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Thermal optimization of branched-fin heat sinks subject to a parallel flow
International Journal of Heat and Mass Transfer 77 (2014) 278–287
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Technical Note
Thermal optimization of branched-fin heat sinks subject to a parallel flow Dong-Kwon Kim ⇑ Department of Mechanical Engineering, Ajou University, Suwon 443-749, Republic of Korea
a r t i c l e
i n f o
Article history: Received 24 February 2014 Received in revised form 9 May 2014 Accepted 11 May 2014
Keywords: Heat sink Thermal optimization Branched-fin Forced convection Parallel flow
a b s t r a c t In the present study, thermal optimization of a heat sink with fins branched in the direction normal to fluid flow was conducted using a model based on volume averaging theory (VAT). For a water-cooled system, it was shown that, compared to a rectangular-fin heat sink, the thermal resistance of the optimized branched-fin heat sink decreased by up to 30%. The extent of the reduction increased with an increase in the pumping power and a decrease in the length of the heat sink. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the electronic equipment industry, the trend toward denser and more powerful products has led to the use of electronic devices with higher power densities [1,2]. However, high power densities result in increased junction temperatures, which adversely affect the reliability of electronic devices. Therefore, an effective cooling mechanism is essential for reliably operating electronic components [3,4]. Many ideas pertaining to cooling methods have been proposed. Among the various types of cooling systems that have been developed, the plate-fin heat sink is the most widely used because of its simple design and ease of fabrication. As summarized in the review by Liu and Garimella [5], several studies have focused on optimizing the size of plate-fin heat sinks. Many optimization methods have been proposed based on the fin model [6,7], three-dimensional numerical models [8,9], and volume averaging theory (VAT) [10,11]. The layout of the heat sink used in these studies is shown in Fig. 1(a). The heat sink was optimized by determining the fin thickness and channel width that minimize the thermal resistance for a heat sink of a given size. This procedure was based on the assumption that fin thickness and channel width are constant along the directions parallel and normal to fluid flow. However, there is no guarantee that a fin with a rectangular cross-section is the most thermally efficient. For example, Morega and Bejan demonstrated that the thermal
⇑ Tel.: +82 31 219 3660; fax: +82 31 219 1611. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.05.010 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
resistance of an air-cooled heat sink can be reduced by approximately 15% by increasing the fin thickness in the flow direction [12]. Kim et al. also showed that, for a water-cooled system, the thermal resistance of a plate-fin heat sink can be reduced by as much as 15% by increasing the fin thickness in the direction normal to fluid flow [13]. Heat transfer through branched fins is a topic of considerable interest [14,15]. Bejan and Almogbel optimized a T-shaped fin assembly to maximize global thermal conductance, which was subject to both total volume and fin material constraints [16]. Lorenzini and Rocha minimized global thermal resistance to optimize a Y-shaped assembly of fins [17]. Lorenzini and Rocha also performed thermal optimization of a T–Y assembly of fins to maximize heat removal [18]. These theoretical studies supported the conclusion that branched fins can reduce total thermal resistance. Contrary to these results, Yu and Li investigated the effective thermal conductivity of composites with embedded self-similar H-shaped fractal-like tree networks, and in their study, they found that fractal-like tree networks can significantly increase the thermal resistance of composites [19]. They showed that this heightened thermal resistance resulted from the use of longer branches. As indicated by these contradictory results, it remains unclear whether branched fins can further reduce the thermal resistance of a plate-fin heat sink. To date, despite these lingering uncertainties, thermal optimization of a heat sink with fins branched in the direction normal to fluid flow under forced convection has not been thoroughly investigated. This paper seeks to determine which type of heat sink (among the three heat sinks shown in Fig. 1) performs better under forced
D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
279
Nomenclature a c Dh,fr h H k K L p Dp P Ppump q00 Q_ R T Tf u U wc ww
wetted area per volume [m1] heat capacity of a fluid [J kg1 K1] hydraulic diameter of the frontal area [m] interstitial heat transfer coefficient based on the onedimensional bulk mean temperature [W m2 K1] height of the heat sink [m] thermal conductivity [W m1 K1] permeability [m2] length of the heat sink [m] pressure [Pa] pressure drop in the heat sink [Pa] fin pitch [m] pumping power [W] heat flux [W/m2] flow rate [m3 s1] thermal resistance [W/K] temperature [K] fluid bulk mean temperature [K] velocity [m s1] unit step function channel width [m] fin thickness [m]
x,y,z Y hi
Cartesian coordinate system [m] dimensionless coordinate (y/H) averaged value
Greek symbols e porosity l viscosity q density Subscripts and superscripts 1 first channel 2 second channel b branched-fin heat sink cap capacitive conv convective e effective f fluid i node number opt optimized r rectangular-fin heat sink s solid
convection. Thermal optimization of branched-fin heat sinks was conducted using a model based on VAT. The results demonstrated that, using Y-shaped fins, the thermal resistance of a plate-fin heat sink can be reduced [i.e., the heat sink shown in Fig. 1(b) is the best choice]. This paper shows that, when using Y-shaped fins, the reduction in thermal resistance increases as the pumping power increases and the length of the heat sink decreases.
and energy equations in the direction of the x-axis over the averaging analysis domain shown in Fig. 2(c).
2. Mathematical formulation
! @ @hTif 1 @hTif 1 s f1 ; kfe1 e1 qf cf hui ¼ h1 a hTi hTi þ @y @z @y ! @ @hTif 2 @hTif 2 kfe2 e2 qf cf huif 2 ¼ h2 a hTis hTif 2 þ ; @y @z @y
! dp lf @ 2 huif 1 f1 ; ¼ e1 hui l dz K 1 @y2 ! dp lf @ 2 huif 2 f2 ; ¼ e2 hui l dz K 2 @y2 f1
In this paper, the topic under consideration relates to forced convection through channels, as shown in Fig. 2(a) and (b). The cover plate is insulated, and the heat sink base is uniformly heated. A coolant passes through several channels and carries the heat away from the heat sink base. For simplicity, the flow is assumed to be laminar and is fully developed both hydrodynamically and thermally in this analysis. Moreover, the pumping power required to drive the fluid through the heat sink is assumed to be constant. It is also assumed that the aspect ratio of the channel is much greater than one and that the conductivity of the solid is higher than the conductivity of the fluid. The channel width and fin thickness vary in the direction normal to the fluid flow [wc1 = wc1(y), wc2 = wc2(y), ww = ww(y)]; fin pitch P, which is equal to the sum of the channel widths and fin thickness, is constant. The momentum and energy equations are given in Eqs. (1)–(3). For the fluid phase,
