energies
Article
Experimental Study of Natural Convection Cooling
of Vertical Cylinders with Inclined Plate Fins
Jong Bum Lee, Hyun Jung Kim and Dong-Kwon Kim *
Department of Mechanical Engineering, Ajou University, Suwon 443-749, Korea; eamice@ajou.ac.kr (J.B.L.);
hyunkim@ajou.ac.kr (H.J.K.)
* Correspondence: dkim@ajou.ac.kr; Tel.: +82-31-219-3660
Academic Editors: Vincent Lemort and Samuel Gendebien
Received: 12 April 2016; Accepted: 17 May 2016; Published: 24 May 2016
Abstract: In this paper, natural convection from vertical cylinders with inclined plate fins is
investigated experimentally for use in cooling electronic equipment. Extensive experimental
investigations are performed for various inclination angles, fin numbers, and base temperatures.
From the experimental data, a correlation for estimating the Nusselt number is proposed. The
correlation is applicable when the Rayleigh number, inclination angle, and fin number are in the
ranges 100,000–600,000, 30˝ –90˝ , and 9–36, respectively. Using the correlation, a contour map
depicting the thermal resistance as a function of the fin number and fin thickness is presented. Finally,
the optimal thermal resistances of cylinders with inclined plate fins and conventional radial plate fins
are compared. It is found that that the optimal thermal resistance of the cylinder with inclined fins is
30% lower than that of the cylinder with radial plate fins.
Keywords: nusselt number; natural convection; inclined plate fin
1. Introduction
Heat dissipation from electronic devices increases considerably with continuous needs for high
performance, cost effective, and miniaturized electronic devices [1–3]. The high power dissipation
from the electronic device raises the junction temperature of the device, which adversely influences
the overall performance and durability of the device. One example is a light-emitting diode (LED), the
efficiency of which is significantly reduced as the junction temperature increases due to an increase in
the nonradiative recombinations through defect states and an increase in the leakage of carriers from
the quantum well [4,5]. Therefore, thermal management of electronic devices is important to maintain
their performance and durability [6,7]. Various cooling techniques for thermal management have been
suggested; among these, natural convective heat sinks have proven to be appropriate because of their
outstanding simplicity, reliability, and low cost [8].
Many studies have focused on natural convective heat sinks, as reviewed in Martynenko and
Khramtsov [9] and in Raithby and Hollands [10]. In particular, many previous researchers investigated
natural convection from cylindrical heat sinks, which are cylindrical bodies with fins attached. These
studies are motivated by the fact that the heat dissipation from the cylinder can be increased by
using the fins. Sparrow and Bahrami investigated heat transfer from square vertical fins attached
to a horizontal tube by using the naphthalene sublimation technique [11]. Chen and Chou also
conducted an experimental study of horizontal cylinders with square vertical fins [12]. Yildiz and
Yüncü suggested a Nusselt number correlation for annular fin arrays mounted on a horizontal cylinder
from their experimental data [13]. Hahne and Zhu also suggested a Nusselt number correlation
for horizontal cylinders with annular fins from measured heat transfer coefficients [14]. Recently,
vertical cylinders with vertically oriented radial plate fins, as shown in Figure 1, are widely used for
Energies 2016, 9, 391; doi:10.3390/en9060391
www.mdpi.com/journal/energies
Energies 2016, 9, 391
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Energies 2016, 9, 391
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coolingused
LEDforlighting.
For lighting.
natural For
convection
from these
cylinders
with radial
plateplate
fins, fins,
an empirical
cooling LED
natural convection
from
these cylinders
with radial
an
correlation
for
estimating
the
Nusselt
number
was
proposed
by
our
group
[15].
empirical correlation for estimating the Nusselt number was proposed by our group [15].
Energies 2016, 9, 391
2 of 15
used for cooling LED lighting. For natural convection from these cylinders with radial plate fins, an
empirical correlation for estimating the Nusselt number was proposed by our group [15].
Figure
1. Cylindricalheat
heat sink
sink with
plate
fins.fins.
Figure
1. Cylindrical
withradial
radial
plate
Heat transfer through inclined fins is currently of considerable interest. Takeishi et al. investigated
Heat
transferheat
through
inclined
fins is currently
considerable
Takeishi
al. investigated
convective
transfer
in rectangular
ducts withofinclined
pin fins.interest.
They showed
thatetducts
with
Figure
1.
Cylindrical
heat
sink
with
radial
plate
fins.
inclined
pintransfer
fins havein
higher
thermal performance
to those
normal
pin fins
because
convective
heat
rectangular
ducts with compared
inclined pin
fins.with
They
showed
that
ducts with
of pin
the low
loss and
the extended
area of inclined
pin fin
surfaces
[16]. normal
Hagote and
inclined
finspressure
have through
higher
thermal
performance
compared
tointerest.
those with
pinDahake
fins because
Heat transfer
inclined
fins is
currently
considerable
al. investigated
investigated
heat transfer from
vertical
plates
with of
inclined
plate fins
underTakeishi
natural et
convection.
They
of the convective
low pressure
loss
andinthe
extendedducts
areawith
of inclined
pin
[16]. that
Hagote
and
Dahake
heatthat
transfer
rectangular
inclined
pin fin
fins.surfaces
They
ducts
with
demonstrated
the average
heat transfer coefficient
is maximized
whenshowed
the inclination
angle
is
investigated
heat
transfer
from vertical
plates
with inclined
plate
finswith
under
natural
convection.
They
inclined
pin
fins
have
higher
thermal
performance
compared
to
those
normal
pin
fins
because
60°[17]. From these studies, we can expect that the thermal performance of vertical cylinders with
of
the
low
pressure
loss
and
the
extended
area
of
inclined
pin
fin
surfaces
[16].
Hagote
and
Dahake
demonstrated
that
the
average
heat
transfer
coefficient
is
maximized
when
the
inclination
angle
is
radial plate fins (Figure 1) will be further enhanced by employing inclined plate fins as shown in
˝
investigated
heat
transfer
from
vertical
plates
with
inclined
plate
fins
under
natural
convection.
They
60 [17].
