The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13) N13P1015 Kanazawa City, Ishikawa Prefecture, Japan, September 27-October 2, 2009.

Modelling of water hammer phenomenon with column separation

Anu Dutta, P Goyal , R. K. Singh , A. K. Ghosh

Reactor Safety Division

Bhabha Atomic Research Centre Mumbai 85

adutta@barc.gov.in , pgoyal10@rediffmail.com , rksingh@barc.gov.in, ccss@barc.gov.in

H. S. Kushwaha Health, Safety & Environment Group

Bhabha Atomic Research Centre Mumbai 85

kushwaha@barc.gov.in

ABSTRACT Column separation is a kind of water-hammer event when the pressure in a pipeline drops to the vapour pressure at specific locations such as closed ends, high points or knees (changes in pipe slope). The liquid columns are separated by a vapour cavity that grows and diminishes according to the dynamics of the system. The collision of two liquid columns, or of one liquid column with a closed end, may cause a large and nearly instantaneous rise in pressure. This pressure rise travels through the entire pipeline and forms a severe load for hydraulic machinery, individual pipes and supporting structures. In one water-hammer event many repetitions of cavity formation and collapse may occur. In this paper a mathematical model which describes the water hammer induced column separation is presented. The classical water hammer equations are not valid in regions of cavity formation. Computation results are favourably compared with the literature quoted experimental results. Consistent agreement has been obtained between numerical prediction and experimental results. The objective of modelling column separation is to predict the pressures that occur when large vapour cavities collapse. In order to improve the performance and reliability of systems, it is important to be able to predict the onset and degree of cavitation taking place. KEYWORDS Cavitation, vapour pressure, water hammer, characteristics method 1. INTRODUCTION The phenomenon of column separation may occur within a piping system when boundary conditions are such that the pressure is reduced at the upstream end of a pipe. The reduction in pressure causes a negative wave pulse to be transmitted down the pipe, thereby reducing its velocity, the fluid down-stream continues at its steady velocity until the wave arrives. This difference in velocities between portions of the pipe flow tends to put the liquid column into tension, which fluids cannot withstand. When vapour pressure is reached, a vapour cavity

1 / 7

The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13) N13P1015 Kanazawa City, Ishikawa Prefecture, Japan, September 27-October 2, 2009.

forms in the pipe. After the cavity is formed it may continue to grow in volume until the flow velocities of the two columns become equal. Many studies on cavitation phenomenon have been investigated. More than a century ago Joukowsky [1] mathematically described physical aspects of wave propagation in the liquid systems. He also observed and explained column separation. Kephart and Davis [2] used a preliminary design technique i.e., rigid-column theory to determine the magnitude of the pressure rise when liquid columns rejoined in pump discharge lines. Escande [3] used cavities and lumped non-linear friction in his calculations. Vaporous cavitation only for a single vapour cavity in a pipeline was presented by Streeter and Wylie [4]. Rupture of a pipeline just below an upstream reservoir was considered. A single cavity was assumed to form at the point in the pipeline where pressure first dropped to the vapour pressure of the liquid. Weyler [5] used a numerical model to describe column separation as a single vapour cavity at the valve. Numerically cavities were not allowed to form at the other computational sections. A semi-empirical “bubbles shear stress” model was proposed to predict the increased momentum loss observed under liquid column separation conditions. The present paper describes the column separation as multiple cavity discrete vaporisation model which is based on the model as described in Wylie and Streeter [6], Simpson and Wylie [7] and Bergant and Simpson [8]. In this model Liquid column separations and regions of vaporous cavitation are both modelled using discrete cavities at all computational sections. Liquid is assumed to be in between all computational sections and the method of characteristics is applied throughout the pipeline, even in vaporous cavitation regions. 2. MATHEMATICAL MODEL 2.1. Governing equation The classical water-hammer equations are applied when the pressure is above vapour pressure. They comprise the continuity equation and the equation of motion:

0sin2

=∂∂

+−∂∂

+∂∂

xV

gaV

xHV

tH θ (1)

