Paper N0: II.05
Some Aspects of Physical and Numerical Modeling of Water Hammer in Pipelines Apoloniusz Kodura, Katarzyna Weinerowska
Abstract: Numerical modelling of water hammer phenomenon in pipelines is still an open problem. Although the matter has been widely analysed by many authors, the proper mathematical description of all aspects of this phenomenon is still not recognized. It is known, that the pressure characteristics obtained by computations are usually significantly different from the results of experiments. However, experiences show that thanks to some special mathematical and numerical treatment one can obtain results of sufficiently good correspondence to observations. Unfortunately this kind of approach proved to be effective if the single pipeline of constant diameter is analysed. For more complicated cases the proper representation of the phenomenon is much more difficult, as many different factors influence and distinctly modify water hammer phenomenon run. In the paper the results of chosen experiments on water hammer phenomenon (for simple positive water hammer run in pressure pipeline of different diameters, water hammer run in pressure pipeline with the local leak) are presented. The results of experiments and numerical analysis of the phenomenon are presented. The possibility and efficiency of numerical simulation of the water hammer phenomenon are discussed and the conformity between calculated and observed (measured) pressure characteristics are analysed. Keywords: water hammer, diameter change, local leak, measurements, numerical errors
1.
Introduction
Rapid changes of flow velocity in pipelines, caused by sadden valve operating (opening or closure), abrupt changes in work of the pumps, pump failures, mechanical vibrations of some elements and other reasons, result in violent change of the pressure value, which is then routed in the pipeline in a form of a rapid pressure wave. The celerity of such disturbance may
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exceed 1000 m/s, and the values of pressure oscillate from very high to very low, often causing noises and serious damages in pipelines, different forms of cavitation and other negative consequences. The phenomenon is widely known as water hammer and has been the subject of various analyses since the end of the nineteen century. Although those first analyses were of the preliminary character, the basic theoretical problems were discussed, the formula for the wave celerity was derived and the description of wave transformation and the influence of pipeline junctions and fixtures were taken into consideration. Since then many other works were presented (e.g. Parmakian 1955, Streeter and Wylie 1967 and others), and a vivid advance in the phenomenon description was observed. However, in spite of a big progress in mathematical modeling and measurements, water hammer is still one of the most interesting problems of pipeline hydraulics, and the subjects of numerous publications (e.g. Brunone et al. 2000, Covas et al. 2004 and others), as one of those problems which still is not recognized in sufficient way. The possibility of measuring and recording the observed values of pressure enabled to compare the measurements with the results of calculations, which proved to be significantly different. Since then the attempts to improve the classical description of water hammer and to recognize the main factors influencing the phenomenon run have been carried out. However, the problem is still the open question. The phenomenon of water hammer may be considered in two ways. From the practical point of view, the main problem is to know the extreme values of the pressure, which are usually considered as the peak of the first amplitude of the pressure wave in pipeline. As it is known, the pressure wave is attenuated in a pipeline due to many reasons, from which one of the most important is flow resistance. The duration of the phenomenon depends on the material of the pipe, initial value of the flow, kind of the medium in pipes and other factors but each time it is usually a question of a few seconds. Thus, for some researchers the basic and most important task is to recognize the highest values of pressure, as the oscillation period, attenuation intensity and other characteristics are of less importance. As the maximal value of pressure wave during simple water hammer run may be calculated from the theoretical formula, which usually leads to the values close to the observed ones, problem of water hammer is sometimes considered to be solved. However, the situation is relatively easy if the simple case of a single pipe of constant diameter is considered. However, in most practical cases the situation is usually more complicated and additional factors influencing water hammer run occur. If the pipeline network is considered, not only the maximal value of pressure can not be easily calculated from the simple formulas, but also the place of its occurrence is not easy to recognize with simple theoretical approach. The pressure wave may be routed in complicated network of pipes, waves may superimpose or reflect from obstacles and the prediction of the extreme values of the pressure is not easy. Thus, no matter if the problem is considered from practical point of view only or from theoretical point of view as well, there are still some aspects that are not recognized in sufficient degree. From the theoretical point of view the coincidence of the first amplitude of pressure wave is not enough for proper description of the phenomenon. To achieve this, the consistency of the calculated and measured values of pressure should be obtained for the
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whole duration of the water hammer run. From scientific point of view it is important to model the phenomenon in the proper way, to obtain the result close to observations and to recognize the factors modifying the simple water hammer in pipeline. It is not only interesting from cognitive point of view, but also important for more detailed analyses, in which the observations and prediction of pressure characteristics may be applied to solve practical problems of different nature, e.g. localization of leaks in pipeline. However, in most cases it is very difficult or impossible to obtain good coincidence between numerical solution and measurements, and the calculation results are often significantly different from observations, not only as to the values of pressure of next amplitudes of the wave, but for the oscillation period and the time of the phenomenon duration as well.
