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

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other factor such as different ball density may be a signiﬁcant factor.

The wafer-type split-disk valve with torsional springs to promote

closure has received a considerable amount of attention. The majority

of studies have been conducted for the most common case of a

horizontal valve installation. Figure 21.27 depicts results from different

tests carried out at Delft Hydraulics Laboratory and reported by

442

2.5

2.0

1.5

1.0

0.5

0.0

Demag

DRV-B

Measured DN 800

medium spring

Velocity when fully

open = 1.5 m/s

Magnitude of reversed velocity at closure (m/s)

Deceleration

radient |dV/dt|

(

m/s

)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

DN 600

DN 300

DN 700

DN 800

DN 500

DN 400

Fig. 21.28. Performance of Demag DRV-B valves

Pressure transients in water engineering

Provoost (1983), by Collier and Hoerner (1983) and by Mualla (1969).

Also included are computer predictions by Ellis and Mualla (1986) (Fig.

21.27) for different spring stiffnesses and valve sizes. Pake (1983) has

provided results for the DN 150 Duo-chek valve for more unusual

cases of vertically upwards ﬂow and for a valve with one of the torsional

springs removed (broken). The improvement in performance in verti-

cally upward ﬂow reﬂects the inﬂuence of buoyant weight of the

valve doors. Vertically downward ﬂow would be expected to show a

poorer performance than the horizontal installation, as the door

weight moment then acts against closure. The curves of Fig. 21.27

show a clear deterioration of performance with increasing diameter

and as would be expected an improved closure performance as the

spring stiffness is increased.

Another valve type which has received considerable attention is the

Demag nozzle pattern. Several variations of this valve are available and

results for DRV-B and DRVg types are included. Figure 21.28 depicts a

set of computer predictions by Perko (1986) for the DRV-B and also

experimental data for the DN 800 valve from Koetzier et al. (1986).

For the DRV-B DN 300, Fig. 21.29 shows how closure performance

443

Stiff spring, V

= 3.0 m/s

Demag DRV-B DN 300

Magnitude of reversed velocity at closure (m/s)

Deceleration

radient |dV/dt|

(

m/s

)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 2

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

No spring, V

= 0.0 m/s

Normal spring, V

= 1.5 m/s

Fig. 21.29. Performance of Demag DRV-B DN 300 valve

Check valve characteristics

changes with varying spring stiffness. In this ﬁgure the initial velocity is

representative of the linear spring stiffness, with a broken or missing

spring requiring zero velocity to maintain the valve in the open position.

A stiff spring by contrast needs a ﬂow velocity of 3.0 m/s to fully open

the valve. Not surprisingly, the stiffer spring is able to close the valve

quickly but the absence of a spring requires differential pressure

444

Magnitude of reversed velocity at closure (m/s)

Deceleration

radient |dV/dt| (m/s

)

0 2 4 6 8 10 12 14 16 18 20 22 24 26

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Gestra RK 56

DN 200 weak spring

Demag DRV-B

DN 300 strong spring

Alsthom Clasar

DN 800

Demag DRVg

DN 200 weak spring

Demag DRV-B

DN 300 weak spring

Fig. 21.30. Performance of different valves with spring-assisted closure

Pressure transients in water engineering

across the valve door to achieve movement and closure resulting in a

poor performance.

Figure 21.30 shows results from Delft Hydraulics for a number of

valves having a linear spring closure action. The Gestra RK 56 is of

the globe type as illustrated in Fig. 21.9b. The Clasar type is a wafer

pattern as depicted in Fig. 21.9c. Despite its size (DN 800), the

Clasar valve still retains a very good closure performance.

21.13 Use of the charts

To use the charts requires a preliminary hydraulic transient analysis to

be completed. For the free-acting valve types this can be done without

reference to any speciﬁc valve pattern. The primary task of this initial

analysis is to yield a value of the deceleration gradient jdV=dtj at the

proposed check valve site. This value can then be used to enter any

chart at a point on the horizontal axis. Choice of a velocity gradient

may be relatively straightforward in many cases where the coast-down

of velocity follows a relatively familiar pattern. There is an initial

curved transition from steady ﬂow to an almost uniform gradient.

