Slender fins with variable cross sectional area
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Slender Fins with Variable Cross-Sectional Area. P M V Subbarao Professor Mechanical Engineering Department IIT Delhi. Geometry Decides the Volume of Material …. L. q b. x. b. x=b. x=0. LONGITUDINAL FIN OF TRIANGULAR PROFILE. The differential equation for temperature excess :.

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Slender Fins with Variable Cross-Sectional Area

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Slender Fins with Variable Cross-Sectional Area

P M V Subbarao

Professor

Mechanical Engineering Department

IIT Delhi

Geometry Decides the Volume of Material …


L

qb

x

b

x=b

x=0

LONGITUDINAL FIN OF TRIANGULAR PROFILE

The differential equation for temperature excess :


For a slender fin:


Define Fin Factor m

Define Temperature excess:


The particular solution for

is:

The differential equation for temperature excess is a form

of Bessel’s equation of 0th order:

The fin heat dissipation is:

The fin efficiency is:


Optimum Shapes : Triangular Fin

L=1

With profile area

This makes


For Least material


Optimum Shapes

Iterative solving yields bT=2.6188 and


Performance of Optimally Designed TRIANGULAR PROFILE (L=1)

Heat dissipated:

For optimum fin width


And solve for Ap


Comparison of Strip Fins

Rectangular Profile:

Triangular Profile:

For the same material, surrounding conditions and

which is basically the user’s design requirement.

Triangular profile requires only about 68.8% as much metal as rectangular profile.


Selection of Material

Rectangular Profile:

Triangular Profile:

Consider three materials:

Steel

Aluminum

Copper

7249

2704

8895

43.3

202.5

389.4


Comparison of Longitudinal Fin (cont.)

Fin mass is proportional to Ap & r.

Apis inversely proportional to thermal conductivity.

For given h, qb, and qb:


L

qb

x

b

x=b

x=a=0

STRIP FIN OF CONCAVE PARABOLIC PROFILE

The differential equation for

temperature excess is an

Euler Differential Equation


nth Order Strip Fins


L

qb

x

b

x=b

x=a=0

Optimum Shape of Parabolic Profile

Solution of Euler equation:

The temperature excess is linear if p=1.

For p=1:


The heat dissipated will be:

And the efficiency will be:


Performance of Strip Fins

h

mb


Size of A FinVs Number of Fins

In both (strip and triangular) fins, profile area varies as :

To double the heat flow make one fin eight times as large

or you use two fins !!!

In pin fin, profile volume varies as

To double the heat flow, make one fin 3.17 times as large or use two fins.

More number of small fins are better than …...


Pentium III

Pentium IV

Pentium II


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