2.
Main factors influencing water hammer modeling
The problems of the inconsistence between calculated and observed pressure characteristics are of various nature. The main aspect is connected with mathematical description of the phenomenon. As the significant difference between calculations and measurements is observed for most cases of water hammer – especially as to the intensity of wave attenuation, wave smoothing and the duration of the phenomenon, many researchers focused on the improvement of the form of the governing equations. The traditional description (Chaudhry 1979, Streeter and Wylie 1967, and others) of the phenomenon run: 8 λ ∂H 1 ∂Q + + Ro Q Q = 0 , where Ro = ∂x gA ∂t gπ 2 D w 5
(1a)
∂H c 2 ∂Q =0 + ∂t gA ∂x
(1b)
(where Q is a rate of discharge, H – water head, g – acceleration due to gravity, a – wave celerity, A – cross-section area, Di – internal pipe diameter and λ is the linear friction factor), was often replaced by more complicated description, in which the main emphasis was put on the modification of friction term in momentum equation. One can find many different approaches in which the friction factor is increased in relation to steady friction term on different ways, from the simplest idea of multiplying it by some constant (even up to 10 and more), to more complicated ones, based on introducing to the friction formula additional terms dependent from space and/or time velocity derivatives. The difference in calculations and observations is particularly vivid for pipes made of polymers. The reason of this fact is the viscoelastic behavior of this material as the reaction on the stress. The equations (1a,b) describing elastic model may be applied for steel pipes and for preliminary calculations for plastic pipes. If more accurate calculations are needed, it is necessary to develop the form of mathematical description with taking into account viscoelastic character of pipe walls deformations (Brunone et al 2000, Covas et al. 2004).
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Examples presented in literature may suggest the problem of water hammer can be effectively solved by applying the approaches mentioned above. However, as the experiences of Mitosek and Szymkiewicz (2005) prove, this way can not lead to successful solution, from formal point of view. As long as the mathematical description (1a,b) is improved by modification of the friction term only, the results of calculations will not be consistent with the observations. The problem is connected with the nature of the phenomenon and the type of the equations. The observations, especially the shape of the pressure waves, prove that the phenomenon run is influenced by some additional mechanism of diffusive character, which should be represented in mathematical description. As long as the set of equations (1a,b) or its modified version is of hyperbolic type, the results of calculations will differ from the observations. Thus, the modification of the friction term may influence the values of pressure during the phenomenon duration but it will not lead to the characteristic smoothing of the wave shape observed in measurements. This can be done only by a term of diffusive type. Many authors present the results of the calculations obtained for the models with modified friction term, which seem to be very close to the observations. This is often connected with another question, which is numerical aspect of mathematical modeling. Unfortunately, some features of the pressure characteristics obtained from calculations are the result of pure numerical effects. This may lead to the wrong interpretation of the quality of calculations and mathematical description. The very good coincidence between calculations and observations is very often the result of numerical errors introduced to equations with the numerical scheme applied in solution. The most vivid example is numerical diffusion, which is nearly always introduced to the scheme and which leads to smoothing of the solution. This smoothing is of unphysical character and the coincidence of such solution with observations should not be interpreted as properly solved water hammer problem. The problem with proper mathematical description of the phenomenon and numerical effects influencing the solution are not the only aspects making the solving of water hammer problem difficult. The remarks presented above were connected with single pipeline of constant diameter, but the situation becomes even more difficult if more complicated systems of pipes are taken into consideration. In many practical cases additional factors should be taken into account, such us changes of pipe diameters, changes of the pipe material, junctions of pipelines and local leaks in pipeline. As the universal way of proper solving of water hammer problem for single pipeline is still not recognized, it is obvious that it is impossible to consider big complicated systems of pipelines. However, some simple examples of different situations in pipeline systems may be analyzed in order to recognize the main factors and effects influencing the accuracy of the solution. Thus in the paper some simple examples of observed and calculated pressure characteristics for different scenarios are presented. The aim of the research is to compare measured values of pressure with those calculated with use of the simple water hammer model (1a,b), and to recognize the difficulties in modeling and main factors affecting the results.
Some Aspects of Physical and Numerical Modeling of Water Hammer in Pipelines
3.
