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209
Turbulent Flow Heat Transfer
x
y
r
R
ΔT b
Δx
q" w
FIGURE 10.7
Turbulent flow heat transfer in a circular tube.
For a fully developed flow (v = 0) and assuming (∂ 2 T/∂x 2 ) « (1/r)
(∂/∂r)[r(α + ε H )(∂T/∂r)], the energy equation becomes
∂T
1
∂
∂T
u
=
(R − y)(α + ε H )
∂x
R − y (−∂y)
(−∂y)
Assume
u = V
∂T
∂T b
=
∂x
∂x
From energy balance,
""
q 2πR dx = ρC p VA c dT b
w
""
""
dT b
q 2πR
2q
V
=
w
=
w
≡ C
dx
ρC p πR 2
ρC p R
And applying BCs,
y = 0 T = T w
y = R
∂T
= 0
∂y
The energy equation becomes
dT b
∂T
V
(R − y) d(−y) = d (R − y)(α + ε H )
dx
−∂y
dT b 1
dT
V
(R − y)
2
= (R − y)(α + ε H )
+ C
dx 2
−dy
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209
Turbulent Flow Heat Transfer
x
y
r
R
ΔT b
Δx
q" w
FIGURE 10.7
Turbulent flow heat transfer in a circular tube.
For a fully developed flow (v = 0) and assuming (∂ 2 T/∂x 2 ) « (1/r)
(∂/∂r)[r(α + ε H )(∂T/∂r)], the energy equation becomes
∂T
1
∂
∂T
u
=
(R − y)(α + ε H )
∂x
R − y (−∂y)
(−∂y)
Assume
u = V
∂T
∂T b
=
∂x
∂x
From energy balance,
""
q 2πR dx = ρC p VA c dT b
w
""
""
dT b
q 2πR
2q
V
=
w
=
w
≡ C
dx
ρC p πR 2
ρC p R
And applying BCs,
y = 0 T = T w
y = R
∂T
= 0
∂y
The energy equation becomes
dT b
∂T
V
(R − y) d(−y) = d (R − y)(α + ε H )
dx
−∂y
dT b 1
dT
V
(R − y)
2
= (R − y)(α + ε H )
+ C
dx 2
−dy