Hydraulic Transients and Computations 3030402320, 9783030402327

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Zh. Zhang

Hydraulic Transients and Computations

Hydraulic Transients and Computations

Zh. Zhang

Hydraulic Transients and Computations

123

Zh. Zhang Institute of Energy Systems and Fluid Engineering (IEFE) Zurich University of Applied Sciences (ZHAW) Winterthur, Switzerland

ISBN 978-3-030-40232-7 ISBN 978-3-030-40233-4 https://doi.org/10.1007/978-3-030-40233-4

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Mechanical engineers in hydraulics and hydromechanics are often required to compute hydraulic transients which occur in flow systems with process controls and frequent shutoff and restart. Especially in hydropower stations, hydraulic transients are always related to the operational reliability of hydraulic machines and the system security. Despite advanced computer technology, transient computations, however, can still not be simply performed. One often experiences that in practical applications, indeed, there are great confusions between quite many and different computational methods and algorithms. One reason for such great confusions is found in an insufficient understanding of fundamentals. It often happens with inexperienced people who invest great effort into “accurate” computations of the wave propagation speed, however, without knowing how and how much the wave propagation speed will finally affect computations. Another reason is found in complex fluid mechanics of coupled hydraulic transients and hydraulic characteristics of hydraulic machines like pumps and turbines. For these reasons, hydraulic transients represent highly complex fluid flow processes. Their computations, therefore, decisively rely on effective computational methods. In the past, the most widely applied computational method is the method of characteristics (MOC). It particularly requires generating so-called characteristic grids and demands computations of both pressures and flow velocities at each node of the time–space grids. Because of the large amount of computations, applications of the method are all limited to the use of either individual or commercial large software programs. On the other hand, most hydraulic systems in hydropower stations are highly complex, which additionally hinders accurate computations. The author of this book has been engaged in the field of hydropower for about two decades. He has continuously experienced that in the hydromechanical discipline great potentials exist in extending and completing fundamentals, improving design and computational methods, and enhancing hydraulic performances of related machineries. At least in hydromechanics of Pelton and Francis turbines, the existences of such great potentials have been proved. The author began to compute hydraulic transients in hydropower stations in the year 2008 when he used to work at the Oberhasli Hydroelectric Power Company (KWO Ltd.). In the followed years, v

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he carried out a great number of hydraulic transients in different hydraulic systems. During this time and till now, he has seen and felt great confusion in the world of computational methods of hydraulic transients. He especially worked out and completed the wave tracking method (WTM) for accurate computations. The method appears to be quite simple and easily accessible so that he could accurately complete all computations simply by using MS Excel. As a matter of fact, hydraulic transients reflect accurate concepts of fluid mechanics and therefore can be accurately computed by correctly specifying all related initial and boundary conditions, without using any controversial hypotheses. This implies that computation results generally do not need any experimental validation. Because of this feature, methods for transient computations are significantly superior to CFD, which relies on different turbulence models and always needs experimental validation—mostly inaccessible to engineers. All computational examples presented in this book demonstrate this fact. The author of this book explains that for different reasons, including lack of time, most part of the contents of this book has not been published in any form. It appears, in fact, to be rather difficult to separately publish individual topics, because almost all fundamental topics and computational algorithms are tightly connected with one another. Thus, this book gathers all research results achieved by the author through the last 10 years. The book is written for providing a useful reference for engineers and scholars who work in this area. Many computation examples with detailed computational algorithms are shown. The author, therefore, hopes that an engineer could make transient computations by his own programming. He is also hopeful that he could contribute to the final standardization of methods for transient computations. Finally, the author highly esteems and particularly thanks his lovely wife Nan for her great spiritual support in the author’s nonstop research activities in engineering science since decades and especially for the great patience she has shown in the last difficult years. Zurich, Switzerland October 2019

Zhengji Zhang

Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 History of Development of Computational Methods . . . 1.1.1 Main Contributions . . . . . . . . . . . . . . . . . . . . . 1.1.2 Confusion in Computations and Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Rigid and Elastic Water Column Theories . . . . . . . . . . 1.3 Wave Tracking Method (WTM) . . . . . . . . . . . . . . . . . 1.4 Method of Characteristics (MOC) . . . . . . . . . . . . . . . . 1.5 CFD and Its Restrictions . . . . . . . . . . . . . . . . . . . . . . . 1.6 Design Aspects of Hydraulic Systems . . . . . . . . . . . . . 1.7 Objectives and Main Content of This Reference Book . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stationary Flows and Flow Regulations . . . . . . . . . . . . . . . . . . 2.1 Laws of Flow Friction and Pressure Drop . . . . . . . . . . . . . 2.1.1 Laminar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Turbulent Flows in Hydraulically Smooth Pipelines 2.1.3 Turbulent Flows in Hydraulically Rough Pipelines . 2.1.4 Resistance Constants . . . . . . . . . . . . . . . . . . . . . . 2.2 Flow Resistance Constants in Pipeline Systems . . . . . . . . . 2.2.1 Pipeline Network of Pipes in Series Connection . . . 2.2.2 Pipeline Network of Pipes in Parallel Connection . . 2.2.3 Shock Losses and Borda-Carnot Formula . . . . . . . 2.3 Hydraulic Characteristics of Regulation Organs . . . . . . . . . 2.3.1 Characteristic of the Injector Nozzle of the Pelton Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Characteristic of the Spherical Valve . . . . . . . . . . . 2.3.3 Characteristic of the Butterfly Valve . . . . . . . . . . . 2.3.4 Characteristic of the Gate Valve . . . . . . . . . . . . . .

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Flow Regulations in Pipeline Systems . . . . . . . . . . . . . . 2.4.1 Simple Pipeline Systems with One Regulation Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Pipeline Systems with Two or More Regulation Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Transient Flows and Computational Methods . . . . . . . . . . . . . . . 3.1 Occurrence of Hydraulic Transients in Hydropower Stations . 3.2 Method of Rigid Water Column Theory . . . . . . . . . . . . . . . . 3.2.1 Restrictions of Application . . . . . . . . . . . . . . . . . . . 3.2.2 Flows in Pipelines of Constant Cross-Sectional Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Flows in Specially Curved Pipelines of Constant Cross-Sectional Area . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Flows in Stepped Pipes . . . . . . . . . . . . . . . . . . . . . . 3.3 Method of Elastic Water Column Theory . . . . . . . . . . . . . . . 3.3.1 Joukowsky’s Equation . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Primary and Reflected Shock Waves . . . . . . . . . . . . 3.3.3 Influence of Wave Speed on Accuracies of Transient Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Self-stabilization of Transient Flows by Regulation Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Pipe Elasticity and Size Response to the Pressure Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Momentum Equations in Pipe Flows . . . . . . . . . . . . 3.3.7 Continuity Equation in Pipe Flows . . . . . . . . . . . . . 3.3.8 Wave Propagation Speed . . . . . . . . . . . . . . . . . . . . 3.4 Transverse Relation Between Rigid and Elastic Water Column Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Rigid Water Column Theory and Applications . . . . . . . . . . . . . . . . 77 4.1 Flow Oscillations in Open Pipeline Systems . . . . . . . . . . . . . . . 78 4.1.1 Damped Flow Oscillation in a U-Tube . . . . . . . . . . . . 78 4.1.2 Damped Flow Oscillation Between a Lake and a Surge Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Flow Regulation and Computations of Shock Pressures . . . . . . 91 4.2.1 Flow Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2.2 Computational Algorithms and Simplifications . . . . . . . 92 4.2.3 Pressure Response by Closing the Injector Nozzle . . . . 97 4.2.4 Pressure Response by Opening the Injector Nozzle . . . . 100 4.2.5 Pressure Response by Stepped Pipelines . . . . . . . . . . . 101

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Surge Tank Functionality and System Stability . . . . . . . . . . . 5.1 Functionalities of the Surge Tank . . . . . . . . . . . . . . . . . . 5.2 Momentum Equations and Numerical Solutions . . . . . . . . 5.2.1 Basic Computations . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Simplifications of Computations . . . . . . . . . . . . . 5.2.3 Reaction of the Surge Tank on the Turbine Start . 5.2.4 Reaction of the Surge Tank on the Turbine Shutdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Damping Effect of the Surge Tank Throttle Area . 5.3 System Stability Performance and the Thoma Criterion . . . 5.3.1 System Instability Owing to External Stimulations 5.3.2 Thoma Criterion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Elastic Water Column Theory and Fundamentals . . . . . 6.1 Transient Flow Mechanics and Differential Equations 6.2 Wave Equation and Wave Parameters . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Wave Tracking Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Fundamental Equations of the Wave Tracking Method . . . . . 7.1.1 Joukowsky’s Equation in Upstream Flows . . . . . . . . 7.1.2 Joukowsky’s Equation in Downstream Flows . . . . . . 7.1.3 Joukowsky’s Equation in Spatial Scale . . . . . . . . . . 7.2 Multiple Initial and Boundary Conditions . . . . . . . . . . . . . . . 7.3 Generation of Primary Shock Waves . . . . . . . . . . . . . . . . . . 7.3.1 Regulation Mechanism . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Upstream Shock Waves F of Primary Order . . . . . . 7.3.3 Downstream Shock Waves f of Primary Order . . . . . 7.3.4 Shock Waves by Predefined Flow-Rate Regulation . . 7.3.5 Connection of Shock Waves on Both Sides of a Hydraulic Machine . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Application Cases with a Pump and a Francis Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conservation and Traveling Laws of Shock Waves at Series Junctions of Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Conservation Laws of Shock Waves . . . . . . . . . . . . 7.4.2 Traveling Laws of Shock Waves . . . . . . . . . . . . . . . 7.4.3 Traveling Laws of Using Volume Flow Rate . . . . . . 7.5 Conservation and Traveling Laws of Shock Waves at T-Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 T-Junction of Diverging Flows . . . . . . . . . . . . . . . . 7.5.2 T-Junction of Converging Flows . . . . . . . . . . . . . . . 7.6 Traveling of Shock Waves Through an Orifice . . . . . . . . . . .

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7.7

Full Reflection of Shock Waves at the Reservoir . . . . . . . . 7.7.1 Entrance Velocity Effect . . . . . . . . . . . . . . . . . . . . 7.7.2 Quantification of the Entrance Velocity Effect . . . . 7.8 Total Reflection of Shock Waves at Closed Valve . . . . . . . 7.8.1 Valve at the Downstream End of a Pipe . . . . . . . . 7.8.2 Valve at Upstream of a Pipe . . . . . . . . . . . . . . . . . 7.9 Reflection of Shock Waves on the Moving Surface of Water in the Surge Tank . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Friction Effect on the Propagation of Shock Waves . . . . . . 7.10.1 General Friction Effect and Computations . . . . . . . 7.10.2 Overall Friction Effect in a Round Trip of a Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Local Resistance Effect on the Propagation of Shock Waves 7.12 Throttle Resistance at the Entrance of the Surge Tank . . . . . 7.13 Two Regulation Organs and Origins of Wave Generations . 7.14 Shapes of Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 7.15 Computational Examples and Algorithms . . . . . . . . . . . . . . 7.15.1 Closing of the Injector Nozzle . . . . . . . . . . . . . . . . 7.15.2 Opening of the Injector Nozzle . . . . . . . . . . . . . . . 7.15.3 Stepped Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . 7.15.4 Flow Oscillation in Surge Tanks . . . . . . . . . . . . . . 7.15.5 Flow Regulation Between Two Reservoirs . . . . . . . 7.16 Evaporation of Flows and Restrictions of Computations . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Characteristic Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Computational Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Generation of Shock Waves and CB at an Injector Nozzle . 8.4 Flow State at a Reservoir of Constant Height . . . . . . . . . . 8.5 Pressure Head at the Closed Valve . . . . . . . . . . . . . . . . . 8.6 Traveling Laws of Shock Waves at Series Junctions of Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Traveling Laws of Shock Waves at T-Junction . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Method of Direct Computations and Transient Conformity . . . 9.1 Method of Direct Computations (MDC) . . . . . . . . . . . . . . . 9.1.1 Remarks on the Method of Direct Computations . . 9.1.2 Numerical Computational Algorithms . . . . . . . . . . 9.2 Validation of the Direct Method and Computation Examples 9.2.1 Closing of the Injector Nozzle . . . . . . . . . . . . . . . . 9.2.2 Opening of the Injector Nozzle . . . . . . . . . . . . . . . 9.2.3 Stepped Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . .

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Pressure Jumps at the Beginning and End of Each Flow Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Pressure Jumps at the Beginning of Flow Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Pressure Jumps at the End of Flow Regulations . . . Pressure Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Pressure Fluctuations After Flow Regulations . . . . 9.4.2 Pressure Fluctuations During Flow Regulations . . . Transient Conformity of Rigid and Elastic Fluid Flows . . . . Simplifications of Computations . . . . . . . . . . . . . . . . . . . . Application to the Two-Step Closing of an Injector Nozzle . 9.7.1 Comparison Between MDC and WTM . . . . . . . . . 9.7.2 Time Increment Effect and Computational Bias . . . 9.7.3 Explicit Explanation of the Viscous Friction Effect . 9.7.4 Explicit Explanation of Overlapping Pressure Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks to the Method of Direct Computations . . . . . . . . .

10 Hydraulic Characteristics of Pumps and Turbines . . . . . . . . . . 10.1 Hydraulic Characteristics of the Pump . . . . . . . . . . . . . . . . 10.1.1 Characteristics in Terms of Coefficients of Discharge and Head . . . . . . . . . . . . . . . . . . . . . 10.1.2 Four-Quadrant Diagrams and Operation Map . . . . . 10.1.3 Unification of the Pump and the Valve Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Hydraulic Characteristics of the Pelton Turbine . . . . . . . . . 10.2.1 Characteristic of the Injector . . . . . . . . . . . . . . . . . 10.2.2 Power Output and Flow Regulations . . . . . . . . . . . 10.2.3 Linear Closing Law of Injectors . . . . . . . . . . . . . . 10.2.4 Parabolic Closing Law of Injectors . . . . . . . . . . . . 10.2.5 Unification of Characteristics of the Injector and the Spherical Valve . . . . . . . . . . . . . . . . . . . . 10.3 Hydraulic Characteristics of the Francis Turbine . . . . . . . . . 10.3.1 Characteristics in Terms of Unit Parameters . . . . . . 10.3.2 Master Equation of the Francis Turbine . . . . . . . . . 10.3.3 Reconstruction of the Master Equation of the Francis Turbine . . . . . . . . . . . . . . . . . . . . . 10.3.4 Unification of the Francis Turbine and a Spherical Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Application Examples of Complex Transient Computations . . . 11.1 Shut-Down of a Pelton Turbine . . . . . . . . . . . . . . . . . . . . . 11.2 Pump Emergency Stop with Simultaneous Closing of a Spherical Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Unified Characteristics and Rotor Dynamics . . . . . 11.2.2 Connection of Shock Pressures on Both Sides of the Pump Unit . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Determination Equation and Primary Shock Waves 11.2.4 Tracking the Shock Waves . . . . . . . . . . . . . . . . . . 11.2.5 Numerical Computations . . . . . . . . . . . . . . . . . . . . 11.2.6 Computational Results . . . . . . . . . . . . . . . . . . . . . 11.3 Pump Start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Computational Specifications and Algorithms . . . . . 11.3.2 Computational Results . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A: Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Appendix B: Characteristics of Regulation Organs . . . . . . . . . . . . . . . . . 301 Appendix C: Computation of the Pump Start by Neglecting the Water Hammer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Chapter 1

Introduction

Hydraulic transients represent highly complex fluid flow processes. They are encountered at almost all hydropower stations as well as at water supply networks, when the hydraulic systems are getting regulated, started-up or shut-down. Each time and under certain conditions, hydraulic transients often lead to various undesirable occurrences like rapid pressure rises, cavitations with noises, system instability, and cumulative fatigue of pipe materials. All these phenomena are usually well controlled based on optimized system designs and operations. Uncontrolled hydraulic transients in hydropower stations, for instance, take place at load rejections or emergency shutdowns of hydraulic machines like pumps and turbines. Because each occurrence of hydraulic transients leads to remarkable and rapid pressure rise in considered hydraulic systems, the phenomenon is in engineering applications also called pressure shock or water hammer. It represents a very important sub-discipline of fluid mechanics and has drawn great attention in related fields. Figure 1.1 illustrates the pressure response in a hydraulic system of a hydropower station, while the Pelton turbine was shut down and the injector nozzle was closed (Zhang 2009). Rapid pressure rise at the injector nozzle has been detected. In addition, high-frequency pressure fluctuations clearly indicate the elasticity of fluid flows. Here, one is usually particularly interested in the maximum pressure rise in the flow and the capacity of the installed surge tank. As a result of hydraulic transients, the maximum pressure rise in the flow could exceed the allowable maximum of the corresponding hydraulic systems. In worst cases, it could even lead to ruptures of both the pressurized pipelines and the housing of hydraulic machines like pumps and turbines. Examples of historical great accidents related to hydraulic transients can be found for instance in Chaudhry (2014) and Popescu et al. (2003). Another undesirable outcome of hydraulic transients is the flow oscillation in the considered hydraulic system after each flow regulation or startup of one or more installed hydraulic machines. Such a flow oscillation, which is comparable to that in Fig. 1.1, leads to corresponding oscillations of either the power input at a pump or the power output at a turbine. In general, hydraulic transients represent a highly harmful flow form in the fields of hydropower and water supply systems. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_1

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1 Introduction

Fig. 1.1 Pressure fluctuations at the injector of a Pelton turbine when the injector is closed (Zhang 2009), see also Fig. 3.1

The only purposeful utilization of the water hammer effect in pipe flows is found at so-called ram pumps. The phenomenon of the pressure rise associated with hydraulic transients was known already in the 18th century. In 1772, John Whitehurst invented a manually controlled pulsation engine called hydraulic ram. The machine utilizes the hammer effect in the flow to raise the water, without using any mechanical energy. The first self-acting ram pump was invented by Joseph Michel Montgolfier later in 1796. Since then, the hydraulic ram pump has been continuously improved and broadly applied, see, for instance, Najm et al. (1999). Hydraulic transients and the associated water hammer in flow systems generally behave as negative occurrences and cause operational restrictions, for instance in hydropower stations. During the period of occurrences, resultant changes in all related operational parameters, like the rapid pressure rise and the amplitude of flow oscillations, must be kept below predefined limits. These are substantial aspects one must be concerned with, both in the early construction phase as well as later in test and operational phases. They also need to be considered when modifying or extending a hydraulic system for different purposes. On the one hand, therefore, transient computations are indispensible for hydraulic designs of each hydraulic system with components like pipelines, surge tanks, and shut-off valves. They also provide the references for sizing the system and choosing materials of pipelines. On the other hand, both the operation and regulation restrictions of hydraulic machines in each given hydraulic system are all based on transient flow analyses and verifications. Because of the high complexities in fluid mechanics of hydraulic transients, the main task in the related field basically covers three categories, namely • Basics of the physics of hydraulic transients, • Development of reliable and efficient computational methods, • Case studies and application specifications.

1 Introduction

3

Regarding the physics of hydraulic transients, rigorous research works can be traced back to the 19th century, when the flow phenomena were found to be related with the elasticity of water. To the method of transient computations, basically, two essential theories or methods are available: rigid and elastic water column theories. The former is based on Newton’s second law of motion for solid bodies. The latter considers the compressibility property of water which represents a reasonable cause for both the pressure rise and pressure fluctuations in hydraulic transients. It is found in most case studies and applications. Both the fluid mechanics of hydraulic transients and the computational methods form the main contents of this book.

1.1 History of Development of Computational Methods 1.1.1 Main Contributions Under consideration of the theory of elasticity of the 19th century, investigations were mainly focused on the determination of theoretical values of the propagation speed of shock waves caused by a local disturbance in a fluid such as water. It had been known that the wave propagation speed is directly related to the Young’s modulus of elasticity. Even the influence of elasticity of pipe walls on the wave propagation in the flow had been examined. At water flows in pipelines, the wave propagation speed could reach 1476 m/s. Notable researchers as well as their activities and contributions can be found, for instance, in review articles of Tijsseling and Anderson (2004, 2006, 2007) and Chaudhry (2014). On the other hand, the fundamental equation in water hammer theory, which relates the pressure rise to the change of flow velocity in a pipeline, has been found as follows a h = − c, g

(1.1)

where a is the wave propagation speed. In the case of slowing the flow, for instance, by closing the valve installed in a pipeline, it follows from c < 0 straightforwardly h > 0. In the contrary case there is h < 0. Because of the large value of the wave propagation speed, the resultant pressure rise is usually very high. It is hence also called the shock pressure which, as a type of wave, propagates in the flow. The above equation is known as Joukowsky’s equation, a distinctive name also known in the field of aerodynamics. At that time, Joukowsky was involved in a series of incidents with pipeline ruptures in the water supply network of Moscow. He conducted in 1897 extensive experiments with hydraulic transients. He especially investigated the transmission performance of shock waves in pipelines and developed a formula of wave propagation speed by additionally accounting for the elastic modulus of pipe walls. Based on these experimental and theoretical studies, he published

4

1 Introduction

his classic report on the basic theory of water hammer (Joukowsky 1898, English translation in 1904 by Simin). As stated, the first explicit statement of Eq. (1.1) about the water hammer is commonly attributed to Joukowsky. According to the reviews of Tijsseling and Anderson (2004, 2006, 2007), Eq. (1.1) has also been known by Johannes von Kries in 1883, even likely earlier, by Rankine in 1870 in a context which is more general than water hammer. Because Eq. (1.1) has also been derived by Frizell (1898) and Allievi (1902), unaware of the achievements by Joukowsky, it is also referred to as either Joukowsky-Frizell or frequently Allievi equation. Worth mentioning is the restriction of application of the Joukowsky formula in Eq. (1.1). A transient flow is assumed to be caused by closing the valve which is found at the downstream end of a simple pipeline with a length L. The shock wave generated at the valve propagates upstream towards the upper end of the pipe, gets reflected there, and travels back to the valve again. The time used for such a round trip is T 2L = 2L/a. The Joukowsky formula in Eq. (1.1) is applicable only within this time, as remarked in Fig. 1.2a. In the case of opening the valve, Eq. (1.1) is subjected to the same application restriction, as shown in Fig. 1.2b. Obviously, as the fundamental equation in water hammer theory it is highly restricted in practical applications. It is even unable to estimate the maximum pressure rise in a considered hydraulic system. Fig. 1.2 Application restriction of Joukowsky’s equation in the case of closing (a) and opening the flow (b)

1.1 History of Development of Computational Methods

5

This circumstance precisely points out that the main topic in transient computations is the computational method. Allievi (1902) published the “general theory” of water hammer. It is substantial only as a mathematical tool of the theory of water hammer, as Allievi later stated by himself. To represent shock pressures, i.e., for obtaining solutions of water hammers as functions of the steady-state velocity and the time, respectively, he introduced two dimensionless parameters for the flows in a simple pipeline as follows: ρ=

aTc ac0 and θ = 2g H0 2L

(1.2)

The definition is restricted to a simple pipeline of length L and of constant crosssectional area. c0 is the steady-state velocity and Tc the closing time of the flow. The available pressure head is given as H0 . The parameter θ is simply the dimensionless closing time related to the time for a round trip of the shock wave, T 2L = 2L/a. Obviously, the case considered is the simplest form of hydraulic transients caused by linear closing of the flow. By using two parameters defined above, it is conjectured that the maximum pressure rise, for instance, at the downstream end of the pipeline can be represented as Hmax − H0 = f(ρ, θ ). 2H0

(1.3)

