Revista Facultad de Ingeniería, Universidad de Antioquia, No.110, pp. 86-98, Jan-Mar 2024
Experimental and numerical investigation of
flat plate fins and inline strip fins heat sinks
Investigación experimental y numérica de disipadores de calor de aletas continuas y en tiras
alineadas
William Denner Pires-Fonseca 1*Carlos Alberto Carrasco-Altemani 1
1Energy Department, School of Mechanical Engineering. Cidade Universitária, Campinas. R. Mendeleyev, 200 Cidade
Universitária, 13083-860. Campinas, Brasil.
CITE THIS ARTICLE AS:
W. D. Pires-Fonseca and C. A.
Carrasco-Altemani.
”Experimental and numerical
investigation of flat plate fins
and inline strip fins heat
sinks”, Revista Facultad de
Ingeniería Universidad de
Antioquia, no. 110, pp. 86-98,
Jan-Mar 2024. [Online].
Available: https:
//www.doi.org/10.17533/
udea.redin.20230417
ARTICLE INFO:
Received: August 18, 2020
Accepted: April 10, 2023
Available online: April 10, 2023
KEYWORDS:
Experimental analysis;
numerical analysis; heat
sinks; cooling of electronic
equipment
Análisis experimental; análisis
numérico; disipadores de
calor; enfriamiento de equipos
electrónicos
ABSTRACT: The flow and heat transfer characteristics of two analogous heat sinks were
obtained from laboratory experiments and compared to each other and to numerical
simulations. One contained continuous straight fins, and the other inline strip fins, both
cooled by forced airflow parallel to their base. The average airflow velocity in the interfin
channels ranged from 4 to 20 m/s, corresponding to Reynolds numbers from 810 to 3,800.
The measurements indicated that despite its smaller heat exchange area, the strip fins
heat sink convective coefficient was larger enough to obtain a thermal resistance smaller
than that of the continuous fins. Numerical simulations were performed to compare their
results with the experiments. Two distinct fin treatments were used: one considered fins
with no thickness, isothermal with the fins base temperature. The other considered the
fins thickness and perfect thermal contact with the heat sink base. The Nusselt number
simulation results for the continuous fins agreed within 3% with the measurements, but
larger deviations were observed for the strip fins heat sink.
RESUMEN: Las características de flujo y transferencia de calor de dos disipadores de calor
análogos se obtuvieron a partir de experimentos de laboratorio y se compararon entre
sí y con simulaciones numéricas. Uno contenía aletas rectas continuas y el otro, aletas
de tira en línea, ambas enfriadas por un flujo de aire forzado paralelo a su base. La
velocidad promedio del flujo de aire en los canales entre aletas osciló entre 4 y 20 m/s,
lo que corresponde a números de Reynolds de 810 a 3800. Las mediciones indicaron
que a pesar de su menor área de intercambio de calor, el coeficiente de convección
del disipador de calor de las aletas de tira era lo suficientemente grande como para
obtener una resistencia térmica menor que la de las aletas continuas. Se realizaron
simulaciones numéricas para comparar sus resultados con los experimentos. Se
utilizaron dos tratamientos distintos para las aletas: uno consideró aletas sin espesor,
isotérmicas con la temperatura base de las aletas. El otro consideró el grosor de las
aletas y un perfecto contacto térmico con la base del disipador de calor. Los resultados
de la simulación del número de Nusselt para las aletas continuas coincidieron en un 3%
con las medidas, pero se observaron desviaciones mayores para el disipador de calor de
las aletas en tira.
1. Introduction
The continuous increase of the heat flux from electronic
components and devices presents a significant challenge
to the industry in order to keep their temperature within
the manufacturers’ specifications for a reliable life cycle
[1–3]. In order to overcome any electronic failures that
may emerge from an undesirable temperature increase,
a series of innovative cooling techniques have been
developed, including the use of phase change materials,
thermoelectric cooling, liquid cooling techniques, high
conductive fillings and thermal interface materials [4–8].
In spite of all these efforts, forced convection air cooling
still remains the most employed mechanism for the
general cooling of electronic equipment. It is mainly
so when the temperature control must be attained with
86
* Corresponding author: William Denner Pires-Fonseca
E-mail: fonsecawdp@gmail.com
ISSN 0120-6230
e-ISSN 2422-2844
DOI: 10.17533/udea.redin.20230417
86
Experimental and numerical investigation of
flat plate fins and inline strip fins heat sinks
Investigación experimental y numérica de disipadores de calor de aletas continuas y en tiras
alineadas
William Denner Pires-Fonseca 1*Carlos Alberto Carrasco-Altemani 1
1Energy Department, School of Mechanical Engineering. Cidade Universitária, Campinas. R. Mendeleyev, 200 Cidade
Universitária, 13083-860. Campinas, Brasil.
CITE THIS ARTICLE AS:
W. D. Pires-Fonseca and C. A.
Carrasco-Altemani.
”Experimental and numerical
investigation of flat plate fins
and inline strip fins heat
sinks”, Revista Facultad de
Ingeniería Universidad de
Antioquia, no. 110, pp. 86-98,
Jan-Mar 2024. [Online].
Available: https:
//www.doi.org/10.17533/
udea.redin.20230417
ARTICLE INFO:
Received: August 18, 2020
Accepted: April 10, 2023
Available online: April 10, 2023
KEYWORDS:
Experimental analysis;
numerical analysis; heat
sinks; cooling of electronic
equipment
Análisis experimental; análisis
numérico; disipadores de
calor; enfriamiento de equipos
electrónicos
ABSTRACT: The flow and heat transfer characteristics of two analogous heat sinks were
obtained from laboratory experiments and compared to each other and to numerical
simulations. One contained continuous straight fins, and the other inline strip fins, both
cooled by forced airflow parallel to their base. The average airflow velocity in the interfin
channels ranged from 4 to 20 m/s, corresponding to Reynolds numbers from 810 to 3,800.
The measurements indicated that despite its smaller heat exchange area, the strip fins
heat sink convective coefficient was larger enough to obtain a thermal resistance smaller
than that of the continuous fins. Numerical simulations were performed to compare their
results with the experiments. Two distinct fin treatments were used: one considered fins
with no thickness, isothermal with the fins base temperature. The other considered the
fins thickness and perfect thermal contact with the heat sink base. The Nusselt number
simulation results for the continuous fins agreed within 3% with the measurements, but
larger deviations were observed for the strip fins heat sink.
RESUMEN: Las características de flujo y transferencia de calor de dos disipadores de calor
análogos se obtuvieron a partir de experimentos de laboratorio y se compararon entre
sí y con simulaciones numéricas. Uno contenía aletas rectas continuas y el otro, aletas
de tira en línea, ambas enfriadas por un flujo de aire forzado paralelo a su base. La
velocidad promedio del flujo de aire en los canales entre aletas osciló entre 4 y 20 m/s,
lo que corresponde a números de Reynolds de 810 a 3800. Las mediciones indicaron
que a pesar de su menor área de intercambio de calor, el coeficiente de convección
del disipador de calor de las aletas de tira era lo suficientemente grande como para
obtener una resistencia térmica menor que la de las aletas continuas. Se realizaron
simulaciones numéricas para comparar sus resultados con los experimentos. Se
utilizaron dos tratamientos distintos para las aletas: uno consideró aletas sin espesor,
isotérmicas con la temperatura base de las aletas. El otro consideró el grosor de las
aletas y un perfecto contacto térmico con la base del disipador de calor. Los resultados
de la simulación del número de Nusselt para las aletas continuas coincidieron en un 3%
con las medidas, pero se observaron desviaciones mayores para el disipador de calor de
las aletas en tira.
1. Introduction
The continuous increase of the heat flux from electronic
components and devices presents a significant challenge
to the industry in order to keep their temperature within
the manufacturers’ specifications for a reliable life cycle
[1–3]. In order to overcome any electronic failures that
may emerge from an undesirable temperature increase,
a series of innovative cooling techniques have been
developed, including the use of phase change materials,
thermoelectric cooling, liquid cooling techniques, high
conductive fillings and thermal interface materials [4–8].
In spite of all these efforts, forced convection air cooling
still remains the most employed mechanism for the
general cooling of electronic equipment. It is mainly
so when the temperature control must be attained with
86
* Corresponding author: William Denner Pires-Fonseca
E-mail: fonsecawdp@gmail.com
ISSN 0120-6230
e-ISSN 2422-2844
DOI: 10.17533/udea.redin.20230417
86
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
minimum cost. In this case, atmospheric air is the cooling
fluid, due to its availability, handling facility and dielectric
properties.
Because of its thermal properties (relatively small specific
mass and thermal conductivity), the forced convection
air cooling is usually employed with finned heat sinks,
in order to prevent higher temperatures. The fins
increase the convective heat transfer area and they may
increase the heat transfer coefficient, as well as the
conductive and radiative heat losses from the electronic
components. Among the heat sinks, that with straight fins
of constant cross section is the simplest and the most
widely encountered [9–17]. It is usually employed with a
parallel forced airflow in the interfin channels and the base
plate. From the flow entrance in these channels, there is a
development of the velocity and thermal boundary layers,
which increase the heat sink convective thermal resistance
along the flow. This increase may be reduced by a partition
of the continuous fins into strip fins of smaller length. The
strip fins heat exchange area is reduced in comparison to
the heat sink of the same size with continuous straight
fins. This area reduction may, however, be compensated
by an increase in the average heat transfer coefficient of
the strip fins.
This eventual heat transfer enhancement has led to
several studies presented in the literature. Sparrow et
al. [18] proposed a numerical investigation of the laminar
flow and heat transfer from channels with periodical strip
fins along the flow direction. The numerical results were
obtained for a range of Reynolds numbers and for several
values of a dimensionless geometrical parameter. They
indicated that the partition of a single longitudinal fin into
smaller strip fins reduces the boundary layers growth and,
thus, the strip fins convective thermal resistance.
Sparrow and Liu [19] compared numerical results for the
convective heat transfer and the pressure drop for the
laminar flow along continuous fins and strip fins with
inline and staggered arrangements. Their results showed
that for constant mass flow rate and heat transfer area,
the strip fins thermal efficiency is considerably larger than
that of the continuous fins. Their results also indicated
that the staggered strip fins arrangement presented a
better thermal performance than the inline arrangement.
A numerical analysis of the comparative performance
of the inline and staggered geometrical distribution of
strip fins was performed by Al-Sallami et al. [20]. Their
investigation included the effects of perforated fins to
enhance convective heat transfer. Their results also
indicated that the staggered fins arrangement gives a
larger heat transfer rate, at the expense of a larger flow
pressure drop. Ozturk and Tari [21] performed a numerical
investigation of heat sinks with strip fins applied to the
CPU of a computer. They investigated the effects of the
number of fins and their distribution, as well as the fins
material and the heat sink base thickness. They found
that although they had different geometries, all of the heat
sinks studied had similar thermal resistances. However,
replacing aluminum with copper as the heat sink material
improved the performance.
Other investigations were also performed with the purpose
to optimize the geometrical parameters of the strip fins
heat sinks. Teertstra et al. [22] developed an analytical
model for predicting the heat transfer rate from heat
sinks with inline strip fins. They indicated that for a
range of Reynolds numbers (40 < Re < 180), the best
configuration for the inline strip fins is with a ratio of fins
spacing (S) to pitch (P) equal to (S/P) = 0.5 and for a ratio
of fins pitch to baseplate length in the range 0.059 < P/L
< 0.44. Hong and Cheng [23] also presented numerical
results to optimize the geometrical parameters of strip
fins heat sinks. They indicated that there is an optimal
strip length to minimize the flow pressure drop and that
it is independent of both the heat transfer distribution on
the heat sink base and its maximum temperature.
The numerical investigations, as those reported previously,
are useful to predict the heat sink performance from their
detailed information about the flow and temperature
fields, for a specific heat sink configuration and operating
conditions. Additional numerical results may also be
easily obtained to represent the effects of changes in the
heat sink size and operating conditions. Then, a heat
sink prototype, considering distinct design purposes, may
be selected from the numerical simulations. The actual
thermal behavior of the prototype may then be measured
in laboratory experiments.
This paper aims to present the results of an experimental
investigation to compare the thermal and flow
characteristics of two heat sinks, one with continuous
fins and the other with inline strip fins, both cooled by
forced airflow in the laboratory. Most investigations report
results for the interfin flow in the laminar regime. There
are two main reasons for this. One is that a larger flow
pumping power is required as the interfin flow velocity
increases and the other is that as the velocity increases in
the interfin channels, there may appear an inconvenient
noise, like a whistle, and this is not recommended for
several applications due to hearing discomfort. In the
present experimental investigation, the range of airflow
velocities in the interfin channels was larger than it is
usually found in the literature. The experimental results
include the airflow pressure drop through the heat sinks,
the average Nusselt number in the interfin channels,
the convective thermal resistance and the effectiveness
of both heat sinks. The numerical simulations were
87
minimum cost. In this case, atmospheric air is the cooling
fluid, due to its availability, handling facility and dielectric
properties.
