AN ANALYSIS OF POROUS MEDIA HEAT SINKS FOR NATURAL CONVECTION
COOLED MICROELECTRONIC SYSTEMS.
by
Eric R. Savery
Engineering Project submitted in partial fulfillment of
the requirements for the degree of
Master of Engineering
In Mechanical Engineering
Approved by ___________________________________________________
Ernesto Gutierrez-Miravete, Engineering Project Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2011
i
Table of Contents:
List of Tables .................................................................................................................... iii
List of Figures ................................................................................................................... iv
Nomenclature .................................................................................................................... v
Acknowledgement ............................................................................................................ vi
Abstract ............................................................................................................................ vii
1. Introduction ............................................................................................................... 1
3. Methodology/Approach............................................................................................. 5
3.1. Assumptions ........................................................................................................ 5
3.2. Theory ................................................................................................................. 6
3.3. Finite Element Modeling ................................................................................. 10
4. Results and Discussion ............................................................................................ 14
5. Conclusion ................................................................................................................ 26
6. Appendix 1................................................................................................................ 27
7. Appendix 2................................................................................................................ 28
8. Appendix 3................................................................................................................ 29
9. Appendix 4................................................................................................................ 30
10. Reference .............................................................................................................. 31
ii
List of Tables
Table 1: Porous Heat Sink Characteristics........................................................................ 11
Table 2: Block Form ........................................................................................................ 12
Table 3: Control Block Heat Sink Results ....................................................................... 14
Table 4: Porous Heat Sink Results................................................................................... 18
Table 5: Number of Element for Element Pre-set ........................................................... 25
iii
List of Figures
Figure 1: Fin Heat Sink ....................................................................................................... 1
Figure 2: Schematic Representation of Microelectronic Heat Sink.................................... 2
Figure 3: No Heat Sink ...................................................................................................... 3
Figure 4: Solid Block Heat Sink ......................................................................................... 3
Figure 5: Porous Heat Sink ................................................................................................ 4
Figure 6: Schematic Representation of system .................................................................. 5
Figure 7: Temperature vs. Density..................................................................................... 7
Figure 8: Different Porosity, Same Permeability ............................................................... 8
Figure 9: Same Porosity, Different Permeability............................................................... 8
Figure 10: No Heat Sink Mesh ........................................................................................ 12
Figure 11: Solid Block Mesh ........................................................................................... 13
Figure 12: Porous Block Mesh ........................................................................................ 13
Figure 13: No Heat Sink T-V (Isotherms and Velocity Vectors ..................................... 15
Figure 14: Solid Heat Sink T-V (Isotherms and Velocity Vectors)................................. 15
Figure 15: Porous Heat Sink Isotherms and Velocity Field ............................................ 16
Figure 16: Porous Heat Sink Streamlines and Velocity Magnitude ................................ 17
Figure 17: Effect of Porosity Effect on Heat Removal .................................................... 19
Figure 18: Permeability Effect on Heat Transfer ............................................................. 20
Figure 19: Permeability Effect on Volumetric Flow Rate ............................................... 21
Figure 20: Effect of Permeability on Total Heat Flux ...................................................... 22
Figure 21: Effect of Permeability on Volume Flow Rate ................................................ 22
Figure 22: Effect of Mesh on Total Flux .......................................................................... 24
Figure 23: Effect of Mesh on Volume Flow Rate............................................................ 24
iv
Nomenclature
Symbols
Cp
=
K
=
ρ
=
ε
=
ϕ
=
P
=
T
=
k
=
u
=
g
=
t
=
v
=
μ
=
J/kg K
m^2
g/m^3
W/m^2
N/m^2
K
W/(m*K)
m/s
m/s^2
sec
m/s
N*s/m^2
Subscripts
Air
=
Al
=
c
=
eq
=
0
=
p
=
f
=
Air
Aluminum Heat sink
Initial/Cold Temperature
Equivalent
Initial
Porous Base Material
Cooling Fluid
=
=
=
=
=
=
=
=
=
=
=
=
=
Heat Capacity at Constant Pressure
Permeability
Density
Porosity
Heat Flux
Pressure
Temperature
Thermal Conductivity
Fluid Velocity
Gravity
Time
Velocity Field
Fluid Viscosity
v
Acknowledgement
vi
Abstract
This work investigates what effects the use of metallic porous materials has on
heat transfer when it is used as a heat sink numerically using the finite element model.
The study is motivated by the problem of cooling microelectronic components.
Specifically a comparison of how differences in the porous material characteristics
affect the performance of a heat sink is analyzed. In addition a comparison of the
porous media results to a traditional block heat sink and a system with no heat sink at
all is completed. The heat sinks used for the porous and non-porous heat sinks have
