�
�
�
�
�
�
Assuming velocity and temperature profiles to satisfy boundary-layer
conditions,
( )
u = u x, y = u 1 η (1 − η)
2
(9.24)
( )
T − T ∞
2
= (1 − η) = f x, y
(9.25)
T w − T ∞
where η = y/δ(x); u 1 = c 1 x m ; δ (x) = c 2 x n ;
1
1
m = ; n =
2
4
Put this into momentum and energy integral equations:
� � 1/2 �
� 1/2
80
G x
ν
u 1 (x) = 3
(20/21) + Pr
x
� 1/4
� 1/4
δ (x)
240 (1 + (20/21Pr))
x
=
→ δ (x) ∼
x
Pr · G x
T w − T ∞
∂T �
2k (T w − T ∞ )
q w = −k
=
= h (T w − T ∞ )
∂y
δ (x)
0
h x x
2x
Pr
Nu x =
=
=
� 1/4
· Ra
1
x
/4
(9.26)
k
δ (x)
15 ((20/21) + Pr)
∼
Nu x = 0.413Ra x
1/4 for Pr = 0.733
(9.27)
Note: Nu x = 0.359Ra
1
x
/4 for Pr = 0.733 by using the exact similarity solution.
gβ (T w − T ∞ ) x 3
Ra x =
< 10
8
− 10
9
− Laminar natural convection
να
192
Analytical Heat Transfer
Remarks
In the undergraduate-level heat transfer, we have heat transfer correlations
of external natural convection for a vertical plate, an inclined plate, a horizontal plate, a vertical tube, and a horizontal tube as well as heat transfer
correlations of internal natural convection for a horizontal tube, between two
parallel plates, and inside a rectangular cavity with various aspect ratios.
These correlations are important for many real-life engineering applications
such as electronic components.
In the intermediate-level heat transfer, this chapter focuses on how to
analytically solve the external natural convection from a vertical plate at