From
studies,
we the
canfinexpect
thermal
of vertical
Figure
2. Itthese
is mainly
because
surfacethat
areathe
of the
verticalperformance
cylinder with inclined
fins cylinders
is greater with
thatwith
the1)radial
average
transfer
coefficient
is employing
maximized
the is
inclination
is
compared
that
fins
when
the
spaceby
reserved
for thewhen
heat
sink
fixed, fins
i.e., angle
the
radial demonstrated
plate finsto(Figure
will plate
beheat
further
enhanced
inclined
plate
as heat
shown
in
60°
[17].
From
these
studies,
we
can
expect
that
the
thermal
performance
of
vertical
cylinders
with
L and
the height
H are
fixed.area
However,
the best
of our knowledge,
the fins
natural
Figure sink
2. It length
is mainly
because
the fin
surface
of the to
vertical
cylinder
with inclined
is greater
radial
plate from
fins (Figure
1)
will be with
further
enhanced
by
employing
inclined
plate fins as
shown in
convection
vertical
cylinders
inclined
fins
has not been
investigated
compared
to2.that
with
radial
plate
fins
when
theplate
space
reserved
forexperimentally
the inclined
heat sink
isisfixed,
Figure
It
is
mainly
because
the
fin
surface
area
of
the
vertical
cylinder
with
fins
greateri.e., the
yet. As a result, there is no reliable experimental data by which one can obtain the Nusselt numbers,
heat sink
length
L
and
the
height
H
are
fixed.
However,
to
the
best
of
our
knowledge,
natural
compared
withto
radial
plate
when resistance
the space reserved
forcylinders
the heat sink
fixed, i.e.,plate
thethe
heat
and there to
is that
no way
obtain
thefins
thermal
of vertical
withis inclined
fins
convection
from vertical
cylinders
inclined
platein
fins
has
notofbeen
experimentally
investigated
sink
length
L and
the
height
Hwith
are of
fixed.
However,
tothe
the
best
ourof
knowledge,
the inclined
natural
quantitatively.
Therefore,
the degree
improvement
performance
cylinders with
from
vertical
cylinders
with
inclined
plate
fins
has
not
been
experimentally
investigated
yet. Asconvection
a
result,
there
is
no
reliable
experimental
data
by
which
one
can
obtain
the
Nusselt
numbers,
plate fins compared to that of cylinders with conventional radial plate fins, and the conditions under
yet.
As
a
result,
there
is
no
reliable
experimental
data
by
which
one
can
obtain
the
Nusselt
numbers,
and there
is
no
way
to
obtain
the
thermal
resistance
of
vertical
cylinders
with
inclined
plate
fins
which this improvement is achieved, are not clear.
and there is no way to obtain the thermal resistance of vertical cylinders with inclined plate fins
quantitatively. Therefore, the degree of improvement in the performance of cylinders with inclined
quantitatively. Therefore, the degree of improvement in the performance of cylinders with inclined
plate fins compared to that of cylinders with conventional radial plate fins, and the conditions under
plate fins compared to that of cylinders with conventional radial plate fins, and the conditions under
which which
this improvement
is achieved,
are
this improvement
is achieved,
arenot
notclear.
clear.
(a)
(b)
Figure 2. Cylindrical heat sink with inclined plate fins. (a) schematic diagram; (b) top view.
(a)
(b)
Figure 2. Cylindrical heat sink with inclined plate fins. (a) schematic diagram; (b) top view.
Figure 2. Cylindrical heat sink with inclined plate fins. (a) schematic diagram; (b) top view.
Energies 2016, 9, 391
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The purpose of this study is to investigate natural convection cooling of vertical cylinders with
inclined plate fins. Extensive experiments for various inclination angles, fin numbers, and base
temperatures were conducted. From these results, a Nusselt number correlation is suggested. Using
the correlation, a contour map depicting the thermal resistance as a function of the fin number and
fin thickness is presented. Finally, the thermal resistances of cylinders with inclined plate fins and
conventional radial plate fins are compared.
2. Experimental Investigation
The thermal resistances of natural convective heat sinks with inclined plate fins were measured
for various inclination angles, fin numbers, and base temperatures. Schematic diagrams of the heat
sinks are presented in Figure 2a, and the dimensions of the heat sinks are listed in Table 1. Eleven
different heat sinks were used to cover a wide range of inclination angles and fin numbers. In Table 1,
the heat sink with large fin number (N = 36) and large inclination angle (α = 90˝ ) was excluded because
the adjacent fins touch and bend each other in this heat sink because the space required for a single fin
was insufficient for this case. An annular cylinder base and inclined fins for the heat sinks were made
of aluminum alloys 6061 (k = 167 W/mK) and 5052 (k = 138 W/mK), respectively.
Table 1. Geometric configuration of tested heat sinks.
Number
N
α (˝ )
Heat sink 1
Heat sink 2
Heat sink 3
Heat sink 4
9
12
18
36
30
Heat sink 5
Heat sink 6
Heat sink 7
Heat sink 8
9
12
18
36
60
Heat sink 9
Heat sink 10
Heat sink 11
9
12
18
90
H (mm)
D (mm)
L (mm)
T (mm)
30
60
50
1.0
The heat sinks were assembled by interference fitting of the fins and base. Then, a cartridge
heater was inserted into the inner hole of the base (Figure 3a). A thermal interface material (TC 5080;
Dow Corning: Auburn, MI, USA) was used to minimize the contact thermal resistance between the
heat sink base and the heater. The supporting cylindrical blocks, the length of which was 200 mm, were
made of Teflon to minimize the heat loss from the top and bottom sides of the base (Figure 3b,c). Power
was supplied to the heater from a power supply (E3633A; Agilent Technologies: Santa Clara, CA, USA)
in the range 2–45 W. To calculate the actual heat transfer rate to the heat sink, the heat loss through the
supporting blocks (qloss,1 + qloss,2 in Figure 3b) was measured and subtracted from the electric power
supplied to the heater (Appendix A). To measure the base temperature, four T-type thermocouples
were circumferentially attached to the cylinder (Figure 3d) A data acquisition unit (34970A DAQ;
Agilent Technology) was used to acquire the signals from the thermocouples and convert them into
temperature data. The temperatures were measured until the change in the temperature was smaller
than ˘0.1 ˝ C in a 2 min period. The experiments were performed in an isolated and quiescent room.
