Energy and Power En gi neering, 2011, 3, 525-532
doi:10.4236/epe.2011.34064 Published Online September 2011 (http://www.SciRP.org/journal/epe)
Improvement of Free Convection Heat Transfer Rate of
Rectangular Heatsink on Vertical Base Plates
Hamid Reza Goshayeshi1, Mahdi Fahiminia1, Mohammad Mahdi Naserian2
1Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
2Young Researchers Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran
s
Received December 21, 2010; revised February 4, 2011; accepted February 28, 2011
Abstract
In this paper, the laminar heat transfer of natural convection on vertical surfaces is investigated. Most of the
studies on natural convection have been considered constantly whereas velocity and temperature domain, do
not change with time, transient one are used a lot. Governing equations are solved using a finite volume ap-
proach. The convective terms are discretized using the power-law scheme, whereas for diffusive terms the
central difference is employed. Coupling between the velocity and pressure is made with SIMPLE algorithm.
The resultant system of discretized linear algebraic equations is solved with an alternating direction implicit
scheme. Then a configuration of rectangular fins is put in different ways on the surface and heat transfer of
natural convection on these surfaces without sliding is studied and finally optimization is investigated.
Keywords: Natural Convection, Vertical Surfaces, Simple Algorithm, Rectangular Fins
1. Introduction
This document is a natural convection is observed as a
result of the movement of fluid because of density
changes which is caused by heating process. A radiator
used for warming the house is an example of practical
equipment for heat transfer of natural convection. The
movement of fluid, whether gas or liquid, in natural
convection is caused by buoyancy force due to density
reduction beside to heat transfer's surfaces in heating
process. When the domain of an exterior force such as
gravity, has no effect on the fluid then will be no buoy-
ancy force and heat transfer will be in the form of con-
duction. But gravity is not th e only force causing natural
convection. When a fluid is confined in the rotating ma-
chine, centrifugal force is exerted on it and if one surface
or so, with more or less temperature than that of the fluid
are in touch with the fluid, natural convection flows will
be experienced. The fluid which is adjacent to the verti-
cal surface with constant temperature, and the fluid tem-
perature is less than the surface temperature, a velocity
boundary layer forms the natural convection. The veloc-
ity profile in this boundary layer is completely different
with the velocity profile in forced co nvection. Th e veloc-
ity is zero on the wall due to lack of sliding .Then the
velocity goes up and reaches its maximum and finally is
zero on the external border of velocity boundary layer.
Since the factor that causes the natural convection, is
temperature gradient, the heating boundary layer appears
too. The temperature profile has also the same value as
the temperature of wall due to the lack of particles slid-
ing on the wall, and temperature of particles goes down
as approaching to external border of temperature bound-
ary layer and it would reach the temperature of far fluids.
The initial boundary layer enlargement is laminar , but in
the distance from the uplifting edge, depending on fluid
properties and the difference of temperature of wall and
the environment, eddies will b e formed an d movement to
turbulent zon e will be started.
Today being mor e aware abou t energy sources’ limita-
tions and also increase demand for energy consumption
from one side and considerable waste of energy in heat-
ing systems from the other side , have caused the socie-
ties to investigate on energy systems and find solutions
for reducing energy dissipation from these systems.
There are many appliances in engineering in which two
environments with different temperatures are separated
by a wall or a thin surface. The wall on one side is get-
ting warm and on the other side is making it is beside
fluid warm. Heat transfer between the two environments
and the temperature of the wall, depends on the shape of
the boundary layers on the wall surfaces. The amount of

