Hydraulic transients in pumping systems numerical modelling and experimental analysis

Transcription

1 Hydraulic transients in pumping systems numerical modelling and experimental analysis João Nuno Gomes Borga Delgado Thesis to obtain the Master of Science Degree in Civil Engineering Examination Committee Chairperson: Professor António Alexandre Trigo Teixeira Supervisor: Professor Dídia Isabel Cameira Covas Supervisor: Professor António Patrício de Sousa Betâmio de Almeida Member of the Committee: Professor Helena Margarida Machado da Silva Ramos Member of the Committee: Professor Sandra Maria Carvalho Martins October 2013

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3 RESUMO A presente dissertação tem como objectivo a discussão das principais incertezas associadas à modelação de regimes transitórios em sistemas elevatórios com e sem dispositivos de protecção. Foi desenvolvido um modelo matemático para o cálculo de escoamentos variáveis em pressão unidimensionais baseado na teoria clássica do golpe de ariete e resolvido através do Método das Características (MdC). O modelo incorpora o elemento bomba descritos através dos parâmetros de Suter, um reservatório hidropneumático (RHP) e a formulação de Vítkovský et al., (2000) para descrever as perdas de carga em regime transitório. O modelo foi testado utilizando medições de pressão durante a ocorrência de regimes transitórios e de caudal em regime permanente em duas instalações experimentais no Laboratório de Hidráulica do Departamento de Engenharia Civil no Instituto Superior Técnico. As medições de pressão foram realizadas em duas/três localizações (na extremidades de montante, jusante e secção intermédia). Foram realizados ensaios experimentais com e sem o RHP ligado ao sistema para diferentes caudais iniciais. O regime transitório foi gerado através de diferentes manobras nas componentes do sistema (i.e., válvulas de montante e jusante e electro bomba). Os dados recolhidos foram comparados com os resultados numéricos do modelo matemático e foram utilizados para calibrar parâmetros, obtendo-se uma boa concordância. Alguns testes realizados com o RHP ligado ao sistema geraram pressões superiores nos circuitos hidráulicos do que quando o mesmo não se encontrava ligado. Estas análises são importantes para desenvolver modelos numéricos mais sólidos, assim como sensibilizar as entidades das principais incertezas nos modelos numéricos desenvolvidos. Palavras Chave: Golpe de Aríete, órgãos de protecção, sistemas elevatórios, reservatório hidropneumático, válvula de retenção, resistência ao escoamento. i

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5 ABSTRACT The current research work aims to discuss the main uncertainties associated with the hydraulic transient modelling of raising pipe systems with and without surge protection. An one-dimensional hydraulic transient solver was developed based on the classic waterhammer theory and solved by the Method of Characteristics (MOC). The solver incorporates the pump-element described by Suter parameters, a pressurised surge tank (hydropneumatic vessel, HPV) and Vítkovský et al., (2000) formula for unsteadystate friction description. The model was tested using transient pressure data and steady-state flow data from two hydraulic circuits installed at the Hydraulics Laboratory of the Civil Engineering Department, in Instituto Superior Técnico in Lisbon, Portugal. Pressure data were collected at two/three locations (at the upstream end, at a middle section and at the downstream end). Transient tests with and without the HPV connected to the system were carried out for different flow rates. Transients were generated by different manoeuvres in devices of the system (i.e., the upstream and the downstream valves and the pump). Collected data were compared with the results of the numerical modelling and used to calibrate model parameters. Good agreement between data and numerical results was obtained. Some tests with the HPV connected lead to higher pressure surges than when there was no protection in the system. These analyses are important to develop more solid and reliable numerical models as well to create awareness of the main uncertainties of developed models. Keywords: Waterhammer, pumping systems, hydropneumatic vessel, check valve, unsteady-state friction. iii

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7 ACKNOWLEDGMENTS Firstly, I would like to acknowlede my supervisors Professor Dídia Covas and Professor António Betâmio de Almeida for their guidance, support, knowledge transmitted and constructive criticism during the development of this research work that strongly contributed for the final result of this research. Very special gratitude is expressed to Professor Dídia Covas for giving me the opportunity to work with, in the last two years, which strongly contributed for my knowledge. Gratitude is expressed to Fundação para a Ciência e Tecnologia (FCT) for the project PTDC/ECM/112868/2009 Friction and mechanical energy dissipation in pressurized transient flows: conceptual and experimental analysis for funding the current research in terms of experimental work and grants hold. To Dulce Fernandes for her kindness and administrative support. Gratitude is expressed to João Pedro and Nuno Martins for their support and knowledge during the performance of the experimental program. To all my friends of Instituto Superior Técnico, for the support and fellowship during the past year of my master. Especial gratitude is given to Adriana, Álvaro, Bruno, Correia, David, Fabião, Filipa, Fonseca, Francisco, José, Luís, Madeira, Mariana, Manuel, Romeu, Rui, Semedo and Vera. Friends To my friends of Salvaterra de Magos who always supported and advised me in the good and bad moments of my life, helping me to become the person I am nowadays. I express my sincere acknowledgements to all of them, especially to André, Andreia, Catarina Fernandes, Catarina Coelho, Cátia, Cláudia, Coelho, Galamba, Inês, Lopes, Patrícia, Quim, Risso, Rui, Sandro, Sofia, Teresa and Tiago. To Maria for what she represents for me. Thank you so much for your patience, support, motivation and final review of this document. Finally, I would like to express my gratitude to my family, especially my parents, Ana and João, for the unconditional and immeasurable support during my personal and academic life and for helping me while studying in Lisbon during the past five years. Also to my sister, Joana, for her importance in my life and advices in a multitude of aspects. I would like to dedicate this Master Thesis to them. v

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9 PUBLICATION The following paper was published during the development of this research work: (i) Delgado, J., Martins, N., Covas, D. (2013) Uncertainties in Hydraulic Transient Modelling in Raising Pipe Systems: Laboratory Case Studies, Proceedings to the 12th International Conference on Computing and Control for the Water Industry, CCWI2013, Italy. vii

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11 TABLE OF CONTENTS RESUMO... I ABSTRACT... III ACKNOWLEDGMENTS... V PUBLICATION... VII TABLE OF CONTENTS... VIII APPENDICES... XIII LIST OF FIGURES...XV LIST OF TABLES... XXIII NOMENCLATURE... XXVII ABBREVIATIONS... XXXI 1 INTRODUCTION Importance of hydraulic transient analysis Aim and methodology Document outline STATE-OF-THE-ART Introduction Hydraulic transients analyses Simple formulations Classical waterhammer models Advanced simulation models Classic waterhammer problem Basic equations and assumptions Elastic wave speed Friction resistance Numerical methods Characteristic equations and stability condition Computational grids ix

12 2.4 Unconventional waterhammer problems Introduction Unsteady-state friction Pipe material non-elastic and viscoelastic behaviour Transient cavitation Fluid-structure interaction Transient events generated by centrifugal pumps Introduction Sequence of events after the power failure General pump description Abnormal pump characteristics four quadrants of operation Check valves Introduction Dynamics of check valves Avoiding check valve slam design criteria Surge protection devices Introduction Surge tank One-way surge tanks Hydropneumatic vessel Flywheel By-pass lines Valves Summary, main gaps in literature and thesis motivation HYDRAULIC TRANSIENT SOLVER Introduction Method of Characteristics Characteristic equations Friction resistance Compatibility equations x

13 3.3 Boundary conditions Introduction Upstream boundary conditions Downstream boundary conditions Sudden pump stoppage Introduction Pump stoppage due to power failure Solution for single pump failure Solution for parallel pumps sudden failure with equal characteristics Hydropneumatic vessel Air vessel governing equations Numerical solution Model testing with artificial data Introduction Pumping system results Final remarks Summary TRANSIENT DATA COLLECTION AND ANALYSIS Introduction Copper-pipe facility Experimental facility description Data collection Data analysis Steel-pipe facility Experimental facility description Data collection Data analysis Summary MODEL CALIBRATION AND VALIDATION Introduction xi

14 5.2 Copper-pipe facility Steady-state friction calibration Sudden pump stoppage without hydropneumatic vessel Sudden pump stoppage with hydropneumatic vessel Downstream valve closure with hydropneumatic vessel Steel-pipe facility Steady state friction calibration Sudden pump stoppage without hydropneumatic vessel Sudden pump stoppage with hydropneumatic vessel Downstream valve closure with hydropneumatic vessel Summary CONCLUSIONS AND FURTHER RESEARCH Thesis overview Conclusions Suggestions for future research work REFERENCES APPENDICES xii

15 APPENDICES Appendix A Experimental results obtained in copper-pipe facility Appendix B Theoretical rating curves of centrifugal pump KSB Etaline /1502 Appendix C Experimental results obtained in steel experimental facility xiii

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17 LIST OF FIGURES Figure 2.1 Velocity profile inversion for the total closure of the downstream valve (a) along the pipeline and (b) in a particular cross-section. Numbers 1 to 7 refer to the time sequence of the velocity profiles during the inversion of the flow, immediately after the valve closure (Bratland, 1986) Figure 2.2 Velocity profiles measurement: (a) pressure variation in time; (b) and (c) velocity profile variation in time (Brunone et al., 2000) Figure (a) Stress and strain for an instantaneous constant load; (b) Boltzmann superposition principle for two stresses sequentially applied (Covas, 2003) Figure 2.4 Kelvin-Voigt model for viscoelastic materials with n-elements Figure 2.5 Several types of cavitation: (a) downward sloping pipe; (b) increasing upward pipe; (c) horizontal pipe; (d) mild upward pipe. adapted from Bergant et al., (2006) Figure 2.6 Generation of secondary pressure waves due to Poisson coupling (Tijsseling, 1996) Figure 2.7 Pressure waves measured and calculated by classical waterhammer theory: (a) numerical results obtained by Tijsseling (1996); (b) numerical results vs data collected by (Ribeiro, 2010) Figure Sequence of events following the sudden pump stoppage without a check valve adapted from Mays (1999) Figure Pump characteristics for four specific rotational speed values: (a) piezometric head; (b) torque Figure 2.10 Zones and quadrants of pump operation - adapted from Ramos et al., (2005) Figure 2.11 Overpressures caused by two different check valves: (a) swing and (b) nozzle check valve (Thorley, 1983) Figure 2.12 Check valves types: (a) ball valve; (b) swing valve; (c) split disc; (d) nozzle valve Figure 2.13 Dynamic Performance Characteristics of check-valves: (a) dimensional and (b) dimensionless curves Thorley (1991) Figure 2.14 Check valve selection adapted from (Val-Matic, 2003) Figure 2.15 Surge tanks: (a) Ryushima power station; (b) Kendoon Power Station; (c) typical location in rising mains; (d) typical location in gravity mains Figure 2.16 Feed tanks: (a) scheme with check-valve; (b) typical location in rising mains Figure 2.17 Hydropneumatic vessel: (a) and (b) photos of air vessels, (c) typical location in rising mains (d) typical location in gravity mains xv

18 Figure 2.18 Flywheels: (a) pumping group, (b) turbine Figure 2.19 By-pass lines: (a) general layout in a rising main; (b) check valve in the by-pass; (c) control valve in the by-pass Figure 2.20 Valves used to prevent transient events: (a) safety/pressure-relief valve; (b) pressureregulating valve; (c) air inlet-valve Figure 3.1 Method of Characteristics grid: (a) straight characteristic lines; (b) location of calculation sections Figure 3.2 Effect of the decay coefficient value in the surge damping for a hypothetical example Figure 3.3 Location of calculation points for Vítkovský et al., (2000) numerical scheme Figure 3.4 Equations valid in boundary conditions Figure 3.5 Constant-head upstream reservoir Figure Valve discharging to constant-head reservoir Figure 3.7 Local headloss coefficient laws for different types of valves Figure 3.8 Valve discharging to atmosphere Figure 3.9 Notation regarding pump boundary conditions Figure 3.10 Linear interpolation between grid points (Chaudhry, 1987) Figure 3.11 Flowchart for sudden pump stoppage boundary conditions adapted from Chaudhry, (1987) Figure 3.12 Air vessel scheme Figure Hydropneumatic scheme with by-pass for differential headloss Figure 3.14 Flowchart for air vessel numerical solution Simple-Fixed Point Iteration method Figure Pumping system topology Figure 3.16 Results obtained for scenario S-A1: (a) piezometric head envelopes; (b) piezometric head at downstream the pump station; (c) flow rate at the pump station; (d) pump rotational speed Figure 3.17 Results obtained for scenario S-A2: (a) piezometric head envelopes; (b) piezometric head at downstream the pump station; (c) flow rate at the pump station; (d) pump rotational speed Figure Results obtained for the air vessel in scenario S-A2: (a) flow rate; (b) water level Figure 3.19 Comparison of the significant values for all scenarios between the two simulators xvi

19 Figure Copper-pipe facility: (a) general view; (b) side view; (c) upstream view; (d) downstream view Figure 4.2 Possible configurations of flow in copper-pipe facility Figure 4.3 Measurement equipment in copper-pipe facility: (a) upstream transducer T1; (b) middlesection transducer T2; (c) downstream transducer T3; (d) data acquisition system; (e) rotameter Figure 4.4 Flowchart of data acquisition system procedure adapted from Carriço (2008) Figure 4.5 Schematic of the copper-pipe facility Figure 4.6 Measurements of the water level in the air vessel during the steady-state in the copperpipe facility Figure 4.7 Data collected at transducer T1 during the sudden pump stoppage in the copper-pipe facility: (a) with the HPV for flows between 100 and 500 l/h; (b) with the HPV for flows between 600 and 1000 l/h; (c) without the HPV for flows between 100 and 500 l/h; (d) without the HPV for flows between 600 and 1000 l/h Figure 4.8 Data collected at the three transducers during the sudden pump stoppage for initial flow rate 600 l/h: (a) with protection; (b) without protection Figure 4.9 Flow supply from the air vessel after the sudden pump stoppage Figure 4.10 Result analysis of sudden pump stoppage with air vessel experimental tests in copperpipe facility: (a) estimated mean reverse velocity as function of check valve closure time; (b) estimated mean reverse velocity as function of initial flow rate; (c) check valve closure time in function of initial flow rate Figure 4.11 Data collected at transducer T1 during the upstream valve closure in copper-pipe facility: (a) with the HPV for flows between 100 and 500 l/h; (b) with the HPV for flows between 600 and 1000 l/h; (c) without the HPV for flows between 100 and 500 l/h; (d) without the HPV for flows between 600 and 1000 l/h Figure 4.12 Data collected at the three transducers during the upstream valve closure tests for initial flow rate 600 l/h: (a) with HPV and (b) without HPV Figure Flow supply from the air vessel after the upstream valve closure Figure 4.14 Data collected during the upstream valve opening without HPV in copper-pipe facility: (a) flows between 0 and 500 l/h in transducer T1; (b) flows between 600 and 1000 l/h in transducer T1; (c) flows between 100 and 500 l/h in transducer T3; (d) for flows between 600 and 1000 l/h in transducer T xvii

20 Figure 4.15 Data collected at three transducers during upstream valve opening in copper-pipe facility without the HPV for the (a) downstream valve closed; and the final flow rates of (b) 100 l/h; (c) 600 l/h; (d) 1000 l/h Figure 4.16 Data collected at transducer T3 during the downstream valve closure in copper-pipe facility: (a) with the HPV for flows between 100 and 500 l/h; (b) with the HPV for flows between 600 and 1000 l/h; (c) without the HPV for flows between 100 and 500 l/h; (d) without the HPV for flows between 600 and 1000 l/h Figure Data collected at the three transducers during the downstream valve closure tests for initial flow rate of 600 l/h: (a) with HPV and (b) without HPV Figure Maximum overpressure observed for downstream valve closure in copper-pipe facility: (a) with protection; (b) without protection Figure 4.19 Data collected at the three transducers during the downstream valve opening tests: (a) with protection; (b) without protection Figure 4.20 Comparison between the sudden pump stoppage and the upstream valve closure: (a) 300 l/h with air vessel; (b) 300 l/h without air vessel; (c) 600 l/h with air vessel; (d) 600 l/h without air vessel Figure 4.21 Steel experimental facility: (a) plan; (b) detail of reversible area (c); front view (d) side view Figure Experimental facility reversible zone details: (a) troplan inside the reservoir; (b) pump isolated in the wooden box and check valve; (c)variable speed drive; (d) hydropneumatic vessel Figure 4.23 Downstream valves: (a) manual ball valve and pneumatically actuated butterfly valve; (b) trigger; (c) air valve; (d) scour valve Figure Flow paths studied: (a) with air vessel in bifurcation Configuration A; (b) without air vessel Configuration B. Note: red suction and discharge line; blue downstream end of the pipe system and discharge to the reservoir Figure Experimental facility equipment: (a) flow meter; (b) gate valve to control flow rate; (c) pressure transducers location; (d) data acquisition system Figure 4.26 Schematic of the steel-pipe facility Figure 4.27 Measurements of the water level in the air vessel during the steady-state in the steel-pipe facility: (a) sudden pump stoppage; (b) downstream valve closure Figure 4.28 Data collected at transducer T1 during the sudden pump stoppage with air vessel in the steel-pipe facility: (a) 5.0 l/s; (b) 10.0 l/s; (c) 15.0 l/s; (d) 20.0 l/s xviii

21 Figure 4.29 Data collected at the two transducers during the sudden pump stoppage with HPV test for initial flow rate 5.0 l/s Figure 4.30 Result analysis of pump stoppage with air vessel experimental tests in steel-pipe facility: (a) maximum overpressure as a function of valve closure time per flow rate; (b) estimated mean reverse velocity as a function of initial flow rate; (c) check valve closure time as a function of initial flow rate 105 Figure 4.31 Data collected at the two transducers during the sudden pump stoppage without air vessel tests in the steel-pipe facility: (a) 5.0 l/s; (b) 10.0 l/s; (c) 15.0 l/s; (d) 20.0 l/s Figure Data collected at transducer T1 during the sudden pump stoppage without air vessel tests in the steel-pipe experimental facility Figure 4.33 Data collected at the two transducers during the downstream valve closure with air vessel tests: (a) 5.0 l/s; (b) 10.3 l/s; (c) 15.8 l/s; (d) 20.1 l/s Figure 4.34 Data collected at transducer T2 during the downstream valve closure in the steel-pipe facility Figure 4.35 Maximum overpressures observed as a function of initial flow rate Figure 5.1 Collected steady-state pressure data in the copper-pipe facility :(a) total headloss; (b) unit length headloss Figure 5.2 Darcy-Weisbach friction factors comparison: (a) based on measurements between transducers; (b) based on mean unit length headloss Figure 5.3 Model calibration for sudden pump stoppage without air vessel in copper-pipe facility with initial flow rate 600 l/h Simulation 1: (a) piezometric head downstream the pump; (b) piezometric head at middle section; (c) piezometric head downstream the pipe; (d) flow rate downstream the pump 115 Figure 5.4 Model calibration for sudden pump stoppage without air vessel in copper-pipe facility with initial flow rate 600 l/h Simulation 2: (a) piezometric head downstream the pump; (b) piezometric head at middle section; (c) piezometric head downstream the pipe; (d) flow rate downstream the pump Figure 5.5 Model calibration for sudden pump stoppage without air vessel in copper-pipe facility with initial flow rate 600 l/h Simulation 3: (a) piezometric head downstream the pump; (b) piezometric head at middle section; (c) piezometric head downstream the pipe; (d) flow rate downstream the pump and check valve manoeuvre Figure 5.6 Model calibration with unsteady-state friction for sudden pump stoppage without air vessel in copper-pipe facility with initial flow rate 600 l/h Simulation 3: (a) piezometric head downstream the pump; (b) piezometric head at middle section; (c) piezometric head downstream the pipe; (d) flow rate downstream the pump and check valve manoeuvre xix

22 Figure 5.7 Model validation for copper-pipe facility with initial flow rate 300 l/h Simulation 1: (a) piezometric head downstream the pump; (b) piezometric head at middle section; (c) piezometric head downstream the pipe; (d) flow rate downstream the pump and check valve manoeuvre Figure Model calibration for sudden pump stoppage with air vessel in copper-pipe facility with initial flow rate 600 l/h results from JDS: (a) piezometric head downstream the pump; (b) flow rate downstream the pump and check valve calibrated manoeuvre Figure Model calibration for sudden pump stoppage with air vessel in copper-pipe facility with initial flow rate 600 l/h results from AKS: (a) piezometric head downstream the pump; (b) flow rate downstream the pump and check valve calibrated manoeuvre Figure 5.10 Pump head-discharge curve in copper-pipe facility Figure 5.11 Model calibration for downstream valve closure with air vessel in copper-pipe facility with initial flow rate 400 l/h: (a) piezometric head upstream the valve; (b) piezometric head at middle section; (c) piezometric head upstream the pipe; (d) flow rate upstream the valve and downstream valve manoeuvre Figure Main differences between mathematical model results and experimental data in downstream valve closure with air vessel in copper-pipe facility Figure 5.13 Model calibration for copper-pipe facility with initial flow rate 360 l/h: (a) piezometric head upstream the valve; (b) piezometric head at middle section; (c) piezometric head upstream the pipe; (d) flow rate upstream the valve and downstream valve manoeuvre Figure 5.14 Model validation for copper-pipe facility with initial flow rate 450 l/h: (a) piezometric head upstream the valve; (b) piezometric head at middle section; (c) piezometric head upstream the pipe; (d) flow rate upstream the valve and downstream valve manoeuvre Figure 5.15 Total and unit length headloss in steel-pipe facility Figure 5.16 Steady-state friction calibration: (a) friction factor; (b) equivalent roughness Figure 5.17 Model calibration for sudden pump stoppage without air vessel in steel-pipe facility with initial flow rate 15.8 l/s Simulation 1: (a) piezometric head downstream the pump; (b) piezometric head downstream the pipe; (c) flow rate downstream the pump Figure 5.18 Model calibration for sudden pump stoppage without air vessel in steel-pipe facility with initial flow rate 15.8 l/s Simulation 2: (a) piezometric head downstream the pump; (b) piezometric head downstream the pipe; (c) flow rate downstream the pump and check valve calibrated manoeuvre Figure 5.19 Model validation for sudden pump stoppage without air vessel in steel-pipe facility with initial flow rate 10.0 l/s: (a) piezometric head downstream the pump; (b) piezometric head downstream the pipe; (c) flow rate downstream the pump and check valve calibrated manoeuvre xx