lf
! @2u @2u dp þ ; ¼ @x2 @y2 dz
qf cf
! @ðuTÞ @2T @2T : ¼ kf þ @z @x2 @y2
ð1Þ
ð2Þ
For the solid phase,
0 ¼ ks
! @2T @2T : þ @x2 @y2
ð3Þ
In the VAT-based model, the governing equations for the average velocity and temperature are obtained by averaging the momentum
@ @hTis ¼ h1 a hTis hTif 1 þ h2 a hTis hTif 2 : kse @y @y
ð4Þ
ð5Þ
ð6Þ
ð7Þ
ð8Þ
Here, average fluid velocities huif1 and huif2, average fluid temperatures hTif1 and hTif2, and average solid temperature hTis are defined as follows:
Z wc1 =2 Z P=2 2 2 u dx; huif 2 ¼ u dx; wc1 0 wc2 ðwc1 þww Þ=2 R P=2 R wc1 =2 Tudx Tudx ðwc1 þww Þ=2 ; hTif 2 ¼ R P=2 ; hTif 1 ¼ R0 w =2 c1 udx udx 0 ðwc1 þww Þ=2 Z ðwc1 þww Þ=2 2 T dx: hTis ¼ ww wc1 =2
huif 1 ¼
ð9Þ
ð10Þ
For the average fluid temperatures in Eq. (10), the one-dimensional bulk means are used to easily calculate the bulk mean temperature of the fluid, Tf. There are two equations for the average fluid velocities [Eqs. (4) and (5)] and two equations for the average fluid temperatures [Eqs. (6) and (7)] because there are two different fluid channels in the analysis domain. The quantities e, a, kfe, kse, K, and h denote the porosity, wetted area per unit volume, effective thermal conductivity of the fluid, effective thermal conductivity of the solid, permeability, and interstitial heat transfer coefficient, respectively. These quantities can be represented as follows:
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wc1 wc2 ; e2 ¼ ; P P kse ¼ ks ð1 e1 e2 Þ;
e1 ¼
K1
e1 wc1 hui
a¼
@u @x
f1
2
2 ; kfe1 ¼ kf e1 ; P
; x¼wc1 =2
!1
h1
@x
q00sf 1 s
f1
hTi hTi
ð12Þ
;
x¼ðwc1 þww Þ=2
h2
;
ð11Þ
!1
e2 wc2 huif 2 @u K2 2
kfe2 ¼ kf e2 ;
q00sf 2 s
hTi hTif 2
ð13Þ
:
Here, e, a, kfe, kse, K, and h are functions of y because they are functions of wc and ww. As noted in Slattery [20], some information is lost in any averaging approach; in the present case, we lose the dependence of both the velocity and the temperature distributions for the fluid phases in the averaging direction. In the averaging approach, we typically replace the lost information with empirical data for K and h. For the present configuration, however, these parameters can be determined analytically through an approximation [21,22]. For this approximation, it is assumed that, along the direction of the z-axis, the fluid’s temperature and velocity profiles are similar to the profiles of the Poiseuille flow between two infinite parallel plates. Based on these assumptions, the velocity and temperature distributions are then determined to be
8 2 f1 > if z 6 w2c1 ; < 32 hui 1 4 wz2 c1 u¼ 2 > w : 32 huif 2 1 4 ðzP=2Þ ; if z P wc1 þw 2 w2
(a) Rectangular-fin heat sink
ð14Þ
c2
8 2 4 > < 175 hTif 1 hTis 1 24z þ 16z if z 6 w2c1 ; þ hTis 136 5w2c 5w4c T¼ : 2 4 > w : 175 þ hTis if z P wc1 þw hTif 2 hTis 1 24ðzP=2Þ þ 16ðzP=2Þ 136 2 5w2 5w4 c
c
ð15Þ
(b) Y-shaped fin heat sink
Using Eqs. (14) and (15) and the definitions in Eqs. (12) and (13), the permeability and interstitial heat transfer coefficient are
K1 ¼
e31 P2 12
K2 ¼
;
e32 P2 12
h1 ¼
;
70kf ; 17e1 P
h2 ¼
70kf : 17e2 P
ð16Þ
Eqs. (4)–(7) can be simplified when the solid conductivity is greater than the fluid conductivity (ks > kf) and when the aspect ratio of the channel is much greater than one (H > wc). Estimating the order of magnitude of each term that appears on the right-hand sides of Eqs. (4)–(7) yields the following:
lf w2c1
huif 1 > >
lf 2
H kf
H2
lf
huif 1 ;
w2c2 ks
hTif 1 ;
H2
huif 2 >
hTis >
lf H
kf H2
2
huif 2 ;
hTif 1 ;
ks H2
hTis ð17Þ
respectively. The following equation is also satisfied for constant surface heat flux: f1
f2
@hTi @hTi dT ¼ ¼ : @z @z dz
dp lf ¼ e1 huif 1 ; dz K 1
dp lf ¼ e2 huif 2 ; dz K 2 f dT e1 qf cf huif 1 ¼ h1 a hTis hTif 1 ; dz f f 2 dT e2 qf cf hui ¼ h2 a hTis hTif 2 : dz
Fig. 1. Plate-fin heat sinks.
ð18Þ
Here, Tf is the fluid bulk mean temperature. Using the relations shown in Eqs. (17) and (18), Eqs. (4)–(7) can be simplified as follows:
(c) Inverted Y-shaped fin heat sink
f
ð19Þ
Similar to the boundary conditions presented in another study [22], the appropriate boundary conditions are given as follows:
huif 1 ¼ huif 2 ¼ 0;
hTis ¼ hTif 1 ¼ hTif 2 ¼ T w s
huif 1 ¼ huif 2 ¼ 0; ð20Þ
f1
at y ¼ 0;
ð21Þ
f2
@hTi @hTi @hTi ¼ ¼ ¼ 0 at y ¼ H: @y @y @y
ð22Þ
The velocity and temperature distributions can be obtained by solving Eqs. (8), (19) and (20):
D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
(a) Schematic diagram
(b) Boundary conditions
(c) Analysis domain Fig. 2. Heat sink with branched fins.
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
e2 P2 hui ¼ 1 12lf e2 P2 huif 2 ¼ 2 12lf f1
hTis ¼ T w þ
Z
dp ; dz dp ; dz
Y
e31 þ e32 dY dY;
ð25Þ
! Z 2 f dT qf cf P dp H2 Y 1 hTif 1 ¼ T w þ dz 12lf dz ks 0 ð1 e1 e2 Þ ! Z 1 3 17e41 P2 e1 þ e32 dY dY þ ; 140kf Y ! Z 2 f dT qf cf P dp H2 Y 1 hTif 2 ¼ T w þ dz 12lf dz ks 0 ð1 e1 e2 Þ ! Z 1 3 17e42 P2 3 e1 þ e2 dY dY þ ; 140kf Y
ð26Þ
ð27Þ
where Y y/H. Total flow rate Q_ and fluid bulk mean temperature Tf can be obtained by integrating the average fluid velocity and average temperature, respectively.
2W Q_ ¼ P ¼
Z
Z
H
0
P=2
WHP2 dp 12lf dz
R H R P=2 0
0
Z
1
0
Tudxdy
T f ¼ R H R P=2
udxdy
Z
2W P
udxdy ¼ 0
R1 0
H2 ks
e31 þ e32
R Y
1 0 ð1e1 e2 Þ
R 1 0
ð24Þ
!Z 2 f Y dT qf cf P dp 1 dz 12lf dz 0 ð1 e1 e2 Þ
H2 ks 1
ð23Þ
00 q Tf ¼ Tw H
H
e1 huif 1 þ e2 huif 2
P
0
2
e1 hTif 1 huif 1 þ e2 hTif 2 huif 2 RH
e1 huif 1 þ e2 huif 2
17 e71 þe72 ÞP2 dY e31 þ e32 dYdY þ ð140k f
e31 þ e32 dY
2
: ð33Þ
The thermal performance of the heat sink is evaluated using the concept of thermal resistance. In the present study, thermal resistance is defined as the difference between the heat sink’s base temperature at the outlet and the fluid bulk mean temperature at the inlet per unit heat flow rate. Thermal resistance R consists of capacitive resistance Rcap (which is responsible for the coolant’s temperature increase from the inlet to the outlet) and convective resistance Rconv (which relates to the heat transfer from the fins to the coolant):
R ¼ Rcap þ Rconv ;
Rcap ¼
1
qf cf Q_
;
Rconv ¼
Tw Tf : q00 LW
ð34Þ
From Eqs. (31), (33) and (34), the thermal resistance is derived. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12lf L 1 R¼ qf cf WHP2 Ppump R01 e31 þ e32 dY R R1 3 R R 2 H e1 þ e32 0Y ð1e11 e2 Þ Y1 e31 þ e32 dYdYdY þ 140k17Pf HLW 01 e71 þ e72 dY ks LW 0 þ : R 2 1 e31 þ e32 dY 0 ð35Þ
ð28Þ
RH ¼
Y
dy
e31 þ e32 dY;
0
R1
P dy 2
P dy 2 ! f dT qf cf P dp ¼ Tw þ dz 12lf dx R R 1 H2 3 R 17 e71 þe72 ÞP 2 dY e1 þ e32 0Y ð1e11 e2 Þ Y1 e31 þ e32 dYdY þ ð140k 0 ks f ; R1 3 e1 þ e32 dY 0 0
0
0
2
ð29Þ In Eqs. (28) and (29), the total flow rate and fluid bulk mean temperature are functions of dp/dz and dTf/dz. To complete the solutions, the values of dp/dz and dTf/dz must be determined. In this study, the pumping power is assumed to be constant. Therefore,
dp L: Ppump ¼ Q_ Dp ¼ Q_ dz
ð30Þ
Combining Eqs. (28) and (30), we find
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 1 3 12lf Ppump dp 3 ¼ e þ e ; dY 1 2 dz WHLP 2 0 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 3 WHP2 Ppump e1 þ e32 dY : Q_ ¼ 12lf L 0
ð31Þ
For a fully developed flow, the longitudinal temperature gradient satisfies the following equation:
dT f q00 W ¼ ¼ dz qf cf Q_
q00 W rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : R 2 1 3 3 qf cf WHp12lPpump e þ e dY 1 2 0 L
ð32Þ
f
Therefore, fluid bulk mean temperature Tf can be determined from Eqs. (29), (31) and (32).