Because of its thermal properties (relatively small specific
mass and thermal conductivity), the forced convection
air cooling is usually employed with finned heat sinks,
in order to prevent higher temperatures. The fins
increase the convective heat transfer area and they may
increase the heat transfer coefficient, as well as the
conductive and radiative heat losses from the electronic
components. Among the heat sinks, that with straight fins
of constant cross section is the simplest and the most
widely encountered [9–17]. It is usually employed with a
parallel forced airflow in the interfin channels and the base
plate. From the flow entrance in these channels, there is a
development of the velocity and thermal boundary layers,
which increase the heat sink convective thermal resistance
along the flow. This increase may be reduced by a partition
of the continuous fins into strip fins of smaller length. The
strip fins heat exchange area is reduced in comparison to
the heat sink of the same size with continuous straight
fins. This area reduction may, however, be compensated
by an increase in the average heat transfer coefficient of
the strip fins.
This eventual heat transfer enhancement has led to
several studies presented in the literature. Sparrow et
al. [18] proposed a numerical investigation of the laminar
flow and heat transfer from channels with periodical strip
fins along the flow direction. The numerical results were
obtained for a range of Reynolds numbers and for several
values of a dimensionless geometrical parameter. They
indicated that the partition of a single longitudinal fin into
smaller strip fins reduces the boundary layers growth and,
thus, the strip fins convective thermal resistance.
Sparrow and Liu [19] compared numerical results for the
convective heat transfer and the pressure drop for the
laminar flow along continuous fins and strip fins with
inline and staggered arrangements. Their results showed
that for constant mass flow rate and heat transfer area,
the strip fins thermal efficiency is considerably larger than
that of the continuous fins. Their results also indicated
that the staggered strip fins arrangement presented a
better thermal performance than the inline arrangement.
A numerical analysis of the comparative performance
of the inline and staggered geometrical distribution of
strip fins was performed by Al-Sallami et al. [20]. Their
investigation included the effects of perforated fins to
enhance convective heat transfer. Their results also
indicated that the staggered fins arrangement gives a
larger heat transfer rate, at the expense of a larger flow
pressure drop. Ozturk and Tari [21] performed a numerical
investigation of heat sinks with strip fins applied to the
CPU of a computer. They investigated the effects of the
number of fins and their distribution, as well as the fins
material and the heat sink base thickness. They found
that although they had different geometries, all of the heat
sinks studied had similar thermal resistances. However,
replacing aluminum with copper as the heat sink material
improved the performance.
Other investigations were also performed with the purpose
to optimize the geometrical parameters of the strip fins
heat sinks. Teertstra et al. [22] developed an analytical
model for predicting the heat transfer rate from heat
sinks with inline strip fins. They indicated that for a
range of Reynolds numbers (40 < Re < 180), the best
configuration for the inline strip fins is with a ratio of fins
spacing (S) to pitch (P) equal to (S/P) = 0.5 and for a ratio
of fins pitch to baseplate length in the range 0.059 < P/L
< 0.44. Hong and Cheng [23] also presented numerical
results to optimize the geometrical parameters of strip
fins heat sinks. They indicated that there is an optimal
strip length to minimize the flow pressure drop and that
it is independent of both the heat transfer distribution on
the heat sink base and its maximum temperature.
The numerical investigations, as those reported previously,
are useful to predict the heat sink performance from their
detailed information about the flow and temperature
fields, for a specific heat sink configuration and operating
conditions. Additional numerical results may also be
easily obtained to represent the effects of changes in the
heat sink size and operating conditions. Then, a heat
sink prototype, considering distinct design purposes, may
be selected from the numerical simulations. The actual
thermal behavior of the prototype may then be measured
in laboratory experiments.
This paper aims to present the results of an experimental
investigation to compare the thermal and flow
characteristics of two heat sinks, one with continuous
fins and the other with inline strip fins, both cooled by
forced airflow in the laboratory. Most investigations report
results for the interfin flow in the laminar regime. There
are two main reasons for this. One is that a larger flow
pumping power is required as the interfin flow velocity
increases and the other is that as the velocity increases in
the interfin channels, there may appear an inconvenient
noise, like a whistle, and this is not recommended for
several applications due to hearing discomfort. In the
present experimental investigation, the range of airflow
velocities in the interfin channels was larger than it is
usually found in the literature. The experimental results
include the airflow pressure drop through the heat sinks,
the average Nusselt number in the interfin channels,
the convective thermal resistance and the effectiveness
of both heat sinks. The numerical simulations were
87
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
performed with the computational software PHOENICS
[24]. They were carried out for a three-dimensional
domain consisting of a heat sink channel, assuming
uniform velocity and temperature profiles at the channel
entrance. The heat sink base was assumed to be an
isothermal plate in perfect thermal contact with the fins.
For the numerical simulations in the laminar regime,
the equations of conservation of mass, momentum and
energy were solved under steady-state conditions. For
the turbulent regime, such equations were solved using
an additional zero equation turbulence model (LVEL). The
experimental results were compared with those from the
numerical simulations.
2. Experimental investigation
Two similar aluminum heat sinks, one with a flat plate
fins and the other with inline strip fins, as indicated in
Figure 1, were tested in a laboratory. Each heat sink had
16 lines of fins with a height of 0.0113 m on a rectangular
base of length L = 0.09 m and width W = 0.0415 m (other
dimensions are indicated in Table 1). The tests were
performed with a heat sink assembled in a rectangular
duct in such a way that the fins base was flush mounted to
the inner surface of one duct wall. The heat sink fins tips
then touched the rectangular duct opposite wall, so that
there was no top bypass. On both sides of the assembled
heat sink, there was a lateral bypass with rectangular duct
walls equal to one fins’ spacing. The rectangular duct
cross section was equal to (0.0452 x 0.0113) m, and its
length was equal to 0.170 m. It was made from four 0.005
m thick Plexiglas plates connected by screws and sealed
with silicone rubber.
Each heat sink was assembled 0.010 m from the duct
entrance. The duct was connected, as indicated in Figure 2,
to the front wall of a plenum box with a partition wall where
a calibrated nozzle was installed to measure the airflow
rate during the experimental tests. The airflow in the duct
was provided in suction mode by a fan located outside the
laboratory and connected by a duct with a flow control valve
to the end wall of the plenum box.t
bb t
Figure 1 (a) flat plate fins heat sink and (b) strip fins heat sinkInflow
Figure 2 Illustration of an experimental apparatus
Table 1 Geometric parameters of heat sinks
Parameters Values [mm]
L 90
H 14
W 41.5
P 19
S 5
b 1.86
t 0.86
Two distinct experimental tests were performed with each
heat sink. One was to obtain the airflow pressure drop
through the heat sink and the other was to obtain the
thermal results. The flow pressure drop through the heat
sinks was measured under steady, isothermal ambient
conditions. The airflow rate in the duct was varied by
means of the flow control valve in the tube downstream of
the plenum box and it was measured by the flow pressure
drop in the nozzle using an inclined manometer. The
airflow pressure drop through the heat sink was measured
by a differential pressure transducer (PX750-06DI, Omega
Eng.). This pressure drop was that from the ambient air in
the laboratory to a pressure tap located in the rectangular
duct, 30 mm downstream of the heat sink. Two similar
pressure drop measurements of the flow in the duct
were performed. First, with the heat sink in the duct and
then in the duct without the heat sink, that is, just the
flow pressure drop in the plain duct. The results to be
presented for the airflow pressure drop through the heat
sink were obtained by subtracting the later measurement
from the former. Thus, the present pressure drop results
indicate the effect of the presence of the heat sink in the
duct.
The thermal tests were also performed for each heat sink
mounted in the same rectangular duct, under steady-state
88
performed with the computational software PHOENICS
[24]. They were carried out for a three-dimensional
domain consisting of a heat sink channel, assuming
uniform velocity and temperature profiles at the channel
entrance. The heat sink base was assumed to be an
isothermal plate in perfect thermal contact with the fins.
For the numerical simulations in the laminar regime,
the equations of conservation of mass, momentum and
energy were solved under steady-state conditions. For
the turbulent regime, such equations were solved using
an additional zero equation turbulence model (LVEL). The
experimental results were compared with those from the
numerical simulations.
2. Experimental investigation
Two similar aluminum heat sinks, one with a flat plate
fins and the other with inline strip fins, as indicated in
Figure 1, were tested in a laboratory. Each heat sink had
16 lines of fins with a height of 0.0113 m on a rectangular
base of length L = 0.09 m and width W = 0.0415 m (other
dimensions are indicated in Table 1). The tests were
performed with a heat sink assembled in a rectangular
duct in such a way that the fins base was flush mounted to
the inner surface of one duct wall. The heat sink fins tips
then touched the rectangular duct opposite wall, so that
there was no top bypass. On both sides of the assembled
heat sink, there was a lateral bypass with rectangular duct
walls equal to one fins’ spacing. The rectangular duct
cross section was equal to (0.0452 x 0.0113) m, and its
length was equal to 0.170 m. It was made from four 0.005
m thick Plexiglas plates connected by screws and sealed
with silicone rubber.
Each heat sink was assembled 0.010 m from the duct
entrance. The duct was connected, as indicated in Figure 2,
to the front wall of a plenum box with a partition wall where
a calibrated nozzle was installed to measure the airflow
rate during the experimental tests. The airflow in the duct
was provided in suction mode by a fan located outside the
laboratory and connected by a duct with a flow control valve
to the end wall of the plenum box.t
bb t
Figure 1 (a) flat plate fins heat sink and (b) strip fins heat sinkInflow
Figure 2 Illustration of an experimental apparatus
Table 1 Geometric parameters of heat sinks
Parameters Values [mm]
L 90
H 14
W 41.5
P 19
S 5
b 1.86
t 0.86
Two distinct experimental tests were performed with each
heat sink. One was to obtain the airflow pressure drop
through the heat sink and the other was to obtain the
thermal results. The flow pressure drop through the heat
sinks was measured under steady, isothermal ambient
conditions. The airflow rate in the duct was varied by
means of the flow control valve in the tube downstream of
the plenum box and it was measured by the flow pressure
drop in the nozzle using an inclined manometer. The
airflow pressure drop through the heat sink was measured
by a differential pressure transducer (PX750-06DI, Omega
Eng.). This pressure drop was that from the ambient air in
the laboratory to a pressure tap located in the rectangular
duct, 30 mm downstream of the heat sink. Two similar
pressure drop measurements of the flow in the duct
were performed. First, with the heat sink in the duct and
then in the duct without the heat sink, that is, just the
flow pressure drop in the plain duct. The results to be
presented for the airflow pressure drop through the heat
sink were obtained by subtracting the later measurement
from the former. Thus, the present pressure drop results
indicate the effect of the presence of the heat sink in the
duct.
The thermal tests were also performed for each heat sink
mounted in the same rectangular duct, under steady-state
88
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
conditions. The heating was accomplished by electric
power dissipation in a plate resistance (16 Ω) with almost
the same dimensions of the heat sink base. This plate was
attached to the heat sink base with good thermal contact,
employing a commercial thermal paste (Implastec). In
these tests, the rectangular duct with one heat sink and
the heating plate were thermally insulated by a 25 mm
layer of polystyrene (k = 0.033 W/m.K). All the thermal
tests were performed with the heat sink base temperature
around 40◦C. The electric power dissipation in the heating
plate was provided by a dc power supply (Instrutherm
FA-3005). It was obtained from the dc electric current
and the voltage drop across the heating plate, measured
by a digital multimeter (HP 34401A). The temperature
measurements were obtained from type E thermocouples
(Teflon-coated 0.254 x 10−3 m diameter wires from Omega
Eng.). They were read by a digital temperature indicator
(Omega Eng. DP41-TC), with a resolution of 0.1◦C. The
tested heat sinks and the duct were instrumented with
eight thermocouples as follows. The rectangular heat sink
base had three thermocouples located 2 mm below its
surface, along a diagonal. The other five thermocouples
were located at the duct entrance (1), on the heater plate
(1), on the Plexiglas duct wall (2) and on the insulation
layer (1).
Thirteen experimental tests were performed with each
heat sink, encompassing a range of the average airflow
velocities in the interfin channels from 4 m/s to 20 m/s. In
each thermal test, a predetermined flow rate in the duct
with a heat sink was adjusted by the flow control valve.
