1

Hydraulics: theoretical background

2. Water transport through pipes

2.1 IntroductionOne of the basic rules of nature is that water flowsfrom a high energy level to a low energy level. Thehigh energy level can for instance be a storage tankat a high level as a water tower or a tank on a hill. Itcan also be the high pressure induced by pumps.For the drinking, sewerage and irrigation practise wefocus on the flow in open channel or partially filledpipes and flow through surcharged filled closed pipes.These are the most common phenomena in the wa-ter transport in these fields.

Characteristic for water transport through partiallyfilled pipes or open channels is that pressure at thefluid surface is atmospheric. Consequently openchannel flow is always induced by surface fluid leveldifference between the upstream boundary and thedownstream boundary. The upstream level may, forinstance, be an open water surface or an inflow froma house manifold at a sewer system. A downstreamflow can be an outflow over a weir. Flow is inducedby gravity.

Open channel flow allows for storage of water withinthe profile, through changes of water level. Upstreaminflow of water doesn’t necessarily instantaneously

cause an out flow at the down stream end, but watercan be stored in the channels causing a rise in waterlevel. In urban drainage systems a deliberate stor-age is wanted or even needed in the system, for in-stance by putting weirs at the down stream bound-ary.In open channel flow the storage capacity is an im-portant describing factor.

The energy difference in pressurized flow in sur-charged and closed pipes is mostly induced by anenergy input at the upstream boundary by pumps orby an elevated reservoir.

Closed and surcharged pipes don’t have significantstorage capabilities. The storage capability is for in-stance the stretching of the pipe and the compres-sion of the water. This causes high pressures be-cause of the stiffness of the pipe and the highcompressibility of the water and is called water ham-mer and occurs when changes in flow velocity arerelatively rapid. This type of flow is described sepa-rately. In closed pipe flow friction loss is the maindescribing factor for energy losses.In this chapter we first deal with pressurised flow,than with open channel flow and conclude with the

Fig. 2.1 - Water tower

Openreservoir

Openreservoir

Openchannel

Free water surface = hydraulic grade line

Fig. 2.2 - Two open reservoirs with open channel

Openreservoir Open

reservoir

Openchannel

Free water surface = hydraulic grade line

Weir

Fig. 2.3 - Flow with a weir

Fig. 2.4 - Two reservoirs with pipes at the bottom

PumpPipe

OutflowPumpPipe

OutflowFig. 2.5 - Pump and outflow

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CT5550 - Water Transport

phenomenon water hammer.

2.1 Pressurised flowIn pressurised flow through pipes the pressure at thestart of the pipe is higher than atmospheric. An open(high level) reservoir or a pump induces the pres-sure at the beginning of the pipe (see figure XX’s).Flow through the pipe will cause an energy loss dueto friction loss and local losses caused by release offlow lines (entrance and decelaration losses).Schematically all the losses and levels are summa-rized static pressure and dynamic pressure in figureXX.

Mathematical description

Two equations describe the flow through pipes:3 Continuity equation or mass balance4 Motion equation or momentum balance

2.1.1 Mass balance/continuity equationA control volume of pipe is considered with a crosssection A and a length dx The mass balance statesthat ingoing mass equals outgoing mass. Incomingmass in a time frame dt is:

inQ dt uAdtρ ρ= (eq. 2.1)

withQi n : Incoming volume flow [m3/s]ρ : Specific mass [kg/m3]u : Mean velocity [m/s]A : Cross section of the pipe [m2]Outgoing mass is analogue with the ingoing mass

outQ dt uAdtρ ρ= (eq. 2.2)

The mass balance over a time frame dt is

in outQ t Q t A x∂ = ∂ + ∂ ∂ (eq. 2.3)

in words: ingoing mass equals outgoing mass plusstorage within the control volume. The storage is thechanging of the cross section dA.