23 Figure Model calibration for sudden pump stoppage with air vessel in steel-pipe facility with initial flow rate 15.8 l/s results from JDS: (a) piezometric head downstream the pump; (b) flow rate downstream the pump and check valve calibrated manoeuvre Figure Model calibration for sudden pump stoppage with air vessel in steel-pipe facility with initial flow rate 15.8 l/s results from AKS: (a) piezometric head downstream the pump; (b) flow rate downstream the pump and check valve calibrated manoeuvre Figure 5.22 Pump head-discharge curve in the steel-pipe facility Figure 5.23 Model calibration for downstream valve closure with air vessel steel-pipe facility with initial flow rate 5.0 l/s: (a) piezometric head upstream the valve; (b) piezometric head upstream the pipe; (c) flow rate upstream the valve and downstream valve manoeuvre Figure 5.24 Differences between numerical results and experimental data for downstream valve closure with air vessel in steel-pipe facility xxi

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25 LIST OF TABLES Table 2.1 Main assumptions considered in the basic waterhammer equations (Covas, 2003)... 8 Table 2.2 Ψ coefficient expressions (Chaudhry, 1987)... 9 Table 2.3 Considerations regarding the types of flow regime Table 2.4 Friction factor formulas Table 2.5 Numerical methods used to solve 1-D hydraulic transients in pressurized pipes adapted from (Covas, 2012) Table Computational grids used in the MOC Table Assumptions used in the classical theory that are not in agreement with the unsteady-state friction Table 2.8 Unsteady-state friction formulations Table Assumption used in the classical theory that is not in agreement with non-elastic behaviour of pipes Table Assumptions used in the classical theory that are not in agreement with the transient cavitation Table Assumption used in the classical theory that is not in agreement with the fluid-structure interaction Table 2.12 Zones of pump operation Table 2.13 Selection criteria for check valve (Val-Matic, 2011) Table 3.1 Numerical schemes used to integrate the steady-state friction term for rough turbulent flows adapted from Covas (2012) Table Considerations regarding the steady-state friction term integration Table 3.3 Coefficients CP1, CP2, CN1, CN2 (adapted from Covas, 2003 and Soares et al., 2008) Table Pumping groups characteristics Table 3.5 Scenarios used in pumping system simulation Table 3.6 Hydropneumatic vessel characteristics used in the simulations Table Characteristics of the simulation S-A Table 3.8 Comparison between significant values obtained from AKS and JDS for scenario S-A xxiii

26 Table Characteristics of simulation S-A Table Comparison between significant values obtained from AKS and JDS for scenario S-A2 70 Table Characteristics of simulation S-A Table Characteristics of simulation S-B Table Characteristics of simulation S-B Table Characteristics of simulation S-B Table Flow configurations in copper-pipe facility Table Transducer location in copper-pipe facility adapted from Libraga (2011) Table 4.3 Experiments carried out in copper-pipe facility Table Result analysis for the sudden pump stoppage with air vessel experimental tests in copperpipe facility Table Comparison between observed and theoretical maximum overpressures and elastic wave speed estimation in copper-pipe facility for downstream valve closure with air vessel Table Comparison between observed and theoretical maximum overpressures and elastic wave speed estimation in copper-pipe facility for downstream valve closure without air vessel Table Maximum piezometric head observed for downstream valve closure with and without protection, associated to the first upsurge, in copper-pipe facility Table Transducer location in the steel experimental facility Table 4.9 Experiments carried out in steel-pipe facility Table Result analysis of sudden pump stoppage with air vessel experimental tests in the steelpipe facility Table 4.11 Comparison between maximum overpressures observed and theoretical and elastic wave speed estimation in steel-pipe facility with air vessel protection Table 5.1 Collected steady-state pressure in copper-pipe facility Table 5.2 Copper-pipe facility pump characteristics Table 5.3 Parameters used in model calibration for sudden pump stoppage without air vessel in copper-pipe facility with initial flow rate 600 l/h Table 5.4 Parameters used in model validation for sudden pump stoppage without air vessel in copperpipe facility with initial flow rate 300 l/h xxiv

27 Table Hydropneumatic vessel characteristics used in the model calibration in copper-pipe facility Table 5.6 Parameters used in model calibration for sudden pump stoppage with air vessel in copperpipe facility with inital flow rate 600 l/h Table 5.7 Parameters used in model calibration for downstream valve closure with air vessel in copperpipe facility with initial flow rate 400 l/h Table 5.8 Parameters used in model validation for downstream valve closure with air vessel in copperpipe facility with initial flow rate 500 l/h Table 5.9 Collected steady-state pressure data for sudden pump stoppage without hydropneumatic vessel scenario in steel-pipe facility Table Steel-pipe facility pump characteristics Table 5.11 Parameters used in model calibration for sudden pump stoppage without air vessel in steelpipe facility with initial flow rate 15.8 l/s Table 5.12 Parameters used in model validation for sudden pump stoppage without air vessel in steelpipe facility with initial flow rate 10.0 l/s Table Hydropneumatic vessel characteristics used in the model calibration in steel-pipe facility Table 5.14 Parameters used in model calibration for sudden pump stoppage with air vessel in steelpipe facility with initial flow rate 15.8 l/s Table 5.15 Parameters used in model calibration for downstream valve closure with air vessel in steelpipe facility with flow rate 400 l/h xxv

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29 NOMENCLATURE Nomenclature Description Units a Elastic wave speed m/s a 1, a 2, a 3 and a 4 Constants that define the straight lines representing the head and torque characteristics a m Wave speed in the gas liquid mixture m/s a Pressure wave speed m/s b 1, b 2 Constant coefficients of head-discharge curve - C a Parameter that depends upon the pipe properties m 2 /s C V Headloss coefficient in the valve s 2 /m 5 C P, C ap, C N and C an Constants defined for each pipe section and time - C 0 Constant of the polytropic gas equation - D Diameter of the pipe cross-section m D imp Pump impeller diameter m D 0 Pipe inner diameter at time t=0 m D(t) Pipe inner diameter at time t m dv dt Fluid deceleration m/s 2 ( dv dt ) Non-dimensional fluid-deceleration m/s 2 e Pipe-wall thickness m e 0 Pipe-wall thickness at time t=0 m e(t) Pipe-wall thickness at time t=0 m E Young s modulus of elasticity of the pipe N/m 2 f s Darcy-Weisbach friction factor - F h Dimensionless piezometric head parameter that describes pump characteristics F β Dimensionless torque parameter that describes pump characteristics - g Gravity acceleration m/s 2 h Dimensionless parameter of the pump piezometric head - h f Total friction losses m/m h fs Steady-state component of friction losses m/m h fu Unsteady-state component of friction losses m/m H Piezometric head m H b Barometric pressure head, expressed in absolute pressures m H Pair Absolute pressure head at the end of the time-step N/m 2 H pump Pumping head in the end of each time-step m H R Rated pump piezometric head m H resd Downstream reservoir water level above the reference datum m H resu Upstream reservoir water level above the reference datum m H 0 Pump shutoff head m H U,DST Piezometric head upstream/downstream the surge tank m J 0 Instantaneous creep 1/(N/m 2 ) J(t ) Creep function at t time 1/(N/m 2 ) k Pipe-roughness m K Fluid bulk modulus of elasticity N/m xxvii

30 Nomenclature Description Units K L Local headloss coefficient in the singularity - K orf Local headloss coefficient of the orifice - K ST Local headloss coefficient in the surge tank connection - K V Local headloss coefficient of the valve, that depends on the type of valve - and opening degree k 3 Decay coefficient of Vítkovský et al., (2000) formulation - L Pipe length m m Exponent of polytropic gas equation - n Exponent of flow in the friction loss equation - n pumps Number of pumps in the pumping station - N Pump rotational speed rpm N R Rated pump rotational speed rpm N s Pump specific rotational speed SI p Pressure of the gas N/m 2 p 0 Initial pipe pressure N/m 2 p(t) Pipe pressure at time t N/m 2 p Absolute pressure in a cavity N/m 2 p C Vapour pressure N/m 2 p g Absolute pressure of the gas N/m 2 Q Flow rate m 3 /s Q DC Flow rate downstream the cavity m 3 /s Q DST Flow rate downstream the surge tank m 3 /s Q i Initial flow rate m 3 /s Q orf Flow rate through the orifice at the end of the time-step m 3 /s Q Porf Flow rate through the orifice at the beginning of the time-step m 3 /s Q pump Pump flow-rate m 3 /s Q UC Flow rate upstream the cavity m 3 /s Q UST Flow rate upstream the surge tank m 3 /s Q R Rated pump flow rate m 3 /s Q ST Flow in the surge tank m 3 /s Re Reynolds number - R E External diameter of the pipe m R I Internal diameter of the pipe m S Area of the pipe cross-section m 2 S HPV Area of the hydropneumatic vessel cross-section m 2 S orf Area of the orifice cross-section m 2 S ST Area of the surge tank cross-section m 2 S V Area of the valve cross-section m 2 T Pump torque Nm T R Rated pump torque Nm t Time coordinate s t c Valve closure time s t m Time of valve closure/opening s U Mean reverse velocity at check valve slam m/s v Dimensionless parameter of the pump flow rate - xxviii

31 Nomenclature Description Units v r Maximum reverse velocity of the fluid m/s v r Non-dimensional reverse velocity m/s v 0 steady-state velocity of the fluid m/s V air Volume of enclosed air inside the vessel at the beggining of the time-step m 3 V Pair Volume of enclosed air inside the vessel at the end of the time-step m 3 WR 2 Combined polar moment of inertia of the pump, motor, shaft and liquid inside the pump impeller kgm 2 x Spatial coordinate along the pipe axis m Z Water level in the air vessel at the beginning of the time-step m Z P water level in the air vessel at the end of the time-step m Z ST Water level in the surge tank m Z V Valve elevation above the reference datum m α Dimensionless parameter of the pump rotational speed - α 0 pipe-wall constraint coefficient at time t=0 - α(t) pipe-wall constraint coefficient at time t - α Pressure dependent of the volumetric ratio of gas in the mixture N/m 2 β Dimensionless parameter of the pump torque - ε(t) Total strain at a given time m/m ε e Instantaneous elastic strain m/m ε r (t) Retarded viscous strain m/m ΔH Porf Local headloss in the air vessel orifice m H valve Local headloss in the check valve m t Time-step between calculations s θ Relaxation coefficient - θ S Parameter that describes pump characteristics that varies between - /2 and rad /2 μ k Viscosity of the dashpot of k-element Kg/(sm) ν Pipe-wall Poisson ratio - ν Kinematic fluid viscosity m 2 /s ρ Fluid density kg/m 3 τ k Retardation time of the dashpot of k-element s Ψ Non-dimensional coefficient that depends upon the elastic properties of the pipe cross-section and longitudinal movement constrains - Ψ T Numerical weighting factor - C Volume of the vapour cavity m 3 xxix

32 xxx

33 ABBREVIATIONS AKS Almeida and Koelle Simulator CFL Courant-Friedrich-Lewy condition DGCM Discrete Gaseous Cavity Model DVCM Discrete Vapour Cavity Model HPV HydroPneumatic Vessel JDS João Delgado Simulator MOC Method of Characteristics xxxi

34

35 1 INTRODUCTION 1.1 Importance of hydraulic transient analysis A transient-state flow is defined by the temporal variation of pressure conditions that occurs at the transition between two steady-states, caused by a disturbance in the system (e.g., flow rate or velocity change). The disturbance that creates the transient-state arises from planned or accidental actions in the components of the system, which induces spatial and temporal changes in the flow-pressure conditions. These actions can be valve manoeuvres, start-up or stoppage of hydraulic turbo-machines, mechanical failures in protection devices or sudden disruptions in the infrastructure (Covas, 2012). The oscillatory nature of this phenomenon arises from the deformation of the pipe, and compression and expansion of the fluid. After the disturbance, the transient-state dissipates over time due to the effects of friction resistance, mechanical deformation of the pipe due to the pressure waves, existence of surge protection devices, among others. The operation control in pressurized pipe systems is a concern of design engineers and utilities that operate the systems for reasons related with risk, safety and efficiency. Therefore, a careful analysis of transient-state flows that are likely to occur should be carried out. These analyses are highly important during the design and operation phase, and also, during the planning of a possible expansion or rehabilitation in the pipe system. Since the early design phases of the system, hydraulic transients should be considered in order to find the technical specifications and materials corresponding to a more economic and safer layout. The final optimum solution need to conciliate the hydraulic transients with all collateral dynamic effects (Ramos, "Guideline for design of small powerplants", 2000). Maximum overpressures must be correctly predicted to maintain the integrity of the system components and to avoid accidents caused by the disruptions in the system and the interruption of the services (e.g., water supply and production/supply of electricity). Minimum underpressures must also be carefully estimated to avoid the fluid reaching the sub-atmospheric pressure, which could cause deformation buckling and collapse of the pipe, intrusion of pollutants, vapour cavitation or liquid column separation. The re-joining of the separated columns or collapse of the cavities results in a considerable pressure rise, which may damage the pipe systems (Chaudhry, 1987) The hydraulic transient analysis during the design phase of water supply systems is carried out for exceptional operating situations that are unexpected and highly improbable, that may cause severe damage to the system (Ramos, "Guideline for design of small powerplants", 2000). Examples of these situations are the sudden stoppage of the pump by failure of the electric grid, the complete failure of the turbine wicket closure mechanisms with a very fast flow stoppage, the failure of surge protection devices or phenomena like resonance. 1

36 In the other situations, operating modes are defined for the system (e.g., sequence and timing of valve closure or opening) and pump schemes (e.g., input and output of pumping groups) to minimize the expected transient-pressures (Covas, 2012). During the operation phase of water supply systems, the analysis of transient conditions represents a useful tool that supports the diagnosis to identify ruptures and malfunctions in the system due to excessive pressure variations. These problems occur from the not compliance of design (e.g., reservoir levels, pipe profiles) and operation (e.g., input and output of pumping groups) specifications, the condition of the system components (e.g., valves) or the changing in the operating conditions without an analysis of the transient-state induced (e.g., removal of surge protection devices or adding an additional pumping group). These conditions require the evaluation and diagnosis of the problems source, using whenever possible measurements carried out in the facilities (Covas, 2012). Results obtained by commercial models, although based on classical waterhammer theory, are satisfactory in terms of design of pressurized pipe systems, as maximum and minimum pressures that occur in the systems can be reasonably well estimated and surge protection devices can be designed whenever necessary to control transient pressures. However, these models are not accurate enough for the diagnosis and analysis of existing systems, mainly because these models cannot describe unconventional phenomena, such as cavitation, two-phase flow, unsteady friction and nonlinear rheological behaviour of the pipe. In these cases, it is necessary to use more complex models incorporating the full description of these phenomena, which are not commercially available and which are much more model calibration-dependent. Hydraulic transients can be described by means of 1-D, 2-D or 3-D models depending on the importance of the analysed phenomenon. In general, hydraulic transient models in pipe systems are based on the assumption of one-dimensional flow and are usually solved by Method of Characteristics. 2-D and 3-D models are used when the focus is the description of a particular phenomenon. Examples are the rapid propagation of a crack when a pipe is instantaneously loaded (Venizelos et al., 1997; Ivankovic and Venizelos, 1998), the description of the velocity profile during transient events in the cross-section (Bratland, 1986; Vardy and Hwang, 1991) and the calculation of unsteady-shear stress (Silva-Araya and Chaudhry, 1997). In these cases, the finite-volume method, finite-difference method or hybrid methods are used to solve fluid equations (Covas, 2003). 1.2 Aim and methodology The current research work aims at the development and validation of a mathematical model for the calculations of hydraulic transients in pumping systems based on the comparison of obtained numerical results with data collected from two experimental facilities. To achieve this goal, the following approach is carried out: 2

37 (i) Review of the state-of-the-art regarding hydraulic transients in pumping systems. (ii) Development of a mathematical model for the description of transient events in raising and gravity pipe systems, with different boundary conditions and incorporating the unsteady-state friction effect. (iii) Comparison between results obtained from the mathematical model and a validated classical model based on a semi-real case-study. (iv) Development of an experimental programme to collect steady and transient-state data in two experimental facilities in the Laboratory of Hydraulics and Water Resources, in Department of Civil Engineering, Architecture and Georesources (DECivil), at the Instituto Superior Técnico in Lisbon, Portugal. (v) Mathematical model calibration and validation based on the data collected from the experimental programme carried out. (vi) Establishment of conclusions and recommendations for design purposes and suggestions for future works. 1.3 Document outline The thesis is organized in six chapters. Three of these chapters (Chapters 3, 4 and 5) contain the research work developed, regarding the mathematical model, experimental data collected and model calibration and validation. The current chapter introduces the background that supports the need for the development of a solid numerical model capable of describing maximum and minimum pressures observed as well as the surge damping. This chapter also introduces the objectives and methodology followed, as well as the document outline. Chapter 2 contains the state-of-the-art review about the principal aspects of hydraulic transient events in gravity and pumping single-pipe systems. Aspects regarding classical waterhammer theory and unconventional dynamic phenomena (unsteady-state friction, pipe material non-elastic behaviour, transient cavitation and fluid-structure interaction) are presented. Centrifugal pumps behaviour during transient-state and check valve dynamics is reviewed. Finally, main gaps and thesis motivation are presented. Chapter 3 presents the description of the mathematical model developed. The chapter includes the description of governing equations and numerical solutions of the Method of Characteristics (MOC), simple and complex boundary conditions, such as the sudden pump stoppage and the hydropneumatic vessel, and a formulation used to describe the friction losses during the transient-state. This chapter also includes the comparison between results obtained from the numerical model implemented, and a mathematical model validated used in transient-state design, based on a semi-real case-study. 3

38 Chapter 4 presents the experimental facilities assembled in the Laboratory of Hydraulics and Water Resources, in Department of Civil Engineering, Architecture and Georesources (DECivil), at the Instituto Superior Técnico in Lisbon, Portugal, and the experimental data collected for the transient events generated by different manoeuvres in devices (upstream and downstream valves and pumps) of the system. Chapter 5 focuses on the mathematical model calibration and validation. Differences obtained between the numerical results and the experimental data collected are shown and explained based on the simplifications considered in the development of the mathematical model. Chapter 6 contains a summary of the research work developed. Main conclusions achieved are outlined, and finally, suggestions for future work and investigation are presented. 4

39 2 STATE-OF-THE-ART 2.1 Introduction This chapter includes the review of the research work in the thematic of hydraulic transient-states in pumping systems. This review is imperative to assess the state-of-the-art required for the development of the numerical model and clarifying the main gaps in knowledge and uncertainties that are observed in real pipe systems that, sometimes, are not taken into account by the design engineers and operational entities. Different types of hydraulic transients analyses are presented with particular emphasis to the numerical models based on the classical waterhammer theory. Fundamentals of this theory are presented as well as the Method of Characteristics, which is the method most widely used to solve the main equations of the transient-flow in pressurized pipes. Phenomena that affect real pipe systems and are not covered by the classical theory are also revisited, namely the unsteady-state friction losses, the pipe-material non-elastic behaviour, the transient cavitation and the fluid-structure interaction. This chapter also includes the review of the hydraulic transients caused by the sudden pump stoppages due to power failure of the electrical grid, the dynamic behaviour of check valves, and the main surge protection devices used in raising mains. Sometimes, the most severe transient-state pressures occurs when these three components interact. Finally, the chapter concludes with the main gaps in research in this thematic and the motivations for the development of this work. 2.2 Hydraulic transients analyses The analysis of pressure variation induced in hydraulic systems by the operation of control valves or by any other device that causes a significant variation in the flow rate and, thus, in the pressure, can be carried out by using simple formulae to complex numerical models, based on the fluid mechanics and hydrodynamic principles of transient flows. Different models can be used depending on the degree of detail required and the importance of a given phenomenon for the analyses. The hydraulic analysis can range from simple formulations (e.g., Frizell-Joukowsky, 1898 and Michaud, 1878), to classical mathematical models of simulation (e.g., FLOWMASTER, WANDA, HAMMER commercial models) or to advanced simulation models that incorporate unconventional dynamic phenomena (e.g., Covas, 2003). Considerations regarding the types of analysis, mentioned in the previous paragraph, are presented in the following sub-sections. Other types of transient analyses could be carried out (e.g., frequency analysis), however, these are outside of the scope of this research. 5

