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where
Af
Af is the total, including the tip, fin surface area. Substitution of
Equation (9) into Equation (12) would yield Equation (11):
The second tip condition, case B, corresponds to the assumption that
the convective heat loss from the fin tip is negligible, in which case the tip may
be treated as adiabatic and
dθ
dx
x =L
=0
-------- (13)
Substituting from Equation (5) and dividing by m, we then obtain
C
mL −mL
1
e −C2e
=
Using this expression with Equation (8) to solve for C 1 and C 2 and
substituting the results into Equation (5), we obtain
θ cosh m(
L −x)
=
θ cosh mL
b
0
--------- (14)
Using this temperature distribution with Equation (10), the fin heat
transfer rate is then
q
f
= hPkA
cθb
tanh mL ----- (15)
In the same manner, we can obtain the fin temperature distribution
and, heat transfer rate for case C, where the temperature is prescribed at the
fin tip. That is, the second boundary condition is θ ( L)
= θ , and the resulting
expressions are of the form
θ
=
θ
b
( θ θ )
L
/
b
sinh mx + sinh m(
L − x)
sinh mL
L
------- (16)
q
f
=
cosh mL −θ
L
/ θb
hPkA
cθb
sinh mL
The very long fin, case D, is an interesting extension of these results.
In particular, as L →∞, θ →0
and it is easily verified that
L
θ
=e −
θ
b
mx
q
f
=
hPkA
c
θ
b
Fin Performance
Fins are used to increase the heat transfer from a surface by
increasing the effective surface area. However, the fin itself represents a
conduction resistance to heat transfer from the original surface. For this