Dividing the mass balance by δxδt gives

0A Qt x

∂ ∂+ =

∂ ∂ (eq. 2.4)

Momentum equationReferring to Battjes (CT3310, chapter 2) the momen-tum equation is

2

0f

Q QQ Q pgA c

t x A x AR ∂ ∂ ∂

+ + + = ∂ ∂ ∂ (eq. 2.5)

withg : GravitationR : Wet perimetercƒ : friction coefficient

The system of the continuity equation and the massbalance are known as the equations of De Saint-Venant (1871)

The dimensionless coefficient cf can be expressedin the Chézy coefficient as cƒ  = g/C2. With Q=uA themomentum equation becomes

222 0u A u A p g AA u Au u gA u u

t t x x x C R∂ ∂ ∂ ∂ ∂+ + + + + =∂ ∂ ∂ ∂ ∂

(eq. 2.6)

2.1.3 Rigid column approachWater transport through pipes is characterised withslow changing boundary conditions. When the so-

Hstatic

Hdynamic

Hstatic

Hdynamic

Fig. 2.6 - Two different energy levels

dx

A1

Qin

Qout

dx

A1

Qin

Qout

A1

Qin

Qout

Fig. 2.7 -Mass balance

3

Hydraulics: theoretical background

called rigid column simplification is applied the pre-sumptions are made:

o Uniform and stationary flowo Prismatic pipe: The cross section of the pipe

doesn’t change over the length of the pipe re-

sulting in 0Ax

∂=

∂

o Water is incompressible

o Elasticity of the pipe is negligible: 0At

∂=

∂o The fluid meets Newton’s criteria being that vis-

cosity is constant and only dependent on tem-perature.

The continuity equation then transforms in

0Qx

∂=

∂ which becomes:

0 0 0 0Q uA u A u

A ux x x x x

∂ ∂ ∂ ∂ ∂= → = → + = → =

∂ ∂ ∂ ∂ ∂(eq. 2.7)

And the momentum balance than becomes:

2 0u p g A

A gA u ut x C R

∂ ∂+ + =

∂ ∂(eq. 2.8)

substituting Q

uA

= and dividing by gA gives

2 2

10

Q QQ pgA t x C A R

∂ ∂+ + =

∂ ∂(eq. 2.9)

Considering a piece of pipe with length L and inte-grating the equations over this pipe length gives

2 20 0 0

2 1 2 2

1

1

x L x l x l

x x x

Q Qp Qdx dx dx

t gA t C A R

Q QQp p L L

gA t C A R

= = =

= = =

∂ ∂= − − ⇒

∂ ∂

∂− = − −∂

∫ ∫ ∫

(eq. 2.10)

In stationary flow, the term Qt

∂∂

becomes zero ornegligible: flow will only slowly change over time.For the roughness of the pipe wall in this equationthe Chézy-coefficient is used. Often this is replaced

by the Darcy-Weissbach friction coefficient 2

8gC

λ = .

Different formulas are applied to calculate the fric-tion coefficient λ, which is referred to in the next para-graph.Combined with the substitution of the Darcy-Weissbach friction coefficient the socalled Darcy-Weissbach equation remains:

Picture of de Saint-Venant

Jean Cleade deSaint-Venant (1797-1886) graduated atthe Ecole Polytech-nique in 1816. He hada fascinating careeras a civil engineerand mathematicsteacher at the Ecoledes Ponts et Chaus-sées where he suc-

ceeded Coriolis. Seven years after Navier’s death,Saint-Venant re-derived Navier’s equations for aviscous flow, considering the internal viscousstresses, and eschewing completely Navier’s mo-lecular approach. That 1843 paper was the first toproperly identify the coefficient of viscosity and itsrole as a multiplying factor for the velocity gradi-ents in the flow. He further identified those prod-ucts as viscous stresses acting within the fluidbecause of friction. Saint-Venant got it right andrecorded it. Why his name never became associ-ated with those equations is a mystery. certainly itis a miscarriage of technical attribution.It should be remarked that Stokes, like de Saint-Venant, correctly derived the Navier-Stokes equa-tions but he published the results two years afterdecSaint-Venant.In 1868 de Saint-Venant was elected to succeedPoncelet in the mechanics section of the Académiedes Sciences. By this time he was 71 years old,but he continued his research and lived for a fur-ther 18 years after this time. At age 86 he trans-lated (with A Flamant) Clebsch’s work on elastic-ity into French and published it as Theorie del’élasticité des corps solides and Saint-Venantadded notes to the text which he wrote himself.

