Natural convection around a radial heat sink

International Journal of Heat and Mass Transfer 53 (2010) 2935–2938

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Technical Note

Natural convection around a radial heat sink Seung-Hwan Yu, Kwan-Soo Lee *, Se-Jin Yook School of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea

a r t i c l e

i n f o

Article history: Received 15 October 2009 Received in revised form 11 February 2010 Accepted 11 February 2010 Available online 19 March 2010 Keywords: Natural convection Heat sink Circular base Correlation

a b s t r a c t This paper presents the details of an experimental and numerical investigation of natural convection in a radial heat sink, composed of a horizontal circular base and rectangular fins. The general flow pattern is that of a chimney; i.e., cooler air entering from outside is heated as it passes between the fins, and then rises from the inner region of the heat sink. Parametric studies are performed to compare the effects of three geometric parameters (fin length, fin height, and number of fins) and a single operating parameter (heat flux) on the thermal resistance and the average heat transfer coefficient for the heat sink array. In addition, a correlation is proposed to predict the average Nusselt number for a radial heat sink. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Light-emitting diode (LED) lights have recently attracted the attention of the illumination industry, due to their lower power consumption, longer life, and smaller, more durable structure compared to other light sources. However, their use presents a thermal problem, since about 70% of their total energy consumption is emitted as heat. An efficient heat sink design is essential to solve this problem. Natural convection heat sinks are appropriate for LED lights, considering their overall advantages. However, natural convection heat sinks commonly have rectangular bases, whereas LED lights are generally circular. It is therefore desirable to investigate natural convection heat transfer via a heat sink with a circular base. Numerous experimental [1–4] and numerical [5] studies of rectangular fin or pin fin heat sinks have been carried out. Starner and McManus [1] experimentally investigated natural convection heat transfer from four heat sinks of differing dimensions, with the heat sinks oriented vertically, at a 45° angle, and horizontally. Welling and Woolbridge [2] conducted an experimental study of vertically oriented rectangular fins of constant length attached to a vertical base. They found that there exists an optimal fin height, corresponding to a maximum rate of natural convection heat transfer, for any given fin spacing. Harahap and Mcmanus [3] performed experiments to calculate the average heat transfer coefficients for two different fin lengths, and established a correlation with non-dimensional parameters and relevant fin dimensions. However, most of these studies were concerned with heat sinks

* Corresponding author. Tel.: +82 2 2220 0426; fax: +82 2 2295 9021. E-mail address: [email protected] (K.-S. Lee). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.02.032

with rectangular bases, which might be inefficient for cooling circular LED lights. In this study, natural convection from a heat sink with a circular base and rectangular fins is numerically and experimentally analyzed, and the thermo-flow pattern is observed. The effects of the number of fins, fin length, fin height, and heat flux on the thermal resistance and the average heat transfer coefficient are investigated. A correlation is proposed to predict the average heat transfer coefficient for this type of heat sink, as a function of heat sink dimensions and heat flux.

2. Mathematical modeling Fig. 1 shows a radial heat sink consisting of a circular base and rectangular fins. The fins were arranged radially at regular intervals. The heat sink base was oriented horizontally. The heat sink was made of aluminum, whose properties are listed in Table 1.

2.1. Governing equations For the numerical analysis, the following assumptions were imposed. (1) The flow was steady, laminar, and three-dimensional. (2) Aside from density, the properties of the fluid were independent of temperature. (3) Air density was calculated by treating air as an ideal gas. (4) Radiation heat transfer was negligible. The governing equations were as follows.

2936

S.-H. Yu et al. / International Journal of Heat and Mass Transfer 53 (2010) 2935–2938

Nomenclature b cp F h H k L Mw Nu n Pr p q_ Rc RTH Ra* r T

t u v w

spacing between fins, mm coefficient of heat capacity, J/(kg °C) view factor heat transfer coefficient, W/m2 K fin height, mm thermal conductivity, W/m °C fin length, mm gas molecular weight, kg/kmol Nusselt number, hL/k number of fins in the normal direction Prandtl number pressure, N/m2 heat flux, W/m2 universal gas constant thermal resistance, °C/W _ 3 q2 gbcp pðr2o r2i ÞqL modified Rayleigh number, lLk2 radius, mm temperature, K or °C

fin thickness, mm x-component of velocity, m/s y-component of velocity, m/s z-component of velocity, m/s

Greek symbols e emissivity l dynamic viscosity, N/m2 s h angle, ° q density, kg/m3 r Stefan–Boltzmann constant, 5.67  108 W/m2 K4 Subscripts avg average f fluid (air) i inner o outer s solid (heat sink)

2.1.1. Air side Continuity equation:

2.1.2. Solid side Energy equation:

@ðquÞ @ðqv Þ @ðqwÞ þ þ ¼0 @x @y @z

ð1Þ

Momentum equations:

