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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…. 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

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
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

slide7
Construction of a Fourier series expansion of the boundary values is facilitated by rewriting previous equation as:

where

slide11
And hence

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

slide13
Isotherms and heat flow lines are

Orthogonal to each other!

sinusoidal temperature b c
Sinusoidal Temperature B.C.

q = Cx

H

q = 0

q = 0

y

0

W

x

principle of superposition

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…

slide21
For the statement of above case, consider a new boundary condition as shown in the figure. Determine steady-state temperature distribution.
slide25
If m is a total number of the heat flow lanes, then the total heat flow is:

Where S is called Conduction Shape Factor.

slide26
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
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
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
Multi-dimensional Conduction in Die

The die layer is divided into blocks that correspond to the microarchitectural blocks of interest and their floorplan.

slide37
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
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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