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### Two Dimensional Steady State Heat Conduction

### 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…

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

Solution domain is a superset of geometric domain !!!

Recognizing that

where the constants have been combined and represented by Cn

Using the final boundary condition:

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!

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

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