CAVITATION IN WATER-HAMMER CALCULATION

7/28/2019 CAVITATION IN WATER-HAMMER CALCULATION.

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Scientific Bulletin of the

Politehnica University of Timisoara

Transactions on Mechanics Special

issue

The 6th International Conference on

Hydraulic Machinery and Hydrodynamics

Timisoara, Romania, October 21 - 22, 2004

CAVITATION IN WATER-HAMMER CALCULATION

Gabriel TATU, Prof., Ph.D.*

Hydraulics and Environmental Protection

Department, Technical University of

Civil Engineering Bucharest

*Corresponding author: Bd. Lacul Tei No.124, Sector 2, Bucuresti, Romania

Tel.: (+40) 21 2433660, Fax: (+40) 21 2433660, Email: [email protected]

ABSTRACT

The usual model for hater-hammer calculation in

the presence of the cavitation is presented first. In

this model, the cavitation bubbles are considered as

big cushions concentrated in the calculation nodes of

the finite differential scheme. The study aims to

analyze the behavior of the model as a function of

the number of elements chosen to obtain the finite

differential scheme (the number of the calculation

nodes). The main remark is that, for a too small

number of finite elements, the computing results are

not real, i.e. the pressure oscillations have extremely

high amplitudes, which are not real at all. Finally, arecommendation is made: when applying this model, a

great attention must be paid to take practical

conclusions.

1. INTRODUCTION

The present paper was inspired by the paper [7]

where the cavitation is calculated within a water-

hammer program when analysing the behaviour of a

hydraulic system and giving a diagnosis verdict. The

author is risking (in my opinion) to emit a verdict on

that system, based on the results of running thatprogram. In my opinion, it is a wrong position and I

will try bellow to demonstrate it.

When pressure drops under the cavitation limit, as

in the boiling phenomenon, the water turns

suddenly into vapour. Cavitation bubbles (filled with

vapour and other gases, formerly dissolved in the

water) appear in the whole volume of water while

the pressure remains constant at the value of the

cavitation limit. When after, the pressure skips

above this limit, the cavitation bubbles are disap-

pearing (suddenly also) in the mass of water and the

continuity of the fluid is re-established.

Since the water-hammer phenomenon is calculated

using the finite elements method and all the variables(pressures, flow rates etc.) are calculated in the

computing nodes, the only solution to calculate the

cavitation phenomenon is to consider the whole

cavitation volume as big vapour cushions concentrated

in such computation nodes.

This way, a great simplification of the real situation

is made: in the real phenomenon that (vapour) volume

is spread along a long distance nearby the computation

node and not as a big cushion but as very numerous

and very small (even microscopic) bubbles.

On another side, the volume of such a vapour

cushion will depend on the number of the calculation

nodes. The bigger is the number of the calculation

nodes,thesmalleristhevolumeofthevapourcushions

(concentrated in the computing nodes).

So, it is expected that the calculation results will

depend on the number of the calculation nodes and

this paper aims to demonstrate this hypothesis.

A very simple case was considered (see figure 1).

A pipe of 1000 m length, having an ascending-

descending profile was considered between two

constant level reservoirs. This way, for the normal

steady state regime, in the highest point of the profilethe pressure is close to zero (the hydraulic line touches

the profile).

Starting from this initial situation, a water-hammer

phenomenon was provoked by suddenly closing the

downstream valve V, placed just before the corre-

sponding reservoir.

The propagation of the over-pressures and then of

the under-pressures created by reflection at the up-

streamreservoirallowstodraw,asaresult,amaximum

enveloping line and, respectively, a minimum envel-

oping one, giving the maximum-maximorum and the

minimum-minimorum pressures along the pipe.


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550

50 m

10 m

V

500 m 500 m

10 m

Hydraulic line

Pipe axis

Cavitation line

30 m

0 m

Figure 1

The shape of the longitudinal profile was specially

conceived to facilitate the apparition of the low

pressures and, consequently, of the cavitation.

Considering the cavitation limit at a vacuum of 10

m.w.c., the cavitation line will be a parallel to and

under the longitudinal profile (pipe axis), at a distance

of 10 m. In the points where cavitation appears, the

enveloping line for the minimum-minimorum pres-

sures is super-posing over the cavitation line.

