Finned Metal Foam Heat Sinks
for Electronics Cooling in Forced
Convection
A. Bhattacharya
Research Assistant
e-mail: bhattach@colorado.edu
R. L. Mahajan
Professor
e-mail: mahajan@spot.colorado.edu
CAMPmode,
Mechanical Engineering Department,
University of Colorado, Campus Box 427,
Boulder, CO 80309–0427
I
In this paper, we present recent experimental results on forced convective heat transfer in
novel finned metal foam heat sinks. Experiments were conducted on aluminum foams of 90
percent porosity and pore size corresponding to 5 PPI (200 PPM) and 20 PPI (800 PPM)
with one, two, four and six fins, where PPI (PPM) stands for pores per inch (pores per
meter) and is a measure of the pore density of the porous medium. All of these heat sinks
were fabricated in-house. The forced convection results show that heat transfer is significantly enhanced when fins are incorporated in metal foam. The heat transfer coefficient
increases with increase in the number of fins until adding more fins retards heat transfer
due to interference of thermal boundary layers. For the 20 PPI samples, this maximum
was reached for four fins. For the 5 PPI heat sinks, the trends were found to be similar to
those for the 20 PPI heat sinks. However, due to larger pore sizes, the pressure drop
encountered is much lower at a particular air velocity. As a result, for a given pressure
drop, the heat transfer coefficient is higher compared to the 20 PPI heat sink. For example, at a ⌬p of 105 Pa, the heat transfer coefficients were found to be 1169 W/m 2 -K
and 995 W/m 2 -K for the 5 PPI and 20 PPI 4-finned heat sinks, respectively. The finned
metal foam heat sinks outperform the longitudinal finned and normal metal foam heat
sinks by a factor between 1.5 and 2, respectively. Finally, an analytical expression is
formulated based on flow through an open channel and incorporating the effects of
thermal dispersion and interfacial heat transfer between the solid and fluid phases of the
porous medium. The agreement of the proposed relation with the experimental results is
promising. 关DOI: 10.1115/1.1464877兴
Introduction
The large-scale integration of electronic circuits has resulted in
a continuous increase in chip power dissipation requirements. According to the SIA roadmap 关1兴, in the ‘‘cost performance’’ and
‘‘high performance electronic products,’’ a chip power dissipation
level of 30 W/cm2 is anticipated in the next five years. For these
products, air-cooling is still the preferred and maybe the only
option to keep cost within bounds. Existing technology solutions
for cooling applications become ineffective beyond 50W/chip,
where the typical chip size is of the order of 520 mm2 . This has
necessitated the development of new and innovative thermal management solutions capable of removing heat from the chip and
dissipating it to the environment. Reviews of previous studies
undertaken in this field can be found in 关2– 4兴.
With air cooling, the two most commonly used techniques for
enhancing heat dissipation are increasing the air flow rate and
increasing the surface area for heat exchange by introducing fins
or baffles. However, both these techniques have disadvantages.
Increasing the airflow rate is easy to implement, but the associated
acoustic noise of the coolant air poses a severe limitation. Extended surfaces, in the form of fin arrays, have been extensively
implemented. Studies include those on flow through channels
关5,6兴, optimization of fin arrays in terms of fin spacing, channel
gap, fluid properties and tip clearance 关7–12兴, and novel designs
based on stringent heat transfer and fluid flow constraints 关13–
15兴. However, the extended surfaces result in increase in weight,
size, and pressure drop due to flow resistance. If the pressure drop
is excessive, air may bypass the heat sink. A solution to this probContributed by the Electronic and Photonic Packaging Division for publication in
the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received by the EPPD September 5, 2000. Associate Editor: A. Ortega.
Journal of Electronic Packaging
lem lies in placing the heat sink in a ducted arrangement in conjunction with a high power fan capable of handling higher pressure drops.
High porosity metal foams 共see Fig. 1兲 are potentially excellent
candidates for high heat dissipation. These foams provide extended surface area and also serve to enhance the heat transfer
coefficient due to local thermal dispersion caused by the eddies
that are shed in the wake of the flow past the metal fibers. More
detailed information on the geometric configuration of these
foams is given in 关16 –22兴. These metal foams have porosities of
around 90 percent and come in different pore sizes where the pore
size is characterized by the parameter, PPI 共pores per inch兲.
Early work by Lee et al. 关21兴 demonstrated the feasibility of
such a metal foam heat sink in reducing the chip-to-ambient thermal resistance to 0.4°C/W, when used in a ducted arrangement
with a conventional muffin fan. Calmidi 关16兴 verified the claims
made in 关21兴 and laid the theoretical foundation for understanding
the underlying transport mechanisms in such high porosity media.
The study included the measurement and modeling of the effective thermal conductivity of such materials, an empirical correlation for pressure drop, and a preliminary model for thermal dispersion. The data indicated significant enhancement in heat
dissipation for these high porosity metal foams. However, this
gain comes at the cost of a much higher pressure drop when
compared to standard finned heat sinks. In order to overcome this
limitation, Calmidi et al. 关17兴 proposed a novel finned metal foam
sink that incorporates normal longitudinal or pin fins in the pure
metal foam heat sink. It was claimed that such a heat sink would
combine the advantage of thermal dispersion in metal foams with
that of extended surface area of finned heat sinks, thereby increasing the total heat dissipation without appreciable increase in ⌬ P.