40 2.2.1 Simple formulations Simple formulations are based on the elastic theory and allow a simple and fast prediction of the pressure variation in a pipe for a given idealized manoeuvre in a system with the simple configuration of type: reservoir pipe valve. These formulations consider a uniform pipe with negligible headlosses and velocity head, and consider also linear valve manoeuvres. These formulations depend upon the valve manoeuvre time. For fast manoeuvres (i.e., t c < 2L a), maximum overpressure at the upstream of the valve can be estimated based on Frizell-Joukowsky equation: ΔH J = a Q i g S (2.1) where a = elastic wave speed (m/s); g = gravity acceleration (m/s 2 ); Q i = initial flow rate (m 3 /s); and S = area of the pipe cross-section (m 2 ). For slow manoeuvres (i.e., t c > 2L a), maximum overpressure is estimated based on Michaud formulation: ΔH M = 2L t c Q i gs (2.2) where L = pipe length (m); and t c = valve closure time (s). These formulations allow the evaluation of the maximum overpressures order of magnitude and the expected layout of piezometric head envelopes. However, since they are based on assumptions which are not compatible with most of the real systems, they should only be used to validate results obtained from numerical models or experimental data, or used in the early design phases or in the preliminary and feasibility studies. This type of analysis must be used with precaution since most systems do not have simple configuration of the type reservoir pipe valve, the valve manoeuvres are not linear and the friction losses are not negligible, which can create higher overpressures that the ones estimated. On the one hand, these formulas underestimated overpressures; on the other hand, they can lead to economically non-optimized solutions obtained by the excessive conservativeness inherent to these expressions Classical waterhammer models Classical waterhammer models are based on the elastic theory and are widely used in the hydraulic transient analysis. Several commercial models are available in the market. Classical waterhammer theory is based on the equations of continuity and motion widely presented in literature, Chaudhry (1987), Almeida and Koelle (1992) or Wylie and Streeter (1993). The theory settles on some assumptions regarding the fluid, the flow and the pipe characteristics and can be solved, for 6

41 instance, by the Method of Characteristics. Maximum and minimum pressures can be reasonably well estimated, if the boundary conditions are correctly described, and surge protection devices can be designed, whenever necessary to control transient pressures. Although this analysis is adequate for design purposes, it cannot describe the behaviour observed in real systems, where the assumptions made in the classical theory are not valid, leading to the need of more solid and reliable numerical models Advanced simulation models Sometimes, results obtained by the use of classical simulators are not sufficiently robust and accurate to describe the physical behaviour observed in real systems, particularly for diagnostic purposes. This is because some of the simplifying assumptions inherent to the classical theory of waterhammer are not valid in those pipe systems. Examples of phenomena not described by the classical theory are: the unsteady-state friction, the pipe-material non-elastic behaviour, the presence of air and transient cavitation and the pipe axial movement. For these cases, classical theory is replaced with different types of models to describe these unconventional phenomena, sometimes assuming bi or tri-dimensional flows. These models are not available in the market and require a more complex parameter definition and calibration. Usually, without these types of models, the hydraulic transient diagnosis is not correctly carried out, and the causes of problems are not properly identified. 2.3 Classic waterhammer problem Basic equations and assumptions The classic waterhammer problem can be analysed using the continuity and momentum equation of pressurized pipes. These equations can be derived from the Reynolds Transport Theorem (Chaudhry, 1987; Almeida and Koelle, 1992; Wylie and Streeter, 1993): Continuity Equation Momentum Equation dh dt + a2 Q gs x = 0 (2.3) H x + 1 dq gs dt + h f s = 0 (2.4) where H = piezometric head (m); Q = flow rate (m 3 /s); h fs = steady-state component of friction losses (m/m); t = time coordinate (s); and x = spatial coordinate along the pipe axis (m). Assumptions regarding the derivation of previous equations are presented in Table

42 Table 2.1 Main assumptions considered in the basic waterhammer equations (Covas, 2003) Fluid assumptions (i) Fluid is one-phase, homogenous and compressible. The compressibility of the fluid is incorporated in the elastic wave speed. (ii) Changes in fluid density and temperature during transients are negligible compared to pressure and flow-rate variations. Flow assumptions (iii) The flow is one-dimensional (1-D) with a pseudo-uniform velocity profile in each cross-section of the pipe. Consequently, momentum and Coriolis coefficients, α and α, are assumed constant and equal to 1, during the transient events. (iv) Head losses during transient flow are calculated similarly to steady-state by common friction formulae (e.g., Colebrook-White, Manning-Strickler or Hazen-Williams). This simplification is not valid for fast transients, unless a correction is introduced in the momentum equation. Pipe assumptions (v) The pipe material has a linear elastic behaviour. (vi) The dynamic fluid-pipe interaction is neglected assuming a constrained pipe without any axial movement; otherwise, additional pipe motion equations should to be considered. (vii) The pipe is straight and uniform, with a constant cross-section, without lateral in/out flow; otherwise, cross-section variation should be incorporated into basic equations, and lateral in/outflow should be described as a boundary condition or, in case of distributed flow, a term should be added to the main equations. The total derivative of a given variable B may be expressed as follows: db dt = B B + u t x B B + v + w y z (2.5) since the flow is considered one-dimensional, and considering second-order terms, v H x and v Q x, negligible (hypothesis valid for most of the engineering applications since its order of magnitude is inferior than the other terms), main equations that describe the transient flow in pressurized pipes may be written as follows: H t + a2 Q gs x = 0 (2.6) H x + 1 Q gs t + h f s = 0 (2.7) previous equations must be satisfied along the space-time domain and additional equations to boundary conditions must be considered. There are two independent variables, x and t, and two dependent variables, H and Q, meaning that this set of equations has a solution. 8

43 2.3.2 Elastic wave speed Wave speed represents the velocity that a disturbance travels in a given fluid inside the pipe. Considering a single-phase fluid, the wave speed, a, depends upon the fluid properties (e.g., fluid bulk modulus of elasticity, fluid density), pipe properties (i.e., pipe-wall material, diameter and wall-thickness) and the type of supports and constraints of the pipe. For linear-elastic pipes and single-phase fluids, the elastic wave speed can be estimated by the following expression (Halliwell, 1963): K a = ρ (1 + K E Ψ) (2.8) where K = fluid bulk modulus of elasticity (N/m 2 ); ρ = fluid density (kg/m 3 ); E = Young s modulus of elasticity of the pipe (N/m 2 ); and Ψ = non-dimensional coefficient that depends on the elastic properties of the conduit cross-section and longitudinal movement constrains (-). The Ψ coefficient depends on the type of the pipe-wall: if the ratio between the diameter and the pipe-wall thickness is less than 25, D e < 25, the pipe-wall is considered thick; otherwise the pipe wall is considered thin. Three types of longitudinal movement constrain can be considered. Expressions for the coefficient Ψ calculation are presented in Table 2.2. Table 2.2 Ψ coefficient expressions (Chaudhry, 1987) Rigid conduits Ψ = 0 (2.9) Thick-walled elastic conduits Conduit anchored against longitudinal movement throughout its length Conduit anchored against longitudinal movement at the upper end Ψ = 2(1 + ν) R E R I R 2 2 E R 2νR 2 I I R 2 2 E R (2.10) I Ψ = 2 [ R E R I 2 R E 2 R I 2 ν(r E 2 3 R 2 I ) R 2 2 ] (2.11) E R I Conduit with frequent expansion joints Ψ = 2 ( R E R I R ν) (2.12) E R I Thin-walled elastic conduits Conduit anchored against longitudinal movement throughout its length Conduit anchored against longitudinal movement at the upper end Ψ = D e (1 ν2 ) (2.13) Ψ = D (1,25 ν) (2.14) e Conduit with frequent expansion joints Ψ = D e (2.15) Note: R E = pipe external diameter (m); R I = pipe internal diameter (m); D = diameter of the pipe cross-section (m); e = pipe-wall thickness (m); and ν = pipe-wall Poisson ratio (-). 9

44 Expressions used to determine the Ψ coefficient for tunnels through rock, reinforced concrete pipes, wood-stave pipes, polyvinyl chloride and reinforced plastic pipes, and non-circular pipes are presented in specialized literature (e.g., Chaudhry, 1987 and Wylie and Streeter, 1993). Equation (2.8) is applied when no transient data are available and a prediction for this value is required. However, it only provides an estimate for the real wave speed, due to the uncertainties related with the percentage of free-air in the fluid, pipe s physical characteristics and constraints. When transient data in two different sections are available, a better estimate of the wave speed can be obtained based on the observed traveling time between those two sections Friction resistance Classical waterhammer theory considers that the friction resistance during the transient event is determined as if the flow behaves like successive steady-flows between each time-step. There are two main approaches: one approach considers a constant friction factor and equal to the one obtained in the steady-state; the other considers that transient flow behaves in a quasi-steady-state manner, and the friction factor is determined at each time-step (Reynolds-dependent friction factor). Main formulas to determine the friction factor are presented in this section, whereas the numerical schemes used to integrate the steady-state friction term are presented in section The steady-state component of the friction losses can be evaluated based on Equations (2.16) and (2.17), for turbulent or laminar conditions, respectively: h fs = f s Q Q n 1 2gD S n, n ε ]1,2] (2.16) h fs = 32 ν Q gd 2 S (2.17) where f s = Darcy-Weisbach friction factor (-); n = exponent of flow in the friction loss equation (-); and ν = kinematic fluid viscosity (m 2 /s). Three types of flow regimes can be defined depending on the flow and pipe characteristics. Some considerations regarding each type of flow are presented in Table 2.3, whereas the expressions used to determine the friction factor, for each type of flow regime are presented in Table 2.4. White (1999) presents more detailed information regarding the steady-state friction. 10

45 Table 2.3 Considerations regarding the types of flow regime Type of flow regime Laminar Smooth turbulent Rough turbulent Considerations Friction factor depends only upon a linear relationship with the Reynolds number. Friction factor is determined based on the Hagen-Poiseuille expression (exact solution) Equation (2.18). For smooth walled-pipes, flow is independent of pipe roughness and fluid viscosity is the major source of energy losses. The friction factor depends only upon the Reynolds number. Several authors developed expressions for this type of flow regime Equations (2.19) to (2.21) were developed by Blasius, Nikuradse and Karman-Prandtl, respectively. Friction factor depends only upon the ration between pipe-roughness and pipe-diameter. For a given value of the latter ratio, the friction factor is constant for all the Reynolds number and given by the von Karman expression Equation (2.22). Table 2.4 Friction factor formulas Type of flow regime Equations Laminar Hagen-Poiseuille f s = 64 Re (2.18) Blasius f s = Re 0.25 (2.19) Smooth turbulent Nikuradse f s = Re (2.20) Rough turbulent All domains of turbulent flow Karman-Prandtl Von Karman Colebrook-White 1 = 2log 10 ( 2.51 ) (2.21) f s Re f s 1 = 2 log 10 ( k f s 3.7D ) (2.22) 1 = 2 log 10 ( k f s 3.7D ) (2.23) Re f s Note: k = pipe-roughness (m); and Re = Reynolds number (-). In the transition between laminar to smooth turbulent conditions, the friction factor has an unstable behaviour (i.e., Re = ). Currently, there are no reliable formulations to calculate the friction factor in this flow regime. Regarding the transition between smooth turbulent and rough turbulent conditions, the friction factor depends on the ratio between pipe-roughness and pipe-diameter and the Reynolds number, simultaneously. Colebrook and White (1937) developed experiments that lead to the formulation of Equation (2.23), which is the most widely used in the friction factor determination (accuracy of ± 15 %) for the turbulent zone. Colebrook-White formula is an implicit equation that depends on f s in both members of the equation, which leads to an iterative process to calculate that parameter. There are some approximate explicit 11

46 formulae in literature, with results very similar to the previous equation. Zygrang and Sylvester (1982) presented an expedite formulation with an error of 0,114% (acceptable in engineering applications):. 1 = 2 log [ k f s 3,7D 5,02 Re log ( k 3,7D 5,02 Re log ( k 3,7D + 13 ))] (2.24) Re Local headlosses in the pipe system are given by the following expression: Q 2 H L = K L 2gS (2.25) 2 where K L = local headloss coefficient in the singularity (-), which depends on the type of singularity and Reynolds number (White, 1999) Numerical methods The two differential Equations (2.6) and (2.7) can be solved by several numerical methods. In this research work five different methods are presented. These methods are the Method of Characteristics, Finite-Difference Method, Finite-Element Method, Lagrangian Methods and Spectral Methods. The reason for the use of any numerical method relies on its ability to solve problems by means of computer implementation. The main goal is to obtain a method that describes accurately the pressure-flow conditions for a given transient event with the maximum efficiency (i.e., less computation effort and time). Table 2.5 presents a summary of the advantages and the disadvantages of the referred numerical methods used to solve 1-D hydraulic transients in pressurized pipes. In the following text, it is presented a brief description of the five methods. Method of Characteristics Method of Characteristics is the most popular and widely used method for the solution of 1-D hydraulic transients. The method converts the two partial differential equations into two ordinary differential equations, which can be solved by finite-differences techniques. This method has enormous vantages, such as, accurate simulation of steep wave fronts and simplicity of programming and efficient computations, even for complex pipe systems, since each boundary condition and pipe-section are analysed separately, for a given time-step. This method is stable as long as the Courant-Friedrich-Lewy (CFL) condition is respected. Interpolations are necessary in multi-pipe systems, or in gas-liquid mixtures in which the wave speed is pressure-dependent. Depending on the Courant number, interpolations may lead to numerical dispersion and damping (Covas, 2003). To incorporate unconventional phenomena such as unsteady-state friction or pipe rheological behaviour, an additional term is introduced in the differential equations, and the solution is still given by simple sum of algebraic terms. 12

47 Finite-Difference Method The Finite-Difference Method replaces the two partial differential equations by finite differences. This method can be implicit or explicit. The implicit finite-differences scheme solves simultaneously the algebraic equation for all the entire system (Chaudhry, 1987). Simulations require a considerable computation time, in systems with several and complex boundary conditions, since the boundary conditions have to be solved by an iterative technique. The implicit method has the main advantage of being unconditionally stable, leading to the use of larger time steps to shorten the simulation time. However, the time-steps cannot be increased randomly and must respect the CFL condition, because it results in the smoothing of pressure peaks, or in the occurrence of high-frequency oscillations behind steep wave fronts (Covas, 2012). The explicit finite-difference scheme calculated the variables in terms of their values in the precious time-steps. This scheme is conditionally stable. Lagrangian methods The Lagrangian approach to transient analysis is based on tracking the movement and transformation of pressure waves as they propagate with time throughout the water distribution system in an event-oriented environment (Wood et al., 2005; Jung et al., 2009). The Wave Characteristic Method is one example of Lagrangian methods and was firstly presented in the literature as the wave plan method (Wood et al., 1966). According to the users of this method (Wood et al., 2005), has the advantage of allowing a faster calculation with lesser variables than the traditional Method of Characteristics and allows an easy implementation of different components. Spectral Methods These methods allow the differential equations resolution of the transient events in the amplitude-frequency domain. Two examples are the Impedance Method and the Transfer Matrix Method that are based on the linearized equations and on sinusoidal flow and pressure variations. However, according to (Chaudhry, 1970), the latter method is simpler and more systematic, among others. This type of analysis is only favorable to describe of steady oscilatory flows and phenomena that are time-dependent (e.g., unsteady-state friction and pipe-wall viscoelastic behaviour), or to calculate natural frequencies of the pipe. Main disavantages are related to the difficulty in modelling complex boundary conditions (e.g., turbo-machines), complex multi-pipe systems and uncertainty inherent to the linearization of the equations. Finite-Element Method This method does not have any advantage in 1-D flows, creating more complex the resolution of the two partial differential equations. 13

48 Table 2.5 Numerical methods used to solve 1-D hydraulic transients in pressurized pipes adapted from (Covas, 2012) Numerical method Advantages Disadvantages Limited by the stability condition, Simplicity of programming and efficient computation in multi-pipe affecting space and time calculation steps. Method of Characteristics systems with complex boundary conditions. Interpolations needed in multi-pipe systems and two-phase fluids. Accurate simulation of steep wave fronts. Need to incorporate additional terms in the main equations to describe unconventional phenomena. High calculation steps leads to Implicit Finite- Difference Method Explicit Finite- Difference Method Lagrangian Methods Spectral Methods Finite-Element Method Unconditionally stable for any space and time calculation step. Easy implementation. Fast calculation with lesser variables. Easy implementation of different components. Faster calculation with lesser variables than the Method of Characteristics. Time-dependent parameters are easy to incorporate in the wave speed. No advantage is observed in this method. artificial perturbations. Low computation efficiency. Difficult to implement complex boundary conditions. Limited by the stability condition, affecting space and time calculation steps. Difficult implementation of components that modify the classical theory. Loss of precision caused by the linearization of equations. Different implementation of some components Characteristic equations and stability condition The Method of Characteristics (MOC) requires the verification of the stability condition known by the Courant-Friedrich-Lewy stability condition, i.e., C N 1.0, where C N represents the Courant number, defined as the ratio between real travelling wave speed (U ± a) and the numerical wave speed (dx dt). Accordingly, the CFL condition is given by the following expression (Chaudhry, 1987; Wylie and Streeter, 1993): dx dt ±a (2.26) Despite the CFL condition restricting the space and time calculation steps, it allows the transformation of Equations (2.3) and (2.4) (if the second order terms are not neglected) or Equations (2.6) and (2.7) (if the second order terms are neglected) into a set of two ordinary differential equations valid along the characteristic lines Equations (2.27) and (2.28) - known as the characteristic equations: 14

49 C + : dq dt + gs dh a dt + gsh f + s = 0 C : dq dt gs dh a dt + gsh f s = 0 if dx = +a (2.27) dt if dx = a (2.28) dt The detailed analysis of these equations is presented in section The characteristic lines represent two independent paths that propagate flow and piezometric head information through the space-time domain. These characteristic lines are referred as C + or C, whether the propagation of information is carried out from the previous time at the upstream section or downstream section, respectively. These lines become curves if the second order terms are not neglected. These characteristic lines form a computational grid in the space-time domain, allowing the calculation of the transient events at a given hydraulic system Computational grids Different computational grids can be used in the space-time domain depending on the verification of the CFL satibily condition, accuracy required and computational time. The type of grid used depend on the type of condition imposed by the CFL stability condition ( x t = a or x t > a). Table 2.6 presents the conditions and considerations regarding the type of grids used in the MOC. 15

50 Table Computational grids used in the MOC Condition Type of grids Considerations Image x t = a Rectangular double-grid without interpolations It is the most commonly used grid in the MOC. It corresponds to the duplication of the time-step for a specified degree of accuracy. The specified time grid allows an orderly calculation process, which is extremely important in multi-pipe systems. This grid with the condition of x t = a only uses interpolations if required (e.g., two-phase fluids, pressure-dependent wave speed, time-dependent wave speed due to unsteady friction). Diamondshape staggered cross grid This grid avoids the duplication of the time-step, reducing calculation time. However, it requires an even number of reaches, which may be a disavantage (e.g., short pipes). x t > a Rectangular double-grid with interpolations Characteristic grid When this condition is observed, the characteristic lines cross at a point outside the grid and numerical interpolations are necessary. The interpolations induce numerical damping and dispersion of the pressure wave. The interpolations may be carried out in space or time, and can be linear or higher-order chemes, using the known conditions at the previous time-step. This grid is used for the condition of x > a t; otherwise, this grid this grid becomes a particular case of the diamond-shaped staggered grid ( x = a t). This grid avoids the interpolation errors of the rectangular grid (for x > a t). A free-floating grid develops in the (x,t) plane. Programming details are similar to the rectangular grid (Covas, 2003). 2.4 Unconventional waterhammer problems Introduction Usually, conditions of real pipe systems are far from the assumptions considered in the classic waterhammer theory. Accordingly, collected data in real systems cannot be accurately described by classical waterhammer simulators. This sub-section focuses on four types of unconventional waterhammer problems that, under certain circumstances can increase or decrease maximum and minimum pressures observed or create faster surge damping. The unconventional waterhammer problems presented in the following sub-sections are: (i) unsteady-state friction; (ii) non-linear elastic rheological behaviour of pipe-wall; (iii) presence of air and transient cavitation; (iv) fluid-structure interaction. 16

51 2.4.2 Unsteady-state friction Unsteady-state friction is one of the main causes of surge damping and wave-phase shifting in pipes with elastic behaviour. This pressure attenuation is due to friction resistance in the pipe-wall, and turbulence generated by the inversion of the velocity profile. Table 2.7 presents the assumptions used in the classical theory that are not in agreement with this phenomenon. Table Assumptions used in the classical theory that are not in agreement with the unsteady-state friction Flow assumptions (iii) The flow is one-dimensional (1-D) with a pseudo-uniform velocity profile in each cross-section of the pipe. Consequently, momentum and Coriolis coefficients, α and α, are assumed constant and equal to 1, during the transient events. (iv) Head losses during transient flow are calculated similarly to steady-state by common friction formulae (e.g., Colebrook-White, Manning-Strickler or Hazen-Williams). During a fast transient event (e.g., fast downstream valve closure), the velocity profile, in each section of the pipe, changes from the typical uniform profile and begins to invert near the pipe-wall (Figure 2.1). Accordingly, the friction resistance cannot be calculated by steady-state resistance formulas, which considers a velocity profile unidirectional during the transient event (energy losses are caused by shear stress, which is proportional to the velocity gradient near the pipe-wall). Additionally, the inversion of the velocity profile increases turbulence in the flow with further energy dissipation. (a) (b) Figure 2.1 Velocity profile inversion for the total closure of the downstream valve (a) along the pipeline and (b) in a particular cross-section. Numbers 1 to 7 refer to the time sequence of the velocity profiles during the inversion of the flow, immediately after the valve closure (Bratland, 1986) This feature was observed by Brunone et al., (2000), whom measured velocity profiles in a polyethylene pipe facility with 300 m length and nominal diameter of 110 mm, using an acoustic velocity meter. Furthermore, the author measured an asymmetry relatively to the pipe-axis (Figure 2.2) 17

