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