In Chaudhry (2014), corresponding diagrams are also given. For θ ≤ 1, for instance, the maximum pressure rise at the downstream end of the pipe is independent of the closing time and is computed to be Hmax − H0 = 2ρ H0 . This relation, with respect to the definition of ρ, goes immediately back to the Joukowsky equation. For θ > 1, unfortunately, both the significance of using two parameters (ρ and θ ) and the related conclusions cannot be verified. One significant reason is that the maximum pressure head rise Hmax − H0 in the flow is basically independent of the static pressure head H 0 . It is, therefore, unreasonable to limit this maximum to the pressure head H 0 . Another main reason is the erroneous neglect of the so-called self-stabilization effect by using predefined velocity variations. More about this rather significant and non-negligible effect will be presented in Sect. 3.3.4 of this book. It is, however, worth mentioning that Allievi made good use of the general solution of the hyperbolic wave equation and suggested a progressive computational procedure. It is about a concept of using two co-existent dynamic parameters F and f, which in form of waves propagate along the pipeline in opposite directions. Although the applicability of the concept has been shown only for the simplest hydraulic system (reservoir-conduit-valve) with frictionless flow, the concept, indeed, describes a highly useful computational approach of hydraulic transients. The “wave plan method” which is initiated later by Wood et al. (1966, 2005) and also called Wave Characteristic Method (WCM) is clearly of the same computational structure as

6

1 Introduction

Allievi’s. Unfortunately, the suggested concept has not widely been and further followed in practical applications, although corresponding computational algorithms have been outlined, for instance, by Dubs (1947), Jaeger (1949), Sharp (1981), and Wylie and Streeter (1993). The reason of this undesirable situation might be the lack of effective computational tools which could enable complex hydraulic transients to be numerically solved. Only recently, Zhang (2016, 2018) demonstrated the extension and high applicability of the mentioned concept by presenting applications in complex hydraulic systems. It is about the wave tracking method (WTM), which will be further explained below in Sect. 1.3. Obviously, development of methods for transient computations has always been a significant issue. In the earlier time, when the complex hydraulic transients could not be numerically computed, they were mainly solved by the graphical method which was developed by Schnyder-Bergero. Because this method is highly inconvenient, it is basically only applicable to flows in simple pipelines with neglect of viscous friction effects. In today’s computer age, it is no longer applied. Both its algorithms and related developments will, therefore, not be further considered in this book. Another topic in transient computations is the specification of friction effects and the establishment of corresponding computational models. In the first approximation, the Darcy-Weisbach friction coefficient for stationary or quasi-stationary flows in pipes can be used for hydraulic transients which are commonly caused by flow regulations. For fast transients which is caused by instantaneous closing of the valve, however, the approximation will lead to remarkable computational inaccuracies, if compared with measurements (Zielke 1968; Lambert et al. 2001). For this reason, different friction models for transient computations have been developed in the past (Zielke 1968; Hino et al. 1977; Brunone et al. 1991, 1995; Vardy 1992). Some of them are based on the wall shear stress models and turbulence models considering the longitudinal turbulence nonuniformities. More complex are the considerations of two-dimensional (2D) velocity distributions in the pipeline and the use of a 2D turbulence water hammer model (Vardy and Hwang 1991; Pezzinga 1999, 2000; Zhao and Ghidaoui 2004). The method of modelling the transient frictions by using a convolution integral (Zielke 1968) is still further considered (Urbanowicz 2017) and even in more and more complex form. The updated explanations and computations of transient frictions can be found in Bergant and Simpson (1994), Bergant et al. (1999, 2001), Ghidaoui et al. (2005), Adamkowski and Lewandowski (2006), and Shamloo et al. (2015). In general, the transient frictions and friction models produce more damping compared to the quasi-steady friction model. The topic remains up to date and is subject to further investigations (Vardy and Brown 1995, 2003, 2004; Vitkovsky et al. 2000, 2006; Rathore et al. 2015). Although the unsteady friction in transient pipe flows affects transient processes differently than the steady friction, it is mostly of less engineering significance. First, the fast transients originating from the instantaneous closing of the valve in a hydraulic system does not often occur and is usually well controlled and protected. Second, the maximum shock pressure rise in fast transients is always restricted within the first period of each resultant pressure fluctuation and is therefore independent of the used friction models. The unsteady friction only causes an additional attenuation

1.1 History of Development of Computational Methods

7

of pressure fluctuations in the pipeline system. For this reason, the issue of unsteady frictions and the related friction models will not be further evaluated here. Throughout the current book, the Darcy-Weisbach friction coefficient for stationary flows is simply used in transient computations. This means that the transient flow is assumed to be quasi-stationary.

1.1.2 Confusion in Computations and Computational Methods The computational methods of hydraulic transients represent the main object in transient computations. The related computational concept, on the one hand, refers to the clarification and specification of all relevant boundary conditions. These include, for instance, the behavior of shock waves at the pipe entrance/end with constant pressure head because of the presence of reservoirs, at the closed regulation valve, in the surge tank with moving surface of the water, at series and T-junctions etc. The extended boundary conditions also include the transient friction models which are reviewed above. On the other hand, in relying on developed computer technologies, numerical methods become the standard of solving all complex hydraulic transients encountered in practice. Most transient computations are directly integrated into the CFD software or other software programs; it is the time of using advanced computational technologies in applied hydraulic transients. Unfortunately, the author recognizes the lack of understanding of fundamentals of hydraulic transients and therefore a great confusion in both the considerations of diverse boundary conditions and the employed computational algorithms. It occurs very often that simple fluid mechanics are made very complex by using different physical and mathematical models. This can be found not only in the employment of quite different friction models, as described above and more informatively in Ghidaoui et al. (2005). It is also related to different assumptions and computational algorithms used in computations. Among diverse initial and boundary conditions which are often poorly specified, a predefined velocity variation c = f(t) through a regulation valve has very often been used, which, however, does not represent the reality and thus always leads to significantly mistaken results. In addition, the moving surface of the water in the surge tank, as a special boundary condition in transient computations, remained indescribable for a long time. Furthermore, one still exercises great effort to accurately determine the wave propagation speed in the flow (Chaudhry 2014; Giesecke and Mosonyi 2009); in most applications, as in hydropower stations, this seems to be unnecessary. On the other hand, the great confusion in transient computations lies in quite different concepts of considering the hydro-mechanical performances of pumps, turbines, regulation devices and other components integrated in a hydraulic system. This second aspect also represents a great challenge for most engineers and scientists, because

8

1 Introduction

solid knowledge in hydro-mechanical characteristics of such hydraulic machines is indispensable for performing reliable transient computations. The common feature of all currently available computational methods is the complexity of computational algorithms. A brief review of part of them can be found, for instance, at Abuiziah et al. (2014) and Popescu et al. (2003). To simulate hydraulic transients which occur in a certain hydraulic component like the pump, for instance, even an equivalent electrical circuit as modularity in software has been used (Nicolet et al. 2007). In fact, most of and quite different computational algorithms for hydraulic transients have been implemented in either individual software programs or commercial software of using CFD. They are all based on developments through years and mostly function as a black box for their end-users like engineers and scientists. Besides the great confusion regarding the methods of transient computations, developments of computational methods are subject, to some extent, to considerable stagnations. One may confirm, for instance, that diverse FORTRAN programs, which were written 40 years ago for solving special problems, still find their applications in the leading literatures (Chaudhry 2014). It can be concluded that large computational expenses associated with all available computational algorithms might be the main reason why there is still no standard procedure for computing hydraulic transients. Moreover, it seems to be still impossible to perform transient computations based on the engineers’ own and real-time programming. Even the most applied method of characteristics (MOC, see below) is hardly accessible because it demands a fairly great amount of computations in both the time and spatial domains. The great variety and complexity of computational methods especially prevent engineers from entering this special field. It can, therefore, be expected that a standard method for computing hydraulic transients will only be given, if all related computational algorithms can be simplified and standardized. Fortunately, as the author of this book firmly believes, this will become much possible soon. By using the wave tracking method (WTM), one is able to compute all types of complex hydraulic transients, even simply with the aid of MS Excel, as already demonstrated in completed computation examples (Zhang 2016, 2018).

1.2 Rigid and Elastic Water Column Theories In terms of hydrodynamics, hydraulic transients can be computed by employing either the rigid water column theory for approximations or by the elastic water column theory under consideration of the compressibility of water. The former basically only applies to fluid flows in a simple and relatively short pipeline,1 when aiming to

1 The notion “relatively short” designates the boundary between two theories, beyond which the rigid

water column theory does not reflect the real flow conditions and will lead to computation errors, see Sect. 3.2.1. In practical applications, the condition “short pipes” is usually always fulfilled.

1.2 Rigid and Elastic Water Column Theories

9

compute the time-dependent pressure rise in the flow. The fluid in a pipeline of constant cross-sectional area is simply considered as a solid body which is dynamically balanced by all active forces in line with Newton’s second law of motion. Basically, the rigid water column theory is as well capable as the elastic water column theory in computing low-frequency flow oscillations which occur, for instance, between the lake and a surge tank in the hydraulic system of a hydropower station (Fig. 1.1). Such computations usually aim to determine the necessary size of the surge tank during the design phase. It should be indicated that the neglect of the compressibility of the flow does not always contribute to the simplification of computations. At a pipeline network which, for instance, consists of N pipes of different diameters and connections, then N momentum equations must be simultaneously solved. This feature of computations inevitably requires a great number of iterative procedures in numerical computations and always leads to high computational expenses. The fact to be mentioned is that the rigid water column theory is unable to resolve high-frequency pressure fluctuations like those in Fig. 1.1. At the elastic water column theory, the compressibility of a fluid flow is expressed by the bulk or elastic modulus which is basically a thermodynamic quantity. In the context of hydraulic transients and computations, the elastic modulus of water is a parameter which determines the propagation speed of shock waves in the flow. The elastic modulus of water at a temperature of T = 20 °C, for instance, is equal to E = 2.18 GPa, with which the maximum wave propagation speed is computed to be a = 1476 m/s. As a matter of fact, both the elastic modulus of pipe wall materials and the geometrical size of pipes also affect the wave propagation speed so that the real wave speed is somewhat smaller than that maximum, see Sect. 3.3.8. There are two special aspects with regard to the influence of the wave propagation speed on transient computations. To the first, the fact that the wave propagation speed in pipe flows also depends on the pipe material and its size has led to great efforts in analyzing the related dependences. The efforts seem to be justified, because according to Joukowsky’s formula in Eq. (1.1) the pressure rise in the flow is directly proportional to the wave speed. In fact, this proportionality is only restricted to a time period t ≤ T2L , see Fig. 1.2. The fact is that the wave speed affects the frequency of high-frequency pressure fluctuations proportionally (Fig. 1.1). In most cases, however, it does not influence the maximum pressure rise in the flow. This statement and circumstance will be frequently confirmed by computations in the current book. To the second, there is apparently a paradox in transient computations between the rigid and the elastic water column theories. The incompressibility of water, which is assumed in the rigid water column theory, conceptually means infinite wave propagation speed. Because under the condition of “short pipes” the rigid water column theory also provides correct computational results (without including high-frequency pressure fluctuations), the maximum pressure rise must be independent of the wave propagation speed. This is an indirect validation of the first special aspect above. In reality, an intrinsic physical relation between the rigid and elastic water column theories exists, which is called the conformity of hydraulic transients and will also be treated in the current monograph.

10

1 Introduction

The compressibility of water as an additional phenomenological aspect in transient computations, actually, behaves as a favorable property and hence contributes to the simplification of transient computations. This is simply so because, regarding the limited wave propagation speed, the flows at different locations in a pipeline system do not instantly affect each other. At each time-step when carrying out numerical computations, the flows at different locations as well as in different pipelines can be separately computed in the way that each time one only needs to solve one momentum equation. Such a remarkable attribute of the elastic water column theory reinforces the position of the theory in applied hydraulic transients. In practice, almost all transient computations have been relying on fundamentals of the elastic water column theory. The related history is in effect the history of creating and improving advanced computational methods. In this process, the advanced computer technology has played an important role.

1.3 Wave Tracking Method (WTM) The limited wave propagation speed in water permits the propagation of pressure shock waves in the flow to be tracked in the time series. Such pressure shock waves are basically represented by two co-existent dynamic parameters F and f which are solutions of the first order hyperbolic wave equation. Commonly, F is denoted as the primary and f the reflected wave (in the current book, they are redefined as upand downstream waves, respectively). Both F and f involve information about the flow rate and the static pressure in different forms. The concept of tracking these two opposite waves has been suggested by Allievi (1902) in his study, already mentioned in Sect. 1.1. In reality, it represents a Lagrangian approach in fluid mechanics and provides a much capable and hence much applicable method in transient computations. Especially, it does not require any equidistant space gridding. Transient computations in the time and space domains, as at MOC, become simplified by basically performing computations only in the time series. The wave plan method which is due to Wood et al. (1966) is a comparable concept. The method of tracking wave parameters F and f according to Allievi’s suggestion conceptually appears to be simple and easily applicable. Methodologically, however, it lacks a clear procedure how it works. This might be the main reason that the method has hardly been further followed. Basically, two significant issues associated with the method have to be concerned: • Generation of primary waves • Tracking of wave propagations. In many conceptual descriptions of the method of tracking waves, no procedures have been shown how the primary wave F can be generated at all, for instance, at a regulation valve or, more complex, at a pump. Obviously, the generation of primary waves essentially depends on hydraulic characteristics of each included hydraulic

1.3 Wave Tracking Method (WTM)

11

machines and components. This must be considered as additional difficulties in general transient computations. In fact, the described issues also exist in all other methods including the method of characteristics (MOC). Another issue associated with the method of tracking shock waves is the mechanism how a pressure shock wave to be tracked will behave while passing through a discontinuous section like a series junction or a T-junction. Each approaching wave is subject to reflection and transmission. For a long time, it was not explicitly indicated how all departure waves can be computed from known approaching waves. This actually represents a key point for further tracking them. Recently, the concept of tracking shock waves in transient computations has been extended and further developed by Zhang (2016, 2018). It represents a highly advanced method which has been validated by solving hydraulic transients in highly complex hydraulic systems. In relation with two significant issues listed above, significant points in Zhang (2016, 2018) are summarized as follows: First, the so-called wave tracking method (WTM), as expected, does not require dividing the pipeline into equidistant grids. For computing the pressure rise at the section of the regulation valve or T-junction, one only needs to track two wave parameters F and f in the time series until reaching that location. Against the method of characteristics, the wave tracking method can highly reduce the number of calculations. Second, the significant generation mechanism and computation procedures of primary shock waves have been outlined and given. One of presented computation examples is related to a complex hydraulic system with pumps and spherical valves. An hydraulic transient arising from an emergency shut-down of one pump with simultaneous closing of the spherical valve have been computed and compared with experiments. Furthermore, for tracking wave propagation in a pipeline network, both the wave reflection and transmission at each discontinuous section like a stepped series junction or a T-junction need to be computed. For a single approaching wave, reflection and transmission parts, respectively, are known, as can be found, for instance, in Löwy (1928) and Chaudhry (2014). For three given waves approaching a T-junction according to Fig. 1.3, Zhang (2016, 2018) derived two wave conservation laws, which enable departure waves to be computed as follows: f1 =

A1 − A2 − A3 2 A2 2 A3 F1 + f2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(1.4)

F2 =

2 A1 A2 − A1 − A3 2 A3 F1 + f2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(1.5)

f3 =

2 A1 2 A2 A3 − A1 − A2 F1 + f2 + F3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(1.6)

In the current book, these relations are denoted traveling laws of shock waves. For stepped series junction, one only needs to use the first two equations by setting A3 = 0.

12

1 Introduction Lake

Surge tank A3, Q3 A2, Q2

A1, Q1

to turbines

F3

f3

F1

F2

f1

f2

Fig. 1.3 Transmissions and reflections of pressure shock waves while passing through a T-junction in a pipeline system

The easy access and applicability of the wave tracking method have especially been demonstrated in the two given references of Zhang by performing entire computations simply with MS Excel sheet. For one computation example, the technique of unifying characteristics of the pump and the spherical valve has been demonstrated and applied. The entire computations cover all quantities like flow rate in the system, reverse flow through the pump, pressure response, deceleration of rotational speed and system oscillations. Both high accuracy and reliability of computations have all been reached. Another computation example is related with the shut-down of a Pelton turbine. Figure 1.4 shows the comparison of computations and measurements. Because of its outstanding advantages against all other methods, the wave tracking method constitutes the main object of the current book. Fig. 1.4 Pressure response in the flow by non-linearly closing the injector of a Pelton turbine (Zhang 2018)

1.4 Method of Characteristics (MOC)

13

1.4 Method of Characteristics (MOC) Among diverse available computational methods in connection with advanced computer technology, the most widely applied method for computing hydraulic transients up to now is the method of characteristics (MOC). The method relies on considerations of characteristic lines of the wave equation and further on the finite difference techniques (Gray 1953; Streeter and Lai 1962; Giesecke and Mosonyi 2009; Horlacher and Lüdecke 2012). In details, the characteristic lines of the wave equation are represented by discretization of both the time and the space, which is usually expressed as “characteristic grids” (Chaudhry 2014). Hence, each computation, i.e., numerical approach of hydraulic transients is quasi of two-dimensional nature. The method basically requires an equidistant space grid along the pipeline or even in the entire pipeline network. On each node of such a characteristic grid (space and time), both the pressure and the flow velocity must always be computed, because they will be used for further computations of flows at all other space nodes in all next time steps. In many cases, actually, a fine equidistant space grid appears to be unnecessary, if e.g. only local pressure rises at pipe cross-sections with valves or at T-junctions are interested. For this reason, the method of directly tracking shock waves (WTM) appears to be much more applicable.

1.5 CFD and Its Restrictions The advanced progress in applied hydraulic transients in the past decades is impressed by numerical simulations of fluid flows. A great number of investigations rely on computational fluid dynamics (CFD) and therefore only restricted to case studies. One could easily find hundreds of application examples in both journals and conference papers, so that no representative references are given here. Because for engineering applications, all transient flows in pipeline networks can be considered to be one-dimensional, direct applications of continuity, momentum and energy laws are sufficiently accurate. This is the concept, based on which different methods including the method of characteristics (MOC) and the wave tracking method (WTM) are developed. Therefore, it is basically not necessary to solve the Navier-Stokes equations under transient conditions. In addition, as presented and clarified in this book, hydraulic transients reflect accurate fluid mechanics and thus are expected to generate accurate solutions. For computations, therefore, no controversial assumptions and hypotheses are required, so that the computational results generally do not need any experimental validation. In contrast, the numerical simulation of transient flows by solving the Navier-Stokes equations in CFD under different turbulence models always requires much more computational time. Furthermore, the computational results still require experimental validation. All these features indicate that CFD for transient computations is not only inaccurate but also very expensive.

14

1 Introduction

1.6 Design Aspects of Hydraulic Systems Each hydraulic system is designed for performing a series of special tasks. Thus, the functionality and the operational safety of the system are two significant aspects. Often, as for economic reasons, a hydraulic system may have multiple functionalities and thus complex structures. This also includes the extension and the optimization of an existing system. The operational safety of a hydraulic system must therefore be validated each time, at least by detailed computations of hydraulic transients. Sometimes, experimental validations are also necessary if they are available. Figure 1.5 shows, for instance, a complex hydraulic system in a hydropower station. The existing hydraulic system consists of two surge tanks in the high-head part and five hydraulic machine groups (turbines and pumps) in the machine house 1. If the existing hydraulic system is going to be extended by further installing a group of hydraulic machines, a great number of operational varieties must be accounted for. Especially, for dimensioning a new surge tank, the impact of emergency shut-down of the entire system must be included (Zhang 2012). Among many components, the hydraulic surge tank behaves as a significant component at almost all hydropower stations. It is designed towards its multiple functionalities. For the turbine mode of operation, for instance, the surge tank has to be capable to provisionally provide water flow for turbines which are getting started. By stopping the turbine flow or reducing the flow rate, it then absorbs remaining kinetic energy which is included in the flow between the surge tank and the lake. For this reason, it is one of the main tasks in transient computations to correctly size each surge tank. Another hydraulic aspect related to the surge tank size is the stability performance of the hydraulic system with regard to flow oscillations between the surge tank and Surge tank

Surge tank Surge tank

Lake

Surge tank Machine house 1 to be extended Surge tank Machine house 2

Fig. 1.5 A complex hydraulic system in a pumped storage power station, to be extended with the machine house 2 and additional surge tanks

1.6 Design Aspects of Hydraulic Systems

15

the lake. Such a low-frequency flow oscillation occurs after each flow regulation including the start and the stop of the hydraulic machines, as shown in Fig. 1.1. As a natural process, it is always attenuated. Under certain condition with synchronized external stimulations, however, the described flow oscillation might become unstable. According to Thoma (1910), a minimum diameter of surge tank exists, below which the system could become unstable. The so-called Thoma criterion has, thus, very often been used for determining the minimum size of the surge tank. Actually, flow regulations at turbines with the mentioned synchronized external stimulations do not occur in any hydropower stations for a long time. For sizing the surge tank, basically the rigid water column theory can be applied. At complex hydraulic systems like that in Fig. 1.5, however, the elastic water column theory can be preferably applied because at any arbitrary location the flow state is then instantaneously independent of the flow states elsewhere. To the planed extension of an existing hydraulic system according to Fig. 1.5, the author has carried out all transient computations in relying on both the rigid water column theory and the wave tracking method (WTM), as reported in Zhang (2012). All computations are performed with the use of MS Excel.

1.7 Objectives and Main Content of This Reference Book The objectives of this book are, on the one hand, to describe and reveal most relevant fluid-mechanical features of hydraulic transients and, on the other hand, to present related computational methods. These are two categories or aspects which hydraulic transients represent. Regarding the aspects of fluid mechanics of hydraulic transients, significant fundamentals including new knowledge and findings will be presented. This comprises, for instance, the role of the wave propagation speed in transient computations, the self-stabilization effect of transient flows through valves, and the conservation and traveling laws of shock waves in pipeline networks. For the last see Eqs. (1.4)– (1.6). While the compressibility of water has been accounted for in almost all transient computations, the rigid water column model for transient computations will be demonstrated to be comparably applicable. Especially, the conformity of hydraulic transients will be presented; it represents the inherent connection between the rigid and elastic water column theories. Regarding computational methods, which indeed represent the main object of transient computations, the extended wave tracking method (WTM) will be presented in details. This method, when compared with the method of characteristics (MOC), shows great advantages in computational simplicity and reliability, supported by exact specifications of a great number of initial and boundary conditions. These include, for instance, computational algorithms of primary shock waves, traveling performances of shock waves at T-junctions, and reflection behaviors of waves on the moving free surface of water in the surge tank.

16

1 Introduction

Furthermore, the method of direct computations (MDC) will be demonstrated which is derived from the wave tracking method (WTM) and based on conformity of hydraulic transients. It is an extremely simple method, is as accurate as WTM, although it is only applicable to relatively simple hydraulic systems. With respect to applications in hydropower stations with pumps and turbines, hydraulic characteristics of machines will be presented and given in appropriate forms for easy applications. To simplify computations, the method of unifying characteristics of respective hydraulic machine and control valves will be described. Since in the field of hydraulic transients there is still a great confusion regarding different computational methods and algorithms, the current book tries to provide a useful reference for engineers and scientists who work with designs and evaluations of hydraulic systems under transient flow conditions. The author also hopes to be successful in the standardization of methods of transient computations. For this purpose as well as for guiding practical applications, diverse application examples are shown in details in form of tabulated, i.e., numerical computations.

