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International Journal of Heat and Mass Transfer 127 (2018) 32–40
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Efficient three-dimensional topology optimization of heat sinks in
natural convection using the shape-dependent convection model
Younghwan Joo, Ikjin Lee, Sung Jin Kim ⇑
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
a r t i c l e
i n f o
Article history:
Received 27 June 2018
Received in revised form 3 August 2018
Accepted 3 August 2018
Keywords:
Three-dimensional topology optimization
Shape-dependent heat transfer coefficient
Heat sink
Natural convection
Constructal design
a b s t r a c t
In this study, heat sinks in natural convection are thermally optimized using the method of threedimensional topology optimization. In order to perform three-dimensional topology optimization with
low computational cost, a shape-dependent convection model is proposed. This model accounts for the
variation of the heat transfer coefficient depending not only on the local shape of the fins but also on
the development of the thermal boundary layer. The physical validity of the proposed model is confirmed
by the fin geometry of the topology-optimized design that matches the multiscale structures proposed
previously by the constructal theory. For further validation, the effective heat transfer coefficient evaluated by the proposed model is compared to that obtained from numerical simulations. Because the new
topology-optimized design has a complicated fin geometry, design simplification is performed to yield a
more manufacturable design. The thermal performance of the topology-optimized heat sink is compared
to that of the radial plate-fin heat sink optimized analytically using an existing correlation. It is found that
the topology-optimized heat sink has 13% lower thermal resistance and 48% less mass than the optimized
radial plate-fin heat sink. This implies that the three-dimensional topology optimization method suggested in this study can provide a heat sink design with improved thermal performance and reduced
mass for various practical applications.
Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction
Natural convection heat sinks are widely used because of their
simplicity and reliability [1,2]. Thermal engineers are interested in
finding the optimum geometry of a heat sink that dissipates the
maximum amount of heat under a given physical volume. Conventionally, heat sinks with a given fin shape such as plates or pins
have been optimized experimentally, numerically, or analytically
[3–7]. Apart from conventional methods, topology optimization,
a method commonly used in structural problems [8], has been
recently applied to the thermal optimization of heat sinks [9–13].
Since this method does not impose any constraint on the fin shape,
completely new designs for heat sinks can be suggested.
To optimize a heat sink using the topology optimization
method, the convective heat transfer rate from the heat sink surface needs to be accurately evaluated. Alexandersen et al. [14]
solved full conjugate heat transfer problems numerically in a 2-D
computational domain to account for the local variation of the heat
transfer coefficient, and they extended the method to a more complex 3-D problem [15]. Recently, they applied their 3-D topology
⇑ Corresponding author.
E-mail address: sungjinkim@kaist.ac.kr (S.J. Kim).
https://doi.org/10.1016/j.ijheatmasstransfer.2018.08.009
0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
optimization method to a passive cooler for light-emitting diode
lamps, and obtained new designs in the given physical domain
[16]. Their method generally requires high computational cost
because the velocity field and the corresponding distribution of
the heat transfer coefficient need to be calculated each time the
shape of the heat sink fins changes at every iteration. The supercomputing equipment had to be utilized to cope with a significantly increased computational load caused by the 3-D full
conjugate heat transfer problem. Hence, for the practical use of
3-D topology optimization, it is necessary to develop a simple
and efficient method that enables 3-D topology optimization with
fairly accurate predictions of the shape-dependent heat transfer
coefficient.
In our previous work [13], the shape-dependent convection
model for topology optimization of heat sinks in natural convection was proposed for a 2-D computational domain. In this model,
a geometric parameter that reflects the local shape of the fins was
obtained, and then the heat transfer coefficient was evaluated by
substituting this geometric parameter into the existing Nusselt
number correlation. Since it was not necessary for this model to
solve the full conjugate heat transfer problem, the computational
load could be greatly reduced. Even if this model is extended to
3-D topology optimization, the computational load for this
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
33
Nomenclature
A
cp
d
El
f
g
H
h
k
L
Ne
Nu
Pr
Q
R
Rth
Sh
Sv
T1
Tb
wc
xi
surface area [m2]
specific heat [kJ/kg-K]
fin diameter [m]
Elenbaas number [–]
volume fraction [–]
standard acceleration of gravity [m/s2]
fin height [m]
convective heat transfer coefficient [W/m2-K]
thermal conductivity [W/m-K]
heat sink length [m]
adjacent elements [–]
Nusselt number [–]
Prantl number [–]
total heat input [W]
outer radius of domain [m]
thermal resistance [K/W]
horizontal spacing [m]
vertical spacing [m]
ambient temperature [K]
base temperature [K]
channel spacing [m]
spatial location of element i [–]
shape-dependent convection model is expected to be manageable
by a PC. However, this model developed for the 2-D computational
domain cannot be directly applied to 3-D topology optimization
because the variation of the heat transfer coefficient along the
direction of gravity, which is caused by the development of the
thermal boundary layer, was neglected in the model for the 2-D
domain. Therefore, for efficient 3-D topology optimization, it is
necessary to develop a new convection model in which the heat
transfer coefficient can simply be evaluated depending not only
on the shape of the fins but also on the development of the thermal
boundary layer.
