1. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 1

Abstract-
The optimization of a standard parallel plate heat sink was
studied using numerical methods for three different test
scenarios in natural convection. A baseline case of just a
stand-alone heat-sink arrangement, suspended in air, was
tested for its heat transfer performance, and then non-
dimensionalized for data analysis using Nusselt and Elenbaas
numbers. Numerical results from the un-shielded heat-sink
were compared with theory and correlated so that the shielded
case could be evaluated for its heat transfer enhancement.
Radiation theory was also considered for a single data point,
which was studied more comprehensively for the shielded
case. This was then modified to include a chimney structure,
so that mass flow around the heat-sink was increased, and
compared to the baseline case. Further analysis was performed
on a modified heat-sink, where a cross-cut section from the
plate-fins was removed and tested using the same physics
conditions for the previous two cases. The straight fin shielded
arrangement marked a 50% improvement in Nusselt number
when compared with the baseline case. Radiation was
included in the physics models, which was subsequently
removed and compared with literature resulting in a 5% error.
V2f and Realizable k-ԑ turbulence models were used for
analysis on the cross-cut modification, showing an average
increase in the Nusselt number of 17% when both turbulence
models were correlated and compared to the shielded straight
fin parallel plate case, justifying the potential a cross-cut heat
sink has for improvement in cooling performance through
enhanced heat dissipation.
1. INTRODUCTION
Tower mounted radio-head devices and active pico
and fimto cellular data stations are in greater demand
due to the increase in mobile data communication. As a
result, the electronic components within these
communication devices are prone to overheating at peak
operational times. Strict criteria govern their design,
such as: weight limitations (~17-20kg), ingress
protection (IP) rating and a rigid passive cooling
Date of submission: 5/5/15’, Contact: 14055309@studentmail.ul.ie
requirement (due to exposure to harsh environmental
conditions) [1]. The heat dissipation experienced can be
very high and can remain functional in ambient air
conditions of up to 55°C with wind speed as little as
0m/s. Heat-sink specimens are placed within a solar
cabinet (to act as a solar shield) for thermal management
of communication devices through design optimization.
Experimentation with different geometries is common,
with particular focus on the cooling element (heat sink),
which will form the basis of the study to follow. In
recent years, the advancement of passive cooling
techniques has become necessary to meet the inherent
demand for continual operation of the equipment in
question.
The relevance of utilizing a plume based heat-sink
assembly was studied in O’Flaherty and Punch [2] to
determine the optimal design parameters. The harsh
environment of Kuwait city in the middle-east was
represented using computational fluid dynamics (CFD)
with the solar shield length, heat sink position and the
gap distance between the heat-sink and the shield
analyzed. The heat-sink was simulated at different
distances from the base of the shield, which showed that
heat dissipation reached its maximum capacity (~95W)
at 0mm from the base. 900mm and 2mm were found to
be the most suitable dimensions for the shield length
and gap distance, respectively [2]. Agonafer et al. [3]
similarly chose to investigate a chimney enclosure with a
series of modifications made to the geometry. A number
of external slats and louvers were applied to the vent
design at different locations to enhance cooling
capabilities. Radio-head electronic components (Power
amplifier and a transmission device) were studied in
isolation to analyze cooling performance for each CFD
testing scenario. These involved a combination of
internal and external slats and louvers at the top, bottom
and front sections of the chimney. Agonafer et al. found
that internal and external slats at 4mm wide performed
better than the louvered counterpart, with mass flow
Numerical analysis of parallel plate heat-
sink design for electronic cooling
applications
Peter McGibney
Department of Mechanical, Aeronautical & Biomedical Engineering, University of Limerick

2. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 2
rate at its highest and overall electronic circuitry
temperature at its lowest, thus proving this to be the best
arrangement [3]. In a later paper by O’Flaherty and
punch [4] an apparatus that consisted of a parallel plate
heat-sink, placed at the base of a chimney structure was
tested experimentally and numerically. A constant heat
load was applied to the heat sink using cartridge
heaters, at various temperature increments. CFD input
conditions were taken from data recorded through
experimentation (ambient and surface temperatures), to
help simulate the testing environment as precisely as
possible. A heat-sink in isolation was also considered as
a baseline case when correlating results obtained from
experimentation and CFD, as a function of Nusselt and
Elenbaas numbers. The results showed the chimney
based arrangement improved the Nusselt number by up
to 64%, proving the potential a chimney structure has
when used for convective heat transfer enhancement [4].
In addition to this, it has been well documented in the
literature that optimization of parallel plate heat sinks –
through cross cut design alterations - can help improve
the heat transfer performance [5-7], by varying the
geometrical layout and individual plate fin
proportionality. Kim and Kim [8] more interestingly
fabricated a series of cross-cut parallel plate heat-sinks
(for experimentation), placed within an enclosure at a
central location, similar in shape to the plume used in
O’Flaherty and Punch [4]. A negative pressure pump
was used to increase the mass flow of air over the heat
sink structure by analyzing pressure drop at the cross
cut region, the thermal resistance and the mass flow
through the vent. Cross-cut location, single and multiple
cross cut cases were tested at different pumping power
increments – which was determined as a product of the
mass flow and the pressure drop for each case. Kim and
Kim found that multiple cross-cuts performed poorly in
comparison to the single cross-cuts. The length of the
cross-cut was also considered to be the design parameter
that affected the thermal performance the most, for pin-
fin, single and multiple cross-cut arrangements with
constant pumping power conditions [8].
CFD analysis is conducted on the heat sink
combinations identical to the numerical investigation
performed in O’Flaherty and Punch [4] using CD-
Adapco Star CCM+ version 9.02.005. A parallel plate
heat-sink with fins on either side of the base plate (to
counteract heat loss error from a single sided heat-sink)
made from aluminum alloy with an emissivity of 0.97,
thermal conductivity of 180 W/m.k and a density of
2700 Kg/m3, is considered. The chimney structure is
made from polycarbonate, with an emissivity of 0.95,
thermal conductivity of 0.19 W/m.k and a density of
1200 Kg/m3. Fig. 1 in section 1.2 illustrates the different
variations (for the initial test phase) along with the
characteristic dimensions of the heat sink. Simulations
are setup in relation to the input data from experiments
performed in [4], for the unshielded and shielded cases.
Analytical models are then used to confirm the viability
of the data obtained, using theory adopted from Bar-
Cohen et al. [10]. A cross-cut modification is performed
on the original heat-sink and its heat transfer capabilities
compared to the straight fin parallel plate, placed at the
base of the plume. The addition of a pump (at the top
opening of the plume), to the cross-cut assembly
instigates turbulent flow conditions where turbulence
models (applicable to convective heat transfer) are tested
for each cross-cut scenario. This will be expanded on in
greater detail at a later stage in the paper.