02

=+∂∂

+∂∂

+∂∂

DVfV

xVV

tHV

xHg (2)

where H is the piezometric head, t the time, V the flow velocity, x the distance along the pipeline, θ the pipe slope, a the pressure wave speed, g the gravitational acceleration, f the Darcy – Weisbach friction factor, D the inner diameter. For most engineering applicatios, the

convective terms and xHV∂∂ and

xVV∂∂ very small compared to the other terms and

therefore neglected. The method of characteristics (MOC) use as the numerical method to solve eqn (1) and (2) the MOC uses the compatibility relations that are valid along the poisitive ( atXC +=ΔΔ+ /: ) and negative ( atXC −=ΔΔ− /: ) characteristics line. The x-t grid shown in Fig. 1

APupcPUpcpcp QQRQBCH +−= (3)

BpmcPmcmcp QQRQBCH ++= (4)

where Cpc, Bpc and Rpc are constants for the positive or plus characteristics(pc) and Cmc, Bmc and Rmc are constants for the negative minus characteristics (mc). Where Δx is the reach length and Δt is the time step as indicated in Fig. 6

2 / 7

The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13) N13P1015 Kanazawa City, Ishikawa Prefecture, Japan, September 27-October 2, 2009.

Fig. 1 staggered grid for the method of characteristics

2.2. Multiple cavity model By starting with steady state conditions in a piping system with no cavities are present, transients are inciated by giving suitable boundary conditions and cavities are allowed to form at any of the computational sections (grid points) if the pressure is computed to be below the vapor pressure. Vapor cavities are thus confined to computational sections and pure liquid with a constant pressure wave speed is assumed in between. The absolute pressure in a cavity is set equal to the vapor pressure:

**

vpp =

The upstream and downstream discharges QP and QPu at a cavity are computed from the compatibility relations (3)and (4). The cavity volume ∀ follows then from

( dtQQorQQdtd t

int PuPPuP ∫ −=∀−=∀ , )

}

(5)

The numerical integration of eqn (5) in the MOC with the staggered grid (Fig.1) is usually given by

( ) ( ){ tQQQQ tt

Pu

tt

P

t

Pu

t

P

ttt Δ−Ψ−+−Ψ+∀=∀ Δ−Δ−Δ− 2)1( 222 (6) in which and are the volumes at the current and at 2Δt earlier , and Ψ is a numerical weighing factor. If the cavity volume becomes zero or negative , the cavity disappears and the computation returns to the standard water hammer procedure

t∀ tt Δ−∀

3. RESULTS AND DISCUSSION

3.1. Experimental verification Two experiments on cavitation conducted by Sanada et al. [9] using a horizontal acrylic pipeline are used to validate the numerically developed model. Experiments was done with water as the working fluid. In first case (schematic diagram of test appratus shown in Fig. 1) water is supplied from a large header tank to a reservoir with steady state water surface level at constant surface pressure. This is joined to quick closing valve at downstream end by a horizontal acrylic pipe. The transient flow was initiated by the sudden closing of the vave. The essential parameters are listed in Table 1. Fig. 3 shows the comparison between the numerical and experimental results for the first test rig. When the downstream valve is suddenly shut in a flowing liquid line, the liquid will accumulate in the vicinity of the valve. A high pressure at the vicinity of the valve cause flow reversal and several cycles of cavity

3 / 7

The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13) N13P1015 Kanazawa City, Ishikawa Prefecture, Japan, September 27-October 2, 2009.

formation and collapse will consequently occur as aresult of fluid accumulation and dissipation. The comparison result shows that the maximum head developed just at the initiation of transient at the valve end is accurately predicted by the numerical model. Time span due to cavity formation and collapse are also in good agreement with experiment and numerical result . But it is observed that mumerical result does not capture rhe damping of pressure wave with time accurately.

Table. 1 Essential test parameters

Parameters Experiment 1 Experiment 2 Length 200 m 200 m

Diameter 15.2 mm 15.2 mm Acoustic Velocity 820 m/s 820 m/s

Vapor Pressure 0.02 3 bar 0.02 3 bar Liquid Density 1000 kg/m3 1000 kg/m3 Vapor Density 0.8 kg/m3 0.8 kg/m3

Upstream Pressure 5.58971 bar 4.90325 bar Downstream

Pressure 0.98065 bar 0.98065 bar

Initial velocity 1.5 m/s 1.45 m/s

Fig. 2 Single pipeline with valve at the downstream end

4 / 7

The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13) N13P1015 Kanazawa City, Ishikawa Prefecture, Japan, September 27-October 2, 2009.