Then, the electric power dissipation was set so that the
heat sink base was around 40◦C and all the thermocouples
were read in time intervals of half an hour. The results from
each test were obtained under steady-state conditions,
assumed when all the temperature readings were within
0.1◦C during a time interval of 30 minutes. The time
interval to attain this condition was around three hours for
each test.
2.1 Data reduction
In each experimental test, the air mass flow rate in the
rectangular duct ( ˙md) was the same as that in the long
radius nozzle located in the plenum box. This nozzle
was built according to ISO 5167, with an inner radius of
0.017 m and it had been previously calibrated to obtain a
nozzle flow coefficient (Kb). From this coefficient, the mass
flow rate was obtained by a standard procedure from the
measurement of the flow pressure drop across the nozzle.
When each heat sink (16 lines of fins) was inserted in the
rectangular duct, the airflow was subdivided into 17 interfin
channels with a rectangular cross section equal to the fins
length and spacing. It was assumed that in each of these
channels, the mass flow rate ( ˙mc) was equal to 1/17 of the
duct mass flow rate ( ˙md). From ˙mc, the average airflow
velocity in the interfin channels (V ) was then obtained.
It was used to define a Reynolds number based on the
rectangular interfin channels hydraulic diameter (Dh), as
in Equation (1).
Rec = V Dh
ν (1)
The forced convective heat transfer from the heat sink to
the airflow (qcv ) was evaluated by an energy balance on
the heat sink under steady-state conditions. The electric
power dissipation (Pd) in the plate resistance in contact
with the heat sink base was not fully transferred from
the heat sink to the airflow. Several thermal losses (ql)
were evaluated and their sum was subtracted from Pd.
These losses included thermal radiation from the heat sink
base and fins, conduction through the thermal insulation
covering the rectangular duct, and conduction through
the power and thermocouple wires. In all the tests, the
estimated thermal losses were estimated to be less than
3% of the power dissipation Pd in the plate resistance. The
energy balance to obtain qcv was expressed by Equation (2).
qcv = Pd − Σql (2)
From the convective heat transfer rate qcv , the heat sink
thermal resistance was obtained as in Equation (3).
Rth = (Tb − T∞)
qcv
(3)
In this equation, Tb and T∞ represent the heat sink
base temperature and the airflow inlet temperature in the
heat sink, respectively. The thermal resistance can be
associated with a simple model [25] based on an average
heat transfer coefficient, as in Equation (4).
Rth = 1
h∗(Ab + ηf Af ) (4)
The heat transfer coefficient (h∗) is based on the flow
inlet temperature (T∞) and the efficiency (ηf ) was
obtained considering the standard adiabatic fins tips
model. The experimental values of Rth were obtained
from Equation (3) and they were used in Equation (4) to
obtain the heat transfer coefficients (h∗). This was an
iterative process because the fin efficiency ηf depends
on h∗. The results were expressed in non-dimensional
form by a Nusselt number based on the interfin channels
hydraulic diameter (Dh), as in Equation (5).
N u = h∗Dh
k (5)
The results of the present investigation have also
considered the heat sinks as heat exchangers [26, 27]. In
this approach, the heat sink convective heat transfer to the
airflow was described as that of a heat exchanger by the
effectiveness (ϵ) method, as presented in Equation (6).
89
conditions. The heating was accomplished by electric
power dissipation in a plate resistance (16 Ω) with almost
the same dimensions of the heat sink base. This plate was
attached to the heat sink base with good thermal contact,
employing a commercial thermal paste (Implastec). In
these tests, the rectangular duct with one heat sink and
the heating plate were thermally insulated by a 25 mm
layer of polystyrene (k = 0.033 W/m.K). All the thermal
tests were performed with the heat sink base temperature
around 40◦C. The electric power dissipation in the heating
plate was provided by a dc power supply (Instrutherm
FA-3005). It was obtained from the dc electric current
and the voltage drop across the heating plate, measured
by a digital multimeter (HP 34401A). The temperature
measurements were obtained from type E thermocouples
(Teflon-coated 0.254 x 10−3 m diameter wires from Omega
Eng.). They were read by a digital temperature indicator
(Omega Eng. DP41-TC), with a resolution of 0.1◦C. The
tested heat sinks and the duct were instrumented with
eight thermocouples as follows. The rectangular heat sink
base had three thermocouples located 2 mm below its
surface, along a diagonal. The other five thermocouples
were located at the duct entrance (1), on the heater plate
(1), on the Plexiglas duct wall (2) and on the insulation
layer (1).
Thirteen experimental tests were performed with each
heat sink, encompassing a range of the average airflow
velocities in the interfin channels from 4 m/s to 20 m/s. In
each thermal test, a predetermined flow rate in the duct
with a heat sink was adjusted by the flow control valve.
Then, the electric power dissipation was set so that the
heat sink base was around 40◦C and all the thermocouples
were read in time intervals of half an hour. The results from
each test were obtained under steady-state conditions,
assumed when all the temperature readings were within
0.1◦C during a time interval of 30 minutes. The time
interval to attain this condition was around three hours for
each test.
2.1 Data reduction
In each experimental test, the air mass flow rate in the
rectangular duct ( ˙md) was the same as that in the long
radius nozzle located in the plenum box. This nozzle
was built according to ISO 5167, with an inner radius of
0.017 m and it had been previously calibrated to obtain a
nozzle flow coefficient (Kb). From this coefficient, the mass
flow rate was obtained by a standard procedure from the
measurement of the flow pressure drop across the nozzle.
When each heat sink (16 lines of fins) was inserted in the
rectangular duct, the airflow was subdivided into 17 interfin
channels with a rectangular cross section equal to the fins
length and spacing. It was assumed that in each of these
channels, the mass flow rate ( ˙mc) was equal to 1/17 of the
duct mass flow rate ( ˙md). From ˙mc, the average airflow
velocity in the interfin channels (V ) was then obtained.
It was used to define a Reynolds number based on the
rectangular interfin channels hydraulic diameter (Dh), as
in Equation (1).
Rec = V Dh
ν (1)
The forced convective heat transfer from the heat sink to
the airflow (qcv ) was evaluated by an energy balance on
the heat sink under steady-state conditions. The electric
power dissipation (Pd) in the plate resistance in contact
with the heat sink base was not fully transferred from
the heat sink to the airflow. Several thermal losses (ql)
were evaluated and their sum was subtracted from Pd.
These losses included thermal radiation from the heat sink
base and fins, conduction through the thermal insulation
covering the rectangular duct, and conduction through
the power and thermocouple wires. In all the tests, the
estimated thermal losses were estimated to be less than
3% of the power dissipation Pd in the plate resistance. The
energy balance to obtain qcv was expressed by Equation (2).
qcv = Pd − Σql (2)
From the convective heat transfer rate qcv , the heat sink
thermal resistance was obtained as in Equation (3).
Rth = (Tb − T∞)
qcv
(3)
In this equation, Tb and T∞ represent the heat sink
base temperature and the airflow inlet temperature in the
heat sink, respectively. The thermal resistance can be
associated with a simple model [25] based on an average
heat transfer coefficient, as in Equation (4).
Rth = 1
h∗(Ab + ηf Af ) (4)
The heat transfer coefficient (h∗) is based on the flow
inlet temperature (T∞) and the efficiency (ηf ) was
obtained considering the standard adiabatic fins tips
model. The experimental values of Rth were obtained
from Equation (3) and they were used in Equation (4) to
obtain the heat transfer coefficients (h∗). This was an
iterative process because the fin efficiency ηf depends
on h∗. The results were expressed in non-dimensional
form by a Nusselt number based on the interfin channels
hydraulic diameter (Dh), as in Equation (5).
N u = h∗Dh
k (5)
The results of the present investigation have also
considered the heat sinks as heat exchangers [26, 27]. In
this approach, the heat sink convective heat transfer to the
airflow was described as that of a heat exchanger by the
effectiveness (ϵ) method, as presented in Equation (6).
89
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
qcv = ˙mcpϵ(Tb − T∞) (6)
From Equations (3) and (6), the heat sink effectiveness may
be expressed as in Equation (7).
ϵ = 1
Rth ˙mcp
(7)
The heat sink base temperature (Tb) was considered
isothermal, and thus, it was associated with the hot side
of a heat exchanger with an infinite thermal capacity. In
this case, the heat capacity ratio is equal to zero and the
heat sink effectiveness (ϵ) is related [25] to the number of
transfer units (NTU) by Equation (8).
ϵ = 1 − exp(−N T U ) (8)
For the finned heat sinks, the number of transfer units NTU
= (hη0At)/( ˙mcp). In this definition, h is the average, over
the channel length, heat transfer coefficient based on the
flow local mixed mean temperature, η0 is the overall finned
surface efficiency and At is the total heat sink convective
heat transfer area. Distinct heat sink designs may lead to
distinct NTU values under similar flow conditions. Larger
NTU values give rise to larger effectiveness, resulting in
smaller heat sink thermal resistances.
2.2 Uncertainty analysis
A spreadsheet was developed in the EES (Engineering
Equation Solver - F-Chart Software) program to evaluate
the uncertainties of the experimental results based on the
method of uncertainty propagation [28]. Considering a
result S, obtained by n experimental measurements xi,
the uncertainty of S (denominated σS ) can be evaluated by
Equation (9):
σS =
√
√
√
√ N∑
i=1
( ∂S
∂xi
∆xi
)2
(9)
The absolute or relative uncertainties of all measurements
were specified, along with the nominal measurements
of each test. The results and their uncertainties were
then obtained for each experimental test performed. The
absolute uncertainties for the readings of the inclined
manometer, U-manometer and thermocouples were
estimated according to Table 2. For the aluminum and air
properties, relative uncertainties of 1% of the tabulated
values were assumed. These uncertainties were inserted
in the EES worksheet to obtain the uncertainty values
for the air mass flow rate, the Reynolds number, the
convective heat transfer rate and the airflow pressure
drop.
Table 2 Values adopted for uncertainties
Variable Uncertainties
Inclined manometer height [in alcohol] 0.005
U-Manometer height [mmca] 0.5
Thermocouple temperature [◦C] 0.1
Barometric pressure [hPa] 5.0
Coefficient K of the nozzle [-] 0.02
Aluminum properties [%] 2.5
Air properties [%] 1.0
3. Numerical investigation
The experimental results of the average Nusselt number
in the channels of both heat sinks were compared not only
to each other, but also to results obtained from numerical
simulations performed under steady-state conditions.
Additional information obtained from the simulations will
also be presented, in order to illustrate the flow and heat
transfer in the heat sinks. Several physical simplifications
were adopted in these simulations, as will be indicated
in the following, and their effect will be reflected in the
results to be presented.
For both heat sinks, i.e., for that with straight fins and also
for that with strip fins, the simulations were performed
for a single heat sink channel. It was assumed that the
heat sink airflow rate was equally distributed into all
the heat sink channels. It was also assumed, for the
strip fins channel, that the flow at the entrance of the
channel did not migrate to a neighbor channel, although
this mixing may actually occur at the end of each strip
fin. That was a necessary assumption to perform the
simulations considering a single heat sink channel.
Additional assumptions were a uniform flow velocity and
temperature at the entrance of the numerical domain.
Atmospheric air was assumed as the cooling fluid for both
heat sinks, flowing parallel to their bases.
The conservation equations were solved by the finite
volumes method to obtain the air velocity and temperature
distributions in one interfin channel of each heat sink.
The numerical results were compared to the experimental
results previously obtained. In order to have a broader
perspective on the present investigation, two distinct fin
models were used in the simulations. The first fin model
considered thin plate fins with no thickness, isothermal
with the heat sink base temperature. The second model
considered the actual fins’ thickness and material, in
perfect thermal contact with the heat sinks isothermal
base. For the strip fins heat sink, for example, the
numerical domain used for the simulations with the fin
model with no thickness is indicated in Figure 3, and for
the model with thick fins, it is indicated in Figure 4.
90
qcv = ˙mcpϵ(Tb − T∞) (6)
From Equations (3) and (6), the heat sink effectiveness may
be expressed as in Equation (7).
ϵ = 1
Rth ˙mcp
(7)
The heat sink base temperature (Tb) was considered
isothermal, and thus, it was associated with the hot side
of a heat exchanger with an infinite thermal capacity. In
this case, the heat capacity ratio is equal to zero and the
heat sink effectiveness (ϵ) is related [25] to the number of
transfer units (NTU) by Equation (8).
