Pipeline start up with entrapped air iahr by Civile

This article was downloaded by: [UPVA] On: 5 August 2010 Access details: Access Details: [subscription number 919660885] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK

Journal of Hydraulic Research

Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t916282780

Pipeline start-up with entrapped air

J. Izquierdoa; V. S. Fuertesa; E. Cabreraa; P. L. Iglesiasa; J. Garcia-Serraa a Grupo Mecánica de Fluidos, Universidad Politécnica de Valencia, Camino de Vera, Spain Online publication date: 08 January 2010

To cite this Article Izquierdo, J. , Fuertes, V. S. , Cabrera, E. , Iglesias, P. L. and Garcia-Serra, J.(1999) 'Pipeline start-up with

entrapped air', Journal of Hydraulic Research, 37: 5, 579 — 590 To link to this Article: DOI: 10.1080/00221689909498518 URL: http://dx.doi.org/10.1080/00221689909498518

PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

Pipeline start-up with entrapped air Mise en fonctionnement d'une conduite avec poches d'air emprisonnées

Downloaded By: [UPVA] At: 15:22 5 August 2010

IZQUIERDO J., FUERTES V.S., CABRERA E., IGLESIAS P.L., GARCIA-SERRA J. Grupo Mecanica de Fluidos, Universidad Politécnica de Valencia, Camino de Vera, s/n - 46022 Valencia (SPAIN)

ABSTRACT A mathematical model for the assessment of the pressure head maxima that air pockets within a pipeline can originate on start-up is presented. This model is based on a general model addressing the filling of a pipeline with several air pockets published by the authors. Here the simulation of the operation of a discharge valve in order to control the peak pressure following the pump start-up is included. Also, in order to correctly model reverse flow through the pump use is made of the dimensionless Suter curves. Water movement is modelled through rigid column theory and air - water straight cross section moving interfaces are considered. Because of the huge pressure values that the very rapid compression of the air can cause, and in order to avoid pipe breaks, practical engineers must lend great attention to this problem. RÉSUMÉ [1 s'agit de présenter un modèle mathématique qui permet d'évaluer les pies de pression créés par des poches d'air emprisonnées dans une tuyauterie, lors de sa mise en fonctionnement. Ce modèle est basé sur un modèle general pour Ie remplissage d'une conduite qui a plusieurs poches d'air, publié par les auteurs. On inclue dans ce cas la simulation de 1'operation d'une valve de décharge pour contröler Ie maximum de pression après la mise en fonctionnement de la pompe qui nourrit 1'installation. Aussi, les courbes adimensionelles de Suter sont utilisées pour modéliser correctement la vitesse de flux opposée au travers de la pompe. Le mouvement de l'eau est modélisé grace a la theorie du modèle rigide et l'interface mobile air-eau est consideré plane. Les ingénieurs de terrain doivent être conscients de ce problème car la rapide compression de l'air peut causer des surpressions qui risquent d'endommager les installations. 1

Introduction

Prudent engineers must be aware of the problems that entrapped air pockets can cause. Reliability and safety of systems suffering proclivity towards this problem are at evident risk. The presence of air within a pipe is manifold. For one thing, in order to avoid cavitation, air is sometimes admitted into the pipe. For another, after service or programmed interruptions, as occur in irrigation systems or for repairing purposes or at a long ago closed valve near a fire hydrant, air pockets develop natu rally at high points due to very small air leaks into the pipe or simply because air comes out of solu tion. If air is not suitably vented out at the installation start-up (and frequently, as in the case of fire demand, there is no possibility to do it) it is rapidly compressed triggering high peak pressures to develop. The behaviour of these air pockets must be analysed in order to assess potential risks. This old and well-known problem has been widely treated in the literature. Gandenberger (1950) describes different cases of important peak pressures derived from the presence of entrapped air pockets. Recently, Van Vuuren (1998) gives recommendations for the correct location of air valves to de-aerate the pipeline. Jönsson (1985) refers to an interesting high peak of pressure due to the presence of entrapped air. Sharp (1996) compares the effect of initiating flow against certain air cushions to this of acting against a closed valve. Martin (1976), in a widely quoted paper, takes into Revision received May 20, 1999. Open for discussion till April 30, 2000.

