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To: Dr. Hosni Abu-MulawehFrom: Viraj VermaSubject: Chip Cooling DesignDate: March 4, 2010
The goal was develop a final design a heat sink or finned attached such that a siliconchip temperature can be maintained at 65 r C at the lowest cost. The silicon chip wasa square with sides measuring 10 mm with a thickness of 2.0 mm. The chip wasmounted on a glass/epoxy board that had a thickness of 2.5 mm and a thermalconductivity of 5 W/m-K. The card cage has a nominal board spacing of 20 mm. Also,the surrounding air temperature was 25 r C with a heat transfer coefficient of 30W/m 2-K.
Before proceeding with the analysis, some major assumptions were made. The chipwas assumed to have uniform temperature. As stated in the design statement,steady state was assumed along with one-dimensional heat flow. No contact resistance was assumed between the fins and the chip. The temperature in-betweenthe fins was assumed to stay constant. There was no convection from the sides of the chip. Thermal conductivity remained constant, radiation was neglected, heat generation was absent, and the convective heat transfer over the surface areas of the fin was assumed.
Due to the complexity of the problem, MATLAB was used. The program enabled oneto run multiple iterations to determine the lowest cost and the total heat lost. Usinga thermal circuit, the heat loss from the chip due to convection and conduction wasfirst calculated. This resultant value showed the heat loss that would need to take
place through the use of fins.
MATLAB was used to determine the optimal fin size and the lowest cost. The finaldesign was determined to have 8 fins. Each fin measured 1 mm thick, 6.44 mm inlength, and made of Material B. The total cost for the finned design was found to be$ 1.71.
After further analyzing this design, it was found to be not conservative. The initialassumptions made propagate uncertainties. It is also recommended that a bettermanufacturing method be found so that the cost of making the fins can be lowered.Finally, if high performance is necessary, increasing fin length and the introduction
of a fan would lower the chip temperature.
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Objective:
The final design goal was to design a finned attachment to maintain the maximum
chip temperature of 65r
C while minimizing the cost. A square silicon chip with thesides measuring 10mm with a thickness of 2.0 mm was dissipating 1.5 Watts. The
chip was mounted on a 2.5 mm thick glass/epoxy board that had a thermal
conductivity of 5 W/m-K and contact resistance of 6.5x10 -4 m2K/W. The chip was
also mounted horizontally in a card cage with a nominal board spacing of 20 mm.
The board was cooled by air with a temperature of 25 r C with a convective heat
transfer coefficient of 30 W/m 2-K flowing parallel to the board.
Design requirements were such that the fin profile had to be rectangular and a
decision between two materials: Material A and Material B must be made. Material
properties of Material A and Material B can be seen in Table 1. Figure 1 shows the
system at its initial condition.
Ta ble 1: M a teri a l Properties
Material Properties Material A Material B
Density (kg/m 3) 2700 8800
Cp (kj/kg-k) 0.95 .387
k (W/m-k) 230 400
Diffusivity (m 2/s) 97x10 -6 120x10 -6
Material Cost ($) 6/kg 10
Manufacturing Cost ($) 0.15+.0016a 2.2 .18+.0014a 1.6
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Figure 1: System a t Initi a l Condition
Assumptions:
To reach a final design so that the chip can be maintained at 65 r C, the following
assumptions were made. First, the chip has uniform temperature and does not vary
with its thickness. As stated in the design statement, steady state was assumed;
implying that properties did not vary with time. One-dimensional heat flow was
assumed; detonating that temperature is a function of the y-coordinate only, and
heat is transferred exclusively in this direction. The area of the glass/epoxy board is
assumed to be the same as the area of the chip. This ensures that the heat flows only
in one-direction.
Next, it was assumed that there is no contact resistance between the fin and the
chip. If more than one fin would be required to cool the chip, then it was assumed
that the temperature between the two fins did not change and remained at 25 r C. It
was also assumed that there was no heat loss from the sides of the chip.
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Finally, for the fins, the following assumptions were made: the thermal conductivity,
k, was constant, the radiation from the surface was negligible, heat generation
effects were absent, and heat transfer coefficient, h, is constant over the surface.
Ana lysis:
To determine the final design, a systematic approach was used. First, the initial
system was modeled as a thermal circuit (see Figure 2). Using this circuit and a
value of 65 r C for the surface temperature of the chip, the heat loss from the chip
through convection, Qconv, and conduction, Qcond, was calculated (see Equations 1
and 2). From the initial analysis, one was able to determine how much heat must be
lost form the addition of fins.
Figure 2: Th erm a l Circuit Represent a tion of Figure 1
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(1)
(2)
Intuitively, one could see that the addition of fins reduced surface of area of the chip.
Thus, the addition of fins reduced the area from which heat loss due to convection
could occur. Also, to maintain the chip at 65 r C, the heat dissipated by the chip must
equal to the total heat loss. Thus, by applying Equation 3, one can solve the total
heat loss in terms of conduction, convection, and fins.
(3)
Cleary, it could be seen that both total heat lost from fins and heat loss from
convection were dependent on the number of fins and thickness of each fin. Thus,
with multiple variables present, it proved to difficult to solve for the optimal size of
the fin while making it cost effective.
To solve for the multiple variables simultaneously, MATLAB was used. It assisted in
calculation of heat loss from convection as a function of the number of fins. To
calculate heat loss from a fin, Equation 4 was used. From the equation, one could
recognize that heat lost from a fin is a function of the fin length, its cross-sectional
area, and perimeter.
