Ellis,J. Pressure transients in water engineering, A guide to analysis and interpretation of behaviour

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and transmit pressure waves to an extent dependent upon the charac-

teristics of the particular hydraulic element. As a general rule other

pipeline features such as tanks and reservoirs, blank ﬂanges, changes

of pipe properties and bifurcations for instance all act passively to

reﬂect and/or transmit pressure waves. A rare exception to this rule

might be where surface waves introduce pressure and ﬂow variations

at the inlet of a submerged pipe.

Other boundaries such as air-valves, pressure-relief valves and check

valves respond to changing ﬂow conditions and may only come into play

at certain stages of a transient event. They will have little if any inﬂu-

ence on propagation of pressure transients at some times while they may

be responsible for producing important secondary pressure surge events

at another stage of the hydraulic transient.

Additional boundaries are also required to represent pressure tran-

sient suppression equipment. Some of the equations used to model

response of such equipment are discussed when considering speciﬁc

forms of protection in later chapters.

A further type of boundary condition is that which is introduced

by the modeller for convenience to simplify or reduce the work of

analysis. This form of boundary is purely a numerical device and need

not correspond with any physical feature of the pipeline network.

A common assumption with most models is that at any instant

of time, ﬂow behaviour at the boundary is identical to that which

would occur under steady ﬂow. For example, the head drop through

a valve or other ﬁtting at some transient ﬂow rate Q ¼ QðtÞ would

be taken as identical to the head loss if Q were a constant ﬂow

rate.

6.2 Reservoirs and tanks

Piezometric level H

at a pipe inlet may be taken as constant and equal

to the free-surface water level as in the case of a large reservoir, or as

relatively slowly varying with time or possibly changing periodically

with surface wave action.

When ﬂow is from the pipeline into the reservoir piezometric level

just inside the pipe, H will effectively be the same as in the reservoir,

thus H ¼ H

Taking that characteristic arriving at the pipe inlet for time t

(Fig. 6.2), then the quasi-invariant relationship gives:

V  g=aH ¼ J or V ¼ J  ðÞg=aH

Pressure transients in water engineering

When ﬂow is into the pipeline from the reservoir, kinetic energy is

developed and losses are incurred so that H

¼ H þð1 þ K

ÞV

=ð2gÞ,

where K

is the entry loss coefﬁcient. The relationships at the pipe

inlet at any time t become:

V  g=aH ¼ J and H ¼ H

ð1 þ K

ÞV

=ð2gÞ

Substituting for H then:

ð1 þ K

Þ=ð2gÞV

 ðÞa=gV þ a=gJH

¼ 0 ð6:1Þ

This quadratic equation can be readily solved for V and, by substituting in

the quasi-invariant relationship, H can be determined. For further details

of the solution see section 6.5 below. In many instances of waterworks

practice, velocity is relatively low so that ð 1 þ K

ÞV

=ð2gÞ is small. For

instance if V ¼ 1 m/s and K

¼ 0:5, ð1 þ K

ÞV

=ð2gÞ¼7:6cm. At

relatively low velocities it is quite acceptable to set H ¼ H

for both

Internal node

Single characteristic

arrives at boundary

Boundary node

Pipeline

Reservoir level = H

Characteristics

Fig. 6.2. ‘Large’ reservoir boundary

Boundaries

inﬂow and outﬂow. Then the solution for velocity V is simply:

V ¼ J ðÞg=aH

ð6:2Þ

In other circumstances, for instance in hydropower applications or

when studying transients in the outﬂow pipelines of high dams, the

velocities can be greater by an order of magnitude or more. If velocity

is 30 m/s — not unusual for dam outlet works — V

=ð2gÞ¼45:9m,

which clearly cannot be neglected and the quadratic form of equation

(6.1) is necessary.

6.3 Branches and changes in pipe properties

Branches are a common feature of many pipeline networks. Consider a

simple connection of three pipes as illustrated in Fig. 6.3a.

Each of the three pipes has a boundary at the junction of the set of

pipelines. The x—t planes for each pipe can be shown as meeting

along a common time axis as depicted in Fig. 6.3b. In general the

positive direction of ﬂow in any pipe may be directed either towards

the junction or away from the junction. Accordingly the Riemann

quasi-invarient travelling along a characteristic towards the junction

may be either of the Jþ or the J type depending upon the assigned

þve direction of ﬂow in that pipe.

From the inﬁnity of characteristics approaching the junction along

any pipe i, that path is chosen which arrives on the common time

axis at the instant t when a solution is to be found. The characteristic

for pipe i will provide a quasi-invarient value J

at time t on the common

time axis. An equation of the form:

 g=a

H ¼ J

is thus obtained for each characteristic at the junction. A common

piezometric head H has been assumed at the junction end of each

pipe — that is, the velocity head in each pipe is considered to be

negligible. In water supply, economic velocities are normally of the

order of 1 m/s while sewage rising mains will usually carry ﬂows at

higher velocities, say up to 3 m/s or more, often to ensure self-cleansing.

