�
�
�
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1
dT b
dT
C
(R − y)V
= (α + ε H )
+
2
dx
−dy R − y
where C = 0 at R − y = 0, (dT/dy) = 0.
Therefore,
1 dT b y − R
dT = V
dy
2 dx α + ε H
T
y
1 dT b
�
0
y − R
α + ε H
dy
dT = V
2 dx
T w
y
y +
""
1 − (y + /R + )
(1/Pr) + (ε H /ν)
1 ∂T b y − R
q
√
dy
+
w
dy =
T − T w = V
2
ρC p
α + ε H
∂x
τ w /ρ
0
0
Therefore,
y +
1 − (y + /R + )
(1/Pr) + (ε H /ν)
T w − T
T
+
≡
(q "" /ρC p u ∗ )
w
=
+
dy
(10.45)
0
The above equation can be integrated if one assumes
ε H
ε m
ε m
≈
or Pr t =
≈ 1
ν
ν
ε H
And from Equation 10.19,
ε m
1 − (y + /R + )
=
− 1
υ
(du + /dy + )
For the laminar sublayer region,
du +
+
+
+
ε m
0 ≤ y
+
≤ 5, y « R
+ , u = y ,
= 1,
= 0
dy +
ν
For the buffer zone,
+
du +
5
5 ≤ y
+
≤ 30, u = 5 ln y
+
− 3.05,
=
dy +
y +
For the turbulent region
y
+
≥ 30, u
+
= 2.5 ln y
+
+ 5.0,
du +
dy + =
2.5
y +
210
Analytical Heat Transfer