Fig. 3. Thermal resistances for plate-fin heat sinks. [L = 1 cm, W = 1 cm, P = 1 mm, lf = 0.000855 kg/(m s), cf = 4179 J/(kg K), qf = 997 kg/m3, kf = 0.613 W/(m K), Ppump = 2.56 W].
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
Fig. 4. Thermal resistances for the rectangular-fin heat sink.
In this study, the thermal resistance from Eq. (35) by isR Ycalculated R1 R1 numerically integrating 0 e31 þ e32 0 1e11 e2 Y e31 þ e32 dYdYdY; R1 7 R e1 þ e72 dY, and 01 e31 þ e32 dY. 0 3. Results and discussion To validate the solution presented in the previous section, the thermal resistances obtained from the proposed model were
compared with the thermal resistances obtained using the results from a two-dimensional direct numerical simulation. The solutions from this simulation were obtained by solving the momentum and energy equations [Eqs. (1)–(3)] using the control volume-based finite difference method. Fig. 3 shows the thermal resistances for heat sinks with rectangular fins, Y-shaped fins, and inverted Yshaped fins. Fig. 3 shows that, within a relative error of 5%, the thermal resistances obtained from Eq. (35) match the thermal resistances obtained from the numerical results when H/wc > 2 and ks/kf > 10. This finding results because, during the derivation of Eqs. (19) and (20), the aspect ratio of the channel was assumed to be greater than one, while the solid conductivity was assumed to be greater than the fluid conductivity. As shown in Fig. 3(a), the thermal resistance as a function of height has a minimum value because the heat transfer area increases, but the fin efficiency decreases as the height becomes larger. Fig. 3(b) shows that thermal resistance decreases as solid conductivity increases because fin efficiency increases as solid conductivity increases. In addition to the comparison based on numerical results, the results from the present model can also be compared with results from existing experimental data for the rectangular-fin heat sink. Using Eq. (35), the thermal resistance of the rectangular-fin heat sink Rr can be calculated:
Rr ¼ Rðe1 ¼ 0; e2 ¼ er Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12lf L 1 H 17P2 er þ ; þ ¼ 2 3 qf cf WHP Ppump er 3ks LWð1 er Þ 140kf HLW
Table 1 Comparison between optimized heat sink with branched fins and optimized heat sink with rectangular fins. Branched fins Constraints Length (L) width (W) Height (H) Pumping power (Ppump) Solid Fluid Results Fin pitch (P) Fin number Porosity
Surface area Flow rate Rcap Rconv R Schematic of fin (not to scale)
Rectangular fins
3 cm 3 cm 3 mm 2.56 W Aluminum [ks = 175 W/(m K)] Water [ks = 0.613 W/(m K)] 0.21 mm 142
0 e1 ¼ 2
0:57 0:52Y þ 0:10Y 2 e2 ¼ 0:80 1:2Y þ 0:52Y 2 0:57 0:52Y þ 0:10Y 0.026m2 0.000044 m3/s 0.0055 K/W 0.0080 K/W 0.014 K/W
ðY ðY ðY ðY
< 0:56Þ P 0:56Þ < 0:56Þ P 0:56Þ
0.33 mm 92 0.31
0.016m2 0.000049 m3/s 0.0049 K/W 0.013 K/W 0.018 K/W
ð36Þ
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
Table 2 Comparison of geometries and thermal resistances of optimized heat sinks. Dimensionless pumping power Ppump
l3 q2 D1 h;fr 11
10
Dimensionless length
Type of fin
Dimensionless fin pitch
Porosity
P Dh;fr
e
Thermal resistance (K/W) R
Rectangular fin
0.060
0.33
0.056
Branched fin
0.040
L Dh;fr
1.0
e1 ¼ e2
1011
5.0
Rectangular fin
0.089
Branched fin
0.094
1.0
Rectangular fin
0.054
0
2
0:56 0:54Y þ 0:11Y 2 ¼ 0:79 1:2Y þ 0:51Y 2 0:56 0:54Y þ 0:11Y
if if if if
Y Y Y Y
< 0:56 P 0:56 < 0:56 P 0:56
0.54
0.24
0.047
0.026
0 2
0:59 0:22Y þ 0:058Y 2 0:28 þ 0:20Y 0:080Y ¼ 0:59 0:22Y þ 0:058Y 2
e1 ¼ e2
1012
Schematic of fin (not to scale)
if if if if
Y Y Y Y
< 0:74 P 0:74 < 0:74 P 0:74
0.025
0.043
(continued on next page)
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287 Table 2 (continued) Dimensionless pumping power Ppump l3 q2 D1 h;fr
Dimensionless length
Type of fin
L Dh;fr
Branched fin
Dimensionless fin pitch
Porosity
P Dh;fr
e
0.025
e1 ¼ e2
1012
5.0
Rectangular fin
0.073
Branched fin
0.067
0
Thermal resistance (K/W) R
2
0:57 0:87Y þ 0:27Y2 ¼ 0:66 1:3Y þ 0:68Y 2 0:57 0:87Y þ 0:27Y
if if if if
Y Y Y Y
< 0:34 P 0:34 < 0:34 P 0:34
0.43
e2
0.029
0.017
0 2
0:56 0:33Y þ 0:059Y2 ¼ 0:69 0:77Y þ 0:36Y 2 0:56 0:33Y þ 0:059Y
e1 ¼
Schematic of fin (not to scale)
if if if if
Y Y Y Y
< 0:56 P 0:56 < 0:56 P 0:56
0.016
[ks/kf = 241, Pr = 5.83,H/W = 0.1, Dh,fr = 2HW/(H + W) = 0.01].
where er denotes the porosity of the rectangular-fin heat sink. Fig. 4 presents the thermal resistances of the rectangular-fin heat sink. As shown in Fig. 4, for low Reynolds numbers, the results obtained from Eq. (36) successfully predict the experimental data obtained by Qu and Mudawar [23]. To design an optimized heat sink, the porosities and fin pitch that minimize the thermal resistance must be determined. In the present study, the optimal values for e1(y), e2 (y), and P that minimize the thermal resistance for a given height, length, and width were numerically determined using the gradient descent method. The numerical simulation was conducted as follows: (1) As shown in the numerical analysis domain in Fig. 2(c), porosities e1(y) and e2(y) were discretized.
N X ði 1ÞH iH U y ðj ¼ 1; 2Þ: ej ðyÞ eji U y N N i¼1 ð37Þ Here, U is the unit step function, which satisfies the following equation:
UðyÞ ¼
0 if y < 0 : 1 if y P 0
ð38Þ
It was found that the results did not change appreciably when N > 50. In the present study, 100 nodes were used to discretize the channel (i.e., N = 100). (2) The initial values of discretized porosities e1i and e2i and fin pitch P⁄ were estimated. It was found that the numerical procedure converged well when e1i ¼ 0 and e2i ¼ 0:5. (3) The partial derivatives of R with respect to e1i, e2i, and P were calculated using Eqs. (35) and (39).