52 (a) (b) (c) Figure 2.2 Velocity profiles measurement: (a) pressure variation in time; (b) and (c) velocity profile variation in time (Brunone et al., 2000). There are several formulations to describe the unsteady-state friction. The mathematical formulation of unsteady-state friction in the MOC consists of the inclusion of an additional term in the momentum equation, as follows: H x + 1 dq gs dt + (h f s + h fu ) = 0 (2.29) where h fu = unsteady-state component of friction losses (m/m). Formulations used to describe the parameter h fu, assume that these losses depend on several factors, namely the weights of past time local accelerations, the instantaneous mean velocity, the instantaneous acceleration, the local and convective acceleration and the velocity profiles. Table 2.8 presents some formulations used to describe the unsteady-state friction term. Table 2.8 Unsteady-state friction formulations Based on Weights of past time local accelerations Formulations Zielke (1968 ); Trikha (1975); Vardy et al., (1993); Vardy and Brown (1996); Vardy and Brown (2003) Instantaneous mean velocity Hino et al., (1976), Hino et al., (1977) Instantaneous acceleration Shuy (1996) Local and convective acceleration Velocity profiles Brunone et al., (1991), Brunone et al., (1995); Vítkovský et al., (2000); Bergrant et al., (2001); Ramos et al., (2004) Bratland (1986); Vardy and Hwang (1991); Silva-Araya and Chaudhry (1997); Pezzinga (1999); Pezzinga (2000); Leite et al., (2011) 18

53 2.4.3 Pipe material non-elastic and viscoelastic behaviour Several pipe-materials present a non-elastic rheological behaviour that changes the pressure wave propagation during a transient event. Two examples are the inelastic behaviour of metals when actuated instantaneously, and the viscoelastic behaviour of plastic pipes, like polyvinyl chloride (PVC) or polyethylene (PE). As the two latter materials have been extensively studied (e.g., Covas, 2003), the first one is still an open theme of investigation. Table 2.9 presents the assumption used in the classical theory that is not in agreement with this phenomenon. The following sub-section presents the pipe viscoelastic behaviour under hydraulic transient events. Table Assumption used in the classical theory that is not in agreement with non-elastic behaviour of pipes (v) The pipe material has a linear elastic behaviour. Pipe assumptions The viscoelastic behaviour of polymers has been extensively studied by several authors (e.g., Ferry, 1970; Aklonis et al., 1972). This rheological behaviour influences the pipe deformation under a transient event, affecting both maximum and minimum pressures observed, as well as the pressure wave shape and damping (Covas, 2003). This behaviour can be incorporated into hydraulic transients modelling either by means of a frequency dependent wave speed (e.g., Suo and Wylie, 1990), or by an additional term added to the continuity equations (e.g., Gally et al., 1979; Covas, 2003). The latter method will be presented. When a given stress, σ 0, is instantaneously applied, polymers have an immediate-elastic response (included in the elastic wave speed) and a retarded viscous response (translated by the additional term). Therefore, the total strain at a given time, ε(t), is given by the sum of the instantaneous elastic strain, ε e and the retarded viscous strain, ε r (t) Equation (2.30) and Figure 2.3. ε(t) = ε e + ε r (t) (2.30) According to Boltzmann superposition principle, for small strains, a combination of stresses that act independently in a system result in strains that can be added linearly. Thus, the total strain generated by a continuous application of a stress σ(t) is given by (Aklonis et al., 1972): t ε(t) = J 0 σ(t) + σ(t t ) J(t ) dt t 0 (2.31) where J 0 = instantaneous creep (m 2 /N) and J(t ) = creep function at t time (m 2 /N). For linear-elastic materials, the instantaneous creep is equal to the inverse modulus of elasticity, J 0 = 1/E, and the second term of the integral is zero. 19

54 Figure (a) Stress and strain for an instantaneous constant load; (b) Boltzmann superposition principle for two stresses sequentially applied (Covas, 2003) Assuming that the pipe material (i) is homogeneous and isotropic, (ii) it has linear viscoelastic behaviour for small strains, (iii) it has a constant Poisson s ratio ν so that the mechanical behaviour is only dependent on creep, and (iv) the hoop stress is directly related to the pressure in the fluid, p as follows σ e = αpd/2e, the circumferential strain ε(t), = (D D 0 )/D 0 is given by: ε(t) = α 0D 0 2e 0 [p(t) p 0 ] J 0 + t α(t t )D(t t ) 2e(t t ) 0 [p(t t ) p 0 ] J(t ) dt (2.32) t being α 0 and α(t) = pipe-wall constraint coefficient at time t=0 and t, respectively; D 0 and D(t) = pipe inner diameter at time t=0 and t, respectively (m); e 0 and e(t) = pipe-wall thickness at time t=0 and t, respectively (m); p 0 and p(t) = initial pressure and pressure at time t, respectively (N/m 2 ). The first term of this equation refers to the elastic strain ε e and the integral part to the retarded strain ε r (t). The creep function, J(t), that characterizes the pipe viscoelastic behaviour needs to be determined experimentally, or calibrated based on transient-pressure data collected. The mathematical representation of this parameter in the numerical model is achieved using the generalized mechanical model of Kelvin-Voigt described by (Figure 2.4): N J(t) = J 0 + J k (1 e t τ k ) k=1 (2.33) where J 0 =creep-compliance of the first spring defined by J 0 = 1/E; J k =creep-compliance of the spring of the Kelvin-Voigt element k defined by J k = 1/E k ; E k = modulus of elasticity of the spring of k-element; τ k = retardation time of the dashpot of k-element,τ k = μ k /E k ; μ k = viscosity of the dashpot of k-element. 20

55 Figure 2.4 Kelvin-Voigt model for viscoelastic materials with n-elements Modelling of this phenomenon in the space-time domain is based on the inclusion of an additional term in the continuity equation whereas the momentum equation remains unchanged (Covas, 2003). The last term of Equation (2.34) refers to the retarded behaviour of the pipe. Equations (2.4) and (2.34) are solved by the Method of Characteristics Transient cavitation dh dt + a2 Q gs x + 2a2 dε r g dt = 0 (2.34) During a severe hydraulic transient event the pressure in the fluid can drop to the vapour pressure, originating a vapour cavity. The liquid column separation occurs if the cavity growths enough to occupy the entire cross-section of the pipe. This occurs specially in pipes with steep slopes and knees. In horizontal and mild slope pipes, the air cavity in the top of the pipe travels along the pipe until it reaches a singularity. There are five basic problems caused by cavitation: noise, vibrations, pressure fluctuations, erosion damage, loss of efficiency and eventually pipe bursts (Tullis, 1989). All these problems are major concerns for design engineers and operational entities. Two main types of transient cavitation occur in fluid systems (Wylie and Streeter, 1993): (i) vaporous cavitation where the liquid phase is completely separated from the vapour phase and the cavity growths as the pressure drops to the vapour pressure; and (ii) gaseous cavitation in which the flow is characterized by the presence of micro-bubbles of free and dissolved gas distributed along the pipe so that the wave speed is pressure-dependent (Soares et al., 2012). In real pipe systems both types of cavitation can occur. Regarding the first type of cavitation, two sub-types may be distinguished: (i) local large vapour cavities and (ii) distributed small vapour cavities (Figure 2.5). The difference between this two types of cavitation depend on the size of the vapour cavities. These types of cavitation depend also on the pipe characteristics (i.e., pipe slope and existence of singularities). The first type is characterized by cavities with a large void fraction, usually formed near singularities such as closed valves and high knees. The second type is characterized by voids with small size and are usually generated in pipes with mild slopes and without considerable singularities. 21

56 (a) (b) (c) (d) Figure 2.5 Several types of cavitation: (a) downward sloping pipe; (b) increasing upward pipe; (c) horizontal pipe; (d) mild upward pipe. adapted from Bergant et al., (2006) Table 2.10 presents the assumptions used in the classical theory that are not in agreement with this phenomenon. An extensive review of transient cavitation and waterhammer with column separation is presented in literature (e.g., Bergant et al., 2006). Table Assumptions used in the classical theory that are not in agreement with the transient cavitation Fluid assumptions (i) Fluid is one-phase, homogenous and compressible. The compressibility of the fluid is incorporated in the elastic wave speed. (ii) Changes in fluid density and temperature during transients are negligible compared to pressure and flow-rate variations. The transient cavitation flow can be described by two models: (i) Discrete Vapour Cavity Model (DVCM), and (ii) Discrete Gaseous Cavity Model (DGCM). The first model assumes that, at the initial instant there is no air in the fluid. The latter assumes the existence of free air at the initial instant and described by the ideal gas law. The DVCM is based on the column separation theory and is the most commonly used model for column separation and distributed cavitation at the present time. It is easily implemented within standard waterhammer software and it reproduces many of the physical features of column-separation events (Bergant et al., 2006). The model consists of the formation of cavities in the sections of the calculation grid when the fluid reaches the vapour pressure. The absolute pressure in the cavity is set equal to the vapour pressure: p = p C (2.35) 22

57 where p = absolute pressure in the cavity (N/m 2 ); and p C = vapour pressure (N/m 2 ). The fluid behaves as a liquid with constant wave speed between the cavities. The piezometric head at the cavity remains constant and its volume varies according to the upstream and downstream flow rates at the cavity: d C dt = Q D C Q UC (2.36) where C = volume of the vapour cavity (m 3 ); and Q UC and Q DC = flow rate upstream and downstream the cavity (m 3 /s). The numerical integration of the previous equation in the MOC and considering a Rectangular double-grid without interpolations (see Table 2.6) is given by the following equation (Bergant et al., 2006): t C = t 2 t t C + [Ψ T (Q DC t Q DC ) + (1 Ψ T )(Q t 2 t DC Q t 2 t DC )]2 t (2.37) Where Ψ T = numerical weighting factor (-); and the superscripts t and t 2 t refer to the current time and 2 t earlier, respectively. The DGCM assumes a quantity of free air dispersed in the fluid along each computational section. The wave speed is affected by the gas concentration and is calculated by the following expression: a m = a α ρa 2 p g (2.38) where a m = wave speed in the gas liquid mixture (m/s); α = pressure-dependent volumetric ratio of gas in the mixture (N/m 2 ); and p g = absolute pressure of the gas (N/m 2 ). Since the wave speed varies, the slopes of the characteristic lines vary also. A constant grid with space and time interpolations can be used to solve this problem. The pressure in a cavity follows the ideal gas law (Bergant et al., 2006): (p p C ) C = (p 0 p C ) C0 = constant (2.39) where the subscript 0 refers to the initial pressure conditions. As in the DCVM, the cavity volume varies according to Equation (2.36) and its integration in the MOC and considering a Rectangular double-grid without interpolations, is given by equation (2.37). 23

58 2.4.5 Fluid-structure interaction In most engineering cases, pipelines are rigidly fixed, either because pipes are buried or due to the existence of solid anchors in pipe singularities (e.g., elbows, knees and tees). However, in some situations, depending on the rigidity of the support, the pressure response can be changed by the pipe movement (Tijsseling, 1996; Tijsseling and Vardy, 1996; Covas et al., 2011). Table 2.11 presents the assumption used in the classical theory that is not in agreement with this phenomenon. Table Assumption used in the classical theory that is not in agreement with the fluid-structure interaction Pipe assumptions (vi) The dynamic fluid-pipe interaction is neglected assuming a constrained pipe without any axial movement; otherwise, additional pipe motion equations should to be considered. There are three types of fluid-structure interaction: (i) friction coupling; (ii) Poisson coupling; (iii) junction coupling. First type refers to the interaction between the friction force between the fluid and the pipe-wall. The second type of interaction is created by the Poisson effect. The overpressures generated by the transient event deform circumferentially the pipe cross-section, and by Poisson effect, the pipe shortens axially, if the pipe it is not rigidly fixed (Figure 2.6). This phenomenon induces precursor waves that are oftentimes confused with vibrations. The latter type is created due to the forces generated by the pressure and momentum in singularities (e.g., elbows and knees). Figure 2.6 Generation of secondary pressure waves due to Poisson coupling (Tijsseling, 1996) The interaction generated by Poisson and junction coupling creates secondary wave pressures that are superposed to the elastic pressure wave generated by the disturbance that creates the transient event. Two examples of fluid-structure interaction are presented in Figure

59 H (m) (a) (b) Tempo (s) medição simulação Q=300 l/h simulação Q= 263 l/h Figure 2.7 Pressure waves measured and calculated by classical waterhammer theory: (a) numerical results obtained by Tijsseling (1996); (b) numerical results vs data collected by (Ribeiro, 2010). 2.5 Transient events generated by centrifugal pumps Introduction Main causes of transient events in rising mains are related with the start-up and the stoppage of pumps. Normal pump start-up is carried out with the downstream control valve closed, and this valve is gradually opened when the pump reaches its rated speed. During pump stoppage the check-valve is slowly closed, and the pump is switched off after the pumping groups are isolated from the piping system. These are the normal procedures used to start and stop a pump. However, accidents occur and piping systems must be designed for emergency situations, such as sudden start-ups due to operation errors or sudden pump stoppages due to power failure of the electrical grid (in the literature this phenomenon is known as pump failure or pump-trip-off or abnormal pump stoppage). Abnormal pump stoppage is the sudden stoppage of the pump, without the opportunity for prior valve adjustments, as when power is interrupted or safety devices on the pump or motor are actuated due to excessive heating or vibration. An operating error in actuating the pump stop button may trigger the same sequence of events (Wylie and Streeter, 1993). Two main undesirable scenarios can occur. If the pump stops fast and the pressure drops to the vapour pressure, transient cavitation and/or column separation may occur. If the check-valve allows a significant reverse flow through it, undesirable overpressures can happen as well in the system. In this section, a review of transient events caused by centrifugal pumps is presented. The sequence of events during a power failure is described as well as the four pump operation modes. Finally, is presented the mathematical means used to describe a centrifugal pump for different modes of operation. 25

60 2.5.2 Sequence of events after the power failure After the abnormal pump stoppage and considering that the pump inertia is usually small compared to the inertia of the fluid inside the pipe, the pump rotational speed reduces and the total developed head decreases. Since the discharge in the pump impeller reduces, negative pressure waves propagate to downstream and positive pressure waves propagate to upstream the suction line. When the flow reverses, several cases can occur depending on the existence of surge protection devices. Two cases will be considered herein: (i) existence and (ii) non-existence of a check-valve. Usually, pumps are protected with check valves to prevent the reverse flow that could cause irreversible damages in the components of the pumping groups If the system is protected with a check-valve, when the flow reverses, the check-valve closes actuated by the fluid momentum and extremely high undesirable overpressures can occur in the system (the check-valves do not close instantaneously, allowing some reverse flow). Section 2.6 presents a detailed review regarding this thematic. If the pump is not protected with a check-valve, the flow reverses through the pump. The pump is said to be operating in the zone of energy dissipation. As the flow reverses, the pump decelerates until the rotational speed reverses too. The pump stars acting as a turbine and increases its rotational speed until runaway conditions are reached (note that the resistant torque of the electrical grid is not acting). Figure 2.8 presents the sequence of events described early. The pump operation modes are explained with higher detail in section Figure Sequence of events following the sudden pump stoppage without a check valve adapted from Mays (1999) General pump description The flow rate of a centrifugal pump, Q, depends on the rotational speed, N, and the pumping head, H (Chaudhry, 1987). The transient-speed changes caused by power failure depend on the torque, T, and the combined inertia of pump, motor and fluid inside the pump impeller. Thus, four variables namely 26

61 Q, H, N, T have to be specified for the mathematical representation of a pump. The curves that describe the relationships between these four variables are referred as pump characteristics. The homologous theory between turbomachines is used to obtain the pump characteristics. Two pumps are considered homologous if the flow pattern through them is similar and if they are geometrically similar. For homologous pumps, the following expressions are valid: H N 2 D imp 2 = Constant ; N QD imp 3 = Constant (2.40) where N = pump rotational speed (rpm); and D imp = pump impeller diameter (m). Since D imp is constant for a particular machine, Equations (2.40) may be written in the following form: H N 2 = Constant; N = Constant (2.41) Q These equations may be written in a dimensionless form by dividing all variables (Q, H, N, T) by their rated conditions: v = Q Q R ; h = H H R ; α = N N R ; β = T T R (2.42) where v, h, α and β = dimensionless parameters for flow rate, piezometric head, rotational speed and torque, respectively (-); N = rotational speed (rpm); and T = the pump torque (Nm). The subscript R refers to pump rated conditions. Equations (2.41) may be written based on Equations (2.42), as follows: h α 2 = Constant; α = Constant (2.43) v The representation of these equations might have some problems when α and v become zero and the expressions result in a division by zero. To avoid this situation, in the first equation the parameter v 2 is added to the denominator. The need to define the pump characteristic curve leads to the definition of a new variable θ S, as presented in the following equations: F h = h (α 2 + v 2 ) (2.44) θ S = tan 1 α v (2.45) where F h = dimensionless piezometric head parameter that describes pump characteristics (-); and θ S = parameter that describes pump characteristics that varies between 0 and 2 (rad). 27

62 Parameter θ S is a positive number defined between 0 and 2, for the four zones of the pump operation, as long as the signs of α and β are taken into account (see Table 2.12). The pumping head for similar pumps with equal specific speed is described by F h as a function of θ S, through Equations (2.44) and (2.45). Similarly, the torque characteristic curve is described by the following expression. F β = β (α 2 + v 2 ) (2.46) where F β = dimensionless torque parameter that describes pump characteristics (-). The pump conditions for all operating points are well defined for a given specific rotational speed: N s = N Q H 3 4 (2.47) where N s = pump specific rotational speed (expressed in SI units). In many design situations, the complete pump characteristic curves are not available from the manufacturer, so the parameters F h and F β must be approximated to others experimentally obtained for similar pumps. Curves tend to have similar shapes for the same specific rotation speeds. If such data are not available, then the curves must be extended by comparison with data for other specific speeds. This is an uncertain procedure, and results of transient studies using such data must be analysed with scepticism (Wylie and Streeter, 1993). The pump head and pump torque characteristic curves are presented in Figure 2.9 for four specific rotational speed values (namely for 25, 85, 161 and 261, expressed in SI units). These plots are knows as the Suter parameters and were firstly presented by Marchal et al., (1965). (a) (b) Figure Pump characteristics for four specific rotational speed values: (a) piezometric head; (b) torque. 28

63 2.5.4 Abnormal pump characteristics four quadrants of operation During a transient event, it is possible that the pump experiences a reversal in flow and/or rotational speed. It is always possible that the pump passes through abnormal modes of operation due to the reversal of piezometric head, or torque. An expected sequence of events was already presented in section 2.5.2, however sometimes, abnormal situations of operation occur. For the complete transient event characterization without loss of information, the characteristics of the pump must be available for all the operation modes. Four quadrants of pump operation can be defined, depending on the sign of the dimensionless flow and rotational speed. The four quadrants are presented below. All the conditions are schematically shown in Figure In this figure, piezometric heads on either side of the pump are depicted by levels of reservoirs. The four quadrants of operation are presented in Table Figure 2.10 Zones and quadrants of pump operation - adapted from Ramos et al., (2005) Table 2.12 Zones of pump operation Quadrant Sign of v Sign of α Range of θ I + + 0⁰ θ < 90⁰ II ⁰ θ < 180⁰ III ⁰ θ < 270⁰ IV ⁰ θ 360⁰ Quadrant I Quadrant I represents the conditions where the turbomachine is acting as a pump. When the piezometric head, flow, rotational speed and torque are all positive, the pump is acting under normal conditions. However, when the pump is being overpowered by another turbo-machine, or the pressure head drops 29

64 fast (e.g., power failure), the pump acts as if it was dissipating energy, or a pump that was designed from a higher reservoir to a lower one (Mays, 1999). Quadrant II Quadrant II is a zone of pure dissipation of energy, since the pump is rotating in the positive direction, but the fluid is flowing upstream. This is typical when the fluid starts to decelerate but the pump is still with positive rotational speed. Quadrant III After the Quadrant II it is expected that the Quadrant III occurs. In Quadrant III, the pump acts as a turbine. The fluid is flowing from a higher to a lower level and the pump is rotating in the same direction. This mode results in zero efficiency. Both the head and the torque are positive in this region. In this zone the turbine accelerates until it reaches the runaway speed. When this occurs, the actuating torque is negative. Quadrant IV Quadrant IV is very unusual and hardly encountered in operation, with the exception of pump/turbines entering reverse rotation pumping during transient operation. One other possible cause is if machine is inadvertently rotated in the wrong direction by improper wiring of an electric motor (Mays, 1999). 2.6 Check valves Introduction An essential element in the design of pumping systems is the proper selection of the check valve. During the forward flow, check valves should remain opened and have a low headloss coefficient. Check valves should return to the closed position when the flow reverses, protecting the pumping group. The design of check valves should be based on five criteria that are presented in Table In this research work only the non-slam characteristics are reviewed, however during the pumping system design all the criteria must been taken into account. Moreover, only the check valves whose movement is only influenced by the motion of the fluid, weight or springs are reviewed in the scope of this research. Check valves actuated, for instance by motors or compressed air systems are outside of the scope of this research work. 30