Energies 2016, 9, 391
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Energies 2016, 9, 391
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(a)
(b)
(c)
(d)
Figure 3. Experimental setup. (a) photograph of a heat sink assembly; (b) schematic diagram of
Figure 3. Experimental setup. (a) photograph of a heat sink assembly; (b) schematic diagram of
experimental apparatus; (c) photograph of experimental apparatus; (d) positions of thermocouples.
experimental apparatus; (c) photograph of experimental apparatus; (d) positions of thermocouples.
3. Results and Discussion
3. Results and Discussion
The difference between the heat sink base temperature (Tw) and the ambient temperature (Tamb)
for various
inclination
angles
numbers
(N), and heat inputs
(q) isthe
shown
in Table
2 and Figure
The
difference
between
the (α),
heatfinsink
base temperature
(Tw ) and
ambient
temperature
(Tamb )
4. From
these experimental
data,
thermal(N),
resistance
of the
heat (q)
sinkiscan
be calculated
for various
inclination
angles (α),
finthe
numbers
and heat
inputs
shown
in Table as
2 and Figure 4.
From these experimental data, the thermal resistance
T of
T the heat sink can be calculated as
Rth 
Rth
w
amb
Tw ´qTamb
”
q
Table 2. Thermal resistances and Nusselt numbers calculated from experimental data.
Table 2. Thermal resistances and Nusselt numbers calculated from experimental data.
Number α (°) N
q (W)
Tw − Tamb (K) Rth (K/W)
NuDh
2.15
±
0.06
10.1
±
0.8
4.69
±
0.4
9.13
± 0.78
Number α (˝ )
N
q (W)
Tw ´ Tamb (K)
R (K/W)
Nu
4.94 ± 0.11
19.9 ± 1.3
4.04 ± th
0.27 10.6 ± 0.71 Dh
2.15
˘
0.06
10.1
˘
0.8
4.69
˘
0.4
9.13
1
30
9
8.26 ± 0.12
31.2 ± 0.8
3.78 ± 0.11 11.32 ± 0.33˘ 0.78
4.94 ˘ 0.11
19.9 ˘ 1.3
4.04 ˘ 0.27
10.6 ˘ 0.71
11.29 ± 0.05
39.6 ± 0.7
3.51 ± 0.07 12.19 ± 0.23
8.26 ˘ 0.12
31.2 ˘ 0.8
3.78 ˘ 0.11
11.32 ˘ 0.33
30
9
1
14.79
50.8
±˘
0.90.7 3.44 3.51
± 0.06
12.45 ±12.19
0.23˘ 0.23
11.29 ±
˘0.07
0.05
39.6
˘ 0.07
2.46 ±˘0.01
9.8
± 0.5
± 0.21
6.77 ±12.45
0.36 ˘ 0.23
14.79
0.07
50.8
˘ 0.9 3.98 3.44
˘ 0.06
6.16
21.2
0.5
± 0.09
0.2 ˘ 0.36
2.46 ±˘0.01
0.01
9.8±˘
0.5 3.44 3.98
˘ 0.217.86 ±6.77
2
30
12 6.16
9.7 ±˘0.01
30
± 0.5
0.05
8.72 ± 0.15
0.01
21.2
˘ 0.5 3.1 ±
3.44
˘ 0.09
7.86 ˘ 0.2
9.7 ˘± 0.01
30±˘1.5
0.5 2.78 ±
3.10.11
˘ 0.059.72 ± 8.72
30
2
12
14.34
0.13
39.8
0.38˘ 0.15
14.34 ±
˘0.19
0.13
39.8
˘ 0.11
18.22
49.7
±˘
0.81.5 2.73 2.78
± 0.05
9.89 ± 9.72
0.19˘ 0.38
18.22 ˘ 0.19
49.7 ˘ 0.8
2.73 ˘ 0.05
9.89 ˘ 0.19
3.77 ± 0.23
10.8 ± 0.6
2.85 ± 0.24
4.7 ± 0.4
3
30
18 3.77 ˘ 0.23
10.8±˘1 0.6 2.34 2.85
˘ 0.245.72 ± 0.4
4.7 ˘ 0.4
8.75 ± 0.45
20.5
± 0.16
8.75 ˘ 0.45
20.5 ˘ 1
2.34 ˘ 0.16
5.72 ˘ 0.4
14.5 ˘ 0.29
30.2 ˘ 0.8
2.08 ˘ 0.07
6.44 ˘ 0.22
3
30
18
20.47 ˘ 0.44
40.2 ˘ 0.8
1.96 ˘ 0.06
6.83 ˘ 0.2
26.91 ˘ 0.07
50.4 ˘ 1
1.87 ˘ 0.04
7.16 ˘ 0.14
(1)
(1)
Energies 2016, 9, 391
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Table 2. Cont.