2. METHODOLOGY
2.1 Geometry Generation
The geometry was created using Star CCM+’s internal
CAD package. Dimensions were taken from reference [4]
for the heat-sink, plume and an air enclosure (to inhibit
air currents causing uncertainty in output data) which
surrounds the experimental apparatus. Each individual
assembly is setup within the CAD modelling
environment, which will be described in detail in the
next subsections.
Fig. 1: Phase one of CAD geometry design with the dimensioned
isolated heat sink (left image) and the heat sink at the base of the plume (right
image).
i) CAD Modelling of Heat-sink and Shield
From the aerial plane of the heat sink, the top face is
first drawn. This is then extruded to complete the full
geometry. Fig. 1 also shows 7 holes at various locations
in the base plate, where the cartridge heaters are
inserted. Observations from experimental analysis

3. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 3
performed in O’Flaherty and Punch [4] revealed a 20mm
section at either side of each cartridge heater without
heat after power was applied. This was considered when
incorporating each hole into the design as a solid
cylindrical section. For both straight parallel plate cases
(shielded and unshielded), an air box surrounding the
heat sink (encompassing the ambient air enclosure), is
added, with an internal volume extract performed (to
isolate the outline of the air volume bounding the shield
and heat-sink), completing this part of the CAD setup.
The shielded assembly was achieved by sketching the
inner and outer outline of the plume with a thickness of
5mm, a width of 161mm and a clearance of 5.8mm (in
addition to the un-shielded assembly), and was
completed by performing an extrude 900mm in length
from the base of the heat-sink. The geometry was tested
as per the temperature settings in O’Flaherty and Punch
[4] as a quarter of the proportion of the full assembly,
which produce identical numerical results according to
[4]. Fig. 2 below displays the sectioned geometries for
both the shielded and unshielded assemblies.
ii) CAD Modelling of Cross-Cut Heat-sink
The inclusion of cross-cuts (at an arbitrary distance
from the edge of the heat-sink) is applied as per the
criteria in Kim and Kim [8], (used to calculate the
optimized cross-cut length) for Lc, which is;
0.0833 =
𝐿 𝑐
𝐿⁄ . (1)
Based on the characteristic length of the heat-sink (L =
300mm), the optimal cross cut length was calculated as
25mm (Lc = 25mm). These were then set as the
dimensions for the cross cut at each parallel plate. As
mentioned in a previous section, multiple cross cuts do
not improve heat transfer performance, according to
Kim and Kim [8]. This meant that single cross-cuts (at
either side of the base plate due to symmetry) were
chosen as the only modification to the geometry. The
right hand image of Fig. 2 is the full cross-cut heat sink
geometry.
(a) (b) (c)
Fig. 2: Sectioned Heat-sink (a), shielded assembly section within air
box (b), and full Cross-Cut heat sink geometry (c)
2.2 Meshing Procedure
Because the measurement of heat-transfer is taken
from the boundary layer between the heat-sink and the
ambient air (otherwise known as film temperature), the
prism layer mesh feature needed to be considered.
tetrahedral/prism and trimmer/prism mesh
combinations were selected first but created bad cell
definition at the cylindrical regions (extruded cut
sections representing the cartridge heaters) and at the
boundary points between the air and heat-sink
geometry. The prism layer sections at the holes were
inadequate and can be seen in Fig. 4 of Appendix A,
where bad cell definition is evident. A polyhedral mesh
was chosen instead, producing an even surface over the
heater sections. Table 1 in Appendix A shows the cell
count for each mesh combination. Fig. 3 shows mesh
scene visualization at particular points in the geometry
for the shielded arrangement. The same mesh procedure
was employed for both the unshielded and cross-cut
setups. 10 prism layers were set for the boundary points
between the air and the heat sink. It is at this point
convective heat transfer was measured, so it was
important to define these settings correctly. The
thickness of the prism layer was set to an absolute
measurement 4mm, which represents the thermal
boundary layer thickness at the surface of the heat sink.
The completed meshed geometry initially contained 1 x
105 cells using an arbitrary base size of 8mm. The
selection of an appropriate mesh definition will be
expanded in later section of this paper.
(a) (b)
(c)
Fig. 3: Prism layer at the boundary between air and shield (a), boundary
between air and heat sink (b), and for the boundary between heaters and heat-
sink (c)

4. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 4
2.3 Numerical Procedure
Physics models for the ambient air volume
surrounding the apparatus were set according to
O’Flaherty and Punch [4]. The flow was assumed to be
laminar with the Boussinesq approximation included
due to the flow being buoyancy driven. Radiation makes
up roughly 20% of the overall heat transfer from the
system, according to Shabany [9], so it was important to
choose the appropriate radiation model. Gray thermal
radiation and surface –to-surface radiation were selected
based on assumptions made in the next section (theory)
of the paper. The time step was also assumed to be
steady and the flow type to be three-dimensional. The
heat-sink and the shield were set as a solid, constant
density, three-dimensional and with segregated solid
energy. Material properties were inputted according to
the parameters outlined in the introduction.
Regions representing all three parts were created
along with each specific boundary condition. For the
ambient air, a pressure boundary is chosen to represent
the air surrounding the heat-sink/shield assembly and
also symmetry boundaries for the planes at the cross
section through the full geometry (for heat sink, shield
and ambient air regions). Wall type boundary points for
both the shield and the heat sink are set according to the
areas of both parts in contact with the surrounding air.
Fig.8 in Appendix B illustrates the different faces where
the boundary conditions are assigned to. Interfaces
(based on particular contact points), between the heat
sink and the ambient air, and also for the contact
between the shield and the ambient air – were created in
each region. These contact points were created in Star
CCM+ to characterize the boundary layers completely,
which can be seen in Fig. 3 for the prism layer meshing.
Temperature settings are inputted in the physics
continua (for ambient air and the cartridge heaters),
taken from O’Flaherty and Punch [4] for 9 different
temperature variations. Heat is applied to the cartridge
heaters within the heat-sink region for the
corresponding boundary condition. Heat transfer is then
measured from the film temperature at the boundary
between the heat sink and the ambient air. This is
monitored continuously when a simulation is run and is
also displayed as a report. A plane section cuts right
through the centre of the geometry in a vertical
direction. Another two plane sections at the top of the
heat-sink and through the top opening of the plume
(which cuts through the plume horizontally), for scalar
and vector scene visualizations, are created. Fig. 9 in
Appendix E shows the visualization scenes within Star
CCM+, used for measuring velocity and temperature
through colour contours for temperature/velocity scale,
which will be described further in the results section.