@ðqu2 Þ @ðquv Þ @ðquwÞ @P þ þ ¼ þl @x @y @z @x @ðqv uÞ @ðqv 2 Þ @ðqv wÞ @P þ þ ¼ þl @x @y @z @y

! @2u @2u @2u þ þ @x2 @y2 @z2 ! @2v @2v @2v þ þ @x2 @y2 @z2

þ gðq  qa Þ @ðqwuÞ @ðqwv Þ @ðqw Þ @P þ þ ¼ þl @x @y @z @z 2

! @ w @ w @ w þ 2 þ 2 @x2 @y @z 2

2

ð6Þ

The density of air was calculated from the ideal gas law,

ð2Þ

ð3Þ

2

ð4Þ

q¼

Patm ðRc =M w ÞT

ð7Þ

where Mw of air is 28.966 kg/kmol. Periodic boundary conditions were adopted in accordance with the geometry of the heat sink (Fig. 1). Because of the number of grids and the computational time involved, only a single fin was considered, as shown in Fig. 2. 2.2. Numerical procedure and validation

Energy equation:

@ðquTÞ @ðqv TÞ @ðqwTÞ k @ 2 T @ 2 T @ 2 T þ þ þ þ ¼ @x @y @z cp @x2 @y2 @z2

@2T @2T @2T þ þ ¼0 @x2 @y2 @z2

! ð5Þ

The numerical simulation was conducted using Fluent V6.3, a commercially available CFD code based on the finite volume method. The grid dependence was investigated by varying the number of grid points from 22,680 to 285,714. We selected 65,016 grid points; additional grid points produced a change of less than 0.5% in the average heat sink temperature for the reference model of n = 20 and ro = 75 mm. The numerical results were validated with experimental data by comparing the differences between the ambient and heat sink temperatures. The geometric parameters of the experimental model were n = 20, ro = 75 mm, L = 55 mm, H = 21.3 mm, and t = 2 mm. Fig. 3 compares the temperature differences between the experimental and numerical results in terms of the heat flux applied to the heat sink base. This implies that the present numerical model can correctly predict the natural convection flow around a radial heat sink. Table 1 Air and heat sink properties. Material

Fig. 1. Radial heat sink with a circular base and rectangular fins.

Air Heat sink (aluminum)

cp (J/ kg °C) 1005.585 2800

l (N/m2 s)

k (W/m °C)

q (kg/ m3)

1.834  10 –

5

2.643  10 193

5

Eq. (7) 880

2937

S.-H. Yu et al. / International Journal of Heat and Mass Transfer 53 (2010) 2935–2938

40

Computational results Experimental results

Tavg− T∞

30

20

10

0

200

400

600

800

2

q(W/m ) Fig. 2. Computational domain and dimensions.

3. Results and discussion Parametric studies were carried out by numerically investigating the effects of the number of fins, fin length, fin height, and heat flux on the thermal resistance and the heat transfer coefficient. Based on these results, a correlation was proposed to predict the Nusselt number for a heat sink with a horizontal circular base and rectangular fins. 3.1. Thermo-flow characteristics There are two flows, i.e., vertical and horizontal flows, around the radial heat sink. The vertical flow is in the upward direction, since air is heated by the heat sink (which is maintained at a higher temperature) and becomes lighter than the surrounding air. The horizontal flow is created by air entering from outside the heat sink to make up for the vertical flow in the inner region. Therefore, the overall flow pattern is chimney-like. The temperature of heat sink maintains almost uniformly high because of high conductivity of aluminum. The heat transfer rate in the outer region of the heat sink was higher than in the inner region. This was because the temperature difference between the air and the heat sink decreased as the cool air proceeded towards the inner region of the heat sink. 3.2. Parametric study The effects of the number of fins, fin length, fin height, and heat flux on the thermal resistance and the heat transfer coefficient were investigated. The reference model is n = 20, ro = 75 mm, L = 55 mm, H = 21.3 mm, t = 2 mm, and q_ ¼ 700 W=m2 . The effect of the number of fins on the thermal resistance and heat transfer coefficient is shown in Fig. 4(a). The average heat transfer coefficient decreased as the number of fins increased, since the flow rate of the cooler air entering the spaces between the fins decreased and the air was heated more quickly on account of the reduced space between fins. However, when the number of fins was less than 36, the thermal resistance of the heat sink decreased with increasing n, since the effect of the increased heat transfer surface area was larger than the effect of the decreased heat transfer coefficient. When the number of fins was greater than 36, the thermal resistance of the heat sink increased with increasing n,

Fig. 3. Comparison of the temperature differences between the experimental and numerical results.