Number
4
5
6
7
8
9
10
11
α (˝ )
30
60
60
60
60
90
90
90
N
q (W)
Tw ´ Tamb (K)
Rth (K/W)
NuDh
36
4.82 ˘ 0.09
11.71 ˘ 0.07
22.32 ˘ 1.53
30.71 ˘ 0.68
41.52 ˘ 0.51
10.8 ˘ 0.5
19.4 ˘ 0.7
30.6 ˘ 1.8
40 ˘ 0.8
49.1 ˘ 0.6
2.25 ˘ 0.12
1.65 ˘ 0.06
1.37 ˘ 0.12
1.3 ˘ 0.04
1.18 ˘ 0.02
1.6 ˘ 0.08
2.17 ˘ 0.08
2.62 ˘ 0.23
2.76 ˘ 0.08
3.04 ˘ 0.05
9
2.56 ˘ 0.01
6.11 ˘ 0.04
9.3 ˘ 0.03
14.09 ˘ 0.05
18.15 ˘ 0.03
10.4 ˘ 0.5
20.5 ˘ 0.5
30 ˘ 0.6
40.8 ˘ 0.9
51 ˘ 0.6
4.07 ˘ 0.2
3.36 ˘ 0.09
3.22 ˘ 0.06
2.89 ˘ 0.06
2.81 ˘ 0.03
7.69 ˘ 0.38
9.31 ˘ 0.24
9.71 ˘ 0.19
10.81 ˘ 0.23
11.13 ˘ 0.13
12
3.05 ˘ 0.02
7.42 ˘ 0.02
11.55 ˘ 0.03
17.14 ˘ 0.04
22.35 ˘ 0.02
10.1 ˘ 0.5
20.3 ˘ 0.6
29.3 ˘ 0.6
40.3 ˘ 0.6
50 ˘ 0.5
3.33 ˘ 0.17
2.74 ˘ 0.08
2.54 ˘ 0.05
2.35 ˘ 0.04
2.24 ˘ 0.02
5.81 ˘ 0.3
7.05 ˘ 0.19
7.61 ˘ 0.15
8.22 ˘ 0.13
8.64 ˘ 0.09
18
4.24 ˘ 0.02
10.67 ˘ 0.21
16.57 ˘ 0.03
23.15 ˘ 0.05
31.6 ˘ 0.04
10.2 ˘ 0.5
20.5 ˘ 0.6
30.2 ˘ 0.6
39.9 ˘ 0.7
50.6 ˘ 0.7
2.42 ˘ 0.13
1.92 ˘ 0.07
1.82 ˘ 0.04
1.72 ˘ 0.03
1.6 ˘ 0.02
3.87 ˘ 0.21
4.88 ˘ 0.17
5.13 ˘ 0.1
5.43 ˘ 0.1
5.83 ˘ 0.08
36
4.91 ˘ 0.06
12.57 ˘ 0.02
21.51 ˘ 0.07
31.78 ˘ 0.43
43.55 ˘ 0.34
10.1 ˘ 0.5
19.9 ˘ 0.5
30 ˘ 0.6
40.1 ˘ 0.5
50.4 ˘ 0.6
2.06 ˘ 0.11
1.59 ˘ 0.04
1.39 ˘ 0.03
1.26 ˘ 0.02
1.16 ˘ 0.02
1.16 ˘ 0.06
1.51 ˘ 0.04
1.72 ˘ 0.03
1.9 ˘ 0.04
2.07 ˘ 0.03
9
2.48 ˘ 0.04
5.68 ˘ 0.02
9.4 ˘ 0.03
13.88 ˘ 0.03
17.95 ˘ 0.02
10 ˘ 0.6
19.5 ˘ 0.5
29.6 ˘ 0.6
40.5 ˘ 0.6
50.7 ˘ 0.5
4.04 ˘ 0.26
3.43 ˘ 0.09
3.15 ˘ 0.06
2.92 ˘ 0.04
2.83 ˘ 0.03
4.8 ˘ 0.31
5.64 ˘ 0.15
6.16 ˘ 0.12
6.63 ˘ 0.1
6.85 ˘ 0.07
12
3.02 ˘ 0.01
7.32 ˘ 0.04
11.23 ˘ 0.06
16.16 ˘ 0.05
21.93 ˘ 0.06
9.6 ˘ 0.5
20.7 ˘ 0.7
28.9 ˘ 0.9
38.7 ˘ 0.8
49.2 ˘ 0.8
3.18 ˘ 0.17
2.83 ˘ 0.1
2.58 ˘ 0.08
2.4 ˘ 0.05
2.24 ˘ 0.04
3.66 ˘ 0.2
4.11 ˘ 0.14
4.52 ˘ 0.14
4.85 ˘ 0.1
5.19 ˘ 0.09
18
3.83 ˘ 0.03
10.42 ˘ 0
16.7 ˘ 0.03
23.1 ˘ 0.18
30.48 ˘ 0.33
9.9 ˘ 0.6
21.9 ˘ 0.5
31.7 ˘ 0.6
41.2 ˘ 0.6
50.9 ˘ 0.6
2.58 ˘ 0.15
2.1 ˘ 0.05
1.9 ˘ 0.04
1.78 ˘ 0.03
1.67 ˘ 0.03
2.1 ˘ 0.12
2.58 ˘ 0.06
2.86 ˘ 0.05
3.04 ˘ 0.05
3.25 ˘ 0.05
In Equation (1), the thermal resistance Rth is defined as the difference between the heat sink
base temperature and the ambient temperature per unit heat transfer rate. The calculated thermal
resistance values are listed in Table 2. The uncertainties of the calculated thermal resistances are
also listed. As shown in Table 2 and Figure 4, the thermal performance of the inclined plate fin heat
sink is minimized when α = 60˝ and N = 36. According to Figure 3 in [15], the thermal resistance of
the conventional radial plate fin heat sink is also minimized when N = 36. In Figure 5, the minimal
resistances measured from the inclined plate fin heat sink (α = 60˝ , N = 36) in the present study are
compared with those from the radial plate fin heat sink (N = 36) in [15]. Figure 5 shows that the
inclined plate fin heat sink has thermal resistances up to 10% lower than those of the radial plate fin
heat sink.
Energies 2016, 9, 391
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Energies 2016, 9, 391
6 of 15
Energies
2016, 9, 391
compared
with those from the radial plate fin heat sink (N = 36) in [15]. Figure 5 shows that the inclined6 of 15
compared
with
those
the radial
plate up
fin to
heat
sink
(N =than
36) in
[15].of
Figure
5 shows
that
inclined
plate fin heat
sink
hasfrom
thermal
resistances
10%
lower
those
the radial
plate
finthe
heat
sink.
plate fin heat sink has thermal resistances up to 10% lower than those of the radial plate fin heat sink.
Figure 4. Temperature differences for various fin numbers, inclination angles, and heat inputs.
Figure 4. Temperature differences for various fin numbers, inclination angles, and heat inputs.
Figure 4. Temperature differences for various fin numbers, inclination angles, and heat inputs.
Figure 5. Comparison of experimental data for minimal resistances between the inclined fin heat sink
Figure
Comparison
experimental
data
and the5.radial
fin heat of
sink
(H = 30 mm,
N =for
36).minimal resistances between the inclined fin heat sink
Figure 5. Comparison of experimental data for minimal resistances between the inclined fin heat sink
and the radial fin heat sink (H = 30 mm, N = 36).
and the
fin heat
sink
= 30 the
mm,thermal
N = 36).resistance can be calculated as follows:
Onradial
the basis
of the
fin (H
model,
On the basis of the fin model, the thermal resistance can be calculated as follows:
1
Rth  h(NAf  Ab )
h(NA1f  Ab )
(2)
(2)
 hp
 
hp 
tanh
H

h
ks 






f
hp
khp
(hpks Ac )0.5 tanhˆ ks Ac H ˙

h
s Ac k s 





f
ˆ
N
  (hpk A )0.5
bk A
b
ks Ahpc  ˙
s c



hp
s c
hA



hpH f ` h hpk A k s

f
tanh
c k  tanh 
phpk s hA
Ac qf 0.5 1   h ks A
hp
hpAs Hc f  ˙
ˆ hNbks Ac k s ˙tanh  ˆb
η“
k
1


 s chpH f 
hA f
hp s 
kskA
1 ` h
k
tanh
s
c

 ks Akc A Hf
A
(3)
(3)
On the basis of the fin model, the thermal
Rth  resistance
1 can be calculated as follows:
Rth “ heat transfer coefficient, fin surface area, and unfinned (2)
where η, h, Af, and Ab are the fin efficiency,
hpηN A f ` Ab q
where
h, Arespectively.
