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HEAT SINK DESIGN PROPOSAL
ME131A
Lex Bayer
Eric Tribbett
Ed Froehlich
“The Boys Down the Hall”
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TABLE OF CONTENTS
1. MODELING
Figure 1: Schematic of testing setup.
1.1 LAB-MODEL
Figure 2: Experimental arrays for lab-model
1.2 COMPUTER-MODEL
Figure 3: Sample calculation using Computer-Model
Figure 4: Thermal Resistance versus number of fins for both fat and thin fins
2. HEAT PIPES
3. RADIATION EFFECTS
4. DESIGN
Figure 5: Plan view of heat sink
5. MANUFACTURE
Figure 6: Aerial 3D view of heat sink
6. A THEORETICAL MODEL OF FINAL DESIGN
Figure 7: Predicted thermal resistance for final design
7. FUTURE IMPROVEMENTS
8. FINAL RESULTS
9. BIBLIOGRAPHY
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1. MODELING
We had two separate approaches to modeling the heat sink—a real life lab-model,
and a computer generated model. The lab-model had the advantage of better
approximating the experimental set up we would be using for testing the heat sink. It
could capture the unique nature of the air flow (the computer model could not account for
angled and non-symmetric flows) as well as allow us to model certain array patterns
which were difficult to model with a computer. The computer model, however, had the
advantage of being able to better approximate the material constraints—the actual rod
diameters and variable fin lengths.
Figure 1 shows a schematic of the lab setup we were required to use to test our
heat sink.
Perspex Box
Circuit Board
1.25m/s
35
Figure 1: Schematic of testing setup. Average flow direction and velocity approximated using anemometer.
FAN
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The air flow over the heat sink was very turbulent and difficult to characterize. An
anemometer was inserted into the box from above to try and determine the nature of the
air flow. The anemometer would only function if it was placed parallel to the air flow.
The only orientation that gave a velocity reading was 1.25m/s at approximately 35 to the
horizontal, as show in Figure 1. This flow seems to make sense since the air is blown into
the box from the right hand side and needs to turn somewhat and exit through the hole
near the heat sink. The only other direction that allowed the anemometer to spin was a
vertical orientation when the anemometer was inserted through the side. This indicated a
downward flow of air onto the heat sink. Some elementary flow visualization was
attempted by inserting a wire into the box with a long, thin hair attached to the end. It
was apparent by the erratic hair motion that the flow was generally turbulent and
unpredictable. The fact that the flow was turbulent weighed heavily in our design
decisions (this will be discussed later).
1.1 LAB-MODEL
A real-life model was generated by using the pre-existing set up in the lab. A base
plate of aluminum had a pre-drilled array of holes into which one could place pins. There
were 4 types of pins:
a) D=3/32”, L=1.25” (Long thin pins)
b) D=3/32”, L=.625” (Short thin pins)
c) D=3/16”, L=1.25” (Long fat pins)
d) D=3/16”, L=.625” (Short fat pins)
Our material constraints where characterized as follows:
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Actual Material
Lab-model
3/16”
3/32”
3/8”
3/16”
15” lengths
15 long pins or 30 short pins
There were two main issues we wished to resolve with our lab-model:
1) Are short or long pins better?
2) How important is the order/orientation of the pins?
Figure 2 displays the various arrays that were tried:
A)
B) 15 x Long fat
Rt=1.72K/W
30 x Short fat
Rt=1.62 K/W
D) 15 x Long fat - centralized
Rt=1.81K/W
Figure 2: Experimental arrays for lab-model
F)
15 x Long fat, 15 x Long thin
Rt=1.36K/W
C) 15 x Long fat –different orientation
Rt=1.71K/W
E)
30 x Short fat, 30 x Short thin
Rt=1.40K/W
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Experiments A and B were an attempt to determine whether 15 long fins or 30
short fins were better. The two experiments yielded a difference in thermal resistance of
0.1K/W in favor of more short pins. However, a similar test was attempted with
experiment F (15 Long fat and 15 Long thin) and E (30 Short fat and 30 Short thin)
which yielded a difference of .04K/W in favor of fewer longer pins. Overall the
difference between using short and long fins was not significant, as can be seen in the
relatively small differences in thermal resistances. The arrays with more short fins had
greater contact resistance, yet still performed comparably to the arrays with fewer long
fins. This would imply that if contact resistances were significantly reduced, the array
with more short fins would perform better. The lab-model—with loose fitting pins—had
greater contact resistance than we expected to achieve in our final design. This would
indicate that using relatively short fins was the better choice for our final design.