(4)
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More importantly, the total cost of fins was calculated by adding the material cost
and the manufacturing cost. But, the manufacturing cost of the fin is based on its
aspect ratio. The aspect ratio was defined as fin length over its thickness. Using
Equations 5 and 6, the material costs of Material A and Material B could be
calculated. Table 1 shows how the manufacturing costs were determined. These
equations were obtained form the company s empirical data.
(5)
(6)
It was instantly evident that the manufacturing cost had more of an effect in total
cost of the fins than the material cost. Thus, the goal became to reduce the
manufacturing cost. Since, the manufacturing cost was directly proportional on the
aspect ratio, it became imperative to reduce it.
Since the aspect ratio is a function of fin length over fin thickness, the easiest way to
reduce it was to maximize the fin thickness and minimize fin length. But, a total heat
loss of 1.5 Watts was still a constraint. Since one-dimensional heat flow assumed, fin
thickness would have to be very thin; but fin thickness would have to be greater
than 0.5 mm due to structural reasons. Thus, a maximum fin thickness of 1 mm was
chosen, as it satisfied both conditions. For a fin to operate properly, there needed to
be spacing between each fin. A fin spacing of 0.25 mm was chosen; therefore
allowing a maximum of 8 fins could be placed on the chip surface.
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Using these new parameters, the material properties, and the initial requirements, a
MATLAB program was written such that it could calculate the total cost as a function
of manufacturing cost and material; the total heat loss as a function of heat loss due
to convection, heat loss due to conduction, and heat loss due to fins (the program
can be seen in the Appendix).
Using MATLAB, one was easily able to run multiple iterations and determine the
most cost effective fin length. Figure 3 and 4 show total heat loss and total cost as a
function of fin length for both materials. Table 2 shows a summary of the findings.
Figure 3: Optim a l Cost for M a teri a l A
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Figure 4: Optim a l Cost for M a teri a l B
Ta ble 1: Summ a ry of Findings
Material A Material B
Number of Fins 8 8
Fin Lengt h (mm) 6.46 6.44
Qconv (W) .0240 0.0240
Qcond (W) .0232 0.0232
Qfin (W) 1.4537 1.4523
Qtot a l (W) 1.5009 1.500
T ot a l Cost ($) 1.98 1.71
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From Table 1, one can see the final results. The optimal price for each metal still
dissipating 1.5 Watts was found. It should be noted that the heat lost from
convection and the heat loss from conduction did not change form Material A to
Material B, since they were dependent on the chip and the board; the heat loss form
the addition of fins changed.
Ot h er Consider a tions
Before a final design recommendation could be made, some finer points needed to
be understood. First, the final design is not a conservative design because of the
assumptions made during the analysis. The design criteria stated that the maximum
temperature of the chip must be maintained at 65 r C. But, in this analysis, the chip
was assumed to be at uniform temperature resulting in the surface temperature
being at 65 r C. Logically it can be deduced that when there is uniform heat
generation, the maximum temperature occurs near the mid-plane and decreases in
temperature near the surfaces. From this it can be concluded that if the surface
temperature is maintained at 65 r C then the temperature near the center of the chip
will be higher than 65 r C.
One source of uncertainty and a reason for why the final design underestimates the
final temperature is due to fin placement. In this analysis, it was assumed that the
temperature in-between the fins stays at a constant 25 r C, but this is not the case in
reality. The temperature of the air in between the fins may increase depending on
how much heat is lost. Thus, it became necessary to estimate the increase in cost if
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there was a 1 r C increase in temperature of air in between the fins. It was concluded
that an increase in 1 r C of the air temperature resulted in a cost increase of 1.5%.
Other sources of error came from the assumptions made such as: k is constant, h isconstant over fin surface, negligible radiation, and not contact resistance between
fins and the chip. Although these assumptions cause uncertainties in the final
design, to what extent was not determinable.
Total cost was one the most important constraints put on this design problem. It
was necessary to reduce cost as much as possible, thus it is commonsense to find
better material manufacturing methods. The manufacturing cost, which was based
on the aspect ratio, had the largest affect on total cost. Thus by finding better
manufacturing methods, the cost can be greatly reduced. Finding a manufacturing
method that decreases the importance of aspect ratio would definitely reduce the
cost associated with the number of fins.
Also, the reliability of electronic components increases as temperature decreases;
for that reason other methods should be considered to increase the heat transfer
from the chip. The obvious way to increase the heat transfer from the chip would be
to increase the length of the fin. Increasing length of the fin poses potential
problems. First, increasing the length of the fin would increase the aspect ratio, thus
greatly increasing the cost. Finally, the length of the fin is also limited by the
nominal board spacing, thus the maximum allowed fin length is 18mm.
To help lower the chip temperature, a fan can be introduced so free and forced
convection can take place. A fan may improve the transfer of thermal energy from
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the heat sink to the air by moving cooler air between the fins. A major problem with
introducing a fan will be the cost of including and running the fan. Next, mounting
the fan near the chip would be difficult. Obtaining a fan of such a small size may be
not cost effective.
Recommend a tion
To maintain the chip temperature at 65 r C of a chip dissipating 1.5 Watts of heat, it is
recommended that 8 fins with spacing of 0.25mm in between, with a length of 6.44
mm, and a thickness of 1.0 mm made from Material B be applied. By running
multiple iterations, Material B proved to be the cheapest to manufacture all 8 fins
and still dissipate 1.5 Watts of heat. Figure 4, below, shows a basic schematic of the
final recommendation. The total cost to cool the chip with 8 fins was found to be
$1.71.
If a more robust and conservative design is required, it is suggested that the finlength should be increased. Also, the implementation of a fan or another cooling
device would help in lowering the chip temperature thus increasing its
performance.
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