X1=Y1=90, mm, X2=Y2=210 mm, Q1=10 W, Q2=15 W, and k=10 W/m·K. The present numerical analysis shows excellent accordance with the previous analytical solutions. Also, from Table 1, the maximum temperature at each heat source agrees within 0.5%.

Table 1 Maximum temperature of each heat source

T1max T2max

Present numerical analysis 84.80°C 108.00°C

Muzychka et al.'s solution 84.97°C 108.43°C

RESULTS & DISCUSSIONS In the present study, we transform the multiple heat

sources to a single equivalent heat source to determine the spreading thermal resistance by using the Lee et al.’s correlation. When four heat sources with the same area are mounted on the square base plate as shown Fig. 8 (a), the four heat sources can be replaced by adopting a single equivalent heat source which has the same spreading thermal resistance as shown Fig. 8 (b). The single equivalent heat source can be presented in terms of three geometric parameters

( ), ,eqA f l m d= (9)

where A is the total area for four heat sources, Aeq is the area of a single equivalent heat source. l and m are the lengths of base plate and heat source, respectively, and d is the central distance between heat sources. The thermal modeling for the numerical analysis is shown in Fig. 8. Here, the length of base plate is 200 mm, thickness is

Fig. 6 Contour plot of the temperature distribution on the top surface of the base plate by the Muzychka et al.'s solution

Fig. 7 Contour plot of the temperature distribution on the top surface of the base plate by the present numerical analysis

261


(a) Four heat sources

(b) Single equivalent heat source

Fig. 8 Thermal modeling for a single equivalent heat source 6 mm, and k=50 W/m·K. The uniform heat flux condition is set at Q=5 W. The convection boundary condition is applied at the bottom surface of the base plate with h=10 W/m2·K and Tf=25°C. The adiabatic conditions are applied at the top and the edge surfaces of the base plate.

The calculated equivalent heat source area is shown in Fig. 9 and Fig. 10. Fig. 9 shows the dimensionless single equivalent heat source area according to the heat source size and the distance between heat sources. As the heat source

0.10 0.15 0.20 0.25 0.300

1

2

3

4

5

6

7

m/l

Aeq

/A

Aeq /A : d/l = 0.3 Aeq /A : d/l = 0.4 Rs : d/l = 0.3 Rs : d/l = 0.4

0.0

0.1

0.2

0.3

0.4

R

s [ C

/W]

Fig. 9 The dimensionless size of a single equivalent heat source for various m/l

0.0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6

7

d/l

Aeq

/A Aeq /A : m/l = 0.1 Aeq /A : m/l = 0.2 Rs : m/l = 0.1 Rs : m/l = 0.2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

R

s [ C

/W]

Fig. 10 The dimensionless size of a single equivalent heat source for various d/l

size increases, the dimensionless equivalent area of a single heat source decreases. In general, as the heat source size increases, for a fixed distance between heat sources, the spreading thermal resistance decreases. The equivalent size of a single heat source increases simultaneously. However, the increasing rate of total four heat source size is greater than that of the single equivalent heat source size. Consequently, the dimensionless size of a single equivalent heat source decreases.

In Fig. 10, the dimensionless size of a single equivalent heat source increases as the distance d/l increases for a fixed

º

º

262


0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Present results Correlation

-10%

Aeq

/A

0.805(m/l)-1.323277(d/l)1.18964

+10%

Fig. 11 Correlation of equivalent size of a single heat sources with m/l and d/l heat source size. The spreading thermal resistance decreases as d/l increases. Thus the dimensionless size of a single equivalent heat source increases as well. The increasing rate reduced as the distance d/l increases. Simultaneously, the decreasing rate of the spreading thermal resistance is mitigated as d/l increases.

In an attempt to provide a correlation for estimating the

size of a single equivalent heat source, the multiple regression method is utilized as shown Fig. 11. The correlation for the dimensionless size is then expressed by

1.32327 1.18964

0.805eq m d

l l

AA

−

= ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(10)

From the equation (10), it is possible to predict the size

of a single equivalent heat source for four heat sources within ±10% accuracy. Assumption is square heat sources and a square base plate, the same heat flux for each heat source, and symmetry conditions. Equation (10) should be used in the following range.

0.5 0.4d m L≤ + (11)

Now, we perform the verification of the proposed correlation. The thermal conductivity of base plate is 50 W/m·K, and the convective heat transfer coefficient is 10 W/m2·K. The heat is 3 W for each heat source, and the external air temperature is 25°C. From Table 2, the spreading thermal resistance of a single equivalent heat source size calculated by equation (10) agrees within 10% compared with spreading thermal resistance for four heat sources and the Lee et al.’s approximate equation. In Table 2, (L) and (F) indicate the results by Lee et al.’s approximate equation and the numerical analysis for four heat sources, respectively. It is also noted that the deviation of estimated spreading thermal resistance is approximately 20% at m=30 mm and d=120 mm because this condition is out of range as depicted in the equation (11).