65 Table 2.13 Selection criteria for check valve (Val-Matic, 2011) Selection criteria Initial costs Maintenance costs Headloss and energy costs Non-slam characteristics Fluid compatibility Sealing ability Significance Valve purchase costs can vary widely and should also include installation costs. The more complex the valve, the higher the maintenance costs. Some valves can create local headlosses that, during their lifetime, can represent higher energy cost, than the valve initial cost. It is essential to match the closing characteristic of the valve with the dynamics of the pumping system. Only certain check valves can tolerate sediment and solids in the flow. Some applications require drop tight sealing. Check valves are sometimes selected without proper thought of their response under transient flow conditions. In reality, check valve slam is caused by valves that are not matched to the system of which they are an integral part (Thorley, 1991). Usually, a low level of engineering effort is carried out in the design of this element, allowing the occurrence of check valve slam causing noise, vibration and significant overpressures. The major damage caused by check valves is related with the delayed closure, allowing a reverse velocity. When it closes, the fluid stops quasi-instantaneously creating undesirable overpressures. This overpressure is proportional to the reverse velocity Equation (2.1). Accordingly, for the same type of check valve, the higher the deceleration of the fluid, the higher the overpressure. The same check-valve has different behaviours in systems with different characteristics and the same system has different behaviours with different check valves. Figure 2.11 shows two responses of different check valves in the same system (Thorley, 1991). Pipe systems that are more susceptible to check valve slam are those where a high head continues to exist after the sudden pump stoppage. The higher the head difference, the higher the fluid deceleration. Therefore, pumping systems protected with air vessels at downstream the check valves and parallel pumping groups where only one pump trips-out out of two or more are vulnerable to this phenomenon. Conditions worsen if those systems have a high head or an initial vertical lift followed by a long horizontal pipeline (Thorley, 1991). 31

66 (a) (b) Figure 2.11 Overpressures caused by two different check valves (Thorley, 1983): (a) swing and (b) nozzle check valve Dynamics of check valves Previous studies at Delft Labs concluded that the valve geometry affected the magnitude of pressure surges and reverse velocities. The conclusions were (Rahmeyer, 1996): (i) Reverse velocities and pressure surges are greater for valves with a larger mass of valve components. (ii) Reverse velocities are greater for valves with larger strokes or travel of components to close. (iii) Reverse velocities are less for valves that were spring assisted to close. These conclusions are justified because of the increased time necessary to accelerate and overcome the inertia of valve internals and the distance they must travel. There are several types of check valves available in the market. Figure 2.12 presents four of those types. Ball valves (Figure 2.12a) are simple and compact and are composed of a flanged body that guides a ball in and out as the flow goes forward, or reverse, respectively. The ball must travel a considerable distance to isolate the pumping group. Swing check valves (Figure 2.12b) consist of a swing that stokes the flow as it reverses. Similarly to the latter, the swing takes a considerable time to close. To reduce the travel time, the swing was replaced by a central pin where two opposing discs rotate split disc check valve (Figure 2.12c). Nozzle check valves (Figure 2.12d) have only one moving part with little mass and a short stroke. This valve allows small reverse flows. 32

67 (a) (b) (c) (d) Figure 2.12 Check valves types: (a) ball valve; (b) swing valve; (c) split disc; (d) nozzle valve. The velocity with which the check valve responds, for a given flow deceleration is represented by the Dynamic Performance Characteristics (Thorley, 1991). These are curves that relate, for each type of valve, the liquid column deceleration with the maximum reverse velocity (Figure 2.13). These curves can be plotted for a specific valve, or in a non-dimensional form. The non-dimensional maximum reverse velocity and deceleration are obtained by Equations (2.48) and (2.49), respectively: ( dv dt ) v r = v r v 0 (2.48) = dv dt D v (2.49) 2 0 where v r = non-dimensional reverse velocity (m/s); v r =maximum reverse velocity of the fluid (m/s); v 0 = steady-state velocity of the fluid (m/s); ( dv deceleration (m/s 2 ). dt ) = dimensionless fluid deceleration (-); and dv dt = fluid 33

68 (a) (b) Figure 2.13 Dynamic Performance Characteristics of check-valves: (a) dimensional and (b) dimensionless curves Thorley (1991) Avoiding check valve slam design criteria The design of check valves should be based on the technical information provided by the manufacturers. Most common technical information provided is the valve Dynamic Performance Characteristics. This one provides only maximum reversal velocity for different flow decelerations. The design of check valves are not based on finding of the fastest closure time valve, but selecting a check valve whose non-slamming characteristics are matched with the system. Thorley (1991) and Provoost (1983) came up with the following design criteria: (i) Estimate the permissible level to which the pressure can be allowed to rise without unacceptable conditions developing in the system. (ii) Based on Frizzel-Joukowsky s expression, calculate the corresponding maximum reverse velocity, v r and convert to the dimensionless form, as in Equation (2.48). (iii) By using a computer model of the system, take the average deceleration of the liquid column downstream of the pump from the results of a transient analysis and convert it to a dimensionless deceleration according to Equation (2.49). (iv) Based on the Dynamic Performance Characteristics, plot lines for these two values as indicated in Figure Valves with curves below the intercepting point are acceptable in the context of minimizing the risk of the check valve slam. Note that the upper proceeding could be carried out with the real deceleration and maximum reverse velocity, instead of by using the dimensionless parameters, depending on the data supplied by the manufacturer. 34

69 Figure 2.14 Check valve selection adapted from (Val-Matic, 2003) 2.7 Surge protection devices Introduction Often, maximum pressure variations caused by a transient event are higher than maximum design pressures, compromising the safety of the system. There are several techniques for controlling transients. Some include design criteria (e.g., increase diameter, reduce velocity, change the pipeline profile and longer valve manoeuvres), while others include the design of surge protection devices. The selection of the best solution for the surge protection devices depends upon the hydraulic characteristics and topography that can influence the pipe system profile of the hydraulic circuit. It is also important the simulation of the integrated system in order to know the best and correct dynamic response and the influence of different devices always kept up with an economical comparison study (Ramos, "Guideline for design of small powerplants", 2000). For reasons of protection efficiency, these devices are placed near the system component that creates the transient event. However, this scenario is not always possible, leading to complex schemes of surge protection. In the following sub-sections some typical surge protection devices used in rising or gravity mains are present, namely: (i) surge tanks; (ii) one-way surge tanks; (iii) hydropneumatic vessel; (iv) flywheel; (v) by-pass lines; and (vi) valves. A brief description of the devices is presented, followed by the main effects in the surge attenuation Surge tank This device is connected to the pipeline and acts as a balancing tank for the flow variation that may occur in the pipeline. The top of the tank is exposed to the atmospheric pressure. Its function is to accumulate the excessive energy provided from the transient event by reflecting totally or partially the pressure waves. As the pressure raises the tank withdraw water from the pipeline and feeds the same 35

70 in the case of pressure drop. The height of the tank must at least be equal to maximum head expected to occur at the tank site in normal operating conditions. Figure 2.15 presents two examples of these structures and the typical layouts used in gravity and rising main. (a) (b) (c) (d) Figure 2.15 Surge tanks: (a) Ryushima power station; (b) Kendoon Power Station; (c) typical location in rising mains; (d) typical location in gravity mains. These devices are limited by the topographic conditions and are generally constructions with significant height, creating, generally, significant environmental problems. This solution requires low maintenance costs, as the operation does not require mechanical or electric components. Precautions must be taken in cold weather environments. The mathematical modelling of these devices is performed considering the continuity and headloss equations in the tank-pipe connection Equations (2.50) and (2.51), and the continuity equation inside the tank Equation (2.52) -, in addition to the positive and negative characteristics equations Equations (2.27) and (2.28). In these equations, the inflow in the surge tank is considered positive: Q UST = Q ST + Q DST (2.50) S ST dz ST dt = Q ST (2.51) H U,DST = Z ST K ST Q ST Q ST (2.52) where Q UST = flow rate upstream the surge tank (m 3 /s); Q ST = flow in the surge tank (m 3 /s); Q DST = flow rate downstream the surge tank (m 3 /s); S ST = area of the surge tank cross-section (m 2 ); Z ST = water level in the surge tank (m); H U,DST = piezometric head upstream/downstream the surge tank (m); and K ST = local headloss coefficient in surge tank connection (-). 36

71 Common design parameters are the height of the tank, the diameter/cross-section and the characteristics of the connection (i.e., diameter, headlosses) One-way surge tanks One-way surge tanks consists of tanks exposed to the atmospheric pressure equipped with a check valve (Figure 2.16a) that only feeds the pipeline when the pressure drops below the water level in the tank. Its function is to prevent the fluid reaching the vapour pressure and potential water-column separation by feeding the pipeline with water after the occurrence of a pressure drop. They are usually placed in the sections with potential occurrence of vapour pressure (e.g., high points of the pipe profile), as in Figure 2.16b. (a) (b) Figure 2.16 Feed tanks: (a) scheme with check-valve; (b) typical location in rising mains. The main disadvantages are the small efficiency in controlling the overpressures (only control indirectly since they limit the underpressures) and the need of a device to refill the tank (e.g., by-pass to the main pipeline). The mathematical modelling of this tank is performed by the same equations presented for the surge tank, as long as the pressure is below the water level in the feed tank. When it is not, Equations (2.27) and (2.28) are used to determine the pressure-flow conditions in the tank section. The usual design parameters are the volume, the characteristics of the connection (i.e., diameter, headlosses) and the refill solution Hydropneumatic vessel The hydropneumatic vessel (or air vessel) can be used when the topographic conditions are not favourable to the use a surge tank. The air vessel have the same working behaviour of the surge tank, however to reduce the dimensions, this solution uses compressed air to accommodate the pressure variations induced by the transient events. The pressure in the vessel decreases gradually when it feeds the pipeline during an underpressure, and increases during an overpressure. Figure 2.17 depicts two air vessel solutions and the typical locations where the devices are placed. The devices are usually 37

72 placed near the system element that creates the disturbance (e.g., pumping groups or turbine) - Figure 2.17c,d. (a) (b) (c) (d) Figure 2.17 Hydropneumatic vessel: (a) and (b) photos of air vessels, (c) typical location in rising mains (d) typical location in gravity mains. The main advantages of these devices are the efficient control of underpressures and overpressures and the small space required for their placement. The main disadvantages are related with the initial and maintenance costs and the frequent use of a compressor to supply the vessel with air. The mathematical modelling of the air vessel is carried out by the same equations of the surge tank, with an additional equation of the polytropic gas equations: H Pair V m Pair = C 0 (2.53) where H Pair = absolute pressure head at the end of the time-step (N/m 2 ); V Pair = volume of enclosed air inside the vessel at the end of the time-step (m 3 ); m = exponent of polytropic gas equation (-); and C 0 = constant of the polytropic gas equation, given by the initial conditions. The values of m are equal to 1.0 or 1.4 for an isothermal and for an adiabatic expansion of the air entrapped, respectively. For design purposes, an average value of 1.2 may be considered. The usual design parameters are the initial and maximum volume of air, the diameter, height and total volume of the vessel, and the characteristics of the connection Flywheel Flywheels are rotating masses of iron/steel that are installed in the rotation axis of turbo-machines. In pumps they are installed between the wheel and the motor and in turbines between the wheel and the generator. With the adding the mass, the turbo-machines have slower stoppage times, that can reduce the pressure variations. 38

73 (a) (b) Figure 2.18 Flywheels: (a) pumping group, (b) turbine. The main advantages of these solutions are the increasing of stoppage time and the low maintenance costs. The main disadvantages are the efficiency only for small pipelines, the increase of power required for turbo-machines to start-up and the reduction of the motor/generator lifetime. The mathematical modelling of this element can be achieved by the increase of inertia in the pumping/turbo-generator group. The design parameter is the flywheel mass By-pass lines This solution consists of the assembly of a by-pass line with a check valve in the pumping groups, between the suction and the discharge line. As the pressure drops below the reservoir level, the by-pass feds the main line with water, reducing the underpressures. The efficiency of the water supply through the by-pass depends on the friction losses (the higher the friction losses the lower the efficiency). For this reason this solution is mostly used in shorter low-head systems against down surges when the reservoir level is above the pumping station (Figure 2.19a). Wylie and Streeter (1993) presents two solutions for a by-pass line: the first considers a control valve on the discharge line and a check-valve in the by-pass between the suction and the main line (Figure 2.19b). As the pressure drops below the reservoir level, the check valve opens, allowing the water supply to the main line (acts as a feed tank). The second configuration reverses the valve locations (Figure 2.19c) and the control valve in the by-pass line opens, as the pump trips-off allowing the water to flow towards the discharge line. Care must be taken during the control valve closure after the sudden pump stoppage. 39

74 (a) (b) (c) Figure 2.19 By-pass lines: (a) general layout in a rising main; (b) check valve in the by-pass; (c) control valve in the by-pass. Another typical application of the configuration presented in Figure 2.19b is in the in-line booster pump. If this pump suddenly stops, it represents an obstruction to the normal flow. A positive pressure builds up on the suction side, whilst a negative pressure change develops downstream (Thorley, 1991). If the pressure upstream the pump is higher than the downstream one, the check valve opens and reliefs the pressure upstream, helping to minimize the pressure drop downstream the pump. Main advantages of this protection scheme are related with the economy and simplicity of the solution. However precautions must be taken with the type of the check and the control valves used in the solution. After some years of use the valves may accumulate rust and compromise the efficiency of the solution Valves Several types of valves can be used as surge protectors. The choice of the valve type depends on a several factors, for instance the pipeline profile and the phenomenon that triggers the transient event (downsurge/upsurge event). A number of commonly used valves used to control transients are (Chaudhry, 1987): (i) Safety and pressure-relief valves (); (ii) Pressure regulating valves; (iii) Air-inlet valves. 40

75 (a) (b) (c) Figure 2.20 Valves used to prevent transient events: (a) safety/pressure-relief valve; (b) pressureregulating valve; (c) air inlet-valve. Safety and pressure-relief valves are a spring or weight-loaded valve that opens as the pressure reaches a given value, withdrawing water from the main pipe and consequently reducing the overpressure. The difference between these two types is related with the opening-closure process. As the first type quasi-instantaneously opens or closes as the pressure in the pipe is higher or lower than a given pressure limit, respectively, the latter valve has an opening proportional to the pressure rise in the main pipe. These valves represent simple and economical solutions, however they are only suitable for controlling overpressures. They require constant maintenance and must not be used in waters with significant solids concentration. Pressure-regulating valves are pilot-controlled valves which are opened or closed by servomotors. These are usually placed downstream pumping-stations or upstream turbo-generator groups. The opening or closing manoeuvres are calibrated to reduce the pressure variations. For instance, following a power failure of the grid that supplies a pumping station, the valve is rapidly opened and then gradually closes to reduce the pressure rise (Chaudhry, 1987). Air inlet valves are used to admit air inside the pipeline to prevent the occurrence of vapour pressures. During the occurrence of a pressure drop, the valve supplies air into the pipeline and the pressure differential between the outside atmospheric pressure and the pressure inside the pipe is reduced, and thus preventing the collapse of the air pockets (Chaudhry, 1987). The increase of the air concentration in the water also reduces the pressure wave speed. After the transient event the air must be expelled of the pipeline preventing the occurrence of high upsurges caused by another transient event. For these reason these valves are usually provided of mechanisms to introduce and expel air from the main pipelines. These valves require constant maintenance and the volume of air admitted into the pipeline is hardly controlled. Thus, it could result in a more severe transient. Accordingly, these valves are not recommended as unique surge protection for waterhammer. 41

76 2.8 Summary, main gaps in literature and thesis motivation A state-of-the-art regarding the hydraulic transient flow in pressurized pipes was presented. This included the presentation of theoretical requirements for the development of a solid mathematical model that describes the 1-D transient-flow in closed conduits in the space-time domain, as well as the identification of the main gaps in this thematic. The classical theory is sufficient to accurately predict the maximum and minimum pressure variations in the pipe systems, offering good results for design purposes. However, this theory is not accurate enough to describe the transient pressure conditions observed in real systems for diagnostic and assessment purposes. The differences between experimental data obtained in real systems and results of a classical theory are due to improperly described boundary conditions or unconventional dynamic phenomena. Most of these phenomena were presented in section 2.4. The main aspects that lead to the development of this research work were the pump dynamic behaviour, the characterization of check-valves behaviour during a transient event and the interaction between them and a hydropneumatic vessel. The lack of information regarding pump characteristics during a transient event and the lack of information for simulating different check valves can compromise the accuracy of transient analysis during the design stage. This thesis aims to clarify the effects of transient events in raising mains with approximately horizontal profiles, where the introducing of a surge protection device, such as a hydropneumatic vessel, could create higher pressure variations, caused by the increased fluid deceleration. 42

77 3 HYDRAULIC TRANSIENT SOLVER 3.1 Introduction This chapter focuses on the description of the main principles, equations and numerical methods used for the implementation of a hydraulic transient solver in the scope of the current research. The one-dimensional differential equations of continuity and motion of the fluids in pressurized pipes are described as well as the Method of Characteristics. The latter method is the most widely used numerical method to obtain the solution of pressure and flow rate conditions in the pipe space-time domain due to its easiness dealing with multi-pipes and complex boundary conditions and due to not having convergence and damping problems, as long as the Courant number is kept close to unity. Simple and complex boundary conditions that represent most of the pumping systems are described namely the upstream and downstream constant-head reservoirs, pump with head-discharge curve known, pump sudden stoppage solution by means of Suter parameters, the hydropneumatic vessel, behaviour of check valves, and valves discharging to reservoirs or atmosphere, controlled by an opening or closure manoeuvre. The model incorporates as well the Vítkovský et al., (2000) formulation to describe the unsteady component of friction losses during transient flow. In the last section of this chapter the hydraulic transient solver developed will be tested with a semi-real case-study and the results will be compared with the ones obtained from a validated simulator. The objective is to assess if the numerical model developed is capable of reproducing accurately the transient-state conditions induced by a sudden pump stoppage. 3.2 Method of Characteristics Characteristic equations Unsteady-state pressurized pipe flow in closed conduits is described by a set of two differential equations, based on mass and momentum conservation principles, independently of initial and boundary conditions (Chaudhry, 1987; Almeida and Koelle, 1992; Wylie and Streeter, 1993). The partial differential equations (2.6) and (2.7) may be rearranged: L 1 = a 2 Q H + gs x t = 0 (3.1) L 2 = Q H + gs t x + gs h f s = 0 (3.2) and consider a linear combination between them obtained, L = L 2 + λl 1 : L = ( Q t + a2 λ Q ) + λgs ( H x t + 1 H λ x ) + gs h f s = 0 (3.3) 43

78 the first and the second terms of Equation (3.3) represent the total derivative - Equation (2.5) - of the flow rate and of the piezometric head, respectively, as long as the following condition is fulfilled. u = dx dt = a2 λ = 1 λ λ = ± 1 a (3.4) Introducing Equation (3.4) into Equation (3.3), and considering Equation (2.5), the following total differential equations may be written as follows: C + : dq dt + gs dh a dt + gsh f + s = 0 C : dq dt gs dh a dt + gsh f s = 0 if dx = a (3.5) dt if dx = a (3.6) dt two real values of λ have been used to convert the original partial differential equations into a set of total differential equations - Equations (3.5) and (3.6) - each with the restriction of being only valid when the respective domain is respected (Wylie and Streeter, 1993). Equations (3.5) and (3.6) represent two straight independent lines that propagate flow and piezometric head information in the space-time domain (Figure 3.1a). The P subscript is relative to the section where the calculations are being made, and subscripts A and B are sections at the previous time-step, on the left and right of the calculation section, respectively. Denominations used herein are relative to the Figure 3.1b. (a) (b) Figure 3.1 Method of Characteristics grid: (a) straight characteristic lines; (b) location of calculation sections The following finite-difference schemes are used to solve the total differential Equations (3.5) and (3.6): P C + : dq A + gs P a dh A P + gs h + fs dt A = 0 (3.7) 44

79 P C : dq B gs P a dh B P + gs h fs dt B = 0 (3.8) Considering that Equations (3.9) and (3.10) are valid along the positive and negative characteristics lines, the integrals can be calculated by finite differences: P C + : dq A P C : dq B P = Q P Q A ; dh A P = Q P Q B ; dh B accordingly, the following numerical schemes are obtained: = H P H A (3.9) = H P H B (3.10) C + (H P H A ) + a P gs (Q P Q A ) + a h + fs dt A C (H P H B ) a P gs (Q P Q B ) a h fs dt B = 0 (3.11) = 0 (3.12) The solution of the previous equations requires a detailed analysis regarding the last term. The numerical schemes used to solve them are presented in the following section Friction resistance Introduction Classic waterhammer theory and corresponding transient solvers only consider energy losses due to steady-state friction. However this assumption is only close to reality in presence of slow transients. For fast transients, classical models cannot accurately describe the physical phenomena observed in real life systems. During an unsteady-state flow, the total friction resistance (i.e., the slope of the energy line) can be divided in two components: h f = h fs + h fu (3.13) where h f = total friction losses (m/m). While the steady-state component can be calculated by resistance formulas, for instance Hazen-Williams or Colebrook-White for turbulent flows and Hagen-Poiseuille for laminar flow, the unsteady-state term requires the use of different formulations (see ). 45

80 Steady-state friction The last term of Equations (3.11) and (3.12) cannot be straightforwardly calculated by using a finite different scheme as the h fs varies with the flow rate along the characteristic lines C + and C. Different approaches can be followed and the respective numerical schemes are presented in Table 3.1, only valid for rough turbulent flows (n = 2). Classical theory considers the velocity profile unidirectional during the transient event. The energy losses are caused by shear stress, which is proportional to the velocity gradient near the pipe-wall. The first approach (Equation 3.16) considers a frictionless flow which corresponds to the perfect-fluid solution. The second approach (steady-state approximation) considers the friction constant and equal to the initial steady-state conditions. The last approach (quasi-steady-state approximation) describes friction as a function of the Reynolds number that varies during the transient event; different approximations can be used to calculate the second order term, Q 2. The K subscript in Table 3.1 refers to the sections of the previous time-step, on the left (A) or on the right (B) of the calculation section, depending on the calculation being carried out along the positive or negative characteristic line, respectively. Table 3.2 presents some considerations regarding the approximations depicted in Table 3.1. In the mathematical model, Equations (3.14), (3.16) and (3.19) were used to describe the steady-state friction. Equation (3.19) was chosen to describe a second-order approximation, since it allows the rearrangement of terms, leading to a simple set of linear algebraic equations (see 3.2.3). 46