Figure 2 shows a finned metal foam heat sink with two fins
共although three fins are shown in the figure, the fins forming the
duct are half the thickness of the central fin兲. The analysis of the
Copyright © 2002 by ASME
SEPTEMBER 2002, Vol. 124 Õ 155
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
where the thermal boundary layers formed on the fins and the base
interfere with each other thereby inhibiting heat dissipation. As an
example, consider heat transfer due to flow in a channel of rectangular cross section. One can calculate h from the empirical
correlation for Nu for flow through a channel 关23兴,
NuD h ⫽
h̄D h
⫽2.98
k
(4)
or approximately by using correlations for flow over an isothermal
flat plate 关23兴 for all the three surfaces 共similar to the methodology followed in Eq. 共2兲兲
Nu⫽
Fig. 1 Picture of metal foam „5 PPI…
heat sink is performed using the theory of extended surface 关23兴.
Flow development is ignored since developing length 共hydraulic
and thermal兲 for flow through a channel filled with metal foam
is small 关16兴. With this assumption, the total heat dissipated is
given by
冋
Q⫽Q b ⫹Q f
Q⫽L⌬T h 共 W⫺nt 兲 ⫹n 冑2hkt tanh
冉冑 冊册
2h
H
kt
(1)
(2)
To evaluate h, the following correlation of Calmidi 关16兴 for flow
through metal foam on a horizontal isothermal surface was used.
h⫽211⫹109.3u
共 0.4m/s⬍u⬍4.0m/s兲
(3)
The thermal resistance of the finned heat sink 共W⫽5 cm, H
⫽4 cm, t⫽4 mm兲 is plotted in Fig. 3 along with a normal metal
foam heat sink in a velocity range of 0.4 m/s⬍u⬍4.0 m/s. A three
to four-fold improvement in heat dissipation capacity is indicated.
It may be noted that Eq. 共2兲 provides an overestimate of Q
since the value of h used is that for flow through simple metal
foam over an isothermal horizontal plate. In reality, flow through
the finned metal foam is analogous to flow through a channel
h̄L
⫽0.664 Re0.5 Pr0.33
k
(5)
The second approach over-predicts the total heat transfer by about
150% and 50% for the 2-finned and 4-finned heat sinks, respectively. Similar argument applies for the finned metal foam heat
sinks suggesting that values of h based on flat surface correlations
cannot be accurately applied for flow through a channel. Thus, the
experimental values of h are expected to be lower than those
calculated from Eq. 共2兲. However, the magnitude of reduction is
expected to be different than in the example above, since in the
metal foam channels, flow and heat transfer mechanisms are quite
different than those in open channels 关23,24兴.
In order to investigate these matters and to establish a generalized formulation for heat transfer from such finned metal foams,
experimental studies were undertaken using aluminum foams of
20 PPI and 5 PPI having a porosity of 90 percent with different
number of fins. The results are discussed in Section IV and an
empirical correlation proposed in Section V. A comparison with
the longitudinal finned heat sinks is provided in Section VI. In
Section VII, we propose a channel based heat transfer relation that
incorporates contributions due to thermal dispersion and interfacial heat transfer that arise in such porous media. The paper concludes with a brief summary in Section VIII.
II
Fabrication of the Samples
Fabrication of the proposed heat sinks forms an important aspect of this work. There are a number of issues involved in the
fabrication of these heat sinks. The two most important are:
共i兲 The foam must be cut into a precise geometry, and
共ii兲 The thermal contact resistance between the fin 共and base兲
surface and the foam must be minimized.
Fig. 2 Schematic of the novel finned metal foam heat sink proposed by Calmidi et al. †17‡
Fig. 3 Thermal resistance as a function of average velocity
156 Õ Vol. 124, SEPTEMBER 2002
The normal aluminum finned heat sinks were first manufactured
in-house. All the heat sinks fabricated had base areas of 62.5
mm⫻62.5 mm 共2.5 in.⫻2.5 in.兲. The base thickness and fin
heights were kept constant at 6.25 mm and 56.25 mm, respectively. The fin thickness for the heat sink with one fin was 6.25
mm, that with two fins was 3.125 mm and so on. For the heat sink
with one fin 共Fig. 4兲, an aluminum block was milled to form the
heat sink with the fin in the middle. However, for two or more
fins, the milling process became more difficult. The fins were
therefore machined separately and attached to the base by means
of screws. In order to reduce contact resistance, a thin layer of
thermal grease was applied at the contact surfaces of the fins and
base. The foams were obtained in the form of blocks from the
manufacturer. They were then cut into precise geometry using a
sharp blade band saw to within tolerance limits of ⫹0.075 mm.
Once the foam pieces were cut to the precise geometry, the lateral
surfaces were smoothed using a sander.
The conductive epoxy Thermaxx® 2600K 共Ablestick Inc.兲,
having a reported thermal conductivity of about 20–25 W/m-K,
was used to bond the foam to the fin and base surfaces. A thin
layer of the epoxy was first applied on the metal and foam surfaces, and the foams were next pushed forcibly into the gaps
between the fins, and then pressed from the top as well to ensure
good contact with base 共Fig. 4兲. The epoxy was then cured in a
Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 4 Schematic of the fabricated 1—finned metal foam heat
sink
value and the fluid velocity was varied in the range of 0.5–2m/s.
