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Thermal optimization of plate-fin heat sinks with fins of variable thickness under natural convection
International Journal of Heat and Mass Transfer 55 (2012) 752–761
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Thermal optimization of plate-fin heat sinks with fins of variable thickness under natural convection 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 19 October 2010 Received in revised form 4 October 2011 Available online 8 November 2011 Keywords: Heat sink Thermal optimization Variable fin thickness Natural convection
a b s t r a c t In this study, thermal performance of a vertical plate-fin heat sink under natural convection was optimized for the case in which the fin thickness varied in the direction normal to the fluid flow. For this optimization, the averaging approach presented in an earlier paper for the case of the heat sinks under forced convection was extended to study the performance of heat sinks under natural convection. In the case of an air-cooled heat sink, the thermal resistance decreases by up to 10% when the fin thickness is allowed to increase in the direction normal to the fluid flow. However, the difference between the thermal resistances of heat sinks with uniform thickness and the heat sinks with variable thickness decreases as the height decreases and as the heat flux decreases. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Because of the trend toward the production of denser and more powerful products in the electronic equipment industry, a higher level of performance of cooling technology is required [1,2]. Many cooling methods have been proposed for ensuring reliable operation of electronic components [3,4]. Owing to their inherent simplicity, reliability, and low long-term cost, natural convection heat sinks have been effectively used in cooling electronic components [5]. Studies on the design of plate fin heat sinks based on natural convection have been extensively carried out, as summarized in an extensive review by Kraus and Bar-Cohen [6]. Elenbaas [7] was the first to document a detailed study of the heat transfer from vertical plate fin heat sinks. Starner and McManus [8] as well as Welling and Wooldridge [5] have studied the heat transfer from heat sinks experimentally. van de Pol and Tiemey [9] developed a correlation between Nusselt number and Elenbass number by using the experimental results of Welling and Woodridge. Bilitsky [10] performed a comprehensive investigation of natural convection heat transfer from several heat sinks that differed in fin height and fin width. Moreover, numerical studies have also been performed by Culham et al. [11,12], Narasimhan and Majdalani [13], and others [14–16]. Several studies that focused on sizing-optimization of plate-fin heat sinks under natural convection have been carried out. Elenbass [7] was the first to suggest a correlation for the optimum ⇑ Tel.: +82 31 219 3660; fax: +82 31 219 1611. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.10.034
channel width of an isothermal fin array; Elenbass found that the Nusselt number for the optimum fin array is 1.25. Bar-Cohen and Rohsenow [17] extended the Elenbass correlation to isoflux plates. Bejan suggested a correlation for calculating the optimal channel width from asymptotic solutions [18]. Later, Bar-Cohen and Jelinek [19] suggested correlations for determining the optimum channel width and fin thickness of the optimally spaced least-material array. However, these studies were focused on the heat sinks with a uniform fin thickness. Recently, my colleagues and I performed an investigation to determine which of the three types of heat sinks shown in Fig. 1 can achieve the best performance under forced convection [20]. We showed that the thermal resistance of a water-cooled platefin heat sink can be decreases by as much as 15% by increasing the fin thickness in the direction normal to the fluid flow. However, to the best of my knowledge, no study on plate-fin heat sinks with a variable fin thickness under natural convection has been carried out. It may be possible to reduce the thermal resistance of natural convection heat sinks by increasing the fin thickness in the direction normal to the fluid flow, as shown in the case of forced convection heat sinks. However, in the case of natural convection, the degree of improvement in the performance of heat sink with fins of variable thickness compared to the performance of heat sinks with fins of uniform thickness and the conditions under which this improvement is achieved are not clear. This is because (1) the flow rate of a buoyancy-induced flow is generally one or two orders of magnitude lower than that of a pressure-driven flow and (2) the flow rate of a buoyancy-induced flow is determined by the fluid temperature in the channel, while the flow rate of a pressure-driven flow is independent of the fluid temperature.
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
Nomenclature a c g Dh f h H k kse K L NuO p P Pheated q q00 q00sf Q_ R Rcap Rconv Ra00 Re T Tf Tf
Wetted area per volume (m1) Heat capacity of a fluid (J kg1 K1) Gravity acceleration (m s2) Hydraulic diameter of the channel Friction factor Interstitial heat transfer coefficient (W m2 K1) Height of the heat sink (m) Thermal conductivity (W m1 K1] Effective thermal conductivity of solid (W m1 K1) Permeability (m2) Length of the heat sink (m) Overall Nusselt number Fin pitch (wc + ww) (m) Perimeter of the channel (m) Heated perimeter of the channel (m) Heat transfer rate (m) Heat flux (W/m2) Heat flux from the fins to the fluid (W/m2) Flow rate (m3/s) Total thermal resistance (W/K) Capacitive thermal resistance (W/K) Convective thermal resistance (W/K) Modified channel Rayleigh number Reynolds number Temperature (K) Fluid bulk mean temperature (K) Total averaged temperature of the fluid in the channel (K)
In this study, thermal performance of a vertical plate-fin heat sink under natural convection was optimized for the case in which the fin thickness varied in the direction normal to the fluid flow. For this purpose, the model based on the volume averaging theory (VAT) presented in an earlier study [20] was extended to study the performance of heat sinks under natural convection.
2. Mathematical formulation The problem under consideration in this paper is related to natural convection through channels as shown in Fig. 2(a). The cover plate is insulated and the heat sink base is uniformly heated. A coolant passes through several channels by natural convection and carries the heat away from the heat sink base. In the analysis of the problem, for simplicity, the flow is assumed to be laminar and to be both hydrodynamically and thermally fully developed; moreover, the flow is assumed to satisfy the Boussinesq approximation. All thermophysical properties are assumed to be constant. In addition, it is assumed that the aspect ratio of the channel is much higher than 1 and that the solid conductivity is higher than the fluid conductivity. The channel width and fin thickness vary in the direction normal to the fluid flow (wc = wc(y), ww = ww(y)), but the fin pitch p, which is equal to the sum of the channel width and fin thickness, is constant. The momentum and energy equations are given as follows: For the fluid phase,
! @ u @ u ¼ qf gbðT f T 1 Þ; þ @x2 @y2 2
lf
2
! @uT @2T @2T : ¼ kf qf cf þ @z @x2 @y2
ð1Þ
ð2Þ
u W wc ww x, y, z X, Y, Z
Velocity (m/s) Width of the heat sink (m) Channel width (m) Fin thickness (m) Cartesian coordinate system Dimensionless Cartesian coordinate system
Greek symbols b Thermal expansion coefficient ( (oq/oT)p/q) (K1) e Porosity (wc/(ww + wc)) l Viscosity (Pa s) q Density (kg m3) Subscripts and superscripts uni Heat sink with fins of uniform thickness f Fluid opt Optimized s Solid w Wall var Heat sink with fins of variable thickness 1 Ambient Special symbols < >f Averaged value for the fluid phase < >s Averaged value for the solid phase
For the solid phase,
0 ¼ ks
! @2T @2T : þ @x2 @y2
ð3Þ
T f is the total volume averaged temperature of the fluid in the channel [17]. The validity of Eqs. (1) and (2) is discussed in Appendix A. In the model based on the VAT, the governing equations for the averaged velocity and temperature are obtained by averaging the momentum and energy equations in the x-direction over the analysis domain shown in Fig. 2(b).
qf gbðT f T 1 Þ ¼
lf K
f
ehui l
! @ 2 huif ; @y2
! @hTif @ @hTif s f ; kfe eqf cf hui ¼ haðhTi hTi Þ þ @y @z @y f
@ @hTis ¼ ha hTis hTif ; kse @y @y
ð4Þ
ð5Þ
ð6Þ
where the average fluid velocity, average fluid temperature, and average solid temperature are defined as shown below:
2 hui ¼ wc f
hTis ¼
2 ww
Z
R ðww þwc Þ=2
ðww þwc Þ=2
udx;
ww =2
Z
Tudx ww =2 ; hTi ¼ R ðw w þwc Þ=2 udx ww =2 f
ww =2
Tdx:
ð7Þ
0
In Eq. (7), for the average fluid temperature, the one-dimensional bulk mean is employed to calculate the bulk mean temperature of the fluid easily. The quantities e, a, kfe, kse, K, and h are the porosity, wetted area per unit volume, effective thermal conductivity of the fluid, effective thermal conductivity of the solid, permeability, and
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
Fig. 2. Heat sink with fins of variable thickness.
