Two Dimensional Steady State Heat Conduction

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# Two Dimensional Steady State Heat Conduction - PowerPoint PPT Presentation

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…. l 2 < 0 or l 2 > 0 Solution. OR. q = C. Any constant can be expressed as

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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…

l2 < 0 or l2 > 0 Solution

OR

q = C

Any constant can be expressed as

A series of sin and cosine functions.

H

q = 0

q = 0

y

l2 > 0 is a possible solution !

0

W

x

q = 0

Construction of a Fourier series expansion of the boundary values is facilitated by rewriting previous equation as:

where

And hence

Substitutingf(x) = T2 - T1into above equation gives:

Isotherms and heat flow lines are

Orthogonal to each other!

Sinusoidal Temperature B.C.

q = Cx

H

q = 0

q = 0

y

0

W

x

### 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.
If m is a total number of the heat flow lanes, then the total heat flow is:

Where S is called Conduction Shape Factor.

Conduction shape factor

Heat flow between two surfaces, any other surfaces being adiabatic, can be expressed by

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 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.
• Rv2in 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 isd