At a given flow rate, the temperatures of the base plate and the
inlet air were monitored until steady state was attained. The temperature and pressure drop readings were recorded. The flow rate
was then increased and the same procedure repeated until the
pressure drop reached 100 N/m2 共One of the preliminary objectives of our study was to compare the performance of finned metal
foam heat sinks with that of the high performance heat sinks reported in Sathe et al. 关13兴. This value of pressure drop was chosen
following the constraint mentioned in that study.兲 Calmidi and
Mahajan 关18兴 had found in their studies that the heat transfer
coefficient was independent of the input power at a given flow
rate. Hence the experiments were not repeated at different input
power levels. The heat transfer coefficient was calculated using
Newton’s Law of Cooling.
high temperature oven at 200°C, as per the instructions given by
the manufacturer. Figure 5 shows pictures of the heat sinks 共20
PPI兲 with one and four fins used in our study.
h⫽
IIIB
IIIA
Experiments
A schematic of the experimental setup is shown in Fig. 6 where
the test section is a part of the in-house fabricated wind tunnel
关16兴. The heat sink was placed in the test section and air was
blown through the wind tunnel by means of the fan-motor assembly. The direction of airflow was from left to right. Patch heaters
were attached to the bottom surface of the heat sinks and were
connected to a DC power supply. Five 0.127-mm T-type thermocouples were attached to the bottom plate at different locations in
order to monitor the base plate temperature. The thermocouples
were connected to the OMEGA DASTC data acquisition system.
The heat sink was insulated on all the sides 共except inlet and
outlet兲 by Styrofoam insulation. Pressure taps were provided at
the inlet and outlet for monitoring the pressure drop across the
sample.
Experiments were conducted with heat sinks having one, two,
four, and six fins, and using aluminum foam with pore sizes corresponding to 5 and 20 PPI and porosity of 0.9. During the experiments, the power supplied to the heaters was kept at a fixed
Q
A b 共 T b ⫺T amb 兲
(6)
Error Analysis
The main uncertainties in this experiment are due to errors in
measurements of power, thermocouple readings and physical dimensions. The maximum error in the multimeter readings for the
voltage and resistance measurements is 0.5% 共manufacturer’s data
sheet兲. This results in an error of 1.5% in the estimation of the
heat input Q. The combined error 共resolution of the data acquisition unit, thermocouple calibration兲 in the estimation of the temperature difference under the given conditions is 0.2°C. For the
temperature differences observed in our experiments 共of the order
of 30°C兲, the error in our measurement is thus 0.67%. The error in
measurement of the cross-sectional area is estimated to be 0.75%.
The heat losses through the Styrofoam insulation 共thermal conductivity is 0.03W/m-K兲, as well as the conduction losses through
the thin thermocouple wires 共diameter of 0.127 mm兲, are neglected. Based on the preceding errors, the total error in the measurement of the effective thermal conductivity can be calculated
using 关25兴
⌬h
⫽
h
冑冉 冊 冉 冊 冉
⌬Q
Q
2
⫹
⌬A m
Am
2
⫹
⌬ 共 ⌬T 兲
⌬T
冊
2
(7)
and was found to be ⫾1.8%. However, it may be noted that the
above analysis does not take into account the error due to thermal
contact resistance introduced by the epoxy used to bond the metal
foam to the fins and the base.
IVA
Fig. 5 Pictures of the metal foam heat sinks with one and four
fins used in our study
Fig. 6 Schematic of the experimental setup
Journal of Electronic Packaging
Results With 20 PPI
The results of the experiments with 20 PPI samples are shown
in Figs. 7 and 8. From Fig. 7, it is seen that the heat transfer
coefficient increases at a given flow rate when fins are incorporated in the metal foams. The variation of h with u is found to be
almost linear in the given velocity range. There are two reasons
for this enhancement. First, the thermal conductivity of aluminum
共218 W/m-K兲 is about 32 times that of the metal foam 共6.9
W/m-K兲 关22,26兴. Thus, replacing the strip of metal foam by solid
aluminum results in an increase in heat dissipation by conduction.
Second, a larger surface area of heated metal is in contact with
metal foam 共base plus fins兲 compared to a normal metal foam heat
sink 共base only兲.
However, we note that the increase in heat transfer coefficient
progressively decreases with the addition of fins. From the curves,
it is seen that the improvement in heat transfer from one fin to two
fins is approximately 65 percent, while that from two to four fins
is about 25 percent. When the number of fins is further increased
to six, the heat transfer coefficient actually decreases.
These trends can be explained in terms of the interference of
the thermal boundary layers. For a given cross-sectional area, the
channel gap between two fins 共fin pitch兲 decreases with the addition of fins. As a result, the thermal boundary layers, which are
SEPTEMBER 2002, Vol. 124 Õ 157
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 7 Plot of heat transfer coefficient as a function of air velocity for the 20 PPI foam samples
material like metal foam in air, where the ratio of the solid to fluid
conductivities is large, it is not advisable to assume LTE. However, in their study on forced convection, Calmidi and Mahajan
关18兴 had shown that in the range of flow velocities and foam
parameters 共Re and Da兲 for their experiments, the interfacial heat
transfer due to local thermal nonequilibrium 共LTNE兲 is negligible.