e¼
wc ; p
K
e wc huif @u
a¼
2 ; p
kfe ¼ kf e; !1
@x
2
¼ x¼ww =2
kse ¼ ks ð1 eÞ;
e3 p2 12
;
h
q00sf s
hTi hTif
¼
70kf : 17ep ð8Þ
Here, e, a, kfe, kse, K, and h are functions of y, because they are functions of wc and ww. Eqs. (4) and (5) can be simplified when the solid conductivity is higher than the fluid conductivity (ks > kf) and the aspect ratio of the channel is much higher than 1(H > wc). Upon estimating the order of magnitude of each term appearing on the right-hand side of Eq. (4), we have
lf w2c
huif >
lf H2
huif :
ð9Þ
By combining Eqs. (5) and (6), we obtain Fig. 1. Plate-fin heat sinks.
interstitial heat transfer coefficient, respectively. These quantities can be represented as
eqf cf huif
@hTif @ @hTis kse ¼ @y @z @y @ @ @ kse @hTis þ kfe hTis : @y @y @y ha @y
ð10Þ
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
Upon estimating the order of magnitude of each term appearing on the right-hand side of Eq. (10), it follows that
ks H
hTis > 2
kf
ks w2c
H
H4
hTis ; 2
hTis :
ð11Þ
By using the relations presented in Eqs. (9) and (11), Eqs. (4) and (5) can be simplified as
lf
qf gbðT f T 1 Þ ¼
eqf cf huif
ehuif ;
K
ð12Þ
@hTif ¼ ha hTis hTif : @z
ð13Þ
The appropriate boundary conditions are similar to those presented in [21], and they are given as follows:
huif ¼ 0 at y ¼ 0; H;
ð14Þ
hTis ¼ hTif ¼ T w
ð15Þ
at y ¼ 0;
@hTis @hTif ¼ ¼ 0 at y ¼ H: @y @y
ð16Þ
The velocity and temperature distributions can be obtained by solving Eqs. (6), (12), and (13):
qf gb 2 2 hui ¼ p e ðT f T 1 Þ; 12lf f
Z
2
hTis ¼ T w C
H ks
Y
0
H2 ks
Z
Y
1
e3 dY dY ;
0
and C ¼
ð18Þ
1
e3 dY dY þ
Y
! 17 2 4 p e ; 140kf
ð19Þ
q2f cf gb @hTif 2 ðT f T 1 Þ p : @z 12lf
ð20Þ
The total flow rate Q_ and the fluid bulk mean temperature Tf can be obtained by integrating the average fluid velocity and average temperature, respectively:
W Q_ ¼ ðwc þ ww Þ ¼
Z
H
0
Z
ww þwc
udxdy ¼ W ww
qf gbHW 2 p ðT f T 1 Þ 12lf R H R wc
Z
Z
H
ehuif dy
Tudxdy
1
e3 dY;
ð21Þ
0
RH
¼
0
ð22Þ
The total flow rate and fluid bulk mean temperature are functions of ðT f T 1 Þ and ohTif/oz. To complete the solutions, the values of ðT f T 1 Þ and @hTif =@z should be determined. For a fully developed flow, the fluid bulk mean temperature varies linearly in the zdirection, and the longitudinal temperature gradient satisfies the following equation:
qf cf Q_
dT f ¼ q00 W: dz
Therefore,
1 L
Z
L
T f dz:
ð25Þ
0
From Eqs. (21), (24), and (25), T f T 1 is given as
Tf T1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 6lf Lq00 : ¼t R q2f cf gbHp2 01 e3 dY
ð26Þ
From Eqs. (21), (23), and (26), @hTif =@z is given as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 24lf q00 u @hTif dT f q00 W ¼t : ¼ ¼ R @z dz qf cf Q_ q2f cf gbHLp2 01 e3 dY
ð27Þ
Finally, the total flow rate Q_ and the fluid bulk mean temperature Tf are given by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u R ugbHLW 2 q00 p2 1 e3 dY 0 Q_ ¼ t 24lf cf
ð28Þ
and
R 1 q00 H 0
ks
e3
R
Y 1 0 1e
R
1 Y
00
17q e3 dY dY þ 140k p2 e7 dY fH
R 1
e3 dY
2
;
ð29Þ
respectively. The thermal performance of the heat sink is evaluated by using the concept of thermal resistance. In this study, thermal resistance is defined as the difference between the base temperature of the heat sink at the outlet and the fluid bulk mean temperature at the inlet per unit of heat flow rate. The thermal resistance consists of the capacitive resistance, which is responsible for the temperature rise of the coolant from the inlet to the outlet, and the convective resistance, which is related to the heat transfer from the fins to the coolant:
R ¼ Rcap þ Rconv ; Rcap ¼
1
qf cf Q_
;
ð30Þ Rconv ¼
Tw Tf : q00 LW
ð31Þ
From Eqs. (28)–(31), the thermal resistance is given as
0
hTif huif wc dy RH f udxdy hui wc dy 0 0 0 R 1 CH2 3 R Y 1 R 1 3 17C 2 7 e 0 1e Y e dY dY þ 140kf p e dY 0 ks ¼ Tw : R1 3 e dY 0
T f ¼ R0H R0 wc
Tf
0
where
Y y=H
Here, the total volume averaged temperature of the fluid in the channel T f is defined as
Y
Z
ð24Þ
f
Tf ¼ Tw Z
1 1e
1 1e
q00 W z þ T1: q cf Q_
ð17Þ
hTif ¼ T w C
Tf ¼
ð23Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24lf cf R1 gbHLW 2 q00 p2 0 e3 dY R1 3 RY 1 R1 3 R1 7 17p2 H ðe ð 0 1e ð Y e dYÞdYÞÞdY þ 140k e dY 0 ks LW 0 f HLW : þ R1 3 2 ð 0 e dYÞ
1 R¼ qf cf
ð32Þ
In this study, the thermal resistance is calculated from Eq. (32) by R1 RY R1 R1 numerically integrating 0 ðe3 ð 0 11 e ð Y e3 dYÞdYÞÞdY, 0 e7 dY and R1 3 e 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 those obtained from the results of a two-dimensional direct numerical simulation. The solutions of this simulation are obtained by solving the momentum and energy equations (Eqs. (1) and (2)) by using the control-volume-based finite difference method. Fig. 3 shows the thermal resistances for the heat sinks
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761 Table 1 Geometrical description of typical heat sinks with fins of variable thickness. Schematic of heat sinks (not to scale)
Fig. 3. Thermal resistances for trapezoidal heat sinks (L = 20 cm, W = 40 cm, p = 1 cm, e = 0.4 + 0.2Y, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, gb = 0.0295 m/s2 K).
with fins of variable thickness; the porosities of these heat sinks vary linearly from 0.4 at the base to 0.6 at the top. Fig. 3 shows that the thermal resistances obtained from Eq. (32) match those obtained from the numerical results, within a relative error of 5% when ks/kf > 50 and H/wc > 1. This is because, to simplify the governing equations, the aspect ratio of the channel was assumed to be higher than 1 and the solid conductivity was assumed to be higher than the fluid conductivity. As seen in Fig. 3, the thermal resistance decreases as the solid conductivity increases because the fin efficiency increases as the solid conductivity increases. As a function of the height, the thermal resistance generally has a minimum value because the heat transfer area increases but the fin efficiency decreases as the height becomes larger. Fig. 4 shows the thermal resistances of heat sinks with fins of variable thickness; the porosities of the heat sinks vary as shown in Table 1. Regardless of the fin shape, the thermal resistances obtained from Eq. (32) were in good agreement with those obtained from the numerical results, with a relative error of less than 5%. When the channel width and fin thickness do not vary in the direction normal to the fluid flow, i.e., when wc and ww are constants, as both the conductivity ratio (ks/kf) and aspect ratio (H/ wc) increase, the characteristics of the fluid flow in the channels and the heat transfer from the fins become similar to those in
Fig. 4. Thermal resistances for heat sink with fins of variable thickness (L = 20 cm, W = 40 cm, p = 1 cm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K).
Porosity (e)
Case A
0.3 + 0.4Y
Case B
0.4 + 0.2Y
Case C
0.5
Case D
0.6 0.2Y
Case E
0.7 0.4Y
the case of natural convection between two isoflux plates. Therefore, in addition to the comparison based on the numerical results, it is also possible to compare the present results with the existing experimental data obtained by Wirtz and Stutzman [22]. The overall Nusselt number is defined as
NuO;wc
q00sf
wc : T w;z¼L T 1 kf
ð33Þ
The amount of heat transferred from the base to the fins is equal to that transferred from the fins to the fluid ðq00 WL ¼ 2q00sf HLW=ðwc þ ww ÞÞ in the steady state. Therefore, as
Fig. 5. Nusselt numbers for parallel plates.