f, and Ab are the fin efficiency, heat transfer coefficient, fin surface area, and unfinned
surfaceη,area,
surface
area,
respectively.
where
η, h, A
, and
A are the fin efficiency, heat transfer coefficient, fin surface area, and unfinned
f
b
surface area, respectively.
s
c
s
(3)
c
h “ Nu Dh k f {Dh
(4)
Ab “ πDL ´ NLt
(5)
A f “ 2H f L ` 2H f t ` Lt
(6)
Energies 2016, 9, 391
7 of 15
Here, Ac and p are the fin cross-sectional area and fin perimeter, respectively.
Ac “ Lt
(7)
p “ 2L ` 2t
(8)
The fin height Hf is given as a function of α, D, and H:
b
Hf “
D2 cos2 α{4 ` H 2 ` HD ´ Dcosα{2
(9)
Dh is the hydraulic diameter of the channel between two adjacent fins, i.e., the yellow region in
Figure 2b, and can be calculated from the area Ach and the wetted perimeter pch of the channel:
Dh “
4Ach
“
pch
´
¯
4 πpH ` D{2q2 {N ´ πpD{2q2 {N ´ H f t
2H f ` 2πR{N ´ t
(10)
The Nusselt number NuDh can be calculated from Equations (2)–(10) once the thermal resistance
is obtained from the experimental data. The calculated Nusselt numbers are listed in Table 2.
In the present study, the Nusselt number correlation, which match best with the experimental
data, is developed. In the case of the heat sinks with vertical fins under natural convection, the Nusselt
number correlations were generally developed as functions of a Rayleigh number RaDh . For example,
Welling and Woodridge [8] suggested a Nusselt number correlation for rectangular vertical fins on the
flat surface in the form of
Nu Dh “ f pRa Dh q
(11)
where the Rayleigh number RaDh is defined as
Ra Dh “
gβ f pTw ´ Tamb qDh4
νf αf L
(12)
Similarly, Karagiozis et al. [18] used the average fin spacing as the hydraulic diameter, and
suggested a correlation for triangular vertical fins on the flat surface in the form of Equation (11).
In the present study, to reflect the effect of the inclination angle on the heat transfer coefficient, the
Equation (11) is modified as
Nu Dh “ f pRa Dh , αq
(13)
Here, g, βf , νf , and αf are the gravitational acceleration, volume expansion coefficient of fluid,
kinematic viscosity of fluid, and thermal diffusivity of fluid, respectively. We tested various functional
forms for the function f in Equation (13), and finally found that the functional form
Nu Dh “ C1 ` C2 lnpRa Dh q ` pC3 ` C4 α ` C5 α2 qlnpRa Dh q2
(14)
is matched best with the experimental data. The best empirical coefficients for predicting the Nusselt
numbers are determined using a least-squares fit on the experimental data:
C1 “ 2.41, C2 “ ´0.926, C3 “ 0.120, C4 “ 7.72 ¨ 10´4 , C5 “ ´8.24 ¨ 10´6
(15)
Finally, the correlation of the Nusselt number for the vertical cylinders with inclined plate fins
under natural convection is given as
Nu Dh “ 2.41 ´ 0.926 lnpRa Dh q ` p0.120 ` 7.72 ¨ 10´4 α ´ 8.24 ¨ 10´6 α2 qlnpRa Dh q2
(16)
Energies 2016, 9, 391
8 of 15
Finally, the correlation of the Nusselt number for the vertical cylinders with inclined plate fins
8 of 15
under natural convection is given as
Energies 2016, 9, 391
Nu  2.41  0.926 ln( Ra )  (0.120  7.72 104   8.24 106 2 )ln( Ra )2
(16)
Dh
In Figure Dh
6, the Nusselt numbersDhcalculated from Equation (16) are compared with
those obtained
In Figure 6,data
the for
Nusselt
numbers
calculated
from Equation
are that
compared
with those
from experimental
various
Rayleigh
numbers.
Figure 6 (16)
shows
the Nusselt
number
obtained
from
experimental
data
for
various
Rayleigh
numbers.
Figure
6
shows
that
the
Nusselt
correlation is in good agreement with the experimental data within a ˘ 10% error. This correlation
number
correlation
is in good
agreement
with the experimental data˝ within a ±˝10% error. This
is valid
in the
ranges 100,000
< Ra
Dh < 600,000, 9 < N < 36, and 30 < α < 90 , within which the
correlation is valid in the ranges 100,000 < RaDh < 600,000, 9 < N < 36, and 30° < α < 90°, within which
experimental data were obtained. In addition, though the correlation is developed only for the inclined
the experimental data were obtained. In addition, though the correlation is developed only for the
fin heat sinks, this correlation can predict the Nusselt numbers of the radial fin heat sinks (α = 0) within
inclined fin heat sinks, this correlation can predict the Nusselt numbers of the radial fin heat sinks
a ˘ 30%
error (Appendix B).
(α = 0) within a ± 30% error (Appendix B).
Figure 6. Nusselt numbers calculated using proposed correlation and those from experimental data.
Figure 6. Nusselt numbers calculated using proposed correlation and those from experimental data.
The thermal resistances of the heat sinks are predicted using the proposed correlations and are
compared
with
the experimental
Figures
and 8. Theusing
resultsthe
from
the correlation
match the
The
thermal
resistances
of the data
heatin
sinks
are 7predicted
proposed
correlations
and are
experimental
data
well
within
a
±15%
error.
Figures
7
and
8
present
the
thermal
resistances
for the
compared with the experimental data in Figures 7 and 8. The results from the correlation match
various
fin
numbers
and
inclination
angles,
respectively.
experimental data well within a ˘15% error. Figures 7 and 8 present the thermal resistances for various
fin numbers
and inclination angles, respectively.