the same volume mass of material, material and dimensional footprint. Where the
heat sink attaches to the outer boundary a constant temperature is applied. By
varying the heat sinks porosity and permeability the total heat flux out of the chip is
calculated. The results of the all case are compared to quantify the overall effect of
porous material characteristics has on heat dissipation.
vii
1. Introduction
Since the discovery of the microprocessor computing power is becoming more
powerful and consequently the chips have been creating more heat. The heat the
most powerful processors create must be dissipated; else the processor will become
less reliable or fail1. For every 10ºC reduction in temperature, the failure rate of the
electronic component is halved1. Therefore the more heat removed from the
transistor, the greater the reliability. To transfer the waste heat from the transistor to
the environment requires the use of a heat sink. A heat sink is a device which
dissipates energy from a component to the ambient environment by use of natural or
forced convection. The heat sink’s ability to dissipate the thermal energy is a
function of the material properties, geometry and the environmental conditions2.
Increasing the amount of thermal energy dissipated by the heat sink allows for higher
processing speeds. Below is an example of a simple fin heat sink used on a typical
integrated circuit.
Figure 1: Fin Heat Sink
An alternative to the standard non-porous heat sinks which are used today is the use
of semi-porous material. The use of porous material as a heat sink base material
allows for the cooling fluid to flow through the heat sink similar to a fin heat sink
with and infinite number of fins. This report describes a study designed to show how
the use of semi-porous media with a significant amount of surface area affects the
efficiency of a heat sink to dissipate heat energy from a computer chip. The
COMSOL Multiphysics finite element modeling program is used to compare the
efficiency of a heat sink that uses a semi-porous material with that of a block heat
sink and also with the absence of a heat sink.
1
2. Problem Description
The design of a heat sink is dependent on the physical space, cooling needs and cost
of the component. In a semiconductor application increasing capabilities results in the
need for additional cooling. The increased cooling requirements can result in larger heat
sinks, increasing the overall size and cost of the final component. For portable
electronics the goal is to provide the most capability, in the smallest package for the
lowest cost. Ideally cooling should occur by natural convection alone and this is the
situation considered in this study. Figure 2 is a schematic representation of the system
investigated. The chip is not included in the figure but it thermal effect is represented by
fixing the temperature at the base of the heat sink. The Figure then shows the heat sink
surrounded by free space occupied by air.
Open Boundary
Insulated Wall
Heat Sink
Open
Space
(air)
Insulated Wall
Chip
Insulated Wall
Open Boundary at Tc
Figure 2: Schematic Representation of Microelectronic Heat Sink
During normal operation the high temperature at the base of the heat sink will produce
heat flow into the surrounding air. As the temperature increases, natural convection will
set in.
2
Current methods of transferring the heat from a component to the environment are with a
heat sink and without them. Below Figure 3 and Figure 4 show the current methods of
remove heat from a system.
Qout
Qin
Qout
Qin
Qout
Qout
u @ Tc
u @ Tc
Figure 3: No Heat Sink
Figure 4: Solid Block Heat Sink
Figure 3 represents a system in which has no heat sink to transfer the heat from the heat
source to the atmosphere. Figure 4 on the other hand has a large block heat sink that
transfer the heat from the heat source to the outer walls of the heat sink and then to the
atmosphere. In both cases the sum of the energy in (Qin) and the sum of the energy out
(Qout) equal each other. If the fluid flowing through the system is not pushed by a pump
or a fan then a natural circulation of the fluid will move the fluid away from the heat
source. In Figure 3 it is apparent that the amount of heat transferred out of the system is
based on the surface area of the heat source and the velocity of the fluid. Alternatively in
Figure 4 the amount of heat that is able to be dissipated is based on the outer surface of
the heat sink and the fluid velocity. Since the heat sink provides a larger area for heat to
transfer to the atmosphere it is accurate to assume that the system with the heat sink will
transfer more heat. Therefore a correlation between an increase in heat sink total surface
area and an increase in heat transfer can made. A way to increase this surface area is to
cut slots or holes into the block. If an infinite number of holes were to be cut the material
would become porous. Figure 4 below provides a general diagram of the porous case.
3
Qout
Qin
Qout
Qout
u @ Tc
Figure 5: Porous Heat Sink
In the porous case the fluid would flow through the heat sink transferring heat to the
cooling fluid as it goes through. Since the interior of the porous material is filled with
voids there is an increase in the surface area and therefore an increase in the total heat
flux3/4. On the other hand too many holes in the porous system would result in the
system becoming similar to the system with no heat sink. In that case the lack of the
porous base material will cause a decrease in heat flux.
Therefore it is desired to determine the optimum characteristics of the porous material
which provide the best heat transfer. To determine the most efficient porous material
heat sink a comparison of the porous material characteristics is required in addition to
comparing the porous material heat sink to the control case i.e. the system with a block
heat sink and the system with no heat sink.
4
3. Methodology/Approach
A systematic approach is used to model and analyze three different heat transfer
systems, one system without a heat sink and two systems with heat sinks. The two
systems with heat sinks will be made with porous and non-porous material. The heat
sinks are made of the same base material and the cooling fluid is the same (air) in all the
systems. Each heat sink is made from the same base material to ensure that the
differences in the results are not due to material thermal properties but due to the porous
characteristics. The solid block heat sink and no heat sink are control cases while the
various porous block heat sinks are subjected to differing porous material characteristics
investigate their effects on the heat sink’s ability transfer heat.
3.1. Assumptions
Figure 6 shows a schematic representation of the system considered in this study.
Open Boundary
Insulated Wall
Air
Heat Sink
Insulated Wall
Open Boundary at Tc
Figure 6: Schematic Representation of system
The following assumptions were used:

2D system

Steady State Conditions

Bounded area is 50mm by 50mm.
5
Insulated Wall
Wall with
Temperature
of Th

Chip Size 10mm

Heat sink size 10mm x 10mm

Air inlet temperature is Tc = 293.15 K.

Chip wall has a constant temperature Th = 310

Laminar incompressible Newtonian flow
3.2. Theory
Heat Transfer: There are three different modes of heat transfer conduction, convection
and radiation. In the case of the model without a heat sink the heat transfer of interest is
pure convection. In addition to convection for the porous and non porous material cases
conduction heat transfer is also modeled. The system without a heat sink results in a pure
convection process where heat from the wall transfers directly into the fluid without an
intermediate step. When a heat sink is used, an intermediate step is present where the
energy from the wall must pass through the block conduction and then into the fluid via
convection. Equation 1 is the governing equation for conduction and convection in nonporous media.
T
 kT   uT  
t
[1]
One might mistakenly think that adding an additional heat transfer step might decrease
the efficiency of removing heat from a chip or heat source but the exact opposite true.
The advantage of the intermediate step is that the heat sink increases the surface area in
which convective heat transfer is able to occur. Increasing the surface area in which
convection can occur can have a substantial increase in the rate in which heat can be
transferred from a heat source. This is the principle behind the widespread use of fins to
assist energy dissipation in thermal engineering.
Natural Convection: In cases where there is not a fan or another motive force to
circulate coolant, a natural force known as buoyancy causes the fluid to move. The
ability of natural convection is a function of the buoyancy of the cooling fluid. This fluid
movement is consequence of the density gradient caused by the differing temperatures
6
between the incoming air and the air which has been heated. As air is heated its density
decreases causing an upward force opposite to that of gravity, this is the buoyancy force.
The force then causes an upward velocity of the cooling fluid. For modeling this
phenomenon the following equation is used to determine the volume force.
Fb   g (   0 )
[2]
Where  0 is a reference value of the fluid density. Equation 2 shows that as the density of
a point decreases below the initial density an upward force is created. Since density is
directly proportional to the temperature of the cooling fluid at a particular point, the force
will also vary from point to point. Figure 7 below show how density of air changes as a
function of air5.
Figure 7: Temperature vs. Density
The force that is caused by the buoyancy force is then introduced into the momentum
balance equation. Equation 3 below is the general equation governing the flow of the air.
As the buoyancy force F increases the mass flow rate increases and therefore the
momentum also increases.
 