@R R eji ¼ eji þ c R eji ¼ eji c ¼ ðj ¼ 1; 2Þ; 2c @ eji @R RðP ¼ P þ cÞ RðP ¼ P cÞ ¼ : @P 2c
ð39Þ
Here, c is a small positive number. (4) The values of e1i, e2i, and P were calculated using the following equations:
e1i ¼ e1i c
@R ; @ e1i
e2i ¼ e2i c
@R ; @ e2i
P ¼ P c
@R : @P
ð40Þ
(5) Using the values of e1i, e2i, and P calculated above as new estimates for e1i ; e2i , and P⁄, steps 2–5 were repeated until converging values for e1i, e2i, and P were obtained. Based on the least-squares fitting, polynomial equations for porosities e1(y) and e2 (y) were subsequently obtained from e1i and e2i.
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
log10
Ppump
!
1 l3f q2 f Dh;fr
2:93log10
L Dh;fr
< 9:55:
ð41Þ
When Eq. (41) was satisfied, the performance of the heat sink with branched fins was only slightly better than the performance of the simple heat sink with rectangular fins. 4. Conclusion
Fig. 5. Contour plots of Ropt,b/Ropt,r. [ks/kf = 241, Pr = 5.83, H/W = 0.1,Dh,fr = 2HW/ (H + W) = 0.01].
Thermal optimization of a plate-fin heat sink was conducted for fins branched in the direction normal to fluid flow. This optimization was performed using a model based on VAT. Compared to heat sinks with rectangular fins, the thermal resistance was reduced by up to 30% by employing branched fins for a water-cooled heat sink. However, the amount of the reduction decreased as the pumping power decreased and as the length of the heat sink increased. Because of their improved thermal performance, heat sinks with branched fins are expected to be a suitable next-generation cooling solution. Conflict of interest
Table 1 lists the channel width, fin thickness, fin pitch, fin number, fin surface area, friction factor, capacitive thermal resistance, convective thermal resistance, and total thermal resistance for the optimized heat sink with branched fins and compares these values with the corresponding values for the optimized heat sink with rectangular fins under the same constraints. As shown in Table 1, the heat sink with Y-shaped fins was selected as the optimized branch-fin heat sink. Compared with the rectangular-fin heat sink, this heat sink has a smaller fin pitch and a larger number of fins. Consequently, the heat sink with branched fins has a greater surface area because the surface area is inversely proportional to the fin pitch and is proportional to the number of fins. Therefore, the convective resistance of the heat sink with branched fins is lower than the convective resistance of the heat sink with rectangular fins. Meanwhile, the capacitive resistances of these two heat sinks are similar because the flow rates are similar. Consequently, heat sinks with branched fins can effectively reduce the convective thermal resistance without compromising capacitive thermal resistance. Finally, the total resistance of the heat sink with branched fins is lower than the total resistance of the heat sink with rectangular fins. For the case presented in Table 1, the total thermal resistance decreases by approximately 22% if the fins are branched in the direction normal to fluid flow. It is also noteworthy that the two porosities, e1 and e2, are identical in the branch side of the optimized branched fin. The optimized geometries and thermal resistances of both heat sinks with fins of variable thickness and heat sinks with fins of uniform thickness for various pumping powers and lengths are listed in Table 2. By comparing the thermal resistances of these two types of optimized heat sinks, a contour map was drawn, as shown in Fig. 5. Fig. 5 shows the ratio of the optimal thermal resistances of heat sinks with branched fins to the optimal thermal resistances of heat sinks with rectangular fins (Ropt,b/Ropt,r) for water-cooled systems. In Fig. 5, in the region where the ratio is less than one, the optimized heat sink with branched fins performs better than the optimized heat sink with rectangular fins. The opposite is true when the ratio is greater than one. Therefore, the contour map indicates that optimized heat sinks with branched fins have lower thermal resistances than optimized heat sinks with rectangular fins. By employing branched fins, the thermal resistance was reduced by up to 30%. However, the difference between the thermal resistances decreased as the length increased and as the pumping power decreased. As shown in Fig. 5, the thermal resistance was reduced by less than 6% when
None declared. Acknowledgments This research was supported by the Nano Material Technology Development Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT, and Future Planning (NRF-2011-0030285). References [1] S. Oktay, R.J. Hannemann, A. Bar-Cohen, High heat from a small package, Mech. Eng. 108 (1986) 36–42. [2] A. Bar-Cohen, Thermal management of electric components with dielectric liquids, Proc. ASME/JSME Therm. Eng. Joint Conf. 2 (1996) 15–39. [3] F.P. Incropera, Convection heat transfer in electronic equipment cooling, J. Heat Transfer 110 (1988) 1097–1111. [4] W. Nakayama, Thermal management of electronic equipment: a review of technology and research topics, Appl. Mech. Rev. 39 (1986) 1847–1868. [5] D. Liu, S.V. Garimella, Analysis and optimization of the thermal performance of microchannel heat sinks, Int. J. Numer. Methods Heat Fluid Flow 15 (2005) 7– 26. [6] R.W. Knight, J.S. Goodling, D.J. Hall, Optimal thermal design of forced convection heat sinks-analytical, J. Electron. Packag. 113 (1991) 313–321. [7] P. Teertstra, M.M. Yovanovich, J.R. Culham, Analytical forced convection modeling of plate fin heat sink, J. Electron. Manuf. 10 (2000) 253–261. [8] J.H. Ryu, S.J. Kim, D.H. Choi, Numerical optimization of the thermal performance of a microchannel heat sink, Int. J. Heat Mass Transfer 45 (2002) 2823–2827. [9] W. Qu, I. Mudawar, Analysis of three-dimensional heat transfer in microchannel heat sinks, Int. J. Heat Mass Transfer 45 (2002) 3973–3985. [10] S.J. Kim, D. Kim, Forced convection in microstructures for electronic equipment cooling, J. Heat Transfer 121 (1991) 639–645. [11] C.Y. Zhao, T.J. Lu, Analysis of microchannel heat sinks for electronics cooling, Int. J. Heat Mass Transfer 45 (2002) 4857–4869. [12] M. Morega, A. Bejan, Plate fins with variable thickness and height for air-cooled electronic modules, Int. J. Heat Mass Transfer 37 (1994) 433– 445. [13] D.K. Kim, J. Jung, S.J. Kim, Thermal optimization of plate-fin heat sinks with variable fin thickness, Int. J. Heat Mass Transfer 53 (2010) 5988–5995. [14] A. Bejan, S. Lorente, Constructal law of design and evolution: physics, biology, technology, and society, J. Appl. Phys. 113 (2013) 151301. 1–20. [15] P. Xu, B. Yu, M. Yun, M. Zou, Heat conduction in fractal tree-like branched networks, Int. J. Heat Mass Transfer 49 (2006) 3746–3751. [16] A. Bejan, M. Almogbel, Constructal T-shaped fins, Int. J. Heat Mass Transfer 43 (2000) 2101–2115. [17] G. Lorenzini, L.A.O. Rocha, Constructal design of Y-shaped assembly of fins, Int. J. Heat Mass Transfer 49 (2006) 4552–4557. [18] G. Lorenzini, L.A.O. Rocha, Constructal design of T–Y assembly of fins for an optimized heat removal, Int. J. Heat Mass Transfer 52 (2009) 1458– 1463. [19] B. Yu, B. Li, Fractal-like tree networks reducing the thermal conductivity, Phys. Rev. E 73 (2006) 066302. 1–8.
D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287 [20] V.C. Slattery, Advanced Transport Phenomena, Cambridge University Press, Cambridge, 1999. pp. 194–197. [21] D.-K. Kim, S.J. Kim, Averaging approach for microchannel heat sinks subject to the uniform wall temperature condition, Int. J. Heat Mass Transfer 49 (2006) 695–706.