References Abuiziah, I., Oulhaj, A., Sebari, K., & Ouazar, D. (2014). Comparative study on status and development of transient flow analysis including simple surge tank. International Scholarly and Scientific Research & Innovation, 8(2). Adamkowski, A., & Lewandowski, M. (2006). Experimental examination of unsteady friction models for transient pipe flow simulation. Journal of Fluids Engineering, 128, 1351–1363. Allievi, L. (1902). General theory of the variable flow of water in pressure conducts, see “Theory of Water Hammer” (E. E. Halmos, Trans.). Typography Riccardo Garoni, Rome, 1925. Bergant, A., & Simpson, A. R. (1994). Estimating unsteady friction in transient cavitating pipe flow. In Proceedings 2nd International Conference on Water Pipeline Systems (pp. 333–342). Edinburg, Scotland. Bergant, A., Simpson, A. R., & Vitkovsky, J. (1999). Review of unsteady friction models in transient pipe flow. In 9th International Meeting on the Behaviour of Hydraulic Machinery Under Steady Oscillatory Conditions. Brno, Czech Republic: IAHR. Bergant, A., Simpson, A. R., & Vitkovsky, J. (2001). Developments in unsteady pipe flow friction modelling. Journal of Hydraulic Research, 39(3), 249–257. Brunone, B., Golia, U. M., & Greco, M. (1991). Some remarks on the momentum equations for fast transients. In IAHR, Proceedings of Meeting on Hydraulic Transients with Column Separation, 9th Round Table. Valencia, Spain. Brunone, B., Golia, U. M., & Greco, M. (1995). Effects of two dimensionality on pipe transients modeling. Journal of Hydraulic Engineering, 121(12), 906–912. Chaudhry, M. H. (2014). Applied hydraulic transients (3rd ed.). New York Inc: Springer-Verlag. Dubs, R. (1947). Angewandte hydraulik. Zurich: City-Druck. Frizell, J. P. (1898). Pressures resulting from changes of velocity of water in pipes. Transactions of the American Society of Civil Engineers, 39, 1–18 (paper 819). Ghidaoui, M. S., Zhao, M., McInnis, D. A., & Axworthy, D. H. (2005). A review of water hammer theory and practice. Applied Mechanics Reviews, Transactions of the ASME, 58, 49–76. Giesecke, J., & Mosonyi, E. (2009). Wasserkraftanlagen, 5. Auflage. Springer-Verlag. Gray, C. A. M. (1953). The analysis of the dissipation of energy in water hammer. Proceeding ASCE, 119, 1176–1194 (paper 274).

References

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Hino, M., Sawamoto, M., & Takasu, S. (1977). Study on the transition to turbulence and frictional coefficient in an oscillatory pipe flow. Transactions of the Japan Society of Mechanical Engineers, 9, 282–284. Horlacher, H., & Lüdecke, H. (2012). Strömungsberechnung für Rohrsystem. Expert Verlag. Jaeger, C. (1949). Technische Hydraulik. Verlag Birkhäuser Basel. Joukowsky, N. E. (1898). Waterhammer, Memoirs Imperial Academy Society of St. Petersburg, (Vol. 9, No. 5) (in Russian). In O. Simin (Trans.). (1904), Proceeding American Water Works Association (Vol. 24, pp. 341–424). Lambert, M. F., Vitkovsky, J. P., Simpson, A. R., & Bergant, A. (2001). A boundary layer growth model for one-dimensional turbulent unsteady pipe friction. In 14th Australasian Fluid Mechanics Conference. Adelaide, Australia: Adelaide University, 10–14 December 2001. Löwy, R. (1928). Druckschwankungen in Druckrohrleitungen. Wien: Julius Springer. Najm, H. N., Azoury, P. H., & Piasecki, M. (1999). Hydraulic ram analysis: A new look at an old problem. Proceedings of the Institution of Mechanical Engineers, Part A, Journal of Power and Energy, 213(2), 127–141. Nicolet, C., Allenbach, P., Simond, J., & Avellan, F. (2007). Modeling and numerical simulation of a complete hydroelectric production site. Powertech, 2007 IEEE Lausanne, Switzerland, July 2007. Pezzinga, G. (1999). Quasi-2D model for unsteady flow in pipe networks. Journal of Hydraulic Engineering, 125(7), 676–685. Pezzinga, G. (2000). Evaluation of unsteady flow resistances by quasi-2D or 1D models. Journal of Hydraulic Engineering, 126(10), 778–785. Popescu, M., Arsenie, D., & Vlase, P. (2003). Applied hydraulic transients for hydropower plants and pumping stations. AA Balkema Publishers. Rathore, V., Ahmad, Z., & Kashyap, D. (2015). Modelling of transient flow in pipes with dynamic friction. In 20th International Conference on Hydraulics, Water Resources and River Engineering. Roorkee, India. Sharp, B. (1981). Water hammer, problems and solutions. Edward Arnold (publishers) Ltd. Shamloo, H., Norooz, R., & Mousavifard, M. (2015). A review of one-dimensional unsteady friction models for transient pipe flow. In Second National Conference on Applied Research in Science and Technology, Science Journal (CSJ), Special Issue (Vol. 36, No. 3). Streeter, V. L., & Lai, C. (1962). Waterhammer analysis including fluid friction. Journal of the Hydraulics Division, ASCE, 88(HY3), 79–112. Thoma, D. (1910). Theorie des Wasserschlosses bei selbsttätig geregelten Turbinenanlagen. Dissertation, Kgl. Technische Hochschule zu München, Oldenburg in München, Germany. Tijsseling, A. S., & Anderson, A. (2004). A precursor in waterhammer analysis, rediscovering Johannes von Kries (RANA: Reports on applied and numerical analysis; Vol. 0402). Eindhoven: Technische Universiteit Eindhoven. Tijsseling, A. S., & Anderson, A. (2006, January). The Joukowsky equation for fluids and solids. Journal of Scientific Computing. Tijsseling, A. S., & Anderson, A. (2007, January). Johannes von Kries and the history of water hammer. Journal of Hydraulic Engineering, ASCE. Urbanowicz, K. (2017). Analytical expressions for effective weighting functions used during simulations of water hammer. Journal of Theoretical and Applied Mechanics, 55(3), 1029–1040. Vardy, A. E. (1992). Approximating unsteady friction at high Reynolds numbers. In Proceedings International Conference on Unsteady Flow and Fluid Transients (pp. 21–29). Durham, England. Vardy, A. E., & Brown, J. M. B. (1995). Transient, turbulent, smooth pipe friction. Journal of Hydraulic Research, 33(4), 435–456. Vardy, A. E., & Brown, J. M. B. (2003). Transient turbulent friction in smooth pipe flows. Journal Sound and Vibration, 259(5), 1011–1036. Vardy, A. E., & Brown, J. M. B. (2004). Transient turbulent friction in fully rough pipe flows. Journal Sound and Vibration, 270(12), 233–257.

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1 Introduction

Vardy, A. E., & Hwang, K. L. (1991). A characteristics model of transient friction in pipes. Journal of Hydraulic Research, 29(5), 669–684. Vitkovsky, J., Bergant, A., Simpson, A., & Lambert, M. (2006). Systematic evaluation of onedimensional unsteady friction models in simple pipelines. Journal of Hydraulic Engineering Division of the American Society of Civil Engineers, 132(7), 696–708. Vitkovsky, J., Lambert, M., Simpson, A., & Bergant, A. (2000). Advances in unsteady friction modelling in transient pipe flow. In 8th International Conference on Pressure Surges. The Hague, The Netherlands: BHR. Wood, D. J., Dorsch, R., & Lightner, C. (1966). Wave analysis of unsteady flow in conduits. Journal of Hydraulics Division, ASCE, 92(HY2), 83–110. Wood, D. J., Lingireddy, S., Boulos, P. F., Karney, B. W., & Mcpherson, D. L. (2005). Numerical method for modeling transient flow. Journal AWWA, 97–7, 104–115. Wylie, E. B., & Streeter, V. L. (1993). Fluid transients in systems. Englewood Cliffs, USA: PrenticeHall. Zhang, Z. (2009). Druckstossberechnung beim Schliessen der Peltonturbine in Grimsel 1. Technical report. Oberhasli Hydroelectric Power Company (KWO), Innertkirchen, Switzerland, Nr. KWO_TB09_00x, 16, Juni 2009. Zhang, Z. (2012). Hydrodynamisches Verhalten von Wasserschlössern im Triebwassersystems Gr2 und Gr3. Technical report. Oberhasli Hydroelectric Power Company (KWO), Innertkirchen, Switzerland, Nr. A000246540. Zhang, Z. (2016). Transient flows in pipe system with pump shut-down and the simultaneous closing of the spherical valve. In IAHR, 28th symposium on hydraulic machinery and systems. Grenoble, France, see also IOP Conference Series: Earth and Environmental Science (Vol. 49, p. 052001). https://doi.org/10.1088/1755-1315/49/5/052001. Zhang, Z. (2018, August). Wave tracking method of hydraulic transients in pipe systems with pump shut-off under simultaneous closing of spherical valves. Journal of Renewable Energy, 132, 157–166. Zhao, M., & Ghidaoui, M. S. (2004). Review and analysis of 1D and 2D energy dissipation models for transient flows. In Proceedings International Conference on Pressure Surges (pp. 477–492). Bedford, UK: BHR. Zielke, W. (1968). Frequency-dependent friction in transient pipe flow. Journal of Basic Engineering, ASME, 90(1), 109–115.

Chapter 2

Stationary Flows and Flow Regulations

Most industrial processes for fluid flows are stationary, even though they are all constructed with unsteady regulation facilities to achieve their control. Such processes are regularly found in hydraulic systems of hydropower stations. They ensure stationary power output of water turbines. At the nominated flow rates, hydraulic machines run under optimized operation conditions and with highest hydraulic efficiency. Stationary water flows in hydropower stations, thus, represent the basic form of all relevant hydro-mechanical processes. They also represent the initial state of all hydraulic transients, such as normal startup and shutdown conditions of machines as well as the system reactions caused by a power outage. For this reason, hydrodynamics of stationary flows and flow regulations will be treated in this chapter. All processes will be considered for fluids exhibiting viscous friction. Significant hydro-dynamical features of stationary flows are pressure drops due to both viscous friction and diverse local disturbances. The latter are generated by abrupt changes in cross-sections of the respective pipelines. This especially refers to regulation organs like control valves. The local pressure drop caused by a valve, for instance, is usually known and given by a resistance constant. Its dependence on the opening degree of the valve is called hydraulic characteristic. In hydropower applications, both spherical and butterfly valves are primarily used as closure organs for shutting off the flow. Their hydro-mechanical characteristics serve not only to regulate flows and flow processes, they are equally indispensable for computing hydraulic transients, which occur during the opening or the closing of valves. Flow regulations in hydraulic machines are usually performed by internal regulation devices of machines. In Francis turbines, for instance, the flow regulation is performed by changing the guide-vane angles. It is directly related to the complex turbine characteristics, commonly only obtainable by use of measurements. Differently, the discharge of Pelton turbines is regulated by the injector and can be expressed in quite a simple form. Because of this simplicity and especially in view of the physical disconnection between the pipeline system and the Pelton wheel, the injector of the Pelton turbine can be well used to simulate flow regulations, for both quasi-stationary flows as well as hydraulic transients. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_2

19

20

2 Stationary Flows and Flow Regulations

2.1 Laws of Flow Friction and Pressure Drop All engineering flows that are bounded by solid surfaces, as in pipelines and hydraulic machines, are subjected to viscous effects. This leads to the shear stress on the bounded solid surfaces, given by 1 τ = cf ρc2 , 2

(2.1)

with cf as the skin friction coefficient, or wall shear stress coefficient. The reference flow velocity c simply refers to the mean flow velocity in the pipeline (Fig. 2.1). The skin friction on the pipe wall leads to the boundary shear stress and thus the pressure drop in each pipe flow. In a circular pipe of constant diameter d, the resulting pressure drop due to the viscous effect is proportional to the pipe length L and can be characterized by the so-called Darcy-Weisbach equation p = λ

L1 2 ρc . d2

(2.2)

In this equation, λ is the Darcy-Weisbach friction coefficient of pipe flows. Sometimes, the resistance coefficient in terms of ζ = λ(L/d) is also used in Eq. (2.2). The relation between the skin friction coefficient cf in Eq. (2.1) and the friction coefficient λ in Eq. (2.2) can be derived from the momentum equilibrium applied to the mass flow in a given pipeline. For stationary flows, the skin friction drag force and the pressure force are found in balance, which, according to Fig. 2.1 with A as the pipe cross-section, is expressible as p A = τ (π d L).

(2.3)

Combining this with Eqs. (2.1) and (2.2), one immediately obtains λ = 4cf .

(2.4)

p1

c

p2

d

τ

τ

1

L

2

Fig. 2.1 Stationary viscous flow in a circular pipe of constant cross-sectional area

2.1 Laws of Flow Friction and Pressure Drop

21

Both the skin friction coefficient and the Darcy-Weisbach friction coefficient depend on the flow state (laminar or turbulent) and thus on the Reynolds number of the flow, defined as Re =

cd , ν

(2.5)

in which v denotes the kinematic viscosity of the fluid. For turbulent flows, the friction coefficient λ depends additionally on the roughness of the pipe surface. Essentially, the following rules for both the laminar and the turbulent flows are applied.

2.1.1 Laminar Flows Laminar flows in a circular pipe are found under the condition of Re < 2300. The viscous effect leads to a parabolic distribution of flow velocity, which is called Hagen-Poiseuille profile. The friction coefficient can in this case be expressed as λ=

64 . Re

(2.6)

It is independent of the roughness of the pipe surface.

2.1.2 Turbulent Flows in Hydraulically Smooth Pipelines Almost all engineering flows, especially in hydropower stations, are turbulent flows with Re > 2300. In hydraulically smooth pipelines and for Re = 3000–100,000, the friction coefficient is, according to Blasius, given by the empirical equation λ=

0.3164 . Re0.25

(2.7)

For flows with Re > 106 , the Prandtl and von Kármán equation can be applied  √  1 √ = 2 lg Re λ − 0.8. λ

(2.8)

22

2 Stationary Flows and Flow Regulations

2.1.3 Turbulent Flows in Hydraulically Rough Pipelines At turbulent flows in circular pipes with rough surface, the friction coefficients are additionally a function of the relative roughness height on the pipe surface. For such flows, the so-called Colebrook and White formula can be used to compute the friction coefficient as a function of both the Reynolds number and the pipe roughness height (k), as follows   1 2.51 k/d . √ = −2 lg √ + λ Re λ 3.71

(2.9)

For turbulent flows in smooth pipelines (k = 0), Eq. (2.9) goes back to Eq. (2.8). On the other hand, at large Reynolds number, the friction coefficient becomes to be independent of the Reynolds number. Correspondingly, Eq. (2.9) can be simplified. Commonly, one applies the Moody diagram (also known as the Moody chart) for friction coefficients in pipe flows.

2.1.4 Resistance Constants The pressure drop according to Eq. (2.2) can also be interpreted in the form of the pressure-head drop. With respect to the flow rate Q = c A, one obtains h = λ

L Q2 = R Q2. 2gd A2

(2.10)

In this equation, R denotes the resistance constant with unit s 2 /m 5 . Its use in the given form, i.e., as the coefficient of the square of the flow rate Q, is quite popular in the field of hydropower plants. For circular pipelines there is A = π d 2 /4, so that R takes the form R = 0.81λ

L . gd 5

(2.11)

For flows in channels of other cross-section forms, the equivalent diameter (deq = 4 A/U ) has often been used for computing the Reynolds number and further the 2 /4, the friction coefficient according to Eq. (2.9), for instance. Because A = π deq above equation for computing the resistance constant R is not applicable. The advantages of using Eq. (2.10) to compute the pressure-head drop in pipe flows are obvious, when the formula is applied, for instance, to the flow in a complex pipeline consisting of pipes of different diameters. Because of constant flow rate (Q = const) in each pipe, the flow resistance constant R plainly represents the total flow resistance in the considered pipeline. Besides the viscous friction effect, it may also involve effects of bends, abrupt changes in the pipe cross-section, and

2.1 Laws of Flow Friction and Pressure Drop

23

regulation organs. It can be easily determined by simply measuring one pressure head-drop and the related flow rate. This is the reason, why Eq. (2.10) has found its popular applications in hydropower plants. For more about its applications, see Sect. 2.2 below.

2.2 Flow Resistance Constants in Pipeline Systems A pipeline network is configured by pipes of different diameters and lengths. It basically consists of serial and parallel connection forms. Each pipe or pipeline is marked, at least, by viscous friction. Other local resistances causing pressure drops may also be included. Thus, for computing the overall pressure-head drop in a pipeline network, two basic connection forms of pipelines need be first considered.

2.2.1 Pipeline Network of Pipes in Series Connection If a pipeline is constructed by pipes of different diameters arranged in series, the flow rate through all pipes remains constant (Q = const). The overall resistance constant is then computed as R = Ri .

(2.12)

One finally obtains the total pressure-head drop in such a pipeline according to h tot = R Q 2 .

(2.13)

2.2.2 Pipeline Network of Pipes in Parallel Connection The parallel connection of two or more pipelines commonly aims to enlarge the flow area and therefore to enhance the flow rate, without causing an excessive pressure drop in the flow. If all resistance constants of the included pipelines are known, then the combined overall resistance constant can be computed. The physical background of the computations is analogous to that for electrical circuits. Figure 2.2 shows, for instance, two pipelines in parallel connection. The pressurehead drops in the two single pipelines as well as in the combined pipeline network are, according to Eq. (2.10), given by h 1 = R1 Q 21 ,

(2.14)

24

2 Stationary Flows and Flow Regulations

Resistance constant R1 Q1 Q

Q A

Q2 Resistance constant R2

B

Fig. 2.2 Parallel connection of two pipes with different geometrical configurations and hence different resistance constants

h 2 = R2 Q 22 ,

(2.15)

h = R Q 2 .

(2.16)

Because Q = Q 1 + Q 2 and h 1 = h 2 = h, one obtains 1 1 1 √ =√ +√ . R R2 R 1

(2.17)

The overall resistance constant of the considered pipeline network is found as R1 R= 2 . √ 1 + R1 /R2

(2.18)

For experimental determination of this resistance constant, one only needs to apply Eq. (2.16) based on one measurement of the pressure-head drop for a given flow rate. The total flow rate, which is given by Q = Q 1 + Q 2 , is divided into two branched pipelines. One finds out, respectively,  Q1 = Q  Q2 = Q

R , R1

(2.19)

R . R2

(2.20)

For parallel connections of three or more pipelines, computation results like Eq. (2.17) are equally obtainable.

2.2 Flow Resistance Constants in Pipeline Systems

25

2.2.3 Shock Losses and Borda-Carnot Formula Shock loss in fluid dynamics is in particular understood as energy loss which locally occurs in the flow at a pipe cross-section with abrupt change in cross-sectional areas. Such a pipe cross-section generally behaves as a local flow resistance, which can be described by its resistance coefficient. In this paragraph, only stepped pipe crosssections and an orifice will be considered. A stepped pipe cross-section conceptually means a section causing both a sudden expansion or sudden contraction of the flow (Fig. 2.3). The flow through an orifice may be considered as the combination of a contraction flow and a subsequent expansion flow. The loss in mechanical energy, in each case, occurs because of flow separation and the energy dissipation in generated vortices. It always takes the form of a pressure-head drop in the flow.

2.2.3.1

Sudden Expansion of the Flow

The shock loss which occurs in the flow with sudden expansion (Fig. 2.3a) is known as Borda-Carnot loss. On the generation mechanism, the Borda-Carnot shock loss actually happens in the flow downstream of the position of section expansion through turbulent vortex mixing and, consequently, energy dissipation in the flow. It can be accurately calculated by applying the momentum and Bernoulli energy equations between the pipe cross-section 1 before and the pipe cross-section 2 after the stepped pipe section. The pipe cross-section 2 has to be selected where the turbulent vortex mixing is completed and the flow distribution becomes again uniform. Usually, the d2, A2, c2, h2

(a) d1, A1, c1, h1 1

2

>10d2

(b) d1, A1, c1, h1 d2, A2, c2, h2 1

Ac

Fig. 2.3 Flow through pipe connections with sudden expansion (a) and sudden contraction (b)

2

26

2 Stationary Flows and Flow Regulations

distance to the extended pipe cross-section takes about 5–10 times of the pipe diameter. With the mean flow velocity before the flow expansion as the reference velocity, the Borda-Carnot shock loss in term of the head is computed as h shock =

  A1 2 2 1 1− c1 . 2g A2

(2.21)

In using the resistance constant and with respect to Q = c1 A1 , the above equation may also be written as h shock =

  A1 2 2 1 1 − Q = Rexpans Q 2 . A2 2g A21

(2.22)

The drop in static pressure head is calculated as h 12

1 1 = h1 − h2 = g A1 A2



 A1 − 1 Q2. A2

(2.23)

It obviously differs from the pressure-head drop according to the Bernoulli energy equation for flows free of losses. Because of A2 > A1 , however, there is still h 12 < 0, i.e., h 2 > h 1 . This indicates that the available kinetic energy in the upstream flow is partly dissipated because of flow separation and partly converted into pressure energy. According to Eq. (2.22), the shock loss associated with a sudden expansion of the flow is always less than the dynamic pressure head in the upstream flow. For this reason and in dealing with flows of hydropower stations, such a shock loss is almost negligible against the friction losses which all happen in long pipelines from hundreds to thousands meters. By the way, pipes with sudden expansion in cross-sections have never been found in hydraulic systems in hydropower stations. Especially, Borda-Carnot shock losses can be anyhow neglected when dealing with hydraulic transients.

2.2.3.2

Sudden Contraction of the Flow

In the case of flow which is subjected to a sudden contraction (Fig. 2.3b), the resultant shock loss in the flow is computed with the reference flow area A2 as h shock = ζ

1 Q2. 2g A22

(2.24)

The shock loss coefficient ζ can be approximated by   A2 . ζ ≈ 0.42 1 − A1

(2.25)

2.2 Flow Resistance Constants in Pipeline Systems

1

27

d

D

2

Fig. 2.4 Flow through an orifice

In all practical applications as in hydropower stations, pipe connections are always in the form of continuous transitions and the area ratio A2 /A1 , additionally, does not much differ from unity. Consequently, all related shock losses are doubtlessly negligible small, especially, if compared with the water hammer effect in transient flows. By neglecting this type of shock losses, the flow with sudden contraction can be simply treated by the Bernoulli equation.

2.2.3.3

Flow Through an Orifice

An orifice is commonly inserted within a pipeline of constant cross-section (Fig. 2.4). The total pressure-head drop in the flow is caused by flow contraction and subsequent flow expansion with total mixing of flow and the downstream redistribution. It depends on the ratio of the orifice diameter to pipe diameter β = d/D. With high accuracy, the resistance coefficient can then be computed by ζ =

2 1 1 + 0.7 1 − β 2 − β 2 . 4 β

(2.26)

In Idelchik (2007), a similar formula can be found. The pressure-head drop in the flow, while passing through an orifice, is then given as h = ζ

1 1 2 c =ζ Q 2 = Rorifice Q 2 . 2g 2g A2

(2.27)

In this last equation, the resistance constant Rorifice is again applied. Like that in Eq. (2.10), its use in hydropower stations significantly contributes to simplifications of all computations including those of hydraulic transients.