In this study, heat sinks in natural convection are thermally
optimized using 3-D topology optimization. To optimize the design
of the fin shape in a computationally efficient way, a simple but
accurate shape-dependent convection model that is applicable to
3-D topology optimization is suggested. All other details about
the optimization method, including the governing equations,
objective function, constraints, and optimization procedure, are
the same as our previous work [13]. In order to validate the proposed model, it is checked whether the fin geometry of the
topology-optimized design matches the multiscale structures that
are known to enhance thermal performance by utilizing the
unheated fluid at the entrance region. For further validation, the
effective heat transfer coefficient obtained by the proposed model
is compared to that obtained from numerical simulations. After the
validation, the complex fin geometry of the topology-optimized
heat sink is simplified to improve its manufacturability while
maintaining the essential features of the fin shape favorable to
the thermal performance. Finally, the thermal performance of the
newly obtained heat sink is compared to that of the radial platefin heat sink optimized using the existing correlation under the
constraint of the same physical volume.
2. The shape-dependent convection model for 3-D topology
optimization
In this study, topology optimization is performed with a new
shape-dependent convection model that is applicable to a 3-D
Greek symbol
a
THERMAL diffusivity [m2/s]
b
volumetric thermal expansion coefficient [1/K]
C
boundary of design domain [–]
c
relative density [–]
g
fin efficiency [–]
l
dynamic viscosity [N-s/m2]
m
kinematic viscosity [m2/s]
q
density [kg/m3]
X
computational domain [–]
Subscripts
conv
convection
eff
effective
f
fluid
fin
fin
ins
insulation
L
heat sink length
q
heat flux
S
solid
sur
surface
computational domain. This model predicts the local heat transfer
coefficient in natural convection without solving a full conjugate
heat transfer problem. In this section, the shape-dependent convection model is suggested based on the existing correlation for a
pin-fin heat sink [3]. This model accounts for the variation of the
heat transfer coefficient along the direction of gravity, which is
caused by developing thermal boundary layers.
To perform topology optimization with low computational cost,
a surrogate model predicting the local heat transfer coefficient for a
2-D computational domain was suggested in our previous study
[13]. The effective channel spacing, which offers a minimum distance to neighboring structures, was introduced to reflect the local
shape in the prediction of the local heat transfer coefficient. In 3-D
topology optimization, the concept of the effective channel spacing
is extended in order to obtain both horizontal and vertical spacing
under an arbitrary shape during optimization.
A staggered pin-fin array under consideration is shown in Fig. 1.
The horizontal spacing, Sh, is the horizontal distance between the
vertical single arrays, while the vertical spacing, Sv, is the distance
between the pin-fins in a single vertical array. For pin-fin heat
sinks in natural convection with a vertically oriented base plate
as depicted in Fig. 1, the correlation of the heat transfer coefficient
was proposed in the previous study, Ref. [3]:
81
8
¼ f ðS ; Sv ; dÞ ¼ ðh1:3 þ h1:3 þ h1:3 Þ1:3 þ h8
h
;
fin
h
fin;1
fin;2
fin;3
fin;4
ð1Þ
hfin;1 ¼
Sh Sv 4Sh Sv pd qf cp gbgfin ðT b T 1 Þ
;
pdL
48
mf
ð2Þ
hfin;2 ¼
kf
Sv
0:3669 0:0494 Gr1=4
L ;
L
d
ð3Þ
hfin;3 ¼
kf
;
½2:132Sh 0:4064Ra0:188
d
d
ð4Þ
kf
;
0:85Ra0:188
d
d
ð5Þ
2
and
hfin;4 ¼
34
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
Fig. 2. Schematic for the calculation of the effective fin diameter and channel
spacing.
expressed in Eq. (7), the value of the relative density at the k-th element is checked to determine whether the condition for reaching
the neighbor structures is satisfied.