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[22] D.-K. Kim, S.J. Kim, Closed form correlations for thermal optimization of microchannels, Int. J. Heat Mass Transfer 50 (2007) 5318–5322. [23] W. Qu, I. Mudawar, Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink, Int. J. Heat Mass Transfer 45 (2002) 2549–2565.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Thermal optimization of branched-fin heat sinks subject to a parallel flow Dong-Kwon Kim ⇑ Department of Mechanical Engineering, Ajou University, Suwon 443-749, Republic of Korea
a r t i c l e
i n f o
Article history: Received 24 February 2014 Received in revised form 9 May 2014 Accepted 11 May 2014
Keywords: Heat sink Thermal optimization Branched-fin Forced convection Parallel flow
a b s t r a c t In the present study, thermal optimization of a heat sink with fins branched in the direction normal to fluid flow was conducted using a model based on volume averaging theory (VAT). For a water-cooled system, it was shown that, compared to a rectangular-fin heat sink, the thermal resistance of the optimized branched-fin heat sink decreased by up to 30%. The extent of the reduction increased with an increase in the pumping power and a decrease in the length of the heat sink. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the electronic equipment industry, the trend toward denser and more powerful products has led to the use of electronic devices with higher power densities [1,2]. However, high power densities result in increased junction temperatures, which adversely affect the reliability of electronic devices. Therefore, an effective cooling mechanism is essential for reliably operating electronic components [3,4]. Many ideas pertaining to cooling methods have been proposed. Among the various types of cooling systems that have been developed, the plate-fin heat sink is the most widely used because of its simple design and ease of fabrication. As summarized in the review by Liu and Garimella [5], several studies have focused on optimizing the size of plate-fin heat sinks. Many optimization methods have been proposed based on the fin model [6,7], three-dimensional numerical models [8,9], and volume averaging theory (VAT) [10,11]. The layout of the heat sink used in these studies is shown in Fig. 1(a). The heat sink was optimized by determining the fin thickness and channel width that minimize the thermal resistance for a heat sink of a given size. This procedure was based on the assumption that fin thickness and channel width are constant along the directions parallel and normal to fluid flow. However, there is no guarantee that a fin with a rectangular cross-section is the most thermally efficient. For example, Morega and Bejan demonstrated that the thermal
⇑ Tel.: +82 31 219 3660; fax: +82 31 219 1611. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.05.010 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
resistance of an air-cooled heat sink can be reduced by approximately 15% by increasing the fin thickness in the flow direction [12]. Kim et al. also showed that, for a water-cooled system, the thermal resistance of a plate-fin heat sink can be reduced by as much as 15% by increasing the fin thickness in the direction normal to fluid flow [13]. Heat transfer through branched fins is a topic of considerable interest [14,15]. Bejan and Almogbel optimized a T-shaped fin assembly to maximize global thermal conductance, which was subject to both total volume and fin material constraints [16]. Lorenzini and Rocha minimized global thermal resistance to optimize a Y-shaped assembly of fins [17]. Lorenzini and Rocha also performed thermal optimization of a T–Y assembly of fins to maximize heat removal [18]. These theoretical studies supported the conclusion that branched fins can reduce total thermal resistance. Contrary to these results, Yu and Li investigated the effective thermal conductivity of composites with embedded self-similar H-shaped fractal-like tree networks, and in their study, they found that fractal-like tree networks can significantly increase the thermal resistance of composites [19]. They showed that this heightened thermal resistance resulted from the use of longer branches. As indicated by these contradictory results, it remains unclear whether branched fins can further reduce the thermal resistance of a plate-fin heat sink. To date, despite these lingering uncertainties, thermal optimization of a heat sink with fins branched in the direction normal to fluid flow under forced convection has not been thoroughly investigated. This paper seeks to determine which type of heat sink (among the three heat sinks shown in Fig. 1) performs better under forced
D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
279
Nomenclature a c Dh,fr h H k K L p Dp P Ppump q00 Q_ R T Tf u U wc ww
wetted area per volume [m1] heat capacity of a fluid [J kg1 K1] hydraulic diameter of the frontal area [m] interstitial heat transfer coefficient based on the onedimensional bulk mean temperature [W m2 K1] height of the heat sink [m] thermal conductivity [W m1 K1] permeability [m2] length of the heat sink [m] pressure [Pa] pressure drop in the heat sink [Pa] fin pitch [m] pumping power [W] heat flux [W/m2] flow rate [m3 s1] thermal resistance [W/K] temperature [K] fluid bulk mean temperature [K] velocity [m s1] unit step function channel width [m] fin thickness [m]
x,y,z Y hi
Cartesian coordinate system [m] dimensionless coordinate (y/H) averaged value
Greek symbols e porosity l viscosity q density Subscripts and superscripts 1 first channel 2 second channel b branched-fin heat sink cap capacitive conv convective e effective f fluid i node number opt optimized r rectangular-fin heat sink s solid
convection. Thermal optimization of branched-fin heat sinks was conducted using a model based on VAT. The results demonstrated that, using Y-shaped fins, the thermal resistance of a plate-fin heat sink can be reduced [i.e., the heat sink shown in Fig. 1(b) is the best choice]. This paper shows that, when using Y-shaped fins, the reduction in thermal resistance increases as the pumping power increases and the length of the heat sink decreases.
and energy equations in the direction of the x-axis over the averaging analysis domain shown in Fig. 2(c).
2. Mathematical formulation
! @ @hTif 1 @hTif 1 s f1 ; kfe1 e1 qf cf hui ¼ h1 a hTi hTi þ @y @z @y ! @ @hTif 2 @hTif 2 kfe2 e2 qf cf huif 2 ¼ h2 a hTis hTif 2 þ ; @y @z @y
! dp lf @ 2 huif 1 f1 ; ¼ e1 hui l dz K 1 @y2 ! dp lf @ 2 huif 2 f2 ; ¼ e2 hui l dz K 2 @y2 f1
In this paper, the topic under consideration relates to forced convection through channels, as shown in Fig. 2(a) and (b). The cover plate is insulated, and the heat sink base is uniformly heated. A coolant passes through several channels and carries the heat away from the heat sink base. For simplicity, the flow is assumed to be laminar and is fully developed both hydrodynamically and thermally in this analysis. Moreover, the pumping power required to drive the fluid through the heat sink is assumed to be constant. It is also assumed that the aspect ratio of the channel is much greater than one and that the conductivity of the solid is higher than the conductivity of the fluid. The channel width and fin thickness vary in the direction normal to the fluid flow [wc1 = wc1(y), wc2 = wc2(y), ww = ww(y)]; fin pitch P, which is equal to the sum of the channel widths and fin thickness, is constant. The momentum and energy equations are given in Eqs. (1)–(3). For the fluid phase,