H. R. GOSHAYESHI ET AL. 526
temperature on the wall will reach a balance of the tem-
perature of both environments. Since the boundary layer
of one side makes the boundary layer of the other side
they are called conjugate boundary layers.
Although a lot of investigations have been done on
natural convection heat transfer, the data the effect of
geometrical complexes on natural convection heat trans-
fer has many deficiencies.
Jofre and Barron have gathered some data on natural
convection heat transfer, which the fluid is air, on the
vertical surface with some swells in the shape of triangle
ribs [1]. They concluded that at RaL = 109. The average
Nusselt number is increased 200% compared to the av-
erage Nusselt of turbulent flow achieved by Ekertand
Jackson. Of course if the fact is considered that the flow
on the flat surface is not turbulent on the entire surface,
and the data are compared with upstream flow that is
assumed laminar flow, the results show 100% increase as
compared to the recent one. We can say that this big dif-
ference is caused by the lack of attention to the correct
way of radiation heat t ransfer [2] .
Bhavnani and Burgles [3] after several experiments
proved that making special changes on vertical surfaces
(horizontal little fins) reduces heat transfer in the natural
convection heat transfer process. This conclusion can
cause changes in the ways of insulating heat repelling
surfaces and in this respect is of great importance. On the
other hand we face many practical technologies causing
natural convection heat transfer on the fins vertical sur-
faces. This matter is more obvious in the electronic cir-
cuit in which segment duty cycle is highly effected by
the duty cycle. Rebert, Green, Chapman and others have
offered five important advantages of algorithmic solving
method [4,5]. Numerical solution of the governing equa-
tions of boundary zones for vertical surfaces has been
done by Helus and Churchil and step changes of surface
temperature has been achieved [6]. Most of the works [7]
in this field are true for low Prandtl and in high Prandtl,
computing errors ca n not be negl igible.
Although th ere are many articles about forced conv ec-
tion gas flow on vertical surfaces, there are quantitative
researches done about natural convection flow on sur-
faces with rectangular fins. In this paper, a complete nu-
merical solution including developing and developed
enlarged hydrodynamic and heating parts is presented to
analyze natural convection flow on the extended vertical
surface.
2. Numerical Modeling
For analyzing the above mentioned matter, first the dif-
ferential equation of the movement of this layer should
be achieved. So y axis is on the horizon and x axis is
erected vertically on the surface As in Figure 1. 2dimen-
tional flow is considered with physical properties, fixed
wall temperature, more than the entrance free flow tem-
perature. So upward flow is produced by buoyancy force
in top of the surfaces. Two ends of the surface are open
towards the environment with free flow temperature.
Compressibility effect in the free flow with low velocity
taken here can be negligible, and just laminar flow re-
gime can be considered [8]. Density conversion caused
by temperature changes are exerted in buoyancy force is
done by boussinesq approximation. The equation of con-
servation of mass, momentum and the energy for stable
and laminar flow is describe d follo w [9 ,1 0] .
Continuity Equation :
0.
uv
xy




 (1)
X-Momentum Equation:
2.
x
uu p
uv uB
xy x


 


 

 (2)
Y-Momentum Equation:
2 .
vvp
uv ug
xy y

  


 

 (3)
Energy Equation:




2
2.
pp
cT cT T
uv
xy y



 



 

(4)
With boundary conditions:
Figure 1. Geometry of coordinate system and velocity Boun-
dary layer.

H. R. GOSHAYESHI ET AL.527
0
0
0
w
x
uv
TT
x
v
TT













(5)
Assuming that there a re not any parameter changes on x
direction, (2) is omitted. Pressure variation on “y” direc-
tion is caused by changes in height of the fluid. Out of
velocity boundary layer, u, v, and their derivative are zero.
.
p
x



 (6)
With putting (5) in (3), we have:

2
2.