As shown in Fig. 3, this channel spacing is decomposed into the
effective horizontal and vertical spacing as
Sh;eff;j ¼ wc;j cos h;
ð10Þ
and
Sv;eff;j ¼ wc;j sin h:
Fig. 1. Configuration and geometric parameters of a pin-fin heat sink.
where qf is the density of the fluid; cp is the specific heat; g is the
standard acceleration of gravity; b is the volumetric thermal expansion coefficient; gfin is the fin efficiency; Tb is the base temperature;
T1 is the ambient temperature; mf is the kinematic viscosity of the
fluid; kf is the thermal conductivity of the fluid; Gr is the Grashof
number; af is the thermal diffusivity of the fluid; and Ra is the Rayleigh number. This correlation requires the fin diameter (d), the horizontal spacing (Sh), and the vertical spacing (Sv) to predict the
average heat transfer coefficient for a fin array. In order to utilize
this correlation in 3-D topology optimization, three new parameters
are introduced under a local shape of a structure: the effective fin
diameter (deff), the effective horizontal spacing (Sh,eff), and the effective vertical spacing (Sv,eff).
The design variable for topology optimization is the relative
density (c) that ranges from 0 to 1. Elements with a relative density
of 1 are considered as solid region, and elements with a relative
density of 0 are considered as void region. For a solid-void interfacial element i, there are 26 neighbor elements named j. The effective fin diameter is obtained by calculating the maximum
thickness defined as
deff ¼ maxfjxk xi jjcðxi Þ cðxj Þ < 0; cðxk Þ < 0:25; j 2 Ne g;
ð6Þ
xk ¼ xi þ lðxj xi Þ;
ð7Þ
l ¼ 1; 2; :::; 1;
The heat transfer coefficient of the i-th element is determined
by choosing the direction where the heat transfer coefficient predicted by Eq. (1) has the minimum value
h
fin;i ¼ minfhfin;j hfin;j ¼ f ðSh;eff;j ; Sv;eff;j ; deff Þ; j 2 N e g
ð8Þ
xk ¼ xi þ lðxj xi Þ;
ð9Þ
l ¼ 1; 2; :::; 1:
Fig. 2 presents the schematic for the calculation of the effective
fin diameter and the channel spacing. The calculation for the distance to the neighbor structures starts in the direction of the j-th
element. While increasing the distance by |xj-xi| step by step as
ð12Þ
which is a function of local geometric parameters.
In 3-D topology optimization, the variation of the heat transfer
coefficient along the direction of gravity needs to be considered.
When there is a vertical plate that has a higher surface temperature than the ambient air, the thermal boundary layer is formed
near the wall with the natural convective flow. It is known that
the local heat transfer coefficient decreases along the flow direction, and the relationship between the local heat transfer coefficient and the location along the vertical direction is given by [5]
hz / z1=4 :
ð13Þ
In order for the shape-dependent convection model to reflect
the variation of the heat transfer coefficient in the z-direction,
the variation profile expressed in Eq. (13) was combined with Eq.
(1) as
hfin ¼
3 1=4 1=4
L hfin z
4
where x is the spatial location; c is the relative density of element;
and Ne is the 26 adjacent elements of the i-th element. According to
Eq. (6), the distances only along the directions where
cðxi Þ cðxj Þ < 0 are considered in determining the maximum distance. Before obtaining the effective horizontal and vertical spacing,
the channel spacing along the (xj-xi) directions, which means the
distance to neighboring structures, is obtained as
wc;j ¼ fjxk xi jjcðxi Þ cðxj Þ > 0; cðxk Þ > 0:25; j 2 Ne g;
ð11Þ
Fig. 3. Decomposition of wc into Sh,eff and Sv,eff.
ð14Þ
35
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
that satisfies
1
L
Z
L
:
hfin dz ¼ h
fin
ð15Þ
0
The average heat transfer coefficient appearing in Eq. (14) is
obtained by Eq. (12). According to Eq. (14), the local heat transfer
coefficient is the function of the effective fin diameter, horizontal
spacing, vertical spacing, and location along the z-axis.
3. Results and discussion
In the previous section, the shape-dependent convection model
accounting for the variation of the heat transfer coefficient in natural convection was proposed. In this section, the proposed model
is used to perform 3-D topology optimization under a specific computational domain. The physical validity of the newly obtained
design is checked by the comparison of the topology-optimized
design to the multiscale structures for convection problems
described in Chapter 6 of Ref. [17]. To estimate the thermal performance of the newly suggested designs, numerical simulations are
performed. Finally, a simplified design for 3-D topology optimization is developed in order to enhance its manufacturability.
