Heat Transfer for Fins (or Extended Surfaces) Handout Table 3.4 Fin temperature distribution,

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θ = T −T∞, and heat loss, qf, for uniform cross section Table 3.5 Fin Efficiency,

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η f , for common fin shapes where

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θb = Tb −T∞

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η f =actual heat transfer rate with fin

maximum heat transfer rate with fin=

qfh Af θb

<1

Fin effectiveness:

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ε f =heat transfer rate with fin

heat transfer rate without fin=

q f

h Ac,b θb

=η f

Af

Ac,b

⎛

⎝ ⎜

⎞

⎠ ⎟

Resistance Analogy: Single fin,

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θb = q f Rt, f , or fin array,

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θb = qt Rt,o , with thermal resistances:

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Rt, f =1

η f h Af

single fin

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Rt,N =1

N η f h Af

N fins in parallel

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Rt,b =1

h Ab

convection from exposed base

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Rt,o =1

Rt,N

+1

Rt ,b

⎛

⎝ ⎜

⎞

⎠ ⎟

−1

overall for fin array without contact resistance

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Rt,o =ʹ ́ R t ,c

N Ac,b

+ Rt,N

⎛

⎝ ⎜

⎞

⎠ ⎟

−1

+1

Rt ,b

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

−1

overall for fin array with contact resistance

Overall Efficiency: ηo =qt

h At θb=

1Rt,o h At

=1− NAf

At

⎛

⎝⎜

⎞

⎠⎟ 1−

η f

C1

⎛

⎝⎜

⎞

⎠⎟ , C1 =1+η f h Af

ʹ́Rt,cAc,b

⎛

⎝⎜⎜

⎞

⎠⎟⎟

A total base area (without fins) Ac fin cross-sectional area Ac,b fin cross-sectional area at base ( Ac = Ac,b for constant area) Af fin surface area Ab area of the exposed base (

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Ab = A − N Ac,b) At total fin array surface area (

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At = N Af + Ab)