The cross-cut heat-sink arrangement was modelled in
its entirety due to the addition of a pumping condition
at the exit of the plume. This represents heat transfer in
forced convection, using a suction pump defined as a
boundary condition at the outlet of the plume. To model
the flow through the plume, another air volume is
created. The heat sink is subtracted from the shield air
volume, identical to the process used for ambient air for
the shielded and unshielded variations. The shield air
volume is combined with the ambient air, so that the
inlet and outlet boundaries for the air flow through the
plume can be defined. A pressure difference between the
cross-cut section of the heat-sink and the outlet is
measured (pressure drop), so that the pumping power
can be computed. This is calculated according to Kim
and Kim [8] using equation (2);
𝑃𝑃𝑜𝑤𝑒𝑟 = 𝑄̇ × ∆𝑃. (2)
Where 𝑄̇ is the mass flow rate and is set as an arbitrary
value initially. The boundary for the outlet is set as a
pressure type representing the negative pressure pump.
The inlet to the shield air flow is set as mass flow and
the outer walls of the fluid domain to a wall type. A
pressure outlet for the ambient air is set as a pressure
outlet. Naturally, more contact points needed to be set
for; the ambient to plume outer wall; plume fluid to the
inside wall of the plume and the heat-sink in contact
with the plume fluid region, in addition to the contacts
used previously. Fig.8 in Appendix B shows the cross-cut
variation in full displayed using transparency feature to
illustrate the plume and heat sink within the ambient air
domain. Standard realizable k-ԑ and a V2F k-ԑ
turbulence models (utilizing the all y+ model feature),
for the fluid physics conditions and are tested using the
same input temperatures and with varying pumping
power.
i) Convergence Criteria
Convergence for each numerical simulation was based
on two physical parameters in the model; the heat
removed at the boundary between the heat sink and
ambient air, and the velocity from a cut plane. Once
both values are observed to have plateaued, then the
simulation results are said to be converged.
Convergence based on residual plots stay within a range
of 1 to .01, meaning the solution has not stabilized and is
therefore inaccurate. However, plots from each
simulation were used to determine the point of
equilibrium, which was evident when monitored
continuously, showing a particular iteration at which
each parameter levels off.

5. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 5
2.4 Theoretical Modelling
i) Radiation and Convection
For natural convection (in relation to the unshielded
geometry), theoretical models are used to isolate
convective heat transfer by first considering radiation.
As mentioned earlier, Shabany [9] iteratively derived
implicit relations for radiative heat transfer from a
standard parallel plate heat sink. The viability of
simulations to follow will be determined based on their
accuracy when compared with theory. Two different
simulation setups are first considered; one with
radiation included in the physics settings and another
without radiation. Radiative heat transfer can be
computed using equation (3);
Qcombined = 𝑄 𝑝𝑏𝑒 + (𝑁𝑓 − 1)𝑄𝑐ℎ𝑎𝑛𝑛𝑒𝑙 (3)
Where 𝑄 𝑝𝑏𝑒 is the radiative heat transfer from the
plan, base and elevation planes of each fin on the heat
sink, 𝜂 𝑓 denoted the number of fins and 𝑄𝑐ℎ𝑎𝑛𝑛𝑒𝑙 is the
heat dissipation from a single channel.
Appendix C includes the preceding theoretical
formulae used to find the total radiation from the heat-
sink. The heat-sink is assumed to be diffuse and gray
while the surrounding areas are assumed to be black
body. Temperature applied to the heat-sink is also
assumed to be constant.
The governing equation for convective heat transfer is;
Qconv = 𝑁𝑓 𝑄 𝑓𝑖𝑛 + (∆𝑇)𝐴 𝑏ℎ 𝑏
̅̅̅, (4)
Where 𝑄 𝑓𝑖𝑛 is the convective heat transfer from a
single fin, 𝑁 is the number of fins that makes up the heat
sink, ∆𝑇 is the temperature difference between the
ambient air and the surface of the heat sink, 𝐴 𝑏 is the
area of heat sink base plate and ℎ 𝑏
̅̅̅ is the heat transfer co-
efficient of the base plate. Detailed calculations for ℎ 𝑏
̅̅̅
and 𝑄 𝑓𝑖𝑛 can be found in Appendix C.
Specific Nusselt and Elenbaas numbers are then
determined using equations (5) and (6) respectively -
which will form the basis of the comparative analysis at
a later section of this paper.
Nus = [
576
𝜂 𝑓(𝑅𝑎 𝑆 𝑆/𝐿)2 +
2.87
𝜂 𝑓(𝑅𝑎 𝑆 𝑆/𝐿)1/2]
−1/2
=
ℎ 𝑓𝑖𝑛 𝑆
₭
(5)
Els = 𝑅𝑎 𝐿 (
𝑆4
𝐿4) (6)
The Nusselt number for both experimental and
theoretical is shown in equation (5). Numerical data will
be interpreted through the equation depicted to the right
hand side of the same equation.
ii) Turbulence Theory
The transport equation that governs ε (turbulent
kinetic energy rate of dissipation) can be computed
using equation (7);
2𝜈
𝜕𝑢 𝑖
𝜕𝑥 𝑗
̅̅̅̅̅̅̅̅ 𝜕
𝜕𝑥 𝑗
(𝑁)(𝑢𝑖) = 0. (7)
Where (𝑁)(𝑢𝑖) is the applicable Navier Stokes
operation, which is presented in detail in Appendix C.
Derivations for exact turbulent kinetic energy
according to Dewan [11] can be found by summing up
the mean kinetic energy 𝐾 𝑚 and 𝑘 𝑇, which describes the
turbulent kinetic energy, using equation (8);
𝐾 + 𝑘 = [1
2⁄ (𝑢̅2
+ 𝑣̅2
+ 𝑤̅2)] + [1
2⁄ (𝑢′̅2
+ 𝑣′̅2
+ 𝑤′̅̅̅2
)]. (8)
The turbulent kinetic energy transport equation can be
derived more comprehensively by considering the trace
of the Reynolds stress tensor 𝜏𝑖𝑗 (see Appendix C for
derivation steps).
A realizable k-ε turbulence model, according to Shih et
al. [12], can be modelled for a variety of different flow
cases – but more importantly can be applied to flow over
a flat plate. The turbulent eddy viscosity is expressed
using equations (9) and (10);
𝑢2̅̅̅ = 2
3⁄ 𝑘 + 2𝜈 𝑇
𝜕𝑢̅
𝜕𝑥
, (9)
𝜈 𝑇 = 𝐶𝜇
𝑘2
𝜀
. (10)
Where 𝐶𝜇 is set at a constant value of 0.09, but can also
be considered as a variable, dependent on the sum of the
mean flow through the system and on the turbulence
properties.