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
presented in [23], the Nusselt number for isoflux parallel plates can be calculated by using Eq. (32) as
NuO;wc ¼
ðwc þ ww Þ wc 1 : 2WHL kf R ks !1; H !1;e¼const kf
ð34Þ
flux channels at low Rayleigh numbers are related to the friction factor as follows:
NuO;Dh
wc
q00sf T w;z¼L T 1
Dh ¼ kf
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ra00Dh 16fReDh ðPheated =PÞ
ð35Þ
: 2
Fig. 5 shows the overall Nusselt number for natural convection between isoflux plates. In this figure, the Rayleigh number is defined 2 as Ra00wc gbq2f cf q00sf w5c =lf kf L: As shown in Fig. 5, the results obtained from Eq. (34) helped to predict the correlation of the experimental data obtained by Wirtz and Stutzman [22] well with a relative error of less than 10% for <1000. As the conductivity ratio (ks/kf) increases, the characteristics of the fluid flow in the channels and heat transfer from the fins become similar to those in the case of natural convection through isoflux channels. Therefore, it is also possible to compare the present results with those involving the correlations for rectangular and triangular channels and parallel plates. The Nusselt number for iso-
In Eq. (35) f, Dh, Pheated, P, ReDh , and Ra00Dh ð gbq2f cf q00sf D5h =lf kf LÞ are the friction factor, hydraulic diameter, heated perimeter, total perimeter, Reynolds number, and Rayleigh number, respectively. On the basis of the fact that the amount of heat transferred from the base to the fins is equal to that transferred from the fins to the fluid (q00 pL ¼ q00sf Pheated L) in the steady state, The Nusselt number for isoflux channels can be calculated by using Eq. (32) as
Dh q00 p Dh ¼ T w;z¼L T 1 kf T w;z¼L T 1 P heated kf p Dh 1 ¼ : WLP heated kf R ks !1
NuO;Dh
q00sf
ð36Þ
kf
Fig. 6 shows the overall Nusselt number for rectangular and triangular isoflux channels and for parallel plates. The results obtained from Eq. (35) match well with those obtained from Eq. (36) with a maximum error of 10%. In summary, on the basis of the results given in Figs. 3–6, it can be confirmed that the proposed model is valid when the channel aspect ratio is high (H/wc > 1), the conductivity ratio is high (ks/kf > 50), and the Rayleigh number is low (Ra00 < 1000). To design an optimized heat sink, the porosity and fin pitch for which the thermal resistance is minimized should be determined. In this study, the optimal values of e(y) and p, for which the thermal resistance is minimized for a given height, length, and width were numerically obtained with the gradient descent method. The numerical simulation is carried out as follows: (1) Approximate the porosity e(y) as
eðyÞ Fig. 6. Nusselt numbers for rectangular and triangular channels and parallel plates.
N X ði 1ÞH iH U y : ei U y N N i¼1
Table 2 Comparison of the optimized heat sinks with fins of variable thickness and optimized heat sinks with fins of uniform thickness. Variable thickness fins Constraints Length (L) width (W) Height (H) Input heat flux (q00 ) Solid Fluid Results Fin pitch (mm) Fin number Porosity Surface area (m2) f Re Rcap (°C/W) Rconv (°C/W) R (°C/W) Schematic of channel (not to scale)
Uniform thickness fins
40 cm 40 cm 50 cm 5 W/cm2 Aluminum (ks = 175 W/m K) Air (kf = 0.028 W/m K) 12.2 32 0.592 0.363Y + 0.144Y2 31.9 22.7 0.0097 0.0132 0.0229
13.5 29 0.458 28.9 24.0 0.0091 0.0153 0.0244
ð37Þ
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761 0.040
Variable-thickness-fin heat sink Uniform-thickness-fin heat sink
0.038
Thermal resistance R (K/W)
0.036 0.034 0.032 0.030 0.028 0.026 0.024 0.022 0.020 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
H/L
(a) Total thermal resistances 0.024
0.020
0.020
0.018
0.018
0.016
0.016
0.014
0.014
0.012
0.012
0.010
0.010
0.008
0.008
conv
(K/W)
Rcap
Convective thermal resistance R
Rconv
0.024 0.022
Variable-thickness-fin heat sink Uniform-thickness-fin heat sink
0.022
0.006
0.006 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Capacitive thermal resistance Rcap (K/W)
0.026
0.026
1.4
H/L
(b) Convective and capacitive thermal resistances Fig. 7. Thermal resistances of the optimized heat sinks (W = L = 40 cm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/ m K, gb = 0.0295 m/s2 K).
Here, U is the unit step function, which satisfies the following equation:
UðyÞ ¼
0; y < 0 : 1; ; y P 0
ð38Þ
(2) Start with the guessed discretized porosity ei and fin pitch p⁄. (3) Calculate the partial derivatives of R with respect to ei and p by using Eq. (32):
@R Rðei ¼ ei þ cÞ Rðei ¼ ei cÞ ’ ; @ ei 2c @R Rðp ¼ p þ cÞ Rðp ¼ p cÞ : ’ @p 2c
ð39Þ ð40Þ
Here, c is a small positive number. (4) Calculate ei and p from the following equations:
ei ¼ ei c
@R @R ; p ¼ p c : @ ei @p
ð41Þ
(5) Treat ei and p as newly guessed ei and p⁄, and return to step 2. Repeat step 2–5 until the converged values for ei and p are obtained. Subsequently, obtain the polynomial equation for the porosity e(y) from ei by least-square fitting. Table 2 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 fins of variable thickness for comparison with the corresponding values for the optimized heat sink with fins of uniform thickness under the same constraints. As shown in Table 2, the friction factor is lower for the heat sink with fins of variable thickness. As a result, a smaller fin pitch and a larger number of fins can be selected for this type of heat sink without increasing the capacitive thermal resistance. When the number of fins increases, the convective resistance decreases because the convective resistance generally decreases as the fin surface area, which is proportional to the number of fins, increases. Consequently, the convective resistance of the heat sink with fins of variable thickness can be lower than that of the heat sink with fins of uniform thickness, while the capacitive resistances of these types of heat sinks are similar. Finally, the total resistance of the optimized heat sink with fins of variable thickness is lower than that of the optimized heat sink with fins of uniform thickness. For the case presented in Table 2, the total thermal resistance decreases by about 6% after the fin thickness is varied in the direction normal to the fluid flow. Fig. 7 shows the thermal resistances of the heat sinks with fins of variable thickness and heat sinks with fins of uniform thickness for various heights. Generally, the total thermal resistances of optimized heat sinks with fins of variable thickness are lower than those of optimized heat sinks of uniform thickness, and the difference between the total thermal resistances increases as the height increases. This is because the convective thermal resistance becomes dominant over the capacitive thermal resistance as the height increases and the heat sinks with fins of variable thickness can reduce the convective thermal resistance effectively without compromising the capacitive thermal resistance, as previously mentioned. Unfortunately, for low heights (H/L < 0.5), the thermal resistances of optimized heat sinks with fins of variable thickness are only slightly lower (<1%) than those of optimized heat sinks with fins of uniform thickness. The optimized geometries and thermal resistances of the heat sinks with fins of variable thickness and heat sinks with fins of uniform thickness for various heat fluxes and various heights are listed in Table 3. By comparing the thermal resistances of the two types of optimized heat sinks, a contour map was drawn, as shown in Fig. 8. Fig. 8 shows the ratio of the optimal thermal resistances of the heat sinks with fins of variable thickness to those of heat sinks with fins of uniform thickness (Ropt,var/Ropt,uni) for aircooled systems. In Fig. 8, in the region where the ratio is less than 1, the optimized heat sink with fins of variable thickness performs better than the optimized heat sink with fins of uniform thickness. The opposite is true when the ratio is greater than 1. Therefore, the contour map indicates that the optimized heat sinks with fins of variable thickness have lower thermal resistances than the optimized heat sinks with fins of uniform thickness. The thermal resistance is reduced by up to 10% by employing fins of variable thickness. However, the difference between the thermal resistances decreases as the height decreases and as the heat flux decreases. As shown in Fig. 8, the thermal resistance is reduced by less than 1% when
!! gbq2f cf q00 L4 H < 18:9 2:67 log10 L lf k2f !!2 gbq2f cf q00 L4 þ 0:0962 log10 : lf k2f
ð42Þ
When Eq. (31) is satisfied, the heat sink with fins of variable thickness is almost not superior to the simple heat sink with fins of uniform thickness. In Table 4, the calculated dimensionless heat fluxes for various lengths and heat fluxes are listed. As shown in this table, the magnitude of the dimensionless flux shown in Fig. 8 can be
D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
759
Table 3 Comparison of geometries and thermal resistances of optimized heat sinks Dimensionless heat flux
Dimensionless height
gbq2f cf q00 L4
H L
lf k2f
5.45 1010
5.45 1011
5.45 1012
5.45 1010
5.45 1011
5.45 1012
1.5
1.5
1.5
1.0
1.0
1.0
Type of fins
Dimensionless fin pitch
Porosity
Thermal resistance (°C/W)
p L
e
R
Uniform thickness
0.0538
0.59
0.0545
Variable thickness
0.0533
0.655 0.192Y + 0.0770Y2
0.0538
Uniform thickness
0.0433
0.488
0.0334
Variable thickness
0.0403
0.604 0.321Y + 0.130Y2
0.0320
Uniform thickness
0.0358
0.385
0.0221
Variable thickness
0.0282
0.577 0.479Y + 0.178Y2
0.0197
Uniform thickness
0.0454
0.674
0.0592
Variable thickness
0.0454
0.7020.0866Y + 0.0315Y2
0.0591
Uniform thickness
0.0347
0.581
0.0342
Variable thickness
0.0344
0.649 0.207Y + 0.0836Y2
0.0336
Uniform thickness
0.0282
0.475
0.0211
Variable thickness
0.0258
0.600 0.339Y + 0.136Y2
0.0201
W = L = 40 cm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K.