lf
! @2u @2u dp þ ; ¼ @x2 @y2 dz
qf cf
! @ðuTÞ @2T @2T : ¼ kf þ @z @x2 @y2
ð1Þ
ð2Þ
For the solid phase,
0 ¼ ks
! @2T @2T : þ @x2 @y2
ð3Þ
In the VAT-based model, the governing equations for the average velocity and temperature are obtained by averaging the momentum
@ @hTis ¼ h1 a hTis hTif 1 þ h2 a hTis hTif 2 : kse @y @y
ð4Þ
ð5Þ
ð6Þ
ð7Þ
ð8Þ
Here, average fluid velocities huif1 and huif2, average fluid temperatures hTif1 and hTif2, and average solid temperature hTis are defined as follows:
Z wc1 =2 Z P=2 2 2 u dx; huif 2 ¼ u dx; wc1 0 wc2 ðwc1 þww Þ=2 R P=2 R wc1 =2 Tudx Tudx ðwc1 þww Þ=2 ; hTif 2 ¼ R P=2 ; hTif 1 ¼ R0 w =2 c1 udx udx 0 ðwc1 þww Þ=2 Z ðwc1 þww Þ=2 2 T dx: hTis ¼ ww wc1 =2
huif 1 ¼
ð9Þ
ð10Þ
For the average fluid temperatures in Eq. (10), the one-dimensional bulk means are used to easily calculate the bulk mean temperature of the fluid, Tf. There are two equations for the average fluid velocities [Eqs. (4) and (5)] and two equations for the average fluid temperatures [Eqs. (6) and (7)] because there are two different fluid channels in the analysis domain. The quantities e, a, kfe, kse, K, and h denote the porosity, wetted area per unit volume, effective thermal conductivity of the fluid, effective thermal conductivity of the solid, permeability, and interstitial heat transfer coefficient, respectively. These quantities can be represented as follows:
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
wc1 wc2 ; e2 ¼ ; P P kse ¼ ks ð1 e1 e2 Þ;
e1 ¼
K1
e1 wc1 hui
a¼
@u @x
f1
2
2 ; kfe1 ¼ kf e1 ; P
; x¼wc1 =2
!1
h1
@x
q00sf 1 s
f1
hTi hTi
ð12Þ
;
x¼ðwc1 þww Þ=2
h2
;
ð11Þ
!1
e2 wc2 huif 2 @u K2 2
kfe2 ¼ kf e2 ;
q00sf 2 s
hTi hTif 2
ð13Þ
:
Here, e, a, kfe, kse, K, and h are functions of y because they are functions of wc and ww. As noted in Slattery [20], some information is lost in any averaging approach; in the present case, we lose the dependence of both the velocity and the temperature distributions for the fluid phases in the averaging direction. In the averaging approach, we typically replace the lost information with empirical data for K and h. For the present configuration, however, these parameters can be determined analytically through an approximation [21,22]. For this approximation, it is assumed that, along the direction of the z-axis, the fluid’s temperature and velocity profiles are similar to the profiles of the Poiseuille flow between two infinite parallel plates. Based on these assumptions, the velocity and temperature distributions are then determined to be
8 2 f1 > if z 6 w2c1 ; < 32 hui 1 4 wz2 c1 u¼ 2 > w : 32 huif 2 1 4 ðzP=2Þ ; if z P wc1 þw 2 w2
(a) Rectangular-fin heat sink
ð14Þ
c2
8 2 4 > < 175 hTif 1 hTis 1 24z þ 16z if z 6 w2c1 ; þ hTis 136 5w2c 5w4c T¼ : 2 4 > w : 175 þ hTis if z P wc1 þw hTif 2 hTis 1 24ðzP=2Þ þ 16ðzP=2Þ 136 2 5w2 5w4 c
c
ð15Þ
(b) Y-shaped fin heat sink
Using Eqs. (14) and (15) and the definitions in Eqs. (12) and (13), the permeability and interstitial heat transfer coefficient are
K1 ¼
e31 P2 12
K2 ¼
;
e32 P2 12
h1 ¼
;
70kf ; 17e1 P
h2 ¼
70kf : 17e2 P
ð16Þ
Eqs. (4)–(7) can be simplified when the solid conductivity is greater than the fluid conductivity (ks > kf) and when the aspect ratio of the channel is much greater than one (H > wc). Estimating the order of magnitude of each term that appears on the right-hand sides of Eqs. (4)–(7) yields the following:
lf w2c1
huif 1 > >
lf 2
H kf
H2
lf
huif 1 ;
w2c2 ks
hTif 1 ;
H2
huif 2 >
hTis >
lf H
kf H2
2
huif 2 ;
hTif 1 ;
ks H2
hTis ð17Þ
respectively. The following equation is also satisfied for constant surface heat flux: f1
f2
@hTi @hTi dT ¼ ¼ : @z @z dz
dp lf ¼ e1 huif 1 ; dz K 1
dp lf ¼ e2 huif 2 ; dz K 2 f dT e1 qf cf huif 1 ¼ h1 a hTis hTif 1 ; dz f f 2 dT e2 qf cf hui ¼ h2 a hTis hTif 2 : dz
Fig. 1. Plate-fin heat sinks.
ð18Þ
Here, Tf is the fluid bulk mean temperature. Using the relations shown in Eqs. (17) and (18), Eqs. (4)–(7) can be simplified as follows:
(c) Inverted Y-shaped fin heat sink
f
ð19Þ
Similar to the boundary conditions presented in another study [22], the appropriate boundary conditions are given as follows:
huif 1 ¼ huif 2 ¼ 0;
hTis ¼ hTif 1 ¼ hTif 2 ¼ T w s
huif 1 ¼ huif 2 ¼ 0; ð20Þ
f1
at y ¼ 0;
ð21Þ
f2
@hTi @hTi @hTi ¼ ¼ ¼ 0 at y ¼ H: @y @y @y
ð22Þ
The velocity and temperature distributions can be obtained by solving Eqs. (8), (19) and (20):
D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
(a) Schematic diagram
(b) Boundary conditions
(c) Analysis domain Fig. 2. Heat sink with branched fins.
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
e2 P2 hui ¼ 1 12lf e2 P2 huif 2 ¼ 2 12lf f1
hTis ¼ T w þ
Z
dp ; dz dp ; dz
Y
e31 þ e32 dY dY;
ð25Þ
! Z 2 f dT qf cf P dp H2 Y 1 hTif 1 ¼ T w þ dz 12lf dz ks 0 ð1 e1 e2 Þ ! Z 1 3 17e41 P2 e1 þ e32 dY dY þ ; 140kf Y ! Z 2 f dT qf cf P dp H2 Y 1 hTif 2 ¼ T w þ dz 12lf dz ks 0 ð1 e1 e2 Þ ! Z 1 3 17e42 P2 3 e1 þ e2 dY dY þ ; 140kf Y
ð26Þ
ð27Þ
where Y y/H. Total flow rate Q_ and fluid bulk mean temperature Tf can be obtained by integrating the average fluid velocity and average temperature, respectively.
2W Q_ ¼ P ¼
Z
Z
H
0
P=2
WHP2 dp 12lf dz
R H R P=2 0
0
Z
1
0
Tudxdy
T f ¼ R H R P=2
udxdy
Z
2W P
udxdy ¼ 0
R1 0
H2 ks
e31 þ e32
R Y
1 0 ð1e1 e2 Þ
R 1 0
ð24Þ
!Z 2 f Y dT qf cf P dp 1 dz 12lf dz 0 ð1 e1 e2 Þ
H2 ks 1
ð23Þ
00 q Tf ¼ Tw H
H
e1 huif 1 þ e2 huif 2
P
0
2
e1 hTif 1 huif 1 þ e2 hTif 2 huif 2 RH
e1 huif 1 þ e2 huif 2
17 e71 þe72 ÞP2 dY e31 þ e32 dYdY þ ð140k f
e31 þ e32 dY
2
: ð33Þ
The thermal performance of the heat sink is evaluated using the concept of thermal resistance. In the present study, thermal resistance is defined as the difference between the heat sink’s base temperature at the outlet and the fluid bulk mean temperature at the inlet per unit heat flow rate. Thermal resistance R consists of capacitive resistance Rcap (which is responsible for the coolant’s temperature increase from the inlet to the outlet) and convective resistance Rconv (which relates to the heat transfer from the fins to the coolant):
R ¼ Rcap þ Rconv ;
Rcap ¼
1
qf cf Q_
;
Rconv ¼
Tw Tf : q00 LW
ð34Þ
From Eqs. (31), (33) and (34), the thermal resistance is derived. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12lf L 1 R¼ qf cf WHP2 Ppump R01 e31 þ e32 dY R R1 3 R R 2 H e1 þ e32 0Y ð1e11 e2 Þ Y1 e31 þ e32 dYdYdY þ 140k17Pf HLW 01 e71 þ e72 dY ks LW 0 þ : R 2 1 e31 þ e32 dY 0 ð35Þ
ð28Þ
RH ¼
Y
dy
e31 þ e32 dY;
0
R1
P dy 2
P dy 2 ! f dT qf cf P dp ¼ Tw þ dz 12lf dx R R 1 H2 3 R 17 e71 þe72 ÞP 2 dY e1 þ e32 0Y ð1e11 e2 Þ Y1 e31 þ e32 dYdY þ ð140k 0 ks f ; R1 3 e1 þ e32 dY 0 0
0
0
2
ð29Þ In Eqs. (28) and (29), the total flow rate and fluid bulk mean temperature are functions of dp/dz and dTf/dz. To complete the solutions, the values of dp/dz and dTf/dz must be determined. In this study, the pumping power is assumed to be constant. Therefore,
dp L: Ppump ¼ Q_ Dp ¼ Q_ dz
ð30Þ
Combining Eqs. (28) and (30), we find
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 1 3 12lf Ppump dp 3 ¼ e þ e ; dY 1 2 dz WHLP 2 0 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 3 WHP2 Ppump e1 þ e32 dY : Q_ ¼ 12lf L 0
ð31Þ
For a fully developed flow, the longitudinal temperature gradient satisfies the following equation:
dT f q00 W ¼ ¼ dz qf cf Q_
q00 W rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : R 2 1 3 3 qf cf WHp12lPpump e þ e dY 1 2 0 L
ð32Þ
f
Therefore, fluid bulk mean temperature Tf can be determined from Eqs. (29), (31) and (32).