A closure co-efficient is then considered next, which is
expanded on in detail in Appendix C.
Modifications to k-ε turbulence models are needed to
include the effects of buoyancy experienced in plume
type flows. Boundary layer assumption is employed, so
that the time-averaged flow is included [11]. Time
averaged equations for continuity, momentum and
thermal energy can be expressed (using the assumptions
just described), in equations (11) to (13), respectively.
𝜕𝑢
𝜕𝑥
+
1
𝑟
𝜕
𝜕𝑟
(𝑟𝜈) = 0 (11)
𝑢
𝜕𝑢
𝜕𝑥
+ 𝜈
𝜕𝑢
𝜕𝑟
=
1
𝑟
𝜕
𝜕𝑟
[𝑟(−𝑢′ 𝑣′̅̅̅̅̅̅)] + 𝑔𝛽(∆𝑇) = 0 (12)

6. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 6
𝑢
𝜕𝑇
𝜕𝑥
+ 𝜈
𝜕𝑇
𝜕𝑟
=
1
𝑟
𝜕
𝜕𝑟
[𝑟(−𝑢′ 𝑣′̅̅̅̅̅̅)] (13)
The buoyancy term can be seen on the right hand side
of equation (12), where the movement of fluid through
the system is trigged. β is the increase in volumetric
expansion (relative to the original volume), per the rise
in temperature measured in Kelvin increments [11].
Transport equations for the k-ε turbulence model was
then derived, using equations (19) and (20) in Appendix
C.
The second choice turbulence model is the k-ε V2f.
This model was first proposed by Durbin [13], which is
based on a root mean square velocity, 𝜈′2
, using a scale
for velocity rather than considering the turbulence
kinetic energy. Velocity fluctuations are noted to be
sensitive to near wall regions, which acts as a damping
function, which means that additional dampers are not
needed, according to Dewan [11]. This means that low
Reynolds numbers are not needed for the wall presence.
A V2f model has four transport equations for turbulent
kinetic energy, turbulent dissipation, a transport
equation (that differs from the standard k-ε model) and
an equation that deals with wall blocking behavior,
found in the 𝑣′2̅̅̅̅ transport equation. The equations from
Dewan [11] are written as;
𝜕𝑘
𝜕𝑡
+ 𝑢̅𝑗
𝜕𝑘
𝜕𝑥 𝑗
=
𝜕
𝜕𝑥 𝑗
(𝑣 + (
𝑣 𝑡
𝜎 𝑘
)
𝜕𝑘
𝜕𝑥 𝑗
) + 𝑃𝑘 − 𝜀, (14)
𝜕𝜀
𝜕𝑡
+ 𝑢̅𝑗
𝜕𝜀
𝜕𝑥 𝑗
=
𝜕
𝜕𝑥 𝑗
(𝑣 + (
𝑣 𝑡
𝜎 𝜀
)
𝜕𝜀
𝜕𝑥 𝑗
) +
𝐶 𝜀1 𝑃 𝑘−𝐶 𝜀2
𝑇
, (15)
𝜕𝜈′2̅̅̅̅̅
𝜕𝑡
+ 𝑢̅𝑗
𝜕𝜈′2̅̅̅̅̅
𝜕𝑥 𝑗
=
𝜕
𝜕𝑥 𝑗
(𝑣 + (
𝑣 𝑡
𝜎 𝑘
)
𝜕𝜈′2̅̅̅̅̅
𝜕𝑥 𝑗
) − 𝜈′2̅̅̅̅ 𝜀
𝑘
+ 𝑘𝑓, (16)
𝑓 − 𝐿2
∇2
𝑓 = (𝐶1 − 1)
2
3⁄ − 𝜈′2̅̅̅̅̅
𝑘
⁄
𝑇
+ 𝐶2
𝑃 𝑘
𝑘
. (17)
The specific derivations for 𝑇 and 𝐿, the turbulent time
a length scales, respectively, are expanded on in detail in
Appendix C, along with set constants that are used for
V2f turbulent modelling.
4. RESULTS
Results from both theoretical and numerical data are
plotted as a function of the non-dimensional formulae
outlined in section 2.4. The first section will deal with a
mesh independence study that was carried out for all
three arrangements. This was so that the data obtained
from each simulation would not been altered by
discrepancy in mesh definition.
4.1 Mesh Independence
To confirm the minimum base size applicable (until
data from each mesh independence simulation is
consistent) a number of different mesh densities are
tested by changing the cell base size. This was tested for
all three cases, stepping up in set increments in
millimeters. The first temperature setting (i.e. 5°C above
ambient), is simulated for each generated mesh, and the
heat transfer from each plotted as function of cell count.
From Fig. 4, the heat dissipation for each case (at the
same temperature setting), reaches a consistent level at
2.09W, 2.98W, and 3.04W, for the unshielded, shielded
and cross-cut variations, respectively. Standard base
sizes selected for all three cases were then set to 8mm for
all three cases. An illustration of the mesh densities at
the boundary points for the shield and heat-sink in
ambient air is illustrated in Fig. 12 of Appendix A.
From Fig. 4, the shielded case produced varied results
for heat dissipation, up until the cell count reaches ~4
million, which corresponds to the base size selection of
8mm as mentioned previously. The corresponding
unshielded independence data reveals an optimum
mesh density of ~2.5 million cells for both the heat sink
and the air volume combined. The combined data from
the mesh sensitivity study can be seen in Table II to IV of
Appendix A.
Fig. 4: Mesh Independence study with heat dissipation as a function of cell
count for all three cases
4.2 Unshielded
The application of heat to the model was incremented
according data points (for nine different temperature
setting), taken from O’Flaherty and Punch [4]. For the
first data point of 5°C, the velocity levels out at 0.17 m/s
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
0 2000000 4000000 6000000
HeatDissipation(W)
Cell Count
Unshielded
Shielded
Cross-Cut

7. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 7
at the exit of the heat-sink as seen from the contour plot
in part (b) of Fig. 9 in Appendix E. The corresponding
temperature plot in (a) of the same figure at the point
thermal equilibrium is reached in the system.
Heat dissipation results from each numerical
simulation were then used to calculate an experimental
Nusselt number, as seen from the RHS of equation (5).
The numerical results for the unshielded case were
compared to the theory calculations (which were carried
out for each data point and compared), and agreed with
an average error of 5% for the Nusselt number and 3.5%
for the Elenbaas number. The theoretical and numerical
data obtained was correlated and compared to
O’Flaherty and Punch [4], which was in error of 6% for
the Nusselt number and 2% for the Elenbaas number.