JOURNAL OF HYDRAULIC RESEARCH. VOL. 37. 1999. NO. 5

579

Downloaded By: [UPVA] At: 15:22 5 August 2010

consideration n entrapped air pockets with static interfaces. This work by Martin is quoted in the standard books on hydraulic transients (Wylie and Streeter, 1993) and used to determine the highest peak of pressure that the compression of an air pocket can generate in the most unfavourable although unrealistic case of a dead end in a pipe. In this paper, the start-up of a pumping station feeding a main with irregular profile is considered. The general layout of m reaches of constant slope with n air pockets separated by n + 1 liquid col umns described with more detail in Fuertes et al. (1997) is addressed. The model herein presented contains, as a particular case, the filling of pipelines with irregular profile without entrapped air addressed by Liou and Hunt (1996), but focuses especially on the more unfavourable case of pres ence of entrapped air. This enriched situation has been previously proposed as a discussion to the Liou and Hunt's paper by Cabrera et al. (1997). In this paper we include the operation of a dis charge valve used to minimise the first peak of pressure and the use of the pump Suter curve to cor rectly model reverse flow through the pump. The model treats the liquid columns by making use of the rigid model (mass oscillation theory), which provides sufficient accuracy for the present problem (see Chaudhry (1989); Abreu et al. (1991); Cabrera et al. (1992)) and considers polytropic evolution of air pockets (Martin (1976)). Yet, unlike Martin, moving boundaries of the liquid columns are considered. In this way, a set of n + 1 non-linear ordinary differential equations describing the rigid behaviour of the liquid col umns, n + 1 integral equations taking account of the moving boundaries and n algebraic equations characterising the evolution of the air pockets describe the model. This set of equations can be transformed into a set of In + 2 ordinary differential equations that can be solved by standard numerical methods. Nevertheless, the rapid variation of the magnitudes involved in some stages together with its steadiness in other phases advise the use of an adaptive step size method. 2 Pipe with n entrapped air pockets. Basic equations The general layout of Figure 1 is considered. The pipe exhibits irregular profile, with m constant slope reaches, and with n entrapped air pockets. A source of energy (tank or pump) is located upstream and a valve has been installed at the delivery in order to control the flow. The filling col umn, of initial length L0, driven by the source of energy begins to invade the pipe when the dis charge valve is opened, thus compressing the air just downstream of the valve. This ensemble formed by the source of energy, the discharge valve and the initial filling column constitutes the upstream boundary condition.

Fig. 1. Pipe with n entrapped air pockets.

580

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 37. 1999. NO. 5

The following assumptions are made: a. Water behaviour is modelled be means of the rigid model approach (Abreu et al. (1991)). b. A polytropic coefficient of 1.2 is used to model entrapped air behaviour (Martin (1976)). c. The Darcy-Weisbach equation with constant friction accounts for loses, as is common in the standard literature (Chaudhry (1987); Wylie and Streeter (1993)). d. Air - water interfaces are well-defined cross sections. Under these hypotheses, the problem is modelled by: Filling column: 1. Mass oscillation equation (rigid column approach) dv _ pl-p\ dt pL

Az fv\v\ L 2D

(1)

Downloaded By: [UPVA] At: 15:22 5 August 2010

2. Interface position

L = L 0 + J W , ( § = V)

(2)

o where v = filling column velocity, t = time, p0* = upstream pipe pressure, P\* = entrapped air pocket pressure, p = water density, L = filling column length, g = gravity acceleration, Az = piezometric head between the ends of the filling column, ƒ = Darcy-Weisbach friction factor and D = pipe diameter (* stands for absolute pressure). n blocking columns / entrapped air: 3. Mass oscillation equation for the blocking column ;' (i = 1, 2, ..., n) dv, _ p'i-p'j+i

pLbJ

Az,,,,

/y,.|y,.[

LhJ

for the last blocking column (i = n), pn + ,* = patm* applies. 4. Entrapped air pocket i (i = 1, 2, ..., n) Pi • Uf - *i-1 - K i -1)" = Pi, o ' (XL o - *f-1,0 - Lb., -1)" = const.