28

2 Stationary Flows and Flow Regulations

2.3 Hydraulic Characteristics of Regulation Organs For quite different reasons of applications, for instance, in a hydropower station, both the hydraulic loads and the operations of all included hydraulic machines need to be regulated. This also includes the opening and the closing of the hydraulic system when the machines are either started or stopped. There are plenty varieties of applied regulation apparatuses. Some of them are directly integrated into the hydraulic machines such as the guide vane apparatus in Francis turbines and the injector nozzles in Pelton turbines. For other separate regulation devices, gate valves, butterfly valves, and rotary spherical valves are commonly used. The choice of regulation organs, basically, depends on expected functionalities in operations and system pressures. To some extent, the injector nozzle of the Pelton turbine can also be considered as a separate regulation device, because it is always found within the hydraulic system, and it is mechanically and hydraulically disconnected with the turbine wheel. Figure 2.5 shows a typical hydraulic flow system with a spherical valve and a Pelton turbine at the downstream end of the pipeline. The injector of the Pelton turbine behaves as an independent regulation organ for regulating the discharge through the injector and hence the flow rate in the system. In particular, the hydraulic characteristics of the Pelton turbine are represented by the hydraulic characteristics of the injector. This, as shown below, is commonly determined through measurements. Worth mentioning here in advance is that the injector of the Pelton turbine especially represents a much favourable regulation organ which can be used to simulate the generation of hydraulic transients in any arbitrary hydraulic system. Corresponding applications will be shown later in other chapters. The spherical valve, as shown in Fig. 2.5, is commonly used to safely close the flow system, for instance, when the turbine is stopped. Therefore, it generally has two states: closed and fully opened. For system safety, it has been widely applied in hydropower stations with pumps and turbines. Under normal operation conditions including load regulations, the spherical valve is fully opened, without causing any energy head loss in the flow. It behaves, however, as a “regulation” organ while it is getting closed from its full open position, for instance, during the emergency shut-down of machines (pumps or turbines). All resultant phenomena like hydraulic transients and water hammer effect in the system, thus, significantly depend on the

Lake

Spherical valve

A, L, R

H0

Pelton turbine

Injector

Fig. 2.5 Closure and regulation organs in a pipeline system of the Pelton turbine

2.3 Hydraulic Characteristics of Regulation Organs

29

hydraulic characteristic of each used spherical valve. Because of its significance, the spherical valve together with its hydraulic characteristic will also be described below. Other regulation and closure organs which have often been found in hydropower stations are butterfly and gate valves. For completeness, hydraulic characteristics of these two types of valves, based on the author’s own developments, will be shown in the subsequent sections as well.

2.3.1 Characteristic of the Injector Nozzle of the Pelton Turbine The injector of the Pelton turbine is used to convert the pressure energy of the flow into the kinetic energy of a compact high-speed free surface jet in the atmosphere. The efficiency of the energy conversion usually reaches up to 98–99%. Based on the principle of energy conversion and the requirement of discharge regulation, the injector is designed, according to Fig. 2.6, to consist of a nozzle and an inbuilt needle (spear). The latter enables the discharge to be regulated by changing the needle position in the injector nozzle. Because the water is injected into air, the jet speed initially depends on the available total pressure head h tot in the flow ahead of the injector nozzle and can be computed from Torricelli’s equation. By neglecting the energy loss which is caused by viscous frictions and amounts only to about 1%, the jet speed is computed as c0 =



2gh tot .

(2.28)

The discharge out of the injector nozzle is obtainable by knowing the jet crosssectional area at the jet waist. Accounting for the contraction effect of the jet after leaving the nozzle, the discharge is computed by Q = c0 Ajet = ϕ AD0 2gh tot .

(2.29)

Fig. 2.6 Injector nozzle of the Pelton turbine and parameter specifications (Zhang 2009, 2016a)

30

2 Stationary Flows and Flow Regulations

The jet contraction is considered by the area ratio ϕ = Ajet /AD0 , with AD0 = π D02 /4 as the aperture of the injector nozzle and used as the reference area. The area ratio ϕ is known as discharge coefficient of the injector in the terminology of the Pelton turbine. In other applications, it is also called contraction factor. The discharge coefficient changes with the spear-needle position in the injector nozzle. This relation between two parameters can, usually, only be determined by calibration measurements. According to Zhang (2009, 2016a), as also found in common applications, the discharge coefficient ϕ of an injector can be accurately parameterized by the following quadratic function with s as the spear-needle stroke, see Fig. 2.6:  2 s s + a2 . ϕ = a1 D0 D0

(2.30)

This equation for the discharge coefficient of the injector is known as the injector characteristic. The constants a1 and a2 depend on the geometrical design of both the spear needle and the injector nozzle. Figure 2.7, based on measurements, shows the discharge coefficient ϕ of a given injector plotted against the needle stroke, s/D0 . For other geometrical configurations of injectors, respective characteristics can be computed according to the method presented by Zhang (2009, 2016a). Under optimized design of injectors, however, injector characteristics never differ much from each other. For applications of hydraulic transients, therefore, different characteristics do not lead to any significant differences in computational results. The opening and closing of the injector nozzle are usually given by the needle stroke as a function of time, s = f(t), which is called the regulation law or regulation dynamics. In almost all cases of flow regulations, the shock pressure, i.e., the water hammer is determined by the changing rate of the flow dϕ/dt and thus by the regulation dynamics ds/dt. For an optimized closing law of injector nozzles and its significance, see Sect. 10.2.4.

ϕ = a1

s ⎛ s ⎞ + a2 ⎜ ⎟ D0 ⎝ D0 ⎠

2

αn

D0

s αs

Fig. 2.7 Discharge coefficient of an injector nozzle according to Zhang (2009, 2016a)

2.3 Hydraulic Characteristics of Regulation Organs

31

Quite often, it appears to be necessary, to represent the discharge in Eq. (2.29) as a function of the static pressure head at the injector. In terms of h tot = h + c2 /2g and c = Q/A, Eq. (2.29) is also written as  Q = ϕ AD0

 Q2 . 2g h + 2g A2 

(2.31)

The discharge through the injector is then resolved as Q=

ϕ AD0



1 − (ϕ AD0 /A)2

2gh.

(2.32)

In almost all hydraulic systems with injector nozzles of Pelton turbines, one has (ϕ AD0 /A)2  1, so that the above equation is simplified to Q ≈ ϕ AD0 2gh.

(2.33)

If compared with Eq. (2.29), one confirms that this simplification is equivalent to the neglect of the dynamic pressure head in pipe flows. More about characteristics of injector nozzles of Pelton turbines can be found in Sect. 10.2 for reference. Under many hydraulic aspects regarding the use of injector nozzles, that section especially shows the parabolic closing law of injectors, which is optimized for flow regulation, without causing rapid pressure rise in the flow. The Pelton injector operates independent of both the Pelton wheel and the rotor system. It represents a regulation organ with well accessible hydraulic performance against other types of regulation devices. Because of these favorable features, the Pelton injector will be preferred throughout the present book, when simulating hydraulic transients and showing examples of applications.

2.3.2 Characteristic of the Spherical Valve The hydraulic characteristic of a spherical valve is represented by the resulting pressure drop in the flow as a function of the valve opening degree β. The fully opened position of the valve is denoted, according to Fig. 2.8a, by β = 90◦ . With the reference flow velocity c = Q/A, the pressure-head drop by a spherical valve is expressed as h sphV = cp

c2 Q2 = cp . 2g 2g A2

(2.34)

When the spherical valve is installed in a pipeline in which reverse flows may take place, then Q|Q| should be used in the above equation.

32

2 Stationary Flows and Flow Regulations

(b)

(a)

Δ h = cp

β

d

d

c

c2 2g

Δhsph

Fig. 2.8 Spherical valve and the valve characteristic

Because of equal flow velocities before and after the spherical valve, the pressurehead drop in the above equation also represents the energy head loss in the flow. The resistance coefficient cp is of purely geometrical character and in effect only a function of the opening degree (β) of a given spherical valve. The fact to be mentioned is that up to now hydraulic performances of spherical valves are obtainable all through measurements. Until few years ago, a most appropriate and accurate computation formula has been presented by Zhang (2016b, 2018), the author of this book, based on hydro-mechanical analyses, viz. 106 cp =  2 − 1. aβ 2 + bβ

(2.35)

The formula is applicable for nearly all types of spherical valves that are used in hydropower stations. This is so because different designs of spherical valves do not have much different hydraulic characteristics. Thus, for a good approximation, both constants in the above equation can be set equal to a = 0.108, b = 1.35 .

(2.36)

In Fig. 2.8b, computed resistance coefficients from Eq. (2.35) are shown. At the closed position (β = 0), the coefficient tends to infinity and at full opening (β = 90◦ ) it vanishes, cp ≈ 0. The formula according to Eq. (2.35) can be well validated by comparison with known data from literatures, see Appendix B.1.

2.3.3 Characteristic of the Butterfly Valve Another often applied regulation organ in hydropower stations is the butterfly valve, as sketched in Fig. 2.9a. Its hydraulic characteristic is depicted again by a pressure

2.3 Hydraulic Characteristics of Regulation Organs Fig. 2.9 Butterfly valve and the valve characteristic

33

(a) β0 β

d

c

Δhbutterfly

(b)

drop in the flow as a function of the valve opening degree β. Two special design concepts should be indicated. As for the first, the rotary disc is sometimes designed with an eccentric axis, in order to ensure self-closing of the valve. As for the second, the disc position at the closed state is more or less tilted towards the pipeline axis by an angle β0 , which can vary up to 45°. For general applications, the pressure-head drop at a butterfly valve is computed according to h butterfly = cp

c2 . 2g

(2.37)

The resistance coefficient cp represents the hydraulic characteristic of the butterfly valve. It is essentially a function of the opening degree of the disc (β) and usually obtainable only by measurements. For a butterfly valve with centered disc axis, the author of current book derived the following accurate formula for the resistance coefficient:

34

2 Stationary Flows and Flow Regulations

cp =

2 √ 1 + 0.42σ − 1 + 0.2, 1−σ

(2.38)

in which the parameter σ is represented by the ratio σ =

cos β . cos β0

(2.39)

The last term, i.e., the constant 0.2 in Eq. (2.38) corresponds to the residual resistance, when the valve is fully opened. It is basically determined by measurements and experiences. Figure 2.9b shows computation results from Eq. (2.38) with the assumption cp = 0.2 for the residual resistance. The computational results have satisfactorily agreed with corresponding values inferred from literatures, see Appendix B.2. If Eq. (2.37) is written as h butterfly = R Q 2 ,

(2.40)

then, the resistance constant R is computed as R=

cp . 2g A2

(2.41)

2.3.4 Characteristic of the Gate Valve In applications of gate valves there is a distinguished difference between the circular and the rectangular forms of the gate plate, as illustrated in Fig. 2.10. For both forms, the author of this book also derived reliable formulas for the computation of the respective resistance coefficients. As usual, it is defined by h gateV = cp

c2 . 2g

(2.42)

For the circular form of the gate plate (Fig. 2.10a), the radius of the circular plate only has a negligible effect, provided the ratio of the radii or diameters dplate /dpipe does not exceed the value 1.3. This condition is practically always fulfilled. Then the following formula of resistance coefficient can be applied ⎛ ⎜ cp,circ = ⎝

⎞2 1 ⎟ 2  − 1⎠ + 0.1.   h h 0.29 dpipe + 0.69 dpipe

(2.43)

2.3 Hydraulic Characteristics of Regulation Organs

(a)

35

(b)

Rplate

h

d/2

h

d/2

Fig. 2.10 Characteristics of gate valves with different forms of gate plates

For rectangular gate plates (Fig. 2.10b), the following similar formula has been derived ⎞2

⎛ ⎜ cp,rect = ⎝

 0.64

h dpipe

2

⎟  − 1⎠ + 0.1.  h + 0.36 dpipe

1

(2.44)

In both cases, the constant 0.1 in the equations approximately or rather formally represents the residual resistance, when the valves are fully opened. It indeed depends on the configuration of the gate valve, that is to say whether the pipe is grooved for the gate plate or not.

36

2 Stationary Flows and Flow Regulations

Figure 2.10c shows computational results for both forms of the gate valve, respectively, according to Eqs. (2.43) and (2.44). Comparison with values from literature again showed satisfactory agreements, see Appendix B.3.

2.4 Flow Regulations in Pipeline Systems Most flow processes, either in pipeline networks or in hydraulic systems of hydropower stations, are specified for stationary operations. Their regulations, thus, aim to change the flow from one to another stable state. In hydropower stations with a group of hydraulic machines, flow regulations depend on the types of installed hydraulic machines. For Pelton turbines, for instance, the injectors are used to regulate the discharge, whereas for Francis turbines the turbine load is regulated by the guide-vane apparatus. In the case of machine groups, the flow regulation in an extended sense also includes turn-on or shut-off of one or more machines, leading to the change in flow rate in the entire hydraulic system. A simple hydraulic system e.g. includes a high-altitude reservoir (or lake), a pipeline flow channel, a surge tank, and a group of turbines at low altitude (Fig. 2.11). The surge tank serves to balance and damp the flow oscillations in the considered hydraulic system while hydraulic machines are started or shut down. For the general case, we assume that the flow in the pipeline is subjected to energy loss, which arises from both the viscous friction effect and all other local flow resistances. As mentioned at the beginning of Sect. 2.3, the injector nozzle of the Pelton turbine can be applied as a quite applicable regulation organ, because it is mechanically and hydraulically disconnected with the turbine wheel. For this reason, the regulation feasibility of the Pelton injector nozzle is apt to be taken into account for performing and simulating flow regulations in a pipeline system. The reader should be reminded that the “flow regulation” in the current context refers to the change of one stable Surge tank Flow channel

Resistance constant R

Pressure shaft Distributor Machine house

Fig. 2.11 A simple hydraulic system with a surge tank and turbines

Q1

Q2

Q3

H0

Q

htot

Lake

2.4 Flow Regulations in Pipeline Systems

37

flow state into another. The flow in the time of duration of this transition can be approximated as quasi-stationary, if the water hammer is not of interest. Below, operations of one injector as well as two injectors for dual flow rates are considered. The purpose of computations is to specify respective stationary flows which, in the cases of transient flows, are always taken as starting points for transient computations.

2.4.1 Simple Pipeline Systems with One Regulation Valve For the pipeline system shown in Fig. 2.11, the total gross hydraulic head is given by H0 , which represents the most significant system parameter. It can be directly used to compute the flow rate in the system. Because of flow resistance and hence energy loss in the pipeline, the effective, i.e., the net hydraulic head at the injector remains h tot = H0 − R Q 2 . It is explicitly a function of the flow rate in the pipeline, i.e., the discharge through the injector nozzle. From Eq. (2.29) and with h tot as the available total pressure head, the discharge out of the injector nozzle is computed as    Q = ϕ AD0 2gh tot = ϕ AD0 2g H0 − R Q 2 .

(2.45)

In explicit form, the discharge is resolved as Q=

ϕ AD0 1 + 2g R(ϕ AD0 )2



2g H0 .

(2.46)

For a given injector, the discharge coefficient ϕ can be computed by Eq. (2.30) as a function of the spear-needle stroke in the injector nozzle. Because of the use of the constant pressure head H0 , the above equation only applies to stabilized or quasistationary flows without water hammer. It clearly represents the way to regulate the discharge and hence the load of the Pelton turbine. Knowledge about the discharge is required, for instance, when computing the hydraulic power contained in the highspeed jet, as obtained by Pjet = ρgh tot Q.

2.4.2 Pipeline Systems with Two or More Regulation Valves To simulate the flow regulation at a hydraulic system with two or more hydraulic machines (or regulation apparatus), a multi-injector Pelton turbine model can be applied. The power output of the system is then regulated by opening or closing additional injectors. According to Fig. 2.11 and again for the common case of stationary operations, each injector operates independent of the others. The total pressure head in the distributor, however, depends on the total flow rate in the main pipeline and thus

38

2 Stationary Flows and Flow Regulations

on the number of opened injectors. When supposing the full-opening of all injectors, then, because of the maximum discharge and thus the maximum pressure drop in the pipeline, the total pressure in the distributor reaches its minimum. For representing the computational procedure only, two independent injectors are assumed to operate. The quantities to be determined are the discharge at each injector with individually given openings and the total discharge of the system. Discharges at two injectors are denoted by Q 1 and Q 2 , respectively. According to Eq. (2.45) and with respect to the equal total pressure head in the distributor, one obtains Q 1 = ϕ1 AD0,1 2gh tot ,

(2.47)

Q 2 = ϕ2 AD0,2 2gh tot .

(2.48)

With respect to the overall resistance constant R of the pipeline and the total discharge Q = Q 1 + Q 2 , the total pressure head in the distributor, i.e., at injectors is computed as h tot = H0 − R Q 2 .

(2.49)

By adding Eqs. (2.47)–(2.48) and subsequently replacing the total pressure head by Eq. (2.49), one obtains     Q = ϕ1 AD0,1 + ϕ2 AD0,2 2g H0 − R Q 2 .

(2.50)

The overall discharge is then obtained as Q=

ϕ1 AD0,1 + ϕ2 AD0,2 2 2g H0 .  1 + 2g R ϕ1 AD0,1 + ϕ2 AD0,2

(2.51)

The remaining total pressure head in the distributor is then obtainable from Eq. (2.49). The discharge at each injector can be computed in turn by Eqs. (2.47) and (2.48), respectively. Similar computations can be carried out, when more than two injectors are in operations.

References Idelchik, I. E. (2007). Handbook of hydraulic resistance (4th Revised ed.). US: Begell House Publishers Inc. Zhang, Z. (2009). Freistrahlturbinen. Springer-Verlag. Zhang, Z. (2016a). Pelton Turbines. Springer Verlag.

References

39

Zhang, Z. (2016b). Transient flows in pipe system with pump shut-down and the simultaneous closing of the spherical valve. In IAHR, 28th symposium on hydraulic machinery and systems. Grenoble, France, see also IOP Conference Series: Earth and Environmental Science (Vol. 49, p. 052001). https://doi.org/10.1088/1755-1315/49/5/052001. Zhang, Z. (2018, August). Wave tracking method of hydraulic transients in pipe systems with pump shut-off under simultaneous closing of spherical valves. Journal of Renewable Energy, 132, 157–166.

Chapter 3

Transient Flows and Computational Methods

Hydraulic transients in a hydraulic system are encountered when the flow is found under regulations, for instance, for process control. In hydropower stations, the occurrence of hydraulic transients is primarily associated with the start-up and the shutdown of hydraulic machines as well as with the load regulation. Direct consequences of such actions are rapid pressure rises in form of water hammer in the flow, highfrequency pressure fluctuations, and low-frequency flow oscillations (surges) in the pipeline system. The first two occurrences could considerably affect the structural safety as well as the functionality and reliability of all related devices in the hydraulic system. The third occurrence, i.e., low-frequency flow oscillations between the reservoir and the surge tank, for instance, determine the capacity of each surge tank used. Other consequences are cavitation and system instability. The worst cases are given at the load rejection and the emergency shut-down of machines like the pump and the turbine. For the sake of both the system reliability and operational safety, each hydraulic system must be assessed with respect to such operation conditions, even in the early design and construction stages. The occurrence of high-frequency pressure fluctuations in transient flows in a water-hydraulic system is associated with the compressibility of water. To assess the extent of hydraulic transients in hydraulic systems, basically, two available computational methods exist which are based, respectively, on the rigid and the elastic water column theory. The first theory and the related computational model ignore the compressibility of the flow. It thus basically represents an approximation in transient computations. The second, i.e., the elastic water column theory takes into account the compressibility of water and, thus, reflects reality more accurately than the first theory. It has become a special field in hydrodynamics. The choice between two methods for transient computations depends on what should be computed and how accurate each computation is expected to be. Both the application limit and application examples of the method based on the rigid water column theory will be shown below in Sect. 3.2. The fact to be mentioned is that

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_3

41

42

3 Transient Flows and Computational Methods

computations based on the rigid water column theory with neglect of compressibility of flows are not always simpler than computations based on the elastic water column theory, especially if it is about a complex pipeline network. For a pipeline network, for instance, with N pipes of different diameters and connections (serial or parallel), then N momentum equations have to be simultaneously solved, when using the rigid water column theory. This feature of computations inevitably requires highly complex and often necessary iterative numerical solution techniques (see Sect. 5.3). In contrast, the method of using the elastic water column theory solves the flow at a given position in a pipeline system and at a given moment then independent of the flow state elsewhere in the system. In any case, both computation models rely on numerical solutions of coupled equations.

3.1 Occurrence of Hydraulic Transients in Hydropower Stations A typical and simple hydraulic system, as sketched in Fig. 2.11, consists of a highaltitude reservoir (lake), a long pipeline for water transport, a surge tank, a pressure shaft and a group of water turbines. For other types of turbines (like the Francis turbine) and pumps, a connection channel to a low-altitude reservoir is necessary. The surge tank primarily serves to absorb the kinetic energy included in the pipeline flow, when the turbine load is reduced or the turbines are shut off. It also serves to provide water for a short time, when turbines are started and the flow in the long pipeline reacts with a large delay because of the inertia of the flow. Hydraulic transients occur, as soon as the hydraulic load in the system changes. Figure 3.1 shows the measured pressure response in the flow, while a Pelton turbine was shut down and the discharge changed from Q = 3.3 m3 /s towards zero. The injector of the Pelton turbine was closed in two steps within a time t close = 27.6 s. Within this closing period, significant high-frequency pressure fluctuations, which are called pressure shock waves, have been recorded at the pipe cross-section just ahead of the Pelton injector (curve 2). The piezometric head, which is used in Fig. 3.1 to represent the pressure response in the flow, is the sum of the static pressure and the relative altitude, i.e., h z = p/ρg + z. The reference altitude is the height where the injector is found. The low-frequency flow oscillation (curve 1), which is often called the pressure surge, represents the system oscillation between the upper lake and the surge tank. The curve, after subtracting an altitude height z = 440 m, is representative of the static pressure head at the pipe section, where the surge tank is connected. It thus approximately reflects the level of the water in the surge tank, if the local flow resistance at the entrance of the surge tank is neglected. The shock pressure in the flow depends on the closing time of the injector. The faster the injector is closed, the higher is the maximum shock pressure in the flow. As will be shown later when using both the rigid and the elastic water column theory,

3.1 Occurrence of Hydraulic Transients in Hydropower Stations

2300

ST, A=7.11 m2

Lake

43

Overflow 2307.7 m.a.s.l. φ1.65, L1442

2211

φ2.6, L4566

φ1.4, L167 Datum z=0

1771 m.a.s.l.

Fig. 3.1 Pressure response and flow oscillation in a hydraulic system after the shutdown of a two-injector Pelton turbine under the partial load (Q0 = 3.3 m3 /s)

the shock pressure including its maximum is actually directly proportional to the changing rate of the discharge. For this reason, the closing of each injector should always be gently performed. In practice, it has also been suggested to close the injector in two steps with different speeds. To specify the closing law of the injector with respect to the occurrence of the pressure shock in the flow, computations of related hydraulic transients are indispensable, if measurements are neither available nor possible. In Zhang (2016), a parabolic closing law with particular characteristics has been suggested. Its favorable applicability will be presented in Sect. 10.2.4.