Fig. 5 illustrates this data comparison for the Elenbaas
number as a function of the Nusselt number.
Fig. 5: Correlation plotted against data from O'Flaherty and Punch [4]
(top), with numerical and theoretical data plotted against O'Flaherty and
Punch [4] (bottom)
The addition of radiation at the initial stages of the
numerical testing was also considered to confirm its
accuracy against radiation theory (outlined in section
2.4). At 5˚C above ambient, radiation accounts for 15% of
the over heat dissipation, working out as 0.1237W.
However, simulation predicted heat dissipation through
radiation was 18% (0.122W) of the overall heat
transferred, which is consequently almost unity when
compared to the theoretical value for radiation.
Radiation theory is used extensively for the shielded
case, where the appropriate radiation models are
included in the physics model definitions for ambient
air.
4.3 Shielded
For the shielded variation, simulations were set up
using the same temperature values. This is to determine
the percentage improvement that the addition of a
chimney has by improving mass flow.
Fig. 6: Numerical data for shielded case, plotted against the unshielded
correlation and shielded numerical from O’Flaherty and Punch [4]
Fig. 6 illustrates the unshielded correlation, compared
to the shielded numerical data from O’Flaherty and
Punch [4], and with the numerical data from obtained
from the shielded investigation used in this paper.
Numerical data obtained from each shielded simulation
(for heat dissipation), had radiation included, so that the
accuracy of radiation theory can be determined more
comprehensively by computing each data point using
the formulae detailed in section 2.4. Theory from
Shabany [9] and O’Flaherty and Punch [4] was
employed for each data point from the numerical
analysis. Comparative analysis showed that the data set
representing radiation theory from reference [4] agreed
with results from O’Flaherty and Punch [4], with a 7%
error, compared to the 9% for radiation theory from
reference [9], for the Nusselt number. The Elenbaas
number was in error of only 2% when compared to the
literature [4]. An improvement in the Nusselt number
was observed to be 50%, which proves the potential that
the application of a chimney has over just a stand-alone
heat-sink. However, the numerical data for the shielded
case and the literature [4], shows that there may be a
discrepancy in the numerical procedure, which would
0
0.5
1
1.5
2
2.5
0 50 100 150 200 250 300
Nusselt(-)
Elenbaas (-)
Numerical
Theoretical
O'Flaherty and
Punch [4]
0
0.5
1
1.5
2
2.5
0 100 200 300
Nusselt(-)
Elenbaas (-)
Correlation
O'Flaherty and
Punch
0
0.5
1
1.5
2
2.5
3
3.5
0 100 200 300
Nusselt(-)
Elenbaas (-)
Shielded (Radiation [9])
O'Flaherty and Punch [4]
Unshielded Correlation
Shielded ( Radiation [4])

8. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 8
be the subject of further investigation. Fig. 6 also shows
an incline in the last two data points for the Nusselt
number, which is uncharacteristic of the expected shape
of the graph.
4.4 Cross-cut
The forced convection variation involved the addition
of an extra fluid domain (within the shield structure,
and subtracted from the air domain), so that the inlet
and outlet conditions for the negative pressure pump
could be applied. The heat sink is suspended within the
fluid domain at the same position used for the shielded
case. The full geometry (with each highlighted region)
can be seen in part (a) of Fig. 8 in Appendix A. The
geometry initially included a forced convection setup,
but was altered to suit a natural convection scenario due
to inconsistency in geometry definition. Two turbulence
models were tested against the shielded correlation for
the Nusselt number, plotted a function of the Elenbaas
number, for both the V2f and realizable k-ԑ variations.
The same temperature settings used in the previous two
cases were employed for each simulation data point,
showing a marked increase in heat transfer performance
as seen in Fig. 10 below.
Fig. 10: V2f and Realizable k-e turbulence models plotted against
straight fin parallel plate shielded correlation
The V2f and Realizable k-ԑ models signify an average
increase in the Nusselt number of 15% and 17%
respectively. This enhances the perception that single
cross-cut modifications can improve heat-sink cooling
performance [8].
Both turbulence models were correlated and
compared with the data obtained from the straight fin
parallel plate analysis and from O’Flaherty and Punch
[4]. From Fig. 11 below the unshielded correlation
showed an increase in 16% from shielded case and 22%
when compared with shielded data from O’Flaherty and
Punch [4].
Fig. 11: Turbulence modelling correlation plotted against O'Flaherty
and Punch [4], and against numerical data for shielded correlation
From inspection of Fig. 11, the first data point shows
no improvement in the Nusselt, which is in fact lower
when compared to data points from the corresponding
data sets at 1.36, an average of 8.4% below the Nusselt
number from shielded correlation and O’Flaherty and
Punch [4]. Inconsistency in the Realizable k-ԑ data from
Fig. 10 can also be observed, levelling off at the third
data point and is markedly lower (11%) when matched
with data from V2f. This is uncharacteristic when the
general shape of the plotted data-set from Fig. 10 is
considered.
5. DISCUSSION
As mentioned in a previous section of the paper,
convergence criteria was based on velocity and heat
transfer plots, monitoring the point at which each levels
off. Relaxation factors for segregated flow - with respect
to velocity and pressure – were reduced to try bring the
residuals to convergence. This had little effect on
bringing them to a reasonable level, which reached a
minimum at ~0.01. This is two orders of magnitude
above the universally acceptable level of 10-4. One reason
for this issue could lie in the definition of the physics
conditions for the ambient air, where a steady flow type
was chosen for each test case. The unsteady nature of
natural convection heat transfer means that a steady-
state model may not be an accurate air flow modelling
assumption.
With radiation models included in the physics
conditions, the time to complete a simulation is almost
triple (7+hrs) that of the models without radiation. This
meant that only one data set with radiation could be
considered due to time constraints.
The cross-cut analysis could be extended by
comparing a multiple cross-cut case, subject to the
criteria outlined in Kim and Kim [8]. This could confirm
the accuracy of the numerical data obtained from each
turbulence setup and help to solidify the numerical
procedure used in this paper. The original forced
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 100 200 300
Nusselt(-)
Elenbaas (-)
V2f
Shielded Correlation
Realizable
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 100 200 300
Nusselt(-)
Elenbaas (-)
Shielded Correlation
Turbulence Correlation
O'Flaherty and Punch [4]

9. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 9
convection scenario was geometrically unsuitable for the
numerical analysis. Non-manifold conditions prevented
the model from being utilised, due to the plume air
region having faces (heat-sink air subtract) that are
incompatible. Using the surface wrapper feature as a
meshing model clears the error by deleting the subtract
feature within the air, which was inadequate for the
numerical analysis i.e. the film temperature at the
boundary of the surface of the heat-sink needs to be
considered, so that the appropriate heat dissipation data
can be captured. A different approach to the geometrical
setup for the air within the shield and the ambient air
would need to be considered, prompting the need for a
more efficient modelling technique in the future, in
addition to the points raised in this section.