 (  )   Fb  p   2
 t

0 
7
[3]
Porous Material Characteristics: There are two characteristics of significant
importance when analyzing the porous heat sinks. Those characteristics are the materials
porosity and permeability. Permeability (κ) is the measure of the ability of a material to
transmit fluid though itself. Porosity (ε) is also known as void fraction, is the measure of
voids or spaces inside a material. This is defined in terms of a fraction of the total
volume of the material. An increase in porosity means the heat sink is has more void
space than the solid material. Figure 8 and 9 helps explain the nature of and the
difference between porosity and permeability.
Figure 8: Different Porosity, Same Permeability
Figure 8 shows the porosity of the block on the left is 4 times the porosity of the block on
the right. However in the figure on the left the three shaded holes are blocked and do not
allow through flow of fluid. The permeability of both blocks will be approximately the
same because only one hole passes all the way through the blocks. Assuming that each
hole equates to a tenth of the total area of the block, the block on the left has a porosity of
.4 while the block on the right has a porosity of .1 but both blocks would have the same
permeability.
Figure 9: Same Porosity, Different Permeability
Figure 9 shows how two blocks could have the same porosity but completely different
permeabilities. The block on the left has one small hole passing through the entire way,
8
while the block on the right has a large hole passing all the way through. Therefore the
mass flow rate of fluid through the block on the right would be greater than that through
the block on the left. This would mean the permeability of the block on the right is
greater than that of the block on the left.
Heat Transfer in Porous Media: Heat transfer through a porous material shows
similarities with heat transfer through a solid block and also with heat flow in a fluid
medium. Equation 4 governs heat transfer in the porous block heat sink:
Cp eq T
t
 CpuT  keq T   
5
[4]
The difference between the simple heat equation and the heat equation for porous
material is associated with the corresponding effect of the volumetric heat capacity and
thermal conductivity. In the case of the porous material both volumetric heat capacity
and thermal conductivity are functions of the material porosity. As the material becomes
more porous both the thermal conductivity and the volumetric heat capacity become more
like that of the cooling fluid. Alternatively, as the porosity nears zero the conductivity
and volumetric heat capacity become more like those of the base material. Equation 5a,b
below are used to determine the equivalent thermal conductivity and volumetric heat
capacity.
keq   p k p   F k F
 * Cp eq   p  pCp p   F  F Cp F
[5]
 p F  1
Equation 5c shows that the fractions of porous base material and fraction of cooling fluid
must add up to 1.
Free and Porous Media Flow: Since in a porous material the cooling fluid is able to
move through the pores of the heat sink, a way to model the flow of liquid through it is
required. For a pure porous system flow the Darcy-Brinkman equation is the accepted
9
method of approximating the flow through the porous media. Since the system modeled
has both free and porous flow the interactions between the porous media and the free
flow region must be taken into account. For this reason a modified Darcy-Brinkman
equation is need. The governing Stokes equation for the fluid flow in porous media is
provide in equation 6 and described in Reference 6.


 
 
