International Journal of Heat and Mass Transfer 123 (2018) 561–568
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International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Natural convection heat transfer of a straight-fin heat sink
Xiangrui Meng a,b, Jie Zhu b,⇑, Xinli Wei a,⇑, Yuying Yan b
a
b
School of Chemical Engineering and Energy, Zhengzhou University, Henan, China
Department of Architecture and Built Environment, The University of Nottingham, Nottingham, UK
a r t i c l e
i n f o
Article history:
Received 30 January 2018
Received in revised form 1 March 2018
Accepted 1 March 2018
Available online 20 March 2018
Keywords:
Natural convection heat transfer
Heat sink
Mounting angle
Stagnation zone
a b s t r a c t
The influence of mounting angle on heat dissipation performance of a heat sink under natural convection
condition is investigated in this paper by numerical simulation and experimental test. It is found that the
heat sink achieves the highest cooling power when its mounting angle is 90°, while it reaches the lowest
when the mounting angle is 15°, which is 6.88% lower than that of 90°. A heat transfer stagnation zone is
the main factor that affects the cooling power of the heat sink, and its location and area vary with the
mounting angle. It is identified that cutting the heat transfer stagnation zone is an effective way to
improve the heat sink performance.
Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction
Heat sink is a passive heat exchanger that transfers heat generated by an electronic or a mechanical device to a fluid medium,
such as air and liquid coolant. Heat dissipation is very important
in the modern electronic industry, according to the statistical data,
high temperature causes more than 55% failures of electronics [1].
The heat sink is also used in other areas, for example, heat dissipation of DSC (Dye-Sensitized Solar Cell) [2]. The heat sink has different structures, and can be classified into active and passive types.
Compared to the active heat sink, the passive heat sink dissipates
thermal energy through the nature convection, and usually it is
made of aluminium finned radiator, so it has high reliability and
low cost characters. The driving force in the passive heat sink is
buoyancy force generated by temperature difference. The natural
convection of heat sink can be divided into limited and infinite
space convections according to the external space.
Most of the passive heat sinks have simple structure and low
cost characters because of their straight fins. Elenbaas [3] carried
out the earliest investigation on natural convective heat dissipation for a parallel fin heat sink, Bodoia and Osterle [4] deduced a
theoretical solution of the natural convection heat dissipation for
the parallel vertical fin heat sink on the basis of theoretical analysis. Other researchers studied and optimised the geometrical
dimensions of parallel fin heat sink, and gave out some formulas
⇑ Corresponding authors.
E-mail addresses: lazjz@nottingham.ac.uk (J. Zhu), xlwei@zzu.edu.cn (X. Wei).
https://doi.org/10.1016/j.ijheatmasstransfer.2018.03.002
0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
for calculating geometrical dimensions [5–11]. Heat dissipation
performance of the parallel straight fin heat sink can be improved
by increasing air turbulence between the fins, such as arranging
staggered cylinders [12], drilling holes on base plate [13], opening
slots [14] or drilling holes on the fins [15].
The above studies are all conducted with the horizontal or vertical heat sink [16–18], nevertheless, the influence of the heat sink
mounting angle on heat dissipation is rarely mentioned. Mehrtash
et al. [19] studied the effect of inclination of fin-plate heat sink on
heat dissipation by numerical simulation with three-dimensional
steady-state natural convection. Based on Mehrtash’s research
results, Tari et al. [20] developed a Nusselt number formula, and
found that the fin spacing is an important parameter affecting heat
sink thermal performance [21]. Shen et al. [22] investigated heat
dissipation properties of the heat sinks placed in eight different
directions, and discovered that the denser the fin arrangement,
the more sensitive the directionality. There are two main factors
limiting the sink natural convection heat dissipation, one is that
the heat transfer direction does not match with natural convection
flow, and the other one is that the convection between the fins is
blocked.
In this paper, the influence of heat sink mounting angle on its
heat dissipation is investigated. A test rig is designed and built to
measure heat dissipation performances of a heat sink at different
mounting angles. The numerical simulation of the heat sink performance is carried out, and the simulation results are compared with
the experimental data. The optimum mounting angle of the heat
sink is obtained, which is useful for heat sink design and
installation.
