February 8, 2008

Pumping stations and water transport

Hydraulics: theoretical backgroundct5550

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 16

Moody diagram

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 18

Local losses

Deceleration

area

guH2

2

ξ=Δ

February 8, 2008 19

Source:Idel’cik

February 8, 2008 20

Source:Idel’cik

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 25

Schematic model sewer system

February 8, 2008 26

Schematic model urban drainage system

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 30

Water hammer

v

February 8, 2008 31

Water hammer

• Further explanation Ivo PothofWL|Delft Hydraulics

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?

February 8, 2008 35

Few questions

• How much will pressure drop with doubling of flow?• What effect has increasing roughness on pressure

drop?• What has more effect: increasing roughness or

decreasing diameter?