Energies 2016, 9, 391
9 of 15
Figure 7. Thermal resistances for various fin numbers.
Figure 7. Thermal resistances for various fin numbers.
Energies 2016, 9, 391
9 of 15
Figure 7. Thermal resistances for various fin numbers.
Figure
8. 8.
Thermal
variousinclination
inclination
angles.
Figure
Thermalresistances
resistances for
for various
angles.
In Figure
7, as
thethefin
increases,the
thethermal
thermal
resistance
decreases.
It is mainly
In Figure
7, as
finnumber
number increases,
resistance
decreases.
It is mainly
becausebecause
the
the fin
surface
area
increases
as
the
fin
number
increases.
As
shown
in
Figure
8,
the
thermal
resistance
fin surface area increases as the fin number increases. As shown in Figure 8, the thermal resistance
is minimized
when
thethe
inclination
resistanceofofthe
the
inclined
is minimized
when
inclinationangle
angleisis60°.
60˝ . Thermal
Thermal resistance
inclined
fin fin
heatheat
sinksink
withwith
˝
˝
the inclination
angle
of
60°
is
smaller
than
that
of
30°,
because
the
fin
surface
area
is
greater.
the inclination angle of 60 is smaller than that
30 , because the fin surface area is greater. The The
˝
inclined
heatsink
sink with
inclination
angleangle
of 60 of
has60°
slightly
thermal
resistance
compared
inclined
fin fin
heat
withthe
the
inclination
hassmaller
slightly
smaller
thermal
resistance
˝ . It is because the amount of heat transfer suppression caused by the boundary layer
to
that
of
90
compared to that of 90°. It is because the amount of heat transfer suppression caused by the boundary
decreases
withwith
an increase
in the in
fin-to-fin
spacing spacing
[15] and the
of the
inclined
layeroverlap
overlap
decreases
an increase
the fin-to-fin
[15]fin-to-fin
and thespacing
fin-to-fin
spacing
of the
˝ is larger than that of 90˝ . Next, a contour map that
fin
heat
sink
with
the
inclination
angle
of
60
inclined fin heat sink with the inclination angle of 60° is larger than that of 90°. Next, a contour map
depicts the thermal resistance as a function of the fin number and fin thickness by using the proposed
that depicts the thermal resistance as a function of the fin number and fin thickness by using the proposed
correlation is presented in Figure 9. This contour map covers the range of 9 < N < 36, within which the
correlation is presented in Figure 9. This contour map covers the range of 9 < N < 36, within which the
experimental data were obtained. The contour map shows that the thermal resistance decreases as the
experimental
were This
obtained.
The as
contour
map shows
thatthe
theeffective
thermal
resistance
decreases
as
fin numberdata
decreases.
is because,
the fin number
increases,
surface
area, ηNA
f + Ab ,
the fin
decreases.
This
is because,
thewhile
fin number
the effective
surfaceslowly
area, ηNAf
of number
the heat sink
increases
rapidly
(Figureas
10a),
the heatincreases,
transfer coefficient
h decreases
+ Ab,(Figure
of the10b).
heatAt
sink
increases fin
rapidly
(Figure
the
transfer
coefficient
h decreases
the maximum
number
(N = 36,10a),
alongwhile
the line
AA’heat
in Figure
9), the
thermal resistance
slowly
(Figure 10b).
the maximum
number
(N = when
36, along
theasline
in Figure
9), the thermal
is minimized
at a At
certain
fin thickness.fin
This
is because,
N = 36,
the AA’
fin thickness
increases,
the
heat transfer
coefficient
(Figure
10b), whereas
the fin efficiency
and
the effective
surface
resistance
is minimized
at hadecreases
certain fin
thickness.
This is because,
when N
= 36,
as the fin
thickness
area (ηNAf + Ab ) increase (Figure 10a). Finally, Figure 9 shows that a thermal resistance of 1.06 K/W
can be achieved by optimizing the fin number and fin thickness of the inclined plate fin heat sink
in the range of 9 < N < 36. For comparison, a contour map of the thermal resistances of cylinders
with conventional radial plate fins, which are calculated from the Nusselt number correlation [15],
is presented in Figure 11. It shows that the optimal thermal resistance is 1.50 K/W for the heat sink
with radial plate fins.
heat are
sinkoptimized
with radial in
plate
fins,the
which
thefins.
range of 9 < N < 36, are compared in Figure 12. This figure clearly
Finally,
the thermal
resistances
cylinders
with inclined
inclined plate
conventional
plate
shows that
the thermal
resistance
of of
the
optimized
platefins
finand
heat
sink with radial
the inclination
fins,
which
are
optimized
in
the
range
of
9
<
N
<
36,
are
compared
in
Figure
12.
This
figure
clearly
angle of 60° is 29.3% lower than that of the optimized radial plate fin heat sink. Therefore, we can
showsthat
that cylinders
the thermalwith
resistance
of the
optimized
inclined
plate than
fin heat
sinkofwith
the inclination
conclude
inclined
plate
fins perform
better
those
cylinders
with radial
angle
of2016,
60°9,is391
29.3% lower than that of the optimized radial plate fin heat sink. Therefore, we
Energies
10 can
of 15
plate fins.
conclude that cylinders with inclined plate fins perform better than those of cylinders with radial
plate fins.
Figure
9.
map
resistance
for
and
fin
thicknesses
(H
30
Figure
9. Contour
Contour
map
ofthermal
thermal
resistancefor
forvarious
variousfin
fin
numbers
and
finfin
thicknesses
(H ==(H
30mm,
Figure
9. Contour
map
of of
thermal
resistance
various
finnumbers
numbers
and
thicknesses
=mm,
30 mm,
−5
˝ C,
D
==6060mm,
L =L50=mm,
α = 60°,
Tw60
− ˝T,amb
=
50
°C,
k
f = 0.026
W/m
K,
k
s = 220 W/m K, νf = 1.6 × 10 m2/s,
´
D
mm,
50
mm,
α
=
T
T
=
50
k
=
0.026
W/m
K,
k
=
220
W/m
s νf = 1.6 × 10−5K,m2/s,
ambkf = 0.026 W/m
f
D = 60 mm, L = 50 mm, α = 60°, Tw − Tamb =w 50 °C,
K, ks = 220 W/m K,
−5
2/s,
´5
2
´5
2
ανf ==2.23
×
10
m
β
f = 0.0033/K).