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CT5550 - Water Transport

1 2 2 5 5

80,0826

Q QL Lp p Q Q

g D Dλ

λπ

− = =

(eq. 2.11)

Another popular representation of the Darcy Weiss-bach formula is:

2

2L u

HD gλ

∆ = (eq. 2.12)

2.3 Friction coefficients and locallosses

In Battjes (CT2100, chapter 12.4) an extensive elabo-ration on different friction parameters is given, boththeoretically and experimentally determined. Themost used formulas are those of Manning, Chézyand White Colebrook.White and Colebrook (1937) performed experimentsto asses how ë varies in the transition from smoothto rough conditions as a function of the Reynoldsnumber at a constant relative roughness. They foundthat λ in technical rough pipes much smoother var-ies than in experiments of Nikuradse. Colebrook de-veloped an expression with which λ can bedeterimend as a function of the relative roughnessin Nikuradse’s coefficient k/D and the Reynoldsnumber:

1 2,52log 0,27

ReNk

Dλ λ

= − +

(eq. 2.13)

This formula can only iteratively be solved. The vari-able determining the friction coefficient is theNikuradse roughness of the pipe wall. This can varybetween 0,05 mm for smooth pipes like PVC or PEpipes and 20 mm for old cast iron pipes with largeencrustations.In network calculation programs the λ-value is cal-culated automatically as a function of the hydrauliccircumstances expressed in the Reynolds numberand the relative roughness of the pipe expressed ink/D.For convenience diagrams are developed to quicklydetermine the value of λ. The Moody-diagram is themost commonly used diagram. This diagram isshown in figure 2.8.

Local lossesLocal losses are caused by sudden deceleration offlows combined with release of flow lines from thepipe wall. Examples are given in figure 2.9.

In fact local losses are separately addressed becausethis is a violation of the assumption that the pipes

are prismatic 0Ax

∂=

∂.

Local losses depend on the velocity in the pipe andare expressed in an analogue formula as the DarcyWeissbach equation.

2

2localu

Hg

ξ∆ = (eq. 2.14)

For the value of ξ a lot of experiments have been

Release of flow linesRelease of flow lines

Fig. 2.9 - Release of flow lines in sudden increase indiameter and in bends

Fig. 2.8 - Moody diagram

5

Hydraulics: theoretical background

performed resulting in large tables with all kind ofhydraulic situations and the resulting local loss coef-ficient. An extensive handbook is made by I’del cik,which is used all over the world.

Some examples of such tables are in appendix 2.1.

2.4 Summary of pressurized transportThe factors determining the energy losses duringpressurized flow are schematically drawn in figure2.10.Energy losses are mainly friction losses and decel-eration losses. The formula of Darcy Weisbach isone dimensional and there is no time dependency.Network calculations based on the one-dimensionalformulas are so-called steady state calculations.

2.5 Open channel flowWhen turning to open channel flow, occuring in sani-tary engineering in urban drainage networks, exactlythe same principles are applied as in pressurised flow.In this also a 1-dimensional approach is usuallyadopted. This means that the (3 dimensional) veloc-ity vector in the momentum and mass-balance equa-tion are integrated into a one-dimensional form. Thedependent variables are integrated quantities. In or-der to get a closed system of equations simplifyingassumptions are needed. Local effects such ascaused by weirs, are taken into account by estimat-ing their effect on the integrated equations. The re-duction into one space dimension is defensible; whenstudying the geometry of a conduit in an urban drain-age system the flow is to a large extent 1-dimen-sional, except for the velocity field at the direct en-trance. In manholes the velocity field is 3-dimen-sional, but this can be circumvented in a 1-D modelby applying extra terms to account for frictionallosses.

