two dimensional steady state heat conduction
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Two Dimensional Steady State Heat Conduction
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
It is just not a modeling but also feeling the truth as it is…
2 < 0 or 2 > 0 Solution
)sin()cos(,0 43212 kyCkyCeCeCyxT kxkx
kyky eCeCkxCkxCyxT 43212 )sin()cos(,0
OR
0 W
H
x
y
= 0
= 0 = 0
= C
2 > 0 is a possible solution !
Any constant can be expressed asA series of sin and cosine functions.
Substituting boundary conditions :
kyky eCeCkxCkxCyxT 4321 )sin()cos(,
0
00:0,0
1
431
C
eCeCCHyx kyky
4343432
4321
00)sin(
)sin()cos(00:0,0
CCCCCCkxC
CCkxCkxCWxy
0sin
0)sin(0:0, 32
kW
CeekWCHyWx kyky
W
nknkWkW
0sin
Where n is an integer.
yW
ny
W
n
n
eexW
nCCyx
1
32 )sin(,
Solution domain is a superset of geometric domain !!!
Recognizing that
yW
nee
yW
ny
W
n
sinh2
W
yn
W
xnCyx
nn
sinhsin,1
where the constants have been combined and represented by Cn
W
yn
W
xnCTyxT
nn
sinhsin,
11
Using the final boundary condition:
W
Hn
W
xnCTT
nn
sinhsin
112
2:0, TTWxHy
Construction of a Fourier series expansion of the boundary values is facilitated by rewriting previous equation as:
112 sin)(
nn W
xnATTxf
where
W
HnCA nn
sinh
Multiply f(x) by sin(mx/W) and integrate to obtain
Substituting these Fourier integrals in to solution gives:
And hence
Substituting f(x) = T2 - T1 into above equation gives:
W
yn
W
xnCyx
nn
sinhsin,1
Temperature Distribution in A Rectangular Plate
Isotherms and heat flow lines areOrthogonal to each other!
),(),( yxTkyxq
Linearly Varying Temperature B.C.
0 Wx
H
y = 0 = 0
= Cx
2sin
0
WAdx
W
xnxf n
W
WW
n dxW
xnCx
Wdx
W
xnxf
WA
00
sin2
sin2
WHn
Wyn
W
xn
nTxTTyxT
n
n
sinh
sinhsin
12)(,
1
1
121
X/W
Sinusoidal Temperature B.C.
0 W
H
x
y = 0 = 0
= Cx
2sin
0
WAdx
W
xnxf n
W
WW
n dxW
xn
W
x
Wdx
W
xnxf
WA
00
sinsin2
sin2
W
xHx
sin),(
Principle of Superposition
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
It is just not a modeling but also feeling the truth as it is…
For the statement of above case, consider a new boundary condition as shown in the figure. Determine steady-state temperature distribution.
For ith heat tube and jth isothermal block :
11
.
jijj
iiij TT
y
Lxkq
1iT
2iT
ijT
1ijT
Where n is number of isothermal blocks.
1211
.TT
y
Lxkq i
j
iii
321
.ii
j
ii TTy
Lxk
21
1
.TT
y
Lxkq n
jj
iii
1iT
2iT
ijT
1ijT
If m is a total number of the heat flow lanes, then the total heat flow is:
m
iitot qq
1
m
in
jj
iitot TT
y
Lxkq
121
1
.
211
21
1
.TTSkTT
y
Lxkq avg
m
in
jj
iitot
Where S is called Conduction Shape Factor.
Conduction shape factor
Heat flow between two surfaces, any other surfaces being adiabatic, can be expressed by
21 TTSkqtot
where S is the conduction shape factor
• No internal heat generation
• Constant thermal conductivity
• The surfaces are isothermal
Conduction shape factors can be found analytically
shapes
Thermal Resistance Rth
kSRth
1
Shape Factor for Standard shapes
Thermal Model for Microarchitecture Studies
• Chips today are typically packaged with the die placed against a spreader plate, often made of aluminum, copper, or some other highly conductive material.
• The spread place is in turn placed against a heat sink of aluminum or copper that is cooled by a fan.
• This is the configuration modeled by HotSpot.
• A typical example is shown in Figure.
• Low-power/low-cost chips often omit the heat spreader and sometimes even the heat sink;
Thermal Circuit of A Chip• The equivalent thermal circuit is designed to have a direct and intuitive
correspondence to the physical structure of a chip and its thermal package.
• The RC model therefore consists of three vertical, conductive layers for the die, heat spreader, and heat sink, and a fourth vertical, convective layer for the sink-to-air interface.
Multi-dimensional Conduction in Die
The die layer is divided into blocks that correspond to the microarchitectural blocks of interest and their floorplan.
• For the die, the Resistance model consists of a vertical model and a lateral model.
• The vertical model captures heat flow from one layer to the next, moving from the die through the package and eventually into the air.
• Rv2 in Figure accounts for heat flow from Block 2 into the heat spreader.
• The lateral model captures heat diffusion between adjacent blocks within a layer, and from the edge of one layer into the periphery of the next area.
• R1 accounts for heat spread from the edge of Block 1 into the spreader, while R2 accounts for heat spread from the edge of Block 1 into the rest of the chip.
• The power dissipated in each unit of the die is modeled as a current source at the node in the center of that block.
Thermal Description of A chip
• The Heat generated at the junction spreads throughout the chip.
• And is also conducted across the thickness of the chip.
• The spread of heat from the junction to the body is Three dimensional in nature.
• It can be approximated as One dimensional by defining a Shape factor S.
• If Characteristic dimension of heat dissipation is d
dS onconstricti 2
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