Figure 2

The program for water-hammer calculation was

provided with two options: with cavitation, i.e. if cavitation appears the pres-

ence of the vapor cushions in the computing nodes

is considered, together with their increasing and

decreasing phases, while the pressure remains

constant at the cavitation limit;

without cavitation, i.e. if cavitation appears onlythe pressures are fixed at the cavitation limit while

the presence of the vapor cushions is not considered.

Figure 2 presents the results without cavitation.

Cavitation appears on about a half of the pipes

length (nearby the highest point of the profile) and

the maximum pressure is about 116 m.w.c. (just in

front of the valve V).In this figure as in the next ones, DY represents the

vertical dimension of the drawing and it gives the

amplitude of the water-hammer phenomenon: the

bigger is the value of DY, the bigger is that

amplitude, i.e., the bigger is the maximum pressure.

The next figures (from 3 to 7) represent the results

of running the water-hammer program with cavitation

but considering different numbers for the computing

nodes (denoted by N). So, the first number written in

these figures (before DY), represents the number of

the computing nodes, from the upstream reservoir to

the sudden closing valve


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Figure 3

Figure 4

Figure 5

Figure 6


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Figure 7

In the last two figures (6 and 7), because of the

huge amplitude of the pressure variation (maximum

pressures of 1253 m and 1345 m, respectively), the

pipe axis is very close to the cavitation line which is

super-posed with the minimum pressure line (i.e. thecavitation is produced on the whole length of the pipe).

2. CONCLUSIONS

The conclusion is very clear: without any doubt,

the number of the computing nodes has a great

influence on the results.

In this case, if the number of the computing nodes

is under 110, a huge amplification of the maximum

pressures is produced because the big vapor cushions,

when suddenly disappearing, avoid the water columns

to clash each other, producing hydraulic shocks. The

bigger is the vapor cushion volume, the bigger is thehydraulic shock.

But as said before, generally this in not a real

situation. In fact, in most cases, the (very small, even

microscopic) cavitation bubbles are spread in the

whole volume of water and on the whole length

affected by cavitation. Only in very special cases,

the vapor volume will have the shape of a vapor

cushion and will be concentrated in a given point

and only for those special cases, the model for

cavitation calculation will be suitable. Surely, it is

not the case analyzed in the paper [7] where theground is flat and the longitudinal profile is horizon-

tal. The calculated cavities, of 0,4 m3

or 1,4 m3, do

not really exist.

There are then, at least two very strong reasons for

not applying this model:

the results of the model depend strongly on thenumber of the computing nodes, chosen for

applying the finite elements method;

the basic hypothesis, i.e. the existence of theconcentrated cavities is very far from the reality.

Consequently, the given verdict, namely that the

cause of the damages was the water-hammer, has not a

correct basis. In my opinion, the pretended damaging

maximum pressures of more than 15 bars do not really

exist and then, the cause of the damages must beanother, yet not found.

Just in fact, in our department, where water-hammer

has been studied for more than 40 years and a great

experience has been accumulated, cavitation is not

calculated but is avoided by suitable means. We have

adopted that politics from the very beginning of

applying the numerical methods for water-hammer

calculation [6] and still consider it is the best position.

REFERENCES

1.Riemann, B. ber die Fortpflanzung ebenerLuftwellen von endlicher Schwingungsweite, Abb.d. Ges. d. Wiss., Gottingen, 1860.

2.Jukovski, N.E. O Gidravlieskom udare vvodoprovodnh trubah, Bul. Politehn. Obscestva,

nr.5, 1899.

3.Allievi, L. Teoria generale del moto perturbatodellacqua nei tubi in pressione, Ann. Soc. Ing.

Arch. Italiani, Milano, 1903.

4.Bergeron, L. Du coup de blier en hydrauliqueau coup de foudre en lctrici , Dunod, Paris, 1950.

5.Streeter, V.L., Wylie, E.B. Hydraulic Transients,

Mc. Graw-Hill, New York, 1967.6.Cioc, D., Tatu, G. - Un programme gnral pour lecalcul du coup de blier et quelques rsultats, XII

Convegno di Idraulica e Construzioni Idrauliche,

Bari, 23-27 ottobre 1970.

7.Ivetic, M. The Failure of a Desalination PlantHeader Pipeline Causes and Proposed Remedy,

Proceedings of the International Conference on

CSHS03, Belgrade, 29-30 September 2003.

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CAVITATION IN WATER-HAMMER CALCULATION