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as the following format:
(
)
2
2
x
d x
dθ + m x
2
− ν θ = 0
(2.48)
dx
dx
θ = a 0 J ν (mx) + a 1 Y ν (mx), ν = 0, 1, 2, . . .
(2.49)
2. For heat loss problem—Modified Bessel function: The solutions of
Equation 2.47 can be the typical modified Bessel function with heat
loss as the following format:
(
)
2
2
x
d x
dθ − m x
2
+ ν θ = 0
(2.50)
dx
dx
θ = a 0 I ν (mx) + a 1 K ν (mx), ν = 0, 1, 2, . . .
(2.51)
Comparing Equations 2.47 and 2.50, r = x, the general solution of Equation
2.47 is the same format as Equation 2.51 with ν = 0
θ(mr) = a 0 I 0 (mr) + a 1 K 0 (mr)
(2.52)
with the following boundary conditions:
At the fin base, r = r 1 , T = T b , then θ = T b − T ∞ = θ b .
At the fin tip, r = r 2 , there are four possible cases as discussed before,
1. Convective boundary
∂T �
∂θ(r 2 )
−h
−k
= h(T r2 − T ∞ ) or
=
θ r 2
∂r
∂r
k
r=r 2
2. The tip fin is insulated
∂T �
∂θ(r 2 )
−k
= 0 or
= 0
∂r
∂r
r=r 2
3. Top tip temperature is given
T| r=r 2 = T r 2 or θ(r 2 ) = T r 2 − T ∞ = θ r 2
4. For a long fin r 2 /t > 10 ∼ 20
T| r=r 2 = T ∞ or θ(r 2 ) = T ∞ − T ∞ = 0
For case 2, the fin tip is insulated, for example:
At r = r 1 , θ = θ b = a 0 I 0 (mr 1 ) + a 1 K 0 (mr 1 ),
∂θ(r 2 )
dI 0 (mr) �
dK 0 (mr) �
At r = r 2 ,
= a 0
+ a 1
∂r
dr
dr
r=r 2
r=r 2
= a 0 mI 1 (mr 2 ) − a 1 mK 1 (mr 2 ) = 0,
29
1-D Steady-State Heat Conduction