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Multi Dimensional Steady State Heat Conduction

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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 ... – PowerPoint PPT presentation

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Title: Multi Dimensional Steady State Heat Conduction


1
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
2
Heat treatment of Metal bars rods
3
Heat Flow in Complex Geometries (Casting Process)
4
Microarchitecture of Pentium 4
5
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.

6
  • 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.

7
Need for Thermal Optimization
8
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.

9
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10
Thermal Optimization of Floor Plan
Initial Model
Low Cooling cost Model
11
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
12
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
13
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
14
Rearranging above equation gives
Both sides of the equation should be equal to a
constant say l2
15
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
16
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.
17
i.e. l2 -k2
Integration of above ODEs gives

18
Solution to the Laplace equation is
Asymptotic variation in x direction and harmonic
variation in y direction
19
i.e. l2 k2
Integration of above ODEs gives

20
Solution to the Laplace equation is
Harmonic variation in x direction and asymptotic
variation in y direction.
21
Summary of Possible Solutions
22
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
23
l20 Solution
24
Simultaneous Equations
The solution corresponds to l20, is not a valid
solution for this set of Boundary Conditions!
25
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
26
Substituting boundary conditions
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28
Where n is an integer.
Solution domain is a superset of geometric domain
!!!
Recognizing that
29
where the constants have been combined and
represented by Cn
Using the final boundary condition
30
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
31
Substituting these Fourier integrals in to
solution gives
32
And hence
Substituting f(x) T2 - T1 into above equation
gives
33
Therefore
34
Isotherms and heat flow lines are Orthogonal to
each other!
35
Linearly Varying Temperature B.C.
q Cx
H
q 0
q 0
y
0
W
x
36
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37
Sinusoidal Temperature B.C.
q Cx
H
q 0
q 0
y
0
W
x
38
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39
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
40
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42
For the statement of above case, consider a new
boundary condition as shown in the figure.
Determine steady-state temperature distribution.
43
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44
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45
Where n is number of block. If we assume Dy Dx,
then
46
If m is a total number of the heat flow lanes,
then the total heat flow is
47
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48
Thermal Resistance Rth
49
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52
Shape Factor for Standard shapes
53
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55
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

56
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.

57
Multi-dimensional Conduction in Die
The die layer is divided into blocks that
correspond to the microarchitectural blocks of
interest and their floorplan.
58
  • 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.

59
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
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