Since our experiments also fall in the same range of Re and Da,
Eq. 共8兲 can be used to obtain a reasonable estimate of the thermal
boundary layer thickness.
The pressure drop data is plotted in Fig. 8. As expected, ⌬p
increases with addition of each fin due to the head losses at entry,
exit and wall friction. However, compared to the enhancement in
heat transfer, the increase in ⌬p is relatively smaller.
From the plots of heat transfer coefficient and pressure drop, it
is seen that the optimum number of fins in a heat sink of our
dimensions is four. With this heat sink, we can obtain a heat
transfer coefficient of 995 W/m2 -K at an air velocity of 1.7 m/s
with a pressure drop of 105 Pa across the heat sink.
IVB
Results With 5PPI
The second set of experiments was conducted on aluminum
foam samples corresponding to 5 PPI with one, two, and four fins.
Figure 9 shows the results of heat transfer coefficient. The pressure drop data is plotted in Fig. 10. Both the heat transfer and
pressure drop data show trends similar to those for the 20 PPI
samples. However, we note that the heat transfer coefficient for a
20 PPI sample at a particular air velocity is higher than that given
Fig. 8 Plot of pressure drop at different air velocities for the 20
PPI foam samples
formed on adjacent fin surfaces, interfere with each other.
This interference results in retardation in heat transfer. Hence,
there is an optimum value for the number of fins for maximum
heat transfer.
For flow through a porous medium over an isothermal surface
共assuming Darcy flow兲, the thermal boundary layer thickness can
be written as 关26兴
␦ T⫽ ␩ 0x
冑
␣ef f
U ⬁x
(8)
where erf(␩0/2)⫽0.99. Using typical values of U ⬁ ⫽1m/s, x
⫽6.25 cm 共2.5 in.兲, the boundary layer thickness is calculated as
0.65 cm. Thus, the value of 2 ␦ T ⫽1.3 cm is greater than the channel gap of 1.12 cm in our four-finned heat sink. As a result, the
thermal boundary layers formed on the adjacent fin surfaces interfere to reduce heat transfer. The reduction becomes more severe
for the heat sink with six fins where the channel gap is only 0.8
cm. However, increase in number of fins also results in two extra
surfaces added per fin across which heat transfer can occur. For
the case of the heat sink with four fins, this enhancing effect is
larger than the reduction in heat transfer due to boundary layer
interference. The net result is an increase in the heat transfer coefficient as we increase the number of fins from two to four.
However, with further increase in the number of fins to six, the
retarding effect dominates and the heat transfer rate goes down.
It may be mentioned that Eq. 共8兲 is valid only if the solid and
fluid phases are locally in thermal equilibrium 共LTE兲. For a porous
158 Õ Vol. 124, SEPTEMBER 2002
Fig. 9 Heat transfer coefficients for the 5 PPI samples as a
function of flow velocity
Fig. 10 Pressure drop characteristics of the 5 PPI samples as
a function of flow velocity
Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 11 Heat transfer coefficients for the samples with four
fins
Fig. 13 Experimental Nusselt numbers versus empirical correlation †Eq. „10…‡ for finned metal foam heat sinks
Nue ⫽
冉 冊
hd p
Dh
⫽C ReKm Prne
ke
dp
p
(9)
where ReK⫽u冑K/ ␯ , Pre ⫽ ␮ C p /k e , and d p is the pore diameter of
the foam sample. This is similar to the turbulent flow correlation
over a flat plate. However, the extra parameter (D h /d p ) is added
to take into account the effect of fin pitch. Using least squares fit,
the best-fit parameters are found to be C⫽5.95, m⫽0.61, n
⫽0.58, p⫽⫺0.66. Noting that the values of m and n are very
close, we replaced ReK and Pre by the Peclet number defined as
PeK,e ⫽ReK Pre . The resulting correlation becomes
3/5
Nue ⫽6.43 PeK,e
Fig. 12 Pressure drop characteristics for the samples with
four fins
y a 5 PPI sample having the same number of fins 共see Fig. 11兲.
This is due to the larger surface area to volume ratio and higher
thermal dispersion effects 共presence of larger number of fibers兲 for
the former. The accompanying pressure drop, however, is also
higher for the 20 PPI sample 共see Fig. 12兲. The implication is that
in many practical applications, where pressure drop or fan power
is a major constraint, a much higher velocity may be attained
using a 5 PPI foam sample, which in turn, can lead to higher heat
transfer rate. For example, for an allowable pressure drop of 100
Pa, a flow velocity of about 2 m/s is achieved for the 5 PPI sample
compared to about 1.6 m/s for the 20 PPI sample. The net result is
a higher h, see Table 1.
V
Empirical Correlation
In this section, we propose a correlation of the form given in
Eq. 共9兲 to estimate h or Nue .
Table 1 Heat transfer coefficients around 100 Pa
Journal of Electronic Packaging
冉 冊
Dh
dp
⫺ 2/3
(10)
The above correlation matches the experimental data with an error
band of ⫾12% 共see Fig. 13兲.
d ⫺1/3
. This means that
From Eq. 共10兲 it can seen that h ␣ D ⫺2/3
h
p
when D h increases or equivalently the number of fins decreases
共all other parmeters held constant兲, the heat transfer coefficient
decreases. The thermal enhancement effects due to conduction
and extended surface area of the fins are, then, small. On the other
hand, when d p increases or equivalently PPI goes down, h decreases. As stated in Section IVB, this is due to the extra specific
surface area of the higher PPI samples as well as enhanced thermal dispersion effect.
