Loading...

PPT – Multi Dimensional Steady State Heat Conduction PowerPoint presentation | free to download - id: 68f184-NGNhO

The Adobe Flash plugin is needed to view this content

Multi 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

Heat treatment of Metal bars rods

Heat Flow in Complex Geometries (Casting Process)

Microarchitecture of Pentium 4

Thermal Optimization of Microarchitecture of an IC

- Microprocessor power densities escalating rapidly

as technology scales below 100nm level. - There is an urgent need for developing

innovative cooling solutions. - The concept of power-density aware thermal floor

planning is a recent method to reduce maximum

on-chip temperature. - A careful arrangement of components at the

architecture level, the average reduction in peak

temperature of 15C. - A tool namely, Architectural-Level Power

Simulator (ALPS), allowed the Pentium 4 processor

team to profile power consumption at any

hierarchical level from an individual FUB to the

full chip. - The ALPS allowed power profiling of everything,

from a simple micro-benchmark written in

assembler code, to application-level execution

traces gathered on real systems. - At the most abstract level, the ALPS methodology

consists of combining an energy cost associated

with performing a given function with an estimate

of the number of times that the specific function

is executed.

- The energy cost is dependent on the design of the

product, while the frequency of occurrence for

each event is dependent on both the product

design and the workload of interest. - Once these two pieces of data are available,

generating a power estimate is simple - multiply the energy cost for an operation

(function) by the number of occurrences of that

function, - sum over all functions that a design performs,
- and then divide by the total amount of time

required to execute the workload of interest.

Need for Thermal Optimization

Thermal Management Mechanism in Pentium 4

- The Pentium 4 processor implements mechanisms to

measure temperature accurately using the thermal

sensor. - In the case of a microprocessor, the power

consumed is a function of the application being

executed. - In a large design, different functional blocks

will consume vastly different amounts of power,

with the power consumption of each block also

dependent on the workload. - The heat generated on a specific part of the die

is dissipated to the surrounding silicon, as well

as the package. - The inefficiency of heat transfer in silicon and

between the die and the package results in

temperature gradients across the surface of the

die. - Therefore, while one area of the die may have a

temperature well below the design point, another

area of the die may exceed the maximum

temperature at which the design will function

reliably. - Figure is an example of a simulated temperature

plot of the Pentium 4 processor.

(No Transcript)

Thermal Optimization of Floor Plan

Initial Model

Low Cooling cost Model

General Conduction Equation

- Conduction is governed by relatively

straightforward partial differential equations

that lend themselves to treatment by analytical

methods if the geometries are simple enough and

the material properties can be taken to be

constant. - The general form of these equations in

multidimensions is

For Rectangular Geometry

Steady conduction in a rectangular plate

Boundary conditions x 0 0 lt y lt H

T(0,y) f0(y) x W 0 lt y lt H T(W,y)

fH(y) y 0 0 lt x lt W T(x,o) g0(x) y H

0 lt x lt W T(x,H) gW(x)

H

y

0

W

x

Write the solution as a product of a function of

x and a function of y

Substitute this relation into the governing

relation given by

Rearranging above equation gives

Both sides of the equation should be equal to a

constant say l2

Above equation yields two equations

The form of solution of above depends on the sign

and value of l2. The only way that the correct

form can be found is by an application of the

boundary conditions. Three possibilities will be

considered

Integrating above equations twice, we get

The product of above equations should provide a

solution to the Laplace equation

Linear variation of temperature in both x and y

directions.

i.e. l2 -k2

Integration of above ODEs gives

Solution to the Laplace equation is

Asymptotic variation in x direction and harmonic

variation in y direction

i.e. l2 k2

Integration of above ODEs gives

Solution to the Laplace equation is

Harmonic variation in x direction and asymptotic

variation in y direction.

Summary of Possible Solutions

Steady conduction in a rectangular plate2D SPACE

All Dirichlet Boundary Conditions

q C

Define

T T2

H

T T1

T T1

Laplace Equation is

q 0

q 0

y

0

W

x

T T1

q 0

l20 Solution

Simultaneous Equations

The solution corresponds to l20, is not a valid

solution for this set of Boundary Conditions!

l2 lt 0 or l2 gt 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 gt 0 is a possible solution !

0

W

x

q 0

Substituting boundary conditions

(No Transcript)

Where n is an integer.

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

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

obtain

Substituting these Fourier integrals in to

solution gives

And hence

Substituting f(x) T2 - T1 into above equation

gives

Therefore

Isotherms and heat flow lines are Orthogonal to

each other!

Linearly Varying Temperature B.C.

q Cx

H

q 0

q 0

y

0

W

x

(No Transcript)

Sinusoidal Temperature B.C.

q Cx

H

q 0

q 0

y

0

W

x

(No Transcript)

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

(No Transcript)

(No Transcript)

For the statement of above case, consider a new

boundary condition as shown in the figure.

Determine steady-state temperature distribution.

(No Transcript)

(No Transcript)

Where n is number of block. If we assume Dy Dx,

then

If m is a total number of the heat flow lanes,

then the total heat flow is

(No Transcript)

Thermal Resistance Rth

(No Transcript)

(No Transcript)

(No Transcript)

Shape Factor for Standard shapes

(No Transcript)

(No Transcript)

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