(4)

where x0 - 0 and Lhn = L for the first entrapped air pocket (i = 1). 5. Position of the water blocking column ;' (/ = 1, 2,..., n)

x, = * />0 +JV,W-^ = vj

(5)

where y, = velocity of the blocking column i, p* = entrapped air pocket pressure i, Lhi = water blocking column length ;', Az,,, = geometric head between the ends of the blocking liquid column / and X; = co-ordinate, refereed to the pipe origin, of the upstream blocking column i,

JOURNAL OF HYDRAULIC RESEARCH. VOL. 37. 1999, NO. 5

581

According to Figure 1, the lengths of the different pipe reaches are noted as L, (J= 1,2, ..., m), and their slopes by 6, (/' = 1,2, ..., m). In summary, a set of 2 + 3n equations describes the whole system. Together with the corresponding boundary and initial conditions, it can be solved for the 2 + 3n unknowns v, L, v„ p,■*, x,- (i = 1, 2,..., n). Initial condition: The system being at rest and the initial length of the filling column and the size of the air pockets defined, the initial condition is, obviously, described by v(0) - 0, L(0) = L(l. v,(0) = 0, p,.*(0) = /?,,„*, x,(0) = xw (/' = 1, 2,..., n) where piQ* = pMm* and x, 0 = initial length of air pocket;'.

Downloaded By: [UPVA] At: 15:22 5 August 2010

Boundary conditions: a. Upstream boundary condition. A pump being the source of energy, (see Figure 2),p0* is given by: HR + P^+Hp Y

P°+S +k(x)f Y 2g

(6) 2g

P* Fig. 2. Upstream boundary condition. The valve loss is described by &(x),where T represents the instantaneous valve opening. HP is the pump head described by the corresponding Suter curve. b. Downstream boundary condition. Water free discharge to the atmosphere, p„ +*= pM,„* (it cor responds to the last blocking column, i = n). Gravity terms: The gravity terms in equations (1) and (3) depend on the position of the corresponding water col umn. Subscripts p and q (see Fig. 1) are used to identify the reaches (1 to m) where the front and the tail, respectively, of the water column are located. a) Phases of the movement of the blocking column i: a. 1) Blocking column i front has not arrived to the end of the pipe (x, + Lhj < Lmlal) - Front and tail of column i are on the same reach {q =p) ^

582

= send,,

(7)

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 37. 1999. NO. 5

Front and tail of column i are on consecutive reaches (q = p + 1) ti = i-1 -

Az„,,

V.;=l

senQp+

(8)

send,,

Front and tail of column i have one or more reaches in between (q> p + 1) /

■

/./ -q-\

V Az,,,

V/=i

LjSeni

-senQ„ +

+ 1+

7=1

send,.

(9)

a.2) Water column i is draining (.r, + Lhi a L,atal) - Tail of blocking column i is located on the last reach (p = m)

Downloaded By: [UPVA] At: 15:22 5 August 2010

Az,,. i

senO„

Lb i = L,„uuli - x-, * const.

(10)

Tail of blocking column *' has not arrived to the last reach (p<m) 2 Az,,

LjsenQj

j = /. + i

-senQp+ L,aual ~ X,

(ID Ll0taa

a.3) Blocking column i has completely drained an no longer exists (x, = Llolal) In this case, the 3 equations corresponding to the blocking column - pocket air i, do not apply, and the pressure of the last entrapped air pocket takes the atmospheric value. This must be con sidered in the corresponding equation. Every time a blocking column drains completely the system of equations is reduced, analogously, by three. b) Filling water column movement: b. 1) The filling water column has not reached the downstream end of the pipe (L < Ltoml) - The front of the filling water column is located at the first reach (q = 1) A:

(12)

senQ,

The front of the filling water is beyond the first reach (q > 1) / = 1- 1

! = 'I - I

LiSenQj

Az _ 2ll L

{ 1

JOURNAL OF HYDRAULIC RESEARCH. VOL. 37. 1999, NO. 5

send,,

(13)

583

b.2)The blocking column arrives at the end of the pipe (L a L,otaï) 2 Az _ £=_; L>

L senQ

j Zdowmtream

L'tBtal

^-upstream _

c o n s

(14)

•-•total

When L = Luml, only one equation, namely the mass oscillation equation for the filling column (now occupying the whole pipe), applies: dv _ Pa - Pan, dt ' pL

Az L

fv\v\ 2D

(15)

Eventually, the acceleration term vanishes and the steady state is established.

Downloaded By: [UPVA] At: 15:22 5 August 2010

3 Application to the case of two entrapped air pockets The most simple case (m = 2 and n - 1) is considered in Cabrera et al. (1997). Here another case with w = 4 and n = 2, as shown in Figure 3, is presented. More complex systems can be addressed exactly the same way since the model is completely general.