The equations applied in a 1-dimensional approachare the well-known De Saint-Venant equations (DeSaint-Venant (1871)):

Momentum balance:

{ {2

0fh

I III IVII

Q QQ Q hgA c

t x A x R Aβ

∂ ∂ ∂+ + + = ∂ ∂ ∂ 1424314243 (eq. 2.15)

Mass balance:

( )( ) 0

Q A h Q hB h

x t x t∂ ∂ ∂ ∂

+ = + =∂ ∂ ∂ ∂

(eq. 2.16)

In which:Q discharge (m3/s)A cross-sectional area (m2)B width of the free water surface (m)g gravitational acceleration (≈9.813 m/s2)Rh hydraulic radius (m)cf resistance constant (-)h water level (m)x location along x-axis (m)t time (s)β Boussinesq’s number (-)

The assumptions applied are:- Hydrostatic pressure.- Velocity components in y and z direction are neg-

ligible compared to the velocity component in xdirection (uy=uz<<ux).

The individual terms in the momentum balance equa-tions (eq. 2.15) are:

I acceleration termII convective termIII gravitational termIV friction term

The third term may also be written as:

{ {bb

IIIbIIIa

zh a agA gA gA i

x x x x

∂∂ ∂ ∂ = + = − ∂ ∂ ∂ ∂

(eq. 2.17)

In which:a water depth (m)zb bottom level (m)ib bottom slope (-)

Velocity head

Entrance loss

Friction lossDeceleration loss

Friction loss

Pressurehead

2u2g

Statichead

Piezo -Metriclevel

Energylevel

Reference level

Energyloss

Energy level=Static pressure

u1

u2

Velocity head

Entrance loss

Friction lossDeceleration loss

Friction loss

Pressurehead

2u2g

Statichead

Piezo -Metriclevel

Energylevel

Reference level

Energyloss

Energy level=Static pressure

u1

u2

Fig. 2.10 - Losses and energy levels

6

CT5550 - Water Transport

Term IIIa is the pressure term and term IIIb is thegravity term.

The De Saint-Venant equations (using the dynamicwave approach) form a hyperbolic system of partialdifferential equations. This implies that in order toobtain a well-posed problem the initial condition (Qand h at t=0) and two boundary conditions have tobe defined.

Several possible simplifications of eq. 2.15 and 2.16are applied depending on the possibility of neglect-ing terms in the momentum balance equation. Forinstance when terms I and II (the inertia terms) canbe neglected, which is the case when the flow variesonly slowly with time and space, either the kinematicor diffusion wave equation is obtained. In Table 1the possible simplifications for the momentum equa-tions are summarised along with the identificationnormally used.

The simplified versions of the momentum equationare mostly used to drive analytical solutions for spe-cial (academic) cases, or to simplify the numerical

calculation for practical cases. In fact the first com-puter programs applied in the field of urban drain-age (end of the 1970’s and the early 1980´s) werebased on the kinematic wave simplification. The mainreason for this was that the inertia terms (especiallythe convective term) are more difficult to handle interms of numerical stability, computational effort andRAM. Nowadays there is a very high degree ofacciability to computational power which has led toa situation in which any network can be dealt withusing the full dynamic wave equations.

Due to the integration of two space dimensions, sev-eral new parameters enter the equations. For in-stance, the geometry of the problem like hydraulicradius, wetted area and hydraulic depth are suchparameters. The parameter β in the convective term(term II) is the Boussinesq number (Boussinesq,(1897)), defined by:

2

02

A

u dA

u Aβ =

∫(eq. 2.18)

The Boussinesq number accounts for the fact thatthe velocity is not uniformly distributed over the crosssection of the flow, in fact it is a correction factor dueto simplifying the 3-dimensional flow equations.

The value for β is >1 and can eventually reach val-ues >1.2 in extreme cases, e.g. when a sedimentbed is present in a channel with a circular cross sec-tion at shallow water depths, see Kleywegt (1992).In practice, however a value of 1 is applied since theeffect is rather limited when compared to the uncer-tainty in other parameters introduce in the equations(for instance the value for the wall roughness en lo-cal loss coefficients).

