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 :.
Slender Fins with Variable Cross-Sectional Area
P M V Subbarao
Mechanical Engineering Department
Geometry Decides the Volume of Material …
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
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
With profile area
For Least material
Iterative solving yields bT=2.6188 and
Performance of Optimally Designed TRIANGULAR PROFILE (L=1)
For optimum fin width
And solve for Ap
Comparison of Strip Fins
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
Consider three materials:
Comparison of Longitudinal Fin (cont.)
Fin mass is proportional to Ap & r.
Apis inversely proportional to thermal conductivity.
For given h, qb, and qb:
STRIP FIN OF CONCAVE PARABOLIC PROFILE
The differential equation for
temperature excess is an
Euler Differential Equation
Optimum Shape of Parabolic Profile
Solution of Euler equation:
The temperature excess is linear if p=1.
The heat dissipated will be:
And the efficiency will be:
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 …...