�
temperature profiles from the heated vertical wall, for different Prandtl
fluids.
u
T ∞
f
"
= √
2 gx T w − T ∞
ux G
−1/2
=
x
2ν
u
= (
)
(9.13)
G
1/2
x (2ν/x)
From the dimensionless velocity and temperature profiles, the associated heat flux and the heat transfer coefficient (or Nusselt number) can be
y=0
determined.
q
""
w = −k
∂T �
k
Gr x
∂y
x
4
�
�
�
0
= − (T w − T ∞ )
�
� 1/4 dθ �
dη
�
�
�
y=0
(9.14)
where
dθ �
dη
�
�
�
y=0
= θ (0) = f (Pr)
"
∴ h =
q ""
w
T w − T ∞
=
−k(∂T/∂y) 0
T w − T ∞
Nu =
hx
k
= −
� Gr x
4
� 1/4 dθ �
dη
�
�
�
(9.15)
Numerical results: for laminar natural convection:
Pr
0.01
0.733
−
dθ
dη
�
�
�
�
0
0.081 0.508
For air, Pr = 0.733,
1
0.567
2
0.716
10
1.169
100
2.191
1000
3.966
Nu x =
hx
k
= 0.359Gr
1/4
x
Nu x =
h x L
k
=
4
3
Nu L = 0.478Gr
1/4
L
(9.16)
(9.17)
189
Natural Convection