VI
Comparison With Longitudinal Finned Heat Sinks
In an experimental investigation, Lau and Mahajan 关9兴 reported
heat transfer for longitudinal finned heat sinks 共see Fig. 14兲 in a
ducted arrangement. The study showed that by using an appropriate hydraulic diameter, the theories of pipe flow and extended
surfaces could be combined to obtain the net heat transfer rates.
The following empirical correlation was formulated for the nondimensional heat transfer coefficient.
Nu⫽
h oD h
0.8
Pr0.4兲
⫽1.42共 0.023 ReD
h
k
Q
,
where h o ⫽
共 A b ⫹ ␩ A f 兲共 T b ⫺T amb 兲
(11)
uD h
4SH
Re⫽
, D h⫽
␯
S⫹2H
The factor 1.42 takes into account the enhanced heat transfer due
to developing flow.
Figure 15 shows the plots of heat transfer coefficient at different flow velocities for heat sinks having one, two, four and six fins
with base and fin dimensions corresponding to the ones used in
our experimental study along with our own experimental data.
The value of h 共based on base area兲 for the longitudinal finned
SEPTEMBER 2002, Vol. 124 Õ 159
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 16 System curves for the different heat sinks along with
the fan curve
Fig. 14 Schematic of the longitudinal finned heat sink of Lau
and Mahajan †9‡
heat sink with four fins is about 120 W/m2 -K at an air velocity of
1.75 m/s. This is in contrast to 995 W/m2 -K at ⬃1.6 m/s for our
20 PPI heat sink. For the longitudinal heat sink having six fins, the
value is close to 180 W/m2 -K.
Bejan and Sciubba 关7兴 proposed the following expressions for
the optimal spacing of parallel plates for maximum cooling in
forced convection.
冉
S o pt
␮␣
⫽3.033
L
⌬ P.L 2
⬙ ⫽0.479
q max
冉 冊
␳⌬ P
Pr
冊
1/4
(12a)
1/2
WC p 共 T b ⫺T amb 兲
(12b)
The heat transfer coefficients for this optimal array is also plotted
in Fig. 15. It can be seen that the heat transfer coefficients offered
by the optimized plate fin array are consistently lower than those
for our 4-finned metal foam heat sink by almost a factor of 2 in
the entire velocity range. Thus it can be seen that significant enhancements in heat transfer rates can be achieved by replacing
plate fin heat sinks with finned metal foams.
The above comparisons between different heat sinks were
based on a fixed velocity. However, such a comparison is not fair
since, for a given pressure drop constraint a normal longitudinal
finned heat sink can sustain much higher velocities compared to
finned metal foam heat sinks. A better comparison is perhaps
through the operating points of the heat sinks based on the fan
characteristic curves. Figure 16 shows the fan curve for a 12cm
24V DC muffin fan 共a type commonly used in many electronics
applications兲. The system curves for the heat sinks in their operating range are also shown. To compare the heat transfer performance of the two sinks the operating points of the heat sinks
based on the fan curves were identified, and the corresponding
flow velocities 共hence Re兲 were calculated. These velocities were
then used to calculate the heat transfer coefficients. The correlation of Lau and Mahajan 关9兴 was used to calculate h for the
longitudinal finned heat sinks while for the finned metal foam heat
sinks the experimental results were used. The values of the heat
transfer coefficients based on the heat sink base area for these
operating points are presented in Table 2. It is seen in Table 2 that
the maximum heat transfer coefficient for the finned metal foam
heat sinks is much higher than for the normal longitudinal finned
heat sinks. The clear indication is that replacing the air gap between the fins by metal foams can lead to a large enhancement in
heat dissipation
We, however, note that even though the 4 finned metal foam
heat sinks were found to give the best performances from our
experiments, the same number may not represent the optimum
configuration for longitudinal finned heat sinks. To explore this
further, we compared our experimental data with that of Lau and
Mahajan 关9兴. They used three types of heat sinks listed in Table 3.
Using the procedure based on fan characteristic curve, as outlined
above, heat transfer coefficients of about 236.5, 371.2, and
433.7 W/m2 -K were calculated from their results for the 13, 46,
and 56 finned heat sinks respectively. Even for these large numbers of fins, these values are significantly lower than those for the
finned metal foam heat sinks.
Table 2 Heat transfer coefficients based on fan curve
Fig. 15 Comparison of heat transfer coefficients for longitudinal finned and finned metal foam „20 PPI… heat sinks
160 Õ Vol. 124, SEPTEMBER 2002
Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Table 3 Heat sinks of Lau and Mahajan †9‡ Base area of the
heat sinks: 10.16 cmÃ10.16 cm „4 in.Ã4 in.…
Further, the model shows that for flow through cylindrical
packed tubes transition to turbulence takes place at a Red 共Reynolds number based on particle/pore diameter兲 value of about 20.
For our experiments, the values of Red were between 250 and 530
for the 5 PPI samples, and between 150 and 300 for the 20 PPI
samples, clearly indicating that our experiments were in the turbulent flow regime.