u 

2
T


u *       pl 
u  u  
 * u t      F u  Qbr u  F


[6]
 p 
 p 
p
3 p

  k



It should be noted that equation 6 is affected by both permeability and porosity of the
heat sink material. This equation is used to model all cases where porous heat sinks are
used.
3.3. Finite Element Modeling
Using COMSOL Multiphysics, a heat sink is modeled to be 10 mm by 10 mm in a 50mm
by 50 mm system. The heat sink sits on a wall that is heated at the base to 310K. This
wall mimics a computer chip in that it provides a heat flux that must be dissipated. The
chip is designed to run at a maximum temperature of 310K so depending on the
efficiency of the heat removal mechanism the speed of the chip will be determined. The
higher the value of the dissipated heat flux the faster the chip is able to operate. As
mentioned before the heat sink will consist of two forms; a porous block in one case and
a solid non-porous block in the other. A third condition will be modeled with no heat
sink at all. The porous heat sinks will consist of different characteristics; 10 different
porosities and 3 different permeabilities for a total of 30 different porous heat sink
options. In addition one solid block heat sink and one system without a heat sink will be
analyzed for a total of 32 different cases all of which are provided below in Table 1
(Porous Heat Sink) and Table 2 (Block Heat Sink).
The characteristics chosen for the porous heat sinks are those that most easily show the
effect of permeability and porosity on heat transfer. Some of the porous material
characteristics analyzed may not be readily producible in the real world but they provide
a useful understanding of the effect that porous materials has when used as a heat sink.
10
Each heat sink is made of the same base material and each system utilizes the same
cooling fluid (air). All the models are subjected to the same wall temperature of 310K
representing the chip. The models are analyzed to determine the total amount of thermal
energy dissipated for the prescribed wall temperature. In addition plots of the flow path
and temperature gradients are produced. The resulting data is then used to determine
which heat sink has the potential of transferring the most heat.
Table 1: Porous Heat Sink Characteristics
Case Number
Porosity
1a
1b
1c
1d
1e
1f
1g
1h
1i
1j
2a
2b
2c
2d
2e
2f
2g
2h
2i
2j
3a
3b
3c
3d
3e
3f
3g
3h
3i
3j
Permeability
0.05
0.1544
0.2589
0.3633
0.4678
0.5722
0.6767
0.7811
0.8856
0.99
0.05
0.1544
0.2589
0.3633
0.4678
0.5722
0.6767
0.7811
0.8856
0.99
0.05
0.1544
0.2589
0.3633
0.4678
0.5722
0.6767
0.7811
0.8856
0.99
11
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
Table 2: Block Form
Form
Depth (mm)
4-a
0
4-b
10
Mesh control can have a significantly effect on the results of the analysis. To ensure that
meshing is controlled in the same manner between all of the models the same mesh
setting are used. For all three systems the meshing is physics-controlled with normal
sized elements. This resulted in models with between 300 and 1200 total elements. This
set up allows COMSOL to optimize the number elements, the mesh size and the mesh
geometry. This optimization is based on the geometry of the model and the physics
involved. Figures 10, 11 and 12 below are the meshes that COMSOL has determined to
be he most effective based on the system geometry.
Figure 10: No Heat Sink Mesh
12
Figure 11: Solid Block Mesh
Figure 12: Porous Block Mesh
Detailed information about all the models including the material properties used can be
found in Appendix 4.
13
4. Results and Discussion
Control Cases (No Heat Sink and Solid Block Heat Sink): The first analysis that was
completed was the comparison of the use of no heat sink and the use of a solid block heat
sink. (Table 3) Results below provide the important data from the model analysis. Table
3 provides the corresponding heat removal rates and cooling fluid volumetric flow rates
(both per unit width) associated with a non-porous block and the model with no heat sink.
The heat removal rate (per unit width) was determined by integrating the heat flow across
the heated wall (chip). Similarly the volumetric flow rate (per unit width) was
determined by integrating the velocity over the inlet.
Table 3: Control Block Heat Sink Results
Trial
Depth (mm)
No Heat Sink
Solid Heat Sink
0
10
Heat Removal
Rate (W/m)
1.513
2.9278
Volumetric Flow
Rate (m^2/sec)
0.0012
0.001
As expected, the heat removal rate out of the chip with the solid heat sink is larger than
the case without the heat sink. This is due to the fact that there is a greater amount of
surface area for heat to transfer to occur between the source of heat (chip) and the cooling
fluid. Figure 14 shows how the block heat sink has a large area of material at the 310K