1.6
αf = 2.23 ˆ 10 m /s, βf = 0.0033/K).
f ×
αf = 2.23
10ˆ−5 10
m2/s,mβf /s,
= 0.0033/K).
(a)
(a)
Figure 10. Cont.
Energies 2016, 9, 391
11 of 15
Energies 2016, 9, 391
11 of 15
Energies 2016, 9, 391
11 of 15
(b)
(b)
Figure
10.
Contour
maps
of
(a)
effective
surface
area
and
(b) heat
transfer
coefficient
for various
various fin
Figure 10.
10. Contour
Contourmaps
mapsofof(a)(a)effective
effective
surface
area
and
heat
transfer
coefficient
Figure
surface
area
and
(b)(b)
heat
transfer
coefficient
for for
various fin
˝
numbers
and fin and
thicknesses
(H = 30
mm,
= 60Dmm,
= 50 mm,
α = 60°,
Tw −, TTwamb´=T50
°C,
kf ˝=C,0.026
fin numbers
fin thicknesses
30 D
mm,
= 60 LL
mm,
= 50αmm,
=
50
ambk
numbers
and fin thicknesses
(H =(H
30 =
mm,
D = 60
mm,
= 50Lmm,
= 60°,α T=w 60
− Tamb = 50 °C,
f = 0.026
´5
2
´5
2
−5 ν 2= 1.6
−5 α
2=
kK,
W/m
K, K,
ks =νf220
W/m
K,
ˆ 10 m
/s,
ˆ0.0033/K).
10 m /s, βf = 0.0033/K).
W/mW/m
s = 220
W/m
× 10
× 10
10
mf2/s,
/s,2.23
f =
f = k0.026
2/s,α
−5 m
K, ks = 220
W/m
K, =νf1.6
= 1.6
× 10−5mfm/s,
αf f== 2.23
2.23 ×
ββ
f = 0.0033/K).
Figure
Figure 11.
11. Contour
Contour map
map of
of thermal
thermal resistance
resistance of
of the
the cylinder
cylinder with
with conventional
conventional radial
radial plate
plate fins
fins for
for
various
fin
numbers
and
fin
thicknesses
for
(H
=
30
mm,
D
=
60
mm,
L
=
50
mm,
T
w − Tamb = 50 °C,
various fin numbers and fin thicknesses for (H = 30 mm, D = 60 mm, L = 50 mm, Tw ´ Tamb = 50 ˝ C,
Figure
ContourK,map
of thermal
of 2theαfcylinder
with
radial plate fins for
−5
2conventional
kkf ==11.
0.026
220
W/m
K, K,
νresistance
f = 1.6 × 10−5 m´5
2.23
βf =m
0.0033/K).
2=/s,
2 /s, β = 0.0033/K).
0.026W/m
W/m K,ksk= =
220
W/m
ν = 1.6 ˆ 10 /s,m
α ×= 10
2.23m
ˆ /s,
10´5
f
s
f
f
f
various fin numbers and fin thicknesses for (H = 30 mm, D = 60 mm, L = 50 mm, Tw − Tamb = 50 °C,
kf = 0.026 W/m K, ks = 220 W/m K, νf = 1.6 × 10−5 m2/s, αf = 2.23 × 10−5 m2/s, βf = 0.0033/K).
Finally, the thermal resistances of cylinders with inclined plate fins and conventional radial plate
fins, which are optimized in the range of 9 < N < 36, are compared in Figure 12. This figure clearly
shows that the thermal resistance of the optimized inclined plate fin heat sink with the inclination
Energies 2016, 9, 391
12 of 15
angle of 60˝ is 29.3% lower than that of the optimized radial plate fin heat sink. Therefore, we can
conclude that cylinders with inclined plate fins perform better than those of cylinders with radial
plate
fins.
Energies
2016, 9, 391
12 of 15
Figure 12. Comparison of the thermal resistances between the optimized inclined fin heat sinks and
Figure 12. Comparison of the thermal resistances between the optimized inclined fin heat sinks
the optimized radial fin heat sink (9 < N < 36, t < 2 mm, H = 30 mm, D = 60 mm, L = 50 mm, Tw − Tamb =
and the optimized radial fin heat sink (9 < N < 36, t < 2 mm, H = 30 mm, D = 60 mm, L = 50 mm,
50 °C, kf = 0.026 W/m K, ks = 220 W/m K, νf = 1.6 × 10−5 m2/s, αf = 2.23 × 10−5 m2/s, βf = 0.0033/K).
Tw ´ Tamb = 50 ˝ C, kf = 0.026 W/m K, ks = 220 W/m K, νf = 1.6 ˆ 10´5 m2 /s, αf = 2.23 ˆ 10´5 m2 /s,
βf = 0.0033/K).
4. Conclusions
4. Conclusions
In the present paper, natural convection from vertical cylinders with inclined plate fins is
investigated experimentally. Experimental investigations are performed for various inclination
In the present paper, natural convection from vertical cylinders with inclined plate fins is
angles, fin numbers, and base temperatures. Although its applicability is limited to certain specific
investigated experimentally. Experimental investigations are performed for various inclination angles,
ranges (100,000 < RaDh < 600,000, 9 < N < 36, and 30° < α < 90°), the suggested correlation enables easy
fin numbers, and base temperatures. Although its applicability is limited to certain specific ranges
thermal performance calculation. Using the proposed correlation, a contour map depicting the
(100,000 < RaDh < 600,000, 9 < N < 36, and 30˝ < α < 90˝ ), the suggested correlation enables easy
thermal resistance as a function of the fin number and fin thickness is presented. Finally, the optimal
thermal performance calculation. Using the proposed correlation, a contour map depicting the thermal
thermal resistances of cylinders with inclined plate fins and conventional radial plate fins are
resistance as a function of the fin number and fin thickness is presented. Finally, the optimal thermal
compared. It is found that that the optimal thermal resistance of the cylinder with inclined fins is 30%
resistances of cylinders with inclined plate fins and conventional radial plate fins are compared. It is
lower than that of the cylinder with radial plate fins. Therefore, inclined plate fins have potential for
found that that the optimal thermal resistance of the cylinder with inclined fins is 30% lower than that
use in cooling equipment in various thermal systems.
of the cylinder with radial plate fins. Therefore, inclined plate fins have potential for use in cooling
Acknowledgments:
Thisthermal
researchsystems.
was supported by the Nano Material Technology Development Program
equipment
in various
through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT and Future
Planning (NRF-2011-0030285). This work was supported by “Human Resources Program in Energy Technology”
Appendix A. Measurement of the Heat Loss through the Supporting Blocks
of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from
the Ministry
of Trade, Industry
Energy,
of to
Korea.