6. CONCLUSION
A standard parallel plate heat sink was examined as a
stand-alone feature against a heat-sink of the same size
and shape placed within a polycarbonate plume. The
CFD investigation methods used in this paper showed
that the correlated data (from theory and numerical
analysis in relation to the unshielded case), agreed with
the literature with an error of only 4%, an indication of
how viable the results are. As expected, the shielded
variation marked a 50% increase in the Nusselt number,
which is close to the results found in the literature at
61%. The cross-cut geometry showed potential as a
modification to the straight fin parallel plate heat sink,
using the appropriate turbulence models, marking an
average increase 19% compared to the benchmarked
data.
The use of a parallel plate heat sink for cooling
electronic components within radio communication
devices is a broad area, and from the numerical methods
used in this paper, it is something that could be
investigated further when considering the points raised
in the previous sections.
APPENDICES
APPENDIX A: MESHING
Table I. Meshing Model Combinations
Case Meshing Models Cell Count
Unshielded Polyhedral/Prism 1 x 105
Unshielded Tetrahedral/Prism 4 x 105
Unshielded Trimmer’Prism 5 x 105
Table II: Mesh Independence data (shielded)
Base Size (mm) Cell Count Heat Transfer (W)
6 4,540,973 2.85
8 3,310,328 2.90
12
16
24
1,697,004
938,846
416,597
2.85
2.76
3.06
Table III: Mesh Independence data (unshielded)
Base Size (mm) Cell Count Heat Transfer (W)
6 2,787,099 2.09
8 1,287,737 2.09
10
20
25
988,293
125,058
69,831
2.11
2.05
2.08
Table IV: Mesh Independence data (cross-cut)
Base Size (mm) Cell Count Heat Transfer (W)
8 3,310,238 3.03
12
20
24
1,624,831
596,204
315,464
3.03
3.06
3.12
(a) (b)
Fig. 7: Bad mesh cells at prism layer for unshielded case using
Trimmer mesh model (a) and tetrahedral model (b)

10. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 10
(a) (b)
(c)
Fig. 12: Shielded mesh for 20mm (a), 12mm (b) and 8mm (c) at
boundary points between symmetry plane for ambient air, heat sink and shield
APPENDIX B: REGIONS
(a) (b) (c)
Fig. 8: Specific boundary conditions for crosscut (a), unshielded (b),
and shielded (c) layouts
APPENDIX C: TURBULENCE THEORY
The Navier Stokes equation for the rate of turbulent
kinetic energy dissipation is;
(𝑁)(𝑢𝑖) = 𝜌
𝜕𝑢 𝑖
𝜕𝑡
+ 𝜌𝑢 𝑘
𝜕𝑢 𝑖
𝜕𝑥 𝑘
+
𝜕𝑝
𝜕𝑥 𝑖
− 𝜇
𝜕2 𝑢 𝑖
𝜕𝑥 𝑘 𝜕𝑥 𝑘
. (18)
The steps for Reynolds stress tensor derivation are
found in equation (19), where;
𝜕(𝜌𝑢′
𝑖 𝑢′
𝑗
̅̅̅̅̅̅̅̅̅)
𝜕𝑡
+
𝜕
𝜕𝑥 𝑘
(𝜌𝑢 𝑘̅̅̅𝑢𝑖
′ 𝑢𝑗
′̅̅̅̅̅̅) =
−
𝜕
𝜕𝑥
[(𝜌𝑢𝑖̅ 𝑢𝑗
′ 𝑢 𝑘
′̅̅̅̅̅̅ + 𝑃′(𝛿 𝑘𝑗 𝑢𝑖
′ + 𝛿𝑖𝑘
′
𝑢𝑗
′)̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅)] +
𝜕
𝜕𝑥 𝑘
[𝜇
𝜕
𝜕𝑥 𝑘
(𝑢𝑖
′ 𝑢𝑗
′̅̅̅̅̅̅)] −
𝜌 (
𝑢 𝑖
′ 𝑢 𝑘
′̅̅̅̅̅̅̅(𝜕𝑢 𝑗̅̅̅̅)
𝜕𝑥 𝑘
+
𝑢 𝑗
′ 𝑢 𝑘
′̅̅̅̅̅̅̅(𝜕𝑢 𝑖̅̅̅)
𝜕𝑥 𝑘
) + 𝑃′
(
𝜕𝑢 𝑖̅̅̅
𝜕𝑥 𝑗
+
𝜕𝑢 𝑗̅̅̅̅
𝜕𝑥 𝑖
) − 2𝜇
𝜕𝑢 𝑖̅̅̅
𝜕𝑥 𝑘
𝜕𝑢 𝑗̅̅̅̅
𝜕𝑥 𝑘
. (19)
From the equation to the right hand of (19) is the
convection term 𝐶𝑖𝑗, with the left hand side containing
the turbulent diffusion 𝐷𝑖𝑗
𝑇
, molecular diffusion 𝐷𝑖𝑗
𝐿
,
stress production 𝑃𝑖𝑗, pressure strain ∅𝑖𝑗 and a
dissipation term 𝜀𝑖𝑗, respectively. Therefore, the exact
transport equation for turbulent kinetic energy can be
written as;
𝜕𝑢𝑖̅
𝜕𝑥 𝑘
+ 𝑢𝑗̅
𝜕𝑘
𝜕𝑥𝑗
=
𝜕
𝜕𝑥𝑗
(𝑣
𝜕𝑘̅
𝜕𝑥𝑗
)
−𝑣
𝜕𝑢′
𝑖
̅̅̅̅̅
𝜕𝑥 𝑗
𝜕𝑢′
𝑗
̅̅̅̅̅
𝜕𝑥 𝑗
−
𝜕
𝜕𝑥 𝑘
(1
2⁄ 𝑢𝑖
′ 𝑢𝑖
′ 𝑢𝑗
′̅̅̅̅̅̅̅̅ +
𝑢′
𝑗 𝑝′̅̅̅̅̅̅̅̅
𝜌
), (20)
when 𝑖 = 𝑗. For incompressible flow, the pressure
strain term is negligible after i is subbed in for j.