Velocity head will thus be of the order of 0.05—0.5 m. Ignoring the

change in velocity head from one pipe to another in these conditions

will in general not lead to signiﬁcant error. Other applications may

require more careful consideration of velocity changes; for example, in

the pressure pipelines or penstocks of a hydroelectric plant, the ﬂow

velocity can be much higher with velocities through relief valves as

Pressure transients in water engineering

high as 60 m/s. Velocity at the emergency closing valves of a dam can be as

high as 30 m/s. Velocity head V

=ð2gÞ then becomes much more substan-

tial in these circumstances. Supposing a pipe junction is made between

pipes having cross-sectional areas A

¼ 1m

and A

¼ 2m

(Fig. 6.4).

At a ﬂow rate of 20 m

/s the change in piezometric level between the

Piezometric lines for pipes 1, 2 and 3

Common head = H measured

from horizontal datum

Positive direction of flow may be

defined as towards the junction or

away from the junction in any pipe

Pipe No. 2

Pipe No. 1

Pipe No. 3

Area = A

(a)

(

)

Characteristics arriving at the junction

Fig. 6.3. Representation of a pipe junction

Boundaries

pipes, ignoring losses, is 15.3 m — a quantity which cannot be ignored.

Consideration of energy losses and of kinetic energy changes becomes

necessary and calculation of piezometric level changes based on the

assumption of common head at the junction is no longer acceptable.

Returning to the simpler case where it is acceptable to neglect

velocity head changes at the junction then ignoring compressibility of

the relatively small volume of water at the junction, conservation of

volume yields:

ðA

Þ¼0 ð6:3Þ

where the summation is over all contributing pipes. Substituting for V

then:

¼ J

 ðÞg=a

Piezometric line or hydraulic gradient

C+ characteristic C– characteristic

Wave speed = a

Area = A

Common head = H

Fig. 6.4. Closed-end or blank ﬂange boundary

Pressure transients in water engineering

and in the conservation equation:

½A

ðJ

 ðÞg=a

HÞ ¼ 0

ðA

Þ¼g

ðA

ÞH

giving,

H ¼ 1=g

ðA

Þ=

ðA

Þð6:4Þ

If the assigned positive direction of ﬂow in pipe i is towards the junction

then the plus sign is used for that pipe otherwise the minus sign is used.

Through induction, equation (6.4) can be shown to apply to any

number of pipes m, where 1  m 1, meeting at a junction. This

equation (6.4) for the junction of m pipes is the simplest form obtain-

able and this equation can be enhanced to include the presence of

air or gas at the junction, or isolating valves on each branch, which

may be either open or shut.

The corresponding velocity at the junction end of any pipe i at

time t is given by substituting the value of H obtained above, into

the equation for the quasi-invarient value J

at the junction end of

the pipe. Thus:

¼ J

 ðÞg=a

H ð6:5Þ

6.3.1 Speciﬁc cases — number of pipes ¼1

When the number of pipes ¼1 the equation represents the solution at a

closed end or blank ﬂange (Fig. 6.5). Thus:

¼ 1=gðÞA

=ðA

Þ or H

¼ a

=gJ

ð6:6Þ

and

¼ J

 g=a

or V

¼ J

 g=a

 a

=gJ

or V

¼ 0 ð6:7Þ

6.3.2 Speciﬁc cases — change of cross-sectional area

When the number of pipes ¼2 the equation can be used to represent

some change of pipe properties, either a change of cross-section,

material, wall thickness or resistance. With reference to Fig. 6.5, then

ignoring changes of kinetic energy and losses at the junction of pipes:

H ¼ 1=gðþA

 A

Þ=ðA

þ A

Þð6:8Þ

Boundaries

Velocities in pipes 1 and 2 are respectively given by:

¼ J

 g=a

H and V

¼ J

þ g=a

H ð6:9Þ

Any change of pipe material, wall thickness or diameter will lead to a

change of wave speed a. Wave speed may also be altered by the

pressure-dependent gas concentration in a pipe.