since the heat transfer coefficient was very small. Consequently, there exists optimum number of fins that gives the minimum thermal resistance. Fig. 4(b) shows the effect of the fin length. As the fin length increased, the thermal resistance and average heat transfer coefficient decreased. The thermal resistance leveled off and reached a steady value when the fin was longer than 55 mm. This was because the air temperature in the inner region was almost the same as the heat sink temperature, and hence any additional fin length beyond 55 mm did not contribute to the heat transfer rate. Fig. 4(c) indicates the effect of the fin height. A lower thermal resistance resulted from the increased heat transfer surface area created by the incremented fin height. However, the change in the heat transfer coefficient was relatively small, since the velocity of the air entering from outside increased very little with increasing fin height. Fig. 4(d) illustrates the effect of the heat flux applied to the heat sink base. The decrease in thermal resistance due to increasing heat flux resulted in a greater rising air velocity, which in turn increased the flow rate of the cooler air entering from outside. Accordingly, the average heat transfer coefficient increased almost linearly, thanks to the enhanced effect of natural convection. 3.3. Correlation A correlation for predicting the Nusselt number for a heat sink with a horizontal circular base and rectangular fins was derived as a function of the parameters investigated in the previous section, as well as other geometric parameters, and was obtained from numerical data. This formula is based on the correlations for rectangular heat sinks obtained in previous studies [1,3], using average fin spacing and the modified channel Rayleigh number,

Nu ¼ 0:195ðRa Þ0:263 where Ra ¼

 1:35  0:444  0:142  1:4 nbavg ro ro ro H L bavg H

_ 3 q2 gbcp pðr 2o r 2i ÞqL ; bavg lLk2

ð8Þ

= {(2pro/n  t) + (2p(ro – L)/n  t)}/2 ,

and the properties are based on the film temperature. The predicted correlation was consistent with the numerical data, with an error of less than 10%, when t = 2 mm, 20 6 n 6 36,

S.-H. Yu et al. / International Journal of Heat and Mass Transfer 53 (2010) 2935–2938

3.00

7

RTH

2.75

havg

RTH havg

6

4

2

2.75

o

RTH( C/W)

2

havg (W/m K)

2.50

o

RTH( C/W)

6

h avg(W/m K)

2938

5

2.25 32

40

2

2.50

40

n (a) The effect of the number of fins 3.0

(b) The effect of the fin length 6

7

havg

40

4

6

2.5

5

2

3.0

o

2

2.0

RTH( C/W)

havg h avg(W/m K)

RTH

o

RTH( C/W)

3.5

RTH

5

30

4

60

L (mm)

2.5

1.5 20

50

h avg(W/m K)

24

2.0 200

400

600

800

1000

4 1200

2

H (mm)

(c) The effect of the fin height

q (W/m ) (d) The effect of the heat flux

Fig. 4. The results of parametric study. (a) The effect of the number of fins, (b) the effect of the fin length, (c) the effect of the fin height, (d) the effect of the heat flux.

21.3 mm 6 H 6 63.9 mm, 75 mm 6 ro 6 102 mm, 80 mm, and 300 W=m2  q_  1100 W=m2 .

40 mm 6 L 6

creased in proportion to the heat flux applied to the heat sink base. A correlation was proposed to predict the average Nusselt number for a radial heat sink.

4. Conclusions Natural convection from a radial heat sink was experimentally and numerically investigated. The general flow pattern was like that of a chimney; i.e., the cooling air entering from outside was heated as it passed between the fins, and then rose from the inner region of heat sink. Parametric studies were performed to compare the effects of the number of fins, fin length, fin height, and heat flux on the thermal resistance and the heat transfer coefficient. As the number of fins, fin length, and fin height increased, the thermal resistance and heat transfer coefficient generally decreased. However, there existed optimal values of the number of fins and fin length to obtain an effective low heat sink temperature. The thermal resistance decreased and the heat transfer coefficient in-

References [1] K.E. Starner, H.N. McManus, An experimental investigation of free convection heat transfer from rectangular fin arrays, J. Heat Transfer 85 (2) (1963) 273– 278. [2] J.R. Welling, C.B. Wooldridge, Free convection heat transfer coefficients from rectangular vertical fins, Trans. ASME J. Heat Transfer 87 (3) (1965) 439–444. [3] F. Harahap, H.N. McManus, Natural convection heat transfer from horizontal rectangular fin arrays, J. Heat Transfer 89 (1) (1967) 32–38. [4] R.T. Huang, W.J. Sheu, C.C. Wang, Orientation effect on natural convective performance of square pin fin heat sinks, Int. J. Heat Mass Transfer 51 (9–10) (2008) 2368–2376. [5] S. Baskaya, M. Sivrioglu, M. Ozek, Parametric study of natural convection heat transfer from horizontal rectangular fin arrays, Int. J. Therm. Sci. 39 (8) (2000) 797–805.