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X. Meng et al. / International Journal of Heat and Mass Transfer 123 (2018) 561–568
Nomenclature
A
cp
h
k
Qhs
Rth
Rhts
Tave
Tam
DT
heat sink surface area, m2
specific heat capacity of fluid, J kg1 K1
heat transfer coefficient, W m2 K1
thermal conductivity, W m1 K1
heatsink input power, W
thermal resistance, K W1
the ratio of stagnation zone accounts for the fin whole
area, dimensionless
average temperature of heat sink plate, K
ambient temperature, K
temperature difference between heat sink surface and
the ambient temperature, K
2. Experimental apparatus
2.1. General description
The main components of the test rig include a JP1505D DC
power supply, a DC heating plate, an Agilent 34970A Data Acquisition, a number of K-type thermocouples and PT100 RTDs. The schematic of the test rig is shown in Fig. 1. The experimental system is
located in a large closed space without the external interference to
achieve the heat sink natural convection environment. A special
support is designed to ensure the heat sink could rotate 360° freely,
as shown in Fig. 2. The heat sink and heating plate are fastened by
bolts to reduce the contact thermal resistance and prevent the relative displacement between them. The heating plate is controlled
by the JP1505D DC power supply for different heating powers.
The maximum output power of the power supply is 750 W, its output voltage range is from 0 V to 150 V with accuracy ±0.3 V and its
current range is from 0 A to 5 A with accuracy ±0.01 A. The electric
heating power is constant during the testing, the surface temperature of heat sink is measured and used to judge heat dissipation
performance of the heat sink. The lower surface temperature of
the heat sink, the better heat dissipation performance. Assuming
heat is only dissipated by the heat sink when the temperature of
the heat sink substrate became constant, the heat sink performance can be assessed by its surface temperatures.
The data collection system consists of TC, RTD and Agilent
34970A Data Acquisition, the locations of the measuring points
are shown in Fig. 3. TCs are set at Points 1 to 6 to get the heat sink
bottom temperatures, RTDs are set at Points 7 to 9 to measure the
fin surface temperatures. Agilent 34970A Data Acquisition with
module 34902A, which features a built-in thermocouple reference
u, v, w components of velocity, m s1
x, y, z
components of coordinate
x1,x2,x3,. . .xn independent variables
dx1, dx2, dx3. . . dxn errors of independent variables
Greek symbols
b
thermal expansion coefficient, K1
q
air density, kg m3
q0
air density at T0, kg m3
l
viscosity, N s m2
and 16 two-wire channels, has 6 1/2-digit (22-bit) internal DMM
and can scan up to 250 channels per second. The K-type armoured
thermocouple WRNK-191 is used in the experiment. The material
of WRNK-191 is nickel-chromium & nickel-silicon and its measurement temperature range is from 0 °C to 600 °C with accuracy
±0.5 °C. Because of high thermoelectric power, the WRNK-191 TC
has high sensitivity and its thermal response time is 3S. The
measurement temperature range of SMD Pt100 RTD Temperature
Sensor used for the fin surface is from 50 °C to 200 °C with
accuracy ±0.15 °C. It can be directly pasted to the fin surface with
Fig. 2. Schematic of support.
Fig. 1. Schematic of the test rig: (1) DC power; (2) Agilent 34970A; (3) computer;
(4) heat sink; (5) heating plate; (6) thermal insulation.
Fig. 3. Arrangement of measuring points. Points 1–6K-type thermocouples; Points
7–9, PT100 RTDs.
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X. Meng et al. / International Journal of Heat and Mass Transfer 123 (2018) 561–568
Table 1
The geometries of heat sink.
Length (mm)
Width (mm)
Base thickness (mm)
Fin height (mm)
Fin thickness (mm)
Fins pitch (mm)
Number of fins
150
76
5
50
3
9.17
7
thermally conductive glue. The tested heat sink is an aluminium
straight-fin type, its geometries are listed in Table 1.
2.2. Experimental procedure
(1) Coating thermal grease evenly on the heat sink bottom
before fixing it to the heating substrate with screws.
(2) Adjusting the mounting angle of the heat sink to a certain
angle, and then checking the output voltage of DC power
supply to insure the constant heating power.
(3) The data of each measuring point are to be collected after
the heat sink begin to be heated.
(4) The equilibrium between heating and dissipation is reached
as the maximum temperature fluctuate on the bottom
surface of the heat sink substrate is less than 0.5 °C within
20 min.
The least thermal resistance method is used to assess the heat
sink heat dissipation performance. Thermal resistance Rth can be
calculated by following equation [5]:
Rth ¼
T av e T sur
Q hs
ð1Þ
where Tave and Tsur are the average temperatures of the heat plate
and the ambient respectively, Qhs is the heat dissipation power.
With constant input power for the heat sink, the lower the thermal
resistance, the higher the heat dissipation ability. Heat transfer
coefficient h of the heat sink can be calculated from the following:
h¼
Q hs
A DT
3. Numerical simulation model
The CFD simulation is carried out for the tested straight-fin
heatsink using the symmetric model because the object is symmetric. The simulation zone is shown in Fig. 4 and its dimension is
900 mm 456 mm 330 mm. Because of the regular geometric
shape of the simulation zone, the hexahedral grid is adopted in
the meshing procedure which will not only get high-quality grid
but also be easy to modify the meshing strategy. The mesh density
near the fin surface is increased for its Y+ < 1 [23]. The simulation is
carried out with different meshing strategies, the average surface
temperatures of the fin are monitored and compared. The simulation results are shown in Fig. 5, it can be seen that the optimum
meshing strategy is one with the grid number of 464058, and the
optimum mesh model is shown in Fig. 6.
