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 feeling the
truth as it is
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Heat treatment of Metal bars rods
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Heat Flow in Complex Geometries (Casting Process)
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Microarchitecture of Pentium 4
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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.

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• 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.

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Need for Thermal Optimization
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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.

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Thermal Optimization of Floor Plan
Initial Model
Low Cooling cost Model
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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
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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
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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
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Rearranging above equation gives
Both sides of the equation should be equal to a
constant say l2
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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
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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.
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i.e. l2 -k2
Integration of above ODEs gives

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

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Solution to the Laplace equation is
Harmonic variation in x direction and asymptotic
variation in y direction.
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Summary of Possible Solutions
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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
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l20 Solution
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Simultaneous Equations
The solution corresponds to l20, is not a valid
solution for this set of Boundary Conditions!
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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
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Substituting boundary conditions
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Where n is an integer.
Solution domain is a superset of geometric domain
!!!
Recognizing that
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where the constants have been combined and
represented by Cn
Using the final boundary condition
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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
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Substituting these Fourier integrals in to
solution gives
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And hence
Substituting f(x) T2 - T1 into above equation
gives
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Therefore
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Isotherms and heat flow lines are Orthogonal to
each other!
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Linearly Varying Temperature B.C.
q Cx
H
q 0
q 0
y
0
W
x
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Sinusoidal Temperature B.C.
q Cx
H
q 0
q 0
y
0
W
x
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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
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For the statement of above case, consider a new
boundary condition as shown in the figure.
Determine steady-state temperature distribution.
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Where n is number of block. If we assume Dy Dx,
then
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If m is a total number of the heat flow lanes,
then the total heat flow is
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Thermal Resistance Rth
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Shape Factor for Standard shapes
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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

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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.

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

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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