Experiment B and C adjusted the orientation of the pins relative to the air flow. A
mere 0.1K/W difference in thermal resistance was detected, indicating that the orientation
was effectively inconsequential—probably a result of the turbulent nature of the flow.
Experiment D was an attempt at cheating the testing setup. The thermocouple was
set in the middle of the heater. The idea behind design D was that perhaps one could get
better results if the heat was conducted away from the center of the heat sink. The results,
however, indicated that the central location of pins had the opposite effect and increased
the thermal resistance. This may be a result of reduced air flow the denser array.
On the whole, lab-modeling experiments gave us only a general feel for some of
the design choices. The most important piece of information learned from the actual
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testing setup was that the flow around the heat sink was turbulent and hard to
characterize. This was significant in our later design decisions.
1.2 COMPUTER-MODEL
The computer model was based on guidelines from section one of the lab. Each
fin in the array was treated separately as a free standing pin fin. The heat transfer
coefficient was determined using correlations for infinite cylinders in cross flow:
ReD = VD/
NuD=CReD mPr1/3
hf
= NuDk/D
The heat transfer coefficient for the exposed plate around a pin was calculated as follows:
Rex = Vx/
where x was taken as the distance to the center of that pin
Nux=.332Rex 1/2Pr1/3
hx
= Nuxk/x
The equations governing fin behavior tell us that for an infinite fin modeled as having an
adiabatic tip:
m2=hp/kAc
f =tanh(mL)/(mL)
An energy balance on a single fin yields that:
Qi =mcp (Ti- Ti+1)
where Ti is the temperature of the air before encountering the fin
and Ti+1 is the temperature of the air after the fin
If each row of fins on the heat sink is treated as a flow path for a portion of the air—each
fin heats the air that passes. This yeilds the following inter-related equations:
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Ti+1- =Ti + Qi-/mCp
Qi = hf f Af (Ti- Ti+1) + hiAs(Ti- Ti+1)
The total heat dissipated is the sum of Qi for all the pins. Lastly the thermal resistance can
be calculated as follows:
R=T/Qtotal
The challenge was to adapt this model to be able to account for parameters such
as variable pin length or unsymmetrical pin orientations. The challenge of doing this
seemed daunting.
To model all possible pin lengths and orientations was far too
complex. Also, the model was based on the assumption that the fins heated the air as it
passed.
If the first fin is short, does it really heat up all of the air in front of a tall fin
behind it?
In order to avoid these complications we used a highly simplified model. It was
decided that the optimal length for thin and fat pins would be determined separately and
that only square arrays would be modeled.
An example calculation would have looked like the following:
ANALYSIS FOR SINGLE FIN:
air(kg.m^3) V air (m/s)
D fin 
air (Ns/m^2)
Re
1.1614
2.5
0.004763
1.85E-05
749.08
C (constant) m (constant)
Pr
NU
0.683
0.466
0.707
13.29708
k (W/mK)
h
0.0263
73.43
P (m)
Ac (m^2) k al (W/mK)
L fin (m) L fin(inch)
0.014961865 1.7814E-05
237
0.04233 1.666667
f
m^2
260.23
0.8689
A air (m^2)
mdot Cp air (J/KgK)
mdotCp
0.000874
0.002537
1007.0
2.5544
ANALYSIS of FIN ARRAY
L to pin ctr.