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

-25

0

25

50

75

100

125

150

175

H(m

)

Time(sec)

Experimental Numerical

Fig. 3 Pressure history of head at the valve In the second case Sananda studied the vaporous cavitation corresponding to a quick drop down of the upstream pressure in the flowing liquid line. Fig.4 shows the schematic diagram of the test appratus. Due to quick drop down in upstream pressure drops down to a certain level in a flowing liquid line, rarefaction waves will be sent downstream. The pressure at some location in the line may be reduced to the vapour pressure and cavities may be formed. The reflected pressure waves and rarefaction waves in the line will have a dominant effect on the behaviour of the cavities.

Reservoir

Cause of fluid transients cavitation

Fig. 4 Sudden decrease in upstream pressure

5 / 7

The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13) N13P1015 Kanazawa City, Ishikawa Prefecture, Japan, September 27-October 2, 2009.

Fig. 5 Pressure history of head at the middle of the pipe

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

5

10

15

20

25

30H

(m)

Time (sec)

calculated experiment

Fig. 5 shows comparison between experimental results and numerically calculated pressure history at the middle of the pipeline . From this figure it can be concluded that the position of cavitation region evolution is well predicted by numerical analysis. The predictions also show similar trend during the time of cavity formation and cavity collapse though the damping of the waveform is not as smooth as desired.

3. CONCLUSIONS The Column Seperation induced Water hammer in a horizontal pipe flow in a nearly horizontal circular pipe has been studied numerically with different boundary conditions and compared with literature quoted experimental values. The method of characteristics approach has been used to capture column separation hypothesis in a pipe. It can be observed that the the numerical model can not capture the damping of pressure wave with time in case of severe cavitation (valve closure) condition though it is well captured in when the cavitation is not so severe (sudden decrease in upstream pressure). This may be attributed to the fact that the friction factor is not constant in real flow situations (as assumed in the model) but rather dependent on actual flow conditions and it changes with changing flow conditions with timeThe use of the mathematical model provides a powerful and basic tool to understand the mechanism of transient fluid flow with cavitation under column separation hypothesis in transmission lines. However, further work is required to include the effects of frequency dependent friction in the numerical model.

6 / 7

The 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13) N13P1015 Kanazawa City, Ishikawa Prefecture, Japan, September 27-October 2, 2009.

REFERENCES 1. Joukowsky, N., 1900. U¨ ber den hydraulischen Stoss in Wasserleitungsro¨ hren. (On the

hydraulic hammer in water supply pipes.). Me´ moires de l’Acade´ mie Impe´ riale des Sciences de St.-Pe´ tersbourg, Series 8, vol. 9, No. 5 (in German). (English translation, partly, by Simin, 1904.)

2. Kephart, J.T., Davis, K., 1961. Pressure surges following water-column separation. ASME Journal of Basic Engineering 83, 456-460.

3. Escande, L., 1962. Arrêt instantané du débit d'une conduite forcée avec cavitations. (Instantaneous closing of the discharge in a penstock with cavitations.) In: Proceedings of the IAHR Symposium on Cavitation and Hydraulic Machinery, Sendai, Japan, 113-124 (in French).

4. Streeter, V.L., Wylie, E.B., 1967. Hydraulic Transients. New York: McGraw-Hill. 5. Weyler, M.E., 1969. An investigation of the effect of cavitation bubbles on momentum

loss in transient pipe flow. PhD Thesis, The University of Michigan, Ann Arbor, USA. 6. Wylie EB, Streeter VL. Fluid transients in systems. Englewood Cliffs, NJ: Prentice-Hall;

1993. 7. Simpson AR, Wylie EB. Large water-hammer pressures for column separation in

pipelines. J Hydraul Engng 1991;117(10):1310–6. 8. Bergant A, Simpson AR. Pipeline column separation flow regimes.J Hydraul Engng

1999;125(8):835–48. 9. Sanada K, Kitagawa A, Takenaka T. A study on analytical methods by classification of

column separations in a water pipeline. Trans Jpn Soc Mech Engnr, Ser B 1990;56(523):585–93. In Japanese.

7 / 7