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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.



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



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.



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



(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)



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.



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



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



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



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



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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