44

3 Transient Flows and Computational Methods

3.2 Method of Rigid Water Column Theory 3.2.1 Restrictions of Application The rigid water column theory assumes incompressibility of the flow. As an approximation, therefore, it is basically subject to certain application restrictions. Firstly, the theory is clearly unable to resolve high-frequency pressure fluctuations, which are related with the flow compressibility (curve 2 in Fig. 3.1). It, however, can be used to compute the low-frequency flow oscillation in the considered hydraulic system (curve 1 in Fig. 3.1). Knowledge about the low-frequency flow oscillation, usually, serves to determine and evaluate the surge tank capability, see Chap. 5. Secondly, one often says that the rigid water column theory is only applicable to short pipes, however, without simultaneously clearly mentioning what does the attribute “short” mean. The term “short pipe” is in fact a relative concept. Its applicability can be basically proved with comparisons with accurate computations based on the elastic water column theory. The point to be mentioned here is that “short pipe” is always related to the duration of each conducted flow regulation. In other words, the pipe is relatively “short”, if its length is short in comparison with the “length scale” which is given as the product of regulation time and the wave propagation speed. A concrete case of closing the flow in a simple pipeline with an injector nozzle is illustrated in Fig. 3.2a. For revealing the meaning of “short pipe”, comparable computations have been completed by using the rigid and elastic water column theories, respectively. With respect to a mean wave speed a = 1250 m/s in using the elastic water column theory, the pipe length is simply assumed to be L = 1250 m. Thus, the travel time of a pressure shock wave from the injector, upstream, to the reservoir takes 1 s. Computations using the elastic water column theory rely on the wave tracking method which is described in detail in Chap. 7. Computations using the rigid water column theory have been completed based on the method described in Sect. 4.2.2 and by the numerical method with different time increments. The flow is assumed to be partially closed by linearly changing the needle stroke (ds/dt = const) from s1 = 150 to s2 = 20 mm within 20 s. To the closing dynamics of the injector nozzle ds/dt, see Sect. 2.3.1. Computational results are shown in Fig. 3.2b. These results also permit to find out what happed, for instance, in the first two seconds, i.e., if the flow regulation after two seconds would be stopped. In the same way, one also acquires shock pressures after 10 and 20 s. Thus, two conditions with respect to the regulation time should be revealed. First, the closing time of the injector nozzle is sufficiently long, say treg > 2L/a. One then compares the pressure rise in the flow at the end of the regulation time. Obviously, an excellent agreement between computations using the two different theories has been achieved. In this case, the condition of “short pipe” is well fulfilled, because the pipe behaves as relatively “short” against the “length” which is given

3.2 Method of Rigid Water Column Theory

45

(a) H0

L Injector

(b)

Fig. 3.2 Computed pressure responses at the injector in a simple pipeline (L = 1250 m) by partially closing the injector nozzle (s from 150 to 20 mm) within a period t c = 20 s; numerical solution with different time increment t

by the product of regulation time and the wave propagation speed. For quantitative evaluations based on comparison with the elastic water column theory, a measurable and very representative time is defined here T2L =

2L . a

(3.1)

It is the propagation time of a pressure wave for a round trip (from the regulation organ upstream to the reservoir and back). In the computation example shown in Fig. 3.2, it reads T2L = 2 s. The condition of the “short pipe” in the current example, when related to the duration of closing the flow by L < atreg /2, can thus also be formulated as treg > T2L .

(3.2)

In such a case, it is not interesting, as usually, what has happened with the shock pressures within the first few seconds, i.e., t < T2L . Second, the closing time of the flow is sufficiently short, say treg < T2L . This implies that the condition of a “short pipe” has not been fulfilled. As can be confirmed from Fig. 3.2, considerable discrepancies exist between computations of the two

46

3 Transient Flows and Computational Methods

Fig. 3.3 Computed pressure responses at the injector in a simple pipeline (L = 1250 m) by linearly opening the injector nozzle (s = 0–130 mm)

different theories. Even when using the rigid water column theory, computational results depend additionally strongly on the used time increment of the numerical solutions. In any case, it is not true that the smaller the time increment is, the higher the accuracy of computations will be. The above intentions aim to explain the condition of a “short pipe” for applying the rigid water column theory. All computations, which were used for comparisons, have been carried out based on computational algorithms which will be presented later in corresponding chapters. Another concrete case to be considered is the case of opening the valve, as shown in Fig. 3.3 from computations again based on two different theories. In this case, the condition for a “short pipe”, obviously, has to be rescaled. Under the given condition ds/dt = const for opening the injector nozzle in the current case, agreement between the two computations will be given for a regulation time larger than about treg > 6T2L = 12 s. This means, in turn, that the condition for a “short pipe” in the current case is found at L < atreg /2. From the above considerations, the notation “short time” is clearly a relative concept. It basically depends not only on the regulation time but actually also on the regulation dynamics, say ds/dt by using the injector nozzle. In practical applications as in hydropower stations, except for incidents with fast closing and opening of the flow, the condition of a “short pipe” is usually always fulfilled. This is primarily because of slow regulations of flows by slowly regulating valves of all types. In the case of Fig. 3.1, e.g., the closing time of the injector nozzle is t close = 27.6 s. With the aid of both Figs. 3.2 and 3.3, one confirms that the rigid water column theory is unable to resolve high-frequency pressure fluctuations in the flow after opening or closing the valve.

3.2 Method of Rigid Water Column Theory

47

Finally, a further application restriction for the use of the rigid water column theory should be revealed. Because the theory is based on the assumption of incompressibility of the fluid motion, each action at a regulation organ will cause simultaneous reaction of the flows in the entire pipeline network. In a network with, for instance, N pipes of different diameters and connections (serial or parallel), N momentum equations have to be solved simultaneously. This feature of computations inevitably requires highly complex iterative numerical solutions. In contrast, the method of using the elastic water column theory simply solves the flow at a given position in a pipeline network and to a given moment, always independent of flow states elsewhere in the system.

3.2.2 Flows in Pipelines of Constant Cross-Sectional Area A simple flow system, as sketched in Fig. 3.4, consists of a straight pipeline of constant cross-sectional area A and length L. It represents a hydraulic system of a Pelton turbine, where a spherical valve as a closure organ is always found ahead of the Pelton injector. The flow regulation is commonly performed by opening and closing the injector nozzle whose hydraulic characteristic has already been presented in Sect. 2.3.1. To establish the equation of motion, the total mass of water in the pipeline is considered to be a compact column. Each change in the flow rate, i.e., the flow velocity, implies a corresponding change of the inertial force of the compact column. Active forces exerted on the flow are gravitational, frictional and pressure forces, as listed below according to Fig. 3.4, with indices 1 and 2, respectively, denoting the inlet cross-section of the pipeline and the cross-section ahead of the injector. These forces are: Static pressure force at the pipe section to the lake: ρgh 1 A Component of gravitational force along the pipeline: ρ ALg sin α Static pressure force at the section to the injector: −ρgh 2 A Viscous friction force: −cf π d Lρc2 /2

Lake x

α

H0

he

• • • •

1

c

Ball valve Injector

A, L, R 2

Fig. 3.4 A straight pipeline of constant cross-section

48

3 Transient Flows and Computational Methods

For skin friction coefficient cf we refer to Eq. (2.1). For positive flows as shown in Fig. 3.4, the static pressure at the pipe entrance is given by h 1 = h e − c2 /2g. Newton’s second law of motion is applied to the total mass of the water between cross-sections 1 and 2, i.e., in the compact column. One obtains for Newton’s law in the direction of the pipe axis   1 2 c2 dc = ρgh e − ρc A + ρ ALg sin α − ρgh 2 A − cf π d Lρ . ρ AL dt 2 2

(3.3)

With respect to H0 = h e + L sin α and A = π d 2 /4 as well as λ = 4cf from Eq. (2.4), the above equation is simplified to   L dc L c2 − . h 2 = H0 − 1 + λ d 2g g dt

(3.4)

Against the initial flow state (h 2,0 and c0 ), which is given by dc/dt = 0, it follows immediately from above equation h 2 − h 2,0

   L dc L  2 1 1+λ c0 − c2 − . = 2g d g dt

(3.5)

The first term on the r.h.s. of this equation consists of the influence of the dynamic pressure head and the effect of viscous friction. The former is anyhow negligible. By also neglecting the effect of viscous friction, one obtains the approximation h 2 − h 2,0 ≈ −

L dc . g dt

(3.6)

It should be reminded that according to the last section dealing with application restrictions in the term of “short pipe”, the above equation is generally only applicable for t > T2L for throttling flows, see Eq. (3.2). For opening flows, there is even t > 6T2L , as observed in Fig. 3.3. Under this aspect, Eq. (3.6) will later be referred to, when considering Joukowsky’s equation in Sects. 3.3.1 and 3.3.3. When using both the volume flow rate Q = c A for expressing dc/dt and the resistance constant R according to Eq. (2.10), the total pressure head h 2,tot at crosssection 2 is, for Q > 0, computed from Eq. (3.4) as h 2,tot = h 2 +

c2 L dQ = H0 − R Q 2 − . 2g g A dt

(3.7)

Against the pressure head under the condition of stationary flow, the pressure head at the cross-section close to injectors is only determined by the changing rate of volume flow. It is also proportional to the length of the pipeline. Obviously, the faster the flow rate is reduced, the larger is the rise of the pressure head. From this viewpoint, flow regulation in a pipeline system has to be carefully performed. A most

3.2 Method of Rigid Water Column Theory

49

effective measure to limit the changing rate, dQ/dt, of the flow in the main pipeline is to use a surge tank like that which has already been shown in Figs. 2.11 and 3.1. It should be mentioned that from Eq. (3.5) alone, even in the simplified form Eq. (3.6), the pressure head at the injector cannot be computed. The equation involves two variables (h 2 and c = Q/A) which are not independent of each other. First, both variables are connected by the injector characteristic (Sect. 2.3.1). Second, the changing rate of volume flow rate dQ/dt is additionally related to the dynamic flow regulation, i.e., the closing time of the injector nozzle. Corresponding computational algorithms will be presented later in Sect. 4.2.2. In a similar way as for Eq. (3.7), the pressure response at any position x along the pipeline in Fig. 3.4 can be computed from Eq. (3.3) by using x and h x in place of L and h 2 , respectively. One obtains the static pressure head at section x  h x = h e + x sin α −

 x dQ 1 2 2 − . Q − R Q x 2g A2 g A dt

(3.8)

With h x,0 ≈ h e + x sin α − Rx Q 20 and by neglecting the dynamic pressure head, the above equation can also be written as   x dQ . h x − h x,0 ≈ Rx Q 20 − Q 2 − g A dt

(3.9)

Computations of the pressure head at cross-section 2 from Eq. (3.7) also apply to curved pipelines. Relevant parameters are simply the available hydraulic head H0 and the pipe length L. Special care, however, has to be taken, if the pipeline is specially curved with a horizontal part, see the next section.

3.2.3 Flows in Specially Curved Pipelines of Constant Cross-Sectional Area An important special case of a simple pipeline that occurs relatively often is shown in Fig. 3.5. Its first part consists of horizontal flow and the second part is steeply inclined towards the hydraulic machine or its regulation organ. In such a system, undesired operation disturbance may occur when the valve at the end of the pipeline is rapidly opened. While the flow and the flow acceleration in the steeply inclined part of the pipeline always follow the opening of the valve, the flow acceleration in the horizontal part of the pipeline behaves significantly differently. It depends not only on the pressure head difference between both ends of the pipeline, i.e., h 1 − h m , but also on the length of the water column in that part (L 1 ). For a long horizontal pipe with a large amount of water in it, the flow acceleration caused by the pressure head difference may be so low that the flow would not be able to follow the rapid opening of the valve. As a consequence, different flow accelerations in two pipe segments

3 Transient Flows and Computational Methods

he

50

A1, L1 c1 h1

m hm

c2

Fig. 3.5 A pipeline system with a horizontal part of length L 1 and a steeply inclined part to the injector

will lead to a rapid pressure decrease at the highest position m of the inclined pipe. To some extent, the vapor pressure, i.e., the boiling point of water at the ambient temperature can be reached, leading to direct evaporation of water there. In the worst case, rapid alternation between evaporation and condensation of water could lead to a rapid drop of the local temperature and thus to implosion and burst of pipes. Some accidents arising from this mechanism have also been observed in practice, even though they are not well documented.1 To estimate the limit of fluid acceleration in a given pipeline segment, the momentum equation must be applied to the flow in the horizontal part of the pipeline. For simplicity, the viscous friction effect is neglected, because at the beginning of the opening operation of the valve the flow rate is zero. Analogously to Eq. (3.3), the equation of motion for the flow in the horizontal pipe is given by ρ A1 L 1

dc1 = ρg A1 (h e − h m ). dt

(3.10)

With Q = c A it follows then A1 dQ 1 = g (h e − h m ). dt L1

(3.11)

For secure flows and operations, the pressure head h m should be in no case less than a minimum h m,min (for instance 5 m). With this criterion, the allowable changing rate of the flow, when opening the valve, is limited to  A1  dQ 1 < g h e − h m,min . dt L1

(3.12)

1 After a heavy rainy fall in August 2004, the horizontal part of a hydraulic pipeline in a hydropower

station in an Alpine area of Switzerland was blocked by trunks and sediments. While two turbines in the low-altitude turbine house started, the horizontal part of the pipeline, which was made of cast iron that is brittle at low temperature, had imploded after a short time.

3.2 Method of Rigid Water Column Theory

51

The longer the pipe is, the lower will be the allowable changing rate of the flow. With respect to this circumstance and because of dQ 1 /dt = dQ 2 /dt, the opening of the valve at the end of the pipeline has to be controlled. In practical applications with a quite long horizontal pipe, a surge tank often needs to be installed in the pipeline system, as already shown in Fig. 2.11. During the period of opening the valve, the surge tank provisionally provides water flow for steeply inclined or vertical pressure shaft. When the turbine is stopped, the surge tank serves to absorb the energy included in the stream in the horizontal pipe. More about the functionality and applications of surge tanks will be shown in Chap. 5.

3.2.4 Flows in Stepped Pipes In some hydraulic pipeline systems, a pipeline may consist of pipes of different diameters and lengths connected in series. For the situation shown in Fig. 3.6, an equivalent pipe diameter should be determined, with which the equation of motion according to Eq. (3.7), for instance, can directly be applied. To this end, Newton’s second law of motion is applied to each single pipe of constant cross-sectional area Ai . With respect to the friction effect, which is represented by its resistance constant Ri , one obtains   dc1 = ( pa − pb1 )A1 − ρg A1 R1 Q 2 + ρg A1 h 1 , dt

(3.13)

  dc2 = ( pb2 − pc2 )A2 − ρg A2 R2 Q 2 + ρg A2 h 2 , dt

(3.14)

  dc3 = ( pc3 − pd )A3 − ρg A3 R3 Q 2 + ρg A3 h 3 . dt

(3.15)

ρ A1 L 1 ρ A2 L 2

ρ A3 L 3

The static pressure pc2 , e.g., with its subscripts denotes the pressure at the crosssection “c” on the side of pipe 2. In principle, the pressure pc2 differs from the

he a

h1

H0

L1 b

L2

h2 c

h3

L3 d

Fig. 3.6 A pipeline with stepped pipes

52

3 Transient Flows and Computational Methods

  pressure pc3 because of the difference in the dynamic pressures 21 ρ c22 − c32 on the one hand, and of the local energy loss on the other hand. Such a difference, however, can always be ignored because it is insignificant. The three flow velocities, c1 , c2 and c3 , correspondingly, are related by c1 A1 = c2 A2 = c3 A3 = Q. With the approximations of pb1 ≈ pb2 and pc2 ≈ pc3 , Eqs. (3.13)–(3.15) can be shown to reduce to the single equation ρ

3 3 3   dQ  L i = ( pa − pd ) − ρg Q 2 Ri + ρg hi. dt i=1 Ai i=1 i=1

(3.16)

The static pressure pa at the cross-section “a” is given by pa = ρgh e − 21 ρc12 . With pd = ρgh d and h e + h 1 + h 2 + h 3 = H0 , the above equation is further simplified to 3 3  1 dQ  L i 1 2 = H0 − h d − Ri . c1 − Q 2 g dt i=1 Ai 2g i=1

(3.17)

The total length of the pipeline is given by L = L 1 + L 2 + L 3 . An equivalent pipe cross-sectional area is then introduced in the following form 3 1 1  Li = . Aeq L i=1 Ai

(3.18)

With R = R1 + R2 + R3 as the total resistance constant, the total pressure head at the end of the pipeline is computed, with h d from Eq. (3.17), as h d,tot = h d +

 1  2 1 2 L dQ c3 = H0 − R Q 2 − + c3 − c12 . 2g g Aeq dt 2g

(3.19)

Usually, the last term in this equation is small and can thus be neglected. One then confirms that Eq. (3.7) is actually a special case of this last equation. The case considered in Fig. 3.6 can be extended to a pipeline system with N pipes which are connected in series. One only needs to compute the corresponding equivalent pipe cross-sectional area according to Eq. (3.18).

3.3 Method of Elastic Water Column Theory The elastic water column theory takes account of the compressibility of the fluid in a flow system. Computations and computational methods based on this theory then concern all true flow conditions and thus provide accurate results. They are especially able to predict the high-frequency flow fluctuations like that of curve 2 in Fig. 3.1 and hence to evaluate real hydro-mechanical performances of the system.

3.3 Method of Elastic Water Column Theory

53

The compressibility of a fluid, for instance water, is physically represented by the elastic modulus of the medium which is related to the limited propagation speed of the pressure wave in the medium. This is the reason why the wave propagation speed is always included in transient computations based on the elastic water column theory. The main objective when dealing with hydraulic transients, thus, comprises the generation mechanism of pressure shock waves, the travelling performance of waves in the flow and, finally, reliable and efficient computational methods.

3.3.1 Joukowsky’s Equation To reveal the generation mechanism of shock waves in the flow, the flow is assumed to be regulated by a valve at the end of a pressurized pipeline, as illustrated in Fig. 3.7. An instantaneous change in the flow velocity at the valve, c = c − c0 , leads to compression of the fluid in the adjacent upstream region. The stepwise change of the state of the flow thus propagates backwards in negative x-direction by a limited speed because of the compressibility of the flow. This speed is called the wave propagation speed and is commonly denoted by a. Within a finite time interval t, the propagation distance of the front of the shock wave is s = at. The mass increment which is subject to the velocity change is given by m = ρ As. To this mass element, the momentum equation is applied. The resultant change in the momentum is obtained as mc = ρ Asc. Consequently, a pressure increase p = p − p0 arises from the flow stagnation at the valve, i.e., pipe section 2. Based on momentum equation of this element and by neglecting the viscous friction force, the changing rate of the momentum must be equal to the pressure force Fx = −Ap exerted on the mass element in the x-direction. This process yields −a

Δh

c2/2g

h0

Lake Δs

−Δc

c0

2

L

1

c x

Pressure shock

Fig. 3.7 Generation and propagations of pressure shock waves

Valve

54

3 Transient Flows and Computational Methods

−Ap = ρ As

c . t

(3.20)

In terms of the rise of the pressure head, as h = p/ρg, one then obtains with s/t = a a h = − c. g

(3.21)

This equation is called Joukowsky’s or Allievi’s equation, see also Eq. (1.1). It actually also applies to the case of opening the valve, at which the velocity change is positive (c > 0) and, thus, the pressure head drops (h < 0). Because the propagation speed of a pressure shock wave is excessively high (a = 1200–1400 m/s in real pipeline systems), a small change in the flow velocity will straightaway cause a significant change in the pressure. The phenomenon is known as water hammer. The pressure shock wave, which is directly generated by using a regulation valve, has been commonly called direct wave or primary wave. The computation concept leading to Eq. (3.21) assumes that in the upstream flow, i.e., on the upstream side of the wave front the initial undisturbed flow (c0 , h 0 ) is present. Thus, the Joukowsky equation is valid, as long as it is considered within a time t ≤ T2L = 2L/a, with T2L as the time for a round trip of a shock wave traveling from the regulation valve (location 2), upstream to the lake and back to the valve again. The pipe length is L. As long as t < T2L , Eq. (3.21) in its basic form of a temporal change of the flow state is also applicable to the spatial non-uniform change of flow state as a ∂c ∂h =− . ∂x g ∂x

(3.22)

Especially, if the valve is rapidly closed within the time t ≤ 2L/a, the maximum achievable rise in the pressure head at the regulation valve is directly obtained from Eq. (3.21) as a h max = − c0 , g

(3.23)

with c0 as the initial flow velocity at the valve. At the time t = 2L/a, the first primary wave, which travels towards the lake and gets reflected there, turns back to the valve again. The reflected pressure wave has a negative character and will negatively interact with the primary pressure wave at the valve. It, thus, inhibits the further rise of pressures there, as can be confirmed in Fig. 3.2. The related mechanism will be described below in more details (Sect. 3.3.2). One would be interested in the formal difference between Eq. (3.21) of using the wave speed and, respectively, Eq. (3.6) of using the pipe length. As already said, Eq. (3.21) is only valid within the time t ≤ T2L . Within this time, however, Eq. (3.6) is not applicable, because the condition of “short pipe” in the rigid water column theory, see Eq. (3.2), is not fulfilled. Thus, two equations are not comparable.

3.3 Method of Elastic Water Column Theory

55

By the way, the restriction of the Joukowsky equation to the time t ≤ T2L signifies that the equation is only applicable to the “long pipe”. The primary pressure waves then need a long time to travel to the upper reservoir, to be reflected there and to come back again to the regulation valve. Within this long time, the flow regulation would be already completed and the maximum pressure rise, when closing the valve, is given by Eq. (3.23). Because such “long pipes” are hardly found in hydropower stations, the Joukowsky equation indeed has its very restricted applications.

3.3.2 Primary and Reflected Shock Waves One important aspect in dealing with hydraulic transients is the superposition of pressure shock waves, or the interaction between them. This simply occurs between the primary and the reflected waves, as illustrated in Fig. 3.8 for a simple pipeline system. The case to be considered is the shutting down of the flow by progressively closing the valve at the end of the pipeline. The flow rate is assumed to be linearly reduced by closing the valve within a given time tc . According to Eq. (3.21), linear reduction of the flow rate leads to a linear rise of the pressure head at the pipe section close to the valve. This implies that the progressive stagnation of the mass flow within the first finite time interval t1 leads to a progressive pressure rise adjacent to the valve in the region xs0 − xs1 = at1 (Fig. 3.8a). Away from this region, the flow behaves as undisturbed because of the compressibility of water and thus of the finite propagation speed of the pressure shock waves. Further progressive growing of shock pressure and the shock wave propagation at the wave speed are illustrated in Fig. 3.8a for the time 2t1 and 3t1 . At the time 3t1 , the first pressure shock wave reaches the upper reservoir (lake). Because the pressure head there remains constant equal to h 0 , each arrived pressure shock wave will be balanced by an opposite wave. This opposite shock wave, in effect, originates from the mechanism of wave reflection at the lake because of sudden expansion in the flow section. The opposite shock wave is thus of negative nature and propagates backwards against the propagation of the primary wave, as illustrated in Fig. 3.8a by dashed lines. Thus, it follows at the pipe section x = 0 directly h = (h 0 + h) − h = h 0 . Both the propagation and the reflection of the shock waves at the entrance section of a pipeline are accompanied by the velocity change c. At the time t = 3t1 , the flow velocity at the pipeline entrance is still equal to c0 . At the time t = 4t1 , the velocity in the approaching wave changes to c0 + c. In association with the generation of the balanced pressure −h in the time between, an additional change in the velocity c, according to Eq. (3.21), must be given, as shown in Fig. 3.8b by the dashed line. Thus, at the position s3 there is finally c = (c0 + c) + c = c0 + 2c with c < 0. The process continues and the pressure distribution along the pipeline changes with time. At the time 4t1 , for instance, the first reflected negative pressure shock wave travels downstream to position s2 and prevents the pressure there from further rising. Correspondingly, the flow velocity has been diminished by 2dc/dt · t1 .