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Experimental research
Experiments were carried out in the laboratory of Warsaw University of Technology, Environmental Engineering Faculty, Institute of Water Supply and Water Engineering. The physical model is schematically shown in Fig.1. The main element is the pipeline (single straight pipe of the length L , extrinsic diameter D and the wall thickness e or the pipeline consisted of sections of varied parameters (4)). The pipeline was equipped with the valve (1) at the end of the main pipe, which was joined with the closure time register (2). The water hammer pressure characteristics were measured by extensometers (3), and recorded in computer’s memory. The supply of the water to the system was realized with use of hydrophore reservoir which enabled inlet pressure stabilization. The experiments were carried out for four cases: • simple positive water hammer for the straight pipeline of constant diameter (4); the measured characteristics were the basis for estimation the influence of the diameter change and local leak on water hammer run (Fig.1a); • positive water hammer in pipeline with single change of diameter: contraction and extension (Fig.1b); • positive water hammer in pipeline with local leak (Fig.1c) in two scenarios: with the outflow from the leak to the overpressure reservoir and with free outflow from the leak (to atmospheric pressure, with the possibility of sucking in air in negative phase).
a) 2 3
4
3
D, e
1
3
Q
L
b) 3
4
D 1, e1
3
2 5
3
D 2, e2
L1
1
Q
L2 L
c) 2
Qt
4
D, e
3
3 6
L
1
Qo
L1
Ql
Fig. 1 Scheme of the experimental equipment simple pipeline, b) pipeline with diameter change, c) pipeline with local leak
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The experiments for each scenario were carried out in two steps. During the first step the values of steady flow discharges in the outflows were estimated (by volumetric method) and in the second step the unsteady flow pressure characteristics were measured. Next, the analysis of registered pressure characteristics was carried out, with estimation of the characteristic parameters, such as: water hammer wave period, pressure amplitude, number of oscillations, duration of the phenomenon. The purpose of this analysis was to indicate the factors that may influence the energy dissipation in water hammer in pipeline and to compare the run of the phenomenon for different cases. The pipeline parameters for different scenarios are presented in Tab.1.
Tab.1 Pipeline parameters for different experimental scenarios
Type of phenomenon
Experimental
Type of
series
pipeline
[-]
[-] 3
1
2
Simple water
S1
hammer – single
S2
pipe Pipeline with variable diameter Pipeline with local leak
4.
P1 SS1 ( c ) SS2 ( e ) PP1 ( c ) PP2 ( e ) LL
steel PE80 steel PE80 PE80
External
Pipe wall
Theoretical
diameter
thickness
wave celerity
L
D
e
a
[m]
[mm]
[mm]
[m/s]
Pipe length
4
5
6
7
41.00
48.0
3.00
1260
77.80
60.0
3.35
1369
36.00
50.0
4.60
256
26.50/24.60
48.0/27.5
3.00/3.00
1324/1366
18.40/26.50
27.5/48.0
3.00/3.00
1366/1324
24.25/25.00
50.0/25.0
4.60/2.30
256
25.00/24.25
25.0/50.0
2.30/4.60
256
36.00
50.0
4.60
256
Numerical simulation of the phenomenon
For the needs of analyses presented in the paper, the simplified description of the phenomenon, in the form of equations (1a,b), was assumed as a mathematical model. To solve the problem, the Preissmann’s scheme (Cunge 1980) was applied as numerical method. Computational time step was matched in a way enabling to obtain Courant number close to unity for each scenario. Unphysical oscillations were reduced by appropriate choice of the value of θ parameter in Preissmann scheme. For each of the considered scenarios, in the first step of calculations the values of λ and preliminary values of wave celerity a were estimated, which were then corrected in the further phase of calculations. Eventually, the main calculations were carried out for each considered scenario. In calculations the possibility of wave celerity changes due to density variations caused by pressure changes was taken into account, and changes of the friction factors due to velocity variations were allowed. Also the influence of the changes in different parameters were on the calculation result was analyzed.
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As a consequence, the solution in the form of pressure characteristics was obtained. The examples of observed and calculated pressure characteristics for the chosen scenarios are shown in Fig.2 and Fig.3. More detailed analysis of the case of pipeline with diameter changes may be found in the work of Malesinska (2002), and of the case of pipeline with local leak – in the paper of Kodura and Weinerowska (2005). In the presented paper some examples and general remarks are presented.
5.
General remarks and conclusions
The physical and numerical experiments carried out in the presented research and the comparison of observed and calculated pressure characteristics enable to draw some general conclusions and suggest directions of future analyses that could be developed to improve the water hammer modelling. For single pipeline, the experiments confirmed the well known problem of obtaining the coincidence between observations and calculations if the traditional mathematical description in the form if (1a, b) is applied and if no numerical effects are observed in the solution. Thus it is important to introduce to the equations additional factors which can better model water hammer in pipeline, especially the nature of energy dissipation and pipe wall deformation. a)
H [m]
120 100 80 60
observed pressure characteristic
40
pressure characteristic calculated with a=400 m/s adjasted to first amplitude (theoretical value a=256 m/s)
20 0 -20 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
t [s]
b)
observed pressure characteristic pressure characteristic calculated with a=380 m/s adjusted to oscillation period (theoretical value a=256 m/s)
Fig. 2 Example of the measured and calculated pressure characteristics for the pipeline 3 3 with change of diameter: a) contraction (PP1), Q=0,454 dm /s, b) extension (PP2), Q=1,059 dm /s.