ϵ = 1 − exp(−N T U ) (8)
For the finned heat sinks, the number of transfer units NTU
= (hη0At)/( ˙mcp). In this definition, h is the average, over
the channel length, heat transfer coefficient based on the
flow local mixed mean temperature, η0 is the overall finned
surface efficiency and At is the total heat sink convective
heat transfer area. Distinct heat sink designs may lead to
distinct NTU values under similar flow conditions. Larger
NTU values give rise to larger effectiveness, resulting in
smaller heat sink thermal resistances.
2.2 Uncertainty analysis
A spreadsheet was developed in the EES (Engineering
Equation Solver - F-Chart Software) program to evaluate
the uncertainties of the experimental results based on the
method of uncertainty propagation [28]. Considering a
result S, obtained by n experimental measurements xi,
the uncertainty of S (denominated σS ) can be evaluated by
Equation (9):
σS =
√
√
√
√ N∑
i=1
( ∂S
∂xi
∆xi
)2
(9)
The absolute or relative uncertainties of all measurements
were specified, along with the nominal measurements
of each test. The results and their uncertainties were
then obtained for each experimental test performed. The
absolute uncertainties for the readings of the inclined
manometer, U-manometer and thermocouples were
estimated according to Table 2. For the aluminum and air
properties, relative uncertainties of 1% of the tabulated
values were assumed. These uncertainties were inserted
in the EES worksheet to obtain the uncertainty values
for the air mass flow rate, the Reynolds number, the
convective heat transfer rate and the airflow pressure
drop.
Table 2 Values adopted for uncertainties
Variable Uncertainties
Inclined manometer height [in alcohol] 0.005
U-Manometer height [mmca] 0.5
Thermocouple temperature [◦C] 0.1
Barometric pressure [hPa] 5.0
Coefficient K of the nozzle [-] 0.02
Aluminum properties [%] 2.5
Air properties [%] 1.0
3. Numerical investigation
The experimental results of the average Nusselt number
in the channels of both heat sinks were compared not only
to each other, but also to results obtained from numerical
simulations performed under steady-state conditions.
Additional information obtained from the simulations will
also be presented, in order to illustrate the flow and heat
transfer in the heat sinks. Several physical simplifications
were adopted in these simulations, as will be indicated
in the following, and their effect will be reflected in the
results to be presented.
For both heat sinks, i.e., for that with straight fins and also
for that with strip fins, the simulations were performed
for a single heat sink channel. It was assumed that the
heat sink airflow rate was equally distributed into all
the heat sink channels. It was also assumed, for the
strip fins channel, that the flow at the entrance of the
channel did not migrate to a neighbor channel, although
this mixing may actually occur at the end of each strip
fin. That was a necessary assumption to perform the
simulations considering a single heat sink channel.
Additional assumptions were a uniform flow velocity and
temperature at the entrance of the numerical domain.
Atmospheric air was assumed as the cooling fluid for both
heat sinks, flowing parallel to their bases.
The conservation equations were solved by the finite
volumes method to obtain the air velocity and temperature
distributions in one interfin channel of each heat sink.
The numerical results were compared to the experimental
results previously obtained. In order to have a broader
perspective on the present investigation, two distinct fin
models were used in the simulations. The first fin model
considered thin plate fins with no thickness, isothermal
with the heat sink base temperature. The second model
considered the actual fins’ thickness and material, in
perfect thermal contact with the heat sinks isothermal
base. For the strip fins heat sink, for example, the
numerical domain used for the simulations with the fin
model with no thickness is indicated in Figure 3, and for
the model with thick fins, it is indicated in Figure 4.
90
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
Figure 3 Numerical domain for the first fin model - isothermal
plates with no thickness
Figure 4 Numerical domain for the second fin model - thick
non-isothermal fins
3.1 Simulations
The numerical simulations to obtain the heat sinks flow
and heat transfer parameters were performed with the
package PHOENICS [24]. The conservation equations
of mass, momentum and energy were solved under
steady-state conditions by three-dimensional simulations
using the finite volumes method [29]. When the Reynolds
numbers in the interfin channels (Rec) were up to
2,300, the simulations were performed for laminar flows.
For higher Reynolds numbers, the simulations were
performed with the zero equation turbulence model LVEL
[30] incorporated in the PHOENICS package. It employs
a continuous universal velocity profile in the range from 0
< y+ < 100, described in the work of Spalding [31]. The
LVEL model is indicated for the simulation of the flow in
narrow passages and it evaluates the distances from the
nodal points to the walls automatically, in order to obtain
the velocity profile described in this work of Spalding. The
transition to turbulence for rectangular ducts with abrupt
entrance and the aspect ratios from 3 to 10 is reported in
Hartnett et al. [32] for a duct Reynolds number around
2,500 to 2,800. The rectangular ducts of the plate fins heat
sink used in this investigation had channels with an aspect
ratio equal to 6.1. The conservation equations (continuity,
RANS, and energy) were expressed by Equations (10) to
(12):
∂ui
∂xi
= 0 (10)
∂uiuj
∂xj
= − 1
ρ
∂p
∂xi
+ ∂
∂xj
[( ∂ui
∂xj
+ ∂uj
∂xi
)
(ν + ϵM )
]
(11)
∂(uiT )
∂xi
= ∂
∂xi
[ ∂T
∂xi
(α + ϵH )
]
(12)
In these equations, the spatial coordinates are represented
by xi and xj . The velocity components are indicated
by ui and uj and the pressure and temperature are
represented, respectively, by p and T . The fluid properties
are indicated by (ρ) density, (ν) kinematic viscosity, and
(α) thermal diffusivity. The turbulent diffusivities ϵM
and ϵH were obtained from the LVEL turbulence model,
assuming a turbulent Prandtl number (ϵM /ϵH ) equal to
one. The coupling between velocity and pressure in the
transport equations was solved iteratively by the algorithm
SIMPLEST, developed by Spalding [33]. The advective
terms were interpolated by the hybrid scheme [34].
3.2 Modeling of the heat sink channels
The two fin models (thick and no-thickness fins) employed
in this work gave rise to two distinct numerical domains
for each heat sink, as shown in Figures 3 and 4 only for
the strip fins. Figure 3 shows the numerical domain for
the no-thickness fin model, considering a row of strip
fins centered in the channel. The domain dimensions
were equal to 1.85 mm in the x direction, 11.3 mm in
the y direction and 130 mm along the z direction. The
upstream fin was 10 mm downstream the flow entrance in
the numerical domain, the same heat sink position in the
experimental duct.
Figure 4 shows the numerical domain in the case of the
thick non-isothermal fins. In this case, due to symmetry,
the numerical domain comprised only half of the heat
sink channel width. Similar to the experimental setup,
the numerical domain, in this case, was 1.36 mm in the x
direction, 11.3 mm in the y direction and 130 mm in the z
direction. It contained only half of the fins thickness, i.e.,
0.43 mm.
For the no-thickness fin model, a uniform surface
temperature was imposed on the fins surfaces and on the
heat sink base. For the second model, considering the fins
thickness, a temperature was specified on the heat sink
base. In this case, the fins were considered to have perfect
thermal contact with the heat sink base and an adiabatic
condition was imposed on their tips. Uniform velocity and
temperature profiles were adopted for the air inflow in
the numerical domain. The inlet velocity varied from 4
91
Figure 3 Numerical domain for the first fin model - isothermal
plates with no thickness
Figure 4 Numerical domain for the second fin model - thick
non-isothermal fins
3.1 Simulations
The numerical simulations to obtain the heat sinks flow
and heat transfer parameters were performed with the
package PHOENICS [24]. The conservation equations
of mass, momentum and energy were solved under
steady-state conditions by three-dimensional simulations
using the finite volumes method [29]. When the Reynolds
numbers in the interfin channels (Rec) were up to
2,300, the simulations were performed for laminar flows.
For higher Reynolds numbers, the simulations were
performed with the zero equation turbulence model LVEL
[30] incorporated in the PHOENICS package. It employs
a continuous universal velocity profile in the range from 0
< y+ < 100, described in the work of Spalding [31]. The
LVEL model is indicated for the simulation of the flow in
narrow passages and it evaluates the distances from the
nodal points to the walls automatically, in order to obtain
the velocity profile described in this work of Spalding. The
transition to turbulence for rectangular ducts with abrupt
entrance and the aspect ratios from 3 to 10 is reported in
Hartnett et al. [32] for a duct Reynolds number around
2,500 to 2,800. The rectangular ducts of the plate fins heat
sink used in this investigation had channels with an aspect
ratio equal to 6.1. The conservation equations (continuity,
RANS, and energy) were expressed by Equations (10) to
(12):
∂ui
∂xi
= 0 (10)
∂uiuj
∂xj
= − 1
ρ
∂p
∂xi
+ ∂
∂xj
[( ∂ui
∂xj
+ ∂uj
∂xi
)
(ν + ϵM )
]
(11)
∂(uiT )
∂xi
= ∂
∂xi
[ ∂T
∂xi
(α + ϵH )
]
(12)
In these equations, the spatial coordinates are represented
by xi and xj . The velocity components are indicated
by ui and uj and the pressure and temperature are
represented, respectively, by p and T . The fluid properties
are indicated by (ρ) density, (ν) kinematic viscosity, and
(α) thermal diffusivity. The turbulent diffusivities ϵM
and ϵH were obtained from the LVEL turbulence model,
assuming a turbulent Prandtl number (ϵM /ϵH ) equal to
one. The coupling between velocity and pressure in the
transport equations was solved iteratively by the algorithm
SIMPLEST, developed by Spalding [33]. The advective
terms were interpolated by the hybrid scheme [34].
3.2 Modeling of the heat sink channels
The two fin models (thick and no-thickness fins) employed
in this work gave rise to two distinct numerical domains
for each heat sink, as shown in Figures 3 and 4 only for
the strip fins. Figure 3 shows the numerical domain for
the no-thickness fin model, considering a row of strip
fins centered in the channel. The domain dimensions
were equal to 1.85 mm in the x direction, 11.3 mm in
the y direction and 130 mm along the z direction. The
upstream fin was 10 mm downstream the flow entrance in
the numerical domain, the same heat sink position in the
experimental duct.
Figure 4 shows the numerical domain in the case of the
thick non-isothermal fins. In this case, due to symmetry,
the numerical domain comprised only half of the heat
sink channel width. Similar to the experimental setup,
the numerical domain, in this case, was 1.36 mm in the x
direction, 11.3 mm in the y direction and 130 mm in the z
direction. It contained only half of the fins thickness, i.e.,
0.43 mm.
For the no-thickness fin model, a uniform surface
temperature was imposed on the fins surfaces and on the
heat sink base. For the second model, considering the fins
thickness, a temperature was specified on the heat sink
base. In this case, the fins were considered to have perfect
thermal contact with the heat sink base and an adiabatic
condition was imposed on their tips. Uniform velocity and
temperature profiles were adopted for the air inflow in
the numerical domain. The inlet velocity varied from 4
91
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
to 20 m/s, corresponding to a range of Rec between 810
and 3,800. The airflow properties were considered at the
average between the airflow inlet temperature (assumed
at 18◦C) and the average flow mixing temperature at
the outlet section of the numerical domain. This value
was usually around 20◦C, so that the air properties in
the simulations were taken as follows. Density: 1.204
kg/m3, kinematic viscosity: 1.516×10−5m2/s, conductivity
thermal: 0.02514 W/m.K, specific heat: 1,007 J/kg.K
and number of Prandtl: 0.7. No radiation effects were
considered in the simulations.
3.3 Numerical mesh
An investigation of the numerical mesh was carried out
in order to obtain consistent results without a prohibitive
simulation time and memory. The geometry of the inline
strip fins, simulated with the thickness fin model was
used in these tests. The uniform airflow entry velocity in
the numerical domain was specified at 4 m/s, the lowest
simulated value, corresponding to laminar flow. The tests
started with a relatively coarse grid that was gradually
refined, increasing the number of control volumes in
each direction, as well as the mesh concentration near
the walls. This procedure allowed the simulations to be
performed with the largest channel width, so that the
most refined mesh would be needed to obtain converged
numerical results. The lowest velocity was employed in
these tests in order to solve the laminar flow and energy
equations without any additional model in the simulations.
The six investigated meshes are presented in Table 3,
where Nx, Ny ,and Nz represent the number of control
volumes, respectively, in the x, y, and z directions, and
CV indicates the total number of control volumes in the
computational domain. The tested meshes were refined
in the vicinity of the duct walls and fins. To verify the
independence of the numerical grid in the results, the
mean Nusselt number (N u) was used as a parameter.
