Optimal Fin Shapes

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Optimal Fin Shapes Profiles

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• IIT Delhi

Geometrical Optimization is the Basic Goal of
Optimal Design .
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CYLINDRICAL SPINE (Pin Fins)
Pin fin with adiabatic tip and corrected height
T? h?
d
Tb
b
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Pin Fins Profile Optimization
Sonn and Bar-Cohen (1981) developed an
optimization method based on minimization of
the spine volume. The objective function is to
maximize heat dissipation for a given volume.
With
So that
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Optimization Pin Fin Profile
We find the point where
The results is the transcendental equation
Where
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Trial and error method of root finding, gives
Volume of maximum heat dissipating pin fin
Or
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For Strip fin
For pin fin
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LONGITUDINAL FIN OF TRIANGULAR PROFILE
The differential equation for temperature excess

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The differential equation for temperature excess
is a form of Bessels equation
The fin heat dissipation is
The fin efficiency is
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Optimum Shapes Triangular Fin
L1
With
This makes
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Optimum Shapes
Iterative solving yields bT2.6188 and
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Comparison of Longitudinal Fins
Rectangular Profile Triangular Profile
For the same material, surrounding conditions and
which is basically the users design requirement.
Triangular profile requires only about 68.8 as
much metal as rectangular profile.
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Capacity Enhancement of Fins
To double the heat flow, you use two fins or make
one fin eight times as large.
In pin fin, profile volume varies as
To double the heat flow, you use two fins or make
one fin 3.17 times as large.
There is a virtue in using more number of small
fins.
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Design and Optimization of Fin Arrays
P M V Subbarao Mechanical Engineering
Department IIT Delhi
Millions of Ants are more Powerful than a Single
Cobra
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Geometry of Fin Array
tf
S
b
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Determination of Heat Transfer Coefficient
S
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Optimum spacing
b
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Optimum Natural Convection Array

• From Elenbaas (1942)

• For an array of optimally spaced fins

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Towards an Optimum Array of Optimum Fins

• Heat flow from each optimum fin

• With the h for Optimum Spacing

• With the Interfin Spacing

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Industrial Practice

• Define the thermal resistance of the heat sink is
  given by

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Selection Curves
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Forced Convection Heat Sinks

• Analytical modeling
• Maximization of heat dissipation
• Least-material optimization
• Design for manufacture

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Design Calculations for Fin Arrays Thermal
Resistance

• In order to select the appropriate heat sink, the
  thermal designer must
• first determine the maximum allowable heat sink
  thermal resistance.
• To do this it is necessary to know the
• maximum allowable module case temperature, Tcase,
• the module power dissipation, Pmod, and
• the thermal resistance at the module-to-heat sink
  interface, Rint.
• The maximum allowable temperature at the heat
  sink attachment surface, Tbase, is given by

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• The maximum allowable heat sink resistance, Rmax,
  is then given by
• The thermal resistance of the heat sink is given
  by
• parameters the gap, b, between the fins may be
  determined from

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Constant air velocity
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Constant volumetric flow rate
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Heat Sink Pressure Drop

• To determine the air flow rate it is necessary to
  estimate the heat sink pressure drop as a
  function of flow rate and match it to a curve of
  fan pressure drop versus flow rate.
• A method to do this, using equations presented
  here.
• As in the previous article, the heat sink
  geometry and nomenclature used is that shown
  Figure 1.

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Pressure Drop Curves
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Effect of number of fins and fin height
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Thermal Resistance
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Closure

• a fan with a different fan curve is employed, the
  flow rates will change and the optimum heat sink
  design point may change as well.
• The important point is that to determine how a
  heat sink will perform in a given application
  both its heat transfer and pressure drop
  characteristics must be considered in concert
  with the pressure-flow characteristics of the fan
  that will be used.
• It should also be noted that an underlying
  assumption is that all the flow delivered by the
  fan is forced to go through the channels formed
  between the heat sink fins.
• Unfortunately this is often not the case and much
  of the air flow delivered by the fan will take
  the flow path of least resistance bypassing the
  heat sink.
• Under such circumstances the amount of flow
  bypass must be estimated in order to determine
  the heat sink performance.