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H [m]
observed pressure characteristic calculated pressure characteristic
t [s]
Fig. 3 Example of the measured and calculated pressure characteristics 3 3 for the pipeline with local leak (LL), Qt = 0.735 dm /s, Ql =0.08 dm /s.
Very detailed analysis of the results obtained for the case of pipeline with change of the diameter can be found in the work of Malesinska (2004). On the basis of this work and the research presented in the paper one can draw the following conclusions: • for the case of diameter extension in the pipeline: • the vivid and regular envelope of the pressure oscillations can be observed; however, the maximal value of pressure is not always the value of the first amplitude and it is usually higher than theoretical value obtained from Zukowski’s formula; • the ‘equivalent’ value of wave celerity a, chosen to obtain the best fit between calculations and observations, is less than value of a for each pipe separately; • it is possible to obtain good coincidence between observations and calculations if numerical dissipation is introduced to the scheme; however, it is not possible to achieve such consistence for plastic pipes. For polymers in this case it is impossible to obtain simultaneously the fit of amplitude and frequency of oscillations; • for the case of diameter contraction in the pipeline: • less regular envelope of the pressure oscillations can be observed; however, the maximal value of pressure is the value of the first amplitude and it is comparable with theoretical value obtained from Zukowski’s formula; • the ‘equivalent’ value of wave celerity a, chosen to obtain the best fit between calculations and observations, is higher than value of a for each pipe separately. If the pipeline with local leak is considered, the phenomenon run is influenced by some additional factors. Detailed conclusions drawn on the basis of experiments and calculations for the pipeline with a local leak are presented in the paper of Kodura and Weinerowska (2005). The most important effects observed are: • lack of the influence of the ratio of discharge from local leak to total discharge in the pipeline to the values of period of oscillations, and – in a consequence – to the value of wave celerity if the outflow to the overpressure reservoir from the leak was imposed;
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• no influence of the rate of discharge from local leak on the maximal value of pressure, • consistence between observed values of maximal pressure in first amplitude and corresponding values calculated according to Zukowski’s formula, irrespective of the rate of discharge from the leak; • significant influence of the rate of the discharge from the leak on the vivid decrease of duration of the water hammer phenomenon, what suggests the possibility of utilization of this fact to the pipeline leaktightness assessment, especially that the duration time decreases with the increase of the outflow from the leak. The results of the computations show that in some cases it is possible to obtain relatively good consistence with observations. However, it seems advisable to develop the model in order to improve the mathematical description of the phenomenon and to take into account the factors particularly important for each of the cases, modifying the run of simple water hammer.
References Brunone B., Karney B.W., Mecarelli M, Ferrante M. (2000) Velocity Profiles and Unsteady Pipe Friction in Transient Flow, Journ. of Water Res. Planning and Management, vol.126, No.4.,236-244. Chaudhry M.H. (1979) Applied Hydraulic Transients, New York, Van Nostrand Reinhold Company. Covas D., Stoinanov I., Mano J., Ramos H., Graham N., Maksimović C. (2004) The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part I-experimental analysis and creep characterization. Journ. of Hydr. Res. Vol.42. No.5,516-530. Cunge J.A., Holly F.M. Jr., Verwey A. (1980) Practical Aspects of Computational River Hydraulics. Pitman Publishing Limited, Vol. 3, London. Kodura A., Weinerowska K.(2005) The influence of the local pipeline leak on water hammer properties, Materials of the II Polish Congress of Environmenyal Engineering, Lublin, (in print). Malesińska A. (2004) Rozprzestrzenianie sie zaburzenia cisnienia o skonczonej amplitudzie w ciagu przewodow o roznych srednicach, PhD thesis, Warsaw University of Technology, Warszawa. Szymkiewicz R., Mitosek M (2005) Analysis of unsteady pipe flow using the modified finite element method, Communications of Numerical Methods in Engineering ,vol.21,no.4 Parmakian J. (1955) Water hammer analysis. New York: Dover Publications, Inc. Streeter V.L., Wylie E.B. (1967) Hydraulic transients. New York: McGraw-Hill Book Co. Authors Dr Apoloniusz Kodura: Warsaw University of Technology, Environmental Engineering Faculty,
[email protected], Dr Katarzyna Weinerowska, Gdansk University of Technology, Faculty of Civil and Environmental Engineering,
[email protected] 
