Simulation of low pressure water hammer

Transcription

1 IOP Conference Series: arth and nvironmental Science Simulation of low pressure water hammer To cite this article: D Himr and V Haán 2010 IOP Conf. Ser.: arth nviron. Sci View the article online for updates and enhancements. Related content - Investigation of transient cavitating flow in viscoelastic pipes A Keramat, A S Tijsseling and A Ahmadi - Coincidental match of numerical simulation and physics B Pierre and J S Gudmundsson - Impulse pumping modelling and simulation B Pierre and J S Gudmundsson This content was downloaded from IP address on 08/01/2018 at 06:30

2 Simulation of low pressure water hammer 1. Introduction D Himr 1 and V Haán 1 1 Victor Kaplan Department of Fluid ngineering, Brno University of Technology Technická 2896/2, Brno, , Czech Repulic -mail: yhimrd00@stud.fme.vutr.cz Astract. Numerical solution of water hammer is presented in this paper. The contriution is focused on water hammer in the area of low pressure, which is completely different than high pressure case. Little volume of air and influence of the pipe are assumed in water, which cause sound speed change due to pressure alterations. Computation is compared with experimental measurement. Water hammer is caused y fast velocity change in the flowing fluid. Its magnitude depends on the properties of fluid, pipe and of course on the velocity change [5]. Volume of gas in the fluid is also very important, ecause it has influence on the sound speed and form of the pressure pulsations may e quite different for different static pressure [7], [8], [9]. Pulsations have lower frequency when the pressure is low enough. Cavitation can appear during the water hammer, if the pressure is lower than vapour pressure. It requires using program for computation of two-phase flow. Water hammer is very important phenomenon and it is necessary to simulate it computationally. This task is quite difficult and requires sophisticated software. There are different ways of solution, for example [4] and [6]. 2. Mathematical theory 2.1 Physical model There are two fundamental equations, which descrie fluid flow. First of them is momentum equation, which can e used in following shape: Q + t S ρ p + x 2 λ Q Q = S D S g p (1) The other one is continuity equation: p + t k S Q x = k m (2) Fig. 1 Voigt (Kelvin) model of solid ody It is possile to write them as one dimensional along the axis of the pipe, ecause velocity is dominant just in one direction. Right side of eq. (2) descries expansion and contraction of the pipe wall due to pressure c 2010 Ltd 1

3 pulsations. If pipe were assumed as ideally rigid, m would equal zero. Wall of the pipe can e descried as Voigt (Kelvin) model of solid ody. Its scheme is in Fig. 1.Behaviour of thin pipe wall is descried y eq. (3). Assumption that fluid keeps contact with the wall during deformation is considered. m is memory function and descries history of pressure pulsations. ( t τ) t D dp m = + ( ) exp τ dτ p 0 exp t (3) Δ dt Δt At time t k, this function can e computed as series expressed y eq (4) through the whole history of computation, where time t k equals k multiplied y time step. m tk p D = Δ + t0 exp k i= 0 p tk i t k p Δt + tk i 1 exp ( i + 1) Δt exp ( i + 2) Δt (4) Variale k in eq. (2) is an elasticity modulus of system liquid-pipe. Mathematic expression is elow. 1 k 1 D Δt 1 2 = + e ρ c Δ (5) First term in the rackets depends on properties of the fluid; second term depends on properties and geometry of the pipe. Modulus of elasticity is computed from sound speed, which is aout 1500m/s in the water (see equations of IAPWS-IF97 [1]) and is almost independent on the pressure. This speed is reduced y pipe properties (diameter, thickness of the wall, elasticity modulus and damping) on the value aout 1200m/s for steel or 300m/s for plastic, ut it is always independent on the pressure. There are many cases, which can e solved with constant sound speed and they are accurate enough. However, when a volume of gas is contained within the fluid, it causes noticeale decrease of sound speed, if the pressure is low. This phenomenon is descried in paper [2]. ven quite low volume of gas causes strong dependency of sound speed on the pressure. An example is presented in Fig. 2. Fig. 2 Sound speed dependence on the pressure [2] 2

4 2.2 Numerical model The equations (1) and (2) constitute a hyperolic prolem and cannot e solved y analytical mathematics, ecause the solution is not known. However, numerical mathematics provides some tools. For example: method of characteristics, Lax-Friedrichs method, Lax-Wendroff method, Beam-Warming method etc. Some of them are applicale to hyperolic equations of fluid flow, some of them not. See ook [3] for more informations. Lax-Wendroff method was chosen, ecause it enales computing with variale sound speed without any prolems. This method is ased on the Taylor s series expansion and uses three points from previous time step for computation of the current one. The numerical scheme is displayed in Fig. (3). 3. xperiment Fig. 3 Lax-Wendroff numerical scheme The computational model was verified on experimental test rig. It consists from upper tank and lower tank, which are connected y a pipe with a valve, see Fig. 4. Upper tank has overflow, which guarantees constant head. Fig. 4 xperimental device Tale 1 xperimental device in numers Value Unit Distance of valve from upper tank m Distance of arometer from upper tank m Height of upper level from valve 3.6 m Head 7.7 m Pipe diameter mm Pipe wall thickness 2.5 mm Temperature 19 C Water density kg/m 3 Steady velocity m/s Closing time (measured y optical sensor) s 3