In order to reduce the amount of computational resources, the
following assumptions are made:
(1) The air flow is treated as a three-dimensional steady laminar
flow.
(2) Boussinesq model is used in air zone.
(3) The temperature and heat flow of the heat plate are even.
(4) Except density, the properties of air are constant.
(5) Air is nonslip on the fin surface.
(6) The viscous dissipation and radiation heat transfer are not
considered.
ð2Þ
where A is heat sink surface area, DT refers to temperature difference between the heat sink surface and ambient.
2.3. Error analysis
The experimental errors mainly include systematic and accidental errors. The error dy of a variable y can be obtained by the
quadratic equation of the experimental data as the variables are
assumed as y = f(x1, x2, x3,. . .xn):
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
2
2
@y
@y
@y
dy ¼
dx21 þ
dx22 þ þ
dx2n
@x1
@x2
@xn
ð3Þ
where x1,x2,x3,. . .xn are independent variables, dx1, dx2, dx3,. . . dxn are
their errors. The maximum errors of variables and measurement
ranges are listed in Table 2. In addition, each test is repeated several
times in order to minimise accidental errors, so the average data are
likely to be close to the true values.
Fig. 4. Scope of the simulation.
Table 2
Accuracies of sensors and the maximum relative errors of variables.
Temperature
Accuracy
Maximum error
TC
RTD
±0.5 °C
1.823%
±0.15 °C
1.367%
Voltage
Current
Heating power
Thermal resistance
±0.3 V
0.411%
±0.01 A
0.909%
–
0.998%
–
2.08%
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X. Meng et al. / International Journal of Heat and Mass Transfer 123 (2018) 561–568
@ðquTÞ @ðqvTÞ @ðqwTÞ j @ 2 T @ 2 T @ 2 T
þ
þ
¼
þ
þ
@x
@y
@z
cp @x2 @y2 @z2
!
ð5Þ
where T is air temperature, k is air thermal conductivity, Cp is air
specific heat capacity.
The natural-convection flow is driven by the air density change
and gravity force under heating condition. For external natural
convection flow, the momentum equations can be written as:
!
@ðqu2 Þ @ðquv Þ @ðquwÞ
@P
@2u @2u @2u
þ
þ
þ
þ
¼
þl
@x
@y
@z
@x
@x2 @y2 @z2
@ðquv Þ @ðqv 2 Þ @ðqv wÞ
@P
@2v @2v @2v
þ
þ
þ
þ
¼
þl
@x
@y
@z
@y
@x2 @y2 @z2
!
þ gðq q0 Þ
Fig. 5. Grid independence check.
ð6Þ
!
@ðquwÞ @ðqv wÞ @ðqw2 Þ
@P
@2w @2w @2w
þ 2 þ 2
þ
þ
¼
þl
@x
@y
@z
@z
@x2
@y
@z
ð7Þ
ð8Þ
where p is air static pressure, l is air viscosity.
3.2. Model for solid region
There is no internal heat source in the heat sink, so energy equation of solid region can be written as:
@2T @2T @2T
þ
þ
¼0
@x2 @y2 @z2
ð9Þ
For natural-convection flow, the simulation can get quick convergence with the Boussinesq model,
q ¼ q0 ½1 bðT T 0 Þ
where q is air density at temperature T, q0 is air density at T0, b is air
thermal expansion coefficient.
The interface between the fluid and solid domains is treated as
the fluid-solid coupling surface, and no-slip condition is used for
the fluid-solid boundary. The bottom of the heat sink is set as
‘WALL’ with a constant heat flux.
Low Reynolds number k-e Turbulent Model is adopted in the
simulation program because more accurate results can be obtained
compared with the wall function method [24]. Especially, it can
limit the error no more than 2% when coupled with full pressure
outlet boundary condition. The other settings are Pressure based
solver, SIMPLE algorithm, PRESTO! (Pressure Staggered Option)
for pressure, Second order upwind format for other parameters.
The residual value used as convergence indicator is 1e-06. The heat
flux at the heat sink bottom and the fin surface temperature are
also used as ancillary convergence indicators.
Fig. 6. Final meshed model.
3.1. Model for fluid region
Based on the mass conservation principle, the following continuity equation is adopted.