Schematic of channel (not to scale)
760
D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
on the volume averaging theory (VAT) was used for this optimization. The thermal resistance was reduced by up to 10% compared to that of the heat sinks with fins of uniform thickness by employing fins of variable thickness in the case of an air-cooled heat sink. However, the amount of the reduction decreases as the heat flux decreases or as the height of the heat sink decreases. Owing to its better thermal performance, the heat sink with fins of variable thickness is expected to be a suitable next-generation cooling solution. In addition, the heat sink presented in this study can be used for cooling the heat dissipated during irreversible energy conversion processes in power generators, such as nuclear, hydro-electric, and reverse electrodialytic power generators [26]. Acknowledgement This research was supported by Nano R&D program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number:20110030285). Appendix A. Validity of assumptions made in the present study Fig. 8. Contour plots of Ropt,var/Ropt,uni (W/L = 1, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K).
Table 4 Dimensionless heat fluxes for various lengths and heat fluxes. L 0.2 0.3 0.4 0.4 0.6 0.8 0.8 1.2 1.6
q00 100 100 100 10 10 10 1 1 1
log10
gbq2f cf q00 L4
lf k2f
10.98 11.68 12.18 11.18 11.88 12.39 11.38 12.09 12.59
Two major assumptions are made to obtain Eqs. (1) and (2) from the Navier–Stokes equation and the energy equation. First, the body force per unit volume is assumed to be qf 1 gð1 bðT f T 1 ÞÞ, where qf1 is the fluid density at T = T1. Second, the flow is assumed to be both hydrodynamically and thermally fully developed. To check the validity of these assumptions, a numerical solution for natural convection between two isoflux plates (shown in Fig. A1) was obtained by using the energy and continuity equations along with two different momentum equations as follows: _
2u Þu pl r f þ r f ¼ qf g k ; qf ðuf r f
ðA1Þ
W/L = 1, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K.
achieved when the length of the heat sink is greater than 20 cm. Therefore, in some heat sinks used for cooling rooms with computer and telecommunications equipment [24], we can expect enhancement in the thermal performance upon employing fins of variable thickness. However, for compact coolers that are used in small electronic equipment and have the length less than 20 cm, we cannot expect a reduction in resistance. The present study is based on the assumption that the heat sink is fully covered. If the cover plate is removed, there would be significant inflow from the open front edge of the interfin channels, as observed in [25]. In this case, the heat sinks with fins of variable thickness will not perform better than those with fins of uniform thickness. This is because the magnitude of the inflow for the heat sinks with fins of variable thickness would be lower than that for the heat sinks with fins of uniform thickness, since the interfin distance at the front edge, i.e., the channel width at y = H, is smaller for the former type of heat sink, as shown in Table 2. 4. Conclusion In the present study, the thermal optimization of a plate-fin heat sink was conducted for the case in which the fin thickness varied in the direction normal to the fluid flow. A model based
Fig. A1. Schematic diagram for natural convection between two isoflux plates
D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
maximum wall temperatures calculated using Eqs. (A1) and (A2). Fig. A2 shows that the results obtained from Eq. (A2) are in good agreement with those obtained from Eq. (A1) within a relative error of 10%; thus, it can be observed that the first assumption is reasonable. To check the validity of the second assumption, the velocity and temperature profiles in the entrance region are presented in Fig. A3. As shown in Fig. A3, a hydrodynamically and thermally fully developed condition is achieved when the distance from the entrance, x, is greater than 10wc Ra00wc . Therefore, the second assumption is valid when L > 10wc Ra00wc . This region corresponds to the fully developed limit presented in [17].
400
T(x=L,z=0) (K)
350
300
Eq. (A1) Eq. (A2)
References
250 0
20
40
60
80
100
q''sf (W) Fig. A2. Schematic diagram for natural convection between two isoflux plates (L = 20 cm, wc = 5 mm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, kf = 0.028 W/m K).
0.20
Velocity (m/s)
0.15
0.10
x= 0.52mm (= wc Raw '') x= 1.55mm (= 3wc Raw '') x= 2.58mm (= 5wc Raw '') x= 51.7mm (= 10wc Raw '') x= 103.3mm (= 20wc Raw '') c
c
c
0.05
c
c
0.00 0.000
0.001
0.002
0.003
0.004
z (m) (a) Velocity profiles
0.005
310
x= 0.52mm (= wc Raw '') x= 1.55mm (= 3wc Raw '') x= 2.58mm (= 5wc Raw '') x= 51.7mm (= 10wc Raw '') x= 103.3mm (= 20wc Raw '') c
308
c
c
Temperature (K)
761
306
c
c
304
302
300 0.000
0.001
0.002
0.003
0.004
0.005
z (m)
(b) Temperature profiles Fig. A3. Velocity and temperature profiles in the entrance region (L = 20 cm, wc = 5 mm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, kf = 0.028 W/m K).
