dP ∞
dx
g
x
T
T w , ρ w
q″ w
τ w
∫ ρuu dy
x ∫
∫
ρuu dy dx
P dy dx
x
Pdy
∫
∫
∂
∂
+
∫ Pdy
ρuu dy ∂
∂
+
T ∞ , ρ ∞
U ∞ = 0
y
�
�
�
�
�
�
�
�
�
�
�
Net force = Momentum change
−τ w − ρg dy −
∂P
∂ ρuu dy
dy =
∂x
∂x
where ∂P/∂x = ∂P ∞ /∂x = −ρ ∞ g (outside of the boundary layer).
Also ρ − ρ ∞ = −ρ ∞ β (T − T ∞ ) from β ≡ −1/ρ (∂ρ/∂T) P
�
δ
δ
(T − T ∞ ) dy −
d
dx
τ w
μ ∂u
u
2 dy
(9.22)
= gβ
=
ρ ∞
ρ ∞ ∂y 0
0
0
δ T
q
∂T �
d
""
= −k
� =
∂y
dx
0
ρc p u (T − T ∞ ) dy
(9.23)
0
Boundary conditions: u (x, 0) = 0 T (x, 0) = T w
u (x, δ) = 0, T (x, δ T ) = T ∞
∂u (x, δ)
∂T (x, δ T )
= 0,
= 0
∂y
∂y
FIGURE 9.3
Integral method.
191
Natural Convection
dx
g
x
T
T w , ρ w
q″ w
τ w
∫ ρuu dy
x ∫
∫
ρuu dy dx
P dy dx
x
Pdy
∫
∫
∂
∂
+
∫ Pdy
ρuu dy ∂
∂
+
T ∞ , ρ ∞
U ∞ = 0
y
�
�
�
�
�
�
�
�
�
�
�
Net force = Momentum change
−τ w − ρg dy −
∂P
∂ ρuu dy
dy =
∂x
∂x
where ∂P/∂x = ∂P ∞ /∂x = −ρ ∞ g (outside of the boundary layer).
Also ρ − ρ ∞ = −ρ ∞ β (T − T ∞ ) from β ≡ −1/ρ (∂ρ/∂T) P
�
δ
δ
(T − T ∞ ) dy −
d
dx
τ w
μ ∂u
u
2 dy
(9.22)
= gβ
=
ρ ∞
ρ ∞ ∂y 0
0
0
δ T
q
∂T �
d
""
= −k
� =
∂y
dx
0
ρc p u (T − T ∞ ) dy
(9.23)
0
Boundary conditions: u (x, 0) = 0 T (x, 0) = T w
u (x, δ) = 0, T (x, δ T ) = T ∞
∂u (x, δ)
∂T (x, δ T )
= 0,
= 0
∂y
∂y
FIGURE 9.3
Integral method.
191
Natural Convection