Re at pin
Nu at pin
h_plate at pin
q per pin
Tair at pin
fin1
0.012700
1993
13.204
27.345
1.342
25.000
Ts 50
fin2
0.038
5980
22.871
15.787
1.176
25.525
ANALYSIS FOR PLATE:

air(kg.m^3)
V air (m/s)
L plate 
air (Ns/m^2) (Re plate)
1.1614
2.5
0.0762
1.85E-05 11985.20
Pr
(NU average)
0.707
64.75855
K (W/mK)
(h average)
0.0263
22.35
#fin As pin (m^2)
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0.000485
ANALYSIS FOR MATERIAL AND SPACING:
L material " Lfin
15 1.66666667
finspacing
first fin
fins/row
0.025400051 0.01270003
3
Ti 25
fin3
0.064
9966
29.526
12.229
1.113
25.986
Figure 3: Sample calculation for 9 fat fins
fin4
0.089
13953
34.936
10.335
1.071
26.422
fin5
0.114
17939
39.613
9.115
1.038
26.841
q total
R total
fin6
0.140
21925
43.794
8.245
1.010
27.247
3.6312 =q per row
( sum dependant on # fins)
10.8935 W
2.2949 K/W
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The above model was used to generate values for various pin arrays and the results
obtained are presented in Figure 4:
Thin fins
4.5
9
4
8
3.5
7
3
6
2.5
5
Rt
Rt
Fat fins
2
4
1.5
3
1
2
0.5
1
0
0
0
5
10
15
20
25
30
0
10
# fins
d"
#fins
fat
1
4
9
16
25
20
30
# fins
Rt
L fin "
3.9873
2.9074
2.2949
2.4616
3.4275
d"
thin
#fins
1.66
Rt
1
4
9
16
25
36
L fin "
8.2937
4.6495
3.2159
2.9013
3.0421
4.2747
0.9375
Figure 4: Thermal Resistance versus number of fins for both fat and thin fins.
Figure 4 shows that there is an optimal number of fins for each diameter. There
are two main resistances that contribute to the overall thermal resistance of the fin array:
conduction resistance inside the fins and convection resistance from the fin surface. In an
array of few very long fins thermal resistance will suffer because of the increased
resistance associated with conducting along the length of the fin. In an array of many
short fins the thermal resistance will increase because the fins in each additional row heat
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the air behind it, reducing the convection heat transfer. The optimal number of fins was
found to be 9 fat pins (at 1.66” long) and 16 thin pins (at .9375” long).
2. HEAT PIPES
We got really excited about the notion of using heat pipes in our heat sink—we
wanted use latent heat of vaporization if possible. The idea behind heat pipes is the
following. A fluid is evaporated at the base of a pipe absorbing the latent heat of
vaporization. The vapor then travels up the pipe and condenses on a surface at the top and
drips back down to the bottom of the pipe through a wick or grove in the pipe. We
intended to manufacture these with the 3/8” rods drilled down the center and then sealed
with a small piece of the 3/16” rod pressed into the hole. We considered using acetone
because of its low boiling temperature of 55C.
While we were very enthusiastic about this idea, after talking to several people
and reading much1 about it we decided not to use them. We learned that heat pipes are
only advantageous in that they can conduct heat well and are used when conduction is the
limiting factor for heat removal. Since, from our lab experimentation, we had observed
that the pin tips were generally hot, it seems that our heat sinks were not limited by
conduction but rather the ability to convect heat away from the aluminum. Thus, heat
pipes would help little in our design and were not used.
A second solution would have been to simply drill down the rods and fill them
with acetone and then let the acetone evaporate out. This would surely have significantly
reduced the thermal resistance of the heat sink by capturing the latent heat of
vaporization. However, the solution would have been transient, as eventually all of the
1
1
Heat Pipes, Modeling, Testing and Applications, G.P Peterson;
Heat Pipe Science and Technology, A Faghri
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acetone would have evaporated. This solution also seems to go against the spirit of the
project. We decided to abandon this idea in the end.
3. RADIATION EFFECTS
Our computer model neglects heat transfer to the environment by radiation.