_
2u Þu pl r f þ r f ¼ qf 1 gð1 bðT f T 1 ÞÞ k : qf ðu f r f
ðA2Þ
Eq. (A1) is the original Navier–Stokes equation. In the Eq. (A2), the first assumption is applied to simplify Eq. (A1). Fig. A2 shows the
[1] S. Oktay, R.J. Hannemann, A. Bar-Cohen, High heat from a small package, Mech. Eng. 108 (1986) 36–42. [2] Avram Bar-Cohen, Thermal management of electronic components with dielectric liquids, JSME Int J B 36 (1993) 1–25. [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] J.R. Welling, C.B. Wooldridge, Free convection heat transfer coefficients from rectangular vertical fins, J. Heat Transfer 87 (1965) 439–444. [6] A.D. Kraus, A. Bar-Cohen, Design and analysis of heat sinks, John Wiley & Sons Inc., New York, 1995. [7] W. Elenbaas, Heat dissipation of parallel plates by free convection, Physica 9 (1942) 1–28. [8] K.E. Starner, H.N. McManus Jr., An experimental investigation of free convection heat transfer from rectangular fin array, J. Heat Transfer 85 (1963) 273–278. [9] D.W. van de Pol, J.K. Tiemey, Free convection Nusselt number for vertical Ushaped channels, J. Heat Transfer 95 (1973) 542–543. [10] A. Bilitsky, The Effect of Geometry on Heat Transfer by Free Convection from a Fin Array, MS Thesis, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel (1986). [11] J.R. Culham, T.F. Lemcyzk, S. Lee, M.M. Yovanovich, META-a conjugate heat transfer model for air cooling of circuit boards with arbitrarily located heat sources, Heat Transfer Electron. Equip. 171 (1991) 117–126. [12] J.R. Culham, M.M. Yovanovich, S. Lee, Thermal modeling of isothermal cuboids and rectangular heat sinks cooled by natural convection, IEEE Trans. Compon. Packag. Manuf. Technol. A 18 (1995) 559–566. [13] S. Narasimhan, J. Majdalani, Characterization of compact heat sink models in natural convection, IEEE Trans. Compon. Packag. Technol. 25 (2002) 78–86. [14] F.J. Higuera, Yu S. Ryazantsev, Natural convection flow due to a heat source in a vertical channel, Int. J. Heat Mass Transfer 45 (2002) 2207–2212. [15] A.K. da Silva, S. Lorente, A. Bejan, Maximal heat transfer density in vertical morphing channels with natural convection, Numer. Heat Transfer A 45 (2004) 135–152. [16] J.M. Floryan, M. Novak, Free convection heat transfer in multiple vertical channels, Int. J. Heat Fluid Flow 16 (1995) 244–253. [17] A. Bar-Cohen, W. Rohsenow, Thermally optimum spacing of vertical natural convection cooled parallel plates, J. Heat Transfer 106 (1984) 116–122. [18] A. Bejan, Optimal internal structure of volumes cooled by single-phase forced and natural convection, J. Electron. Packag. 125 (2003) 200–207. [19] A. Bar-Cohen, M. Jelinek, Optimum arrays of longitudinal, rectangular fins in corrective heat transfer, Heat Transfer Eng. 6 (1985) 68–78. [20] D.-K. Kim, J. Jung, S. Kim, Thermal optimization of plate-fin heat sinks with variable fin thickness, Int. J. Heat Mass Transfer 53 (2010) 5988–5995. [21] D.-K. Kim, S.J. Kim, Closed form correlations for thermal optimization of microchannels, Int. J. Heat Mass Transfer 50 (2007) 5318–5322. [22] R.A. Wirtz, R.J. Stutzman, Experiments on free convection between vertical plates with symmetric heating, J. Heat Transfer 104 (1982) 501–507. [23] T.H. Kim, K.H. Do, D.-K. Kim, Closed form correlations for thermal optimization of plate-fin heat sinks under natural convection, Int. J. Heat Mass Transfer 54 (2011) 1210–1216. [24] R. Alexandersson, How to create good qualities by means of passive cooling technique, in: International Telecommunications Energy Conference, INTELEC ‘84, November 4–7, New Orleans, Louisiana, 1984, pp. 528–531. [25] T. Aihara, S. Maruyama, Free convective/radiative heat transfer from pin-fin arrays with a vertical base plate, Int. J. Heat Mass Transfer 33 (1990) 1223– 1232. [26] D.-K. Kim, C. Duan, Y.-F. Chen, A. Majumdar, Power generation from concentration gradient by reverse electrodialysis in ion-selective nanochannels, Microfluid. Nanofluid. 9 (2010) 1215–1224.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermal optimization of plate-fin heat sinks with fins of variable thickness under natural convection 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 19 October 2010 Received in revised form 4 October 2011 Available online 8 November 2011 Keywords: Heat sink Thermal optimization Variable fin thickness Natural convection
a b s t r a c t In this study, thermal performance of a vertical plate-fin heat sink under natural convection was optimized for the case in which the fin thickness varied in the direction normal to the fluid flow. For this optimization, the averaging approach presented in an earlier paper for the case of the heat sinks under forced convection was extended to study the performance of heat sinks under natural convection. In the case of an air-cooled heat sink, the thermal resistance decreases by up to 10% when the fin thickness is allowed to increase in the direction normal to the fluid flow. However, the difference between the thermal resistances of heat sinks with uniform thickness and the heat sinks with variable thickness decreases as the height decreases and as the heat flux decreases. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Because of the trend toward the production of denser and more powerful products in the electronic equipment industry, a higher level of performance of cooling technology is required [1,2]. Many cooling methods have been proposed for ensuring reliable operation of electronic components [3,4]. Owing to their inherent simplicity, reliability, and low long-term cost, natural convection heat sinks have been effectively used in cooling electronic components [5]. Studies on the design of plate fin heat sinks based on natural convection have been extensively carried out, as summarized in an extensive review by Kraus and Bar-Cohen [6]. Elenbaas [7] was the first to document a detailed study of the heat transfer from vertical plate fin heat sinks. Starner and McManus [8] as well as Welling and Wooldridge [5] have studied the heat transfer from heat sinks experimentally. van de Pol and Tiemey [9] developed a correlation between Nusselt number and Elenbass number by using the experimental results of Welling and Woodridge. Bilitsky [10] performed a comprehensive investigation of natural convection heat transfer from several heat sinks that differed in fin height and fin width. Moreover, numerical studies have also been performed by Culham et al. [11,12], Narasimhan and Majdalani [13], and others [14–16]. Several studies that focused on sizing-optimization of plate-fin heat sinks under natural convection have been carried out. Elenbass [7] was the first to suggest a correlation for the optimum ⇑ Tel.: +82 31 219 3660; fax: +82 31 219 1611. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.10.034
channel width of an isothermal fin array; Elenbass found that the Nusselt number for the optimum fin array is 1.25. Bar-Cohen and Rohsenow [17] extended the Elenbass correlation to isoflux plates. Bejan suggested a correlation for calculating the optimal channel width from asymptotic solutions [18]. Later, Bar-Cohen and Jelinek [19] suggested correlations for determining the optimum channel width and fin thickness of the optimally spaced least-material array. However, these studies were focused on the heat sinks with a uniform fin thickness. Recently, my colleagues and I performed an investigation to determine which of the three types of heat sinks shown in Fig. 1 can achieve the best performance under forced convection [20]. We showed that the thermal resistance of a water-cooled platefin heat sink can be decreases by as much as 15% by increasing the fin thickness in the direction normal to the fluid flow. However, to the best of my knowledge, no study on plate-fin heat sinks with a variable fin thickness under natural convection has been carried out. It may be possible to reduce the thermal resistance of natural convection heat sinks by increasing the fin thickness in the direction normal to the fluid flow, as shown in the case of forced convection heat sinks. However, in the case of natural convection, the degree of improvement in the performance of heat sink with fins of variable thickness compared to the performance of heat sinks with fins of uniform thickness and the conditions under which this improvement is achieved are not clear. This is because (1) the flow rate of a buoyancy-induced flow is generally one or two orders of magnitude lower than that of a pressure-driven flow and (2) the flow rate of a buoyancy-induced flow is determined by the fluid temperature in the channel, while the flow rate of a pressure-driven flow is independent of the fluid temperature.
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
Nomenclature a c g Dh f h H k kse K L NuO p P Pheated q q00 q00sf Q_ R Rcap Rconv Ra00 Re T Tf Tf
Wetted area per volume (m1) Heat capacity of a fluid (J kg1 K1) Gravity acceleration (m s2) Hydraulic diameter of the channel Friction factor Interstitial heat transfer coefficient (W m2 K1) Height of the heat sink (m) Thermal conductivity (W m1 K1] Effective thermal conductivity of solid (W m1 K1) Permeability (m2) Length of the heat sink (m) Overall Nusselt number Fin pitch (wc + ww) (m) Perimeter of the channel (m) Heated perimeter of the channel (m) Heat transfer rate (m) Heat flux (W/m2) Heat flux from the fins to the fluid (W/m2) Flow rate (m3/s) Total thermal resistance (W/K) Capacitive thermal resistance (W/K) Convective thermal resistance (W/K) Modified channel Rayleigh number Reynolds number Temperature (K) Fluid bulk mean temperature (K) Total averaged temperature of the fluid in the channel (K)
In this study, thermal performance of a vertical plate-fin heat sink under natural convection was optimized for the case in which the fin thickness varied in the direction normal to the fluid flow. For this purpose, the model based on the volume averaging theory (VAT) presented in an earlier study [20] was extended to study the performance of heat sinks under natural convection.
2. Mathematical formulation The problem under consideration in this paper is related to natural convection through channels as shown in Fig. 2(a). The cover plate is insulated and the heat sink base is uniformly heated. A coolant passes through several channels by natural convection and carries the heat away from the heat sink base. In the analysis of the problem, for simplicity, the flow is assumed to be laminar and to be both hydrodynamically and thermally fully developed; moreover, the flow is assumed to satisfy the Boussinesq approximation. All thermophysical properties are assumed to be constant. In addition, it is assumed that the aspect ratio of the channel is much higher than 1 and that the solid conductivity is higher than the fluid conductivity. The channel width and fin thickness vary in the direction normal to the fluid flow (wc = wc(y), ww = ww(y)), but the fin pitch p, which is equal to the sum of the channel width and fin thickness, is constant. The momentum and energy equations are given as follows: For the fluid phase,
! @ u @ u ¼ qf gbðT f T 1 Þ; þ @x2 @y2 2
lf
2
! @uT @2T @2T : ¼ kf qf cf þ @z @x2 @y2
ð1Þ
ð2Þ
u W wc ww x, y, z X, Y, Z
Velocity (m/s) Width of the heat sink (m) Channel width (m) Fin thickness (m) Cartesian coordinate system Dimensionless Cartesian coordinate system
Greek symbols b Thermal expansion coefficient ( (oq/oT)p/q) (K1) e Porosity (wc/(ww + wc)) l Viscosity (Pa s) q Density (kg m3) Subscripts and superscripts uni Heat sink with fins of uniform thickness f Fluid opt Optimized s Solid w Wall var Heat sink with fins of variable thickness 1 Ambient Special symbols < >f Averaged value for the fluid phase < >s Averaged value for the solid phase
For the solid phase,
0 ¼ ks
! @2T @2T : þ @x2 @y2
ð3Þ
T f is the total volume averaged temperature of the fluid in the channel [17]. The validity of Eqs. (1) and (2) is discussed in Appendix A. In the model based on the VAT, the governing equations for the averaged velocity and temperature are obtained by averaging the momentum and energy equations in the x-direction over the analysis domain shown in Fig. 2(b).