5 At first, valve was open and then, after stailization of flow rate, was shut. Water hammer was induced and corresponding pressure record is in the next graph. Start of shutting is at time 0.5s. Record of asolute pressure is in fig. 5. A small cavitation region is apparent at time aout 1s, ut this is cavitation in vicinity of the valve, not in the place of pressure measurement. Pressure pulsations are highest at the end of the pipe and decrease toward the upper tank whereas a mean value is the same (for the same geodetic height). It causes that cut-off of pressure wave due to the cavitation is apparent at higher pressure than 2kPa in the middle of the pipe (in the place without cavitation), see Fig.5. Uncertainties of measurement are ±0.0036m/s (velocity) and ±2550Pa (pressure). 4. Computation Fig. 5 Record of pressure Prolem is solved at 22 points uniformly spread along the pipe length. Inlet condition is constant pressure that corresponds to height of upper level, outlet condition is resistance. Value of resistance is defined y relative opening of the valve, which is continuously changed, and y orifice coefficient, see following tale. Tale 2 Orifice coefficient Relative opening [-] Orifice coefficient [m 3 /h] Fig. 6 Computational model Damping was tuned in the computational simulation to fit the experimental results. Other parameters are presented in tales 1 and 2. Figure 7 presents computation with constant sound speed, which has value 1200m/s, and without influence of the pipe elasticity. First peak of the pressure is the same as measured one, ut regions of negative pressure appear. Frequency of oscillation is high and it may e concluded that this result is meaningless. If static part of 4

6 pressure were high (for example 5MPa), result would agree far etter with measurement, ecause sound speed would only little vary with pressure. This is apparent from Fig. 2. Tale 3 Set up of computational model Value Unit Length m Inlet pressure Pa Outlet pressure Pa lasticity modulus of pipe 2.42e10 Pa Damping e9 Pa s Mass ratio air/water e-7 - Viscosity 1e-6 m 2 /s Pipe wall roughness 1.5 mm Sound speed in the water 1475 m/s Computational time step 7.3e-4 s Fig. 7 Computation with constant sound speed and rigid pipe Fig. 8 Computation with variale sound speed and rigid pipe 5

7 Fig. 9 Computation with variale sound speed and influence of the pipe elasticity Fig. 10 Computation with variale sound speed and influence of the pipe elasticity-detail Computation with variale sound speed, ut without influence of pipe deformation, agrees with measurement etter. There is no region of negative pressure and first two waves match well enough with experimental results. Damping is noticealy lower than in reality. It corresponds with water hammer in the rigid pipe. Last computation is plotted in Fig. 9. It is the case with variale sound speed and with influence of pipe elasticity. Results of computation correspond with measurement quite well. Frequency of pulsations is the same and after 1.9 second is a it lower than the measured frequency. Time of stailization on a new value is the same. Significantly improved results are caused y taking damping into consideration. Pressure waves are narrow on the top and wide in the lower part. Theory shape corresponds with changes of sound speed in interval etween 950m/s and 150m/s. These changes are shown in Fig. 11. Fluctuation of sound speed depends on the pressure. This dependence is displayed in fig. 12 and 13. Influence of the pipe elasticity is considered. 6

8 Fig. 11 Sound speed fluctuations during computation Fig. 12 Pressure-sound speed dependence Fig. 13 Pressure-sound speed dependence, logarithmic scale 7

9 5. Conclusion Computation of water hammer is descried in the contriution. Comparison of three models proves that change of sound speed and influence of the pipe elasticity should e taken into account. Only then the results correspond with experiment well. Changes of sound speed cause lower frequency of pressure oscillations and do not permit negative pressure during computation. This model can e used also for solution of water hammer with cavitation without using state equation. Computation is not very time consuming and results are accurate enough. Acknowledgments Projects of Ministry of ducation MSM and Grant Agency of Czech Repulic GA 101/09/1716 are gratefully acknowledged for support of this research work. Nomenclature Damping [Pa s] t Time [s] c Sound speed in fluid [m/s] x Length coordinate [m] D Pipe diameter [m] Δ Wall thickness [m] lasticity modulus of pipe [Pa] Δt Time step [s] g p Gravity acceleration in the pipe direction Δx Length step [m] [m/s 2 ] k lasticity modulus of system [Pa] ε Relative deformation [-] m Memory function [s -1 ] λ Friction factor [-] L Pipe length [m] σ Stress [Pa] p Pressure [Pa] ρ Fluid density [kg/m 3 ] Q Flow rate [m 3 /s] τ Time [s] S Cross-section of pipe [m 2 ] References [1] International Association for the Properties of Water and Steam (IAPWS) online 2008 Thermodynamic Derivatives from IAPWS Formulations (last revision Septemer 26, cit , Revised Advisory Note No.3. [2] Himr D, Haán V and Pochylý F 2009 Sound Speed in the Mixture Water-Air ngin. Mechanics [3] Randal J L 2007 Finite Volume Methods for Hyperolic Prolems (New York, US, Camridge University Press) [4] Adamowski A and Lewandowski M 2008 Improved numerical modeling of hydraulic transients in pipelines column separation 3 rd IAHR WG on Cavit. and Dyn. Prol. in Hydr. Machin. and Syst. (Brno, Czech Repulic) [5] Wylie, B Streeter V L and Suo L 1993 Fluid Transients in Systems (Prentice Hall) [6] Kaliatka A, Ušpuras and Vaišnoras M 2004 Justification of RLAP5 Code for Modeling Water Hammer Phenomenon y mploying the Umsicht Test Facility Data nergetika (3) 1-6 [7] Burrows R and Qiu D Q 1995 ffect of Air Pockets on Pipeline Surge Pressure J. of Water Maritime and nergy (Proc. of Institution of Civil ngineers) (112) [8] Martin C S 1976 ntrapped Air in Pipelines Proc. of 2 nd Int.l Conf. on Pressure Surges, BHRA F2-15-F2-28 [9] Izquierdo J, Fuertes V S, Carera, Iglesias P L and Garcia-Serra J 2008 Pipeline Startup with ntrapped Air J. of Hydr. Research