This uniform gradient could be used to give a reasonably conservative

estimate of jdV=dtj. In other circumstances, matters may not be so

clear-cut, with the deceleration of ﬂow at the NRV following a series

of steps for example. In these cases, possibly it would be prudent to

choose a gradient which represented the steeper part of the curve.

Having chosen a value of velocity gradient, then travelling vertically

upwards from this point on the horizontal axis, an intercept on any

curve can be obtained. The curve used will be for the particular size

and type of valve being considered. More than one chart may be

used where several possible valves are under consideration.

From the intercept with the curve a horizontal line can be drawn to

meet the vertical axis from which a value of reversed velocity can be

read off. This velocity can be used in a number of ways.

It may be sufﬁcient to examine the magnitude of this velocity and

decide if it is acceptable.

A calculation of jaV=gj can be made using the reversed velocity to

determine inertial head rise. To this can be added the prevailing

head at the time of valve closure and a total head value established.

The computer model can be rerun with the valve set to close when

velocity reaches the value established from the chart. Peak pressures

and transient behaviour can then be modelled in more detail.

445

Check valve characteristics

Flexible pipe

Flexible pipes add another layer of complexity through changes in pipe

geometry over time. Where plastic pipes are concerned the phenom-

enon of ‘creep’ also has a role to play. Deformation of a ﬂexible pipe

causes changes in its ability to resist buckling. Propagation of pressure

waves may also be affected by changes in pipe cross-sectional shape.

Response of ﬂexible pipes laid below ground also requires consideration

of the surrounding ﬁll material properties.

22.1 Review of pipe materials and properties

The equation for conservation of mass commonly used in development of

the quasi-invariant relationships which apply along characteristics,

considers the cross-sectional shape to remain essentially unchanged. It

may expand or contract with varying strain in the pipe wall and the

area within the pipe will vary for this reason. Pipes which may reasonably

be assumed to retain their commonly circular shape usually include those

made of the following materials: grey cast iron, concrete and asbestos

cement. Load redistribution and stress reallocation do not occur to any

great extent in these materials and failure is usually through brittle

fracture. Other pipes may be more ﬂexible by virtue of lower deformation

modulus and/or by having a thinner pipe wall. These more ﬂexible

structures include thin-walled steel pipes, thermosetting materials such

as glass-reinforced plastic (GRP), thermoplastics including polyvinyl

chloride (PVC), polyethylene (PE) and polypropylene (PP), and various

composite pipes involving a combination of plastic materials. For compo-

site pipes such as Permastran which has a PVC core for watertightness

and the glassﬁbre outer wrap for strength, the glassﬁbre wrap gives the

composite eight times the strength of the PVC.

446

Plastic pipes also have the property of ‘creep’ or visco-elasticity, produ-

cing a time-dependent stress—strain relationship while the load is applied.

Some thermoplastics can sustain relatively large deformations without

any difﬁculty for the material. For example, a DN 900 mm high-density

polyethylene (HDPE) pipe may have an allowable deﬂection of 20%

with respect to strain; however, stability considerations limit deﬂection

to a safe value of 7.5%. Long-term diametrical deﬂection limits of 5—

7.5% can be permitted without problems for joints, soil movement, etc.

Basic consideration for designing an installation of ﬂexible conduits is

control of deﬂection. For underground pipes a load distribution can be

assumed by mobilising soil support, and concentrated loadings can be

relieved by this process. As an example, HDPE can endure deﬂections

of up to 30% of diameter without excessive bending stresses. When

buried, such large deﬂections would not be acceptable due to soil

movement. HDPE elongation can be as much as 500% at failure.

Thermoplastic pipes are characterised by relatively large allowable

strains, high ductility, large deﬂection without cracking and low stiff-

ness. Typically thermoplastic pipe stiffness will be 317 kN/m

Other plastic pipes of GRP or reinforced plastic matrix (RPM) can

fail due to a lack of ductility. Failure may be through delamination of

pipe walls due to excessive stress. As pipe stiffness is reduced, ductility

must rise to withstand the stresses caused by ovalisation, soft and hard

spots on bedding and other irregularities. Typical allowable strains for

different plastics have been given by Petroff (1984) (Table 22.1).