qf gbðT f T 1 Þ ¼
lf K
f
ehui l
! @ 2 huif ; @y2
! @hTif @ @hTif s f ; kfe eqf cf hui ¼ haðhTi hTi Þ þ @y @z @y f
@ @hTis ¼ ha hTis hTif ; kse @y @y
ð4Þ
ð5Þ
ð6Þ
where the average fluid velocity, average fluid temperature, and average solid temperature are defined as shown below:
2 hui ¼ wc f
hTis ¼
2 ww
Z
R ðww þwc Þ=2
ðww þwc Þ=2
udx;
ww =2
Z
Tudx ww =2 ; hTi ¼ R ðw w þwc Þ=2 udx ww =2 f
ww =2
Tdx:
ð7Þ
0
In Eq. (7), for the average fluid temperature, the one-dimensional bulk mean is employed to calculate the bulk mean temperature of the fluid easily. The quantities e, a, kfe, kse, K, and h are the porosity, wetted area per unit volume, effective thermal conductivity of the fluid, effective thermal conductivity of the solid, permeability, and
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
Fig. 2. Heat sink with fins of variable thickness.
e¼
wc ; p
K
e wc huif @u
a¼
2 ; p
kfe ¼ kf e; !1
@x
2
¼ x¼ww =2
kse ¼ ks ð1 eÞ;
e3 p2 12
;
h
q00sf s
hTi hTif
¼
70kf : 17ep ð8Þ
Here, e, a, kfe, kse, K, and h are functions of y, because they are functions of wc and ww. Eqs. (4) and (5) can be simplified when the solid conductivity is higher than the fluid conductivity (ks > kf) and the aspect ratio of the channel is much higher than 1(H > wc). Upon estimating the order of magnitude of each term appearing on the right-hand side of Eq. (4), we have
lf w2c
huif >
lf H2
huif :
ð9Þ
By combining Eqs. (5) and (6), we obtain Fig. 1. Plate-fin heat sinks.
interstitial heat transfer coefficient, respectively. These quantities can be represented as
eqf cf huif
@hTif @ @hTis kse ¼ @y @z @y @ @ @ kse @hTis þ kfe hTis : @y @y @y ha @y
ð10Þ
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
Upon estimating the order of magnitude of each term appearing on the right-hand side of Eq. (10), it follows that
ks H
hTis > 2
kf
ks w2c
H
H4
hTis ; 2
hTis :
ð11Þ
By using the relations presented in Eqs. (9) and (11), Eqs. (4) and (5) can be simplified as
lf
qf gbðT f T 1 Þ ¼
eqf cf huif
ehuif ;
K
ð12Þ
@hTif ¼ ha hTis hTif : @z
ð13Þ
The appropriate boundary conditions are similar to those presented in [21], and they are given as follows:
huif ¼ 0 at y ¼ 0; H;
ð14Þ
hTis ¼ hTif ¼ T w
ð15Þ
at y ¼ 0;
@hTis @hTif ¼ ¼ 0 at y ¼ H: @y @y
ð16Þ
The velocity and temperature distributions can be obtained by solving Eqs. (6), (12), and (13):
qf gb 2 2 hui ¼ p e ðT f T 1 Þ; 12lf f
Z
2
hTis ¼ T w C
H ks
Y
0
H2 ks
Z
Y
1
e3 dY dY ;
0
and C ¼
ð18Þ
1
e3 dY dY þ
Y
! 17 2 4 p e ; 140kf
ð19Þ
q2f cf gb @hTif 2 ðT f T 1 Þ p : @z 12lf
ð20Þ
The total flow rate Q_ and the fluid bulk mean temperature Tf can be obtained by integrating the average fluid velocity and average temperature, respectively:
W Q_ ¼ ðwc þ ww Þ ¼
Z
H
0
Z
ww þwc
udxdy ¼ W ww
qf gbHW 2 p ðT f T 1 Þ 12lf R H R wc
Z
Z
H
ehuif dy
Tudxdy
1
e3 dY;
ð21Þ
0
RH
¼
0
ð22Þ
The total flow rate and fluid bulk mean temperature are functions of ðT f T 1 Þ and ohTif/oz. To complete the solutions, the values of ðT f T 1 Þ and @hTif =@z should be determined. For a fully developed flow, the fluid bulk mean temperature varies linearly in the zdirection, and the longitudinal temperature gradient satisfies the following equation:
qf cf Q_
dT f ¼ q00 W: dz
Therefore,
1 L
Z
L
T f dz:
ð25Þ
0
From Eqs. (21), (24), and (25), T f T 1 is given as
Tf T1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 6lf Lq00 : ¼t R q2f cf gbHp2 01 e3 dY
ð26Þ
From Eqs. (21), (23), and (26), @hTif =@z is given as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 24lf q00 u @hTif dT f q00 W ¼t : ¼ ¼ R @z dz qf cf Q_ q2f cf gbHLp2 01 e3 dY
ð27Þ
Finally, the total flow rate Q_ and the fluid bulk mean temperature Tf are given by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u R ugbHLW 2 q00 p2 1 e3 dY 0 Q_ ¼ t 24lf cf
ð28Þ
and
R 1 q00 H 0
ks
e3
R
Y 1 0 1e
R
1 Y
00
17q e3 dY dY þ 140k p2 e7 dY fH
R 1
e3 dY
2
;
ð29Þ
respectively. The thermal performance of the heat sink is evaluated by using the concept of thermal resistance. In this study, thermal resistance is defined as the difference between the base temperature of the heat sink at the outlet and the fluid bulk mean temperature at the inlet per unit of heat flow rate. The thermal resistance consists of the capacitive resistance, which is responsible for the temperature rise of the coolant from the inlet to the outlet, and the convective resistance, which is related to the heat transfer from the fins to the coolant:
R ¼ Rcap þ Rconv ; Rcap ¼
1
qf cf Q_
;
ð30Þ Rconv ¼
Tw Tf : q00 LW
ð31Þ
From Eqs. (28)–(31), the thermal resistance is given as
0
hTif huif wc dy RH f udxdy hui wc dy 0 0 0 R 1 CH2 3 R Y 1 R 1 3 17C 2 7 e 0 1e Y e dY dY þ 140kf p e dY 0 ks ¼ Tw : R1 3 e dY 0
T f ¼ R0H R0 wc
Tf
0
where
Y y=H
Here, the total volume averaged temperature of the fluid in the channel T f is defined as
Y
Z
ð24Þ
f
Tf ¼ Tw Z
1 1e
1 1e
q00 W z þ T1: q cf Q_
ð17Þ
hTif ¼ T w C
Tf ¼
ð23Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24lf cf R1 gbHLW 2 q00 p2 0 e3 dY R1 3 RY 1 R1 3 R1 7 17p2 H ðe ð 0 1e ð Y e dYÞdYÞÞdY þ 140k e dY 0 ks LW 0 f HLW : þ R1 3 2 ð 0 e dYÞ
1 R¼ qf cf
ð32Þ
In this study, the thermal resistance is calculated from Eq. (32) by R1 RY R1 R1 numerically integrating 0 ðe3 ð 0 11 e ð Y e3 dYÞdYÞÞdY, 0 e7 dY and R1 3 e 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 those obtained from the results of a two-dimensional direct numerical simulation. The solutions of this simulation are obtained by solving the momentum and energy equations (Eqs. (1) and (2)) by using the control-volume-based finite difference method. Fig. 3 shows the thermal resistances for the heat sinks
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761 Table 1 Geometrical description of typical heat sinks with fins of variable thickness. Schematic of heat sinks (not to scale)
Fig. 3. Thermal resistances for trapezoidal heat sinks (L = 20 cm, W = 40 cm, p = 1 cm, e = 0.4 + 0.2Y, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, gb = 0.0295 m/s2 K).
with fins of variable thickness; the porosities of these heat sinks vary linearly from 0.4 at the base to 0.6 at the top. Fig. 3 shows that the thermal resistances obtained from Eq. (32) match those obtained from the numerical results, within a relative error of 5% when ks/kf > 50 and H/wc > 1. This is because, to simplify the governing equations, the aspect ratio of the channel was assumed to be higher than 1 and the solid conductivity was assumed to be higher than the fluid conductivity. As seen in Fig. 3, the thermal resistance decreases as the solid conductivity increases because the fin efficiency increases as the solid conductivity increases. As a function of the height, the thermal resistance generally has a minimum value because the heat transfer area increases but the fin efficiency decreases as the height becomes larger. Fig. 4 shows the thermal resistances of heat sinks with fins of variable thickness; the porosities of the heat sinks vary as shown in Table 1. Regardless of the fin shape, the thermal resistances obtained from Eq. (32) were in good agreement with those obtained from the numerical results, with a relative error of less than 5%. When the channel width and fin thickness do not vary in the direction normal to the fluid flow, i.e., when wc and ww are constants, as both the conductivity ratio (ks/kf) and aspect ratio (H/ wc) increase, the characteristics of the fluid flow in the channels and the heat transfer from the fins become similar to those in
Fig. 4. Thermal resistances for heat sink with fins of variable thickness (L = 20 cm, W = 40 cm, p = 1 cm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K).