temperature. Alternatively Figure 13 shows that the boundary wall is the only location
where the temperature is 310K. If the block was to have a smaller width dimension the
heat removal rate would be less than that of the 11mm block but more than the system
without a heat sink.
The associated volumetric flow rate of the model without the heat sink is greater than that
of the model with the block heat sink. The mass flow rate is a function of the buoyancy
force and the drag caused by the heat sink. Increasing the heat flux results in a greater
buoyancy force therefore increase the flow rate. Alternatively the larger the block is the
greater the drag force acting on the fluid becomes. As can be seen in Figure 14 the flow
path for the model with the block heat sink requires the fluid to flow around the heat sink
while Figure 13 shows how with no heat sink flow proceeds right across the chip without
any disruptions in the flow.
14
Figure 13: No Heat Sink T-V (Isotherms and Velocity Vectors)
Figure 14: Solid Heat Sink T-V (Isotherms and Velocity Vectors)
15
Porous Heat Sink: Figures 15, 16 show typical results obtained for a porous heat sink.
Figure 15 shows isotherms and velocity field while Figure 16 shows pressure contours.
Figure 15: Porous Heat Sink Isotherms and Velocity Field
16
Figure 16: Porous Heat Sink Streamlines and Velocity Magnitude
The second analysis compared the porous heat sinks with differing permeabilities and
porosities. Table 4 below provides the results from the model analysis of porous material
heat sinks. Specific data of interest is the total heat removal rate and volumetric flow rate
of the systems.
For ease of analysis the effect of porosity on the heat sinks performance will be first
compared independent of permeability. Figure 17 shows the effect of porosity on the
computed heat flux at various permeabilities. The increase in heat flow with increasing
permeability is clearly shown.
17
Table 4: Porous Heat Sink Results
Form
1a
1b
1c
1d
1e
1f
1g
1h
1i
1j
2a
2b
2c
2d
2e
2f
2g
2h
2i
2j
3a
3b
3c
3d
3e
3f
3g
3h
3i
3j
Porosity
Permeability
0.05
0.1544
0.2589
0.3633
0.4678
0.5722
0.6767
0.7811
0.8856
0.99
0.05
0.1544
0.2589
0.3633
0.4678
0.5722
0.6767
0.7811
0.8856
0.99
0.05
0.1544
0.2589
0.3633
0.4678
0.5722
0.6767
0.7811
0.8856
0.99
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
1.00E-08
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
5.05E-07
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
1.00E-06
18
Heat Removal
Volumetric Flow
Rate (W/m)
Rate (m^2/s)
3.1391
3.1443
3.1456
3.1452
3.1443
3.1429
3.1406
3.1361
3.1233
2.8752
7.6455
11.5721
10.6431
10.04
9.7644
9.6461
9.585
9.5285
9.4072
7.5302
7.9756
14.0391
14.9183
15.1979
15.1892
15.0854
14.9662
14.8254
14.543
10.7887
0.0011
0.0011
0.0011
0.0011
0.0011
0.0011
0.0011
0.0011
0.0011
0.001
0.002
0.0027
0.0025
0.0024
0.0024
0.0024
0.0024
0.0024
0.0023
0.0021
0.002
0.0031
0.0033
0.0033
0.0033
0.0033
0.0033
0.0033
0.0033
0.0028
Porosity Effect on Heat Transfer
16
14
12
Heat Removal Rate (W/m)
10
kappa = 1e-8
8
kappa = 5.05e-7
kappa =1e-6
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
Porosity (epsilon)
Figure 17: Effect of Porosity Effect on Heat Removal
At the extremes of porosity (ie 0-.1 and .9-1) a detrimental effect on the performance of
the heat sink is seen. Appendix 1 shows why the extreme cases produce a decrease in the
sink’s ability to remove heat. In those cases where the porosity is small, the flow path of
the coolant is similar to that of the solid block heat sink case where fluid does not flow
through the heat sink. This is also similar to the case where the heat sink has a low
permeability. As flow through the heat sink diminishes the heat sink acts as a boundary
which doesn’t allow convection to occur inside the wall boundaries.
On the other hand looking at the temperature plots in Appendix 1 it is apparent that when
the porosity increases above 0.9 the propagation of heat through the heat sink is not as
uniform as compared with materials with less porosity. From the governing equation for
heat conduction through a porous media it is apparent that once porosity becomes too
large the equivalent conductivity becomes more like the cooling fluid than the base
material. This causes heat not to propagate though the heat sink as readily and it begins
to approach the “no heat sink” case. This reduction in heat propagation results in a
decrease in the overall heat being dissipated.
19
The second characteristic of importance for heat transfer in a porous material is the
permeability. Figure 18shows the effect of permeability on the heat flux for various
porosity values.
Permeability Effect on Heat Transfer
16
14
12
Heat Removal Rate W/m
W/m(W/m)
10
Porosity 0.6
Porosity 0.795
8
Porosity 0.99
Wall
Block
6
4
2
0
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
