Pipe flow in water supply
Flow in pipes
A lecture by
Gilberto E. Urroz
March 2006
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Pipe flow in irrigation
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Pipe flow in dams
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Pipe flow in dams
Pipes and pumps
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Pipes over rivers
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Pipes in industrial setups
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Laminar & Turbulent Flows
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Flow films
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Laminar flow: orderly, in layers
Turbulent flow: disorderly, eddies
Transitional: intermittently turbulent
Criteria: Reynolds number
Critical Re = 2000
LaminarTurbulentFaucetFlow.MOV
LaminarPipeFlow.MOV
TurbulentPipeFlow.MOV
TransitionalPipeFlow.MOV
LaminarTurbulentCombo.MOV
TurbulentFlowAroundUs.MOV
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Laminar & turbulent flow out of a faucet
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Laminar flow in a pipe
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Turbulent flow in a pipe
Transitional flow in a pipe
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Hydraulic Radius - 1
Laminar and turbulent flow visualization
• A = cross-sectional area
• P = wetted perimeter (length of boundary
in contact with water)
• Hydraulic Radius: Rh = A/P
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Example
Hydraulic Radius - 2
• FOR PIPE FLOW
• A = πD2/4 , P = πD2, Rh = A/P = D/4
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Head losses and shear stress
Analysis of motion of pipe flow
hf = head loss in length L
τo= wall shear stress
γ = specific weight
Rh = hydraulic radius
τo ⋅ L
hf =
γ ⋅R h
Applies to:
• Any x-section
• Laminar or turbulent flow
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Smooth and rough walls
Dimensional Analysis of Pipe Flow -1
• Examples:
– Glass
– Plexiglass
– PVC
• Examples:
– Copper
– Concrete
Rough walls characterized by an absolute roughness (e)
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Dimensional Analysis of Pipe Flow -2
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Head Losses in Pipe Flow
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Head losses in circular conduits
• Start from
hf = C f ⋅
Analysis of motion of pipe flow
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L V
⋅
Rh 2 g
• For a circular pipe, Rh = D/4
• Replace
Cf = f/4
• Darcy-Weisbach equation:
L V2
hf = f ⋅ ⋅
D 2g
• f = Darcy-Weisbach friction factor
• Cf = 4f = Fanning friction factor (used in
gas flow)
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Head losses and shear stress
hf = head loss in length L
τo= wall shear stress
γ = specific weight
Rh = hydraulic radius
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Shear stress distribution
• Equilibrium of forces
on an element of
radius r , steady flow.
τo ⋅ L
hf =
γ ⋅R h
ΣFx = ma x , a x = 0
Applies to:
• Any x-section
• Laminar or turbulent flow
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Shear stress – linear distribution
• From an ealier result:
• Also,
τ ⋅L
hf = o
γ ⋅R h
hf =
Wall shear stress and friction factor
hf =
• Combine the result
2τ (r ) L
γ ⋅r
L V2
hf = f ⋅ ⋅
D 2g
• With Darcy-Weisbach
• With Rh = D/4 = ro/2
r
τ (r ) = τ o ⋅
ro
τo ⋅ L
τo ⋅ L
=
γ ⋅ Rh γ ⋅ (D / 4)
f
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τ o = ⋅ ρ ⋅V 2
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Friction in non-circular conduits
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Laminar flow in a circular pipe
• Use D = 4Rh in Darcy-Weisbach
• With
L V2
L V2
= f⋅
⋅
hf = f ⋅ ⋅
D 2g
4 Rh 2 g
Re =
ρVD ρV ⋅ 4 Rh
=
μ
μ
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Velocity and shear stress distributions
Centerline and mean velocities –
discharge and head losses
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Combining Hagen-Poiseuille with
Darcy-Weisbach
Hagen-Poiseuille law for laminar flow
32νLV
hf =
gD 2
1. hf ~ V
2. Equation involves no empirical coefficient
3. Equation involves only fluid properties, g, and
V
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Exercises - 1
Exercises - 2
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Exercises - 3
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Rounded entrance produces uniform flow (inviscid flow behavior)
Velocities at the wall are zero (no-slip condition)
A viscous boundary layer develops, but an inviscid core remains
before flow is fully developed
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Exercise in flow development
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