Porosity (e)
Case A
0.3 + 0.4Y
Case B
0.4 + 0.2Y
Case C
0.5
Case D
0.6 0.2Y
Case E
0.7 0.4Y
the case of natural convection between two isoflux plates. Therefore, in addition to the comparison based on the numerical results, it is also possible to compare the present results with the existing experimental data obtained by Wirtz and Stutzman [22]. The overall Nusselt number is defined as
NuO;wc
q00sf
wc : T w;z¼L T 1 kf
ð33Þ
The amount of heat transferred from the base to the fins is equal to that transferred from the fins to the fluid ðq00 WL ¼ 2q00sf HLW=ðwc þ ww ÞÞ in the steady state. Therefore, as
Fig. 5. Nusselt numbers for parallel plates.
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
presented in [23], the Nusselt number for isoflux parallel plates can be calculated by using Eq. (32) as
NuO;wc ¼
ðwc þ ww Þ wc 1 : 2WHL kf R ks !1; H !1;e¼const kf
ð34Þ
flux channels at low Rayleigh numbers are related to the friction factor as follows:
NuO;Dh
wc
q00sf T w;z¼L T 1
Dh ¼ kf
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ra00Dh 16fReDh ðPheated =PÞ
ð35Þ
: 2
Fig. 5 shows the overall Nusselt number for natural convection between isoflux plates. In this figure, the Rayleigh number is defined 2 as Ra00wc gbq2f cf q00sf w5c =lf kf L: As shown in Fig. 5, the results obtained from Eq. (34) helped to predict the correlation of the experimental data obtained by Wirtz and Stutzman [22] well with a relative error of less than 10% for <1000. As the conductivity ratio (ks/kf) increases, the characteristics of the fluid flow in the channels and heat transfer from the fins become similar to those in the case of natural convection through isoflux channels. Therefore, it is also possible to compare the present results with those involving the correlations for rectangular and triangular channels and parallel plates. The Nusselt number for iso-
In Eq. (35) f, Dh, Pheated, P, ReDh , and Ra00Dh ð gbq2f cf q00sf D5h =lf kf LÞ are the friction factor, hydraulic diameter, heated perimeter, total perimeter, Reynolds number, and Rayleigh number, respectively. On the basis of the fact that the amount of heat transferred from the base to the fins is equal to that transferred from the fins to the fluid (q00 pL ¼ q00sf Pheated L) in the steady state, The Nusselt number for isoflux channels can be calculated by using Eq. (32) as
Dh q00 p Dh ¼ T w;z¼L T 1 kf T w;z¼L T 1 P heated kf p Dh 1 ¼ : WLP heated kf R ks !1
NuO;Dh
q00sf
ð36Þ
kf
Fig. 6 shows the overall Nusselt number for rectangular and triangular isoflux channels and for parallel plates. The results obtained from Eq. (35) match well with those obtained from Eq. (36) with a maximum error of 10%. In summary, on the basis of the results given in Figs. 3–6, it can be confirmed that the proposed model is valid when the channel aspect ratio is high (H/wc > 1), the conductivity ratio is high (ks/kf > 50), and the Rayleigh number is low (Ra00 < 1000). To design an optimized heat sink, the porosity and fin pitch for which the thermal resistance is minimized should be determined. In this study, the optimal values of e(y) and p, for which the thermal resistance is minimized for a given height, length, and width were numerically obtained with the gradient descent method. The numerical simulation is carried out as follows: (1) Approximate the porosity e(y) as
eðyÞ Fig. 6. Nusselt numbers for rectangular and triangular channels and parallel plates.
N X ði 1ÞH iH U y : ei U y N N i¼1
Table 2 Comparison of the optimized heat sinks with fins of variable thickness and optimized heat sinks with fins of uniform thickness. Variable thickness fins Constraints Length (L) width (W) Height (H) Input heat flux (q00 ) Solid Fluid Results Fin pitch (mm) Fin number Porosity Surface area (m2) f Re Rcap (°C/W) Rconv (°C/W) R (°C/W) Schematic of channel (not to scale)
Uniform thickness fins
40 cm 40 cm 50 cm 5 W/cm2 Aluminum (ks = 175 W/m K) Air (kf = 0.028 W/m K) 12.2 32 0.592 0.363Y + 0.144Y2 31.9 22.7 0.0097 0.0132 0.0229
13.5 29 0.458 28.9 24.0 0.0091 0.0153 0.0244
ð37Þ
758
D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761 0.040
Variable-thickness-fin heat sink Uniform-thickness-fin heat sink
0.038
Thermal resistance R (K/W)
0.036 0.034 0.032 0.030 0.028 0.026 0.024 0.022 0.020 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
H/L
(a) Total thermal resistances 0.024
0.020
0.020
0.018
0.018
0.016
0.016
0.014
0.014
0.012
0.012
0.010
0.010
0.008
0.008
conv
(K/W)
Rcap
Convective thermal resistance R
Rconv
0.024 0.022
Variable-thickness-fin heat sink Uniform-thickness-fin heat sink
0.022
0.006
0.006 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Capacitive thermal resistance Rcap (K/W)
0.026
0.026
1.4
H/L
(b) Convective and capacitive thermal resistances Fig. 7. Thermal resistances of the optimized heat sinks (W = L = 40 cm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/ m K, gb = 0.0295 m/s2 K).
