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February 8, 2008 2
Hydraulics of water transport through pipes: content
• Water transport through pipes• mathematical description• drinking water transport => rigid pipe• Sewerage transport => open channel flow• Water hammer
• Pumps and motors• Network calculation
February 8, 2008 3
Openreservoir
Openreservoir
Openchannel
Free water surface = hydraulic grade line
Water flows from a high level to a lower level
February 8, 2008 4
Water flows from a high level to a lower level
Openreservoir Open
reservoir
Openchannel
Free water surface = hydraulic grade line
Weir
Pump Pipe Outflow
February 8, 2008 5
Water transport through pipesflow types
• Closed pressurised flow• Fully filled closed pipes• Water pressure higher than atmospheric
• Open channel flow• Partly filled pipe• Free water surface
• Main difference: Open channel accommodates storage in profile
February 8, 2008 6
Heads in a pipe-pump system• Static head to compensate level differences• Dynamic head: to compensate
• Friction loss ΔHW
• Deceleration loss ΔHS
• Local losses
Hstatic
Hdynamic
February 8, 2008 7
Mathematical description
• Continuity equation: mass balance: Ingoing mass = outgoing mass over a certain period of time
dx
A1
Qin
Qout
February 8, 2008 8
Mathematical description
• Mass balance:Qin*dt = Qout* dt + dA* dxIncoming = outgoing + storage
• Dividing by ∂x and ∂t with limit transition:
0=∂∂+
∂∂
xQ
tA
February 8, 2008 9
Mathematical description
• Momentum balance
1: Acceleration term2: Convective term3: Gravitational/pressure term4: Friction term
02
=+∂∂+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂+
∂∂
ARQQ
cxpgA
AQ
xtQ
1 2 3 4
February 8, 2008 10
Mathematical description
• Momentum balance
02
A*u Qmet
0
2
2
=+∂∂+
∂∂+
∂∂+
∂∂+
∂∂
=
=+∂∂+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂+
∂∂
uuDcxpgA
xAu
xuAu
tAu
tuA
ARQQ
cxpgA
AQ
xtQ
π
February 8, 2008 11
Drinking water transport: fully filled closed pipes
• Two possible flow types:• Rapid changing boundaries for pressure and/or volume flow:
water hammer• Slow changing boundaries for pressure and/or flow: friction
flow• Rigid column:
• uniform and stationary flow Q/ t = 0• prismatic pipe A/ x = 0• water incompressible• elasticity pipe negligible A/ t = 0• Newton’s fluid
∂ ∂∂ ∂
∂ ∂
February 8, 2008 12
Rigid column: Continuity becomes
00
00
lyconsequent And
000
=∂∂→=
∂∂+
∂∂
→=∂
∂→=∂∂
=∂∂⎯⎯ →⎯=
∂∂+
∂∂ =
∂∂
xu
xAu
xuA
xuA
xQ
xQ
xQ
tA t
A
February 8, 2008 13
Rigid column: momentum balance
01
with ;0
0;0;0
02
22
2
=+∂∂+
∂∂
==+∂∂+
∂∂
⇒=∂∂=
∂∂=
∂∂
=+∂∂+
∂∂+
∂∂+
∂∂+
∂∂
RACQQ
xp
tQ
gA
AQuuuDc
xpgA
tuA
xu
xA
tA
uuDcxpgA
xAu
xuAu
tAu
tuA
π
π
February 8, 2008 14
Rigid column: Darcy Weissbach
QQDL
DQQ
gLpp
RACQQ
pp
RACQQ
dxxp
RACQQ
xp
tQ
gA
Cg
Lx
ox
Lx
ox
tQ
55221
8
2212
22
Lover gintegratin and 0
22
0826,08
01
2
λπ
λ
λ
==−
⎯⎯ →⎯−=−
⇒−=∂∂
⎯⎯⎯⎯⎯⎯⎯ →⎯=+∂∂+
∂∂
=
=
=
=
=
=∂∂
∫∫
February 8, 2008 15
After some mathematical exercises
• Darcy Weissbach: No time dependency
• White-Colebrook
25
2
12
0826,0
2
QDL
gu
DLHHH
λ
λ
=
=−=Δ
⎥⎦
⎤⎢⎣
⎡+−=
λλ Re32,01
3log21
DkN
February 8, 2008 17
Local losses
• Energy loss due to deceleration and release of streamlines
D1D2 u2 u1
Deceleration
area
0≠∂∂xA
guH2
2
ξ=Δ
February 8, 2008 21
Pressurised transport
Pressure drop
Demand
H1
H2
ΔH
QL, D, λPump
Pipe
Pressure
25
2
12
0826,0
2
QDL
gu
DLHHH
λ
λ
=
=−=Δ
February 8, 2008 22
Sewer transport
• Three flow conditions occur:• Open channel flow• Fully filled closed pipe• Transition situation
• Modelling is very challenging
February 8, 2008 23
Sewerage transport: open channel flow
)(*),(:place and in timeon width dependant surface Flow
0
:equation Momentum
0)( :balance Mass
2
thtxBA
ARQQ
cxpgA
AQ
xtQ
xQ
thA
=
=+∂∂+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂+
∂∂
=∂∂+
∂∂
February 8, 2008 24
Sewer transport: open channel flow
• Mass balance
• Continuity equation
0)( =∂∂+
∂∂=
xQ
thhB
02
=+∂∂+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂+
∂∂
ARQQ
cxhgA
AQ
xtQ
February 8, 2008 27
Transition between open channel and fully filled• Physical (and mathematical) instability
Open channel flow
Transition tosurcharged flow
February 8, 2008 28
The Preissmann slot: preserve open channel flow
• δ is 0,1 to 5% ofdiameter
• Keeps opensurface
• Valid throughstreet level
• Introduces syste-matical error
February 8, 2008 29
Water hammer: fully filled flow
• Sudden changes in boundary conditions:• Increase/decrease in velocity• Pump trip
• Take into account compressibility of water and elasticity of the pipe
February 8, 2008 32
Summary
• Drinking water, normal conditions• Rigid column, closed pipes:• Darcy-Weissbach• Time independent
• Sewerage water, normal conditions• Open channel flow• Time dependant• Transition phase: Preissmann slot
• Water hammer• Time dependant• Special analysis• Practical and construction measures
February 8, 2008 33
To get some feeling of dimensions
• Assume a pipe• Length 5 km• Diameter 500 mm• Flow 400 m3/h
• Pressure drop?
February 8, 2008 34
To get some more feeling
• Assume two pipes• Connected parralel• Same lambda’s, pressure drop, length• Volume flow Q• Diameters pipe 1 :D, pipe 2: 2*D
• What is the ratio between the flows through the pipes?