As indicated in Figure 5, a value of the mean Nusselt
number was independent of the computational mesh when
it contained more than 244,800 control volumes (mesh
5). However, with a mesh of 144,000 volumes (mesh
4) there was a deviation of only 1.5% in this results,
which was considered acceptable and then used to obtain
the numerical results. A microcomputer with a FX8350
processor (©Advanced Micro Devices, Inc.) with 8 physical
cores, 16 GB of RAM and a video card with 1 GB of dedicated
memory was used in the simulations. The average
processing time for each test with the adopted mesh was
around 15 minutes for the simulations in the laminar
regime and 50 minutes for those in the turbulent regime.
The corresponding number of iterations to attain the
converged solution was around 600 and 1,800, respectively,
for the laminar and the turbulent airflow regimes.
Table 3 Investigated meshes
Mesh Nx Ny Nz CV
1 10 15 35 5,250
2 20 30 51 30,600
3 30 40 80 96,000
4 30 60 80 144,000
5 40 60 102 244,800
6 80 120 204 1,958,400012345674.574.584.594.604.614.624.63NuComputational mesh
Figure 5 Change of the average Nusselt number with the
computational mesh
4. Results and discussion
The experimental tests were performed with aluminum
heat sinks with plate fins and inline strip fins, with the
dimensions shown in Table 1, comprising each 16 parallel
lines of fins. Each row of strip fins had five fins separated
from each other with uniform spacing. The average airflow
velocity in the interfin channels varied from 4 to 20 m/s.
The corresponding interfin channels Reynolds number
based on their hydraulic diameter was in the range from
810 to 3,800.
4.1 Pressure drop
The measured airflow pressure drop across the heat
sinks is presented in Figure 6 as a function of the average
airflow velocity in the interfin channels. These pressure
drops for each heat sink were obtained from two sets
of measurements of the flow pressure drop from the
laboratory ambient air to a pressure tap located in the
rectangular duct lower wall, 30 mm downstream the
heat sink. The first set was obtained with the heat sink
installed in the duct and the second, with the heat sink
removed from the duct. The results presented in Figure 6
represent the difference between the first and the second
sets for each heat sink. They indicate the flow pressure
drop increase in the rectangular duct associated to the
92
to 20 m/s, corresponding to a range of Rec between 810
and 3,800. The airflow properties were considered at the
average between the airflow inlet temperature (assumed
at 18◦C) and the average flow mixing temperature at
the outlet section of the numerical domain. This value
was usually around 20◦C, so that the air properties in
the simulations were taken as follows. Density: 1.204
kg/m3, kinematic viscosity: 1.516×10−5m2/s, conductivity
thermal: 0.02514 W/m.K, specific heat: 1,007 J/kg.K
and number of Prandtl: 0.7. No radiation effects were
considered in the simulations.
3.3 Numerical mesh
An investigation of the numerical mesh was carried out
in order to obtain consistent results without a prohibitive
simulation time and memory. The geometry of the inline
strip fins, simulated with the thickness fin model was
used in these tests. The uniform airflow entry velocity in
the numerical domain was specified at 4 m/s, the lowest
simulated value, corresponding to laminar flow. The tests
started with a relatively coarse grid that was gradually
refined, increasing the number of control volumes in
each direction, as well as the mesh concentration near
the walls. This procedure allowed the simulations to be
performed with the largest channel width, so that the
most refined mesh would be needed to obtain converged
numerical results. The lowest velocity was employed in
these tests in order to solve the laminar flow and energy
equations without any additional model in the simulations.
The six investigated meshes are presented in Table 3,
where Nx, Ny ,and Nz represent the number of control
volumes, respectively, in the x, y, and z directions, and
CV indicates the total number of control volumes in the
computational domain. The tested meshes were refined
in the vicinity of the duct walls and fins. To verify the
independence of the numerical grid in the results, the
mean Nusselt number (N u) was used as a parameter.
As indicated in Figure 5, a value of the mean Nusselt
number was independent of the computational mesh when
it contained more than 244,800 control volumes (mesh
5). However, with a mesh of 144,000 volumes (mesh
4) there was a deviation of only 1.5% in this results,
which was considered acceptable and then used to obtain
the numerical results. A microcomputer with a FX8350
processor (©Advanced Micro Devices, Inc.) with 8 physical
cores, 16 GB of RAM and a video card with 1 GB of dedicated
memory was used in the simulations. The average
processing time for each test with the adopted mesh was
around 15 minutes for the simulations in the laminar
regime and 50 minutes for those in the turbulent regime.
The corresponding number of iterations to attain the
converged solution was around 600 and 1,800, respectively,
for the laminar and the turbulent airflow regimes.
Table 3 Investigated meshes
Mesh Nx Ny Nz CV
1 10 15 35 5,250
2 20 30 51 30,600
3 30 40 80 96,000
4 30 60 80 144,000
5 40 60 102 244,800
6 80 120 204 1,958,400012345674.574.584.594.604.614.624.63NuComputational mesh
Figure 5 Change of the average Nusselt number with the
computational mesh
4. Results and discussion
The experimental tests were performed with aluminum
heat sinks with plate fins and inline strip fins, with the
dimensions shown in Table 1, comprising each 16 parallel
lines of fins. Each row of strip fins had five fins separated
from each other with uniform spacing. The average airflow
velocity in the interfin channels varied from 4 to 20 m/s.
The corresponding interfin channels Reynolds number
based on their hydraulic diameter was in the range from
810 to 3,800.
4.1 Pressure drop
The measured airflow pressure drop across the heat
sinks is presented in Figure 6 as a function of the average
airflow velocity in the interfin channels. These pressure
drops for each heat sink were obtained from two sets
of measurements of the flow pressure drop from the
laboratory ambient air to a pressure tap located in the
rectangular duct lower wall, 30 mm downstream the
heat sink. The first set was obtained with the heat sink
installed in the duct and the second, with the heat sink
removed from the duct. The results presented in Figure 6
represent the difference between the first and the second
sets for each heat sink. They indicate the flow pressure
drop increase in the rectangular duct associated to the
92
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
presence of the heat sink. The test data show a monotonic
increase of the flow pressure drop with the channels
velocity. When comparing the two heat sinks, the airflow
pressure drop for the strip fins heat sink is increasingly
larger than that for the plate fins, indicating the need for
larger fan power as the airflow velocity increases. It is also
noticed that for low average flow velocities, in the range
of 4 to 8 m/s, the pressure drop in the two heat sinks is
practically the same, with a percentage difference of only
12%.
The experimental data points of the airflow pressure drop
through the heat sinks were adjusted by quadratic
functions, expressed by Equations (13) and (14),
respectively, for the flat plate fins (∆Ppf ) and inline
strip fins heat sink (∆Psf ). These equations, as well as
Equations (15-20), were obtained using the least-squares
method, implemented in the Curve Fitting toolbox and
coded in MATLAB. The quality of the adjustments is
given by the coefficients of determination (R2) that are
presented for each case.
∆Ppf = 1.0833V 2 + 6.6068V + 17.043
R2 = 0.9995 (13)
∆Psf = 0.7651V 2 + 2.239V + 1.779
R2 = 0.9983 (14)468101214161820220100200300400500600 Strip fins Plate fins∆P [ P a ]
V
[
m
/
s
]
Figure 6 Pressure drop of the airflow through the heat sinks
4.2 Average Nusselt number
The rate of heat transfer by forced convection from the
heat sinks tested in the laboratory to the airflow was
obtained as indicated by Equation (2). From this rate,
it was possible to obtain the experimental data for the
average heat transfer coefficient h∗ (using the flow inlet
temperature as a reference) obtained from Equation (4).
This coefficient was used to evaluate the average Nusselt
number, as defined by Equation (5).
The results for the average Nusselt number for both heat
sinks are shown in Figure 7 as a function of the airflow
Reynolds number in the channel. Note that for both heat
sinks, the values increase with the Reynolds number in
the channel. When analyzing the plate fins heat sink, the
results indicate two distinct regions. Up to a Reynolds
number value around 2,400, the flow was characterized
as laminar and for Reynolds values above 3,000, the flow
was assumed turbulent. Equations (15) and (16) show
correlations for the average Nusselt number as a function
of the Reynolds number in each channel for the plate fins
heat sink, respectively for the laminar and turbulent flow
regimes. This subdivision is coherent with the literature
on the critical Reynolds number in rectangular channels
with abrupt entrance, presents more accurate correlations
as compared to a single correlation for all the data and
reflects the physics of the fluid flow in these channels.
For the inline strip fins heat sink, a single correlation
was adjusted for all the experimental data (Equation 17).
This behavior of the average Nusselt number reflects the
complex flow and convective heat transfer in the channels
with inline strip fins. The strip fins in these channels induce
the flow to separate at the downstream end of each fin and
to start new boundary layers at the downstream strip fin,
and so on. They must induce turbulent flow at low channel
Reynolds number, as indicated by the single correlation
encompassing all the data.
N upf = 0.112Re0.52
c ; 810 < Rec < 2, 400
R2 = 0.9949 (15)
N upf = 0.002Re1.03
c ; 3, 000 < Rec < 3, 800
R2 = 0.9946 (16)
N usf = 0.028Re0.77
c ; 810 < Rec < 3, 800
R2 = 0.9947 (17)
The flat plate heat sink channels have a length of about
28 hydraulic diameters. In order to attain a thermally
developed condition in the investigated laminar range of
810 < Rec < 2,400, these channels should have from 36
to 108 hydraulic diameters [25, 32]. Thus, Equation (15)
represents results for the simultaneous entrance region
in laminar flow. For 3,000 < Rec < 3,800, the entrance
length for turbulent flow usually requires a length from 20
to 40 channel hydraulic diameters, so that Equation (16)
93
presence of the heat sink. The test data show a monotonic
increase of the flow pressure drop with the channels
velocity. When comparing the two heat sinks, the airflow
pressure drop for the strip fins heat sink is increasingly
larger than that for the plate fins, indicating the need for
larger fan power as the airflow velocity increases. It is also
noticed that for low average flow velocities, in the range
of 4 to 8 m/s, the pressure drop in the two heat sinks is
practically the same, with a percentage difference of only
12%.
The experimental data points of the airflow pressure drop
through the heat sinks were adjusted by quadratic
functions, expressed by Equations (13) and (14),
respectively, for the flat plate fins (∆Ppf ) and inline
strip fins heat sink (∆Psf ). These equations, as well as
Equations (15-20), were obtained using the least-squares
method, implemented in the Curve Fitting toolbox and
coded in MATLAB. The quality of the adjustments is
given by the coefficients of determination (R2) that are
presented for each case.
∆Ppf = 1.0833V 2 + 6.6068V + 17.043
R2 = 0.9995 (13)
∆Psf = 0.7651V 2 + 2.239V + 1.779
R2 = 0.9983 (14)468101214161820220100200300400500600 Strip fins Plate fins∆P [ P a ]
V
[
m
/
s
]
Figure 6 Pressure drop of the airflow through the heat sinks
4.2 Average Nusselt number
The rate of heat transfer by forced convection from the
heat sinks tested in the laboratory to the airflow was
obtained as indicated by Equation (2). From this rate,
it was possible to obtain the experimental data for the
average heat transfer coefficient h∗ (using the flow inlet
temperature as a reference) obtained from Equation (4).
This coefficient was used to evaluate the average Nusselt
number, as defined by Equation (5).
The results for the average Nusselt number for both heat
sinks are shown in Figure 7 as a function of the airflow
Reynolds number in the channel. Note that for both heat
sinks, the values increase with the Reynolds number in
the channel. When analyzing the plate fins heat sink, the
results indicate two distinct regions. Up to a Reynolds
number value around 2,400, the flow was characterized
as laminar and for Reynolds values above 3,000, the flow
was assumed turbulent. Equations (15) and (16) show
correlations for the average Nusselt number as a function
of the Reynolds number in each channel for the plate fins
heat sink, respectively for the laminar and turbulent flow
regimes. This subdivision is coherent with the literature
on the critical Reynolds number in rectangular channels
with abrupt entrance, presents more accurate correlations
as compared to a single correlation for all the data and
reflects the physics of the fluid flow in these channels.
For the inline strip fins heat sink, a single correlation
was adjusted for all the experimental data (Equation 17).
This behavior of the average Nusselt number reflects the
complex flow and convective heat transfer in the channels
with inline strip fins. The strip fins in these channels induce
the flow to separate at the downstream end of each fin and
to start new boundary layers at the downstream strip fin,
and so on. They must induce turbulent flow at low channel
Reynolds number, as indicated by the single correlation
encompassing all the data.
