�
�
�
�
�
�
�
�
where applying the properties of I and K,
d
d
I 0 (mr) = mI 1 (mr) and
K 0 (mr) = −mK 1 (mr)
dr
dr

Solve for a 0 and a 1

K 1 (mr 2 )

a 0 =
θ b
I 0 (mr 1 )K 1 (mr 2 ) + I 1 (mr 2 )K 0 (mr 1 )

I 1 (mr 2 )

a 1 =
θ b
I 0 (mr 1 )K 1 (mr 2 ) + I 1 (mr 2 )K 0 (mr 1 )
one obtains the temperature distribution as
θ(r)
T(r) − T ∞
I 0 (mr)K 1 (mr 2 ) + I 1 (mr 2 )K 0 (mr)
=
=
(2.53)
θ b
T b − T ∞
I 0 (mr 1 )K 1 (mr 2 ) + I 1 (mr 2 )K 0 (mr 1 )
Therefore, heat transfer through the fin base can be calculated as
dθ �
dθ �
q f = −kA c
= −k(2πr 1 )(t)
dx
dr
r 1
r=r 1
I 1 (mr 1 )K 1 (mr 2 ) − I 1 (mr 2 )K 1 (mr 1 )
q f = 2πkr 1 tm
θ b
(2.54)
I 0 (mr 1 )K 1 (mr 2 ) + I 1 (mr 2 )K 0 (mr 1 )
And the fin efficiency can be determined as
q f
η f =
h2π(r 2
2 − r 1
2 )θ b
It is important to point out that temperature decreases from the fin base
to the fin tip depending on the specified fin tip boundary conditions. The
temperature decay curve again is a combination of Bessel function I 0 and K 0 .
The characteristics of Bessel functions J, Y, I, and K, and their derivatives are
shown in Figure 2.10.
2.4.1 Radiation Effect
""
If we also consider radiation effect, q = constant = positive value, then the
r
""
above solution can be used by replacing θ = T − T ∞ − (q /h). However, if we
r
""
consider q = −εσ(T 4 − T 4 ), and T ∞ = T sur , h r
∞ )(T + T ∞ ), the
= εσ(T 2 + T 2
r
sur
energy balance equation 2.46 can be written as
d
d(T − T ∞ )
(h + h r )P
A c
−
(T − T ∞ ) = 0
dx
dx
k
The solution of the above equation can be obtained by numerical
integration.
30
Analytical Heat Transfer