Fig. 2.11 - Definition sketch

bz

ha

B

A

Q

x

ibbz

ha

B

A

Q

x

ib

Terms taken into account Identification of the simplified equation

main assumptions

I+II+III+IV Dynamic wave no additional assumptions I+II+III Gravity wave friction is sma ll compared to

gravity and inertia terms III+IV Diffusion wave inertia terms are small compared

to gravity and friction IIIb+IV Kinematic wave inertia terms are small compared

to gravity and friction and

bixa

<<∂∂

Table 1 - Possible simplification for the momentum equation

7

Hydraulics: theoretical background

Transition between open channel flow and sur-charged transportA problem is encountered when pipes starts to flowfull, instability-like phenomena evolve. It should bestated however, that in reality when pipes tend toflow full a kind of chaotic behaviour may occur (seeFig 2.12). This is because air escaping between thewater level and the pipe wall influences the waterflow.

It was observed in laboratory experiments (see e.g.de Somer (1984)) that even at steady state flow un-der certain geometrical conditions unpredictable

transitions in flow mode can occur. The origin of nu-merical instabilities at nearly full flow can be char-acterised as a boundary condition problem. Essen-tially, it is so that nowhere in the system the waterlevel is fixed, under pressurised conditions this im-plies that the flow is defined by a difference in pres-sure only. Therefore, the absolute water level or,equivalently, the local pressure is undefined by lackof a boundary condition in this respect.It is accepted practice to circumvent this by applyingthe so-called Preissmann slot(1 ). In fact the cross-sectional geometry of, e.g,. a circular pipe is slightly

changed in order to avoid the occurrence of a transi-tion between free surface flow and pressurised flowby adding a slot with a small width (d) on top of thepipe. In this manner the water level directly followsfrom the mass-balance equation. In fig 2.13 this slotis depicted, the width of the slot d is defined theoreti-cally by setting the celerity of a disturbance at thefree surface in the slot equal to the celerity of a dis-turbance in the full-pressurised case.

The celerity of a surface wave is defined by:

hc u gH= ± in which Hh is the hydraulic depth:

( )( )h

A hH

B h= . Taking a circular cross-section as an

example, A en B are functions of the water depth(see annexe I); when the conduit is almost completelyfilled it is easily seen that the value of c becomesundefined (2):

2 2

2

lim ( ) 0 lim ( )lim ( )

h R h R h

h R

B h h hc h

→ →

→

= ⇒→ ∞ ⇒ → ∞ eq. 2.19

Applying the Preissmann slot the width of this slot(d) is defined by stating: cfree surface =cpressurised

Fig. 2.12 - Physical instability. Due to air entrainment theflow may change between modes I and II inan unpredictable pattern in time.

Fig. 2.13 - The Preismann slot

(1) The idea of this piezometric slot originates from A. Preissmann and was further developed by Cunge & Wagner(1964) for practical application.

(2) In practice, this celerity is limited to

1c

DK Edρ ρ

=+ representing the velocity of a pressure wave travelling through

the system. In which r is the density of water in kg/m3, d is the wall thickness of the pipe in m, D is the diameter of thepipe in m, E is elasticity modulus of the pipe material in N/m 2 and K is the compressibility of water in N/m 2.

8

CT5550 - Water Transport

2

2

11

DR K Ed

u g uD g R

K Ed

ρπ

δρ ρδ π

+ + = + ⇒ =+

(eq. 2.20)

The resulting values for the slot width d though, maybecome very small resulting in high values for theCourant number (i.e. >3 to 4). Therefore, in prac-tice, the slot width is set to a fixed value of e.g. 0.1-5% of the diameter of the conduit. This does intro-duce a discrepancy with reality. As long as no waterhammer problems are studied these, discrepancieshowever, are insignificant.