N upf = 0.112Re0.52
c ; 810 < Rec < 2, 400
R2 = 0.9949 (15)
N upf = 0.002Re1.03
c ; 3, 000 < Rec < 3, 800
R2 = 0.9946 (16)
N usf = 0.028Re0.77
c ; 810 < Rec < 3, 800
R2 = 0.9947 (17)
The flat plate heat sink channels have a length of about
28 hydraulic diameters. In order to attain a thermally
developed condition in the investigated laminar range of
810 < Rec < 2,400, these channels should have from 36
to 108 hydraulic diameters [25, 32]. Thus, Equation (15)
represents results for the simultaneous entrance region
in laminar flow. For 3,000 < Rec < 3,800, the entrance
length for turbulent flow usually requires a length from 20
to 40 channel hydraulic diameters, so that Equation (16)
93
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 202450010001500200025003000350040004681012141618 Strip fins Plate fins NuRec
Figure 7 Heat sinks average Nusselt number
also represents a correlation for the entrance region in a
turbulent flow. For the flow in the strip fins heat sink,
open channels were associated with the space between
consecutive rows of strip fins. Each strip fins acts a
barrier with an abrupt entrance in these channels and
thus, they induce a transition to turbulent flow at smaller
Reynolds numbers. Equation (17) was expressed for the
channels formed by consecutive rows of strip fins, as a
single correlation for the entire range of the Reynolds
number.
4.3 Convective thermal resistance
The experimental results for the heat sinks convective
thermal resistance, defined by Equation (3), are presented
in Figure 8 and they show a typical decrease as the average
airflow velocity in the interfin channels increases. This
result reflects the convective heat transfer increase with
the airflow rate in the interfin channels, as was indicated
in Figure 7.
These results show that in spite of the strip fins heat sink
smaller heat transfer area, the average heat transfer
coefficient is larger enough to give rise to a smaller
thermal resistance in the investigated range of the
Reynolds number. It should be kept in mind, however, that
the strip fins heat sink demands a larger fan power for the
same airflow velocity.
Equations (18), (19) and (20) show, respectively, the
correlations for the plate fins heat sink (laminar and
turbulent flow regimes) and for the inline strip fins heat
sink thermal resistances.
Rth(pf ) = 1.9V −0.5
R2 = 0.9983 (18)
Rth(pf ) = 7.4V −1
R2 = 0.9991 (19)
Rth(sf ) = 2.4V −0.75
R2 = 0.9993 (20)468101214161820220,20,30,40,50,60,70,80,9 Plate fins Strip finsRth [ºC/W]V [m/s]
Figure 8 Heat sinks thermal resistance
4.4 Treatment of the heat sinks as heat
exchangers
In the treatment of heat sinks as heat exchangers, the
thermal resistance is inversely proportional to their
effectiveness, as presented by Equation (7). In the
experimental tests, the two heat sinks were subjected to
the same range of air flow rates and thus, the comparison
of the two heat sinks effectiveness provides a comparison
of their thermal resistances.
The experimental results obtained for the effectiveness
(ϵ) of the two heat sinks, treated as heat exchangers, are
shown in Figure 9. For the same range of airflow rate, the
strip fins heat sink operates with higher effectiveness than
the plate fins heat sink. These results reflect the fact that
the average convective coefficients are higher for the strip
fins heat sink.
For both heat sinks, the NTU decreases with increasing
airflow rate, thus causing a reduction in their effectiveness
(ϵ). Comparatively, the strip fins heat sink presents a more
94
Figure 7 Heat sinks average Nusselt number
also represents a correlation for the entrance region in a
turbulent flow. For the flow in the strip fins heat sink,
open channels were associated with the space between
consecutive rows of strip fins. Each strip fins acts a
barrier with an abrupt entrance in these channels and
thus, they induce a transition to turbulent flow at smaller
Reynolds numbers. Equation (17) was expressed for the
channels formed by consecutive rows of strip fins, as a
single correlation for the entire range of the Reynolds
number.
4.3 Convective thermal resistance
The experimental results for the heat sinks convective
thermal resistance, defined by Equation (3), are presented
in Figure 8 and they show a typical decrease as the average
airflow velocity in the interfin channels increases. This
result reflects the convective heat transfer increase with
the airflow rate in the interfin channels, as was indicated
in Figure 7.
These results show that in spite of the strip fins heat sink
smaller heat transfer area, the average heat transfer
coefficient is larger enough to give rise to a smaller
thermal resistance in the investigated range of the
Reynolds number. It should be kept in mind, however, that
the strip fins heat sink demands a larger fan power for the
same airflow velocity.
Equations (18), (19) and (20) show, respectively, the
correlations for the plate fins heat sink (laminar and
turbulent flow regimes) and for the inline strip fins heat
sink thermal resistances.
Rth(pf ) = 1.9V −0.5
R2 = 0.9983 (18)
Rth(pf ) = 7.4V −1
R2 = 0.9991 (19)
Rth(sf ) = 2.4V −0.75
R2 = 0.9993 (20)468101214161820220,20,30,40,50,60,70,80,9 Plate fins Strip finsRth [ºC/W]V [m/s]
Figure 8 Heat sinks thermal resistance
4.4 Treatment of the heat sinks as heat
exchangers
In the treatment of heat sinks as heat exchangers, the
thermal resistance is inversely proportional to their
effectiveness, as presented by Equation (7). In the
experimental tests, the two heat sinks were subjected to
the same range of air flow rates and thus, the comparison
of the two heat sinks effectiveness provides a comparison
of their thermal resistances.
The experimental results obtained for the effectiveness
(ϵ) of the two heat sinks, treated as heat exchangers, are
shown in Figure 9. For the same range of airflow rate, the
strip fins heat sink operates with higher effectiveness than
the plate fins heat sink. These results reflect the fact that
the average convective coefficients are higher for the strip
fins heat sink.
For both heat sinks, the NTU decreases with increasing
airflow rate, thus causing a reduction in their effectiveness
(ϵ). Comparatively, the strip fins heat sink presents a more
94
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
effective thermal design than the plate fins heat sink. For
this heat sink to operate as effectively as the strip fin heat
sink, it should operate in a lower range of air flow rate, thus
increasing the NTU and the effectiveness values (ϵ).0,51,01,52,02,50,40,50,60,70,80,91,0ε
N
U
T
S
t
r
i
p
f
i
n
s
P
l
a
t
e
f
i
n
s
Figure 9 Effectiveness of the heat sinks as a function of the
number of transfer units
4.5 Uncertainties of the experimental results
The uncertainties of the experimental results were
estimated as described in section 2.2, for the airflow
pressure drop, the average airflow velocity, the average
Nusselt number, the Reynolds number, and the convective
thermal resistance. The results obtained are presented in
Table 4.
Table 4 Propagated values of relative uncertainties
Variable Lowest ˙m [kg/s] Highest ˙m [kg/s]
∆P 5.3% 1.0%
V 4.6% 3.7%
Nu 2.9% 3.0%
Re 5.2% 4.6%
Rth 1.2% 1.2%
5. Numerical results
5.1 Plate fins heat sink
The numerical airflow temperature distribution in a
plane parallel to the base of a flat plate fin is presented
in Figure 10, considering the no-thickness fin model,
when Rec = 3,800. As expected, the airflow temperature
increases both downstream the duct and also in the vicinity
of the fin.
A comparison between the experimental and the numerical
results for the average Nusselt number for the plate fin is
shown in Figure 11. The numerical results were obtained
from both fin models - with negligible fin thickness (1st
model) and with thick fins (2nd model). The experimental
data are those already presented in Figure 7 for the plate
fins and they are reproduced here by the symbols only.
The numerical simulations performed with the thick fin
model are are physically more realistic and they presented
Nusselt number results closer to the experimental data.
There was a deviation of around 2% for the Reynolds
number Rec in the range from 810 to 2,300 (laminar
regime) and about 3% for the range between 2,500 and
3,800 (turbulent regime). For the simulations with the
no-thickness fin model, the results presented similar
trends, but with larger deviations from the experimental
data, as shown in Figure 11. They varied from 3% to 8% in
the laminar regime and were around 9% in the turbulent
regime.
Figure 10 Numerical temperature distribution for the plate fins
heat sink using the first fin model (Rec = 3800)
5.2 Strip fins heat sink
The numerical airflow temperature distribution in a plane
parallel to the base of a row of strip fin is presented in
Figure 12, for the first fin model (no thickness), when Rec
= 3,800. The airflow temperature increases in the vicinity
of the five strip fins and also in the downstream direction.
The simulations were also performed for the thick fins
model.
For the strip fins, the flow simulation is quite more complex
due to the interrupted boundary layers growth at the end
of each fin. In the present simulations, the x-boundaries
of the numerical domain presented in Figure 12 were
95
effective thermal design than the plate fins heat sink. For
this heat sink to operate as effectively as the strip fin heat
sink, it should operate in a lower range of air flow rate, thus
increasing the NTU and the effectiveness values (ϵ).0,51,01,52,02,50,40,50,60,70,80,91,0ε
N
U
T
S
t
r
i
p
f
i
n
s
P
l
a
t
e
f
i
n
s
Figure 9 Effectiveness of the heat sinks as a function of the
number of transfer units
4.5 Uncertainties of the experimental results
The uncertainties of the experimental results were
estimated as described in section 2.2, for the airflow
pressure drop, the average airflow velocity, the average
Nusselt number, the Reynolds number, and the convective
thermal resistance. The results obtained are presented in
Table 4.
Table 4 Propagated values of relative uncertainties
Variable Lowest ˙m [kg/s] Highest ˙m [kg/s]
∆P 5.3% 1.0%
V 4.6% 3.7%
Nu 2.9% 3.0%
Re 5.2% 4.6%
Rth 1.2% 1.2%
5. Numerical results
5.1 Plate fins heat sink
The numerical airflow temperature distribution in a
plane parallel to the base of a flat plate fin is presented
in Figure 10, considering the no-thickness fin model,
when Rec = 3,800. As expected, the airflow temperature
increases both downstream the duct and also in the vicinity
of the fin.
A comparison between the experimental and the numerical
results for the average Nusselt number for the plate fin is
shown in Figure 11. The numerical results were obtained
from both fin models - with negligible fin thickness (1st
model) and with thick fins (2nd model). The experimental
data are those already presented in Figure 7 for the plate
fins and they are reproduced here by the symbols only.
The numerical simulations performed with the thick fin
model are are physically more realistic and they presented
Nusselt number results closer to the experimental data.
There was a deviation of around 2% for the Reynolds
number Rec in the range from 810 to 2,300 (laminar
regime) and about 3% for the range between 2,500 and
3,800 (turbulent regime). For the simulations with the
no-thickness fin model, the results presented similar
trends, but with larger deviations from the experimental
data, as shown in Figure 11. They varied from 3% to 8% in
the laminar regime and were around 9% in the turbulent
regime.
Figure 10 Numerical temperature distribution for the plate fins
heat sink using the first fin model (Rec = 3800)
5.2 Strip fins heat sink
The numerical airflow temperature distribution in a plane
parallel to the base of a row of strip fin is presented in
Figure 12, for the first fin model (no thickness), when Rec
= 3,800. The airflow temperature increases in the vicinity
of the five strip fins and also in the downstream direction.
The simulations were also performed for the thick fins
model.
For the strip fins, the flow simulation is quite more complex
due to the interrupted boundary layers growth at the end
of each fin. In the present simulations, the x-boundaries
of the numerical domain presented in Figure 12 were
95
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024500 1000 1500 2000 2500 3000 3500 4000
3
4
5
6
7
8
9
10
11
Experimental
Numeric - With thickness
Numeric - No thickness
N u
Re
C
Figure 11 Nusselt number results from the experiments and
the simulations for the plate fins heat sink
specified as symmetry planes, but this assumption may
be subject to further investigation in the future. Thus, it
may be anticipated that in this case the numerical results
for the average Nusselt number in the fin channel will not
be as comparable to the experiments as they were in the
case of the flat plate fins.
In the simulations with the first fin model (no thickness),
the flow was assumed in the laminar regime for Rec <
2,300 and in the turbulent regime for Rec > 2,500. When
the second fin model (thick fin) was used, two distinct sets
of simulations were performed over the entire Reynolds
number range. For the first set the flow was assumed in
the laminar regime, while the second set, it was assumed
in the turbulent regime.
The numerical and experimental results for the average
Nusselt number for the strip fins’ channel are compared in
Figure 13. The experimental results are the same as those
from Figure 7 and they are indicated just by the symbols.
The numerical results for the first fin model were obtained
from simulations considering either laminar flow (Rec <
2,300) or turbulent flow (Rec > 2,500) and in both regimes
the predictions were below the experimental values. For
the second fin model (thick fins), the numerical results
from the laminar flow assumption were also below the
experimental values, but closer than those from the first fin
model. The simulations for the turbulent flow assumption
presented N u results above the experimental values.
