Problem 1 (15 Points)
A power electronics package consists of identical composite slab assemblies that are held together
with a copper “rod-like” interconnect. The interconnect allows communication between the two
heat generating devices and serves as a fin to dissipate heat, as shown in the schematic below.
The devices uniformly generate heat through their volume at a rate of 𝑞̇ [W/m3]. They are square
in cross section of width, w, have a thickness Lg, and a thermal conductivity kg. The devices are
connected to a circuit board of the same cross section having thickness La and thermal conductivity
$$
ka. Bonding of the device and circuit board results in a contact resistance of 𝑅!,#
. All sides of the
device and circuit board stack are insulated except the “back” face connected to the interconnect.
The interconnect has diameter D, length L, and thermal conductivity kf. Ambient air at a
temperature T∞ having a convective heat transfer coefficient, h, flows over the interconnect and
the back (i.e., exposed) surface of each circuit board. Heat transport through the system can be
assumed purely 1-dimensional. The exposed portion of the circuit boards and the temperature of
the interconnect where it bonds to the circuit boards are each at an unknown temperature, Tb.
Given Assumptions:
1. Steady state conditions
2. Parameters mentioned in problem statement are known, constant, and uniform.
3. No contact resistance between circuit board and interconnect.
4. Heat transfer through the system is purely one dimensional.
5. Temperature of interconnect and face of circuit board exposed to convection are both at
temperature, Tb.
Any Additional Assumptions:
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Full Name: ____________________________
Problem 1(a) Qualitatively sketch the temperature profile through the system. Be sure to denote
the shape of the curve in each section (i.e., linear, quadratic logarithmic, exponential, etc.) and
ensure that the relative slopes of the curves scale appropriately assuming 𝑘% > 𝑘& ≫ 𝑘' and the
$$
contact resistance 𝑅!#
is comparable to the thermal resistance across the circuit board.
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Full Name: ____________________________
Problem 1(b) Assuming the rod is sufficiently long to treat each half of the system independently,
*
use a thermal resistance network to derive a symbolic expression for the temperature, 𝑇()!
, in the
circuit board at its interface with the heat generating device region. Your expression should be in
terms of some, or all, of the given parameters (𝑘& , 𝑘' , 𝑘% , 𝑞̇ , 𝐿& , 𝐿' , 𝐿, 𝑤, 𝐷, ℎ, and/or 𝑇+ ).
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Full Name: ____________________________
Problem 1(c) The temperature profile within the heat generating region has the form given by the
,̇
expression: 𝑇(𝑥) = − ./ 𝑥 . + 𝐶0 𝑥 + 𝐶. , where 𝐶0 and 𝐶. are the constants of integration.
!
Recognizing this, derive an expression for the temperature, To, at the insulated end of the heat
generating device in terms of some, or all, of the given parameters. (𝑘& , 𝑘' , 𝑘% , 𝑞̇ , 𝐿& , 𝐿' , 𝐿, 𝑤,
*
𝐷, ℎ, 𝑇+ , and/or 𝑇()!
). Recognize that x=0 is defined on the face of the left-most heat generating
device adjacent to the insulation, as shown in the original schematic.
Assumptions: Which of the assumptions (numbered as on the problem statement page) are
required such that the given temperature expression is valid?
□1
□2
□3
□4
□5
□ Other assumptions required (state in assumption box)
Analysis:
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