A specific closure co-efficient for the realizable k-ε
turbulent model, when it is a variable, where;
𝐶1𝜀 = 𝑚𝑎𝑥 (0.43,
𝜑
5+𝜑
); (21)
𝜑 =
𝑆𝑘
𝜀
, and; (22)
𝑆 = (2𝑆𝑖𝑗 𝑆𝑖𝑗)
1/2
(23)
Other closure co-efficients for realizable k-ε turbulence
models are: 𝐶1𝜀 = 1.9, 𝜎𝑘 = 1 and 𝜎𝑘 = 1.2.
The modified transport equations for buoyancy driven
k-ε turbulence models are;
𝑢
𝜕𝑇
𝜕𝑥
+ 𝜈
𝜕𝑇
𝜕𝑟
=
1
𝑟
𝜕
𝜕𝑟
[𝑟 (𝐶 𝑘
𝑣′2
𝑘
̅̅̅
)
𝜕𝑘
𝜕𝑟
] + 𝑃𝑘 + 𝐺 𝑘 − 𝜀, (24)
𝑢
𝜕𝜀
𝜕𝑥
+ 𝜈
𝜕𝜀
𝜕𝑟
=
1
𝑟
𝜕
𝜕𝑟
[𝑟 (𝐶𝜀
𝑣′2
𝑘
̅̅̅
𝜀)
𝜕𝜀
𝜕𝑟
] + 𝐶𝜀1
𝜀
𝑘
(𝑃𝑘 + 𝐺 𝑘) −
𝐶𝜀2
𝜀2
𝑘
. (25)
Where 𝐺 𝑘 is the buoyancy production term for
turbulent kinetic energy. The turbulent kinetic energy
production term, expressed as 𝑃𝑘 in the equation above,
accounts for 30% of the production. For boundary layer
assumptions, 𝑃𝑘 can be conveyed in equation (26);

11. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 11
𝑃𝑘 ≈ −(𝑢′ 𝑣′̅̅̅̅̅̅)
𝜕𝑢
𝜕𝑟
. (26)
In stream wise heat flux and buoyancy production
terms for turbulence, 𝐺 𝑘 is negligible for boundary
gradients, but is important when considering normal
gradients. Dewan [11] proposes the use of a model that
can be applied to a variety of different axial heat flux in
turbulent flows in response to this, as;
𝑢 𝑡
′
𝑡′̅̅̅̅̅ = −
𝑣 𝑡
𝑃𝑟𝑡
𝜕𝑇
𝜕𝑥 𝑖
+ 𝑘ℎ(𝑘𝑡′2)1/2
. (27)
Where 𝑘ℎ is a constant at 0.56. Next, a transport
equation is derived to consider temperature fluctuations.
This is so that the buoyant production term for turbulent
flows can be modelled correctly, which is expressed as;
𝑢
𝜕𝑡′2̅̅̅̅
𝜕𝑥
+ 𝜈
𝜕𝑡′2̅̅̅̅
𝜕𝑦
=
𝜕
𝜕𝑟
[(𝐶𝑡
𝑘2
𝜀
)
𝜕𝑡′2̅̅̅̅
𝜕𝑦
] + 𝑃𝑡 − 𝜀𝑡 (28)
𝑃𝑡 describes the production rate and 𝜀𝑡 the dissipation
rate for the fluctuations in temperature. For boundary
layer assumptions, 𝑃𝑡 can now be expressed as;
𝑃𝑡 ≈ −2(𝑣′ 𝑡′̅̅̅̅̅)
𝜕𝑇
𝜕𝑦
. (29)
𝜀𝑡 is also simplified to get the equation;
𝜀𝑡 = 𝐶𝑡1 𝜀
𝑡′2̅̅̅̅
𝑘
(30)
Where 𝐶𝑡1is a constant used to represent time scale
proportionality for velocity and temperature
fluctuations.
Turbulence time and length scale can be written as;
𝑇 = 𝑚𝑖𝑛 [𝑚𝑎𝑥 (
𝑘
𝜀
, 𝐶𝑡√
𝑣
𝜀
) ,
0.6𝑘
2√3 𝐶µ 𝑣′2̅̅̅̅̅ 𝑆
], (31)
𝐿 = 𝐶𝐿 𝑚𝑎𝑥 [𝑚𝑖𝑛 (
𝑘
3
2⁄
𝜀
,
𝑘
3
2⁄
√3 𝐶µ 𝑣′2̅̅̅̅̅ 𝑆
) , 𝐶 𝜂 (
𝑣
3
4⁄
𝜀
1
4⁄
)] (32)
From equation (16) in the main body of the text, the 𝑘𝑓
term denotes kinetic energy in relation to the stream
wise velocity. This means that the eddy viscosity
component is then written as;
µ 𝑇
𝜌
= 𝑣 𝑇 = (𝐶µ)𝑇𝑣′2̅̅̅̅ (33)
Finally, the constants used that are specific to the V2f
turbulence model are;
𝐶µ = 0.22, 𝐶𝑙 = 0.23, 𝐶 𝜂 = 85, 𝐶 𝑇 = 6, 𝐶1 = 0.4,
𝐶2 = 0.3, 𝐶𝜀2 = 1.9, 𝜎𝑘 = 1, 𝜎𝜀 = 1.3,
𝐶𝜀1 = 1.4 [1 + 0.045 (
𝑘
𝑣′2̅̅̅̅̅)
1
2⁄
]. (34)
APPENDIX D: NATURAL CONVECTION THEORY
The Rayleigh number is considered first and is
modulated based on the characteristic length, L, for the
heat sink, and is calculated using equation (35).
RaL = 𝑔𝛽( 𝑇 𝑠− 𝑇 𝑠𝑢𝑟𝑟) 𝐿3
𝑣2 Pr (35)
Further non-dimensional theory is employed for the
Elenbaas number, which is used extensively in
reviewing the simulation results (see results section for
application of Elenbaas theory).
Els = 𝑅𝑎𝑙 (
𝑆4
𝐿4) (36)
Heat transfer co-efficients need to be determined in
order to compute Nusselt number data points and for
the overall convective heat transfer.
The heat transfer co-efficient for the base plate, ℎ̅ 𝑏 is
evaluated from equation (37).
ℎ 𝑏
̅̅̅ = 0.59𝑅𝑎𝑙
1
4⁄
(
₭
𝐿
)              (37)
The heat transfer from a single fin is then expressed
as;
Qfin = ℎ̅ 𝑓 𝐴̅ 𝑓 + (∆𝑇) (38)
Implicit radiation theory is used to derive an expression for
shape factor, 𝐹̅, for a gray body (radiative heat passed between
heat sink channels), based on the assumptions made in the
theory section, is expressed as the total view factor;
𝐹̅𝑠−𝑠𝑢𝑟𝑟 = 1 −
2𝐻[(1+𝐿2)0.5−1]
2𝐿𝐻+(1+𝐿2)0.5−1
(39)
Radiative heat transfer from a single channel is found using
equation (40).