2015 4010 200820).
The measurement
of the&heat
lossRepublic
is similar
that (Project
writtenNo:
in Reference
[15]. For measuring the
heat
loss,
instead of the
heat
a slim
cylindrical
is sandwiched
supporting
blocks
Author
Contributions:
Jong
Bumsink,
Lee and
Dong-Kwon
Kimheater
conceived
and designedby
thethe
experiments;
Sang
Woo
as
in Figure
A1. In this
situation,
all and
of the
heat produced
by the heater
released to Kim
the
Leeshown
performed
the experiments;
Hyun
Jung Kim
Dong-Kwon
Kim analyzed
the data;is Dong-Kwon
surroundings
wrote the paper.through the supporting blocks. Therefore, the heat loss through the supporting blocks
(qloss,1 + qloss,2 ) can be measured by measuring the power supplied to the heater. In the present study,
Conflicts of Interest: The authors declare no conflict of interest.
the heat loss was measured for various heater temperatures and was used to calculate the actual heat
transfer
rate
the heat sink. of the Heat Loss through the Supporting Blocks
Appendix
A.toMeasurement
The measurement of the heat loss is similar to that written in Reference [15]. For measuring the
heat loss, instead of the heat sink, a slim cylindrical heater is sandwiched by the supporting blocks as
shown in Figure A1. In this situation, all of the heat produced by the heater is released to the
surroundings through the supporting blocks. Therefore, the heat loss through the supporting blocks
(qloss,1 + qloss,2) can be measured by measuring the power supplied to the heater. In the present study,
the heat loss was measured for various heater temperatures and was used to calculate the actual heat
Energies 2016, 9, 391
13 of 15
Energies 2016, 9, 391
13 of 15
Figure
A1.A1.
Schematic
diagram
setupfor
forheat
heat
loss
measurement.
Figure
Schematic
diagramof
ofexperimental
experimental setup
loss
measurement.
Appendix
B. Comparison
ofof
Nusselt
for the
theRadial
RadialHeat
Heat
Sinks
Calculated
Using
Appendix
B. Comparison
NusseltNumbers
Numbers for
Sinks
Calculated
Using
the the
Proposed
Correlation
and
Those
Data
Proposed
Correlation
and
Thosefrom
fromExperimental
Experimental Data
In Figure
B1, Nusselt
the Nusselt
numbers
calculated
from
correlation
(Equation(16))
(16))are
arecompared
compared with
In Figure
B1, the
numbers
calculated
from
thethe
correlation
(Equation
with
those
obtained
from
experimental
data
for
the
radial
plate
fin
heat
sinks
presented
in As
those obtained from experimental data for the radial plate fin heat sinks presented in Reference [15].
Reference
[15].B1,
Asthe
shown
in Figurecan
B1, predict
the correlation
can predict
the Nusselt
the radial
shown
in Figure
correlation
the Nusselt
numbers
of the numbers
radial finofheat
sinks fin
within
heat sinks within a ˘30% error, despite the fact that the correlation is developed only for the inclined
a ±30% error, despite the fact that the correlation is developed only for the inclined fin heat sinks. For
fin heat sinks. For predicting the Nusselt numbers for both of the inclined fin heat sinks and the radial
predicting the Nusselt numbers for both of the inclined fin heat sinks and the radial fin heat sinks,
fin heat sinks, we developed an alternative correlation as follows:
we developed an alternative correlation as follows:
Nu Dh “ 6.21 ´ 1.90 lnpRa Dh q ` 0.19lnpRa Dh q2
NuDh  6.21  1.90 ln( RaDh )  0.19ln( RaDh )2
(B1)
(B1)
In Figure B2, the Nusselt numbers calculated from the correlation (Equation (B1)) are compared
In
Figure
the Nusselt
numbers calculated
from
the plate
correlation
are compared
with those B2,
obtained
from experimental
data for the
radial
fin heat(Equation
sinks and (B1))
the inclined
fin
withheat
those
obtained
frominexperimental
for the can
radial
platethe
finNusselt
heat sinks
and of
the
sinks.
As shown
Figure B2, thedata
correlation
predict
numbers
theinclined
radial fin
heat plate
sinks.fin
Asheat
shown
Figure
B2, the fin
correlation
predict
the Nusselt
numberswe
of suggest
the radial
sinksinand
the inclined
heat sinkscan
within
a ˘15%
error. Therefore,
theplate
use of
Equation
(16)inclined
for the accurate
of Nusselt
for the inclined
fin heatthe
sinks
fin heat
sinks
and the
fin heatprediction
sinks within
a ±15%numbers
error. Therefore,
we suggest
use of
˝ < α < 90˝ ), and Equation (B1) can be used for the prediction of Nusselt numbers for both of the
(30
Equation (16) for the accurate prediction of Nusselt numbers for the inclined fin heat sinks (30° < α < 90°),
radial fin heat
the inclined
heats sinks.
and Equation
(B1)sinks
can and
be used
for thefin
prediction
of Nusselt numbers for both of the radial fin heat
sinks and the inclined fin heats sinks.
Energies 2016, 9, 391
14 of 15
Energies 2016, 9, 391
Energies 2016, 9, 391
14 of 15
14 of 15
Figure B1. Nusselt
numbers
for
radial
fin
heat
sinks
calculated
using
proposed
correlation
and those
Figure
Nusselt numbers
numbers for
for radial
radial fin
fin heat
heat sinks
sinks calculated
calculated using
using proposed
proposed correlation
correlation and
Figure B1.
B1. Nusselt
and those
those
from
experimental
data
[15].
from
from experimental
experimental data
data [15].
[15].
Figure B2. Nusselt numbers calculated using alternative correlation and those from experimental data.
Figure B2. Nusselt numbers calculated using alternative correlation and those from experimental data.
Figure B2. Nusselt numbers calculated using alternative correlation and those from experimental data.
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