1 / 38

# 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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Two Dimensional Steady State Heat Conduction' - jerry

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

Where n is an integer.

Recognizing that

Using the final boundary condition:

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

where

Multiply f(x) by sin(mpx/W)and integrate to obtain

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

Orthogonal to each other!

q = Cx

H

q = 0

q = 0

y

Linearly Varying Temperature B.C.

0

W

x

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.

For i condition as shown in the figure. Determine steady-state temperature distribution.th heat tube and jth isothermal block :

Where n is number of isothermal blocks. 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 total heat flow is:

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 Resistance R total heat flow is: th

Thermal Model for Microarchitecture Studies total heat flow is:

• 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 total heat flow is:

• 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 total heat flow is:

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 model and a lateral model.

• 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