2.6 Water HammerWater hammer is a special flow condition in sur-charged pipes. As stated in section XX the changingof boundary conditions during normal situation areso slow that a rigid column approach is allowed.Certain changes in boundary conditions can be sofast that the flow doesn’t meet the requirements fora rigid column approach and the flow must be ana-lysed in a different way. Most important features arethat the compressibility of the water and the elastic-ity of the pipe is not neglected, but taken into ac-count.Main effect is that storage within the filled pipe ispossible due to the compressibility of the water (morewater in the cross-section) and the elasticity of thepipe (variation in the cross-section).

A schematic picture of the phenomenon Water Ham-mer is given in figure 2.14. A stream of water is sud-denly stopped, resulting in a positive pressure at theupstream side and a negative pressure at the downstream side.

Mathematical description

The original constitution equations (De Saint-Venantare:

Mass balance: 0A Qt x

∂ ∂+ =

∂ ∂(eq. 2.21)

Continuity equation:

2

0Q QQ Q p

gA ct x A x AR

∂ ∂ ∂+ + + = ∂ ∂ ∂

(eq. 2.22)

We now have to find relations between the pressurep and the fluid density ρ and the cross section A.

Relation between pressure and fluid densityThe compression modulus of a fluid is defined as:

d=

dp Kδ δ

(eq. 2.23)

Through the partial differentials of ρ to t and x wefind:

d p pt dp t K t

d p px dp x K x

ρ ρ ρ

ρ ρ ρ

∂ ∂ ∂= =

∂ ∂ ∂∂ ∂ ∂

= =∂ ∂ ∂

(eq. 2.24)

Relation between cross section A and pressure pConsider a circle shaped cross section of a pipe withan internal diameter D and a uniform wall thicknessδ. The wall thickness is relatively small compared tothe diameter (δ << D) (See figure XX). A small pres-sure increase in the pipe (dp) will increase the posi-tive tension of dσ. The increase will meet the equa-tion 2 δ* dσ = D*dp

D

D + 2d ˜ D dsds

dpd

D

D + 2d ˜ D dsds

dpd

Fig. 2.15 - Cross section pressurised pipe

vv

Fig. 2.14 - Schematic of water hammer

9

Hydraulics: theoretical background

The increase in the wall tension dó will induce anincrease of the length of the pipe wall P = πD andthus the diameter D. According to Hooke’s law thiscan be expressed as:

dD dP dD P E

σ= = (eq. 2.25)

This leads to

2dD D dpD Eδ

= (eq. 2.26)

Because A is proportional to D2

2dA dD D dpA D Eδ

= = (eq. 2.27)

and

dA DA

dp Eδ= (eq. 2.28)

With this the partial differentials of A to t and x can

be expressed in pt

∂∂

and px

∂∂

:

A dA p D pA

t dp t E tA dA p D p

Ax dp x E x

δ

δ

∂ ∂ ∂= =

∂ ∂ ∂∂ ∂ ∂

= =∂ ∂ ∂

(eq. 2.29)

Mass balance

The mass balance of a pipe is

( ) ( ) 0A Aut x

ρ ρ∂ ∂

+ =∂ ∂

(eq. 2.30)

Complete elaboration of this equation gives:

0A A u

A u A uAt t x x xρ ρ

ρ ρ ρ∂ ∂ ∂ ∂ ∂

+ + + + =∂ ∂ ∂ ∂ ∂

(eq. 2.31)

Substituting all equations describing the variation ofρ and A with p followed by a division by A gives:

0D p D p u

uK E t K E x xρ ρ ρ ρ

ρδ δ

∂ ∂ ∂ + + + + = ∂ ∂ ∂

(eq. 2.32)

If we define c like: 2

1 Dc K E

ρ ρδ

= + this will become

2 2

10

p u p uc t c x x

ρ∂ ∂ ∂

+ + =∂ ∂ ∂

(eq. 2.33)

The value of c is equal to the speed of pressure wavein the specific circumstances.In a non-elastic pipe E becomes ∞. If the fluid isnon-compressible than also the value k becomes ∞,leaving c to go to infinity as well. Meaning that the

continuity equation derives to 0ux

∂=

∂, as is valid

when the rigid column approach is applied. Theboundaries towards E and K imply a rigid column,both for the fluid and the pipe wall.