The correlation for turbulent flow in a channel with isothermal
walls is given by Eq. 共11兲 and is rewritten here in terms of the
effective properties
It is to be recognized that these comparisons are based on the
characteristic curve of a single fan. The results may be different
for a fan having a different characteristic curve.
VII Finned Metal Foam Heat Sinks: Comparison
With Flow Through a Channel
It was mentioned in Section II that flow through a finned metal
foam heat sink can be conceived of as flow through a channel, in
which the fluid medium in the channel is replaced by our porous
medium. It will be interesting to investigate if the correlation for
flow through a channel can be applied to predict our experimental
data by replacing the fluid properties by the effective properties of
the porous medium. However, there are several other factors that
need to be considered before making such a comparison. First,
flow through a metal foam gives rise to local disturbance due to
the eddies that are shed in the wake of the flow past the solid
fibers. This causes thermal dispersion, which enhances heat dissipation. Second, due to the presence of these eddies, transition to
turbulence occurs at a much lower Reynolds number compared to
flow through an open channel. Finally, the solid and fluid phases
may not be in thermal equilibrium resulting in interfacial heat
transfer between the fluid and the foam matrix. The result is a
decrease in Nusselt number due to the extra thermal resistance
arising out of this interfacial heat transfer.
Cheng and co-researchers 关27,28兴 have conducted numerical
studies on flow through packed beds in a channel. Their model
predictions were found to be in excellent agreement with the experimental data of Verschoor and Schuit 关29兴 for forced convection of air through a packed bed of glass spheres. However, a
comparison of their model with our experimental data shows that
Nusselt number for metal foams are higher by factors of 5–10.
This can be attributed to the higher effective thermal conductivity
(k e ⫽6.7 W/m-K) of metal foams 关22,26兴 compared to that of
glass spheres (k e ⬃1 W/m-K).
Nu⫽
h oD h
0.8
⫽0.023 ReDh
Pr0.4
e
ke
(13)
uD h
4SH
␮Cp
where ReDh ⫽
, Pre ⫽
D h⫽
␯
ke
S⫹2H
Note that the correction factor of 1.42 used in 关9兴 for developing
flow has been dropped since the developing length for flow
through a porous medium is very small 关16,24,30兴. Replacing the
fluid properties by the properties of the porous medium in Eq.
共13兲, one can conclude that at a given flow velocity, the ratio of
the heat transfer coefficient for flow through a packed channel to
that of an open channel can be written as
冉冊
ho
ke
⫽
hc
kf
0.6
(14)
For our case, the ratio of the effective thermal conductivity of the
metal foam to that of the fluid is 257. As a result the ratio of the
heat transfer coefficients is expected to be 28. Figure 17 shows the
plots of h o /h c for different ReDh 共or equivalently flow velocity兲
for the different finned heat sinks. It is seen that the experimental
values are consistently higher than the theoretical predictions in
the entire range of Reynolds number. However, the overage decreases as ReDh is increased. This behavior can be possibly attributed to the effects of thermal dispersion and interfacial heat transfer due to local thermal nonequilibrium 共LTNE兲 关30兴.
Combining the formulations stated in 关24兴 and 关30兴 for thermal
dispersion and thermal nonequilibrium effects, respectively, one
can postulate a correlation of the form
Nu⫽
冉
冊
冉 冊
h oD h
k e ⫹k d
D hk f
0.8
⫽0.023 ReDh
Pr0.4
⫺Nus f
ke
ke
d fke
(15)
0.37
where k d /k e ⫽0.06 ReK Pre and Nus f ⫽0.52 Re0.5
关18兴. Ded f Pr
pending on the relative contributions of the thermal dispersion
Fig. 17 Heat transfer coefficient ratio for channel flows
Journal of Electronic Packaging
SEPTEMBER 2002, Vol. 124 Õ 161
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 18 Comparison of the experimental and theoretical †Eq. „15…‡ Nusselt
numbers
共second term on RHS兲 and the interfacial heat transfer 共third
term兲, the ratio of the Nusselt numbers can increase or decrease.
The solid line in Fig. 17 shows the ratio of the heat transfer
coefficients h o /h c , as predicted by Eq. 共15兲. This prediciton
shows the same trend as of the experimental data, i.e., h ratio
decreases with ReDh .
The same data is plotted in Fig. 18 in terms of Nu. The agreement is found to be generally good although the experimental
Nusselt numbers are mostly lower than the theoretical predictions.
Recall that in our heat sinks the metal foam had to be bonded to
the base and fins using conductive thermal epoxies than if they
were brazed. The discrepancy between the theoretical and experimental Nu may be attributed to this contact resistance problem.
A conclusion from the above discussion is that perhaps a formulation on the lines of Eqs. 共11兲, 共13兲–共15兲 can be used to predict the heat transfer coefficients in these finned metal foam heat
sinks. However, further refinement of this preliminary model is
necessary to accurately nail down the thermal performance of
these heat sinks, which is a focus of our future research.
VIII
Summary
The performance of novel finned metal foam heat sinks was
investigated under forced convection for potential application in
cooling of electronic components. These heat sinks can be thought
to be similar to longitudinal finned heat sinks where the air gap
between two adjacent fins is replaced by high porosity metal
foams. Experiments were conducted on aluminum foams of 90
percent porosity and pore size corresponding to 5 and 20 PPI. The
forced convection results show that heat transfer is considerably
enhanced when fins are incorporated in metal foams. The heat
transfer coefficient increases with increase in the number of fins
due to additional surface area available for heat transfer. However,
this enhancement comes at the expense of excess pressure drop.