3.1. Problem setup
Fig. 4. Cross-section of the computational domain (x-y plane).
Topology optimization with the shape-dependent convection
model was performed under a 3-D rectangular computational
domain. The 2-D domain in Fig. 4 is a cross-section of the 3-D
domain. To compare the results of 3-D topology optimization with
radial plate-fin heat sinks, the computational domain that is the
same as the radial plate-fin heat sinks was chosen. The center
Table 1
Properties of solid and fluid materials.
Solid
Fluid (Air)
Void
k (W/m-K)
q (kg/m3)
cp (J/kg-K)
l (Pa-s)
b (K1)
220
0.026
0.02
–
1.16
–
–
1007
–
–
1.85 105
–
–
3.34 103
–
Fig. 5. Optimization history: (a) 25th iteration, (b) 45th iteration, (c) 150th iteration, and (d) final iteration (995th iteration).
36
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
cylinder with a radius of 10 mm was set as the fixed structure. The
space beyond the outer radius (R) was set as the void region, so
structures are not designed on this region. The outer radius of
the domain, and the fin height were set to 20 mm and 10 mm,
respectively. The length of the domain was set to 40 mm. The computational domain was discretized using 60 60 40 cubic elements, and computations were performed with Intel Core i5 4core 3.5 GHz processors. The volume fraction was given as 0.084.
The filter radius was set to 2. The adiabatic boundary conditions
were applied at all boundary surfaces of the computational
domain. Uniform heat generation of q_ = 1.59 105 W/m3 was
assumed at the elements belonging to the center cylinder. Table 1
presents the properties of solid and fluid materials used in this
study. A large number of fins was given as the initial design, as
shown in Fig. 4. To reduce the computational time, only one quadrant was used for the computation.
3.2. Optimization history
are observed in the 3-D topology-optimized heat sink, the geometric features are summarized as follows. Fins are not connected in
the direction of gravity, so they look similar to pin-fins. In addition,
fins are branched at their tips, and this shape seems to increase the
surface area exposed to ambient air. These pin-like structures were
found to be maintained at higher volume fractions or lower element sizes. The heat transfer coefficient is large at the edge of
the outer cylinder where flat structures are observed. This is
because the structures near the outer cylinder are directly facing
the fresh air. The shape-dependent convection model accounts
for this effect by using the distance calculation algorithm that
yields a large value of the channel spacing when there is no material outside the outer radius of the domain.
For the analysis on the geometric characteristics, the number of
local fins along the z-direction was investigated, as shown in Fig. 7.
According to Da Silva and Bejan [19], the heat transfer performance
of a vertically oriented plate-fin heat sink in natural convection can
be enhanced by inserting additional fins at the entrance region.
Fig. 5 shows the optimization history. In the earlier stages, most
fins given at the initial design were removed, and wide channel
spacing was formed, except in the entrance region where z is small.
At the entrance region, more fins with irregular shapes were
observed. At the 45th iteration, the initially given rectangular fins
were cross-cut along the z-direction, and branched fin tips were
observed near the outer boundary. Near the entrance region, a
staggered fin array was formed with a larger number of local fins.
At the 150th iteration, most fins were divided into a larger number
of fins and evenly distributed in the computational domain. These
fins were branched near the outer boundary to increase the area in
contact with the ambient air. In the later stages, the number of fins
did not change much, and some fins were straightened along the
radial direction.
3.3. Physical validity of the topology-optimized design
Fig. 6 presents the configurations of the optimized radial platefin heat sink and the 3-D topology-optimized heat sink. The geometry of the radial plate-fin heat sink was optimized by using the
correlation suggested by An et al. [18]. Although complex shapes
Fig. 7. Number of local fins along the z-direction.
Fig. 6. Heat sink configurations: (a) radial plate-fin heat sink and (b) 3-D topology-optimized heat sink.
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
These additional fins utilize unheated air at the entrance region
where the thermal boundary layers have not yet formed. The same
characteristics were observed in the 3-D topology optimization
results. The maximum number of local fins is observed at the
entrance region, and then decreases and converges to a constant
value. Therefore, the 3-D topology-optimized heat sink seems to
utilize unheated air near the entrance region to enhance the thermal performance.