56

3 Transient Flows and Computational Methods 6t1

(a) h

5t1 ,7t1 4t1, 8t1 3t1 ,9t1 2t1,10t1 t1, 11t1 h0

0, 12t1

0s

s2

3

s1

s0

x

4t1 5t1

Negative pressure waves (reflected)

6t1

(b) c t=0 t1 2t1 3t1 4t1 5t1 6t1 7t1

c0 2dc/dt·t1 Uniform distribution

0

x

4t1 5t1 6t1

h0

L Primary wave

Reflected wave

Flow

(c) Q

(d)

Valve

T4L

h h0 tc

t

0

12t1

24t1

36t1

48t1 tc

t

Fig. 3.8 Shock wave propagations and interaction between the primary and reflected shock waves (a graphical method). a Head respond along the pipe, b velocity distribution along the pipe, c predefined linear closing of the flow, d head fluctuations at the valve

3.3 Method of Elastic Water Column Theory

57

At the time 6t1 , the first reflected, i.e., downstream shock wave reaches the valve at the end of the pipeline and acts with a negative effect. This prevents the pressure there from further rising. However, since the sign of the velocity gradient dc/dx at the valve and at time t = 6t1 changes (Fig. 3.8b) which negatively affects the pressure there, the pressure at the valve again decreases afterwards until to the time t = 12t1 . At this last time, uniform flow distribution in pipeline (for both the pressure head h and the velocity c) has been again achieved. Thus, t = 12t1 represents the period of pressure fluctuations in the considered pipeline system. With respect to L = 3at1 , it is also written as T4L =

4L . a

(3.24)

In the current example, the first maximum shock pressure at the valve is obtained at t = 6t1 . Figure 3.8d qualitatively shows the pressure head fluctuations at the valve for frictionless flows. The regular triangular form of the pressure shock waves is only available for t < tc , with tc as the closing time of the valve. Worth mentioning is again that the Joukowsky equation describes the water hammer which is related to primary shock waves without considering the interaction with reflected waves. This means, in the current case, that it only applies for a period up to t = 6t1 = T2L , see Fig. 3.8d. The outlined transient process in Fig. 3.8 represents the mechanism of both the generation and the propagation as well as the superposition of pressure shock waves. The related method is basically a graphical method which exclusively relies on the predefined changing rate of the flow. It was applied for a long time, until the computer technology gradually came to use and computational methods had been widely developed. New investigations show, as described below in Sect. 3.3.4, that the predefinition of the flow rate never represents the reality of hydraulic transients and, thus, computation results are always inaccurate. As shown in Fig. 3.8d, the shock pressure at the closing valve is subject to strong fluctuations as the results of superposition of direct and reflected pressure shock waves. In reality, such a unique form of pressure fluctuations is exactly the consequence of the predefined linear changing rate of the flow. It is thus pseudo-natural and will usually never occur. In effect, the velocity or the volume flow rate itself is additionally determined by the characteristics of valves used. For the Pelton injector, it is determined by Eq. (2.32), at which both the flow rate and the pressure head, unlike the discharge coefficient, are dependent quantities. As a consequence, the wave form in Fig. 3.8d will not occur. The phenomenon reveals an intrinsic flow dynamics, which is denoted as “self-stabilization effect of flows”. It effectively makes flow fluctuations smooth. This notable flow dynamics will be explained below in Sect. 3.3.4 for more details. The pressure fluctuation, as shown in Fig. 3.8d, is described by the period and the amplitude of a wave function. According to Eq. (3.24), the period of the pressure fluctuation is reciprocally proportional to the wave propagation speed. The maximum shock pressure (water hammer) in the flow, however, is commonly independent of

58

3 Transient Flows and Computational Methods

the wave speed. This is apparently against the Joukowsky equation which has been presented in Eq. (3.21). Physical explanation of such an “independence” will be presented below in the next section.

3.3.3 Influence of Wave Speed on Accuracies of Transient Computations 3.3.3.1

Common Cases

According to the Joukowsky Eq. (3.21), the pressure rise in the flow is directly proportional to the wave propagation speed. As a physical parameter, the wave propagation speed is primarily determined by the fluid type. It is additionally a function of the size and the material of the pipelines (see Sect. 3.3.8 below). In almost all literatures dealing with hydraulic transients, great efforts have been invested to accurately determine the wave propagation speed under different geometrical and material arrangements of pipelines. It appears to be quite natural and hence reasonable that the accuracy of transient computations directly depends on the accuracy of applied wave propagation speed. This viewpoint, however, is not well-founded, not to mention that it is also lacking of any comparison computations. In order to reveal the significance of the wave propagation speed in transient computations, the case of Fig. 3.8 is again considered. The closing of the flow is assumed to be performed by dQ/dt = const, i.e., dc/dt = const, see Fig. 3.8c. The time for a round trip of the pressure wave is T2L = 2L/a with a as the wave propagation speed. During this time, the flow velocity has changed by c = (dc/dt)T2L = (dc/dt)2L/a. According to Joukowsky’s formula, i.e., Eq. (3.21), which is only valid within the time t ≤ T2L = 2L/a, the pressure rise is computed as h = −

2L dc . g dt

(3.25)

It is evidently independent of the wave propagation speed, but merely depends on the velocity gradient dc/dt. This circumstance can be well explained with Fig. 3.9 for two fluids with different wave speeds. In the two cases, equal pressure rises have been achieved at t = T2L . Because of its application restriction t ≤ T2L , Eq. (3.25) also differs from Eq. (3.6) by a factor 2. In reality, the wave propagation speed, generally, does not have any remarkable effect on the maximum pressure rise in the flow, not even in real cases of flow regulations by opening or closing the valve. This performance of hydraulic transients can be demonstrated by comparing two computations of using different wave propagation speeds. For this purpose, computations based on the Wave Tracking Method (WTM), which will be described in Chap. 7 for details, have been carried out here in advance, as shown in Fig. 3.10 for the case of fully closing the flow in a simple

3.3 Method of Elastic Water Column Theory h

59

a2 >a1, dc/dt=const

h2L wave speed a2

2 L dc g dt h0

wave speed a1 o

t

T2L,2 T2L,1

Fig. 3.9 Equal shock pressure rise (h2L ) after a round trip (T 2L ) of shock waves with different propagation speeds under the same condition of flow regulation dc/dt = const

H0

(a) L nozzle

(b)

Fig. 3.10 Full-closing of the injector nozzle and the responses of shock pressures to different wave propagation speeds used in computations (wave tracking method)

pipeline. Obviously, the wave propagation speed really only determines the period of the pressure fluctuations in the flow after closing the valve, which indeed is usually not quite interesting. It does not affect the maximum pressure rise in the flow at all. Even the continuous pressure rise during the period of closing the valve is not appreciably affected by the wave speed.

60

3 Transient Flows and Computational Methods

From this fact, one might immediately draw the conclusion that the same maximum of the pressure rise in the flow should be given, if the wave speed is assumed to be infinite. The assumption is nothing else than referring to the condition of the rigid water column theory. The idea is absolutely correct. An intrinsic relation between the rigid and the elastic water column theory really exists. This will be introduced in Sect. 3.4 and exactly revealed in Chap. 9, where a direct method for accurate transient computations will also be derived. The fact that the wave propagation speed does not have any influence on the maximum pressure rise in the flow can be well utilized in practical applications. First, the wave speed variation in a given hydraulic system is commonly, say, less than 2%. There is no need at all to use different wave speeds for different pipes and at different locations within a given hydraulic system. Second, it appears to be simply unnecessary, to compute the wave propagation speed with great efforts and in fully complex dependences on the pipe material and the fixation form at the end of each pipeline, etc. It is, therefore, advisable for engineers, to firstly assess the necessary computational accuracy, before a great effort is made to “accurately” compute the wave propagation speed. In most cases, it is sufficient to use a wave speed of about a = 1300 m/s. For reference purposes, the wave propagation speeds for some fluid are given in Sect. 3.3.8.

3.3.3.2

Special Case: Fast Closing of the Flow

As a special case, rapid and complete closing of the flow (initial velocity c0 ) within the time T2L is considered. Because for this case the Joukowsky formula, i.e., Eq. (3.21) is applicable, the maximum pressure rise in the flow directly depends on the wave propagation speed. Obviously, a 5% inaccuracy in the wave speed will cause a 5% inaccuracy in the computation of the maximum pressure rise h max in the flow. This conceptual viewpoint, however, does not represent much useful information. More meaningful is to first perform the following differential computation from Eq. (3.23) with respect to h max = h max − h 0 dh max da . = h max − h 0 a

(3.26)

In finite difference form, both dh max and da are written as δh max and δa, respectively. The above equation is then further transformed into   h0 δa δh max . 1− = h max a h max

(3.27)

With this expression, the influence of an inaccuracy in the wave speed on the inaccuracy in the transient computations can be estimated. For instance, if a pressure rise of h max / h 0 = 1.5 would be expected, then, a 5% inaccuracy in the wave speed

3.3 Method of Elastic Water Column Theory

61

(δa/a = 0.05) will lead to an inaccuracy in computed pressures of about δh max = ±1.7%. h max

(3.28)

Such a computational precision in dealing with hydraulic transients in hydropower stations is usually well acceptable, not to mention that the assumed fast closing of the valve, as specified, will hardly happen in practical applications. In addition, the use of surge tanks in hydropower stations (Figs. 2.11 and 3.1) ensures further operational security of the systems.

3.3.4 Self-stabilization of Transient Flows by Regulation Valves The self-stabilization effect of transient flows through a regulation valve is a natural feature of fluid dynamics. It makes transient computations quite different, according to whether this effect is accounted for or not. Based on the computations and observations in the forgoing sections, the self-stabilization effect of transient flows should be explained. In the flow model shown in Fig. 3.8, the flow rate variation Q = f(t) has been predefined. It thus ignores the characteristic of the used regulation valve. This also means, when considering the characteristic of the injector given by Eq. (2.32) for instance, that the dependence of the flow rate on the pressure head is consequently ignored. For this reason, the predefined regulation condition Q = f(t) does not represent the reality and the computed significant pressure fluctuations are neither true. In reality, the closing of the flow by closing the valve, for instance, leads to a flow stagnation and thus to a pressure rise at the valve. The latter will again lead to a growth of the discharge through the valve. As a result, the previous flow stagnation at the regulation valve will be immediately relieved. Such a simultaneous self-regulation of flows under transient conditions leads to the smoothening of flow fluctuations. One, in fact, has already confirmed a steady pressure rise in Fig. 3.10 within the period of the closing process. There, the self-stabilization effect of the flow was automatically included in computations. It was active because the hydraulic characteristic of the valve and the hydraulic transients differently react with flow fluctuations. At a regulation valve which is open to air, for instance, the flow rate is commonly proportional to the square root of the pressure head. √ This can be formulated in terms of the flow velocity in the pipeline as c = k h with k > 0 as a quasi-constant. The fluctuation of the flow velocity is obviously determined by the fluctuation of the pressure head in the flow. If the derivative dc/dh is computed and its finite difference form c/h is used, then, one obtains instantaneous pressure changes associated with instantaneous changes in flow velocity as follows

62

3 Transient Flows and Computational Methods

h h = 2 c. c

(3.29)

On the other hand, each instant throttling of discharge by closing the valve leads to an instant pressure rise in form of water hammer at the valve. The corresponding relation can be formulated as a h = − c + f(h i−1 ). g

(3.30)

The first term on the r.h.s. corresponds to that of Joukowsky’s equation, see Eq. (3.21). The second term represents the influence of reflected wave which is a function of pressure head and is related with the previous time t − T2L , denoted by subscript i − 1. It is thus not affected by the current instant flow fluctuations. As the reflected wave, it also disappears within the time from the beginning of the regulation up to t = T2L . The exact expression of this second term will be shown in Sect. 7.1.1. Against Eq. (3.29), the coefficient of c in the above equation is negative. For each velocity variation, the pressure would react differently. This is to say that there would be two trends of different pressure variations, one is positive and another is negative. Because this is impossible and the transient flow must fulfill both equations at each instant, there must be a trend to h = 0 and c = 0. This means that at each instant opening of the valve during a flow regulation the flow will spontaneously and instantaneously tend to be stabilized. The corresponding stabilization mechanism is illustrated in Fig. 3.11. Both Eq. (3.29) for valve characteristics and Eq. (3.30) for water hammer are always satisfied. This performance of hydraulic transients is called self-stabilization of transient flows through valves. It directly leads to a continuous and smooth pressure rise in the flow during the time of closing the valve, see Fig. 3.10 again. After the valve has been completely closed (c = 0), Eq. (3.29) is no longer available, so that the self-stabilization of the flow becomes no longer effective. The pressure then jumps from its current value downwards to the “mirrored” value below the mean pressure head H 0 and falls into fluctuations. Such a fluctuation mechanism will be exactly revealed later in Sects. 9.3 and 9.4. As a matter of fact, as throughout this book, all computations of hydraulic transients are principally based on combined solutions of the above two equations for c and h. As long as the volume flow rate differs from zero, the self-stabilization effect Fig. 3.11 Flow fluctuations and mechanism of self-stabilization of transient flows through a regulation valve

h hs

Δh

s

−Δh cs

Δc

c

3.3 Method of Elastic Water Column Theory

63

is always included. In place of Eq. (3.29) for valves, it may be the characteristic of a hydraulic machine like the pump or the turbine. The self-stabilization effect is of general hydro-mechanical nature of hydraulic transients. It can be better understood by showing the difference between computations with and without considering this effect under otherwise equal conditions. For this purpose, the case of closing the flow by closing the injector nozzle in Fig. 3.10 is again considered. First, the linear discharge coefficient ϕ = ϕ0 + kϕ t is assumed. Because it is a purely geometrical parameter, according to Eq. (2.30), it can be exactly realized by defining the parameter s = f(t). The flow (c and h) through the injector is then subject to the self-stabilization effect. Second, the linear velocity variation is supposed to be directly given as c = c0 +kc t. The self-stabilization effect of transient flows is thus shut off and not included in computations. The method used for such computations is again the Wave Tracking Method (WTM) which in details will be described in Chap. 7. Figure 3.12 shows the comparison between computations under two different concepts. Even though the linear variations of the discharge coefficient ϕ and the flow velocity are quite comparable to each other (Fig. 3.12c), very different shock pressure responses in the flow have been respectively detected. Two conclusions can be drawn. First, the self-stabilization effect of transient flows does really and considerably contribute to smoothening of the pressure rise during the period of closing the flow. Second, for computations of transient flows, one should not predefine the flow rate in form of an enforced assumption c = f(t). This second statement becomes much more significant, when considering the shock pressure fluctuation after the injector is completely closed. The amplitude of computed pressure fluctuations is in the computed example twice higher than the real value. This is completely due to the mechanism that the pressure at the end of the closing time jumps from its current value downwards to its “mirrored” value below the stable pressure head H 0 . Such a pressure jump, however, strongly depends on the closing time of the valve. If, for instance, the closing time is set to be t = 28 s, the jump of the pressure could not occur, because, according to Fig. 3.12b, the current pressure is just equal to the stable pressure head H 0 = 410 m. In other words, the jump potential of the pressure to the pressure H 0 vanishes and the pressure fluctuation afterwards, therefore, must totally disappear. All these occurrences are, in reality, fictional, because c = f(t) can never be predefined. For this reason, computations of hydraulic transients should always be carried out under simultaneous considerations of valve characteristics or characteristics of all other comprised hydraulic machines (pumps and turbines). This clearly means that the graphical method used in the past for transient computations, as in Fig. 3.8, is inaccurate.

64

3 Transient Flows and Computational Methods

H0

(a) L nozzle

(b)

(c)

(d)

Fig. 3.12 Shock pressure responses at the closed injector based on computations with different concepts for closing the flow. a Pipeline and regulation organ considered for computations, b computation based on linear reduction of discharge coefficient ϕ = ϕ0 − kϕ t, c Computation based on linear reduction of flow velocity c = c0 − kc t, d comparison of used velocity reductions with two different concepts

3.3 Method of Elastic Water Column Theory

65

3.3.5 Pipe Elasticity and Size Response to the Pressure Shock The propagation speed of pressure shock waves in pipe flows represents a significant parameter in hydraulic transients. According to the Joukowsky equation, it directly determines the instantaneous pressure rise in the flow. The wave propagation speed, in effect, depends not only on the type of applied fluids, but also on the pipe size and the elastic modulus of applied pipe materials. While the wave speed in water at T = 20 °C is about a = 1475 m/s, the real propagation speed of a pressure wave, as often measured in pipelines, is always lower, and mostly lies in the range a = 1200–1400 m/s. The fact that both the size and the material of pipelines influence the wave propagation speed in water, can be demonstrated by considering the response of the pipe diameter to pressure variations in the flow. For simplicity, a pipeline of both constant cross-sectional area and wall thickness is firstly considered under static load (Fig. 3.13). The pipe is thus geometrically one-dimensional. The wall thickness of the pipeline is assumed to be sufficiently thin against the pipe diameter, as expressed by s  d. Furthermore, a constant pressure load (p) along the pipe axis is assumed. Under these simplified geometrical and pressure load conditions, the pipeline is only subject to transverse expansion. The normal stress σ in the length section of the pipe can be considered to be uniform, simply because of s  d. By considering a unit length of pipeline, one obtains from the force balance 2σ · s = pd.

(3.31)

The normal stress in the length section of the pipe is then given by σ =

(3.32)

(b) σ

(a)

d

p

d=2r

s

pd . 2s

p

σ Fig. 3.13 Pressure load and normal stress in the length section of a pressurized thin-wall circular pipe

66

3 Transient Flows and Computational Methods

Keeping this relation, each variation of the static pressure in pipe flow will lead to variation of the normal stress in the pipe wall. This, in turn, gives rise to variation of the circumferential strain, according to Hook’s law, as dε =

dσ . EM

(3.33)

The constant E M (in Pa, i.e., N/m2 ) is known as the Young’s modulus of the material of the pipe. With respect to Eq. (3.32) and because of dε = dr/r , it follows from Eq. (3.33) 1 d dr = d p. r E M 2s

(3.34)

This relation represents the transverse expansion of the pipe in responding to the pressure rise in the flow, which, for instance, is caused by water hammer. The pressure shock in the flow can then be significantly attenuated, if compared to that in a rigid pipe. Correspondingly, the change in the pipe cross-sectional area, which is given by dA/A = 2 dr/r , is then directly obtained from the above equation as 1 dA 1 d dp = . A dt E M s dt

(3.35)

All computations above are based on constant pressure load along the pipe axis, as this has been demanded by Eq. (3.32) for constant pipe cross-sectional area. This condition basically signifies that dA/dt = ∂ A/∂t and d p/dt = ∂ p/∂t which simply mean ∂ A/∂ x = 0 and ∂ p/∂ x = 0. In addition, free transverse expansion of the pipe must be presumed. All these conditions, however, will never be fulfilled in common hydraulic systems. As for the first, the pipe thickness is commonly not sufficiently thin when compared with the pipe diameter, and thus the normal stress distribution (σ ) within the wall thickness is non-uniform. As for the second, e.g., for vertical pipelines, one always has ∂ p/∂ x = 0. The same occurs, when pressure shock waves propagate along the pipe axis. Third, it very often happens that one or both ends of the pipeline are rigidly fixed, either axially or transversely. The condition of free transverse expansion of the pipe, thus, is not given. Despite all these restrictions, Eq. (3.35), however, clearly represents a good approximation in all cases. This is simply so, because the basic equation Eq. (3.33) is applied in differential form and, thus, the longitudinal effect, which is one order smaller than the transverse effect, can be generally neglected. Equation (3.35) especially reveals the intrinsic physical background of pipe dynamics. In practical applications, one prefers to use an appropriate, i.e., an equivalent elastic modulus (E M,eq ) to account for both the material properties of pipelines and the longitudinal effect. The objective of deriving Eq. (3.35) is to finally reveal the mechanism of the transverse expansion of a pipeline in influencing the propagation speed of pressure shock waves in pipe flows. The details will be shown later in Sect. 3.3.8. On the one

3.3 Method of Elastic Water Column Theory

67

hand, because of non-uniform load of the pipeline, the equivalent elastic modulus and hence the real wave propagation speed in a pipeline network can never be simply and accurately computed. On the other hand, as stated in Sect. 3.3.3, the accurate computation of the wave propagation speed is actually not so important. The global average of the wave propagation speeds in a pipeline system is sufficient for transient calculations.

3.3.6 Momentum Equations in Pipe Flows The main equation which describes the flow-dynamics of hydraulic transients in a pipeline is the momentum equation based on Newton’s second law of motion. It is also called D’Alembert’s principle. The momentum equation expresses the dynamic balance of all active forces exerted on the flow. To its application, an infinitesimal piece (dx) of flow in a pipeline is considered according to Fig. 3.14. The non-constant flow cross-sectional area along the pipe axis is assumed here just for the general case. As a special case, it can be considered to be caused by non-constant pressure distributions along a pipe of initially constant cross-sectional area, when considering the elasticity of the pipe with respect to Eq. (3.35). For this reason, the change of dA is only an infinitesimal value. The positive flow direction is assumed to coincide with x-coordinate. The pressure on the inlet cross-section is denoted as p. The flow rate is Q. On the exit cross-section, the pressure and the flow rate are given by p + d p and Q + dQ, respectively. In order to establish the equation of motion, all active forces exerted on the fluid element, i.e., the infinitesimal volume in Fig. 3.14 must be accounted for. τ

Fig. 3.14 Force balance at an infinitesimal piece of pipe flow; the non-constant flow section is assumed to be caused by non-constant pressure distribution along the pipe

dA

ρ+dρ p+dp ρ p

c

Q+dQ A+dA

d

Q

α

A

ε 0, water in tube leg 1 gradually moves to tube leg 2 with continuous conversion of the potential into the kinetic energy. As soon as the maximum level of water in tube leg 2 is reached, the back flow begins, just like the happening at t = 0 before. To calculate such a repeating flow oscillation, therefore, it is sufficient, only to compute the motion of water in the tube leg 1 towards the tube leg 2. The quantities to be computed also include the achievable maximum height of water in tube leg 2 and the time used for reaching that height. For this purpose, the position of the free surface of water in the tube leg 2 is recorded by the z-coordinate. The positive flow velocity is set to coincide with the positive direction of the z-axis. The head difference 2z as the difference between two water levels behaves as the time-dependent driving force which is given as −2gz Aρ and directed in the negative z-direction. It represents actually the gravitational force. For c > 0 in the present case, the viscous friction force takes −cf 21 ρc2 · π d L and is directed as well in negative z-direction. Thus, the differential equation of motion at time t > 0 is, according to Newton’s second law of motion, given as ρ AL

1 dc = −2gz Aρ − cf ρc2 π d L . dt 2

(4.2)

Obviously, the flow is subjected to acceleration as long as z < 0, i.e., the level of water in tube leg 2 is lower than that in tube leg 1 (exactly for cf = 0). The frictional force, however, always slows down the flow. With A = π/4 · d 2 as well as λ = 4cf from Eq. (2.4), it follows from the above equation 2gz 1 c2 dc =− − λ . dt L 2 d Because of c = written as

dz dt

and thus of

dc dt

=

dc dz dz dt

(4.3)

= c dc , the above equation can also be dz

dc2 λ 4g + c2 + z = 0. dz d L

(4.4)

In principle, the friction coefficient λ in this equation depends on the Reynolds number and thus on the flow velocity. An exact analytical solution of Eq. (4.4) including this friction effect is generally impossible. In reality, this is also unnecessary. With respect to the subordinated role of the friction in influencing the flow, the friction coefficient, while solving the above equation, can be considered to be quasi-constant. With λ = const = 0, Eq. (4.4) is solved as   g 2 1 1z − dλ z . −C ·e − c =4 d L λ2 λd 2

(4.5)

80

4 Rigid Water Column Theory and Applications

The integral constant C must be determined from the initial condition that at z = −h 0 there is c = 0 (Fig. 4.1). One then obtains  C=

 1 h 0 − λ h0 1 e d . + λ2 λ d

(4.6)

Equation (4.5) then becomes         −cf∗ 1+ hz g h0 2 ∗ z ∗ 0 1 − cf − 1 + cf · e , c =4 L cf∗ h0 2

(4.7)

in which the viscous resistance factor is defined as cf∗ = λ

h0 . d

(4.8)

The moving velocity of the water in the U-tube has thus been shown as an explicit function of the height of the water (z) in tube leg 2. For graphical presentations of Eq. (4.7), the motion velocity of water (c) has to be firstly tabulated as a function of the z-coordinate. While doing this, the friction coefficient λ in dependence on Reynolds number can be well accounted for. From the tabulated relation c = f(z), one is further able to compute the maximum reachable height of the water in tube leg 2 and the exact time needed for this. For all these reasons, Eq. (4.7) represents an exact solution of Eq. (4.4). The related numerical solution will be presented below in Sect. 4.1.1.2. At each time and to each computed moving velocity of water confined in the U-tube, the related kinetic energy is simply E kin = 21 ρc2 AL. Its ratio to the initial potential energy, Eq. (4.1), is given as ϕE =

E kin 2 = ∗ E pot,0 cf

      −c∗ 1+ hz z 0 1 − cf∗ − 1 + cf∗ · e f . h0

(4.9)

It is independent of the length (L) of the water column in the U-tube. For an approximation, the resistance factor can be considered as a time average and thus to be independent of the Reynolds number. The resultant computational inaccuracy is usually not significant. For the general case of a varying resistance factor, the energy ratios, as computed from the above equation, have been shown in Fig. 4.2. Obviously, there is ϕE < 1 because of the viscous friction effect. The maximum ϕE , i.e., the maximum moving velocity of the water is found at about z = 0. In addition, the slope of the energy ratio ϕE at z/h 0 = −1, i.e., at t = 0, is computed as dϕE = 2. d(z/h 0 )

(4.10)

4.1 Flow Oscillations in Open Pipeline Systems

81

Fig. 4.2 Ratio of kinetic to potential energies of the water column in a U-tube as a function of water level in the tube leg 2 for different viscous resistance factors

It is expectedly independent of viscous frictions, because at t = 0 there is Q = 0 and thus the friction effect vanishes. The maximum height, which can be reached in tube leg 2, is actually obtainable from Eq. (4.7). One has to be aware, however, that the maximum height depends on the friction in the entire process and thus is a process quantity. By again setting c = 0 in Eq. (4.7) and using the time average of the resistance factor (cf∗ ) for approximation, one obtains 1 − cf∗

 −c∗  z max = 1 + cf∗ · e f h0



1+ zmax h 0



.