The formula can also be written as

2 0p p u

u ct x x

ρ∂ ∂ ∂

+ + =∂ ∂ ∂

With p p dp

ut x dt

∂ ∂+ =

∂ ∂ this becomes:

2 0dp u

cdt x

ρ∂

+ =∂

(eq. 2.34)

Rewriting this using the piezometric level

pz

gϕ

ρ= +

dpg g u

dt t t xϕ ϕ ϕ

ρ ρ∂ ∂ ∂ = = + ∂ ∂ ∂

gives

2

0c u

t g xϕ∂ ∂

+ =∂ ∂

(eq. 2.35)

10

CT5550 - Water Transport

Equation of movementThe equation of movement is

2

1sin 0

u uu u pu g g

t x x C Rβ

ρ∂ ∂ ∂

+ + − + =∂ ∂ ∂

(eq. 2.36)

with β the slope of the pipe. This can be rewrittenusing the piezometric level instead of a pressurelevel: ϕ = z + h = z + p/ρg:

1 pt g tϕ

ρ∂ ∂

=∂ ∂

1sin

px g xϕ

βρ

∂ ∂= −

∂ ∂

leading to

2 0u uu u

u g gt x x C R

ϕ∂ ∂ ∂+ + + =

∂ ∂ ∂(eq. 2.37)

In the case of water hammer the friction can be ne-

glected as well as the convective term ux

∂∂

, which

leaves the equation of movement to

0u

gt x

ϕ∂ ∂+ =

∂ ∂ or

10

ug t x

ϕ∂ ∂+ =

∂ ∂ (eq. 2.38)

Examples of water hammer

The water hammer formulas describe what happenswhen the pressure and/or volume flow boundariesvary so quickly that the rigid column approach is notvalid any more. This is the case when valves areclosed, pumps are started or stopped (planned orunplanned in the case of pump trip) or any other sud-den change in pressure or flow.

The set of equations is:

10

ug t x

ϕ∂ ∂+ =

∂ ∂ and

2

0c u

t g xϕ∂ ∂

+ =∂ ∂

(eq. 2.39)

The general solution of this set of equations is

( ) ( ) 0F x ct f x ctϕ ϕ= + + − + (eq. 2.40)

( ) ( ) 0

gu F x ct f x ct u

c= − + + − + (eq. 2.41)

with ϕ0 and u0 are the initial pressure level and ve-locity at t = 0.These equations are called the characteristic equa-tions. Basically these equations describe the move-ment of the pressure wave during the time step ∆tover a distance c* ∆t along the pipe. The functionϕ=f(x-ct) moves in positive direction and the func-tion ϕ=F(x+ct) moves in negative direction.

These formulas can be used to estimate the maxi-mum and minimum pressures as a result of waterhammer and give an indication whether specialmeasures are necessary to avoid too high or too lowpressures. Too low pressure might induces cavita-tion or when thin wall pipes are used a implosion ofthe pipe. An example of this is given in figure 2.16.

An example for calculation of the maximum andminimum pressure in a pipe as result of closure of avalve will be elaborated

Starting point are the characteristic equations:

( ) ( ) 0F x ct f x ctϕ ϕ= + + − + (eq. 2.42)

Fig. 2.16 - Imploded tank

11

Hydraulics: theoretical background

( ) ( ) 0

gu F x ct f x ct u

c= − + + − + (eq. 2.43)

At the moment the valve closes the forward movingfunction becomes zero: f(x-ct) = 0. This transformsthe equation to:

( ) 0F x ctϕ ϕ= + + (eq. 2.44)

( ) 00g

u F x ct uc

= = − + + (eq. 2.45)

or

00

c ug

ϕ ϕ⋅

− = (eq. 2.46)

If the pump would have been stopped, meaning thatthe backward moving function becomes zero wouldhave resulted in

00

c ug

ϕ ϕ⋅

− = − (eq. 2.47)

Both these functions are called the Joukovsky equa-tions. The values for the pressure that are calculatedusing this function is the absolute maximum or mini-mum that can occur during valve closure or pumptrip. In practise these pressures will be lower, be-cause valves don’t close instantaneously and pumpshave an inertia causing them to continue workingduring some seconds. Another aspect is the value ofc, the speed of pressure wave movement. When thewater contains gas as is possible in sewerage, thevalue will be much lower. Also the characteristics ofthe pipe determine the value of c. A less flexible pipewill induce a higher maximum pressure.