37
fication was performed. Fig. 8 depicts the geometry of the simplified 3-D heat sink. A T-shaped fin, which reflects the branched
fin shape observed in the 3-D topology-optimized heat sink, was
modeled with the minimum fin thickness of 1 mm. Except for
some L-shaped fins near the entrance region, these uniformly sized
T-shaped fins were arranged around the center cylinder. The staggered fin array with a larger number of local fins near the entrance
region was reflected. Fins were aligned at specific positions in the
z-direction while maintaining the total number of fins.
3.4. Design simplification
The 3-D topology-optimized heat sink shown in Fig. 6 has very
complex shapes that seem to be not manufacturable. To improve
manufacturability while maintaining the geometric characteristics
that have advantages in heat transfer performance, design simpli-
Fig. 8. Simplified heat sink based on 3-D topology-optimized heat sink.
3.5. Evaluation of the thermal performance
To evaluate the thermal performance of the 3-D topologyoptimized heat sinks numerically, ICEPAK [20], a commercial software provided by ANSYS, Inc., was used. The full-blown 3-D fluid
models are used in this simulation tool. Thus, the thermal performance evaluated by this simulation tool reflects the actual fluid
field caused by the complex geometries of the 3-D topologyoptimized heat sink. To obtain a 3-D physical model of the heat
sink from a 3-D relative density field, the STL writer code written
by Liu and Tovar [21] was used with a threshold value of the relative density 0.35. This value was chosen empirically to have the
volume fraction maintained while transforming the relative density field into a 3-D physical model. This means the value of 0.35
properly represents the solid-void interface, so the threshold value
for the condition for reaching the neighbor structures was chosen
near 0.35. The threshold value of 0.25, which was used in Section 2,
was chosen in order to lower the possibility of ignoring the existence of neighbor structures in the convection model by using a
value that is lower than 0.35. However, the results of topology
optimization were virtually unaffected by this threshold value
within the range of 0.25–0.35. After generating a mesh geometry,
the geometry was refined by using MeshLab, an open-source
mesh-processing tool, in order to obtained a 3-D model which is
more suitable for numerical simulations. This 3-D model was
directly imported to the simulation tool.
Fig. 9 shows the computational domain for numerical simulations. The size of the domain is 10R 10R 10L, which is large
enough to exclude the domain size effect on the performance
evaluation. Numerical simulations were performed only for the
Fig. 9. Computational domain for numerical simulations.
38
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
quadrant by applying symmetric boundary conditions at min-x and
min-y walls. On the remaining boundaries, the pressure was set to
the ambient pressure and the velocities were not predetermined.
Uniform heat generation was given at the center cylinder. Grid
tests were conducted for the topology-optimized heat sink. The
number of grids changed from 22,855 to 513,671 nodes. The
changes in the average base temperature divided by the base-toambient temperature difference did not exceed 1% after the
number of grids exceeded 267,305 nodes. Thus, the grid of
267,305 nodes was used in the numerical simulations. Other
details about the numerical simulations are the same as in the
method suggested by Joo and Kim [22].
The thermal resistance given by
Rth ¼
Tb T1
;
Q
ð16Þ
where Q is the total heat input, was used as the performance index.
Table 2 presents the results of the numerical simulations. The thermal resistance of the radial plate-fin heat sink in Fig. 6(a) was also
evaluated numerically under the same computational domain for
the comparison. The volume fraction (f) in Table 2 reflects the mass
of a heat sink, which is another performance index for the comparison. The 3-D topology-optimized heat sink has 13% lower thermal
resistance and 48% less mass than the optimized radial plate-fin
heat sink. After the design simplification, the thermal resistance
increased by 1.2%, but it is still much lower than that of the radial
plate-fin heat sink.
3.6. Accuracy and efficiency of the shape-dependent convection model
In order to validate the proposed shape-dependent convection
model, the effective heat transfer coefficient obtained from this
model was compared to that obtained from the numerical simulations, as shown in Table 2. The effective heat transfer coefficient is
defined as
¼
h
eff
Q
AðT b T 1 Þ
ð17Þ
where A is the surface area of a heat sink. The difference in the
effective heat transfer coefficient is about 11%. Even with the simple
shape-dependent convection model, the convective heat transfer
from complex 3-D shapes is fairly well predicted.