 
 
 
uv y
uv
xy
x
g (7)
Difference of density can be defined according to
Thermal Expansion Coefficient.
1.
P
T




 


(8)
1.
TT






 

2
2.
uv y
uv TT
xy
x



 
 
  (9)
Above mentioned equation defines the momentum in
natural convection velocity boundary layer, this momen-
tum is caused by temperature gradient. Since temperature
depends on the velocity and vice versa, momentum
Equation (9) and energy Equation (4) should be solved
simultaneously. Now considering boundary conditions,
temperature a nd velocity distributions are guessed and p ut
in equations. Governing equations are solved using a
finite volume approach. The convective terms are discre-
tized using the power-law scheme, whereas for diffusive
terms the central difference is employed. Coupling be-
tween the velocity and pressure is made with SIMPLE
algorithm. The resultant system of discretized linear al-
gebraic equations is solved with an alternating direction
implicit scheme [11].
3. Result
In this paper, first, vertical surface is considered, and
thenheat sinksare put on the surface. Dimensions, dis-
tances between fins, the type of fins and the location they
are put in, are considered. The dimensions are considered
as shown in Table 1.
The mentioned heat sinks have shown in Figure 2.
Table 1. Dimensions of the fin configurations.
Fin length
L(mm) Fin width W(mm) Fin thickness t(mm) Base thickness d(mm)
80 59.8 1 1.4
Fin height
h(mm) Fin spacing s(mm) Number of fin n
29.2 2.1 20
29.2 3.9 13
29.2 7.4 8
29.2 8.8 7
29.2 13.7 5
29.2 18.6 4
Figure 2. Fin configuration geometry.
The studies on different surfaces show that in fins with
low distances, air flow between the fins is fully devel-
oped due to boundary layers interference, t and ther efore
heat transfer rate is achieved by
1 .
c single channelP
QmC
T (10)
3.
sT
m

H



 (11)
If n = w/s, then,
3
1.
c
s
TW
QHk
S




T (12)
3
1.
c
s
T
QHkT
S




W
(13)
Unlike the previous way, if the fin distances are great,
the boundary layer thickness is so much less than the
distance between the fins and the space between the fins
is like the flow in parallel surfaces in which boundary

H. R. GOSHAYESHI ET AL. 528
T
layers have no interference. Natural convection heat
transfer from two surfaces of a fin is as follow:
1 .
cFin
QhA


 (14)
in which heat transfer coefficient is defined as:
1
34.
LT k
hL






(15)
If A = H  L, then:
1
34
2.
c
LT W
QkHT
s


 




2
(16)
112
,
cc
QsQ s



is observed.
As it has shown in Figure 3, the investigations show
that in the same fin length and height and constant dif-
ference of temperature, heat transfer rate first increases
with increasing the fins distance, until reaches the
maximum point and then decreases. The maximum point
is called optimum distance between the fins, is the dis-
tance in which there is the maxim um heat transfer. We can
have an equation for the optimum distance, by interfering
Equations (12) and (16).
1
4
opt
L
SRa
L

. (17)
For dimensionless presentation of the order of magni-
tude of optimum fin spacing, the Rayleigh number is
employed according to its definition.
3.
L
LT
Ra


 



(18)
And according to our investigations on mentioned di-
mensions the optimum distance equation calculated as
follow:
1
4
2.91 .
opt
L
SRa
L

 (19)
As a point of departure for the presentation of heat
transfer rates from fin-arrays are plotted as a function of
temperature differences as in Figure 4.
From these Figures it can also be seen that, at a given
fin height and temperature difference, the convection
heat transfer rates increases with increasing fin spacing
and reaches a maximum. With further increases of fin
spacing, heat transfer rate starts to decrease. The occur-
rence of this maximum has significant practical applica-
tions for optimum performance of fin-arrays. It would be
appropriate tomanufacture the fin array with aforemen-
tioned fin height and spacing.The values of heat transfer
coefficient obtained for an ambient air temperature of
27ºC and plate surface temperatures of 77, 102, 127 and
157ºC app ear in Figures 5-7, As may be seen, the natural
Figure 3. Variation of convective rate with base-to-ambient
temperature difference at H = 29.2 mm and L = 80 mm.
Figure 4. Variation of convective heat transfer rate with fin
spacing (fin height 29.2 mm).
Figure 5. Natural convection heat transfer coefficients for
different heat sinks.