Qchannel =
𝜎𝐿(𝑆+2𝐻)(𝑇𝑠
4−𝑇𝑠𝑢𝑟𝑟
4 )
(
1−𝜀 𝑒𝑚𝑚
𝜀 𝑒𝑚𝑚
⁄ )+(1
𝐹𝑠−𝑠𝑢𝑟𝑟
⁄ )
(40)
Finally, the radiation from the plan, base and end view for
all appropriate fin surfaces is calculated using equation (41);
Qpbe = 𝜂 𝑓 𝑡(𝐿 + 2𝐿𝐻) + 2𝐿𝐻 + 𝑡 𝑏𝑎𝑠𝑒(𝑊 + 𝐿) (41)

12. ME6062 - Advanced Computational Fluid Dynamics - Spring 2014-2015 12
APPENDIX E: RESULTS
Table IV: Data for Unshielded variation
(a) (b)
Fig. 9: Temperature contour plot (a) and velocity plot (b) for
unshielded case at 5° C above ambient
NOMENCLATURE
𝐿 𝑐 Length of cross cut 𝑚
𝐿 Heat sink length 𝑚
𝑡 Fin thickness 𝑚
𝑊 Heat sink width 𝑚
𝐻 Fin height 𝑚
𝐴 𝑓 Area of a fin 𝑚
𝐴 𝑏 Area of base plate 𝑚
𝑡 𝑏𝑎𝑠𝑒 Base thickness 𝑚
𝑠 Channel spacing 𝑚
𝛽 Co-efficient of thermal expansion 𝑘−1
𝑔 Gravity 9.81 𝑚/𝑠2
𝜎 Boltzmann constant 1.4 × 10−23
𝑚2
𝑘𝑔𝑠−2
𝐾−1
𝜀𝑡 Dissipation rate 𝑚2
/𝑠2
µ Dynamic viscosity 𝑤/𝑚. 𝑘
𝑣𝑡 Turbulent eddy viscosity 𝑤/𝑚. 𝑘
₭ Thermal conductivity of air 𝑤/𝑚. 𝑘
𝑄̇ Mass flow rate 𝑘𝑔/𝑠
∆𝑃 Pressure drop 𝑃𝑎
𝐾 Mean kinetic energy 𝐽
𝑘 Turbulent kinetic energy 𝐽
𝑢, 𝑣, 𝑤 Local velocity 𝑚/𝑠
𝑢̅, 𝑣̅, 𝑤̅ Local mean velocity 𝑚/𝑠
𝑢′
, 𝑣′
Fluctuating velocity component 𝑚/𝑠
𝑄𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 Radiative heat transfer 𝑊
𝑄 𝑝𝑏𝑒 Heat transfer at different planes 𝑊
𝑄𝑐ℎ𝑎𝑛𝑛𝑒𝑙 Heat transfer from a channel 𝑊
𝑄𝑐𝑜𝑛𝑣 Convective heat transfer 𝑊
𝑄 𝑓𝑖𝑛 Heat transfer from a fin 𝑊
𝛥𝑇 Change in temperature °𝐶
𝑇𝑠𝑢𝑟𝑟 Ambient air temperature °𝐶
𝑇𝑠 Surface temperature °𝐶
𝜂 𝑓 Fin efficiency %
𝑁𝑓 Number of fins -
𝜀 𝑒𝑚𝑚 Emmisivity -
𝐶 Constant -
𝑃𝑟 Prandtl number -
ℎ̅ Heat transfer co-efficient -
𝑁𝑢 𝑠 Nusselt Number -
𝑅𝑎 𝑠 Rayleigh number for fin spacing -
𝐸𝑙 𝑠 Elenbaas number for fin spacing -
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Parameter 5˚C Above 10˚C Above 15˚C Above 20˚C Above 25˚C Above 35˚C Above 45˚C Above 55˚C Above 65˚C Above
Ambient temp (˚C) 19.2 18 20 19.2 19.3 19.4 19.4 20.3 20.5
Heat sink temp (˚C) 24.2 28 35 39.2 44.3 54.3 64.2 75.1 85.2
Delta T (˚C) 5.1 10 15 20 25 34.9 44.8 54.8 64.7
Film temp (˚C) 21.7 23 27.5 29.2 31.8 36.85 41.8 47.7 52.85
kinematic Viscosity (m^2/s) 1.553E-05 1.5523E-05 1.59478E-05 1.596E-05 1.6205E-05 1.6682E-05 1.7154E-05 1.84171E-05
Density of air (kg/m^2) 1.1981 1.1929 1.175 1.168 1.158 1.139 1.1211 1.0765 1.0449
Thermal conductivity 0.025727 0.025902 0.026155 0.026282 0.026474 0.026845 0.027207 0.028146 0.02885
Area of base (m) 0.2112579 0.21125794 0.211257943 0.2112579 0.21125794 0.21125794 0.21125794 0.211257943 0.21125794
Convection (W) sim 2.09E+00 6.84E+00 1.10E+01 1.77E+01 2.38E+01 3.67E+01 4.56E+01 5.94E+01 7.39E+01
Convection (W) with rad 2.21E+00 n/a n/a n/a n/a n/a n/a n/a n/a
Convection (W) theory 1.7325369 4.17829043 6.561466065 13.268479 25.21517 41.8930827 50.487645 5.39E+01 53.6974757
Convection (W) theory w/out rad 1.8562447 n/a n/a n/a n/a n/a n/a n/a n/a
Elenbaas 29.390891 55.9458549 80.56855512 108.56432 131.648654 167.693608 203.591449 216.0403998 243.993665
h 1.9398273 3.23883728 3.462481258 4.1986516 4.51310085 4.97650844 4.82238209 5.132781578 5.40445554
Nussselt expt. 0.8671051 1.43798273 1.522406212 1.8371697 1.96043891 2.13186243 2.0383502 2.097171468 2.15428904
Nusselt Theory 0.8976665 1.32972479 1.557493667 1.74793 1.86437145 2.01771763 2.14096356 2.178972303 2.25749531
Radiation sim 1.22E-01 n/a n/a n/a n/a n/a n/a n/a n/a
Radiation theory 0.1237078 n/a n/a n/a n/a n/a n/a n/a n/a
%rad theory 15.005072 n/a n/a n/a n/a n/a n/a n/a n/a
%rad sim 1.81E+01 n/a n/a n/a n/a n/a n/a n/a n/a