�
�
�
where
L
�
�
� 1/4
�
1
4 k Gr L
dθ �
4
h x =
h x dx = −
= h L
L
3 L
4
dη
3
0
0
gβ(T w − T ∞ )L 3
Gr L =
(9.18)
ν 2
� 1/4
3
2Pr
1/4
Nu x =
(
)
(Gr x Pr)
(9.19)
4 5 1 + 2 Pr 1/2 + 2 Pr
Nu x = 0.6(Gr x Pr
2 )
1/4 , if Pr → 0
(9.20)
Nu x = 0.503(Gr x Pr)
1/4 , if Pr → ∞
(9.21)
In general,
b
Nu x = a(Gr x Pr) = a Ra
b
x
where Rayleigh number,
gβ(T w − T ∞ )x 3
Gr x Pr = Ra x =
να
Compared to forced convection
Nu x = a Re
m Pr
n
x
Note: The following is a simple guideline whether the problem can be solved
by forced convection, natural convection, or mixed (combined forced and
natural) convection.
If Gr x /Re 2 < 1, the problem can be treated as forced convection.
x
∼
If Gr x /Re 2 = 1, the problem can be treated as mixed convection.
x

If Gr x /Re 2 > 1, the problem can be treated as natural convection.

x
190
Analytical Heat Transfer
9.2 Laminar Natural Convection on a Vertical Wall: Integral
Method
Integral approximate solution by Pohlhausen 1921: We apply the momentum and
energy balance to the control volume across the boundary layer as shown in
Figure 9.3.