81 Table 3.1 Numerical schemes used to integrate the steady-state friction term for rough turbulent flows adapted from Covas (2012) Type of Approach Numerical scheme P Frictionless I K = h fs dt = 0 K Steady-state approach I K = h fs dt = a 2gDS 2 f s (Q i )Q i Q i t K P (3.14) (3.15) Quasi-steady-state approach (1 st order numerical scheme) Quasi-steady-state approach (2 nd order numerical scheme) P I K = h fs dt = K a 2gDS 2 f s (Q K )Q K Q K t P I K = h fs dt = a 2gDS 2 f s (Q K ) Q K Q K + Q P Q P t 2 K P I K = h fs dt = K P I K = h fs dt = K a 2gDS 2 f s (Q K ) Q K + Q P Q K + Q P t 2 2 a 2gDS 2 f s (Q K )Q P Q K t (3.16) (3.17) (3.18) (3.19) Note: t = time-step between calculations (s). Table Considerations regarding the steady-state friction term integration Type of Approach Frictionless Steady-state approach Quasi-steady-state approach (1 st order numerical scheme) Quasi-steady-state approach (2 nd order numerical scheme) Considerations This approach considers a perfect-fluid. Results do not reproduce reality. The flow rate and the corresponding friction factor term (f s ) is considered constant during all the transient event, and equal to the initial steady-state value. This approach is simple, stable but less exact, despite leading to better results than the precious one. The flow is assumed constant and equal to one of the previous time-step. The scheme tends to be unstable for higher values of friction resistance. If the space step x is not small enough, the approximation may not be satisfactory (and not advisable for long conduits) - Chaudhry (1987). The friction resistance depends on the flow-rate of both the current and the previous time steps. Implicit schemes are unconditionally stable, which is extremely important when interpolations are carried out or when other derivative based terms (e.g., unsteady friction or retarded strain variation) are added to the main equations (Covas, 2003). 47

82 During the transient event, in each space and time-step, the steady-state friction is calculated as if the flow behaves like a sequence of successive steady-states. For slow transients the results lead to satisfactory results. However, in the presence of fast transients and high pulsating frequencies, steady-state and quasi-steady-state leads to unsatisfactory results and do not describe the pressure wave damping and dispersion (Covas, 2003) Therefore, a different formulation used to calculate the unsteady-state friction must be considered. A new additional term is introduced in the dynamic equation that considers the effects of inertia and friction resulting from the velocity profile inversion in time (Carriço, 2008) Unsteady-state friction Vítkovský et al., (2000) formula Several formulations have been presented (see section 2.4.2), being the one implemented in the transient solver described herein. Vítkovský et al., (2000) developed a formulation for the unsteady-state friction factor based on the convective and the local accelerations, considering that these terms were directly related to the effects of local inertia and unsteady wall shear stress. This is an improvement of the formulation proposed by Brunone et al., (1991) and given by the following expression: h fu = k 3 gs ( Q Q + a t Q Q ) (3.20) x where k 3 = decay coefficient of Vítkovský et al., (2000) formulation (-). Parameter k 3 has the disadvantage of being an empirical coefficient, which can only be obtained by comparing the numerical results and collected transient pressure data. Higher values of the k 3 coefficient lead to faster surge damping and higher phase shifting, as shown in Figure 3.2, for a hypothetical example of a transient event described by a sudden pump stoppage. Figure 3.2 Effect of the decay coefficient value in the surge damping for a hypothetical example 48

83 This formulation requires a finite-difference scheme to solve local and convective accelerations. Covas (2003) presented a first order explicit scheme used to calculate the flow-space derivative and a first order implicit scheme for the flow-time derivative, defined, for each characteristic line C + and C. The convective acceleration can be calculated by the following equations: C + : Q x = Q P Q A x C : Q x = Q P Q B x (3.21) (3.22) The local acceleration can be calculated by the following equations: C + : Q t = θ Q P Q P + (1 θ) Q A Q A t t C : Q t = θ Q P Q P + (1 θ) Q B Q B t t (3.23) (3.24) where θ = relaxation coefficient (-). If θ =0, the time derivative becomes explicit and less stable under certain combinations of space and time steps; if θ >0, the numerical scheme is implicit and unconditionally stable. To minimize computer storage and increase calculation speed, θ was set equal to1, Covas (2003). The location of calculation sections used in the previous numerical schemes is presented in Figure 3.3. The superscript represent the location of a given point, at the previous time-step. Figure 3.3 Location of calculation points for Vítkovský et al., (2000) numerical scheme The sign of flow parameter in the Equation (3.20) was assumed equal to the previous time step, as shown in the following equations: 49

84 C + : Q Q = Q A Q A (3.25) C : Q Q = Q B Q B (3.26) Compatibility equations The characteristic equations can be rewritten in a simplified numerical form as a sum of algebraic terms. The compatibility equations forms are presented below: C + : Q P = C P C ap H P (3.27) C : Q P = C N + C an H P (3.28) where C P, C a +, C N and C a = constants defined for each pipe section and time (-), applied in location P. These coefficients depend on the numerical scheme used to describe the steady-state friction and the unsteady-state friction model adopted. These coefficients are defined by the following expressions: C P = Q A + C a H A + C P1 + C P1 (3.29) 1 + C P2 + C P2 C N = Q B C a H B + C N1 + C N1 (3.30) 1 + C N2 + C N2 C ap = C a 1 + C P2 + C (3.31) P2 C an = C a 1 + C N2 + C (3.32) N2 where C a = parameter that depends upon pipe properties (m 2 /s). Coefficients superscripts and refer to steady-state friction and to unsteady-state friction, respectively. The numerical solutions for each coefficient are presented in Table 3.3. Parameter C a is defined by the following expression: C a = gs a (3.33) Equations (3.27) and (3.28) describe the transient flow conditions in the pipe interior sections. A set of boundary conditions must be established to define the entire transient state along the hydraulic circuit. The boundary conditions used in the mathematical model are presented in the following section. 50

85 Table 3.3 Coefficients CP1, CP2, CN1, CN2 (adapted from Covas, 2003 and Soares et al., 2008) Steady-state friction ( ) Frictionless C P1 = C P2 = 0 C N1 = C N2 = 0 1st order numerical scheme 2nd order numerical scheme C P1 = R A t Q A Q A C N1 = R B t Q B Q B C P1 = 0 C N1 = 0 C P2 = 0 C N2 = 0 C P2 = R A t Q A C N2 = R B t Q B Unsteady-state friction ( ) No unsteady-state friction Vítkovský et al., (2000) formulation C P1 = C P2 = 0 C N1 = C N2 = 0 Q A C P1 = k 3 θq P k 3 (1 θ)(q A Q A ) k 3 Q Q P Q A A Q B C N1 = k 3 θq P k 3 (1 θ)(q B Q B ) k 3 Q Q P Q B B C P2 = C N2 = k 3 θ Note: R A = f sa 2DS ;and R B = f sb 2DS where f s A and f = Darcy-Weisbach friction factor calculated on section A and B, respectively. sb 3.3 Boundary conditions Introduction Boundary conditions are specific equations that represent the hydraulic frontiers. At upstream boundary, only the negative characteristic equation - Equation (3.28) - is valid, whereas at the downstream end only the positive one is - Equation (3.27) -, as shown in Figure 3.4. Thus, another equation or set of equations are necessary at the boundaries to solve the transient-state in this location. The terms presented in Table 3.3 must also be changed according to the flow conditions in the boundaries. Figure 3.4 Equations valid in boundary conditions 51

86 Simple boundaries are developed in this section, whereas complex boundary conditions, such as the sudden pump stoppage, are derived in further sections due to the need of a more detailed analysis Upstream boundary conditions Constant-head reservoir The water level in a reservoir may be assumed constant during the transient-state event when the entrance headlosses and velocity are negligible (Chaudhry,1987). These hypotheses are valid for large tanks, during short-duration transients (Figure 3.5). The additional boundary is: H P = H resu (3.34) where H resu = upstream reservoir water level above the reference datum (m). Figure 3.5 Constant-head upstream reservoir Pump with head-discharge curve known A centrifugal pump operating at constant rotational speed has a head-discharge curve known. Thus, the piezometric head at downstream the pump is given by the following expression: H P = H resu +H 0 + b 1 Q P + b 2 Q P 2 (3.35) where H 0 = pump shutoff head (m); and b 1 and b 2 = constant coefficients of head-discharge curve (-), normally negative. Solving simultaneously Equations (3.28) and (3.35) and eliminating H P, from both equations, it is possible to obtain a quadratic expression whose solution is obtained by the equation below. Q P = (b 1 1 ) ± (b C ) 4b an C 2 (H resu + H 0 + C N ) an C an (3.36) 2b 2 To obtain a positive flow, the negative root must be selected, since the constant b 2 is usually negative. The piezometric head in the boundary may now be obtained by Equation (3.28). 52

87 Sudden pump stoppage The sudden pump stoppage is a complex boundary condition that requires a detailed analysis. This subject will be treated on a specific section of this chapter (see section 3.4) Downstream boundary conditions Constant-head reservoir As in upstream constant-head reservoir, the water level may be assumed constant. The additional boundary condition is given by: H P = H resd (3.37) H resd = downstream reservoir ester level above the reference datum (m) Valve discharging to a constant-head reservoir Considering a valve discharging in to a constant-head reservoir, the following expression may be written according to Figure 3.6: H 1 H 2 = K V 2gS V 2 Q P Q P = H P H resd (3.38) where K V = local headloss coefficient of the valve, that depends on the type of valve and opening degree (-); and S V = area of the valve cross-section (m 2 ). Figure Valve discharging to constant-head reservoir Figure 3.7 presents the local headloss coefficients as a function of valve opening for different types of valves. These curves have been included in the current transient solver. 53

88 Figure 3.7 Local headloss coefficient laws for different types of valves Substituting Equation (3.38) in Equation (3.27), it yields: K V 2 2gS Q P Q P + 1 Q V C P + H resd C P = 0 (3.39) ap C ap This expression can be approached to a special quadratic equation of the following type: aq P Q P + bq P + c = 0 (3.40) whose coefficients are given by: a = K V 2 ; b = 1 ; c = H 2gS V C resd C P (3.41) ap C ap the solution for the previous Equation (3.40) is given by Equation (3.42). 2c Q P = b + b 2 + 4a c (3.42) After the flow rate being calculated, the piezometric head is given by Equation (3.27) Valve discharging to the atmosphere Similarly to the valve discharging to a constant-head reservoir, the following expression defines this boundary condition (Figure 3.8): K V Q 2 P H 1 H 2 = 2 2gS Q P Q P = H P (Z V + 2 V 2gS ) (3.43) V where Z V = valve elevation above the reference datum (m). 54

89 Figure 3.8 Valve discharging to atmosphere Substituting Equation (3.43) in Equation (3.27), the following equation is obtained: ( 1 + K V 2gS V 2 ) Q P Q P + 1 C ap Q P + Z V C P C ap = 0 (3.44) Similarly to Equation (3.40), the solution of this equation is given by Equation (3.42) considering the following coefficients: a = 1+K V 2gS V 2 ; b = 1 C ap ; c = Z V C P C ap (3.45) 3.4 Sudden pump stoppage Introduction The most severe hydraulic transient, generated by emergency pump operations is the sudden power failure of the electrical grid. In this section only physical and mathematical aspects regarding the pump sudden stoppage are presented, whereas the qualitative aspects have been described in section 2.5. The mathematical representation of a single pump and the boundary conditions imposed by the sudden pump stoppage are presented. Finally, the numerical solution for a single sudden pump stoppage as well as for a parallel pumping group, with a short suction line between the reservoir and the pump, are described Pump stoppage due to power failure The equations derived in this section will be applied to a single pump connected to a constant-head reservoir with a small suction line with negligible headlosses. Boundary conditions depend on the conditions imposed by the pump, and a differential equation that defines the pump speed variation after the power failure. As the pump is located in the upstream boundary of the pipe, the negative characteristic equation Equation (3.28) - is applied. Notations used herein are depicted in Figure

90 Figure 3.9 Notation regarding pump boundary conditions Continuity equation and conditions imposed by the pump Since the suction line is relatively small with no storage, the flow rate through the pump is equal to the flow rate at the first section of the pipe (describing the continuity equation): Q P = Q pump (3.46) where Q pump = pump flow-rate (m 3 /s). Assuming that calculations are at a given time-step, the values at the beginning of this one, v i, h i, α i and β i are known, and the objective is to determine this variables value, v f, h f, α f and β f, at the end of the time-step. To achieve this purpose, the equations of a segment of the pump characteristics corresponding to v f and α f must be determined. Since these variables are initially unknown, an estimate of them is required and it is determined by extrapolation from the know values of the previous time-step. This extrapolation may be achieved by the following expressions: v e = v i + v i 1 (3.47) α e = α i + α i 1 (3.48) where v e and α e = estimated values at the end of the time-step, v i 1 and α i 1 = variation of these values during the previous time-step. To initialize the procedure at each time step, it is assumed that the variation in the current time-step is similar to the one of the previous time-step. Considering that the pump conditions are given by θ S = tan 1 α e v e, that t is sufficiently small and that pump rotational speed and flow vary gradually, a linear interpolation between the points on either side of θ S is acceptable (Figure 3.10). Assuming that the points corresponding to v f, h f, α f and β f are in these straight lines, then Equations (3.49) and (3.50) are valid. h f (α f2 + v f2 ) = a 1 + a 2 tan 1 α f v f (3.49) 56

91 β f (α f 2 + v f 2 ) = a 3 + a 4 tan 1 α f v f (3.50) where a 1, a 2, a 3 and a 4 = constants that define the straight lines representing the head and torque characteristics. Figure 3.10 Linear interpolation between grid points (Chaudhry, 1987) Having the pump conditions defined, it is necessary to describe the hydraulic conditions at the pump location, where the following equation may be written, for the piezometric head: H P = H resu + H pump H valve (3.51) where H pump = pumping head in the end of each time-step (m); and H valve = local headloss in the check valve (m). The local headloss in the check valve can be evaluated by the following expression: H valve = K V 2gS 2 Q P Q P = C V Q P Q P (3.52) where C V = headloss coefficient in the valve (s 2 /m 5 ). This local headloss coefficient depends upon the valve opening and type of check valve, similarly to Figure 3.7. According to Ramos et al., (2005), a conservative and economical simplification in the check valve modelling is to consider that the valve closure occurs quasi-instantaneously when the flow reverses. Despite this solution is acceptable for design purposes, it may not be rigorous enough to calibrate the numerical results to experimental data collected. For this reason, an experimental analysis on the check valve coefficients related to the inertial effects of accelerating and decelerating flow through the valve closing should be carried out. Since this is outside of the research scope, a check valve manoeuvre is calibrated for the adjustment between the numerical results and experimental data collected. 57

92 Differential equations of rotating masses Since there are no exterior forces acting on the pump after power failure, the decelerating torque is described by the following expression: T = WR 2π dn 2 60 dt (3.53) where WR 2 = combined polar moment of inertia of the pump, motor, shaft and liquid inside the pump impeller (kgm 2 ). Considering the non-dimensional parameters presented in Equation (2.42) and using an average value of β during the time-step, the upper equation may be written in a finite-difference form presented below: α f α t = 60T R β + β f 2πWR 2 N R 2 (3.54) which may be simplified by the following two equations: α f C 6 β f = α + C 6 β (3.55) C 6 = 15T R t πwr 2 N R (3.56) Solution for single pump failure To obtain the solution for the governing equations of the sudden pump stoppage, Equations (3.28), (3.46), (3.49) to (3.52) and Equation (3.55) have to be simultaneously solved. Substituting Equations (3.46), (3.51) and (3.52), into Equation (3.28) and taking into consideration Equation (2.42), the resulting equation, at the end of a given time-step may be written as follows: Q R v f = C N + C an H resu + C an H R h f C an C V Q R 2 v f v f (3.57) substituting Equation (3.49) in Equation (3.57), and Equation (3.50) in Equation (3.55), it yields: F 1 = C an H R a 1 (α f 2 + v f 2 ) + C an H R a 2 (α f 2 + v f 2 ) tan 1 α f v f Q R v f C an C V Q R 2 v f v f + C N + C an H resu = 0 (3.58) F 2 = α f C 6 a 3 (α f 2 + v f 2 ) C 6 a 3 (α f 2 + v f 2 ) tan 1 α f v f α C 6 β = 0 (3.59) previous expressions are nonlinear equations with two unknowns, α f and v f, which may be solved for instance by the Newton-Raphson method. This method firstly guesses a solution, and afterwards refines 58

93 it by using successive iterations until a given degree of accuracy is reached. The method followed herein is presented in (Chaudhry, 1987). (1) (1) Let α f and v f be the initially estimated values of the solution, which may be taken equal to α e and v e as determined for Equations (3.47) and (3.48). Then a better estimation of Equations (3.58) and (3.59) are given by: α f (2) = α f (1) + δα f (3.60) v f (2) = v f (1) + δv f (3.61) where the superscript (1) indicates estimated values before iteration and the superscript (2) indicates values after the iteration. δα f and δv f may be determined by the following expressions: δα f = F 2F 4 F 1 F 6 F 3 F 6 F 4 F 5 (3.62) δv f = F 2F 3 F 1 F 5 F 4 F 5 F 3 F 6 (3.63) in which functions F 3, F 4, F 5 and F 6 are derivative functions of F 1 and F 2, and are presented below. These derivatives are evaluated for the values of α f (1) and v f (1). F 3 = F 1 α f = C an H R (2a 1 α f + a 2 v f + 2a 2 α f tan 1 α f v f ) (3.64) F 4 = F 1 v f = C an H R (2a 1 v f a 2 α f + 2a 2 v f tan 1 α f v f ) Q R 2C an C V Q R 2 v f (3.65) F 5 = F 2 α f = 1 C 6 (2a 3 α f + a 4 v f + 2a 4 α f tan 1 α f v f ) (3.66) F 6 = F 2 v f = C 6 ( 2a 3 v f + a 4 α f 2a 4 v f tan 1 α f v f ) (3.67) (2) (2) If δα f and δv f are less than a given tolerance, ε, then α f and v f are solutions of equations (3.58) (1) (1) (2) (2) and (3.59). Otherwise, α f and v f are assumed equal to α f and v f and the above procedure is repeated. Having determined α f and v f, it is necessary to verify if the segment of the pump characteristics used in the computations corresponds to α f and v f. If it does not, then α e and v e are assumed equal to α f and v f and the above procedure is repeated (Chaudhry, 1987). Assuming that the correct values for α f and v f are determined, then h f and β f are calculated by Equations (3.49) and (3.50), respectively. Q pump and H pump are determined by Equation (2.42). Q P and H P may be determined by Equations (3.46) and (3.51), respectively. 59

94 Having determined the values of the flow rate and piezometric head in the boundary section, calculations may proceed to next time-step, assuming α = α f and β = β f. To enable a better understanding for the above procedure, the flow chart that represents this iterative process is depicted in Figure Figure 3.11 Flowchart for sudden pump stoppage boundary conditions adapted from Chaudhry, (1987) 60

95 3.4.4 Solution for the sudden failure of parallel pumps with equal characteristics In many engineering solutions, a parallel pumping group layout is used to pump more flow, to the same head. Considering that all pumps in the parallel pumping group are identical (most common situation in engineering practice) and that the pumping station is coupled to a reservoir, with negligible headloss in the suction line, than the continuity equation is represented by the following expression: Q P = n pumps Q pump (3.68) where n pumps = number of pumps in the pumping station (-). Equation (3.57) may now be written, based on Equation (3.68) as follows. n pumps Q R v f = C N + C an H resu + C an H R h f C an C V Q R 2 v f v f (3.69) Proceeding similarly as in section 3.4.3, the expression that represents F 1 will assumes the following form: F 1 = C an H R a 1 (α f 2 + v f 2 ) + C an H R a 2 (α f 2 + v f 2 ) tan 1 α f v f n pumps Q R v f C an C V Q R 2 v f v f + C N + C an H resu = 0 (3.70) since n pumps is only affecting a term with the v f parameter, only expression F 4 - Equation (3.65) - will be changed. Expressions regarding F 2, F 3, F 5 and F 6 remain unchanged. Expression (3.65) may now be written as follows: F 4 = F 1 v f = C an H R (2a 1 v f a 2 α f + 2a 2 v f tan 1 α f v f ) n pumps Q R 2C an C V Q R 2 v f (3.71) note that if n pumps assumes the value one, the equations derived in this section are equal to the ones obtained in section

96 3.5 Hydropneumatic vessel Air vessel governing equations Only the mathematical modelling of the hydropneumatic vessel is presented, as the qualitative aspects were already introduced in the section The mathematical modelling is based on the following set of equations, referring to Figure l Figure 3.12 Air vessel scheme The continuity equation at the air vessel connection, the positive and the negative characteristic equations may be written as follows: Q 1 = Q Porf + Q 2 (3.72) C + : Q 1 = C P C ap H 1 (3.73) C : Q 2 = C N + C an H 2 (3.74) where Q Porf = flow rate through the orifice at the end of the time-step (m 3 /s), and considered positive if the flow enters into the vessel. If the losses at the junction are neglected, then: H 1 = H 2 = H P (3.75) The orifice losses may be described by the following expression: ΔH Porf = C orf Q Porf Q Porf (3.76) where ΔH Porf = local headloss in the air vessel orifice (m); K orf = local headloss coefficient of the orifice (-); and S orf = area of the orifice cross-section (m 2 ). Note that the value of K orf may be different for inflow and outflow, if an orifice of differential type (as shown in Figure 3.12) or if there is a by-pass with a check valve scheme(figure 3.13). When the liquid is flowing towards the air vessel, the check valve closes, forcing the liquid to flow through the by-pass, 62