(4.11)

One confirms that the height ratio z max /h 0 is equally independent of the length of the water column in the considered U-tube. Its implicit dependence on cf∗ has to be computed iteratively. Figure 4.3 shows the computational result. For cf∗ ≤ 1, the following approximation based on regression computation can be well applied z max = 1 − 0.6cf∗ + 0.2cf∗ 2 . h0 Fig. 4.3 Relative maximum height of water in tube leg 2 as a function of the viscous resistance factor

(4.12)

82

4 Rigid Water Column Theory and Applications

This equation, in principle, can be utilized to determine the mean value of the skin friction coefficient cf for the motion of the water column in a given U-tube. As the mean value, it can also be an “equivalent friction coefficient”, if local resistances in the U-tube are present. The equation, however, covers through the Reynolds number from zero to a maximum and thus does not distinguish between laminar and turbulent flows. Further, it should be mentioned that all above computations are carried out under the condition of vanishing surface tensions on the surfaces of the water columns in both tube legs.

4.1.1.1

Simplification by Neglecting Wall Friction (C f = 0)

By neglecting the viscous friction between the water column and the wall of the U-tube, it follows from Eq. (4.7) with cf∗ → 0 c2 =

 2g  2 h0 − z2 . L

(4.13)

One obtains this result also from Eq. (4.4) with λ = 0. Obviously, the maximum velocity is found at z = 0. The ratio of the kinetic energy to the initially available potential energy, as defined in Eq. (4.9), is simply computed as ϕE = 1 −

z2 . h 20

(4.14)

This relation has already been shown in Fig. 4.2. At z = 0 there is ϕE = 1, i.e., E kin,z=0 = E pot,0 . This means that the initially available potential energy is completely converted into the kinetic energy. With respect to c = dz/dt, Eq. (4.13) is further written as dz = dt



 2g  2 h0 − z2 . L

(4.15)

The solution of this differential equation is as follows 

z = −h 0 cos

2g t . L

(4.16)

It represents a harmonic oscillation with a period as

T = 2π

L . 2g

(4.17)

4.1 Flow Oscillations in Open Pipeline Systems

83

The oscillation period is only determined by the length of the water column in the U-tube. With the use of Eq. (4.16), one finally obtains from Eq. (4.13) the periodic velocity oscillation as 



2g 2g 2g h 0 sin t = cmax sin t . (4.18) c= L L L Both z = f(t) and c = f(t) have been shown in Fig. 4.1 qualitatively. Between c and z there is a phase shift of π/2.

4.1.1.2

Numerical Solutions

For the general case of Reynolds-dependent flow friction cf = f(Re), both the velocity c = f(t) and the height z = f(t) of the water column in a U-tube can be numerically computed. These two functions, in reality, are related to each other by c = dz/dt and can be resolved either basically from Eq. (4.4) or determinately from Eq. (4.7). To the former, for instance, Eq. (4.4) is written in the finite element form as follows  2 − ci2 = ci−1

 λi 2 4g z i−1 + z i ci−1 + z. d L 2

(4.19)

This equation enables the flow velocity to be numerically computed as a function of z-coordinate. Departing from the relation c = f(z) in the tabulated form, the time in the time series is obtained as ti+1 = ti +

z i+1 − z i . (ci + ci+1 )/2

(4.20)

Here, (ci + ci+1 )/2 is the mean velocity within the time increment t = ti+1 − ti . Correspondingly, z = z i+1 − z i is the movement of the water column in the U-tube. With the known velocity and thus the Reynolds number at the i-step, the ∗ in the next step can be friction coefficient λi+1 and further the resistance factor cf,i+1 computed. Table 4.1 shows an example of such a computational algorithm. At i = 0 there is obviously c = 0. At i = 1, the square of velocity, i.e., c2 is obtained from Eq. (4.10) as c12 =

4 gh 0 (z 1 − z 0 ). L

(4.21)

84

4 Rigid Water Column Theory and Applications

Table 4.1 Example of numerical computations to resolve functions z = f(t) and c = f(t) of damped flow oscillation in a U-tube, z0 = −0.6 i

z

Re

λ

cf *

c2

c

t

0

−0.60

–

–

–

0.000

0.000

0.000

1

−0.58

–

–

–

0.094

0.307

0.130

2

−0.56

9E + 04

0.018

0.036

0.183

0.428

0.185

3

−0.54

1E + 05

0.017

0.034

0.270

0.519

0.227

4

−0.52

2E + 05

0.016

0.032

0.352

0.594

0.263

5

−0.50

2E + 05

0.016

0.032

0.432

0.657

0.295

:

:

:

:

:

:

:

:

Eqs.

–

ci−1 d/ν

f(Re)

(4.8)

(4.19) or (4.7)

–

(4.20)

Figure 4.4 shows the graphical presentation of the computations in Table 4.1. The time of reaching the maximum height of the water in tube leg 2 is computed as tmax = 1.58 s. It is approximately about half of the oscillation period of the frictionless motion T = 3.17 s, which is obtained from Eq. (4.17). Fig. 4.4 Oscillation of water in a U-tube (d = 0.3 m) under the viscous friction effect (water column length L = 5 m, initial height h0 = 0.6 m)

4.1 Flow Oscillations in Open Pipeline Systems

85

4.1.2 Damped Flow Oscillation Between a Lake and a Surge Tank In this section, a type of flow oscillations in a rather realistic hydraulic pipeline system should be considered, which consists of a lake and a surge tank, as illustrated in Fig. 4.5. The lake has a constant height of water. The case to be considered is non-stationary flow from the lake to the surge tank, after shutting off the flow in the pipeline downstream of the surge tank, for instance, by closing the valve. Transients during the closing of the valve and the computational methods will be shown in Sect. 4.2. For simplicity as well as for specifying the initial flow state, the time is set to zero, as soon as the downstream pipeline is completely closed. The initial flow state is, thus, designated by a volume flow rate Q0 and a water height z0 in the surge tank. Depending on the pipeline system and the closing time of the downstream flow, there may be z0 > 0 or z0 < 0. In both cases, the available potential energy against z = 0 is computed as E pot,0 =

1 ρg AST z 02 , 2

(4.22)

in which AST is the cross-sectional area of the surge tank. With the initial velocity c0 in the main pipeline, the related initial kinetic energy is computed as E kin,0 = ρ AL

c02 . 2

(4.23)

he

AST

Surge tank

z0

Lake

z

zmax

z

A, L, c at t=0, c=c0

Q=0

Spherical valve

Fig. 4.5 Flow oscillation in a simple hydraulic system between the surge tank and the lake after closing the valve

86

4 Rigid Water Column Theory and Applications

The sum of initially available potential and kinetic energy represents the total mechanical energy in the flow. For frictionless flow, this sum remains constant. Otherwise, it must decay. The flow between the lake and the surge tank is again assumed to be subjected to the viscous friction effect. For simplicity, the flow resistance at the throttle section beneath the surge tank is neglected. In addition, the flow velocity in the surge tank is rather low and the height of the water column against the length of the pipeline is usually much shorter. Because of this, both the related dynamic pressure and the pressure force for flow acceleration in the surge tank can also be neglected. This implies that the pressure beneath the surge tank is simply equal to the height of the water column in the surge tank. A further assumption is made at the pipe entrance to the lake. Because of the non-zero velocity and thus non-zero dynamic pressure, the static pressure there is somewhat smaller than h e . The difference can still be ignored against the viscous friction loss in the long pipeline. The flow from the lake is considered to be positive. Relying on the force equilibrium on the mass flow in the pipeline, the momentum equation is written as ρ AL

1 dc = −ρgz A − cf ρc2 (π d L). dt 2

(4.24)

In principle, this equation is obtained in the same way as Eq. (3.3). dz Owing to dc = dc = cST dc , cST AST = c A and A = π d 2 /4 as well as λ = 4cf dt dz dt dz according to Eq. (2.4), it follows from Eq. (4.24) c

g AST 1 1 AST 2 dc =− z− λ c , dz L A 2 d A

(4.25)

or   d c2 1 AST 2 2g AST +λ c + z = 0. dz d A L A

(4.26)

This equation has the same form as Eq. (4.4). By considering the friction coefficient λ to be quasi-constant, therefore, one immediately obtains the solution c2 =

  λ AST 2g d 2 A 1 1 − z + C · e−λ d 2 L λ AST d A

AST A

z

.

(4.27)

The integral constant C can be determined from the initial condition according to Fig. 4.5 that at z = z 0 the velocity takes c = c0 , so that  C=

c02

  λ AST 2g d 2 A 1 1− z 0 eλ d − 2 L λ AST d A

AST A z0

.

(4.28)

4.1 Flow Oscillations in Open Pipeline Systems

87

Equation (4.27) can now be written as      AST z 0 AST z AST z 0 2gd 2 A 2gd 2 A z 2 1−λ 1−λ + c0 − 2 eλ A ( d − d ) . c = 2 λ L AST A d λ L AST A d (4.29) 2

The flow velocity in the pipeline is thus represented as an explicit function of z/d. Like in the last section dealing with flow oscillations in a U-tube, computations of flows shown in Fig. 4.5 can be simplified by neglecting the friction effect. The timedependent flow oscillations, i.e., c = f(t) and z = f(t), however, must be obtained through numerical solutions.

4.1.2.1

Simplification by Neglecting Wall Friction (C f = 0)

The above computations can be simplified for inviscid conditions. From Eq. (4.26), one obtains for λ = 0 dc2 2g AST + z = 0, dz L A

(4.30)

and further with respect to the initial conditions of both c = c0 and z = z 0 c2 = c02 −

 g AST  2 z − z 02 . L A

(4.31)

At z = 0, the maximum velocity is reached and given as 2 cmax = c02 +

g AST 2 z . L A 0

(4.32)

The related kinetic energy of total water in the pipeline is computed as E kin,z=0 = ρ

  2 2 c 1 g AST 2 cmax AL = ρ AL 0 + z0 . 2 2 2L A

(4.33)

It is equal to the sum of initially available potential and kinetic energies in the flow, see Eqs. (4.22) and (4.23). This maximum kinetic energy will be progressively and completely converted into the potential energy which is stored in the surge tank. The maximum reachable height of water in the surge tank is obtained from Eq. (4.31) by setting c = 0, yields

z max =

z 02 +

L A 2 c . g AST 0

(4.34)

88

4 Rigid Water Column Theory and Applications

With this maximum height, the maximum potential energy of water in the surge tank can be computed in a similar way as in Eq. (4.22). If then compared with Eq. (4.33), one again confirms E kin,z=0 = E pot,z=max .

(4.35)

The available kinetic energy in the pipeline at the moment of reaching z = 0 is then completely converted into the potential energy by reaching z = z max in the surge tank. Furthermore, the time-dependent flow oscillation should be computed. Because of Ac = AST cST , i.e., Ac = AST dz/dt, Eq. (4.31) is further written as dz = dt



A c0 AST

2

  − k 2 z 2 − z 02 ,

(4.36)

with k2 =

g A . L AST

(4.37)

With respect to Eq. (4.34) for z max , it further becomes  dz 2 = k z max − z2. dt

(4.38)

Integration of this equation under the condition of t = 0 and z = z0 leads to  arcsin

z z max



 − arcsin

z0



z max

= kt.

(4.39)

To further simplify this equation, the condition of z = 0 is considered. By reaching this flow state, the time passed is computed as tz=0

  z0 1 . = − arcsin k z max

(4.40)

With this time, one obtains from Eq. (4.39) z = z max sin k(t − tz=0 ).

(4.41)

Correspondingly, the flow velocity in the pipeline is computed from Ac = AST dz/dt as c = cmax cos k(t − tz=0 ),

(4.42)

4.1 Flow Oscillations in Open Pipeline Systems

89

with cmax = k

AST z max . A

(4.43)

The relation between cmax and z max in this last equation can be actually also obtained from Eqs. (4.32) and (4.34). Both equations (z and c) clearly represent harmonic oscillations of the flow in the considered hydraulic system. Between the height and the velocity oscillations there is a phase difference equal to π/2. The period of the flow oscillation is computed as

2π = 2π T = k

L AST . g A

(4.44)

This is the well-known formula for computing the period of oscillations in a hydraulic system with a surge tank which is built into most hydropower stations. At t = 0 there is z = z0 and c = c0 . It then follows from Eqs. (4.41) and (4.42) with respect to Eq. (4.43) tan ktz=0 = −k

AST z 0 . A c0

(4.45)

This equation can be used to determine tz=0 from z0 and c0 . For z 0 > 0, the time will be negative. This corresponds to the case of slowly closing the flow downstream of the surge tank, so that at the moment (t = 0) of completely closing the flow, there is already z 0 > 0. It should be mentioned that in such a case both z and c at time tz=0 are virtual, because they are simply obtained from extrapolation of Eqs. (4.41) and (4.42) which are only applicable to t > 0, see Fig. 4.6. The maximum reachable height of the water in the considered surge tank is computed by Eq. (4.34). The time used to reach this maximum is given by sin k(t − tz=0 ) = 1, i.e., k(tz=max − tz=0 ) = π/2, as obtained from Eq. (4.41). With respect to Eq. (4.44) for the period of flow oscillation, one obtains Fig. 4.6 Virtual time t z=0 for representing the harmonic oscillation of the height z of water in the surge tank

z0

z

c0 tz=0

c t=0

t

90

4 Rigid Water Column Theory and Applications

tz=max = tz=0 +

T . 4

(4.46)

It should be mentioned that, in reality, no constant flow oscillation can be expected, even if the viscous effect in the pipeline is neglected. First, the flow through the throttle section beneath the surge tank always results in a pressure drop as in the case of flows through an orifice. This flow property considerably diminishes the maximum height of water in the surge tank, which is given by Eq. (4.34). Second, the backflow towards the lake always means that the related kinetic energy in the flow will be dissipated in the lake and cannot be recovered back into the system. Later in Sect. 4.2.3, a real case of closing the flow in a simple hydraulic pipeline will be considered. Detailed computational algorithms and computation results will be shown. The maximum height of the water in the surge tank specifies the necessary hydraulic capacity of the surge tank in absorbing the kinetic energy which is connected with the flow in the pipeline. According to Eq. (4.34), it depends on the initial flow condition at t = 0, when the pipe downstream of the surge tank is completely closed. An extreme case with the maximum reachable height in the surge tank is given when for both c0 and z0 in Eq. (4.34) the respective values for the stationary flow are applied, which is found prior to the flow regulation. As z0 for instance, 2 /2g under stationary viscous flow conditions can be applied. This z 0,st = −Rc0,st estimation simply supposes that the shutdown of the flow downstream of the surge tank abruptly takes place and thus represents a most serious case. From Eq. (4.34), one also confirms how the surge tank size (AST ) will influence the surge tank capacity. Concrete computation examples regarding the start and stop of the turbine in a hydraulic system will be presented in Sects. 5.2.3 and 5.2.4.

4.1.2.2

Numerical Solutions

A numerical computation for the current case with a surge tank is only necessary, when the effect of viscous friction should be considered. For this purpose, the same numerical computational scheme as in Sect. 4.1.1.2 can be implemented. Basically, the above analyses are only for a defined positive flow, i.e., from the lake to the surge tank. In most cases, a computation of such a positive flow is sufficient for evaluating the hydraulic performance and the capacity of the surge tank. When computing the backflow, after reaching the maximum height z max in the surge tank, all terms regarding the viscous effect in the pipeline flow must be recalculated, because the frictional force is then reversely directed.

4.2 Flow Regulation and Computations of Shock Pressures

91

4.2 Flow Regulation and Computations of Shock Pressures In the last section, low-frequency flow oscillations both in a simple U-tube and a simplified surge tank system have been computed based on the rigid water column theory. It was not shown how the flow oscillations are caused by flow regulations and how the shock pressures during the flow regulations look like. In the current section, therefore, the mechanics of generating hydraulic transients and further the shock pressure responses will be outlined. The cases to be considered are opening and closing of the flow in a hydraulic pipeline for a Pelton turbine. The regulation organ is the injector nozzle. It should be again mentioned that the rigid water column model is only applicable for simple pipeline systems. For complex hydraulic systems which, for instance, consist of five different pipes in series, one would have to simultaneously solve five momentum equations. This is almost impossible or unrealistic in practical applications. For such complex hydraulic systems, the elastic water column theory appears to be favorable, see Chaps. 6 and 7 for theories and Chap. 11 for applications.

4.2.1 Flow Regulations Relevant hydrodynamic fundamentals of incompressible flows under transient conditions have been treated in Sect. 3.2. The equation of motion in its basic form has been presented as applicable to both the simple pipelines of constant cross-sectional area and the stepped pipelines of pipes in different diameters. As indicated there, the equation of motion, regarding Eq. (3.7), represents an implicit function, in which Q and h 2 are actually related to each other. In other words, it is one equation for two unknown parameters. An additional condition for coupling the discharge and the pressure is related with flow regulation itself. It is therefore specified by both the valve characteristic and the regulation dynamics.

he

lake

1

L, A

α

H0

x

turbine

c 2

Fig. 4.7 A simple hydraulic system with an injector nozzle of a Pelton turbine for regulating the discharge

92

4 Rigid Water Column Theory and Applications

In order to demonstrate the computational algorithm, a simple pipeline system is considered which operates with a Pelton turbine, as shown in Fig. 4.7. For the purposes of demonstration, the surge tank, which is commonly present, is first ignored. At the Pelton turbine, only the injector is connected to the pipeline, so that the dynamical rotation of the Pelton wheel can be left out of considerations. The flow regulation is achieved simply by regulating the spear-needle stroke in the injector nozzle, of which the detailed hydraulic characteristic has been outlined in Sect. 2.3.1, see Eq. (2.30). The discharge can be regulated down to zero. In the next sub-section, computational algorithms for directly resolving the discharge and the pressure head, respectively, will be firstly explained. Then, three application cases of the rigid water column theory will be shown in Sects. 4.2.3, 4.2.4 and 4.2.5, respectively. Both the reliability and the limitation of the theory will be demonstrated through comparisons with accurate computations based on the elastic water column theory.

4.2.2 Computational Algorithms and Simplifications Basically, there are two available algorithms for numerically computing transient flows and the related shock pressures in the pipeline shown in Fig. 4.7. The discharge related algorithm begins with the computation of the discharge, while the pressure related algorithm begins with the computation of the shock pressure. In both cases, as shown below, the momentum equation has to be combined with the characteristic of the injector nozzle.

4.2.2.1

Discharge-Related Algorithm

As a first step, Eq. (2.29) for the flow through the injector is combined with Eq. (3.7) by eliminating the pressure head h tot 

 1 L dQ . + R Q 2 = H0 − 2 2 g A dt 2gϕ AD0

(4.47)

This equation can be well numerically solved, as this is the common way in all transient computations. With dQ/dt = (Q i − Q i−1 )/t one obtains 

   1 L L 2 = 0, Q Q + R Q + − H + i 0 i−1 i g At g At 2gϕi2 A2D0

(4.48)

or in the short form of kA Q 2i + kB Q i + kC = 0.

(4.49)

4.2 Flow Regulation and Computations of Shock Pressures

93

This is a polynomial equation for resolving the volume flow rate Q i at the time ti with the known flow rate at the last time node ti−1 . In the coefficient kA , the discharge coefficient ϕ is a function of the spear-needle stroke s of the injector and thus of time, see Eq. (2.30). Computation must be stopped as soon as the flow regulation is over. For closing the injector, the end position is denoted by ϕ = 0. One should simply set Q = 0 because it cannot be resolved from the above equation. After the volume flow rate Q i as a function of time has been computed, the shock pressure at the injector can then be determined from either Eq. (2.32) or Eq. (3.7). For reasons of simplification, Eq. (4.47) can also be written in another finite difference form for numerical solutions. While dQ/dt = (Q i − Q i−1 )/t is applied, all other parameters are denoted by the subscript “i − 1” referring to the last time step. One obtains Q i ≈ Q i−1 − g A

t L



1 2 2gϕi−1 A2D0

 t H0 . + R Q 2i−1 + g A L

(4.50)

This is the equation for direct computation of the volume flow rate Q i at the time ti from the known flow rate at the last time step ti−1 .

4.2.2.2

Pressure-Related Algorithm

Actually, as shown above, the discharge-related algorithm for computing the shock pressure in transient flows represents a well applicable and simple method. For the later purpose of demonstrating the intrinsic relations between the rigid and the elastic water column theory in Chap. 9, however, the pressure-related computational algorithm should be outlined here. From Eq. (2.32) with the approximation 1−(ϕ AD0 /A)2 ≈ 1, which simply means the neglect of the dynamic pressure head and hence h ≈ h tot , one obtains  ϕ AD0 dQ dϕ = AD0 2gh + dt dt 2



2g dh . h dt

(4.51)

This equation is inserted into Eq. (3.7) of the momentum equation. Again neglecting the dynamic pressure head, one obtains the shock pressure at the injector section

2 AD0 dϕ √ AD0 ϕ dh h = H0 − R Q 2 − L h− L√ . (4.52) g A dt A 2gh dt The discharge coefficient ϕ and further its rate of change dϕ/dt can be computed from Eq. (2.30) with predefined regulation rule s = f(t).