ExampleThe maximum pressure for two types of pipe will beconsidered at the event of quick closure of a valve.Assume a pipe with a length of 5000 meter a diam-eter of 1000 mm, made of steel with a wall thicknessδ of 15 mm. The elasticity of steel is 2,2*105 N/m2

and the compressibility of water Kwater = 2,05 * 105 N/m2.

2

1 1 1390 m/s

1 1

Dc

c K E DK E

ρ ρδ

ρ δ

= + ⇒ = = +

Fig. 2.17 - Pressure relieve valves

With an initial velocity of the water of 1 m/s the addi-tional pressure will be

0 1390*1139 mWc

10c u

g⋅

= =

Probably this pressure is inadmissible and countermeasures have to be taken. One of the most simplemeasures is to avoid the quick closure of the valve.The deceleration of the fluid is smoother and waterhammer can be avoided. To quantify this the follow-ing calculation can be made. The pressure wave isformed during the last 20% of the actual closing ofthe valve. In this period the deceleration is largest.High pressures (water hammer) can be avoided ifthe pressure wave is allowed to travel from the valveto the upstream boundary and back, during the last20% of the closure interval of the valve. In the last20 % the actual deceleration will take place, causingthe possible water hammer.With a length of 5000 meter between the valve andan upstream boundary this will take 2* 5000/1390 =7 seconds. An upstream boundary is for instance awater tower or a connection to a more looped net-work. Essential is that the pressure can be relievedby relatively small storage capacity, a pressure re-lieve valve (figure 2.17) or a larger network.

Consider another pipe of a more flexible materiallike PVC. The length is 5000 meter; the diameter is630 mm (largest common available diameter) and awall thickness of 25 mm (PN10, see also design ex-ercise). The elasticity of the material is 2,0*109 N/m2

and the compressibility of the water is 2,05*105 N/m2. The pressure wave speed is in this case 270 m/s and the maximum extra pressure is 27 mWc withan initial velocity of 1 m/s.Although this extra pressure is probably allowable inthis situation, also the negative pressure wave at thedownstream side of the valve has to be considered.

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CT5550 - Water Transport

This negative pressure is larger than atmosphericand can cause cavitation. In this case the damp ten-sion is sub seeded and the water ‘boils’ with all nega-tive implications of cavitation (see paragraph XX).Closing the last 20% of the pipe in a time the pres-sure wave can travel from the valve to the upstreamboundary ands back again will be a solution as well.This will take 2*5000/270 = 37 seconds.

Water hammer preventionTo prevent the high prssure as result of water ham-mer for instance a buffer tower can be used. Theprinciple is the same as the wind kettel. Figure 2.18shows the effect of a buffer tower in combinationwith a pump trip. Often water towers are used likethis. Figure 2.19 shows the effect of a buffer towerwhen a valve is quickly closed.

2.7 SummaryThree types of one-dimensional flow can be distin-guished within the field of water transport throughpipes:o Surcharged pressurised flow:§ Rigid column approach§ Darcy Weissbach§ Time independent§ Friction loss dominant§ Drinking water transport, sewerage water

Fig. 2.18 - Buffer tower and pump trip

Fig. 2.19 - Buffer tower and valve

o Open channel flow§ Free water surface§ Time dependant§ Storage dominant§ Storm water, urban drainage and sewerage

o Water hammer§ Special case of pressurised flow§ Fast changing boundaries, time dependent§ Special analysis§ Risk approach

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Hydraulics: theoretical background

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CT5550 - Water Transport