1.00E-06
1.20E-06
Permeability (1/m^2)
Figure 18: Permeability Effect on Heat Transfer
The increase heat flux is due to the ability of the fluid to flow through the heat sink and to
extract heat from the surfaces of pores. This differs from decreasing porosity in that the
optimum flow pattern is able to be achieved without changing the thermal conductivity of
the heat sink. That flow pattern contributes to the volumetric flow rate being so high. By
looking at Appendix 1 plots of flow it is apparent that increasing the permeability causes
the flow to be similar to that of the control case with no heat sink at all.
Figure 19 shows the effect of permeability and porosity on the volumetric flow rate. As
expected higher permeability results in higher coolant flow rate.
20
Permeability Affect on Volumetric Flow Rate
0.0035
Volumetric Flow Raty (m^s/sec)
0.003
0.0025
Porosity 0.6
0.002
Porosity 0.99
Porosity 0.795
Block
0.0015
No Heatsink
0.001
0.0005
0
1.00E-08
5.05E-07
1.00E-06
Permeability (1/m^2)
Figure 19: Permeability Effect on Volumetric Flow Rate
The greater the permeability less flow is diverted around the heat sink. The less diverting
of flow results in less drag and an increase in the volumetric flow rate. For the same
amount of buoyancy force a system with no drag due to the high permeability in the
block will have more flow rate than one with a block with low permeability. The reason
for the increase of volumetric flow rate due to the increase in permeability can be clearly
seen in Appendix 1 flow rate plots.
On the other hand a decrease in porosity decreases the volumetric flow rate. Since a low
porosity has a negative effect on heat flux there will be also a negative effect on
volumetric flow. This is due to the fact that flow is a function of buoyancy force and
buoyancy force is a function of heat flux. Since the heat flux associated with the lower
permeabilities is lower the volumetric flow rate is also lower.
It should also be noted that there is a diminishing effect on the heat sink’s ability to
transfer heat. Once the flow is able to pass through the heat sink with out being disturbed
21
increasing permeability has little effect. Figure 20 and Figure 21 below shows how once
permeability gets above a certain point, further increase does not have much of an effect.
At the permeability value where velocity and heat flux stop increasing the flow path is
similar to that of the control system with no heat sink. This is due to the fact that once
flow of the coolant become unobstructed there is no further increase in its velocity. Since
Heat Removal Rate (W/m)
velocity does not increase, heat transfer will not increase either.
Figure 20: Effect of Permeability on Total Heat Flux
Figure 21: Effect of Permeability on Volume Flow Rate
22
Comparison of Porous and Control Conditions:
From the analysis above it is evident that the porous heat sinks have a district advantage
over the non-porous heat sink in regards to heat dissipation capability. This was expected
due to the ability of the cooling fluid to flow through the heat sink in addition to the
increased surface area between the cooling fluid and the heat sink. The typical flow
through the system without a heat sink can be seen in Figure 13 (see also Appendix 3)
which shows the fluid flowing straight through the system heating up as it passes the
chip. Figure 14 (see also Appendix 2) shows the fluid flow of the block heat sink and
how the fluid rises from the inlet with cool air continues to flow around the heat sink
while the fluid temperature rises until it exist from the top of the system. Alternatively
Figures 15, 16 (see also Appendix 1) show how the cooling air flows through the porous
heat sink.
The increase in volumetric flow rate with porosity is due to two reasons. First the drag
produced by the porous material versus the solid block is less. Since the air can pass
through the block instead of around it the fluid can flow with less pressure drop.
Secondly the increased buoyancy force due to a greater heat flux results in an increase in
velocity and therefore an increase in the volumetric flow rate.
The results of the modeling are comparable with research which has been done in the
field already. In particular other research has shown that increasing permeability while
limiting porosity has improved the heat transfer7. The conclusion is that increasing
permeability while keeping porosity between .1 and .9 improves the heat sink efficiency.
System Meshing:
As described in prior sections it was determined to use a normal mesh setting for analysis
of the systems. This helped to ensure that the models were able to achieve a result in a
reasonable amount of time while obtaining accurate results. To ensure that the results are