All ﬂexible pipes display the ability to deform from a purely circular

cross-section under load. Plastic pipes also exhibit ‘creep’ or plastic

deformation. For instance, the shape can change under gravity from a

purely circular form to an elliptical shape merely by virtue of being

stacked while awaiting use. The shape of a deformed cross-section

depends upon the pipe surroundings. Flexible pipes which are above

ground, or in a trench backﬁlled with only slightly compacted or

dumped bedding or soft clay, or laid under water as in an outfall, will

447

Table 22.1. Allowable pipe strain (Petroff (1984))

Material Allowable strain (%)

FRP 0.3

RPM 0.3

PVC 1.5

HDPE 4.0

Flexible pipe

have a generally elliptical cross-section (Fig. 22.1a), with the ratio of

horizontal/vertical deﬂection h=v  0:91. When a pipe bedding is

compacted to a moderate or high degree then the initial deformation

may show an increase in vertical diameter by up to 5% as sideﬁll is

compacted. Backﬁlling above the pipe then reduces this vertical

elongation in diameter. The resulting shape may either be elliptical as

in Fig. 22.1a or a more ‘rectangular’ form may be adopted (Fig.

22.1b), when pipe stiffness is small compared to sideﬁll stiffness. The

resultant ratio of horizontal/vertical deﬂection may be around

 h=v  0:2. The inﬂuence of the increase in horizontal diameter

extends to about 2.5 diameters so that trench sidewall strength may

also have a signiﬁcant effect.

22.2 Pressure transient effects

As far as pressure transient behaviour and prediction are concerned,

several signiﬁcant implications stem from the departure of the cross-

section from its original circular form. Flattening or ‘ovalisation’ of a

circular section will decrease resistance to buckling with an elliptical

or other deformed shape being more prone to collapse than a purely

circular shape in otherwise identical circumstances. If compressive

stress in the pipe wall is too great, failure can occur due to instability.

For underground pipe there is usually a stabilising effect from

surrounding soil, particularly granular soils. Deﬂection and irregularities

of shape, for example ﬂattening of the pipe crown, reduces stability

considerably. Both long-term buckling and elastic short-term buckling

are possible.

As pressure within a pipe changes with time, the shape will continue

to deform. For example, if pressure falls, an elliptical section will ﬂatten

448

DN 450 (18") steel pipe

EI/D m

= 8.618 kN/m

elliptically deformed

21% maximum diametrical deflection

DN 450 (18") steel pipe

EI/D m

= 1.724 kN/m

rectangularly deformed

18% maximum diametrical deflection

(a) (b)

Fig. 22.1. Observed deformed shapes

Pressure transients in water engineering

further and if internal pressure increases, re-rounding of the section will

tend to occur. Changing shape as a consequence of ovalisation is

accompanied by change of cross-sectional area and a corresponding

variation in pressure wave speed. In the case of thermoplastic pipes,

the phenomenon of ‘creep’ or plastic deformation continues to act in

the long term to produce changes in cross-section shape. The behaviour

of pressure transients will depend upon the cross-sectional form at the

time when the surge occurs. If some reduction in buckling pressure is

considered due to ovalisation while retaining the wave speed for a

purely circular section, the effect may be to predict a lower minimum

transient pressure than will actually occur, resulting in a conservative

design. Ideally, adoption of a reduced resistance to buckling should

be accompanied by the corresponding inﬂuence upon wave speed or

damping of the pressure wave.

For non-circular shapes such as square ducts and other polygonal

forms, deformation from the original shape can occur as straight sides

are deformed by bending as described by Thorley and Enever (1979).

The following questions are pertinent for ﬂexible pipelines.

(a) How does the deformed shape inﬂuence the propagation of pressure

waves and the Reimann quasi-invariants along characteristics?

(b) How is the ability of the pipe to withstand transient pressures

inﬂuenced by the changing shape of the pipe in both short and

long term?

of a plastic pipe?

(d) How does possible scratching of the outer surface of the pipe

inﬂuence pipe strength?

22.3 Strain and deﬂection

Plastic pipes are probably best analysed by strain rather than stress.

Critical strain for a perfectly round pipe "

is given by equation

(22.1):

¼ 1=ð1 

Þ1=SDR

ð22:1Þ

the pipe remaining round till this point. SDR ¼ðoD

=s  1Þ, where oD

is the mean outside pipe diameter.