These results indicate a complex flow around the strip
fins, where turbulence may arise in the flow separations
and re-attachments at each strip fin. The deviations
with the first fin model were from 7% to 21% when the
flow was in the laminar regime and from 17% to 21% in
the turbulent regime. When the second fin model (thick
fin) was adopted in the simulations, the deviations from
the experimental results were from 1% to 6% below the
experimental results when the entire flow was simulated
in the laminar regime and they were between 22% and
37% above the experimental results when the flow was
assumed turbulent over the entire Rec range.
Figure 12 Numerical temperature distribution for the strip fins
heat sink using the first fin model (Rec = 3800)500 1000 1500 2000 2500 3000 3500 4000
4
6
8
10
12
14
16
18
20
22
Experimental
Numeric - With thickness (LVEL)
Numeric - With thickness (Laminar)
Numeric - No thickness
N u
Re
C
Figure 13 Nusselt number results from the experiments and
the simulations for the strip fins heat sink
96
3
4
5
6
7
8
9
10
11
Experimental
Numeric - With thickness
Numeric - No thickness
N u
Re
C
Figure 11 Nusselt number results from the experiments and
the simulations for the plate fins heat sink
specified as symmetry planes, but this assumption may
be subject to further investigation in the future. Thus, it
may be anticipated that in this case the numerical results
for the average Nusselt number in the fin channel will not
be as comparable to the experiments as they were in the
case of the flat plate fins.
In the simulations with the first fin model (no thickness),
the flow was assumed in the laminar regime for Rec <
2,300 and in the turbulent regime for Rec > 2,500. When
the second fin model (thick fin) was used, two distinct sets
of simulations were performed over the entire Reynolds
number range. For the first set the flow was assumed in
the laminar regime, while the second set, it was assumed
in the turbulent regime.
The numerical and experimental results for the average
Nusselt number for the strip fins’ channel are compared in
Figure 13. The experimental results are the same as those
from Figure 7 and they are indicated just by the symbols.
The numerical results for the first fin model were obtained
from simulations considering either laminar flow (Rec <
2,300) or turbulent flow (Rec > 2,500) and in both regimes
the predictions were below the experimental values. For
the second fin model (thick fins), the numerical results
from the laminar flow assumption were also below the
experimental values, but closer than those from the first fin
model. The simulations for the turbulent flow assumption
presented N u results above the experimental values.
These results indicate a complex flow around the strip
fins, where turbulence may arise in the flow separations
and re-attachments at each strip fin. The deviations
with the first fin model were from 7% to 21% when the
flow was in the laminar regime and from 17% to 21% in
the turbulent regime. When the second fin model (thick
fin) was adopted in the simulations, the deviations from
the experimental results were from 1% to 6% below the
experimental results when the entire flow was simulated
in the laminar regime and they were between 22% and
37% above the experimental results when the flow was
assumed turbulent over the entire Rec range.
Figure 12 Numerical temperature distribution for the strip fins
heat sink using the first fin model (Rec = 3800)500 1000 1500 2000 2500 3000 3500 4000
4
6
8
10
12
14
16
18
20
22
Experimental
Numeric - With thickness (LVEL)
Numeric - With thickness (Laminar)
Numeric - No thickness
N u
Re
C
Figure 13 Nusselt number results from the experiments and
the simulations for the strip fins heat sink
96
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
6. Conclusions
Laboratory experiments were carried out to obtain the
airflow pressure drop and the forced convection heat
transfer from two aluminum heat sinks, one with flat
plate fins and the other with inline strip fins. Both heat
sinks were mounted with no top bypass, but with a
lateral bypass equal to one fin spacing on both sides in a
rectangular duct. The experiments were carried out under
steady-state conditions for an average airflow velocity
in the interfin channels in the range from 4 to 20 m/s.
This corresponded to a Reynolds number based on their
hydraulic diameter in the range from 810 and 3,800, a
range for which experimental data are scarce and the
physics of the flow is not completely understood for the
strip fins heat sink. In addition, numerical simulations
for both heat sinks were performed in the PHOENICS
package, assuming two fin models, one with isothermal
fins of negligible thickness and the other with the actual
fin thickness, for comparisons of the numerical with the
experimental results. All the simulations were performed
under steady-state conditions.
The experimental results indicated that the airflow
pressure drop across the inline strip fins heat sink was
always higher than that across the plate fins heat sink.
The difference was small for the lowest flow velocities in
the fins channels, but increased sharply with the velocity.
This showed that the flow through the inline strip fins heat
sink requires a greater fan power than through the flat
plate fins heat sink.
The results of the thermal tests showed that the strip
fins heat sink presented larger average heat transfer
coefficients than the plate fins heat sink. They were
about 25% larger for the lowest channel flow Reynolds
number and about 41% larger for the highest. Despite the
strip fins smaller heat transfer area, their heat transfer
coefficient was larger enough so that they presented a
smaller thermal resistance than the plate fins heat sink.
When the heat sinks were considered as heat exchangers,
the strip fins heat sink effectiveness was distinctively
higher than that of the plate fins heat sink. This result
indicated that the strip fins heat sink presented a better
thermal design under the tested conditions. It required,
however, larger airflow pumping power, mainly in the
highest range of the tested airflow velocity in the interfin
channels.
The present experimental investigation was important to
obtain reliable and useful information to compare the
thermal behavior of two similar aluminum heat sinks, one
with flat plate fins and the other with inline strip fins. The
investigation was carried out in a higher airflow velocity
in the interfin channels than it is usually found in the
literature. The numerical thermal results were obtained
employing two fin models with a commercial CFD package.
There was a reasonable agreement with the experiments
for the flat plate fins, but for the strip fins, with more
complex flows, the comparison was not as good, indicating
that further investigation is still needed.
7. Declaration of competing interest
We declare that we have no significant competing interests
including financial or non-financial, professional, or
personal interests interfering with the full and objective
presentation of the work described in this manuscript.
8. Funding
The support of FAPEMA (Fundação de Amparo à Pesquisa e
ao Desenvolvimento Científico e Tecnológico do Maranhão),
in the form of a Scholarship to the first author, is deeply
appreciated.
9. Author contributions
Conceptualization: William Fonseca and Carlos Altemani.
Experiments and data processing: William Fonseca and
Carlos Altemani. Paper preparation: William Fonseca and
Carlos Altemani.
All authors participated fully in the preparation of this
manuscript.
10. Data available statement
The authors confirm that the data supporting the findings
of this study are available within the article [and/or] its
supplementary materials.
References
[1] S. P. Gurrum, S. K. Suman, Y. K. Joshi, and A. G. Fedorov, “Thermal
issues in next-generation integrated circuits,” IEEE Transactions on
device and materials reliability, vol. 4, no. 4, pp. 709–714, 2004.
[2] H. E. Ahmed, B. Salman, A. S. Kherbeet, and M. Ahmed,
“Optimization of thermal design of heat sinks: A review,”
International Journal of Heat and Mass Transfer, vol. 118, pp. 129–153,
2018.
[3] Z. He, Y. Yan, and Z. Zhang, “Thermal management and temperature
uniformity enhancement of electronic devices by micro heat sinks:
A review,” Energy, vol. 216, p. 119223, 2021.
[4] M. Hajmohammadi, V. A. Abianeh, M. Moezzinajafabadi, and
M. Daneshi, “Fork-shaped highly conductive pathways for maximum
cooling in a heat generating piece,” Applied thermal engineering,
vol. 61, no. 2, pp. 228–235, 2013.
97
6. Conclusions
Laboratory experiments were carried out to obtain the
airflow pressure drop and the forced convection heat
transfer from two aluminum heat sinks, one with flat
plate fins and the other with inline strip fins. Both heat
sinks were mounted with no top bypass, but with a
lateral bypass equal to one fin spacing on both sides in a
rectangular duct. The experiments were carried out under
steady-state conditions for an average airflow velocity
in the interfin channels in the range from 4 to 20 m/s.
This corresponded to a Reynolds number based on their
hydraulic diameter in the range from 810 and 3,800, a
range for which experimental data are scarce and the
physics of the flow is not completely understood for the
strip fins heat sink. In addition, numerical simulations
for both heat sinks were performed in the PHOENICS
package, assuming two fin models, one with isothermal
fins of negligible thickness and the other with the actual
fin thickness, for comparisons of the numerical with the
experimental results. All the simulations were performed
under steady-state conditions.
The experimental results indicated that the airflow
pressure drop across the inline strip fins heat sink was
always higher than that across the plate fins heat sink.
The difference was small for the lowest flow velocities in
the fins channels, but increased sharply with the velocity.
This showed that the flow through the inline strip fins heat
sink requires a greater fan power than through the flat
plate fins heat sink.
The results of the thermal tests showed that the strip
fins heat sink presented larger average heat transfer
coefficients than the plate fins heat sink. They were
about 25% larger for the lowest channel flow Reynolds
number and about 41% larger for the highest. Despite the
strip fins smaller heat transfer area, their heat transfer
coefficient was larger enough so that they presented a
smaller thermal resistance than the plate fins heat sink.
When the heat sinks were considered as heat exchangers,
the strip fins heat sink effectiveness was distinctively
higher than that of the plate fins heat sink. This result
indicated that the strip fins heat sink presented a better
thermal design under the tested conditions. It required,
however, larger airflow pumping power, mainly in the
highest range of the tested airflow velocity in the interfin
channels.
The present experimental investigation was important to
obtain reliable and useful information to compare the
thermal behavior of two similar aluminum heat sinks, one
with flat plate fins and the other with inline strip fins. The
investigation was carried out in a higher airflow velocity
in the interfin channels than it is usually found in the
literature. The numerical thermal results were obtained
employing two fin models with a commercial CFD package.
There was a reasonable agreement with the experiments
for the flat plate fins, but for the strip fins, with more
complex flows, the comparison was not as good, indicating
that further investigation is still needed.
7. Declaration of competing interest
We declare that we have no significant competing interests
including financial or non-financial, professional, or
personal interests interfering with the full and objective
presentation of the work described in this manuscript.
8. Funding
The support of FAPEMA (Fundação de Amparo à Pesquisa e
ao Desenvolvimento Científico e Tecnológico do Maranhão),
in the form of a Scholarship to the first author, is deeply
appreciated.
9. Author contributions
Conceptualization: William Fonseca and Carlos Altemani.
Experiments and data processing: William Fonseca and
Carlos Altemani. Paper preparation: William Fonseca and
Carlos Altemani.
All authors participated fully in the preparation of this
manuscript.
10. Data available statement
The authors confirm that the data supporting the findings
of this study are available within the article [and/or] its
supplementary materials.
References
[1] S. P. Gurrum, S. K. Suman, Y. K. Joshi, and A. G. Fedorov, “Thermal
issues in next-generation integrated circuits,” IEEE Transactions on
device and materials reliability, vol. 4, no. 4, pp. 709–714, 2004.
[2] H. E. Ahmed, B. Salman, A. S. Kherbeet, and M. Ahmed,
“Optimization of thermal design of heat sinks: A review,”
International Journal of Heat and Mass Transfer, vol. 118, pp. 129–153,
2018.
[3] Z. He, Y. Yan, and Z. Zhang, “Thermal management and temperature
uniformity enhancement of electronic devices by micro heat sinks:
A review,” Energy, vol. 216, p. 119223, 2021.
[4] M. Hajmohammadi, V. A. Abianeh, M. Moezzinajafabadi, and
M. Daneshi, “Fork-shaped highly conductive pathways for maximum
cooling in a heat generating piece,” Applied thermal engineering,
vol. 61, no. 2, pp. 228–235, 2013.
97
W. D. Pires-Fonseca, Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 86-98, 2024
[5] P. Hopton and J. Summers, “Enclosed liquid natural convection as
a means of transferring heat from microelectronics to cold plates,”
in 29th IEEE Semiconductor Thermal Measurement and Management
Symposium. IEEE, 2013, pp. 60–64.
[6] H. M. Ali, A. Arshad, M. Jabbal, and P. G. Verdin, “Thermal
management of electronics devices with pcms filled pin-fin heat
sinks: a comparison,” International Journal of Heat and Mass Transfer,
vol. 117, pp. 1199–1204, 2018.
[7] A. Arshad, M. Jabbal, P. T. Sardari, M. A. Bashir, H. Faraji, and Y. Yan,
“Transient simulation of finned heat sinks embedded with pcm
for electronics cooling,” Thermal Science and Engineering Progress,
vol. 18, p. 100520, 2020.