Fig. 3. Two entrapped air pockets within a pipeline.

The corresponding equations are: Filling column: 1. Mass oscillation equation applied to the filling column

S-!(»-!H'-(,+tw,£)-£-'T-$

(16)

2. Filling column position L = Ln+ VC fvdt

584

dL dt

(17)

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. M, 1999, NO. 5

Blocking column / entrapped air pocket 1: 3. Mass oscillation equation applied to the blocking column 1 dv\ dt

P\-Pi pLlK |

^.i LK,

fv\\v\\ 2D

(,8)

4. Evolution of the first entrapped air pocket p\-(xx-Vf

= p].n-(xU)-Ln)"

= const.

(19)

5. Blocking column 1 position

Downloaded By: [UPVA] At: 15:22 5 August 2010

x, =x,, 0 +jv,<fc o

( ^ 7 = v ')

<2°)

Blocking column / entrapped air pocket 2: 6. Mass oscillation equation applied to the blocking column 2 dv2 _ Pi - pi,,,, Azfc-, df " pL,,,2 '? L,, :

/v2]v2l 2D

7. Evolution of the entrapped air pocket 2 P J - ( X 2 - X , - L , , ,)" = pl„ • (x 2 . ( ) -x,. ( ) - Z.,,.,)" = const.

(22)

8. Blocking column 2 position

—

-hdt

(I1 = ^)

(23)

This set of 8 equations together with the initial condition given by v(0) = 0, L(0) = L0, v,(0) = 0, p,*(0) =ƒ>,,„„*, x,(0) =x, „, v2(0) = 0,p2*(0)=Amn*an(i-*2(0) = X2.0, constitutes a well posed problem whose solution describes the system behaviour. The goal is to determine the 8 unknown functions: v, L, V,, pt*, X,, v2, Pi* and x2. If pt* and p2* are obtained from equations (19) and (22), respec tively, and substituted into equations (16), (18) and (21), a set of 6 ordinary differential equations is obtained. To solve for the unknown functions an adaptive variable step fifth-order Runge-Kutta method (see Stoer and Bulirsch, 1980) is used. In the general case, n entrapped air pockets, 2 + 2n simultaneous ordinary differential equations can be solved the same way. Practical application After a long service interruption, two entrapped air pockets have developed in the four-reach pipe, shown in Figure 4. From the data, it follows that the steady state flow is 101.6 1/s (v = 1.438 m/s).

JOURNAL OF HYDRAULIC RESEARCH. VOL. 37. 1999. NO. 5

585

Z=150m

Downloaded By: [UPVA] At: 15:22 5 August 2010

Fig. 4. Practical application: two entrapped air pockets are considered. In this scenario, the pump is started against a closed valve at the pump delivery. The valve is opened after the pump rotational speed has reached its nominal value N0. For a rotating speed of 2900rpm the pump specific speed ns turns to be 0.46 approximately, in SI units. Thus, the Suter curve for a centrifugal pump is considered to model generalised pump behaviour. The pump and the discharge valve together constitute the upstream boundary.

22 20 18 16

12 ÂŁ U

10 2

8 * -6 - 4 2 0 0

100 200 300 400

500

600 700 800

900 1000 1100 1200

time (sec)

Fig. 5. Full transient analysis of the system under consideration. Some relevant results are shown in Fig. 5. The first peak pressure, due to the quick compression of the first entrapped air pocket, is 276.98 m at t = 5.22 sec, and the maximum velocity of the filling column is 18.25 m/s at t = 0.10 sec (see Fig. 6). As can be seen, the first blocking column does not start moving until important pressures have been reached in the first entrapped air pocket. As a consequence, the compression of the second air pocket suffers an important delay with respect to the compression of the first one, giving a much less severe pressure value than in the case of the first air pocket. This pocket and the first blocking column are the only relevant elements as far as peak pressures are concerned.

586

JOURNAL DE RECHERCHES HYDRAULIQUES. VOL. 37. 1999, NO. 5

Downloaded By: [UPVA] At: 15:22 5 August 2010

In order to reinforce with figures the preceding conclusion, the same installation (Figure 5) is again analysed, but now just the first blocking column is considered. The peak pressure is obtained at the same time (276.98 m at t = 5.22 sec), being the maximum velocity of the filling column (18.25 m/s at t = 0.10 sec), the same as before. The full results for this case also match with those shown in Figs. 5 and 6. In summary, the results for the first air pocket are the same, no matter the presence of a second blocking column.