of significance Figures 22 and 23 below compare the results of a model with a coarser
mesh, a normal mesh, a finer mesh and an extremely fine mesh.
23
Mesh Affect on Total Energy Flux
Meshing
Effect on Total
Energy Flux
12
10
8
Heat Removal Rate (W/m))
6
4
2
0
Coarser
Normal
Finer
Extremely Fine
Mesh
Mesh Affect on Total Energy Flux
Figure 22: Effect of Mesh on Heat Removal Rate
Affect of Mesh On Volume Flow Rate
0.0026
Volume Flow Rate (m^2/sec)
0.0025
0.0024
0.0023
0.0022
0.0021
0.002
0.0019
Coarser
Normal
Finer
Mesh
Epsilon .9 Kappa 5.05e-7
Figure 23: Effect of Mesh on Volumetric Flow Rate
24
Extremely Fine
The results of normal mesh setting and the extremely fine setting are within 10% of each
other. In addition the graphs show that the results begin to flatten out right past the
normal mesh criteria. Therefore the results determined in this analysis are considered to
be of numerically significant. Table 5 shows that an increase in accuracy of 10% would
require 20 times as many elements. It is thus determined that the results obtained using a
mesh setting of normal are numerically significant
Table 5: Number of Element for Element Pre-set
Element Pre-set
Number of Elements
Coarser
416
Normal
1170
Finer
4776
Extremely Fine
21434
25
5. Conclusion
The comparison between the use of porous and non-porous heat sinks confirmed the
hypothesis that there is a substantial benefit of using porous material in lieu of nonporous material for heat sinks. The benefit between porous and non-porous material heat
sinks is only relevant to the initial efficiency of the heat sink. Over time fouling and
material degradation may have a greater effect on the porous material versus the nonporous heat sink. In addition neither a cost comparison nor material availability research
was completed which may diminish the porous material overall advantage.
To compare the advantages of porous material versus non-porous one, an analysis of the
effects of permeability and porosity characteristics was performed. Unlike the simple
comparison between porous and non-porous material, the evaluation of the porous
material characteristics was a bit less intuitive. The first characteristic analyzed was
permeability, which is the ease with which fluid can pass through the heat sink. It was
determined that the more permeability the porous media has the more efficient a heat sink
becomes. Although the increase of permeability helped initially, once there was little
resistance for the cooling fluid to pass through the heat sink there was little advantage in
increasing the permeability any further. At that point increasing and decreasing porosity
had a greater effect.
The second material characteristic that was analyzed was the effect of porosity on the
performance of the porous heat sink. It was determined that if the porosity was too high
or too low the performance of the heat sink was adversely affected. Performance of the
heat sink diminished with increase in porosity due to the fact that the equivalent thermal
conductivity of the heat sink diminishes and approaches the thermal conductivity of the
cooling fluid. On the other hand, decreasing the porosity causes surface area for heat
transfer to diminish. Once porosity is diminished enough the flow path of coolant
becomes similar to that around a solid block. Overall it has been determined a heat sink
with high permeability and porosity not near the extremes of the porosity range would
perform best.
26
6. Appendix 1
Data from Porous Material Analysis
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27
7. Appendix 2
Data from Block Analysis
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28
8. Appendix 3
Data from No Heat Sink Analysis
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29
9. Appendix 4
Modeling Characteristic and Heat Sink Properties
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30
10. Reference
1
Fundamentals of Thermal-Fluid Sciences; Second Edition, Yunus A. Cengel & Robert H. Turner, © 2005
Wikipedia (December 9, 2011) Heat Sink. Received From http://en.wikipedia.org/wiki/Heat_sink
3
An analytical study of local thermal equilibrium in porous heat sinks using fin theory. Tzer-Ming Jeng,
Sheng-Chung Tzeng, Ying-Huei Hung; © January 10, 2006
4
Medal Foam and Finned Metal Foam Heat Sinks for Electronic Cooling in Buoyancy-Induced
Convection. A. Bhattacharya & R.L Mahajan, © 2006
5
Porous Heat Transfer; Multiphysics Modeling, Finite Element Analysis, and Engineering Simulation
Software, © 1998-20011 COMSOL inc
6
Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy
solidification. M. Le Bars and M. GRAE Worster, © August 2005.
7
Sintered porous medium heat sink for cooling of high-power mini-devices. G Hetsroni, M Gurevich, R
Rozenblit. © 18 August 2005.
2
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