Here, U is the unit step function, which satisfies the following equation:
UðyÞ ¼
0; y < 0 : 1; ; y P 0
ð38Þ
(2) Start with the guessed discretized porosity ei and fin pitch p⁄. (3) Calculate the partial derivatives of R with respect to ei and p by using Eq. (32):
@R Rðei ¼ ei þ cÞ Rðei ¼ ei cÞ ’ ; @ ei 2c @R Rðp ¼ p þ cÞ Rðp ¼ p cÞ : ’ @p 2c
ð39Þ ð40Þ
Here, c is a small positive number. (4) Calculate ei and p from the following equations:
ei ¼ ei c
@R @R ; p ¼ p c : @ ei @p
ð41Þ
(5) Treat ei and p as newly guessed ei and p⁄, and return to step 2. Repeat step 2–5 until the converged values for ei and p are obtained. Subsequently, obtain the polynomial equation for the porosity e(y) from ei by least-square fitting. Table 2 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 fins of variable thickness for comparison with the corresponding values for the optimized heat sink with fins of uniform thickness under the same constraints. As shown in Table 2, the friction factor is lower for the heat sink with fins of variable thickness. As a result, a smaller fin pitch and a larger number of fins can be selected for this type of heat sink without increasing the capacitive thermal resistance. When the number of fins increases, the convective resistance decreases because the convective resistance generally decreases as the fin surface area, which is proportional to the number of fins, increases. Consequently, the convective resistance of the heat sink with fins of variable thickness can be lower than that of the heat sink with fins of uniform thickness, while the capacitive resistances of these types of heat sinks are similar. Finally, the total resistance of the optimized heat sink with fins of variable thickness is lower than that of the optimized heat sink with fins of uniform thickness. For the case presented in Table 2, the total thermal resistance decreases by about 6% after the fin thickness is varied in the direction normal to the fluid flow. Fig. 7 shows the thermal resistances of the heat sinks with fins of variable thickness and heat sinks with fins of uniform thickness for various heights. Generally, the total thermal resistances of optimized heat sinks with fins of variable thickness are lower than those of optimized heat sinks of uniform thickness, and the difference between the total thermal resistances increases as the height increases. This is because the convective thermal resistance becomes dominant over the capacitive thermal resistance as the height increases and the heat sinks with fins of variable thickness can reduce the convective thermal resistance effectively without compromising the capacitive thermal resistance, as previously mentioned. Unfortunately, for low heights (H/L < 0.5), the thermal resistances of optimized heat sinks with fins of variable thickness are only slightly lower (<1%) than those of optimized heat sinks with fins of uniform thickness. The optimized geometries and thermal resistances of the heat sinks with fins of variable thickness and heat sinks with fins of uniform thickness for various heat fluxes and various heights are listed in Table 3. By comparing the thermal resistances of the two types of optimized heat sinks, a contour map was drawn, as shown in Fig. 8. Fig. 8 shows the ratio of the optimal thermal resistances of the heat sinks with fins of variable thickness to those of heat sinks with fins of uniform thickness (Ropt,var/Ropt,uni) for aircooled systems. In Fig. 8, in the region where the ratio is less than 1, the optimized heat sink with fins of variable thickness performs better than the optimized heat sink with fins of uniform thickness. The opposite is true when the ratio is greater than 1. Therefore, the contour map indicates that the optimized heat sinks with fins of variable thickness have lower thermal resistances than the optimized heat sinks with fins of uniform thickness. The thermal resistance is reduced by up to 10% by employing fins of variable thickness. However, the difference between the thermal resistances decreases as the height decreases and as the heat flux decreases. As shown in Fig. 8, the thermal resistance is reduced by less than 1% when
!! gbq2f cf q00 L4 H < 18:9 2:67 log10 L lf k2f !!2 gbq2f cf q00 L4 þ 0:0962 log10 : lf k2f
ð42Þ
When Eq. (31) is satisfied, the heat sink with fins of variable thickness is almost not superior to the simple heat sink with fins of uniform thickness. In Table 4, the calculated dimensionless heat fluxes for various lengths and heat fluxes are listed. As shown in this table, the magnitude of the dimensionless flux shown in Fig. 8 can be
D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
759
Table 3 Comparison of geometries and thermal resistances of optimized heat sinks Dimensionless heat flux
Dimensionless height
gbq2f cf q00 L4
H L
lf k2f
5.45 1010
5.45 1011
5.45 1012
5.45 1010
5.45 1011
5.45 1012
1.5
1.5
1.5
1.0
1.0
1.0
Type of fins
Dimensionless fin pitch
Porosity
Thermal resistance (°C/W)
p L
e
R
Uniform thickness
0.0538
0.59
0.0545
Variable thickness
0.0533
0.655 0.192Y + 0.0770Y2
0.0538
Uniform thickness
0.0433
0.488
0.0334
Variable thickness
0.0403
0.604 0.321Y + 0.130Y2
0.0320
Uniform thickness
0.0358
0.385
0.0221
Variable thickness
0.0282
0.577 0.479Y + 0.178Y2
0.0197
Uniform thickness
0.0454
0.674
0.0592
Variable thickness
0.0454
0.7020.0866Y + 0.0315Y2
0.0591
Uniform thickness
0.0347
0.581
0.0342
Variable thickness
0.0344
0.649 0.207Y + 0.0836Y2
0.0336
Uniform thickness
0.0282
0.475
0.0211
Variable thickness
0.0258
0.600 0.339Y + 0.136Y2
0.0201
W = L = 40 cm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K.
Schematic of channel (not to scale)
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D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
on the volume averaging theory (VAT) was used for this optimization. The thermal resistance was reduced by up to 10% compared to that of the heat sinks with fins of uniform thickness by employing fins of variable thickness in the case of an air-cooled heat sink. However, the amount of the reduction decreases as the heat flux decreases or as the height of the heat sink decreases. Owing to its better thermal performance, the heat sink with fins of variable thickness is expected to be a suitable next-generation cooling solution. In addition, the heat sink presented in this study can be used for cooling the heat dissipated during irreversible energy conversion processes in power generators, such as nuclear, hydro-electric, and reverse electrodialytic power generators [26]. Acknowledgement This research was supported by Nano R&D program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number:20110030285). Appendix A. Validity of assumptions made in the present study Fig. 8. Contour plots of Ropt,var/Ropt,uni (W/L = 1, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K).
Table 4 Dimensionless heat fluxes for various lengths and heat fluxes. L 0.2 0.3 0.4 0.4 0.6 0.8 0.8 1.2 1.6
q00 100 100 100 10 10 10 1 1 1
log10
gbq2f cf q00 L4
lf k2f
10.98 11.68 12.18 11.18 11.88 12.39 11.38 12.09 12.59
Two major assumptions are made to obtain Eqs. (1) and (2) from the Navier–Stokes equation and the energy equation. First, the body force per unit volume is assumed to be qf 1 gð1 bðT f T 1 ÞÞ, where qf1 is the fluid density at T = T1. Second, the flow is assumed to be both hydrodynamically and thermally fully developed. To check the validity of these assumptions, a numerical solution for natural convection between two isoflux plates (shown in Fig. A1) was obtained by using the energy and continuity equations along with two different momentum equations as follows: _
2u Þu pl r f þ r f ¼ qf g k ; qf ðuf r f
ðA1Þ
W/L = 1, lf = 0.00002 kg/m s, cf = 1008 J/kg K, qf = 1.06 kg/m3, kf = 0.028 W/m K, ks = 175 W/m K, gb = 0.0295 m/s2 K.
achieved when the length of the heat sink is greater than 20 cm. Therefore, in some heat sinks used for cooling rooms with computer and telecommunications equipment [24], we can expect enhancement in the thermal performance upon employing fins of variable thickness. However, for compact coolers that are used in small electronic equipment and have the length less than 20 cm, we cannot expect a reduction in resistance. The present study is based on the assumption that the heat sink is fully covered. If the cover plate is removed, there would be significant inflow from the open front edge of the interfin channels, as observed in [25]. In this case, the heat sinks with fins of variable thickness will not perform better than those with fins of uniform thickness. This is because the magnitude of the inflow for the heat sinks with fins of variable thickness would be lower than that for the heat sinks with fins of uniform thickness, since the interfin distance at the front edge, i.e., the channel width at y = H, is smaller for the former type of heat sink, as shown in Table 2. 4. Conclusion In the present study, the thermal optimization of a plate-fin heat sink was conducted for the case in which the fin thickness varied in the direction normal to the fluid flow. A model based
Fig. A1. Schematic diagram for natural convection between two isoflux plates
D.-K. Kim / International Journal of Heat and Mass Transfer 55 (2012) 752–761
maximum wall temperatures calculated using Eqs. (A1) and (A2). Fig. A2 shows that the results obtained from Eq. (A2) are in good agreement with those obtained from Eq. (A1) within a relative error of 10%; thus, it can be observed that the first assumption is reasonable. To check the validity of the second assumption, the velocity and temperature profiles in the entrance region are presented in Fig. A3. As shown in Fig. A3, a hydrodynamically and thermally fully developed condition is achieved when the distance from the entrance, x, is greater than 10wc Ra00wc . Therefore, the second assumption is valid when L > 10wc Ra00wc . This region corresponds to the fully developed limit presented in [17].
400
T(x=L,z=0) (K)
350
300
Eq. (A1) Eq. (A2)
References
250 0
20
40
60
80
100
q''sf (W) Fig. A2. Schematic diagram for natural convection between two isoflux plates (L = 20 cm, wc = 5 mm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, kf = 0.028 W/m K).
0.20
Velocity (m/s)
0.15
0.10
x= 0.52mm (= wc Raw '') x= 1.55mm (= 3wc Raw '') x= 2.58mm (= 5wc Raw '') x= 51.7mm (= 10wc Raw '') x= 103.3mm (= 20wc Raw '') c
c
c
0.05
c
c
0.00 0.000
0.001
0.002
0.003
0.004
z (m) (a) Velocity profiles
0.005
310
x= 0.52mm (= wc Raw '') x= 1.55mm (= 3wc Raw '') x= 2.58mm (= 5wc Raw '') x= 51.7mm (= 10wc Raw '') x= 103.3mm (= 20wc Raw '') c
308
c
c
Temperature (K)
761
306
c
c
304
302
300 0.000
0.001
0.002
0.003
0.004
0.005
z (m)
(b) Temperature profiles Fig. A3. Velocity and temperature profiles in the entrance region (L = 20 cm, wc = 5 mm, lf = 0.00002 kg/m s, cf = 1008 J/kg K, kf = 0.028 W/m K).
_
2u Þu pl r f þ r f ¼ qf 1 gð1 bðT f T 1 ÞÞ k : qf ðu f r f
ðA2Þ
Eq. (A1) is the original Navier–Stokes equation. In the Eq. (A2), the first assumption is applied to simplify Eq. (A1). Fig. A2 shows the
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