Fig. 3. Thermal resistances for plate-fin heat sinks. [L = 1 cm, W = 1 cm, P = 1 mm, lf = 0.000855 kg/(m s), cf = 4179 J/(kg K), qf = 997 kg/m3, kf = 0.613 W/(m K), Ppump = 2.56 W].
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
Fig. 4. Thermal resistances for the rectangular-fin heat sink.
In this study, the thermal resistance from Eq. (35) by isR Ycalculated R1 R1 numerically integrating 0 e31 þ e32 0 1e11 e2 Y e31 þ e32 dYdYdY; R1 7 R e1 þ e72 dY, and 01 e31 þ e32 dY. 0 3. Results and discussion To validate the solution presented in the previous section, the thermal resistances obtained from the proposed model were
compared with the thermal resistances obtained using the results from a two-dimensional direct numerical simulation. The solutions from this simulation were obtained by solving the momentum and energy equations [Eqs. (1)–(3)] using the control volume-based finite difference method. Fig. 3 shows the thermal resistances for heat sinks with rectangular fins, Y-shaped fins, and inverted Yshaped fins. Fig. 3 shows that, within a relative error of 5%, the thermal resistances obtained from Eq. (35) match the thermal resistances obtained from the numerical results when H/wc > 2 and ks/kf > 10. This finding results because, during the derivation of Eqs. (19) and (20), the aspect ratio of the channel was assumed to be greater than one, while the solid conductivity was assumed to be greater than the fluid conductivity. As shown in Fig. 3(a), the thermal resistance as a function of height has a minimum value because the heat transfer area increases, but the fin efficiency decreases as the height becomes larger. Fig. 3(b) shows that thermal resistance decreases as solid conductivity increases because fin efficiency increases as solid conductivity increases. In addition to the comparison based on numerical results, the results from the present model can also be compared with results from existing experimental data for the rectangular-fin heat sink. Using Eq. (35), the thermal resistance of the rectangular-fin heat sink Rr can be calculated:
Rr ¼ Rðe1 ¼ 0; e2 ¼ er Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12lf L 1 H 17P2 er þ ; þ ¼ 2 3 qf cf WHP Ppump er 3ks LWð1 er Þ 140kf HLW
Table 1 Comparison between optimized heat sink with branched fins and optimized heat sink with rectangular fins. Branched fins Constraints Length (L) width (W) Height (H) Pumping power (Ppump) Solid Fluid Results Fin pitch (P) Fin number Porosity
Surface area Flow rate Rcap Rconv R Schematic of fin (not to scale)
Rectangular fins
3 cm 3 cm 3 mm 2.56 W Aluminum [ks = 175 W/(m K)] Water [ks = 0.613 W/(m K)] 0.21 mm 142
0 e1 ¼ 2
0:57 0:52Y þ 0:10Y 2 e2 ¼ 0:80 1:2Y þ 0:52Y 2 0:57 0:52Y þ 0:10Y 0.026m2 0.000044 m3/s 0.0055 K/W 0.0080 K/W 0.014 K/W
ðY ðY ðY ðY
< 0:56Þ P 0:56Þ < 0:56Þ P 0:56Þ
0.33 mm 92 0.31
0.016m2 0.000049 m3/s 0.0049 K/W 0.013 K/W 0.018 K/W
ð36Þ
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287
Table 2 Comparison of geometries and thermal resistances of optimized heat sinks. Dimensionless pumping power Ppump
l3 q2 D1 h;fr 11
10
Dimensionless length
Type of fin
Dimensionless fin pitch
Porosity
P Dh;fr
e
Thermal resistance (K/W) R
Rectangular fin
0.060
0.33
0.056
Branched fin
0.040
L Dh;fr
1.0
e1 ¼ e2
1011
5.0
Rectangular fin
0.089
Branched fin
0.094
1.0
Rectangular fin
0.054
0
2
0:56 0:54Y þ 0:11Y 2 ¼ 0:79 1:2Y þ 0:51Y 2 0:56 0:54Y þ 0:11Y
if if if if
Y Y Y Y
< 0:56 P 0:56 < 0:56 P 0:56
0.54
0.24
0.047
0.026
0 2
0:59 0:22Y þ 0:058Y 2 0:28 þ 0:20Y 0:080Y ¼ 0:59 0:22Y þ 0:058Y 2
e1 ¼ e2
1012
Schematic of fin (not to scale)
if if if if
Y Y Y Y
< 0:74 P 0:74 < 0:74 P 0:74
0.025
0.043
(continued on next page)
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D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287 Table 2 (continued) Dimensionless pumping power Ppump l3 q2 D1 h;fr
Dimensionless length
Type of fin
L Dh;fr
Branched fin
Dimensionless fin pitch
Porosity
P Dh;fr
e
0.025
e1 ¼ e2
1012
5.0
Rectangular fin
0.073
Branched fin
0.067
0
Thermal resistance (K/W) R
2
0:57 0:87Y þ 0:27Y2 ¼ 0:66 1:3Y þ 0:68Y 2 0:57 0:87Y þ 0:27Y
if if if if
Y Y Y Y
< 0:34 P 0:34 < 0:34 P 0:34
0.43
e2
0.029
0.017
0 2
0:56 0:33Y þ 0:059Y2 ¼ 0:69 0:77Y þ 0:36Y 2 0:56 0:33Y þ 0:059Y
e1 ¼
Schematic of fin (not to scale)
if if if if
Y Y Y Y
< 0:56 P 0:56 < 0:56 P 0:56
0.016
[ks/kf = 241, Pr = 5.83,H/W = 0.1, Dh,fr = 2HW/(H + W) = 0.01].
where er denotes the porosity of the rectangular-fin heat sink. Fig. 4 presents the thermal resistances of the rectangular-fin heat sink. As shown in Fig. 4, for low Reynolds numbers, the results obtained from Eq. (36) successfully predict the experimental data obtained by Qu and Mudawar [23]. To design an optimized heat sink, the porosities and fin pitch that minimize the thermal resistance must be determined. In the present study, the optimal values for e1(y), e2 (y), and P that minimize the thermal resistance for a given height, length, and width were numerically determined using the gradient descent method. The numerical simulation was conducted as follows: (1) As shown in the numerical analysis domain in Fig. 2(c), porosities e1(y) and e2(y) were discretized.
N X ði 1ÞH iH U y ðj ¼ 1; 2Þ: ej ðyÞ eji U y N N i¼1 ð37Þ Here, U is the unit step function, which satisfies the following equation:
UðyÞ ¼
0 if y < 0 : 1 if y P 0
ð38Þ
It was found that the results did not change appreciably when N > 50. In the present study, 100 nodes were used to discretize the channel (i.e., N = 100). (2) The initial values of discretized porosities e1i and e2i and fin pitch P⁄ were estimated. It was found that the numerical procedure converged well when e1i ¼ 0 and e2i ¼ 0:5. (3) The partial derivatives of R with respect to e1i, e2i, and P were calculated using Eqs. (35) and (39).