0 I 0

1—-4—'T—I

1—M

10 16

time (sec)

Fig. 6. Details of the initial transient evolution. It is worth underscoring here the great influence of the size of the entrapped air pocket on the peak pressure. Table 1 summarises the different results of this problem taking different air pocket sizes in the system shown in Fig. 4. The smallest the air pocket, the highest the pressure. Table 1. Influence of air pocket size in the peak pressure (system shown in Fig 4). Li,.\ (m)

xi,o(m)

Lair pocket (IÏ1)

p Umax ( m )

t(s)

630

277.0

5.22

600

100

243.7

6.72

570

120

119

215.3

8.30

540

140

139

191.2

9.97

Finally, as can be expected, a controlled valve opening after the pump has reached its nominal speed, instead of the sudden one simulated so far, reverts in lower peak pressures. Obviously, the slowest the manoeuvre the lower the peak pressure. Values of the main variables are shown in Figure 7 for differ ent opening times, Tm. A synthetic manoeuvre with valve loss coefficient k = kn{tlTmy]Aior 0 < f < Tm and k = kQ for t a Tm has modelled the opening. Note that for this chart xU) = 10m has been considered, what means a really small air pocket: Luirpm.kel = 9m. In this extreme scenario reverse velocities through the pump are observed, since the high compression of the air pocket forces the filling column backwards. Then, use is made of Suter curves to properly model this abnormal pump operation. Only for sufficient slow valve manoeuvres reverse velocities do not show up.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 37. 1999, NO. 5

587

ioo I

1—Ll

[

—.

■ .»-... ;.i -

' I

Opening time (s)

Fig. 7. Main values for different opening times.

Downloaded By: [UPVA] At: 15:22 5 August 2010

700

/"~~~ j

— pressure

■ \

WM I

f -.

"""^

200 ■

100

0 1

Filling column length (m)

Fig. 8. Air pressure and filling column velocity versus filing column length for sudden valve opening and small air pocket. Also, as can be appreciated, head and velocity maxima can relatively be controlled. It is worth to observe here that for small pockets even long opening times do not show able to maintain air pres sure under 200m, about 120% of the steady state pressure for the present example. Reverse velocity can easily be appreciated in Figure 8, showing the phase plane for air pressure and filling column velocity versus filling column length for Tm = 0 and L.,ir p(Kkct = 9m, a really severe scenario. The closed loop of the velocity graph and the way the pressure graph turns back after the peak clearly reveal the rearward behaviour of the filling column after the maximum. 4 Conclusion A mathematical model for accurate determination of the peak pressures that can develop on the start-up of a pipeline with irregular profile with n entrapped air pockets has been presented. Because of the small inertia of air relative to water, the induced peak pressures are, usually, very

588

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 37. 1999, NO. 5

Downloaded By: [UPVA] At: 15:22 5 August 2010

important and can originate the break of the pipe if a correct manoeuvre and/or suitable protection devices are not foreseen. To conclude, some remarks can be made: a. The most interesting case is the simplest one with just one air pocket (n = 1). New blocking colÂ umns do not change at all the maximum pressure of the first pocket. b. Because important pressure values are achieved with small pockets, shorter interruptions of service will cause major problems. c. For small air pockets very long opening times are necessary to maintain air pressure under levels which can be considered safe for the pipeline. d. Pipelines with entrapped air pockets must be filled with very slow manoeuvres, in order to miniÂ mise pressure head. For instance, by controlling the power of the energy source with a slow opening valve. e. The installation of air valves does not automatically guarantee that the problem has been solved, since air can remain entrapped due to valve bad selection or even more frequently to incorrect maintenance. f. The best solution, when possible, is to drain the water of the blocking columns. This amounts to absence of air pockets and the consequent alleviation of some problems. In fact, the filling of pipelines in such conditions can generate cavitation at high points as has been shown in Liou and Hunt (1996). g. In order to perform a faster assessment of the peak pressures, the authors have identified up to six non-dimensional system parameters. Only two, in some cases three, of them are relevant enough as to condition the performance of the system (Fuertes et al., 1999). h. Due to the influence of the polytropic exponent n on the final results, laboratory tests are being performed in order to determine experimentally its most adequate value. Notations A I)

f 8 II HR HP k{x) ki)