97 dissipating more energy. This solution has the advantage of dissipating faster the overpressures, reducing the air vessel volume. Figure Hydropneumatic scheme with by-pass for differential headloss The following equations may be written for the enclosed air volume: H 1 = H Pair H b + Z P + ΔH Porf (3.77) V Pair = V air S HPV (Z P Z) (3.78) Z P = Z (Q Porf + Q orf ) t A HPV (3.79) in which H b = barometric pressure head, expressed in absolute pressures (m); Z P = water level in the air vessel at the end of the time-step (m); Z = water level in the air vessel at the beginning of the time-step (m); V Pair volume of the air inside the air vessel at the end of the time-step (m 3 );V air = volume of the air inside the air vessel at the beginning of the time-step (m 3 ); S HPV = area of the hydropneumatic vessel cross-section (m 2 ); and Q orf = flow rate through the orifice at the beginning of the time-step (m 3 /s), positive if liquid enters into the vessel, and negative otherwise Numerical solution The hydropneumatic vessel is described by a set of nine equations - Equations(2.53) and (3.72) to (3.79) - with nine unknowns, respectively. In the following paragraphs, the numerical solution is presented in a simplified form. Substituting Equations (3.73) to (3.75) in to Equation (3.72) it is possible to obtain the following expression: Q Porf = (C P C N ) (C a + + C a )H P (3.80) Substituting Equations (3.75) and (3.76) to (3.78) in Equation (2.53), it yields: 63

98 K orf (H P + H b Z P 2 2gS Q P orf Q Porf ) [V air A HPV (Z P Z)] m = C 0 (3.81) orf At this point there are three equations, (3.79) to (3.81) for three different unknowns (Q Porf, H P and Z P ). Since Equation (3.81) is non-linear, an iterative process may be used to solve these equations. Two numerical methods have been used: the Simple Fixed-Point Iteration method, which uses an iterative process substituting Q Porf until a given tolerance is reached, and the Newton-Raphson method. The first one was faster and numerically more stable. The second one tended to be numerically unstable, when the pump incorporated the check valve. Figure 3.13 describes the numerical method used to solve the air vessel boundary condition by the Simple-Fixed Point Iteration method. To maintain the calculations stable, the relative error for tolerance considered is very small (ε < 10 5 ). Figure 3.14 Flowchart for air vessel numerical solution Simple-Fixed Point Iteration method 64

99 3.6 Model testing with artificial data Introduction In this section the numerical model, hereby called by JDS (João Delgado Simulator), will be tested with a semi-real case-study and the results will be compared with the ones obtained from a simulator developed and validated by Almeida and Koelle (1992), referred as AKS (Almeida and Koelle Simulator). The objective is to assess if the numerical model developed (JDS) is capable of accurately describing the steady and the transient flow-pressure conditions considering only the classical waterhammer theory (design situation). For this purpose, a semi-real reversible pumping-hydro system will be used. The system is described and obtained results by AKS and by JDS are compared and discussed. The hydraulic system has a simple configuration of reservoir pumping station pipeline reservoir. The pipeline is made of steel, with approximately a total length of m, an inner diameter of 1400 mm and 20 mm thickness. The water intake is located in a constant-head reservoir with m of water level. The pipeline discharges into a reservoir with approximately constant-head level of m. The pipeline is anchored against longitudinal movement throughout its length and the profile is characterized by a steep slope. The schematic of the pumping system topology is presented in Figure Figure Pumping system topology The raising main is composed of four pumping groups installed in parallel, all connected to the main pipe by a branching junction. The characteristics of the groups are presented in Table 3.4. The specific rotational speed of calculated based on equation (2.47). Table Pumping groups characteristics Number of pumps (-) 4 Rated flow rate (m 3 /h ; m 3 /s) 4500 ; 1.25 Rated pumping head (m) Rated rotational speed 985 Rated power (kw) 3500 Efficiency (%) 87.5 Pump-motor inertia (kgm 2 ) 300 Specific rotational speed (SI units) 18 65

100 Six different simulation scenarios were considered to verify if the numerical results from both simulators are consistent. Two main scenarios - pumping system with and without check valve - were combined with the possibilities of no surge protection, surge protection without differential headloss coefficients and with differential headloss coefficients (Table 3.5). Table 3.5 Scenarios used in pumping system simulation Scenarios No Air vessel protection Kin = Kout Yes Kin Kout Without check valve S-A1 S-A2 S-A3 With check valve S-B1 S-B2 S-B3 These set of simulations were carried out to verify if the numerical results with and without check valve are stable, and if the interference between two numerical routines (i.e., pump sudden failure and air vessel) are numerically stable or not, for the methods used to solve the governing equations. Several approaches for the check valve modelling can be considered, but according to Ramos et al., (2005), a quasi-instantaneously check valve closure, when the flow reverses may be used during early design stages. This check valve approach is the one used in AKS. Table 3.6 presents the air vessel characteristics used in the simulations. Table 3.6 Hydropneumatic vessel characteristics used in the simulations Distance from pumping station 22.4 m Inlet orifice diameter (m) 0.7 Inlet/Outlet orifice local headloss coefficient Kin/Kout (-) 2/2 ; 20/2 Initial head of water (m) Vessel diameter (m) 3.0 Initial air volume (m 3 ) 20 Before each transient-state simulation, the steady-state flow must be determined. For steady-state friction resistance, Equations (2.16) and (2.24) were used with an estimated roughness k of 0.1 mm. The initial flow rate at steady-state is 5.1 m 3 /s. The elastic wave speed was estimated based on Equations (2.8) and (2.13) and a theoretical value of m/s was obtained. For this parameter a value of 1120 m/s was used. The piezometric head at downstream the pumping station is m. The pump station is located at 0.0 m, whereas the downstream reservoir is located at m Pumping system results This section includes a comparison between the results from both simulators. The parameters that define each simulation are presented, followed by the results obtained for piezometric head envelopes; 66

101 piezometric head and flow rate at downstream the pumping station; and the pump rotational speed. The flow rate and the water level in the hydropneumatic vessel are compared for the scenarios with this protection device. Afterwards, a comparison between significant values for design purposes is performed (Figure 3.19). Differences between values from both simulators are calculated as follows: Difference = Results AKS Results JDS Results AKS (3.82) Scenario S-A1: Without check valve and no air vessel Parameters that define this simulation are presented in Table 3.7. Results obtained for this scenario are presented in Figure Table 3.8 presents a comparison between significant values obtained from both models. Table Characteristics of the simulation S-A1 Element Pipelines Pumping station Air vessel Reservoirs Simulation parameters Characteristics D=1400 mm; k= m; Q=5.1 m 3 /s; a = 1120 m/s; LPIPE 1=22.4 m; LPIPE 2=403.2 m. Four groups in parallel without check valve (Simulation with Suter parameters for Ns = 25); Pump sudden stoppage at 2.0 s. Each group: QR=1,25 m 3 /s; HR=248.0 m; PR=3500 kw; NR=985 rpm; I =300.0 kg.m 2 No air vessel H resu =281.0 m; H resd =520.0 m AKS: Δt=0.02 s; tmax =30 s. JDS: Δt= s; tmax =30 s. 67

102 (a) (b) (c) (d) Figure 3.16 Results obtained for scenario S-A1: (a) piezometric head envelopes; (b) piezometric head at downstream the pump station; (c) flow rate at the pump station; (d) pump rotational speed Table 3.8 Comparison between significant values obtained from AKS and JDS for scenario S-A1 Parameters AKS JDS Difference Maximum piezometric head (m) % Minimum piezometric head (m) % Minimum flow rate at the pump station (m 3 /s) % Minimum pump rotational speed (rpm) % Scenario S-A2: without check valve and air vessel without differential losses Parameters that define this simulation are presented in Table 3.9. Results obtained for this scenario are presented in Figure Figure 3.18 presents the results obtained for the air vessel. Table 3.10 presents a comparison between significant values obtained from both models. 68

103 Table Characteristics of simulation S-A2 Element Pipelines Pumping station Air vessel Reservoirs Simulation parameters Characteristics D=1400 mm; k= m; Q=5.1 m 3 /s; a = 1120 m/s; LPIPE 1=22.4 m; LPIPE 2=403.2 m. Four groups in parallel without check valve (Simulation with Suter parameters for Ns = 25); Pump sudden stoppage at 2.0 s. Each group: QR=1,25 m 3 /s; HR=248.0 m; PR=3500 kw; NR=985 rpm; I =300.0 kg.m 2 Dorf = 0.7 m; Vair = 20 m 3 ; DHPV = 3.0 m; Z = m. Kin/Kout = 2/2 H resu =281.0 m; H resd =520.0 m AKS: Δt=0.02 s; tmax =30s. JDS: Δt= s; tmax =30s. (a) (b) (c) (d) Figure 3.17 Results obtained for scenario S-A2: (a) piezometric head envelopes; (b) piezometric head at downstream the pump station; (c) flow rate at the pump station; (d) pump rotational speed 69

104 (a) (b) Figure Results obtained for the air vessel in scenario S-A2: (a) flow rate; (b) water level Table Comparison between significant values obtained from AKS and JDS for scenario S-A2 Parameters AKS JDS Difference Maximum piezometric head (m) % Minimum piezometric head (m) % Minimum flow rate ate the pump station (m 3 /s) % Minimum pump rotational speed (rpm) % Maximum flow rate in the air vessel (m 3 /s) % Minimum flow rate in the air vessel (m 3 /s) % Maximum water level in the air vessel (m) % Minimum water level in the air vessel (m) % Scenario S-A3: without check valve and air vessel with differential losses Table 3.11 presents the parameters that define this simulation. Results obtained for this scenario are presented in Appendix A.1. Table Characteristics of simulation S-A3 Element Pipelines Pumping station Air vessel Reservoirs Simulation parameters Characteristics D=1400 mm; k= m; Q=5.1 m 3 /s; a = 1120 m/s; LPIPE 1=22.4 m; LPIPE 2=403.2 m. Four groups in parallel without check valve (Simulation with Suter parameters for Ns = 25); Pump sudden stoppage at 2.0 s. Each group: QR=1,25 m 3 /s; HR=248.0 m; PR=3500 kw; NR=985 rpm; I =300.0 kg.m 2 Dorf = 0.7 m; Vair = 20 m 3 ; DHPV = 3.0 m; Z = m. Kin/Kout = 20/2 H resu = m; H resd =520.0 m AKS: Δt=0.02 s; tmax =30s. JDS: Δt= s; tmax =30s. 70

105 Scenario S-B1: with check valve and no air vessel Table 3.12 presents the parameters that define this simulation. Results obtained for this scenario are presented in Appendix A.2. Table Characteristics of simulation S-B1 Element Pipelines Pumping station Air vessel Reservoirs Simulation parameters Characteristics D=1400 mm; k= m; Q=5.1 m 3 /s; a = 1120 m/s; LPIPE 1=22.4 m; LPIPE 2=403.2 m. Four groups in parallel with check valve (Simulation with Suter parameters for Ns = 25); Check valve closes quasi-instantaneously at the flow reversal; Pump sudden stoppage at 2.0 s. Each group: QR=1,25 m 3 /s; HR=248.0 m; PR=3500 kw; NR=985 rpm; I =300.0 kg.m 2 Dorf = 0.7 m; Vair = 20.0 m 3 ; DHPV = 3.0 m; Z = m. Kin/Kout = 2/2 H resu =281.0 m; H resd =520.0 m AKS: Δt=0.02 s; tmax =30s. JDS: Δt= s; tmax =30s. Scenario S-B2: with check valve and air vessel without differential losses Table 3.13 presents the parameters that define this simulation. Results obtained for this scenario are presented in Appendix A.3. Table Characteristics of simulation S-B2 Element Pipelines Pumping station Air vessel Reservoirs Simulation parameters Characteristics D=1400 mm; k= m; Q=5.1 m 3 /s; a = 1120 m/s; LPIPE 1=22.4 m; LPIPE 2=403.2 m. Four groups in parallel with check valve (Simulation with Suter parameters for Ns = 25); Check valve closes quasi-instantaneously at the flow reversal; Pump sudden stoppage at 2.0 s. Each group: QR=1,25 m 3 /s; HR=248.0 m; PR=3500 kw; NR=985 rpm; I =300.0 kg.m 2 Dorf = 0.7 m; Vair = 20 m 3 ; DHPV = 3.0 m; Z = m. Kin/Kout = 2/2 H resu =281.0 m; H resd =520.0 m AKS: Δt=0.02 s; tmax =30s. JDS: Δt= s; tmax =30s. Scenario S-B3: with check valve and air vessel with differential losses Table 3.14 presents the parameters that define this simulation. Results obtained for this scenario are presented in Appendix A.4. 71

106 Table Characteristics of simulation S-B3 Element Pipelines Pumping station Air vessel Reservoirs Simulation parameters Characteristics D=1400 mm; k= m; Q=5.1 m 3 /s; a = 1120 m/s; LPIPE 1=22.4 m; LPIPE 2=403.2 m. Four groups in parallel with check valve (Simulation with Suter parameters for Ns = 25); Check valve closes quasi-instantaneously at the flow reversal; Pump sudden stoppage at 2.0 s. Each group: QR=1,25 m 3 /s; HR=248.0 m; PR=3500 kw; NR=985 rpm; I =300.0 kg.m 2 Dorf = 0.7 m; Vair = 20 m 3 ; DHPV = 3.0 m; Z = m. Kin/Kout = 20/2 H resu =281.0 m; H resd =520.0 m AKS: Δt=0.02 s; tmax =30s. JDS: Δt= s; tmax =30s. Figure 3.19 presents the differences between the results obtained from both models for all the parameters compared. The maximum difference between the results is approximatelly 5.2%. Figure 3.19 Comparison of the significant values for all scenarios between the two simulators Final remarks Results obtained from both simulators are very similar. Main differences between the results can be explained by the code implementation and time-step used. Regarding the first argument, AKS is a pipe network mathematical model, whereas JDS is a simple-pipe numerical model, which explains the different approaches of solving the governing equations of the elements that compose the pipe system. 72

107 Regarding the difference in calculation time-step, the increase of this parameter can lead to sufficient loss of information for the following time-steps, leading to different results. The major difference observed is approximately 5.2% for the scenarios check valve closure time with air vessel protection. However the difference between results is only from 0.38 s to 0.36 s, and can be explained by the difference of models used to describe the air vessel. While JDS considers a model with quasi-instantaneously response, with only an orifice between the pipe and the air vessel, AKS considers a model with a secondary pipe, connecting the main pipe and the air vessel, which gives a certain inertia, leading the fluid to take longer to outflow from the vessel, and generating a higher check valve closure time. Despite this difference, the maximum piezometric head reached by both simulators are very similar. The simulations for this hydraulic system lasted approximately 76 s for AKS and approximately 2 s with JDS, despite the Δt parameter is much lower in JDS (Δt = s) than in AKS (Δt = 0.02 s). 3.7 Summary This chapter presents the procedures used to the development of a 1-D transient-state flow numerical model in a simple system of upstream boundary middle points downstream boundary. The mathematical model was developed in Microsoft Visual Studio C This solver solves the transient-state fluid equations in the time domain by the Method of Characteristics, using a frictionless, first and second order approach, where the last approach leads to better results. Several boundary conditions were implemented at downstream and upstream end trying to reach the maximum situations possible in pipe systems. The sudden pump stoppage boundary condition assumes the most important role in this mathematical model, and it was solved using the Suter parameters. A hydropneumatic vessel was also included in the numerical model. The mathematical model also incorporates one formulation to describe the dynamic effects of unsteady friction. It was used Vítkovský et al., (2000) formulation. This last formulation was described by a first-order explicit scheme in space and a first-order implicit scheme in time. The comparison of results between the hydraulic transient solver and the validated model shown that the first one describes accurately the transient events triggered by the failure of the power grid that supplies the pump with or without the protection of a hydropneumatic vessel, assuming a simple model (closure when the flow reverses) for the check valve. 73

108 74

109 4 TRANSIENT DATA COLLECTION AND ANALYSIS 4.1 Introduction The current chapter focuses on the description of the experimental data collection programmes carried out in two experimental facilities assembled at the Laboratory of Hydraulics and Water Resources, in the Department of Civil Engineering, Architecture and Georesources (DECivil), at the Instituto Superior Técnico in Lisbon, Portugal. The first facility consists of a 105 m coiled-copper pipeline with 20 mm diameter, specially constructed in 2008 for research and teaching purposes. The second facility is a steel pipeline with 115 m length and 200 mm diameter, installed along the perimeter of the laboratory since the construction of the laboratory in 1992, aiming at supplying by gravity all the existing experimental facilities. This pipeline was hardly used during the last 20 years and it was adapted with the construction of a pumping station at the upstream end in Both facilities are raising mains with approximately horizontal profile. Transient and steady-state pressure data were collected under controlled laboratory conditions. This chapter includes the description of each experimental facility and of the instrumentation used for data collection, as well as the experimental tests carried out. Collected data will be used in Chapter 5 for the hydraulic transient solver validation. Conclusions are drawn regarding the experimental data collected. 4.2 Copper-pipe facility Experimental facility description An experimental copper-pipe facility was assembled for analysis of transient events both for teaching (i.e., lecturing/postgraduate courses) and research purposes. The pipe rig has a configuration of the type reservoir - pump - hydropneumatic vessel - pipe - valve. System is composed of a coiled copper pipe m of length (between the HPV and the downstream valve), 20 mm of inner diameter and 1 mm of pipe-wall thickness. The rig was assembled in a portable metal frame (with four wheels) with 1.0 m depth, 2.0 m length and 1.6 m height (Figure 4.1a,b). The system is supplied upstream from a storage tank with 125 l of capacity by a pump with nominal flow rate of 1 m 3 /h and nominal head of 32.0 m. Downstream the pump there is a needle check valve used to prevent the reverse flow. There is a hydropneumatic vessel (HPV) made of stainless steel, with 60 l of capacity and designed for the nominal pressure of 6.0 bar. This HPV is provided with an outer piezometer to allow water level measurement (Figure 4.1c). The HPV is supplied with air by the existing compressed air system in the laboratory. At the downstream end there is a ball valve with DN 3/4 that allows the control of flow rate and the generation of transient events. This downstream valve is located at 1.4 m of elevation (Figure 4.1d). 75

110 (a) (b) (c) (d) Figure Copper-pipe facility: (a) general view; (b) side view; (c) upstream view; (d) downstream view At the upstream end there is a set of ball valves for changing the system configuration, which allows three different configurations (Figure 4.2). Only Configurations I and II are studied in the current research. Table Flow configurations in copper-pipe facility Configuration I II III Type Tank - pump - pipe - valve Tank - pump - HPV (side-connection) - pipe - valve Tank - pump - HPV (in-line) - pipe - valve 76

111 Figure 4.2 Possible configurations of flow in copper-pipe facility The facility is equipped with instrumentation for collecting both steady and transient-state data. Steady and transient-state pressures were measured by three strain-gauge type pressure transducers, located in three different sections (Table 4.2 and Figure 4.3a,b,c) Pressure transducers measure absolute pressures up to 25 bar with a maximum error of 0.25% (i.e., 0.6 m of maximum error). Table Transducer location in copper-pipe facility adapted from Libraga (2011) Transducer T1 T2 T3 Location Upstream Middle Section Downstream Distance (1) (m) Elevation (2) (m) (1) Distance between pump downstream section and the transducer middle section (2) Elevation measured from the timber frame upper surface (reference level) and the transducer interior membrane Transducers measure the mechanical pressure deformation in the sensor generating an electrical signal which is processed and used by the acquisition equipment. The electrical signal is sent to the oscilloscope that converts it in potential difference and sends it to a computer that interprets the results and converts them into pressure values, saving them in to a data record (Figure 4.3d). The software used to reproduce the results in the computer is PicoScope 6. The flowchart of the data acquisition system procedure is presented in Figure 4.4. Data was acquired with a frequency of 200 Hz At the downstream end of the pipe rig there is a rotameter for measuring steady-state flow rate up to 1000 l/h (Figure 4.3e). 77

112 (a) (b) (c) (d) (e) Figure 4.3 Measurement equipment in copper-pipe facility: (a) upstream transducer T1; (b) middle-section transducer T2; (c) downstream transducer T3; (d) data acquisition system; (e) rotameter Figure 4.4 Flowchart of data acquisition system procedure adapted from Carriço (2008) Figure 4.5 presents the schematic of the copper-pipe experimental facility topology. 78

113 Figure 4.5 Schematic of the copper-pipe facility Data collection Transient tests were carried out to collect steady and transient pressure data. Five different set of tests were performed using different devices (pump and upstream and downstream valves) and manoeuvres (sudden stoppage, opening and closure) to generate the transient event (Manoeuvres A to E) with and without air vessel connected to the system (Configurations I and II). These manoeuvres were realized for different flow rates. All the valve manoeuvres carried out were fast (i.e., t m < 2L a, being t m the time of valve closure/opening). Table 4.3 presents the set of tests carried out. Table 4.3 Experiments carried out in copper-pipe facility Tests ID Manoeuvre type Initial low rate (l/h) HPV (Configuration) A1 Yes (II) A - Sudden pump stoppage (1) A2 No (I) B1 Yes (II) B - Upstream valve fast closure (1) B2 No (I) C2 C - Upstream valve fast opening (2) No (I) D1 Yes (II) D - Downstream valve fast closure (1) D2 No (I) E1 Yes (II) E - Downstream valve fast opening Total (3) E2 No (I) (1) 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 (2) 0,100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 (3) Valve fully opened The manoeuvres with air vessel connected were all carried out with the same initial volume of air. The water level measurements made in the outer piezometer during the steady-state are presented in Figure