@R R eji ¼ eji þ c R eji ¼ eji c ¼ ðj ¼ 1; 2Þ; 2c @ eji @R RðP ¼ P þ cÞ RðP ¼ P cÞ ¼ : @P 2c
ð39Þ
Here, c is a small positive number. (4) The values of e1i, e2i, and P were calculated using the following equations:
e1i ¼ e1i c
@R ; @ e1i
e2i ¼ e2i c
@R ; @ e2i
P ¼ P c
@R : @P
ð40Þ
(5) Using the values of e1i, e2i, and P calculated above as new estimates for e1i ; e2i , and P⁄, steps 2–5 were repeated until converging values for e1i, e2i, and P were obtained. Based on the least-squares fitting, polynomial equations for porosities e1(y) and e2 (y) were subsequently obtained from e1i and e2i.
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log10
Ppump
!
1 l3f q2 f Dh;fr
2:93log10
L Dh;fr
< 9:55:
ð41Þ
When Eq. (41) was satisfied, the performance of the heat sink with branched fins was only slightly better than the performance of the simple heat sink with rectangular fins. 4. Conclusion
Fig. 5. Contour plots of Ropt,b/Ropt,r. [ks/kf = 241, Pr = 5.83, H/W = 0.1,Dh,fr = 2HW/ (H + W) = 0.01].
Thermal optimization of a plate-fin heat sink was conducted for fins branched in the direction normal to fluid flow. This optimization was performed using a model based on VAT. Compared to heat sinks with rectangular fins, the thermal resistance was reduced by up to 30% by employing branched fins for a water-cooled heat sink. However, the amount of the reduction decreased as the pumping power decreased and as the length of the heat sink increased. Because of their improved thermal performance, heat sinks with branched fins are expected to be a suitable next-generation cooling solution. Conflict of interest
Table 1 lists the channel width, fin thickness, fin pitch, fin number, fin surface area, friction factor, capacitive thermal resistance, convective thermal resistance, and total thermal resistance for the optimized heat sink with branched fins and compares these values with the corresponding values for the optimized heat sink with rectangular fins under the same constraints. As shown in Table 1, the heat sink with Y-shaped fins was selected as the optimized branch-fin heat sink. Compared with the rectangular-fin heat sink, this heat sink has a smaller fin pitch and a larger number of fins. Consequently, the heat sink with branched fins has a greater surface area because the surface area is inversely proportional to the fin pitch and is proportional to the number of fins. Therefore, the convective resistance of the heat sink with branched fins is lower than the convective resistance of the heat sink with rectangular fins. Meanwhile, the capacitive resistances of these two heat sinks are similar because the flow rates are similar. Consequently, heat sinks with branched fins can effectively reduce the convective thermal resistance without compromising capacitive thermal resistance. Finally, the total resistance of the heat sink with branched fins is lower than the total resistance of the heat sink with rectangular fins. For the case presented in Table 1, the total thermal resistance decreases by approximately 22% if the fins are branched in the direction normal to fluid flow. It is also noteworthy that the two porosities, e1 and e2, are identical in the branch side of the optimized branched fin. The optimized geometries and thermal resistances of both heat sinks with fins of variable thickness and heat sinks with fins of uniform thickness for various pumping powers and lengths are listed in Table 2. By comparing the thermal resistances of these two types of optimized heat sinks, a contour map was drawn, as shown in Fig. 5. Fig. 5 shows the ratio of the optimal thermal resistances of heat sinks with branched fins to the optimal thermal resistances of heat sinks with rectangular fins (Ropt,b/Ropt,r) for water-cooled systems. In Fig. 5, in the region where the ratio is less than one, the optimized heat sink with branched fins performs better than the optimized heat sink with rectangular fins. The opposite is true when the ratio is greater than one. Therefore, the contour map indicates that optimized heat sinks with branched fins have lower thermal resistances than optimized heat sinks with rectangular fins. By employing branched fins, the thermal resistance was reduced by up to 30%. However, the difference between the thermal resistances decreased as the length increased and as the pumping power decreased. As shown in Fig. 5, the thermal resistance was reduced by less than 6% when
None declared. Acknowledgments This research was supported by the Nano Material Technology Development Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT, and Future Planning (NRF-2011-0030285). References [1] S. Oktay, R.J. Hannemann, A. Bar-Cohen, High heat from a small package, Mech. Eng. 108 (1986) 36–42. [2] A. Bar-Cohen, Thermal management of electric components with dielectric liquids, Proc. ASME/JSME Therm. Eng. Joint Conf. 2 (1996) 15–39. [3] F.P. Incropera, Convection heat transfer in electronic equipment cooling, J. Heat Transfer 110 (1988) 1097–1111. [4] W. Nakayama, Thermal management of electronic equipment: a review of technology and research topics, Appl. Mech. Rev. 39 (1986) 1847–1868. [5] D. Liu, S.V. Garimella, Analysis and optimization of the thermal performance of microchannel heat sinks, Int. J. Numer. Methods Heat Fluid Flow 15 (2005) 7– 26. [6] R.W. Knight, J.S. Goodling, D.J. Hall, Optimal thermal design of forced convection heat sinks-analytical, J. Electron. Packag. 113 (1991) 313–321. [7] P. Teertstra, M.M. Yovanovich, J.R. Culham, Analytical forced convection modeling of plate fin heat sink, J. Electron. Manuf. 10 (2000) 253–261. [8] J.H. Ryu, S.J. Kim, D.H. Choi, Numerical optimization of the thermal performance of a microchannel heat sink, Int. J. Heat Mass Transfer 45 (2002) 2823–2827. [9] W. Qu, I. Mudawar, Analysis of three-dimensional heat transfer in microchannel heat sinks, Int. J. Heat Mass Transfer 45 (2002) 3973–3985. [10] S.J. Kim, D. Kim, Forced convection in microstructures for electronic equipment cooling, J. Heat Transfer 121 (1991) 639–645. [11] C.Y. Zhao, T.J. Lu, Analysis of microchannel heat sinks for electronics cooling, Int. J. Heat Mass Transfer 45 (2002) 4857–4869. [12] M. Morega, A. Bejan, Plate fins with variable thickness and height for air-cooled electronic modules, Int. J. Heat Mass Transfer 37 (1994) 433– 445. [13] D.K. Kim, J. Jung, S.J. Kim, Thermal optimization of plate-fin heat sinks with variable fin thickness, Int. J. Heat Mass Transfer 53 (2010) 5988–5995. [14] A. Bejan, S. Lorente, Constructal law of design and evolution: physics, biology, technology, and society, J. Appl. Phys. 113 (2013) 151301. 1–20. [15] P. Xu, B. Yu, M. Yun, M. Zou, Heat conduction in fractal tree-like branched networks, Int. J. Heat Mass Transfer 49 (2006) 3746–3751. [16] A. Bejan, M. Almogbel, Constructal T-shaped fins, Int. J. Heat Mass Transfer 43 (2000) 2101–2115. [17] G. Lorenzini, L.A.O. Rocha, Constructal design of Y-shaped assembly of fins, Int. J. Heat Mass Transfer 49 (2006) 4552–4557. [18] G. Lorenzini, L.A.O. Rocha, Constructal design of T–Y assembly of fins for an optimized heat removal, Int. J. Heat Mass Transfer 52 (2009) 1458– 1463. [19] B. Yu, B. Li, Fractal-like tree networks reducing the thermal conductivity, Phys. Rev. E 73 (2006) 066302. 1–8.
D.-K. Kim / International Journal of Heat and Mass Transfer 77 (2014) 278–287 [20] V.C. Slattery, Advanced Transport Phenomena, Cambridge University Press, Cambridge, 1999. pp. 194–197. [21] D.-K. Kim, S.J. Kim, Averaging approach for microchannel heat sinks subject to the uniform wall temperature condition, Int. J. Heat Mass Transfer 49 (2006) 695–706.
287
[22] D.-K. Kim, S.J. Kim, Closed form correlations for thermal optimization of microchannels, Int. J. Heat Mass Transfer 50 (2007) 5318–5322. [23] W. Qu, I. Mudawar, Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink, Int. J. Heat Mass Transfer 45 (2002) 2549–2565.