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…
l2 < 0 or l2 > 0 Solution
l2  0  T x, y   C1e  kx  C2e kx C3 cos( ky)  C4 sin( ky) 
OR
l  0  T x, y   C1 cos( kx)  C2 sin( kx) C3e
2
Any constant can be expressed as
A series of sin and cosine functions.
q=0
l2 > 0 is a possible solution !
 ky
 C4 e
q=C
H
q=0
y
0
x
q=0
W
ky

Substituting boundary conditions :

T x, y   C1 cos( kx)  C2 sin( kx)  C3e

x  0,0  y  H : q  0  0  C1  C3e
 ky
 ky
 C4 e
ky
 C4 e
ky


 C1  0
y  0,0  x  W : q  0  0  C1 cos( kx)  C2 sin( kx) C3  C4 
 C2 sin( kx) C3  C4   0  C3  C4   0  C3  C4

x  W ,0  y  H : q  0  C2 sin( kW )  e
 ky
 sin kW   0
n
sin kW   0  kW  n  k 
W
Where n is an integer.

 e C3  0
ky
Solution domain is a superset of geometric domain !!!
 n 
 n 


y



y 
n   W 
q x, y   C2C3  sin(
x) e
 e W  

W 
n 1


Recognizing that
  nW  y  nW  y 
 e    e  



  sinh  n y 


2
W 
 nx 
 ny 
q x, y    Cn sin 
 sinh 

 W 
 W 
n 1

where the constants have been combined and represented by Cn
 nx 
 ny 
T x, y   T1   Cn sin 
 sinh 

 W 
 W 
n 1

Using the final boundary condition:
y  H ,0  x  W : T  T2
 nx 
 nH 
T2  T1   Cn sin 
 sinh 

 W 
 W 
n 1

Construction of a Fourier series expansion of the boundary
values is facilitated by rewriting previous equation as:
 nx 
f ( x)  T2  T1   An sin 

 W 
n 1

where
 nH 
An  Cn sinh 

 W 
Multiply f(x) by sin(mx/W) and integrate to obtain
Substituting these Fourier integrals in to solution gives:
And hence
Substituting f(x) = T2 - T1 into above equation gives:
Temperature Distribution in A Rectangular Plate
 nx 
 ny 
q x, y    Cn sin 
 sinh 

 W 
 W 
n 1

q( x, y )  kT ( x, y )
Isotherms and heat flow lines are
Orthogonal to each other!
Linearly Varying Temperature B.C.
W

0
q = Cx
W
 nx 
f x sin 
dx  An
2
 W 
H
q=0
q=0
y
0
2
An 
W
W

0
2
 nx 
 nx 
f x sin 
dx   Cx sin 
dx
W 0
W 
W 
W
x
W
X/W
ny

sinh
n 1
2    1
n

x


W
T x, y   T1  T2 ( x)  T1  
sin 

  n1 n
 W  sinh nH
W






Sinusoidal Temperature B.C.
W

0
 x 
q ( x, H )  sin  
q = Cx  W 
W
 nx 
f x sin 
dx  An
2
 W 
H
q=0
q=0
y
0
2
An 
W
W

0
x
2
 nx 
 x   nx 
f x sin 
dx   sin   sin 
dx
W 0 W   W 
W 
W
W
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.
Tij 1
Tij
Ti 2
Ti1
For ith heat tube and jth isothermal block :
ki xi .L
Tij  T1 j 1 
qij 
y j
ki xi .L
Ti1  T12 
qi1 
y j
ki xi .L
Ti 2  Ti3 

y j 1
qi 
ki xi .L
n
 y
j 1
T1  T2 
j
Where n is number of isothermal blocks.
Tij 1
Tij
Ti 2
Ti1
If m is a total number of the heat flow lanes, then the
total heat flow is:
m
qtot   qi
i 1
m
qtot  
i 1
ki xi .L
n
 y
j 1
m
qtot  
i 1
ki xi .L
n
 y
j 1
T1  T2 
j
T1  T2   Skavg T1  T2 
j
Where S is called Conduction Shape Factor.
Conduction shape factor
Heat flow between two surfaces, any other surfaces being
adiabatic, can be expressed by
qtot  Sk T1  T2 
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
1
Rth 
kS
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-toair 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
Sconstriction  2 d