Further, the relative enhancement in heat transfer coefficient progressively decreases with addition of fins. This can be attributed to
the interference of boundary layers that results due to reduced
channel gap with increase in the number of fins. Based on the heat
transfer and pressure drop curves, the metal foam heat sink with
four fins was found to be the optimum configuration for 20 PPI
samples for our geometry. For the 5 PPI heat sinks, experiments
were conducted with one, two, and four fins. Trends are found to
be very similar to those for the 20 PPI heat sinks. Compared to
different heat sink designs reported in literature, the performance
of the finned metal foam heat sinks© was found to be superior. If
162 Õ Vol. 124, SEPTEMBER 2002
pressure drop is not a constraint, these heat sinks can enhance the
heat transfer coefficient by about a factor of 6 compared to the
commercially available finned heat sinks used for electronics
cooling. However, if constraint is imposed on ⌬ P, the performance enhancement factor can be 1.5–2. Further, the results indicate that the 5 PPI samples result in higher heat transfer coefficients compared to the 20 PPI samples for a given pressure drop
or fan power.
Based on the experimental data, an empirical correlation for
Nusselt number in terms of Peclet number and dimensionless ratio
of hydraulic diameter to pore diameter, is proposed. A generalized
Nusselt number formulation for the finned metal foam is also
proposed that incorporates the effects of thermal dispersion and
interfacial heat transfer between the solid and fluid phases. The
agreement with the experimental data is promising.
We finally note that our preliminary experimental results on
natural convection in these heat sinks in horizontal configuration
have also shown considerable enhancement in heat transfer compared to commercially available finned heat sinks or normal metal
foam heat sink. These results will be presented in a follow-up
paper.
Acknowledgments
We gratefully acknowledge the financial support received from
CAMPmode, the Center for Advanced Manufacturing and Packaging of Microwave, Optical and Digital Electronics at the University of Colorado and IBM Co-operative Fellowship for the first
author. We would specially like to thank Dr. V.V. Calmidi and Dr.
Sanjeev Sathe of IBM Inc. for the many stimulating discussions
and their valuable advice, and Dr. Brian Leyda 共Energy Research
and Generation Inc.兲 and Ablestick Inc. for providing us with the
metal foam and thermal epoxy samples, respectively.
Nomenclature
A
C
Cp
D, d
Da
h
H
K
k
L
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
area 关 m2 兴
empirical constant
specific heat 关J/kg-K兴
diameter 关m兴
Darcy number, K/H 2
average heat transfer coefficient 关 W/m2 -K兴
fin height 关m兴
permeability 关 m2 兴
thermal conductivity 关W/m-K兴
length of the heat sink 关m兴
Transactions of the ASME
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
n
Nu
P
Pe
Pr
Q
q⬙
Rex
S
t
T
U, u
W
x
⌬p
␣
⌬
␦
␩
␮
␯
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
number of fins
average Nusselt number, hL/k
fan power 关W兴
Peclet number, ud p / ␣
Prandtl number, ␯ / ␣
heat input or dissipated 关W兴
heat flux 关 W/m2 兴
Reynolds number, ux/ ␯
channel gap 关m兴
fin thickness 关m兴
temperature 关°C or K兴
air velocity 关m/s兴
width of heat sink 关m兴
distance 关m兴
pressure drop 关 N/m2 兴
thermal diffusivity 关 m2 /s兴
difference
boundary layer thickness 关m兴
fin efficiency
viscosity 关kg/m-s兴
kinematic viscosity of air 关 m2 /s兴
Subscripts
amb
b
c
d
e, eff
f
h
K
o
p
sf
T
⬁
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
ambient
base
cross section, channel
pore diameter, dispersion
effective
fins, fluid
hydraulic
permeability
overall
pore
interfacial 共solid-fluid兲
thermal
free stream
References
关1兴 Semiconductor Industry Association, 1998, The national technology roadmap
for semiconductors, SIA, California.
关2兴 Kraus, A. D., and Bar-Cohen, A., 1983, Thermal analysis and control of electronic equipment, Hemisphere, New York.
关3兴 Incropera, F. P., 1988, ‘‘Convection heat transfer in electronic equipment cooling,’’ ASME J. Heat Transfer, 110, pp. 1097–1111.
关4兴 Peterson, G. P., and Ortega, A., 1990, ‘‘Thermal control of electronic equipment and devices,’’ Adv. Heat Transfer, 20, pp. 181–314.
关5兴 Tuckerman, D. B., and Pease, R. F. W., 1981, ‘‘High Performance heat sinking
for VLSI,’’ IEEE Electron Device Lett., EDL-2, No. 5, pp. 126 –129.
关6兴 Goldberg, N., 1984, ‘‘Narrow channel forced air heat sink,’’ IEEE Trans. Compon., Hybrids, Manuf. Technol., 7, No. 1, pp. 154 –159.
Journal of Electronic Packaging
关7兴 Bejan, A., and Sciubba, E., 1992, ‘‘The optimal spacing of parallel plates
cooled by forced convection,’’ Int. J. Heat Mass Transf., 35, pp. 3259–3264.
