dx
hA s (T−T ∞ )
h, T ∞
q f
q x
x
x
dq
q x
dx
+ dx
(a)
t
r 1
r 2
dr
(b)
h, T ∞
q f
(c)
(d)
dx
0
x
r o
t
o
r
h, T ∞
h, T ∞
q f
A c = 2πr · t. The fin perimeter including the top and the bottom, P = 2 · 2πr.
Let θ = T − T ∞ , Equation 2.46 becomes
d
dr
�
2πrt
dθ
dr
�
−
h · 4πr
k
θ = 0
d
dr
�
r
dθ
dr
�
−
2 hr
kt
θ = 0
r
d
dr
�
r
dθ
dr
�
−
2 hr
2
kt
θ = 0
(2.47)
r
d
dr
�
r
dθ
dr
�
− m
2 r
2
θ = 0
where m 2 = 2h/kt.
1. For heat generation problem—Bessel function: The solutions of
Equation 2.47 can be a typical Bessel function with heat generation
28
Analytical Heat Transfer
FIGURE 2.9
One-dimensional heat conduction through thin fins with variable cross-section area. (a) Conical
fin; (b) annular fin; (c) taper fin; (d) disk fin.
hA s (T−T ∞ )
h, T ∞
q f
q x
x
x
dq
q x
dx
+ dx
(a)
t
r 1
r 2
dr
(b)
h, T ∞
q f
(c)
(d)
dx
0
x
r o
t
o
r
h, T ∞
h, T ∞
q f
A c = 2πr · t. The fin perimeter including the top and the bottom, P = 2 · 2πr.
Let θ = T − T ∞ , Equation 2.46 becomes
d
dr
�
2πrt
dθ
dr
�
−
h · 4πr
k
θ = 0
d
dr
�
r
dθ
dr
�
−
2 hr
kt
θ = 0
r
d
dr
�
r
dθ
dr
�
−
2 hr
2
kt
θ = 0
(2.47)
r
d
dr
�
r
dθ
dr
�
− m
2 r
2
θ = 0
where m 2 = 2h/kt.
1. For heat generation problem—Bessel function: The solutions of
Equation 2.47 can be a typical Bessel function with heat generation
28
Analytical Heat Transfer
FIGURE 2.9
One-dimensional heat conduction through thin fins with variable cross-section area. (a) Conical
fin; (b) annular fin; (c) taper fin; (d) disk fin.