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For thermally fully developed flow,
∂
T − T w = 0
(8.20)
∂x T b − T w
T − T w = f (r) = f (x)
T b − T w
∂T
∂T w
T − T w ∂T b
∂T w
−
−
−
= 0
∂x
∂x
T b − T w ∂x
∂x
'
-v
'
=0
∂T
∂T w
∂T b
=
=
= constant
(8.21)
∂x
∂x
∂x
From energy equation 8.4,
∂T
1 ∂
∂T
ρC p u
= k
r
= T(r) only
(8.22)
∂x
r ∂r
∂r
�
�
( ) 2
�
""
� �
�
ρC p
r
q
dT
r
2V 1 −
w
dr = d r
k
R
mC p /πD
˙
dr
'
-v
' ' -v '
u
∂T
∂x
And thermal BCs are
∂T
r = 0, T = T c or
= 0
∂r
r = R, T = T w
ρC p dT
r 2
r 4
T − T c =
−
U max
k dx
4
16R 2
ρC p ∂T
R 2
R 4
ρC p ∂T
3
T w − T c =
U max
−
=
U max R
2
k ∂x
4
16R 2
k ∂x
16
where
� R
0 ρuT · 2πr dr
7 ρC p dT
T b = � R
= T c +
U max R
2
96 k dx
0 ρu · 2πr dr
ρC p dT
3
7
T w − T b =
U max
R
2
− R
2
(8.23)
k dx
16
96
174
Analytical Heat Transfer