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The bulk mean velocity is defined as
R
1
u2πr dr
(8.9)
V = πR 2
0
R �
�
1
πR 2
) 2
( r
1 −
u max 2πr dr
=
R
0
2u max 1
1 4 � R
2
−
�
=
r
r
R 2
2
4R 2
0
1
V = U max
(8.10)
2
8.2.1 Fully Developed Flow in a Tube: Friction Factor
∂u �
2r �
τ w = μ
= μU max −
(8.11)
∂y
R 2
y=0
r=R
where u/U max = 1 − (r/R)
2
−2
τ w = μU max R
τ w
μU max (−2/R)
16μ
16
1
f =
=
(
) =
=
∼
(8.12)
2
(1/2)ρ (1/2)U
2
ρUD
Re D
Re D
(1/2)ρU
max
where Re D = ρUD/μ.
Figure 8.3 shows the force balance in the fully developed flow region:
ΔPA c + τ w πDΔx = 0
τ w πDΔx
ΔP = −
(1/4)πD 2
ΔP
−4τ w
4 1 2
=
= − f ρU
(8.13)
Δx
D
D 2
4 1 2 16
32μU
32μ ρ D μ
32μ 2 ρDU
= −
ρU
= −
= −
U = −
D 2
Re D
D 2
D 2 ρ D μ
D 3 ρ μ
32μ 2
= −
Re D ∼ Re D
(8.14)
D 3 ρ
And pumping power can be obtained as P ∼
= ΔP (volume flow). Figure 8.4
shows that the friction factor decreases and the pressure drop increases with
Reynolds number.
171
Internal Forced Convection
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�
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�
�
The bulk mean velocity is defined as
R
1
u2πr dr
(8.9)
V = πR 2
0
R �
�
1
πR 2
) 2
( r
1 −
u max 2πr dr
=
R
0
2u max 1
1 4 � R
2
−
�
=
r
r
R 2
2
4R 2
0
1
V = U max
(8.10)
2
8.2.1 Fully Developed Flow in a Tube: Friction Factor
∂u �
2r �
τ w = μ
= μU max −
(8.11)
∂y
R 2
y=0
r=R
where u/U max = 1 − (r/R)
2
−2
τ w = μU max R
τ w
μU max (−2/R)
16μ
16
1
f =
=
(
) =
=
∼
(8.12)
2
(1/2)ρ (1/2)U
2
ρUD
Re D
Re D
(1/2)ρU
max
where Re D = ρUD/μ.
Figure 8.3 shows the force balance in the fully developed flow region:
ΔPA c + τ w πDΔx = 0
τ w πDΔx
ΔP = −
(1/4)πD 2
ΔP
−4τ w
4 1 2
=
= − f ρU
(8.13)
Δx
D
D 2
4 1 2 16
32μU
32μ ρ D μ
32μ 2 ρDU
= −
ρU
= −
= −
U = −
D 2
Re D
D 2
D 2 ρ D μ
D 3 ρ μ
32μ 2
= −
Re D ∼ Re D
(8.14)
D 3 ρ
And pumping power can be obtained as P ∼
= ΔP (volume flow). Figure 8.4
shows that the friction factor decreases and the pressure drop increases with
Reynolds number.
171
Internal Forced Convection