关8兴 Knight, R. W., Goodling, J. S., and Hall, D. J., 1991, ‘‘Optimal thermal design
of forced convection heat sinks–analytical,’’ ASME J. Heat Transfer, 113, pp.
313–321.
关9兴 Lau, K. S., and Mahajan, R. L., 1989, ‘‘Effects of tip clearance and fin density
on the performance of heat sinks for VLSI packages,’’ IEEE Trans. Compon.,
Hybrids, Manuf. Technol., 12, No. 4, 757–765.
关10兴 Sparrow, E. M., Baliga, B. R., and Patankar, S. V., 1978, ‘‘Forced convective
heat transfer from a shrouded fin array with and without tip clearance,’’ ASME
J. Heat Transfer, 100, pp. 572–579.
关11兴 Kadle, D. S., and Sparrow, E. M., 1986, ‘‘Numerical and experimental study
on turbulent heat transfer and fluid flow in longitudinal fin arrays,’’ ASME J.
Heat Transfer, 108, pp. 16 –23.
关12兴 Jelenik, M., ‘‘Optimum array of cooling fins,’’ MS Thesis, Ben-Gurion University of Negev, Beer Sheva, Israel.
关13兴 Sathe, S. B., Sammakia, B. G., Wong, A. C., and Mahaney, H. V., 1995 ‘‘A
numerical study of a high-performance air-cooled heat sink,’’ National Heat
Transfer Conference–Volume 1, HTD ASME, Vol. 303, pp. 43–53.
关14兴 Hilbert, C., Sommerfeldt, S., Gupta, O., and Herell, D. J., 1990, ‘‘High performance air cooled heat sinks for integrated circuits,’’ IEEE Trans. Compon.,
Hybrids, Manuf. Technol., 13, No. 4, pp. 1022–31.
关15兴 Hwang, L. T., Turlik, I., and Reisman, A., 1987, ‘‘A thermal module design for
advanced packaging,’’ J. Electron. Mater., 16, No. 5, pp. 347–355.
关16兴 Calmidi, V. V., 1998, ‘‘Transport Phenomena in High Porosity Metal Foams,’’
Ph.D. thesis, University of Colorado.
关17兴 Calmidi, V. V., Calmidi M., and Mahajan, R. L., 2000, ‘‘Finned Metal Foam
Heat Sink,’’ U.S. patent filed, University of Colorado.
关18兴 Calmidi, V. V., and Mahajan, R. L., 2000, ‘‘Forced Convection in High Porosity Metal Foams,’’ ASME J. Heat Transfer, 122, pp. 557–565.
关19兴 Hunt, M. L., and Tien, C. L., 1988, ‘‘Effects of Thermal Dispersion on Forced
Convection in Fibrous Media,’’ Int. J. Heat Mass Transf., 31, pp. 301–309.
关20兴 Lu, T. J., Stone, H. A., and Ashby, M. F., 1998, ‘‘Heat transfer in open-cell
metal foams,’’ Acta Mater., 46, pp. 3619–3635.
关21兴 Lee, Y. C., Zhang, W., Xie, H., and Mahajan, R. L., 1993 ‘‘Cooling of a FCHIP
package with 100 W, 1 cm2 chip,’’ Proceedings of the 1993 ASME Int. Elec.
Packaging Conference, Vol. 1, pp. 419– 423.
关22兴 Bhattacharya, A., Calmidi, V. V., and Mahajan, R. L., 2002, ‘‘Thermophysical
Properties of High Porosity Metal Foams,’’ Int. J. Heat Mass Transf., 45, pp.
1017–1031.
关23兴 Bejan, A., 1993, Heat Transfer, Wiley, New York.
关24兴 Nield, D. A., and Bejan, A., 1999, Convection in porous media, 2nd ed.,
Springer-Verlag, New York.
关25兴 Taylor, T. R., 1980, An Introduction to Error Analysis—The Study of Uncertainties in Physical Measurements, University Science Books, Mill Valley,
California.
关26兴 Calmidi, V. V., and Mahajan, R. L., 1999, ‘‘The effective thermal conductivity
of high porosity metal foams,’’ ASME J. Heat Transfer, 121, pp. 466 – 471.
关27兴 Hsu, C. T., and Cheng, P., 1990, ‘‘Thermal Dispersion in a porous medium,’’
Int. J. Heat Mass Transf., 33, pp. 1587–1597.
关28兴 Cheng, P., Chowdhury, A., and Hsu, C. T., 1990, ‘‘Free Convection in packed
tubes and channels with variable porosity and thermal dispersion effects,’’
Convective Heat and Mass Transfer in Porous Media, NATO ASI Series, Series E, Applied Sciences—Vol. 196, Kakac, S. et al., eds. pp. 625– 653.
关29兴 Verschoor, H., and Schuit, G. C. A., 1952, ‘‘Heat Transfer to fluids flowing
through a bed of granular solids,’’ Appl. Sci., 42, pp. 97–119.
关30兴 Nield, D. A., 1998, ‘‘Effects of Local Thermal Nonequilibrium in Steady Convective Processes in a Saturated Porous Medium: Forced Convection in a
Channel,’’ Journal of Porous Media, 1共2兲, pp. 181–186.
SEPTEMBER 2002, Vol. 124 Õ 163
Downloaded From: http://electronicpackaging.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use