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Fins: Overview• Fins

– extended surfaces that enhance fluid heat transfer to/from a surface in large part by increasing the effective surface area of the body

– combine conduction through the fin and convection to/from the fin

• the conduction is assumed to be one-dimensional

• Applications– fins are often used to enhance convection when h is

small (a gas as the working fluid)– fins can also be used to increase the surface area

for radiation– radiators (cars), heat sinks (PCs), heat exchangers

(power plants), nature (stegosaurus) Straight fins of (a) uniform and (b) non-uniform cross sections; (c) annularfin, and (d) pin fin of non-uniform cross section.

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Fins: The Fin Equation• Solutions

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Bessel Equations

with solution

Form of Bessel equation of order

J = Bessel function of first kind of order

Y = Bessel function of second kind of order

with solution

Form of modified Bessel equation of order

I = modified Bessel function of first kind of order

K = modified Bessel function of second kind of order

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Bessel Functions – Recurrence Relations

OR

AND

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Fins: Fin Performance Parameters• Fin Efficiency

– the ratio of actual amount of heat removed by a fin to the ideal amount of heat removed if the fin was an isothermal body at the base temperature

• that is, the ratio the actual heat transfer from the fin to ideal heat transfer from the fin if the fin had no conduction resistance

• Fin Effectiveness– ratio of the fin heat transfer rate to the heat transfer rate that would exist

without the fin

• Fin Resistance– defined using the temperature difference between the base and fluid as

the driving potential

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Fins: Efficiency

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Fins: Efficiency

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Fins: Arrays• Arrays

– total surface area

– total heat rate

– overall surface efficiency

– overall surface resistance

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Fins: Thermal Circuit• Equivalent Thermal Circuit

• Effect of Surface Contact Resistance

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sinh Function

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Fourier SeriesConsider a set of eigenfunctions ϕn that are orthogonal, where orthogonality is defined as

for m ≠ n

An arbitrary function f(x) can be expanded as series of these orthogonal eigenfunctions

or

Due to orthogonality, we thus know

all other ϕnAmϕm integrate to zero because m ≠ n

Thus, the constants in the Fourier series are