[8] R. Kalbasi, “Introducing a novel heat sink comprising pcm
and air-adapted to electronic device thermal management,”
International Journal of Heat and Mass Transfer, vol. 169, p. 120914,
2021.
[9] S. Churchill and R. Usagi, “A general expression for the correlation
of rates of transfer and other phenomena,” AIChE Journal, vol. 18,
no. 6, pp. 1121–1128, 1972.
[10] P. Teertstra, M. Yovanovich, and J. Culham, “Analytical forced
convection modeling of plate fin heat sinks,” Journal of Electronics
Manufacturing, vol. 10, no. 04, pp. 253–261, 2000.
[11] H. Jonsson and B. Moshfegh, “Modeling of the thermal and hydraulic
performance of plate fin, strip fin, and pin fin heat sinks-influence
of flow bypass,” IEEE Transactions on Components and Packaging
Technologies, vol. 24, no. 2, pp. 142–149, 2001.
[12] M. Shaeri and M. Yaghoubi, “Thermal enhancement from heat sinks
by using perforated fins,” Energy conversion and Management, vol. 50,
no. 5, pp. 1264–1270, 2009.
[13] W. Al-Sallami, A. Al-Damook, and H. Thompson, “A numerical
investigation of the thermal-hydraulic characteristics of perforated
plate fin heat sinks,” International Journal of Thermal Sciences, vol.
121, pp. 266–277, 2017.
[14] S. Chingulpitak, H. S. Ahn, L. G. Asirvatham, and S. Wongwises,
“Fluid flow and heat transfer characteristics of heat sinks with
laterally perforated plate fins,” International Journal of Heat and Mass
Transfer, vol. 138, pp. 293–303, 2019.
[15] A. A. Hussain, B. Freegah, B. S. Khalaf, and H. Towsyfyan,
“Numerical investigation of heat transfer enhancement in plate-fin
heat sinks: Effect of flow direction and fillet profile,” Case Studies in
Thermal Engineering, vol. 13, p. 100388, 2019.
[16] B. Freegah, A. A. Hussain, A. H. Falih, and H. Towsyfyan, “Cfd
analysis of heat transfer enhancement in plate-fin heat sinks with
fillet profile: Investigation of new designs,” Thermal Science and
Engineering Progress, vol. 17, p. 100458, 2020.
[17] A. Tariq, K. Altaf, S. W. Ahmad, G. Hussain, and T. Ratlamwala,
“Comparative numerical and experimental analysis of thermal and
hydraulic performance of improved plate fin heat sinks,” Applied
Thermal Engineering, vol. 182, p. 115949, 2021.
[18] E. Sparrow, B. Baliga, and S. Patankar, “Heat transfer and fluid
flow analysis of interrupted-wall channels, with application to heat
exchangers,” 1977.
[19] E. Sparrow and C. Liu, “Heat-transfer, pressure-drop and
performance relationships for in-line, staggered, and continuous
plate heat exchangers,” International Journal of Heat and Mass
Transfer, vol. 22, no. 12, pp. 1613–1625, 1979.
[20] W. Al-Sallami, A. Al-Damook, and H. Thompson, “A numerical
investigation of thermal airflows over strip fin heat sinks,”
International Communications in Heat and Mass Transfer, vol. 75, pp.
183–191, 2016.
[21] E. Ozturk and I. Tari, “Forced air cooling of cpus with heat sinks:
A numerical study,” IEEE transactions on components and packaging
technologies, vol. 31, no. 3, pp. 650–660, 2008.
[22] P. Teertstra, J. Culham, and M. Yovanovich, “Analytical
modeling of forced convection in slotted plate fin heat sinks,”
ASME-PUBLICATIONS-HTD, vol. 364, pp. 3–12, 1999.
[23] F. Hong and P. Cheng, “Three dimensional numerical analyses
and optimization of offset strip-fin microchannel heat sinks,”
International Communications in Heat and Mass Transfer, vol. 36, no. 7,
pp. 651–656, 2009.
[24] D. B. Spalding, “Phoenics. cham,” Multiphase Science and Technology,
vol. 6, no. 1-4, 2009.
[25] T. L. Bergman, F. P. Incropera, D. P. DeWitt, and A. S. Lavine,
“Fundamentals of heat and mass transfer,” John Wiley & Sons, 2011.
[26] R. L. Webb, “Heat exchanger design methodology for electronic heat
sinks,” 2007.
[27] R. J. Moffat, “Modeling air-cooled heat sinks as heat exchangers,” in
Twenty-Third Annual IEEE Semiconductor Thermal Measurement and
Management Symposium. IEEE, 2007, pp. 200–207.
[28] H. W. Coleman and W. G. Steele, “Experimentation, validation, and
uncertainty analysis for engineers,” John Wiley & Sons, 2018.
[29] H. K. Versteeg and W. Malalasekera, “An introduction to
computational fluid dynamics: the finite volume method,” Pearson
education, 2007.
[30] D. Agonafer, L. Gan-Li, and D. Spalding, “The lvel turbulence model
for conjugate heat transfer at low reynolds numbers,” Application of
CAE/CAD Electronic Systems, ASME, vol. 18, pp. 23–26, 1996.
[31] D. Spalding, “A single formula for the law of the wall,” Journal of
Applied Mechanics, vol. 28, no. 3, pp. 455–458, 1961.
[32] J. P. Hartnett, J. C. Y. Koh, and S. T. McComas, “A comparison of
predicted and measured friction factors for turbulent flow through
rectangular ducts,” Journal of Heat Transfer, vol. 84, pp. 82–88, 1962.
[33] D. Spalding, “Mathematical modelling of fluid-mechanics,
heat-transfer and chemical-reaction processes,” A Lecture Course,
Imperial College CFDU Report, 1980.
[34] D. B. Spalding, “A novel finite difference formulation for
differential expressions involving both first and second derivatives,”
International Journal for Numerical Methods in Engineering, vol. 4,
no. 4, pp. 551–559, 1972.
98
[5] P. Hopton and J. Summers, “Enclosed liquid natural convection as
a means of transferring heat from microelectronics to cold plates,”
in 29th IEEE Semiconductor Thermal Measurement and Management
Symposium. IEEE, 2013, pp. 60–64.
[6] H. M. Ali, A. Arshad, M. Jabbal, and P. G. Verdin, “Thermal
management of electronics devices with pcms filled pin-fin heat
sinks: a comparison,” International Journal of Heat and Mass Transfer,
vol. 117, pp. 1199–1204, 2018.
[7] A. Arshad, M. Jabbal, P. T. Sardari, M. A. Bashir, H. Faraji, and Y. Yan,
“Transient simulation of finned heat sinks embedded with pcm
for electronics cooling,” Thermal Science and Engineering Progress,
vol. 18, p. 100520, 2020.
[8] R. Kalbasi, “Introducing a novel heat sink comprising pcm
and air-adapted to electronic device thermal management,”
International Journal of Heat and Mass Transfer, vol. 169, p. 120914,
2021.
[9] S. Churchill and R. Usagi, “A general expression for the correlation
of rates of transfer and other phenomena,” AIChE Journal, vol. 18,
no. 6, pp. 1121–1128, 1972.
[10] P. Teertstra, M. Yovanovich, and J. Culham, “Analytical forced
convection modeling of plate fin heat sinks,” Journal of Electronics
Manufacturing, vol. 10, no. 04, pp. 253–261, 2000.
[11] H. Jonsson and B. Moshfegh, “Modeling of the thermal and hydraulic
performance of plate fin, strip fin, and pin fin heat sinks-influence
of flow bypass,” IEEE Transactions on Components and Packaging
Technologies, vol. 24, no. 2, pp. 142–149, 2001.
[12] M. Shaeri and M. Yaghoubi, “Thermal enhancement from heat sinks
by using perforated fins,” Energy conversion and Management, vol. 50,
no. 5, pp. 1264–1270, 2009.
[13] W. Al-Sallami, A. Al-Damook, and H. Thompson, “A numerical
investigation of the thermal-hydraulic characteristics of perforated
plate fin heat sinks,” International Journal of Thermal Sciences, vol.
121, pp. 266–277, 2017.
[14] S. Chingulpitak, H. S. Ahn, L. G. Asirvatham, and S. Wongwises,
“Fluid flow and heat transfer characteristics of heat sinks with
laterally perforated plate fins,” International Journal of Heat and Mass
Transfer, vol. 138, pp. 293–303, 2019.
[15] A. A. Hussain, B. Freegah, B. S. Khalaf, and H. Towsyfyan,
“Numerical investigation of heat transfer enhancement in plate-fin
heat sinks: Effect of flow direction and fillet profile,” Case Studies in
Thermal Engineering, vol. 13, p. 100388, 2019.
[16] B. Freegah, A. A. Hussain, A. H. Falih, and H. Towsyfyan, “Cfd
analysis of heat transfer enhancement in plate-fin heat sinks with
fillet profile: Investigation of new designs,” Thermal Science and
Engineering Progress, vol. 17, p. 100458, 2020.
[17] A. Tariq, K. Altaf, S. W. Ahmad, G. Hussain, and T. Ratlamwala,
“Comparative numerical and experimental analysis of thermal and
hydraulic performance of improved plate fin heat sinks,” Applied
Thermal Engineering, vol. 182, p. 115949, 2021.
[18] E. Sparrow, B. Baliga, and S. Patankar, “Heat transfer and fluid
flow analysis of interrupted-wall channels, with application to heat
exchangers,” 1977.
[19] E. Sparrow and C. Liu, “Heat-transfer, pressure-drop and
performance relationships for in-line, staggered, and continuous
plate heat exchangers,” International Journal of Heat and Mass
Transfer, vol. 22, no. 12, pp. 1613–1625, 1979.
[20] W. Al-Sallami, A. Al-Damook, and H. Thompson, “A numerical
investigation of thermal airflows over strip fin heat sinks,”
International Communications in Heat and Mass Transfer, vol. 75, pp.
183–191, 2016.
[21] E. Ozturk and I. Tari, “Forced air cooling of cpus with heat sinks:
A numerical study,” IEEE transactions on components and packaging
technologies, vol. 31, no. 3, pp. 650–660, 2008.
[22] P. Teertstra, J. Culham, and M. Yovanovich, “Analytical
modeling of forced convection in slotted plate fin heat sinks,”
ASME-PUBLICATIONS-HTD, vol. 364, pp. 3–12, 1999.
[23] F. Hong and P. Cheng, “Three dimensional numerical analyses
and optimization of offset strip-fin microchannel heat sinks,”
International Communications in Heat and Mass Transfer, vol. 36, no. 7,
pp. 651–656, 2009.
[24] D. B. Spalding, “Phoenics. cham,” Multiphase Science and Technology,
vol. 6, no. 1-4, 2009.
[25] T. L. Bergman, F. P. Incropera, D. P. DeWitt, and A. S. Lavine,
“Fundamentals of heat and mass transfer,” John Wiley & Sons, 2011.
[26] R. L. Webb, “Heat exchanger design methodology for electronic heat
sinks,” 2007.
[27] R. J. Moffat, “Modeling air-cooled heat sinks as heat exchangers,” in
Twenty-Third Annual IEEE Semiconductor Thermal Measurement and
Management Symposium. IEEE, 2007, pp. 200–207.
[28] H. W. Coleman and W. G. Steele, “Experimentation, validation, and
uncertainty analysis for engineers,” John Wiley & Sons, 2018.
[29] H. K. Versteeg and W. Malalasekera, “An introduction to
computational fluid dynamics: the finite volume method,” Pearson
education, 2007.
[30] D. Agonafer, L. Gan-Li, and D. Spalding, “The lvel turbulence model
for conjugate heat transfer at low reynolds numbers,” Application of
CAE/CAD Electronic Systems, ASME, vol. 18, pp. 23–26, 1996.
[31] D. Spalding, “A single formula for the law of the wall,” Journal of
Applied Mechanics, vol. 28, no. 3, pp. 455–458, 1961.
[32] J. P. Hartnett, J. C. Y. Koh, and S. T. McComas, “A comparison of
predicted and measured friction factors for turbulent flow through
rectangular ducts,” Journal of Heat Transfer, vol. 84, pp. 82–88, 1962.
[33] D. Spalding, “Mathematical modelling of fluid-mechanics,
heat-transfer and chemical-reaction processes,” A Lecture Course,
Imperial College CFDU Report, 1980.
[34] D. B. Spalding, “A novel finite difference formulation for
differential expressions involving both first and second derivatives,”
International Journal for Numerical Methods in Engineering, vol. 4,
no. 4, pp. 551–559, 1972.
98