@ðquÞ @ðqvÞ @ðqwÞ
þ
þ
¼0
@x
@y
@z
ð10Þ
ð4Þ
where q is air density, u, v, w are components of velocity, x, y, z are
components of coordinate.
Energy equation is obtained on the basis of energy balance
characteristics.
Table 3
Fin temperatures at measuring points.
Heating power (W)
5
10
20
30
40
50
60
70
80
Temperature at measuring point (°C)
Average of bottom temperature (°C)
1
2
3
4
5
6
7
8
9
24
33
47
52
62
67
82
87
92
22
35
48
53
61
68
80
88
92
24
35
46
53
61
67
80
86
92
24
34
47
51
61
67
80
86
93
23
34
47
52
62
70
84
87
94
22
34
47
52
63
70
83
88
94
17
30
41
48
57
64
72
81
86
20
33
45
50
60
69
76
84
91
18
30
42
47
55
64
74
82
85
23.2
34.2
47
52.2
61.7
68.2
81.5
87
92.8
X. Meng et al. / International Journal of Heat and Mass Transfer 123 (2018) 561–568
565
4. Results and discussion
4.1. Temperature distribution of heat sink
The fin temperature measurement results are listed in Table 3
at the mounting angle of 0°. It can be found from this table, the
fin bottom temperatures (Points 1–6 in Fig. 3) are almost same,
but the surface temperatures (Points 7–9 in Fig. 3) are different.
The surface temperature at the heat sink centre (Point 8) is obviously higher than those near the heatsink edge. As shown in
Fig. 7, the temperature distributions of the heat sink fin and bottom are not uniform whether the mounting angle is 0° or 45°. At
the mounting angle of 0°, the highest temperature appears at the
heat sink centre while the lowest temperature happens at the fin
corner. The fin temperature in the middle of the heatsink is always
higher than the others. This is consistent with the experiment
results in Table 3. Compared with the mounting angle of 0°, the
highest temperature zone moves to the fin end edge at the mounting angle of 45°.
4.2. Influence of heating power on heat transfer coefficient
Heat transfer coefficients from simulation and experimental
test are shown in Fig. 8 at the heat sink mounting angle of 0°. It
is found that the variation of simulation data is similar to that of
experiment results. The maximum error between them is about
10.5% which is acceptable. Heat transfer coefficient increases
rapidly with the heating power when the heating power is below
50 W, but it increases moderately when the heating power is over
(a) Mounting angle 0o
Fig. 8. Experimental and simulation results of heat transfer coefficient.
50 W. The heat transfer driving force in the heat sink is the air flow,
the air will get more heat from the heat sink as the heating power
increases, and its flow velocity will increases as well, so the heat
transfer coefficient becomes higher. The air flow resistance, however, will increase with the velocity, therefore the increase rate
of heat transfer coefficient will decrease synchronously.
4.3. Influences of heating power and mounting angle on thermal
resistance
The variation of thermal resistance with the mounting angle is
shown in Fig. 9. The thermal resistance increases with the mounting angle at first and reaches the maximum when the mounting
angle is 15°, then it decreases. The variation of thermal resistance
is not significant when the mounting angle is over 60°, it reaches
the minimum when the mounting angle is 90°. When heating
power is higher than 50 W, the fluctuation of thermal resistance
is moderate, especially there is nearly no variation when the
mounting angle is bigger than 60°. The ratios between the maximum and minimum thermal resistances for the heating powers
of 5 W, 30 W, 50 W and 80 W, are 29.78%, 18.12%, 6.88% and
13.98% respectively. So in practice, the mounting angle should be
set as 90°. The variation of thermal resistance with heating power
is shown in Fig. 10. It is found that thermal resistance decreases
with heating power. Thermal resistances at the mounting angle
of 15° are the biggest for all heating powers, while the resistances
(b) Mounting angle 45o
Fig. 7. Temperature contours of fin and bottom of heat sink with mounting angles
0° and 45°.
Fig. 9. Thermal resistance variation with mounting angle (experimental results).
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X. Meng et al. / International Journal of Heat and Mass Transfer 123 (2018) 561–568
airflow is larger than 30 °C, calling it ‘A zone’. According to heat
transfer equation Q = hADT, DT will be the key factor when h is
kept constant or slightly fluctuate. Therefore, heat transfer in the
‘D zone’ will be very small compared to that in the ‘A zone’ and
even can be ignored, so the ‘D zone’ is named as ‘heat transfer stagnation zone’. The bigger the area of heat transfer stagnation zone,
the lower the heat transfer. Then the area ratio (Rhts) of heat transfer stagnation zone to the fin is assessed. Fig. 12 shows Rhts variation with the mounting angle. It is found that the Rhts variation is
similar to that of thermal resistance. They reach the maximum values at the mounting angle of 15°. The Rhts increases at first and th ...

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