The spatial variation of the heat transfer coefficient was also
investigated. Fig. 10 shows how the distribution of the heat transfer coefficient used in topology optimization compares to that
obtained from the numerical simulation. It can be seen that the
large values of the heat transfer coefficient are observed near the
outer radius of the domain in both results. The heat transfer coefficient decreases gradually along the z-direction while having a
large value in the entrance region. This is why the relatively large
number of fins is located near the entrance region. Fig. 11 shows
the temperature distributions from topology optimization and
the numerical simulation. The maximum temperature is observed
at the heat source, and the decrease of the temperature along the
radial direction in a fin structure confirms that the heat generated
within the source flows through the fin structures. Fig. 12 presents
the comparison of the interface temperature distribution obtained
Fig. 10. Distributions of the heat transfer coefficient: (a) the shape-dependent
convection model (b) numerical simulation by ICEPAK.
from topology optimization to that obtained from the numerical
simulation. In the result of the numerical simulation, the lower
part of the heat sink has lower temperatures than the upper part.
This means that for the same geometry in this lower part, the full
Table 2
Thermal resistances and effective heat transfer coefficients of the topology-optimized heat sinks and the optimized radial plate-fin heat sink at Q = 2 W.
Case
f (-)
Rth,numerical (K/W)
2
h
eff;numerical (W/m -K)
2
h
eff;topology (W/m -K)
Original topology-optimized HS (Fig. 6(b))
Simplified topology-optimized HS (Fig. 8)
Optimized radial plate-fin HS (Fig. 6(a))
0.084
0.093
0.163
10.44
10.57
11.76
6.82
6.96
5.19
6.13
–
–
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
39
Fig. 11. Distributions of the temperature at the entrance region (min-z): (a)
topology optimization (b) numerical simulation by ICEPAK.
conjugate heat transfer model predicted the higher cooling performance than the shape-dependent convection model.
It was confirmed that the shape-dependent convection model
fairly well predicts the spatial variation of the heat transfer coefficient. However, it should be noted that this convection model is
not generally applicable to all types of 3-D design problems in its
present form. By extending the convection model to cover the
shapes of vertical fins and horizontal plates, the presented method
will be able to deal with various types of 3-D design problems.
The computational time required for 500 iterations was about
12 h in this study with 4 cores of processors. The shapedependent convection model took 38% of the total computation
time, and this is lower than the time taken by the finite element
analysis solving only the conduction equation. It was reported that
the 3-D topology optimization using the full-blown fluid model
took about 10 h, with 1280 cores of processors at a mesh resolution
of 160 320 160 [15]. With the same mesh resolution and the
number of cores, the computational time for using the shapedependent convection model is expected to be about 1/5 of that
using the full-blown fluid model. The computation time for solving
the full conjugate heat transfer problem can be reduced by implementing the local meshing. On the other hand, the shapedependent convection model presented in this study is going to
Fig. 12. Distributions of the interface temperature: (a) topology optimization (b)
numerical simulation by ICEPAK.
be combined with a more efficient topology optimization method
currently being studied, so that the computational time can be
even further reduced.
4. Conclusion
In this study, heat sinks in natural convection were thermally
optimized using the three-dimensional topology optimization
method. A new shape-dependent convection model that provides
accurate predictions of the heat transfer coefficient while having
a significantly less computational load was developed. This model
accounts for the variation of the heat transfer coefficient
depending not only on the local shape of the fins but also on the
development of the thermal boundary layer. The effective heat
transfer coefficient obtained from the proposed model was in close
40
Y. Joo et al. / International Journal of Heat and Mass Transfer 127 (2018) 32–40
agreement with that obtained from numerical simulations, within
an error of 11%. The new design obtained from three-dimensional
topology optimization had the multiscale structures that enhanced
thermal performance by utilizing the unheated fluid at the
entrance region. Based on the original topology-optimized heat
sink that had complex shapes, a more manufacturable design
was suggested via design simplification. In order to compare the
thermal performance of the topology-optimized designs against
that of an existing optimized design, the fin geometry of a heat sink
was optimized using two methods: the method proposed in this
study for the former, and an analytical method based on the existing correlation for the latter. From the comparison, it was found
that the topology-optimized heat sink has 13% lower thermal
resistance and 48% less mass than the optimized radial plate-fin
heat sink. Therefore, three-dimensional topology optimization
suggested in this study is expected to provide novel designs with
improved thermal performance and reduced mass for various
applications.
Conflict of interest
None declared.
Acknowledgement
This work was supported by the National Research Foundation
of Korea (NRF) grant funded by the Korea government (MSIT) (No.
2012R1A3A2026427).
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