1. ME 430 – Final Project
M. Flukey
Saint Martin’s University, Lacey, WA, 98503
Abstract
This project was to design chip placement on a composite board to minimize
the average temperature distribution of the board. It was found that the
optimal distance was approximately 5.1 mm away from the edge, when only
moving one chip at a time. While moving both simultaneously, the best
distance is 3 mm.
Nomenclature
h Convection Heat Transfer Coefficient, W/𝑚2
K
i Subscript for the placement on the x-axis
j Subscript for the placement on the y-axis
k Thermal Conductivity of Material, W/𝑚K
q Power Generation, W
𝑞′′
Heat Flux, W/𝑚2
T Temperature, K
I. Introduction
The purpose of this project is to determine the best positioning of two heat generating chips
to mitigate the total heat inside the composite board holding the chips. One rectangular chip, chip
A, is 45 by 15 mm while producing 2.75 W. The other rectangular chip, chip B, is 75 by 30 mm
generating 7.5 W. The composite board is made up of two materials, copper and epoxy glass.
Epoxy glass layers of 0.5 mm thick layer the top and bottom of the copper core, 2 mm thick.
Thermal conductivities of the epoxy glass and copper are 1 and 200 W/mK, respectively. The
board is shaped into a rectangular of length 200 mm (L in figure 1) and width (H in figure 1) of
125 mm. Along the width of the board liquid cooling occurs. On the top and bottom of the board
there is heat convection through air. These heat convection rates will be varied to show their
effects on the system. For clarity, this design is shown in figure 1 below. Chip A and B are to be
moved for design purposes.

2. Figure 1. Description of System
II. Assumptions
There are numerous assumptions that went into the chip design. Radiation can be
neglected due to its minimal impact on the dissipation of heat. Assumptions about the heat
transfer coefficient can be made as well. For instance, since the h of air varies in the design it can
be assumed that at 0 W/𝑚2
K is no air flow in the system and 20 W/𝑚2
K is air flow of high
speeds. The outside temperature and coolant, 𝑇∞ , is assumed to be at 300 Kelvin.
The chip position can be simplified due to the coolant flow. Since both chips are longer
than they are wide, it can be assumed that placing them lengthwise against the sides will reduce
the heat in the composite board. This assumption reduces the design into a two-dimensional
finite-difference design. The chips are positioned on the top of the system and will move
individually with the first solutions and then simultaneously.
III. Modeling
Using Table 4.21, the following equations are taken. For all interior nodes,
𝑇 𝑚,𝑛 =
𝑇 𝑚,𝑛+1+𝑇 𝑚,𝑛−1+𝑇 𝑚+1,𝑛+𝑇 𝑚−1,𝑛
4
(1)
Where 𝑇 𝑚.𝑛 is the temperature of a node at the position (m,n). For all nodes touching the coolant,
𝑇 𝑚,𝑛 =
𝑇 𝑚,𝑛−1+2𝑇 𝑚±1,𝑛+𝑇 𝑚,𝑛+1+
2ℎ 𝑙Δ𝑥
𝑘
𝑇∞
2(
ℎ 𝑙Δ𝑥
𝑘
+1)
(2)
Where ℎ𝑙 is the heat transfer coefficient of the coolant, Δ𝑥 is the change in x for the nodal
positioning, and the node multiplied by 2 varies depending on the boundary it is at. For example,
on the right boundary the ± turns into a −. However, on the bottom layer, the term multiplied by
the 2 is 𝑇 𝑚,𝑛+1 and the h term turns into ℎ 𝑎. For all heat generating boundaries,

3. 𝑇 𝑚,𝑛 =
2𝑇 𝑚,𝑛−1+𝑇 𝑚+1,𝑛+𝑇 𝑚−1,𝑛+
2𝑞′′Δ𝑥
𝑘
4
(3)
Where 𝑞′′
is the heat flux of the chips, which is determined by their production of watts divided
by the area of the chip.
To accurately obtain the steady state of the system, the code was run to 100,000
iterations. It was shown that it converges at ~50,000 iterations.
IV. Results
For the movement of chip A and chip B, individually it was shown that 5.1 mm from the edge
was the best positioning of the chip, depicted in figures 5 and 6. The movement of both chips
simultaneously produced 3 mm from the edge and a lower maximum temperature. The heat
transfer coefficient of air will vary to show the positioning’s dependence and effects on the
maximum temperature. Figure 2 shows the air flow at a lower speed and figure 3 shows it at a
higher speed.
V. Discussion
Figure 1. Maximum Temperature 4 h of air and 25 h of coolant
Figure 2 depicts a local minima near the start of the graph. This means that both chips
have moved in a slight distance from the edge and the cooling has increased significantly. This
movement of the chips allows the copper to be cooled at the edges closer to the temperature of
the coolant. Since the copper core has a heat transfer coefficient of 200 W/mK, heat is taken

4. away and transferred across the material more easily. With this ease of transfer, the cooling of
the chip is increased when both chips are moved inward. The top epoxy layer has a large part in
this systems heat distribution. Since the epoxy has a heat transfer coefficient of 1 W/mK, it is
easier for heat to transfer downward into the copper than across the epoxy layer.
Figure 2. Maximum Temperature 12 h of air and 25 h of coolant
Figure 3 depicts a local minima similar to figure 2. Figure 3’s local minima seems to be
more of a collection of points rather than a singular point. This is due to the increased heat
transfer coefficient of air. From the modeling of the system, each point is dependent on its
neighboring points. Since h of air is increased the points neighboring the chip are cooler
effectively negating the transfer of heat between the chips. There is a minimum maximum heat
due to chip A, it has a large 𝑞′′
. Its heat flux forces the point directly under the center of the chip
to be at the temperature near 332 K. With this method, the minimum maximum core temperature
is 332 K. When the heat transfer coefficient of the coolant is increased the overall board
temperature will cool down, beside the previously stated point. This effect is more prevalent in
figure 4.
This analysis is also done when the air and coolant temperature is 300 K or 80 ºF. This
assumption is simplifying the temperatures. Depending on where the air is drawn, the
temperature can vary with time and location. The coolant would be a lower temperature than 𝑇∞
and remain a lower temperature than the air would. Since the coolant would be at a lower
temperature and the air varying over time, the minimum maximum temperature wouldn’t change

5. but the point at which it does will. The coolant would be more effective at cooling the copper,
the main heat sink.
Even though only the maximum temperature was taken, the chip placement can still be
applied. For the two-dimensional analysis, the chips being 3 mm away from the edge is ideal. If
translating that into three-dimensional, 3 mm away from all edges and placed in opposite corners
on top of the board is the ideal positioning. Opposite sides for the least amount of heat transfer
between the chips and the distance from the edge due to the 2-D analysis.
Figure 4. Maximum Temperature 20 h of air and 25 h of coolant
Figures 5 & 6. Chip A and Chip B Movement (Independent)

6. References
1Bergman, T. L. and Lavine, A. S., Fundamentals of Heat and Mass Transfer, 7th ed., John
Wiley & Sons, New Jersey, 2011, Chap. 4.
Appendix
Figures of Max Temperature starting from 2 W/mK to 20 W/mK, going left to right.
2 4