Radiative heat transfer may, in fact, be significant. The complex geometry of the heat
sink makes the radiative heat transfer difficult to analyze quantitatively. The emissivity
of the aluminum used is probably less than 0.2. Covering the surface (with paint or some
other substance) could reasonably raise the emissivity as high as 0.7. We considered
using black paint, soot and black permanent marker. A layer of paint would have been
thick enough to significantly insulate the fin, reducing heat transfer due to convection,
and was, therefore, not used. Coating the heat sink with a thin layer of carbon (soot) was
given serious consideration, due to the high emissivity of soot. This would have been
achieved by setting an oxyacetylene welding torch to an overly rich mixture and then
letting the excess carbon condense onto the heat sink. This was our ideal solution but we
did not feel we had enough experience with this method to adequately control the
thickness of the carbon layer. In the end, we decided to use permanent black marker.
This solution increases the emissivity while not significantly insulating the fins. Only the
outer fins were colored since they play the greatest role in radiative heat exchange with
the environment.
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4. DESIGN
We decided use the predicted optimal number of fins from our computer model—
9 fat fins and 16 short fins. Since we modeled the short and fat thins separately, it was
decided to offset the fat and thin rows to minimize the interaction between the fat and
thin pins.
The usage of the aluminum sheet metal was still to be determined. Since we were
limited by the total amount of aluminum we could use, we thought it best to use as much
of the material as possible for heat dissipation. Thus we wanted to use our sheet metal as
fins. Plate-fins are useful when one has forced convection and is sure that the air will
flow over the plate. Pin fins are chosen when one is unsure about the nature of the air
flow2. In our case, we were highly uncertain about the air flow pattern and thus did not
want to risk diverting air flow away from our heat sink with large plate-fins. We aimed
at creating “pin fins” from our sheet aluminum so as not to disrupt the flow. The sheet
metal was cut into thin strips that were then attached to the side of the base plate. We did,
however, use two slightly larger plate-fins on the two extremities of the heat sink in order
to divert air flow that would have obviously missed the heat sink. The position of these
larger plates can be seen in Figure 5:
2
Design and Analysis of Heat Sinks, Kraus and Bar-Cohen
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1.75”
3”
3”
1.25”
.25”
3/16” Aluminum Rod
16 pieces @ .94” long
Horizontal Aluminum Sheet
Metal
3/8” Aluminum Rod
9 pieces @ 1.67” long
Vertical Rising Aluminum Sheet
Metal
Figure 5: Plan view of heat sink
The dimensions of the rectangular-fins were determined by the optimal way to
divide and cut the sheet aluminum that was supplied to us. We decided to orientate the
rectangular-fins on the top and right sides (see Figure 5) parallel to the base plate since
we did want to disrupt the arriving air flow. Vertical rectangular-fins would have acted
as bluff bodies and significantly blocked the air flow. The rectangular-fins in the rear
were placed vertically since disrupting the flow out of the heat sink is less important to
the heat sink performance. Additionally, if these fins were placed horizontally they
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would not be exposed to a significant air flow. The rectangular-fins were attached with
screws that ensured good contact between the fins and the heater plate. An added bonus
to the rectangular-fin design was that these fins could be easily bent. This allowed us to
tweak the heat sink to best interact with the air flow.
We decided to press fit the pins only half way into the plate and not all the way
through. While press fitting them all the way through the base plate would have
decreased the contact resistance it would probably have increased the total thermal
resistance—less of the fin would be exposed to the convecting air as it is buried in the
base plate.
5. MANUFACTURE
Special care was taken in machining the heat sink to reduce thermal resistance. It
was essential that the contact between the pins and the base plate was as tight as possible.
The base plate holes were first drilled with a regular drill bit, then squared out with a
milling bit, and then reamed to provide a smooth and precise finish. Thermal grease was
applied to the base of each pin before it was press fitted. The base plate was tapped for
the screws that were used to attach the rectangular-fins. The hole taps were made as
short as possible so as to also minimize the amount of air trapped in the base plate.
Again, the contact surfaces and screws were lightly covered with thermal grease before
they were attached.
Figure 6 shows a 3D model of the manufactured heat sink:
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Figure 6: Aerial 3D view of heat sink. Air flow enters from the top right.