114 Figure 4.6 Measurements of the water level in the air vessel during the steady-state in the copper-pipe facility Data collected for each manoeuvre are presented and analysed in the following section Data analysis Sudden pump stoppage (Manoeuvre A) In this set of tests, the transient event was generated by the sudden stoppage of the pump by failure of the electric grid (Manoeuvre A). These tests were carried out to analyse the effect of the air vessel protection during the sudden pump stoppage, with particular emphasis to its effect on pipes with approximately horizontal profile. Twenty transient tests have been carried out: ten with the HPV connected (as a surge protection device) and ten without the HPV, each for a different initial flow-rate (Q i ). Collected data at the transducer located at the upstream end of the pipeline (T1 upstream end of the pipe rig) are presented in Figure 4.7. The pump stoppage was set at t = 2.0 s. 80

115 (a) (b) (c) (d) Figure 4.7 Data collected at transducer T1 during the sudden pump stoppage in the copper-pipe facility: (a) with the HPV for flows between 100 and 500 l/h; (b) with the HPV for flows between 600 and 1000 l/h; (c) without the HPV for flows between 100 and 500 l/h; (d) without the HPV for flows between 600 and 1000 l/h Data collected at the three transducers for the initial flow-rate of 600 l/h with and without the HPV are presented in Figure 4.8. The effect of the pressure wave travelling time is observed in Figure 4.8b where the delay in the pressure head change can be observed. 81

116 (a) (b) Figure 4.8 Data collected at the three transducers during the sudden pump stoppage for initial flow rate 600 l/h: (a) with protection; (b) without protection Maximum pressures are extremely high when the HPV is connected to the system (Figure 4.7a,b and Figure 4.8b). The reason for this is that, as the pump fails, the HPV starts supplying the pipeline at downstream as well as the pipe for upstream, between the vessel and the pump (Figure 4.9); because the check valve does not instantaneously close, allowing some reverse flow, when it actually closes, the reverse flow is quite high, inducing an extremely high upsurge. This shows that, under certain circumstances, surges protection devices can create higher pressures than when they are not installed in the system. During these manoeuvres, when the air vessel was connected to the hydraulic system, considerable vibrations were induced in the pipeline, caused by the check-valve slam, which may affect the maximum overpressures observed, caused by the vibration in the pipe. Figure 4.9 Flow supply from the air vessel after the sudden pump stoppage When the air vessel was not connected in the system, the piezometric head drops slowly and the upsurge created by the check valve slam almost is not noticed, since the flow deceleration is much lower than when the vessel is connected to the system. For flow rates of 100 and 200 l/h exists a small upsurge, while the piezometric head is dropping. This feature that is not observed in the other tests might be related with the check valve closure, since the initial flow rate is very small, or due to the pump being far from the rated conditions during the steady-state. Differences between results obtained with and without the HPV connected are related with the pressure gradient, which is considerably higher when the air vessel is connected in the system. After the pump stoppage, the total developed head severely reduces and the air vessel starts to supply the pipeline 82

117 upstream, resulting in a severe check valve slam. When the air vessel is not connected, the fluid within the pipeline reverses slower and the pressure gradient between the pump and the discharge valve is not sufficient to overcome the check valve inertia, leading a negligible overpressure. A brief analysis regarding the results obtained for the mean reverse velocity at the check valve slam is presented in Table 4.4. Piezometric heads H1 and H2 were measured before and after the first upsurge, whilst ta and tb were measured before the first upsurge in the upstream and the middle section of the pipeline (see Figure 4.8a). The first two values (H1 and H2) were determined to obtain the overpressure generated by the check valve slam, whereas the second ones (ta and tb), were determined to estimate the elastic wave speed. The valve closure time, tm is the difference between the instant of pump stoppage and check valve slam instant. The pressure wave speed, a,can be estimated based on the travel time of the elastic transient wave between sections A and B, as follows: a = x A x B t A t B (4.1) where x A and x B represents the middle section and downstream transducers location, respectively (see Table 4.2). Since the check valve has a fast closure, the mean reverse velocity at the check valve slam, U can be estimated by Equation (2.1). Table Result analysis for the sudden pump stoppage with air vessel experimental tests in copper-pipe facility Q (l/h) H1 (m) H2 (m) ΔH (m) ta (m) tb (m) tm(s) c* (m/s) U* (m/s) The following conclusions can be drawn from the analysis of results presented in Table 4.4: (i) For these set of tests, the wave speed is not test-dependent, as a constant value has been obtained for all tests. According to previous analyses carried out in this facility for the 83

118 downstream valve closure (Ribeiro, 2010; Libraga, 2011; Libraga et al., 2011 and Covas et al., 2012a,b) the wave speed varied from test to test which indicated the presence of free-air in the fluid, probably introduced by the cavitation in the downstream flow control valve. (ii) Concerning the mean reverse velocity, this parameter is test-dependent. A complementary analysis was carried out to verify whether it was correlate the check valve closure or the flow rate, as presented in Figure 4.10a,b. A major dispersion on the results was observed. For further analysing this phenomenon, a new set of experiments should be carried out repeating the test several times for the same flow to evaluate if the tests are being affected by the vibrations introduced in the pipeline resulting from the check valve slam, or by any other phenomenon (e.g., air dissolved in the fluid). Another possible cause could be the check-valve unstable behaviour due to the turbulence generated by the fluid fast deceleration, as it is a needle plastic valve that only closes when a certain pressure difference is observed between the two. It should be highlighted that it was not intended to study the check valve behaviour, when the experimental facility was designed and assembled (iii) On the other hand, the check valve closure time varies with the initial flow rate as presented in Figure 4.10c. Two laws between the closure time and the flow rate were estimated: a quadratic and a linear law. Both present a good fitting as the correlation coefficient is close to one. These laws may allow for the simplified description of check valve manoeuvre in the hydraulic transient solver assuming that the manoeuvre is a linear closure of the check valve (which may not be a rough simplification of reality). Further research is necessary to better characterize the check valves behaviour. 84

119 (a) (b) (c) Figure 4.10 Result analysis of sudden pump stoppage with air vessel experimental tests in copper-pipe facility: (a) estimated mean reverse velocity as function of check valve closure time; (b) estimated mean reverse velocity as function of initial flow rate; (c) check valve closure time in function of initial flow rate Upstream valve fast closure (Manoeuvre B) In this set of tests, the transient event was generated by the fast manual closure of the upstream valve (Manoeuvre B). The aim of these was to compare the transient pressure conditions obtained between the sudden pump stoppage and the fast (almost instantaneous) upstream valve closure (see ). The latter is used as a simplified model of this phenomenon for design purposes when the pump modelling is not available. If a good agreement between data from these two manoeuvres is observed, then a simplified model for design purposes could be safely used. Data collected at transducer T1 from a set of twenty transient tests (ten with HPV connected and ten without the HPV) are plotted in Figure Data collected at the three transducers for the initial flow rate of 600 l/h with and without the HPV are presented in Figure The valve closure was set at t = 2.0 s. 85

120 (a) (b) (c) (d) Figure 4.11 Data collected at transducer T1 during the upstream valve closure in copper-pipe facility: (a) with the HPV for flows between 100 and 500 l/h; (b) with the HPV for flows between 600 and 1000 l/h; (c) without the HPV for flows between 100 and 500 l/h; (d) without the HPV for flows between 600 and 1000 l/h (a) (b) Figure 4.12 Data collected at the three transducers during the upstream valve closure tests for initial flow rate 600 l/h: (a) with HPV and (b) without HPV 86

121 A very gradual pressure decrease during the valve closure is observed, when the air vessel is connected to the system (Figure 4.11a,b and Figure 4.12a). This decrease is due to the HPV supplying the system only for downstream after the valve closure, resulting in a decrease of the water level and of the air pressure inside de vessel. The HPV behaves as a small tank with a variable water level. No check valve slam is observed because the air vessel only supplies the pipeline downstream. Figure Flow supply from the air vessel after the upstream valve closure When the air vessel is not connected to the system, the fluid reaches the saturated vapour pressure (approximately -9.2 m) and cavitation occurs flow initial flow rates higher than 500 l/h (Figure 4.11d). A sudden overpressure is observed due to the implosion of the air cavity close to the valve section. This upsurge is higher and takes longer to arises, as the initial flow rate increases (Figure 4.11d). The highest piezometric head is observed at the steady-state flow, which means that the only problem associated with this type of manoeuvre is the occurrence of cavitation and possible formation of air pockets, leading to problems related with the system efficiency Upstream valve fast opening (Manoeuvre C) In these tests the transient flow was generated by the fast manual opening of the upstream valve (Manoeuvre C). The upstream valve was opened with the pump operating. The final steady-state flow rate was controlled by the downstream valve opening, which was previously established during a former steady-state. This manoeuvre was carried out to analyse the effects of a sudden fast upstream valve opening without the air vessel protection, since the transient-state flow generated by the upstream valve closure is controlled by the air vessel (see ). The upstream valve opening with air vessel connected was not performed, since it would produce similar results to Figure 4.12a. However, the pipeline would supply the air vessel until it reached the steady-state pressure (the opposite results to the ones shown in Figure 4.12a the pressure will rise until steady-state pressure is reached). Figure 4.14a presents the collected data at transducers T1, whereas Figure 4.14a depicts the collected data at transducer T3. Collected data at the three transducers for four final flow rates (0, 100 and 600 and 1000 l/h) are presented in Figure The valve opening was set at t = 2.0 s. 87

122 (a) (b) (c) (d) Figure 4.14 Data collected during the upstream valve opening without HPV in copper-pipe facility for: (a) final flows between 0 and 500 l/h in transducer T1; (b) final flows between 600 and 1000 l/h in transducer T1; (c) final flows between 100 and 500 l/h in transducer T3; (d) final flows between 600 and 1000 l/h in transducer T3 88

123 (a) (b) (c) (d) Figure 4.15 Data collected at three transducers during upstream valve opening in copper-pipe facility without the HPV for the (a) downstream valve closed; and the final flow rates of (b) 100 l/h; (c) 600 l/h; (d) 1000 l/h The upstream valve opening generates a transient event that is dissipated as the liquid is discharged by the downstream valve. The amplitude and duration of this transient event decreases with the downstream end valve opening: the more closed is the valve, the higher and longer is the transient event, which makes sense since the higher valve opening, higher the discharge and higher the pressure relief, increasing the surge dissipation. The highest transient is observed when the downstream valve is completely closed (0 l/h) (see Figure 4.14a,c and Figure 4.15a). Accordingly, the piezometric head in the downstream section reaches higher values than the steady state ones, when the downstream valve is fully or partially closed (e.g.,100, 200 and 300 l/h) (see Figure 4.14c,d). The secondary pressure fluctuations observed in the piezometric head are related with the hydraulic circuit movement. These movements were observed during the tests performance and were also reported by Ribeiro (2010) and Libraga (2011). 89

124 Downstream valve fast closure (Manoeuvre D) Transient events were generated by the fast manual closure of the downstream (manoeuvre D). This manoeuvre was carried out to simulate a flow control valve closure when the upstream boundary has a pump working at constant rotational speed. Data collected at transducer T3 from a set of twenty transient tests (ten with HPV connected and ten without the HPV) are depicted in Figure Data collected at the three transducers for the initial flow rate of 600 l/h with and without the HPV are presented in Figure The valve closure was set at t = 2.0 s. (a) (b) (c) (d) Figure 4.16 Data collected at transducer T3 during the downstream valve closure in copper-pipe facility: (a) with the HPV for flows between 100 and 500 l/h; (b) with the HPV for flows between 600 and 1000 l/h; (c) without the HPV for flows between 100 and 500 l/h; (d) without the HPV for flows between 600 and 1000 l/h 90

125 (a) (b) Figure Data collected at the three transducers during the downstream valve closure tests for initial flow rate of 600 l/h: (a) with HPV and (b) without HPV From the analysis of these figures the following conclusions are drawn: (i) For this type of manoeuvre, cavitation occurs for flows between 600 and 1000 l/h when the system has the HPV connected (Figure 4.16b), and for flow rates between 800 and 1000 l/h when the HPV is off the system (Figure 4.16d). When this phenomenon occurs, the second overpressure is higher than the first one, when the HPV is connected, which is related with the vapour cavities implosion (Figure 4.17b). However, this feature is not observed when the air vessel is not operating (Figure 4.17d). (ii) During the experimental tests, the pipe support structure movement was higher when the cavitation phenomenon occurred, which is consistent with the data obtained. The movements on the structure were also higher when the air vessel was not connected, than when it was, which may explain the higher secondary oscillations during those set of tests. (iii) The surge damping is more relevant when the air chamber is not connected as well. When the air vessel is not connected in the system, the elastic wave reflects on the pump, and not on the air vessel, and the turbomachine acts as a source of energy dissipation. In summary, the effects of transient flow for the fast downstream valve closure are more severe when the air vessel is connected to the hydraulic system at upstream end, than when it is not connected. However, this device stabilizes the flow-rate and protects the pump which can be relevant for operating conditions and lifetime of the turbomachine, respectively. Maximum overpressures are plotted as a function of the initial flow rate in Figure A linear relationship between these two parameters is observed. The relating constants H Q = a are and gs which lead to estimated wave speeds, a, of 1072 and 1089 m/s with and without HPV connected, respectively 91

126 (a) (b) Figure Maximum overpressure observed for downstream valve closure in copper-pipe facility: (a) with protection; (b) without protection A comparison between observed and theoretical maximum overpressures with and without air vessel protection is presented in Table 4.5 and Table 4.6, respectively. Maximum overpressures based on Frizell-Joukowsky formula were obtained considering the estimated elastic wave speed (1072 and 1089 m/s with and without HPV, respectively). The elastic wave speed, for each test was estimated based on the observed maximum overpressure (ΔH) and mean velocity (U). Table 4.7 presents a comparison between maximum piezometric head observed in the data collected with (HWITH) and without (HWITHOUT) the HPV connected. Table Comparison between observed and theoretical maximum overpressures and elastic wave speed estimation in copper-pipe facility for downstream valve closure with air vessel Q (l/h) U (m/s) HSTEADY (m) HUNSTEADY (m) ΔH (m) ΔHJ (m) Difference (%) a* (m/s) % % % % % % % % % %

127 Table Comparison between observed and theoretical maximum overpressures and elastic wave speed estimation in copper-pipe facility for downstream valve closure without air vessel Q (l/h) U (m/s) HSTEADY (m) HUNSTEADY (m) ΔH (m) ΔHJ (m) Difference (%) a* (m/s) % % % % % % % % % % Table Maximum piezometric head observed for downstream valve closure with and without protection, associated to the first upsurge, in copper-pipe facility Q (l/h) Max HWITH (m) Max HWITHOUT (m) Difference (%) % % % % % % % % % % The maximum piezometric head (related with the first overpressure) increases with the increase of the initial flow rate, independently of the HPV being connected or not with the system (Table 4.7 and Figure 4.18). The difference between these data, which is relatively small, may be related to the transducer error Downstream valve fast opening (Manoeuvre E) In this set of tests, the transients were generated by the downstream valve fast manual opening (valve completely opened). This manoeuvre was performed to analyse the effects of the air vessel protection effect in the experimental facility. Collected data at the three transducers for the maximum final flow rate (not measured by the rotameter) are presented in Figure The valve opening was set at t = 2.0 s. 93

128 (a) (b) Figure 4.19 Data collected at the three transducers during the downstream valve opening tests: (a) with protection; (b) without protection Figure 4.19 shows that this manoeuvre does not create any upsurge and the downsurge does not reach negative pressures. The air vessel only controls the pressure drop at the upstream section. The surge damping is higher when the air vessel is not connected with the system, which is consistent with the results obtained for Manoeuvre D (the pump acts as a source of energy dissipation) Final remarks Figure 4.20 presents the comparison between the manoeuvres of the sudden pump stoppage and the upstream valve closure (Manoeuvres A and B, respectively) in the copper pipe facility with and without air vessel for the flow rate of 300 l/h (Figure 4.20a) and 600 l/h (Figure 4.20b). This figure shows that the behaviour of the system due to the sudden pump stoppage and due to the valve closure does not have the same features. Accordingly, it is not recommended the use of a simplified upstream valve model to simulate the pump behaviour during power failure. Firstly, the discharge variation on the pump is relatively difficult to predict, since it depends on multiple parameters. Secondly, when an upstream valve is closed, there is no opportunity for the occurrence of reverse flow, leading to no check valve slam, which is the responsible for undesired transient overpressures (see Figure 4.9 and Figure 4.13). 94

129 (a) (b) (c) (d) Figure 4.20 Comparison between the sudden pump stoppage and the upstream valve closure: (a) 300 l/h with air vessel; (b) 300 l/h without air vessel; (c) 600 l/h with air vessel; (d) 600 l/h without air vessel 95

130 4.3 Steel-pipe facility Experimental facility description A new reversible pumping system was assembled at the Laboratory of Hydraulics and Water Resources, in Department of Civil Engineering, Architecture and Georesources (DECivil), at the Instituto Superior Técnico in Lisbon, Portugal. The system is composed of a pipeline made of steel, with a nominal pressure of 10 bar, a total length of 115 m, an inner diameter of 206 mm and a wall thickness of 6.3 mm. The pipeline is installed along the internal perimeter of the Laboratory (Figure 4.21). (a) (b) (c) (d) Figure 4.21 Steel experimental facility: (a) plan; (b) detail of reversible area (c); front view (d) side view The system is supplied from a storage tank, whose maximum level determined by the troplan upper end is 3 m (Figure 4.22a), through a centrifugal pump KSB Etaline /1502 (Figure 4.22b) with a nominal flow rate of 30.3 l/s, a nominal head of 32.0 m and a installed power of 15 kw, for a nominal rotational speed of 2955 rpm. To minimize the problems of vibration and noise, caused by the pump operation, an 96

131 insulating wooden box was constructed around the pump (Figure 4.22b and Figure 4.25b). The pump is equipped with a variable speed drive that allows the changing of rotational speed (Figure 4.22c). The pump rating curves - theoretical and experimental (obtained by direct measurement in this facility) rating curves are presented in Appendix B. Immediately at downstream the pump there is a ball check valve used to prevent the reverse flow (Figure 4.22b). A 1 m 3 hydropneumatic vessel is installed at downstream the pump; this device can be connected in-line or as a side element connected through a branch, or totally disconnected from the system by the opening/closing of a set of gate valves (Figure 4.21b and Figure 4.22d). (a) (b) (d) (c) Figure Experimental facility reversible zone details: (a) troplan inside the reservoir; (b) pump isolated in the wooden box and check valve; (c)variable speed drive; (d) hydropneumatic vessel 97

132 At the downstream end, there are two valves with 50 mm diameter each: a manual ball valve used to control the flow and a butterfly valve used to generate transient events (Figure 4.23a). A butterfly valve actuated pneumatically by a trigger is used to perform identical manoeuvres for each test, with the same closure time. The trigger is controlled by the computer and is used to initiate the valve closure (Figure 4.23b). There are several side discharge valves with ½ and ¾ for connecting pressure transducers and draining the accumulated air (Figure 4.23c), respectively, and three 1 scour valves for draining the pipeline (Figure 4.23d). (a) (b) (c) (d) Figure 4.23 Downstream valves: (a) manual ball valve and pneumatically actuated butterfly valve; (b) trigger; (c) air valve; (d) scour valve The facility has a set of gate and ball valves that allow the flow circulation in two different directions, reason why it is called a reversible system (see detail in Figure 4.21b). The hydropneumatic vessel can be connected in-line or as a side element connected through a branch, or totally disconnected from the system by the opening/closing of a set of gate valves, whether the flow circulates clockwise or counter clockwise. However, the possibility of the flow circulating through the air vessel is not studied herein. Only the counter-clockwise flow direction was studied. The manoeuvres were performed with air vessel 98

133 as a side-element (Configuration A) and with the air vessel disconnected (Configuration B), as presented in Figure (a) (b) Figure Flow paths studied: (a) with air vessel in bifurcation Configuration A; (b) without air vessel Configuration B. Note: red suction and discharge line; blue downstream end of the pipe system and discharge to the reservoir The equipment used for collecting steady-state flow data is an electromagnetic flow meter ABB Processmaster FEP with 65 mm and accuracy of 0.4% of measured values (Figure 4.25a). The flow rate is controlled by the ball valve installed downstream end of the pipeline. The transient pressure data are measured by WIKA pressure transducers with an absolute pressure range from 0 to 25 bar and accuracy of 0.5% of full range. The transducers are located immediately downstream the check-valve, and upstream the valve used to generate transient-state flow at pipe downstream end (Figure 4.25c). The pressure transducers location and elevation are presented in Table 4.8. The data acquisition system is the same as the one used in copper-pipeline facility (Figure 4.25d). Table Transducer location in the steel experimental facility Transducer T1 T2 Location Upstream Downstream Distance (1) (m) Elevation (2) (m) (1) Distance between pump downstream section and the transducer middle section (2) Elevation measured from the timber frame upper surface (reference level) and the transducer interior membrane 99

134 (a) (b) (c) (d) Figure Experimental facility equipment: (a) flow meter; (b) gate valve to control flow rate; (c) pressure transducers location; (d) data acquisition system Figure 4.26 presents the schematic of the experimental steel-pipe facility topology. Appendix C presents some technical draws regarding the reversible area. Figure 4.26 Schematic of the steel-pipe facility 100