A special case of a change in cross-sectional property is where a pipe-

line terminates at a reservoir (Fig. 6.2). Suppose the þve direction of

ﬂow is into the reservoir. Within the reservoir V

¼ 0 ¼ constant

and H

¼ H

ðtÞ where H

ðtÞ is a slowly varying function of time or

may be constant. The reservoir cross-section A

is very large com-

pared to pipe area A

and wave speed in the reservoir can be taken

as the speed in an unconﬁned body of water — that is, a

ðK=Þ,

then:

¼ V

 g=a

¼g=a

H ¼ 1=gðþA

 A

Þ=ðA

þ A

H ¼ 1=gðþA

 A

g=a

Þ=ðA

þ A

Single characteristic

at boundary

Blank flange or closed endPipeline

Boundary node

Internal node

Fig. 6.5. A change of cross-section

Pressure transients in water engineering

since

 A

and jA

jjA

g=a

H ¼ 1=gðA

g=a

Þ=ðA

H ¼ H

6.4 Response of a large pipe or trunk main

It is not uncommon for work to be carried out on parts of an extensive

network of interconnected pipelines. Pressure transients are no

respecters of contract boundaries and will spread throughout a system

of pressure pipes until a shut valve/closed end or a reservoir is

encountered. From each of these the pressure wave may be taken as

reﬂected 100% in one form or another. If the entire network were to

be modelled then the workload could be very considerable, not least

in the effort required to obtain adequate information on other parts

of a large system outwith the contract area of immediate interest. It

may be acceptable, in certain circumstances, to limit the extent of

modelling by an approximation at some important point of wave reﬂec-

tion within the system as a whole.

Suppose, for example, a large pipeline, trunk main or aqueduct supplies

a number of smaller branch networks as illustrated in Fig. 6.6. Suppose

work is to be carried out on one of these branched systems including a

hydraulic transient investigation. Pressure transients created within the

branched system will propagate upstream towards the connection of

the branch with the main aqueduct. Assume an initial steady velocity

exists in the branch main just downstream of the connection. If the

initial steady-state piezometric level at the junction is H

then setting

the datum for head measurements at this level H

¼ 0. Should the

initial velocity be reduced to zero by valve closure in the branch

network then the inertial head rise H

in the branch main will be:

¼ aV

=g ð6:10Þ

Ignoring changes due to resistance and pipe elevation variation, then

the invariant value J leaving the shut valve just after its closure, is

given by:

J¼V  g=aH ¼ 0  g=aða=gV

Þ¼V

ð6:11Þ

In the upstream section of the aqueduct, assuming all ﬂow entering

the branch comes from this direction and that no other ﬂows are

Boundaries

present:

Jþ¼V

þ g=a0 ¼ V

ð6:12Þ

And from the downstream section of aqueduct where zero ﬂow is taking

place:

J¼0 þ g=a0 ¼ 0 ð6:13Þ

Due to valve closure in the branch main, the head change H

at the

junction of the branch main with the aqueduct is given by equation

(6.4). Substituting in this equation then:

¼ 1=g½ðV

ÞA

þ V

 0A

=ðA

þ A

¼ 2=gA

=ðA

þ 2A

Þð6:14Þ

Head rise at the junction as a proportion of the initial head rise at the

valve in the branch is then:

¼ 2=½1 þ 2ðA

Þ=ðA

Þ

J+ = V

J– = –V

J– = 0

Fig. 6.6. Small branches connected to trunk main

Pressure transients in water engineering

or for

a ¼ constant H

¼ 2=ð1 þ 2A

Inserting values for area and wave speed into the above equation it

can be shown that as aqueduct area/branch area increases the response

at the connection tends more and more towards that of a simple reser-

voir. This relationship allows the investigator to assess the implication

of replacing the aqueduct connection by a simpler boundary such as a

reservoir.

6.5 Actuated valves and pipeline ﬁttings

In the case where isolating valves such as gate valves or butterﬂy valves

are distributed along a pipeline at intervals to facilitate maintenance

and these valves remain open throughout a transient event, the head

drop through each valve is relatively small and so there is little to be

gained by modelling each valve as an internal boundary. It is usually

sufﬁcient to establish the overall head loss coefﬁcient for all of the

valves and to distribute this along the pipeline together with other

minor losses from bends, tees, etc., to yield an overall resistance

coefﬁcient. Overall head loss H

is given by:

ðK

ÞþfL=D

=ð2gÞ¼f

L=DV

=ð2gÞð6:15Þ

where f

¼ f þ

D=L.

Other valves or ﬁttings may impose signiﬁcant head losses on the

ﬂow even when fully opened and so warrant inclusion in a model as a

discrete head loss. Where a valve may operate to open or close

during a transient event it is essential to include this as a discrete

head drop in the model.

In general a manufacturer will provide data on the relationship

between ﬂow rate and head loss through a valve in the form of a

head loss coefﬁcient curve as a function of valve position. The head

loss coefﬁcient K

is documented as a function of valve stroke between

the fully open position and some point near to the shut position.

Figure 6.7 depicts typical head loss coefﬁcients for a range of valve

types. Due to the very non-linear relationship between K

and stroke,

particularly as a valve nears the closed position, it is not advisable to

interpolate values of K

directly. Rather the head loss relationship

should be represented in a way that endeavours to reduce the degree

of non-linearity. A simple approach is to interpolate values of 1=

Boundaries