6. A THEORETICAL MODEL OF FINAL DESIGN
Generating a computer model for the behavior of the above design was difficult.
How exactly was one to model our rectangular-fins? How exactly could we account for
the uncharacterized air flow?
To avoid these complications, we again chose a highly simplified model. The pinfins were modeled according to our previous computer model, and the rectangular-fins
were modeled as convection from a plate parallel to the air flow. The convection
coefficient for the front rectangular-fins was found at the middle position of the fins,
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where as the rear fins’ convection coefficient was found at the rear point of the base
plate. Figure 7 presents the results:
h front plate- h back plate-fins
fins
20.673
9.52
q fat
q thin
q front-plate q back plate- q total
fins(W) fins(W) fins (W)
fins (W)
(W)
13.4
10.16
3.5
1.61
28.67
Rtotal
(K/W)
0.87
Figure 7: Predicted thermal resistance for heat sink – based
on previously discussed computer model.
Our model results of 0.87 K/W was about 16% less than the value we found when
testing-1.01 K/W. The difference between these two values can be accounted for in
numerous ways.
When we designed our computer model, we included several
assumptions and approximations about the heat sink in order to simplify the model.
Assuming that our model was vertically homogenous allowed us to model our heat sink
in just two dimensions. Thus we ignored vertical air flows as well as temperature
gradients along the length of the fins. We also assumed that while each fin was affected
by the fins in front of it, each fin was independent of fins and objects to either side.
When calculating the optimal number of fins for a given diameter we used two models
that were separately optimized and then overlaid to give a summed maximum. This
theory of superposition stemmed from prior assumptions of the idealized air flow. Other
assumptions like constant fluid properties, one direction air flow, averaged convection
coefficients, simplified models for convection from both the fins and the plate
contributed in part to our error. Another significant source of error was that our model
did not account for contact resistances.
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By more closely modeling each of these parameters we could have produced a
more accurate model with a smaller margin of error. However, for the time constraints of
this experiment and the utilization of a physical model to support and compliment our
theoretical model, we consider this degree of error to be acceptable. This is primarily
because the models were used to predict trends in performance, rather than to accurately
predict thermal resistance values.
7. FUTURE IMPROVEMENTS
We decided to push-fit the pins only halfway through the base plate because we
did not believe contact resistance would be greatly affected. In manufacturing, we found
that this was not true. To minimize contact resistance between the fin and the base plate
solid contact between the two surfaces is required. In order to achieve solid contact, one
must drill holes that have square profile. In practice, this is very tough to achieve since
most drill bits have a tapered point and most milling bits are slightly rounded. Thus in
hindsight, we think less contact resistance would be achieved if the fins were press-fitted
all the way through the plate and their tips sanded to merge with the bottom of the base
plate. Since more of each fin is buried in the base plate, fewer, longer fins would be
better. Fewer fins also have the added benefit of reducing contact resistance. We had 25
fins – each fin having some contact resistance with the surface. Had we opted for fewer
fins, our contact resistance would have been reduced. The theory does not seem to
suggest that fewer longer fins (perhaps 16 fins) would have significantly reduced the fins'
ability to convect heat away. However, fewer fins would certainly have decreased contact
resistance.
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8. FINAL RESULT
Our heat sink placed second in the class. This was, hopefully, an indication that
we correctly identified the major trends and factors influencing heat sink performance.
9. BIBLIOGRAPHY
-Design and Analysis of Heat Sinks, Kraus and Bar-Cohen, John Wiley and Sons,
New York, 1995
-An Introduction to Heat Pipes, Modeling, Testing, and Applications, G.P
Peterson, John Wiley and Sons, New York, 1994
-Heat pipe Science and Technology, A. Faghri, Taylor and Francis, 1995
-Fundamentals of Heat and Mass Transfer, Incoropera and DeWitt, John Wiley
and Sons, New York, 1996
-Fluid Mechanics, M. White, McGraw-Hill, Boston, 1999
-Background Reading Material for Heat Sink Thermal Design Competition, Lab
#3, K. Goodson, Mechanical Engineering Department, Stanford University, 1999