H. R. GOSHAYESHI ET AL.529
convection heat transfer coefficient increases substan-
tially as the gap between fins increases from 2.1 to 18.6
mm, and then flattens out with further increases in gap.
The temperature at the upper surface of the heat sink
base, the surface exposed to the operating fluid can then
be found from (20), knowing the temperature at the bot-
tom of the base by thermocouple measurement and the
heat flux through the base.
.
Wbpc
d
TT Q
kA






(20)
The heat flux at the base in W/m2 is found with (20),
knowing that the exposed base area is 4768 mm2.
.
c
c
Q
q
A

 (21)
Finally, the heat transfer coefficient is calculated using
(22) and the thermal resistance using (23).

.
c
w
q
hTT


 (22)

w
c
TT
RQ


 (23)
Refer to (22), (23), thermal resistance has reverse rela-
tion to heat transfer coefficient.
As in Figure 8, average Nusselt number is increased
with increasing average Rayleigh number and at a certain
average Rayleigh number, average Nusseltnumber is
increased by increasing the distance between the fins.
4. Conclusions
From these Figs it can also be seen that, at a given fin
height and temperature difference, the convection rates
increases with increasing fin spacing and reaches a
maximum. With further increases of fin spacing, rate
starts to decrease. The occurrence of this maximum has
significant practical applications for optimum perform-
ance of fin-arrays. It would be appropriate to manufac-
ture the fin array with aforementioned fin height and
spacing. The values of coefficient obtained for an ambient
air temperature of 27ºC and plate surface temperatures of
77, 102, 127 and 157ºC appear in Figure 5-7, As may be
seen, the natural convection coefficient increases sub-
stantially as the gap between fins increases from 2.1 to
18.6 mm, and then flattens out with further increases in
gap. It may be noted that the values shown for a gap of
18.6 mm are within a few per cent of those obtained us-
ing a correlation for an individual vertical plate.
Figures 9-11 show that, the convective heat transfer
rate from fin arrays depends on fin height, fin length, fin
spacing and base-to-ambient temperature difference. The
Figure 6. Variation of heat flux rate with Average Rayleigh
number at H = 29.2 mm and L = 80 mm.
Figure 7. Variation of thermal resistancewith heat transfer
coefficient.
Figure 8. Variation of average nusselt number with average
Rayleigh number.
convective heat transfer rates from the fin arrays in-
creases with fin height, fin length and base-to-ambient
temperature difference. The heat transfer rate increases
monotonously with temperature difference between fin
base and surroundings, Tw-Ta. If the distance between
the fins is selected properly, there will be no interference
beween boundary layers of two adjacent fins and the t

H. R. GOSHAYESHI ET AL. 530
Figure 9. Variation of temperature contour of the heatsink (s = 3.9 mm).
Figure 10. Variation of temperature contour of the heatsink (s = 7.4 mm).
Figure 11. Variation of temperature contour of the heatsink (s = 18.6 mm).

H. R. GOSHAYESHI ET AL. 531
surfaces. For stating the magnitude of fin spacing for
having the best convection rates of fins, we choose the-
distance between the fins big enough so that the thick-
ness of boundary layer is smaller than the distance be-
tween the fins and increases without any interfering.
5. References
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[6] S. W. Churchill and S. Usagi, “A General for the Correla-
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H. R. GOSHAYESHI ET AL. 532
Nomenclature (List of Symbols)
Symbol Quantity Unit
A area m2
h convection heat transfer coefficient w/m2k
T temperature k
Cp specific heat at constant pressure kJ/kgk
g gravitational acceleration m/s2
H fin height m
k thermal conductivity w/mk
L fin length m
m
 mass flow rate kg/s
β volumetric thermal expansion coefficient 1/k
n number of fins -
s fin spacing m
t fin thickness m
W base plate width m
1.
c
Q
 convection rate f rom fins in small-s limit w
2
c
Q
 convection rate from fins in large-s limit w
c
Q
 convection rate w
Ra Rayleigh number -
Tw base plate temperature k
Ta ambient temperature k
ν kinematic viscosity m2/s
α thermal diffus ivity m2/s