Table 2 Verification of correlation

[mm] [mm] meq [mm] Tmax [°C] Tc,ave [°C] Ts,ave [°C] Rs [°C/W] error Four heat sources - 48.77 48.386 45.832 0.213

Equivalent heat source 102 49.85 48.568 45.836 0.228 m=30, d=70 Lee et al.'s approximate equation - - - - 0.212

+6.68%(L) +6.52%(F)

Four heat sources - 48.51 48.013 45.757 0.188 Equivalent heat source 108 49.52 48.279 45.833 0.204 m=34,

d=70 Lee et al.'s approximate equation - - - - 0.195

+4.38%(L) +7.77%(F)



+4.71%(L) +8.19%(F)



+4.40%(L) +8.62%(F)

Four heat sources - 47.72 47.329 45.825 0.125 Equivalent heat source 141 48.07 47.093 45.539 0.105

l=240

m=30, d=120 Lee et al.'s approximate equation - - - - 0.100

+4.21%(L) -19.94%(F)

Four heat sources - 52.27 51.818 49.735 0.174 Equivalent heat source 102 53.28 52.014 49.733 0.190 l=220

Lee et al.'s approximate equation - - - - 0.181

+4.86%(L) +8.68%(F)

Four heat sources - 63.73 63.324 61.980 0.112 Equivalent heat source 100 64.67 63.523 62.054 0.122

m=30, d=70

l=180 Lee et al.'s approximate equation - - - - 0.121

+1.32%(L) +8.51%(F)

263


CONCLUSION

The numerical analysis was carried out to predict the spreading thermal resistance in multiple heat sources. From the results of numerical analysis, a correlation which transformed four heat sources to a single equivalent heat source was established. The correlation was expressed as a function of the heat source size, the length of base plate, and the distance between heat sources. As the multiple heat source size increased, the dimensionless size of a single equivalent heat source decreased. In addition, the dimensionless size of a single equivalent heat source increased as the distance d/l increased for a fixed heat source size. The spreading thermal resistance calculated by the present correlation was in agreement within 10% accuracy compared with the direct numerical results with four heat sources.

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parameter for annular contacts on circular flux tubes”, AIAA Journal, Vol. 14, No. 6, pp. 822-824, 1976. [7] M. M. Yovanovich, Y. S. Muzychka, and J. R. Culham, “Spreading resistance of isoflux rectangles and strips on compound flux channels”, Journal of Thermophysics and Heat Transfer, Vol. 13, No. 4, pp. 495-500, 1999. [8] M. M. Yovanovich, J. R. Culham, and P. M. Teertstra, “Analytical modeling of spreading resistance in flux tubes, half spaces, and compound disks”, IEEE Transactions on Components, Packaging, and Manufacturing Technology - Part A, Vol. 21, pp. 168-176, 1998. [9] Y. S. Muzychka, M. Stevanovic, and M. M. Yovanovich, “Spreading thermal resistances in compound annular sectors”, Journal of Thermophysics and Heat Transfer, Vol. 15, No. 3, pp. 354-359, 2001. [10] Y. S. Muzychka, M. M. Yovanovich, and J. R. Culham, “Spreading thermal resistance in compound and orthotropic systems”, Journal of Thermophysics and Heat Transfer, Vol. 18, No. 1, pp. 45-51, 2004. [11] M. M. Yovanovich, “Thermal resistance of circular source on finite circular cylinder with side and end cooling”, Journal of Electronic Packaging, Vol. 125, In Press, 2003. [12] Y. S. Muzychka, M. M. Yovanovich, and J. R. Culham, “Spreading thermal resistances in rectangular flux channels part 1 - geometric equivalences”, 36th AIAA Thermophysics Conference, June 23-26 Orlando, Florida, 2003. [13] Y. S. Muzychka, J. R. Culham, and M. M. Yovanovich, “Spreading thermal resistances in rectangular flux channels part 2 - edge cooling”, 36th AIAA Thermophysics Conference, June 23-26 Orlando, Florida, 2003. [14] Y. S. Muzychka, J. R. Culham, and M. M. Yovanovich, “Spreading thermal resistances of eccentric heat sources on rectangular flux channels”, Journal of Electronic Packaging, Vol. 125, pp. 178-185. 2003. [15] B. B. Mikic, and W. M. Rohsenow, “Thermal contact resistance”, Heat Transfer Lab., Rept. 4542-41, Massachusetts Inst. of Technology, Cambridge, MA Sept, 1966. [16] K. J. Negus, M. M. Yovanovich, and J. V. Beck, “On the nondimensionalization of constriction resistance for semi-infinite heat flux tubes”, Journal of Heat Transfer, Vol. 111, pp. 804-807. 1989.

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Evaluation of Spreading Thermal Resistance for Heat Generating Multi-electronic Components