L U.i

n ns Po P, Pi.O

cross sectional area of pipe inside pipe diameter Darcy-Weisbach friction factor; acceleration due to gravity; piezometric head; reservoir head; pump discharge head; valve loss; valve loss for open valve; length of filling column: length of blocking column i; length of pipe i; nominal rotational speed of the pump; polytropic coefficient; pump specific speed; pressure energy of upstream end; pressure of air pocket;'; initial value of pf,

JOURNAL OF HYDRAULIC RESEARCH, VOL. 37, 1999, NO. 5

589

Downloaded By: [UPVA] At: 15:22 5 August 2010

patm Q T„, t v v, x, x,,„ Z Az Az,,,Y 9, p x *

atmospheric pressure; flowrate; valve maneuvering time; time; velocity of filling column; velocity of blocking column ;'; space coordinate of blocking column i; initial value of x,; elevation; elevation difference between ends of filling column; elevation difference between ends of blocking column /'; water specific weight; slope of pipe /'; water density; instantaneous valve opening; (superscript) absolute value;

References ABREU, J., CABRERA, E., GARCI'A-SERRA, J., IZQUIERDO, J. (1991), Boundary between elastic and inelastic

models in hydraulic transients with entrapped air pockets, Proc. of the 9'1' Round Table on Hydraulic Transients with Water Column Separation, Valencia, Spain. CABRERA, E., ABREU, J., PEREZ, R., VELA, A. (1992), On the influence of liquid length variation in hydraulic transients, Journal of Hydraulic Engineering, ASCE, pp 1639-1650, December 1992. CABRERA, E., IZQUIERDO, J., ABREU, J., IGLESIAS, P.L. (1997), Discussion of the paper "Filling of pipelines

with undulating elevation profiles", Liou, C.P. y Hunt, W.A., Journal of Hydraulic Engineering, ASCE, pp 1170-1173, December 1997. CHAUDHRY, H.M. (1987), "Applied Hydraulic Transients", Van Nostrand Reinhold, New York, USA. CHAUDHRY, H.M. (1989), Application of lumped and distributed approaches for hydraulic transient analysis, Proc. of the International Congress on Cases and Accidents in Fluid Systems, ANAIS, Polytechnic Univer sity of Sao Paulo, Brazil. FUERTES, V.S., IZQUIERDO, J., IGLESIAS, P.L., CABRERA, E„ GARCI'A-SERRA, J. (1997), Llenado de tuberfas

con aire atrapadoA Ingenieria del Agua, Volumen 4, n° 3, pp 57-67, September 1997. FUERTES, V.S., CABRERA, E., IZQUIERDO, J., IGLESIAS, P. (1999), Peak pressures evaluation, by dimensional

analysis, in pipelines with entrapped air pockets. International Symposium on Water Hammer, ASME & JSME Joint Fluids Engineering Conference, San Francisco, USA, July 1999. GANDENBERGER, W. (1950), Grundlagen der graphischen ermittlung der druckschwankungen (Graphical anal ysis of pressure surges in water supply lines), R. Oldenbourg Verlag. JÖNSSON, L. (1985), Maximun transients pressures in a conduit with check valve and air entrainment, Proc. of the International Conference on the Hydraulics of Pumping Stations, Manchester, England. LlOU, C.P., HUNT, A.W. (1996), Filling of pipelines with undulating elevation profiles, Journal of Hydraulic Engineering, ASCE, pp 534-539, October 1996. MARTIN, C.S., (1976), Entrapped air in pipelines, Proc. of the Second International Conference on Pressure Surges, London, England. SHARP, B.B., SHARP D.B. (1996), "Water Hammer. Practical Solutions", Arnold, London-Sidney-Auckland. STOER, J., BULIRSCH, R., (1980), "Introduction to numerical analysis". Springer-Verlag, New York, USA. VAN VUUREN, S.J. (1998), Valves to control air in pipeline design. Water Management International 1998, pp 27-30. WYLIE, E.B., STREETER, V.L. (1993), "Fluid transients in systems", Ed. Prentice Hall, Englewood Cliffs, New Jersey, USA.

590

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 37. 1999. NO. 5