The critical buckling pressure equation can be written to include

strain. Effective maximum strain "

is given by:

¼ð1=D

 1=D

max

Þs

449

Flexible pipe

=s ¼ 1 D

max

giving,

max

¼ 1 "

=s ð22:2Þ

where D

is the original diameter of the round pipe and D

max

is the

maximum deformed diameter.

One of the most difﬁcult tasks is prediction of deformation and strain

over time. A relationship which according to Moser (1981) may be used

for deﬂections of up to 10% in PVC pipes is:

" ¼ðs=D

Þ½3 y=D

=ð1  2 y=D

Þ ð22:3Þ

where y is the vertical change in diameter, D

is mean diameter and "

is the maximum strain. This equation does not include the inﬂuence of

internal pressure. If a cylinder is deformed and therefore not perfectly

circular it is inherently weaker and will collapse at a differential pressure

somewhat below that of a circular pipe in the same circumstances.

When a pipe is relatively unconstrained, for example above ground or

when placed in loose soil, the maximum strain may be:

Design value ¼0.5%

in emergency situations ¼1.5%,

where repeated surging occurs ¼1.3%

The ultimate strain is deﬁned as the initial strain at ﬁrst installation

which would cause failure after 50 years. For PVC this has been

suggested as 1%. Allowable design strain is half the ultimate strain.

Moser could ﬁnd no basis for this 1% ultimate strain value.

The relationship between pipe ring deformation and tangential strain

in the pipe wall has been expressed in a different form by Molin (1981)

as:

¼ pD

=ð2sEÞD

ðy=D

Þðs=D

Þð22:4Þ

where "

is the tangential strain in the pipe wall, y=D

is the deforma-

tion of the pipe ¼ , s is the wall thickness and D

is the deformation

factor.

This equation represents a linear combination of strains due to

internal pressure pD

=ð2sEÞ and strain due to elliptical deformation

ðy=D

ÞðsD

Þ. If the deformation is elliptical then D

¼ 3. In

most practical conditions 4  D

 6, although values of D

in some

circumstances may range from 2.85 to 10 or greater. In circumstances

450

Pressure transients in water engineering

where pipes have been intentionally poorly installed, D

values of

10—20 have been found.

For design purposes for PE pipes, fos=ð1  

Þ¼2 has been proposed

(where fos is factor of safety), giving ﬁnally:

" ¼ pD

=ð2sEÞ6=SDR ð22:5Þ

The value ‘6’ is the initial value for a careful installation.

22.4 Establishing the rate of ovalisation in the longer term

Collapse rate R ¼ rate of increase in % age ovality can be expressed as:

log

ðRÞ¼4:655  2:106  10

5

 p  SDR

ð22:6Þ

where R ¼minutes/(% age decrease in minimum diameter).

Bishop and Lang (1984) provided the following Table 22.2 of pipe

deﬂection and shape values (D

and ) as a function of embedment.

For long-term use D

should be multiplied by a factor of 3.

A buried plastic pipe, after reaching the equilibrium state, is held at

constant deﬂection. For a material such as PVC, which relaxes over

time, once equilibrium is reached internal stress in the pipe wall will

decay with time. Although deﬂection may be less than 5%, equilibrium

state is only slightly inﬂuenced by pipe material properties but is

primarily controlled by initial soil density in the pipe zone.

The vertical diameter of a ﬂexible pipe can increase by up to 5%

due to compaction of bedding soil at the sides of the pipe. When

backﬁlling on top of the pipe, the vertical diameter reduces. For

elliptical deformation, horizontal deﬂection is 91% of vertical deﬂec-

tion. If soil stiffness at the sides of the pipe is large compared to pipe

stiffness, the pipe can deform rectangularly. Horizontal deﬂection can

be vertical deﬂection with horizontal deﬂection 20% of the vertical

deﬂection. Horizontal deﬂections > vertical deﬂections have also been

observed.

451

Table 22.2. Deﬂection and shape v. embedment (Bishop and Lang (1984))

Embedment Deformation factor (D

)  ¼ y=D

Compact sand 5.30 1.50

Compact native soil 10.30 1.13

Loose sand 4.50 7.04

Loose native soil 4.53 9.24

Flexible pipe