94

4 Rigid Water Column Theory and Applications

This last equation has to be solved by a numerical method again. Therefore, the term dh/dt on the r.h.s of equation will be replaced by h/t with h = h i − h i−1 . Within each time increment t, the flow can be considered to be quasi-stationary. This allows to consider the volume flow rate to be that value at the last time step; it is thus denoted by Q i−1 . The approximation is anyhow adequate, because the related frictional term R Q 2 is one order smaller than the pressure head h in the above determination equation. For this reason, Q i−1 always behaves as a known quantity. From Eq. (4.52) one then obtains  h i = H0 − R Q 2i−1 − Bi h i − Ci h,

(4.53)

with

Bi = L

ϕi 2 AD0 dϕi AD0 L and Ci = . √ g A dt A t 2gh i−1

(4.54)

For closing the injector, there is obviously B < 0 because of dϕ/dt < 0. In the parameter C, the pressure head h i−1 is referred to the last time node. This is only for the computational simplicity relying on the assumption of quasi-stationary flows within each finite time step t. This is again allowed, because the last term in the above equation is actually negligible, see below. With respect to h = h i − h i−1 it follows from Eq. (4.53)    (1 + Ci )h i + Bi h i − H0 + Ci h i−1 − R Q 2i−1 = 0.

(4.55)

√ This is a quadratic equation for resolving the unknown parameter h i and further the pressure head h i at each time node. Against Eq. (4.49), this equation also applies to the case Q = 0 at the end of complete closing of the injector. The related flow rate can then be computed, for instance, from Eq. (2.32). On the other hand, when considering the pressure response during the period of flow regulation, the last term in Eq. (4.53) is negligible, if compared, for instance, √ with Bi h i or h i − H0 . By neglecting that last term, one obtains from Eq. (4.53) the following simplification    h i + Bi h i − H0 − R Q 2i−1 = 0.

(4.56)

√ Under the condition of quasi-stationary flow within a time step t, the expression √ h i , against the order of magnitude of h i in equation, can be replaced by h i−1 . In this way, one also obtains the following direct computation  h i = H0 − R Q 2i−1 − Bi h i−1 .

(4.57)

4.2 Flow Regulation and Computations of Shock Pressures

95

For frictionless flow, the pressure head at the injector is only determined by the parameter B and thus by dϕ/dt. Both equations represent an analytical way for computing the pressure head h = f(B). It appears to be mandatory to prove the computational accuracy of the approximated Eq. (4.56) by comparing the equation with Eq. (4.55). For simplicity, the simple flow system in Fig. 4.7 is considered and the injector is assumed to be closed. The design and operation parameters of the considered pipeline are listed in Table 4.2. For a given initial opening of the injector nozzle s0 = 150 mm and a closing time tc = 30 s, the linear closing law of the injector nozzle with ds/dt = −5 mm/s is assumed. The applied computational algorithm for Eq. (4.55) is shown in Table 4.3. For Eq. (4.56), the computation is similar. The computation has to be stopped, as soon as the end of flow regulation is reached.

Table 4.2 An example pipeline system with an injector nozzle of the Pelton turbine Parameter

Symbol

Unit

Value

Available hydraulic head

H0

m

410

Pipeline length

L

m

1250

Pipeline diameter

d

m

1

Viscous friction coefficient

λ

–

0.015

Resistance constant

R

s2 /m5

1.55

Wave propagation speed

a

m/s

1250

Normal flow rate (discharge)

Q0

m3 /s

1.875

Injector nozzle diameter

D0

mm

200

Injector nozzle constant

a1

–

1.536

Injector nozzle constant

a2

–

−0.857

Normal needle stroke (open)

sN

mm

150

Table 4.3 Computational algorithms for solving Eq. (4.55) Time

Injector nozzle

Equation (4.55)

t (s)

s/D0

ϕ

dϕ/dt

B

C

h0.5

h

Q

0.0

0.75

0.67

−0.006

−0.14

0.188

20.1

404.3

1.87

1.0

0.73

0.66

−0.007

−0.17

0.186

20.2

407.3

1.86

2.0

0.70

0.66

−0.008

−0.19

0.183

20.2

408.3

1.84

3.0

0.68

0.65

−0.009

−0.21

0.181

20.2

408.9

1.82

4.0

0.65

0.64

−0.011

−0.24

0.178

20.2

409.6

1.79

5.0

0.63

0.63

−0.012

−0.26

0.174

20.3

410.2

1.76

6.0

0.60

0.61

−0.013

−0.29

0.171

20.3

410.9

1.73

:

:

:

:

:

:

:

:

:

96

4 Rigid Water Column Theory and Applications

Fig. 4.8 Shock pressure response in the flow while closing the injector and comparison of computations, respectively, by Eqs. (4.55) and (4.56)

Figure 4.8 shows the shock pressure response in the flow while closing the injector, and the comparison of computations, respectively, by Eqs. (4.55) and (4.56). Obviously, an excellent agreement between two equations has been given. This clearly demonstrates that the last term in Eq. (4.53) is really negligible and Eq. (4.56) is sufficiently accurate. This statement applies, at least, to the current case of closing the flow. The computation, however, is only valid up to the end of the closing time. At the very beginning (t < 2 s) of closing the flow, a somewhat notable discrepancy between the two computations is confirmed. This discrepancy, however, has no meaning at all, because it corresponds to the range where the condition of “short pipe” has not been fulfilled, see Sect. 3.2.1. Equation (4.56) will be referred to later in Chap. 9 for deriving the conformity between the rigid and the elastic water column theory. There, it will be demonstrated that Eq. (4.55) and even Eq. (4.56) are accurate solutions of hydraulic transients. That is to say that they should not be further considered as “approximations” because of the assumption of rigid water. For this reason, Eq. (4.56) will be further considered and simplified later in Sect. 9.6. As an advanced information, for instance, Eq. (4.57) can be simplified by ignoring the influence of the friction, as this √ at least at the moment of Q ≈ 0. √ is justified Furthermore, the approximation h i−1 ≈ H0 can be assumed if compared with the order of magnitude of h i in the equation. One then obtains from Eq. (4.57)  dϕ L AD0  2g H0 . h ≈ H0 − B H0 = H0 − g A dt

(4.58)

The pressure rise is simply proportional to the changing rate of the discharge coefficient. In the presented computation example in Fig. 4.8, the maximum pressure rise is confirmed at the end of closing the flow (Q = 0). With the specified regulation

4.2 Flow Regulation and Computations of Shock Pressures

97

dynamics ds/dt = −5 mm/s, one obtains dϕ/dt = −0.0384 and further, from the above equation, the maximum achievable pressure head h max = 427.6 m. It satisfactorily agrees with that value which has been shown in Fig. 4.8. The maximum pressure rise in the flow is always obtained in accordance with the maximum gradient dϕ/dt which is included in the parameter B. Thus, for limiting the maximum pressure rise, the flow regulation has to be specified with respect to the regulation dynamics dϕ/dt. The best choice of it is to define a parabolic regulation law; this will be presented in Sect. 10.2.

4.2.3 Pressure Response by Closing the Injector Nozzle For the case of closing the flow, the computational algorithm relying on the rigid water column theory has been outlined in Table 4.3. Corresponding computations shown in Fig. 4.8, however, have not yet been validated, for instance, by comparison with accurate computations relying on the elastic water column theory. Although the comparison of computations in Fig. 3.15 has already demonstrated the computational accuracy of the rigid water column theory, the computation, nevertheless, was based on the use of velocity gradients, which were computed from the elastic water column theory. Therefore, it still seems to be a bit imperfect. For this reason, the same hydraulic system as shown in Fig. 4.7 and Table 4.2 as well as the same case of closing the flow as in Fig. 4.8 are again considered here. This time, the discharge-related computational algorithm according to Eq. (4.49) should be applied, as shown in Table 4.4. It obviously represents quite a simple and rather practical computational procedure. Computational results have been shown in Fig. 4.9. For validation purpose, the accurate computation based on the elastic water column theory, i.e., the wave tracking method (see Chap. 7) has also been shown in the figure. It additionally covers the Table 4.4 Computational steps of using the rigid water column theory by using Eq. (4.49) with kB = 324.6 t (s)

s/D0

kA

kC

Q (m3 /s)

h (m)

0.0

0.750

117

−1019

1.87

404.3

0.5

0.738

118

−1019

1.87

406.0

1.0

0.725

119

−1017

1.86

406.9

1.5

0.713

120

−1014

1.85

407.4

2.0

0.700

122

−1011

1.84

407.9

2.5

0.688

123

−1008

1.83

408.2

3.0

0.675

125

−1004

1.82

408.6

:

:

:

:

:

:

30.0

0.000

–

−428

0.00

427.8

98

4 Rigid Water Column Theory and Applications

Fig. 4.9 Pressure head response in the flow by linearly closing the injector; comparison between computations based on the rigid and the elastic water column theory, respectively

time after closing the flow and shows the true pressure fluctuation in the pipeline. Worth mentioning is that the two computations are independent of each other. They are, therefore, unlike those shown in Fig. 3.15 where the velocity gradient dc/dtfrom the elastic water column theory has been used for the computation by using the rigid water column theory. The linear regulation of the injector nozzle leads to an almost linear increase of the velocity gradient, which has also been shown in the figure. This in turn leads to the almost linear pressure rise. The background of such a linear relation is the relation of Eq. (3.4), at which the non-linear term c2 /2g only plays a subordinate role. It is also determined by Eq. (4.58). The remarkable points which can be confirmed from Fig. 4.9 and from the comparison between two theories are as follows:

4.2 Flow Regulation and Computations of Shock Pressures

99

Remarkable point 1: The rigid water column theory is a good approximation for transient computations. In an expanded sense, it is also an accurate theory, as will be presented later in Chap. 9. According to Eq. (3.4) and like in Fig. 3.15, the shock pressure during the closing time is again mainly determined by the velocity gradient. At the very beginning of closing the flow, the condition of “short pipe” for the rigid water column theory is not fulfilled. Towards the end of the closing period, the flow velocity itself progressively loses its influence and the shock pressure is then purely proportional to the velocity gradient or equivalently to the changing rate of the volume flow rate. From Eq. (3.4), it follows with dQ/dt = −0.11 h = −

L dQ = 17.8 m, g A dt

(4.59)

which has also been confirmed in Fig. 4.9b. Remarkable point 2: According to the above equation or Eq. (3.57), the pressure rise at the injector is directly proportional to the length of the pipeline. It might reach an inacceptable level, if the pipeline is considerably long, for instance, more than 10 km at some hydropower stations. In order to limit the pressure rise in the flow, so-called surge tanks are often used at a short distance ahead of the regulation organs, see Figs. 2.11, 3.1 and 4.5. More about the functionality and computations will be shown in Chap. 5. Another method to reduce the maximum shock pressure in the flow is to optimize the regulation dynamics, for instance s = f(t) at the injector nozzle. The possible ways of doing this can be found in Sect. 10.2. Remarkable point 3: As known, the fundamental difference between the rigid and the elastic water column theory is the different concept about the compressibility of the flow and thus the wave propagation speed in the flow. Disregarding the true wave speed in the water at about a = 1250 m/s, the rigid water column theory simply postulates that the wave speed is infinitely large. Because satisfactory computational agreement between the two theories has been achieved in Fig. 4.9, it can be said that the wave propagation speed used in the elastic water column theory does not affect the computational results for t > T2L (condition for short pipe). Instead of the wave propagation speed, it is the changing rate of the flow velocity, which is responsible for the pressure rise in the flow. This again verifies the statement which has been made in Sect. 3.3.3 regarding Fig. 3.10. It can also be concluded that the Joukowsky equation is only applicable within a very short time t ≤ T2L . For computations, obviously, an inherent relation between the rigid and elastic water column theory must exist. This will be accurately revealed in Chap. 9 where the method of direct computations (MDC) will be derived. Because of the insignificance of the wave speed in influencing the maximum pressure rise in the flow, it is no longer necessary to make a great effort for accurate estimation of the wave propagation speed. This conclusion is valid at least for hydraulic transients in hydropower plants, where the duration of a hydraulic process (regulation of power output, start and shutdown of machines) is usually much longer than the time T2L .

100

4 Rigid Water Column Theory and Applications

Remarkable point 4: Because the rigid water column theory is unable to resolve the high-frequency fluctuations in the flow, computations have to be stopped at the end of each flow regulation. In Fig. 4.9b, the rapid pressure drop at t = 30 s, as it has been confirmed from the elastic water column theory, cannot be detected by the rigid water column theory.

4.2.4 Pressure Response by Opening the Injector Nozzle Corresponding computations can be performed for the flow regulation by opening the injector nozzle of a Pelton turbine. For the same configuration of the pipeline system as treated in the forgoing section referring to Fig. 4.9, the pressure response in the flow at the section of the injector has been computed and shown in Fig. 4.10.

Fig. 4.10 Pressure head response in the flow by linearly opening the injector, and comparison between computations based on the rigid and the elastic water column theory, respectively

4.2 Flow Regulation and Computations of Shock Pressures

101

Again for reasons of comparison, the accurate computation based on the elastic water column theory has also been displayed. Similar remarkable points as in Fig. 4.9 in the last section can be made. Especially at t = 0, the changing rate of discharge is computed as dQ/dt = 0.105, i.e., for the velocity dc/dt = 0.134, which, according to Eq. (3.4) or Eq. (3.57), leads to a pressure-head drop of h ≈ −17.7 m. It is only about half of the pressure drop based on accurate computations. The reason of the factor 2 can be tracked back to the difference between Eq. (3.4) for the rigid water column theory and Eq. (3.25) from the elastic water column theory (Joukowsky). The latter is also valid for the current case of opening the flow. Because for the application of rigid water column theory in the current example, the condition of a short pipe is only fulfilled for t > 8T2L = 16 (Fig. 4.10), the pressure-head drop h ≈ −17.7 at t = 0 is clearly not true. The computational results, however, represent the time-averaged mean of existing pressure fluctuations. In Sect. 3.3.4, the self-stabilization of the high-frequency pressure fluctuations has been outlined. Such a property of hydraulic transients has also been confirmed in Fig. 4.10. It shows its effect in the rapid attenuation of the pressure fluctuations during the progressive opening of the injector. A quantitative description of such an attenuation effect can be found in Sect. 9.4.2.

4.2.5 Pressure Response by Stepped Pipelines If Eq. (4.49) is applied to the flow in a pipeline which comprises pipes of different diameters and lengths (Fig. 4.11), then, the equivalent pipe cross-sectional area Aeq according to Eq. (3.18) should be used. To estimate the related computational inaccuracy, corresponding computations should be compared with accurate computations based on the elastic water column theory. As demonstrated in Fig. 4.11 for the cases of both closing and opening the flow, the computations based on the rigid water column theory have been satisfactorily verified, except for the high-frequency pressure fluctuations. The non-regular pressure fluctuations from the elastic water column theory are due to reflections and transmissions of pressure waves at each non-continuity cross-section. For more about the related computational method see Chap. 7.

102

4 Rigid Water Column Theory and Applications

L=430, d=1.5

L=430, d=1.3

H0

(a)

L=430, d=1.0

(b)

(c)

Fig. 4.11 Stepped pipeline and pressure responses in it by linearly closing and opening the injector; comparison between computations based on the rigid and the elastic water column theory, respectively

Chapter 5

Surge Tank Functionality and System Stability

Each hydraulic system in a hydropower station usually involves one or more surge tanks. Such a surge tank behaves as a significant component. It enables the hydraulic system to be rapidly started and stopped. Its essential functions have been found to relieve the system pressure and to stabilize the system operations under conditions of hydraulic transients. These two functionalities, i.e., hydraulic aspects of the surge tank must be accounted for when designing the surge tank and optimizing the operation conditions of the related hydraulic network. In the old designs of surge tanks regarding their capacity, the concept of the system stability also functioned as a significant hydraulic element. The present chapter exclusively deals with the general hydraulic aspects of the surge tank. The suitable method for performing these tasks is basically restricted to the rigid rather than the elastic water column theory. This is so because all relevant hydraulic aspects of the surge tank in a hydraulic system are only related to lowfrequency flow oscillations (surges). Computations, however, may become rather difficult for complex pipeline networks. This is because, as said many times before, a large number of momentum equations have to be simultaneously solved when using the rigid water column theory. In such cases, one should better perform computations by using the elastic water column theory, with which the low-frequency flow oscillations in the system will be automatically resolved. Corresponding computations will be shown later in other chapters. The simplest surge tank is built in the form of a circular cylinder which is specified by the diameter and the height (see, for instance, Figs. 3.1 and 4.5). Depending on the application situations, the surge tank may take other forms. Quite often, as in pumped storage power stations, a surge tank even on the low-pressure side of the hydraulic system is also required, see Fig. 1.5.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_5

103

104

5 Surge Tank Functionality and System Stability

5.1 Functionalities of the Surge Tank A typical and simple hydraulic system in a hydropower station consists of a lake (upper reservoir), a penstock, a surge tank, a pressure shaft and a group of turbines or pumps, as schematically illustrated in Fig. 5.1. The penstock serves to transport water. It may possess a length of more than 10 km in large hydropower stations. This means that for normal operations a large amount of kinetic energy is included in the flow. This further means that the flow in the penstock cannot be rapidly slowed down when, for instance, the machines (like turbines) are shut off in emergency. There are other aspects which provide the reasons for the use of surge tanks. Basically, a surge tank is required for two main tasks: Task 1: It enables the flow in the long upstream penstock to be gradually slowed down by absorbing and transforming the kinetic energy in the flow into the potential energy of water in the surge tank, when the turbines are shut down or the load is reduced. This ensures that the pressure rise both in the downstream pressure shaft and on the turbine units can be effectively limited. Task 2: The surge tank serves as a water supplier and provisionally provides the water flow while turbines are started or the load is enhanced. Otherwise, if no surge tank is available, one has to carefully set the process of opening and starting the machines with respect to the limited flow acceleration in the upstream pipeline. The related mechanics has been described in Sect. 3.2.3, see Fig. 3.5 and Eq. (3.12). Besides these two tasks of a surge tank, an old aspect in designing a surge tank focuses on the system dynamics and stability. According to Thoma (1910), a surge tank must be of sufficient size (diameter), in order to effectively dampen the system oscillations, which usually takes place between the lake (upper reservoir) and the surge tank in connection with the flow regulation. Otherwise stated, the flow oscillation might become uncontrolled in form of the mechanical resonance. Correspondingly, the Thoma criterion for determining the surge tank size had often been applied. As will be shown below in Sect. 5.3, this criterion is in practice no longer relevant. In practical applications, the surge tanks are built in different forms to meet the various requirements. Besides the circular form of constant cross-sectional area, the surge tanks are also designed in other irregular forms as shown in Fig. 5.2, for Surge tank Lake

Penstock Pressure shaft

Fig. 5.1 Hydropower system with a surge tank

Machines Reservoir

5.1 Functionalities of the Surge Tank

(a)

(b)

105

(c)

Fig. 5.2 Different forms of surge tanks used in hydropower stations; for other forms of the surge tank see Chaudhry (2014), Giesecke and Mosonyi (2009), for instance a surge tank with inlet resistance, b surge tank with a lower chamber, c surge tank with an upper chamber

instance. The throttle with an enlarged flow resistance, as found at the bottom of the surge tank, serves to effectively dampen the low-frequency flow oscillations in the system, i.e., between the lake and the surge tank. The concept of surge tanks with the lower and the upper chambers aims to extend their capacities. The lower chamber, for instance, is able to store more water which is available when starting the turbines. This is especially necessary, according to Fig. 5.1, if the upstream penstock is long and the total mass of water cannot be speeded up to the required flow rate in time. The use of the upper chamber of a surge tank, or a surge tank with free overflow, is to limit the maximum pressure rise in the hydraulic system when a large amount of water is expected to flow into the surge tank. There are many other forms of surge tanks like the differential surge tanks and the surge tanks with compressed air, see for instance Giesecke and Mosonyi (2009) or Chaudhry (2014). The main difference between different forms of surge tanks lies, again, in the capacity and the damping effect of the surge tank. While each hydraulic transient can be computed by numerical methods, the impact of any complex form of the surge tank on the entire flow process can be well captured and accurately evaluated.

5.2 Momentum Equations and Numerical Solutions 5.2.1 Basic Computations With regard to the functionality and the capacity of a surge tank, as outlined above, transient computations are commonly performed by considering the momentum equations and using the numerical computational algorithms. In this section, referring to Fig. 5.3 for a simplified hydraulic system as well as for the reference purpose in further considerations, the basic computational algorithm will be presented. The indices 1, 2, and 3 denote the three branches which are connected at the node point “m” below the surge tank. The surge tank itself is denoted as branch 3.

106

5 Surge Tank Functionality and System Stability

2217 m.a.s.l

2300 m.a.s.l

h2

Q1

Q3 m

d2 L2

d1, L1

z2

Q2

Lake

z3

h3

d3

1760 m.a.s.l

Turbines, Qmax=90 m3/s

Dimensions of the system d1 d2 d3 L1 L2 6.8 3.8 13 3700 1400 R1=0.0003, R2=0, R3=0.001

Fig. 5.3 A simplified hydraulic system in a pumped storage power station (pumps and Francis turbines)

The surge tank performs its functionality in response to the flow regulation in branch 2. The flow regulation may be related to the start or stop of hydraulic machines and can be generally expressed by the given discharge as a function of time as Q 2 = f(t). This implies that it is generally unnecessary to know about what types of hydraulic machines and the regulation organs are available. Therefore, the discharge regulation can be well simulated by supposing a regulation organ like a butterfly valve or a Pelton turbine injector. A spherical valve can also be supposed. However, it is usually only applied as a closure organ. The objective of related transient computations is to compute the flow oscillation between the lake and the surge tank in the hydraulic system shown in Fig. 5.3. It is indeed about the extension and concretized descriptions of computations which have been made in Sect. 4.1.2 together with Fig. 4.5. The cases to be considered are turbine start and stop, which are realized by opening and closing the flows, respectively. For simplicity, the opening and closing of the flows are simulated by using a Pelton turbine injector, of which the characteristic is known according to Eq. (2.33) and given by  Q 2 = ϕ AD0 2gh 2 .

(5.1)

The momentum equation for the volume flow rate Q 2 in branch 2 will now be established. For simplicity, the flow is assumed to be frictionless. This is at least justified for the case of closing the flow, because for the given flow rate Q 2 = f(t) and after closing the injector the interested flow only oscillates between the lake and the surge tank, independent of the pipe length L 2 and thus independent of the flow resistance in it. With respect to the static pressure head h m at the node point “m” below the surge tank, the equation of motion in branch 2 of the constant cross-sectional area is given as

5.2 Momentum Equations and Numerical Solutions

ρ A2 L 2

dc2 = ρg A2 (h m − h 2 ) + ρg A2 z 2 . dt

107

(5.2)

This equation is combined with Eq. (5.1) for eliminating the parameter h 2 . It follows with Q 2 = c2 A2 1 L 2 dQ 2 + Q 2 − z 2 − h m = 0. g A2 dt 2gϕ 2 A2D0 2

(5.3)

This is the first equation of motion in the considered hydraulic system. As will be shown later, the first term in the above equation, which represents the inertial force of the total mass in branch 2, can be neglected. In Fig. 5.3, the variable height of the water in the surge tank is denoted by h 3 . Because the flow Q 3 through the throttle of the surge tank commonly leads to a pressure drop, the pressure head at the bottom of the surge tank is given by h m − R3 |Q 3 |Q 3 , with R3 as the related flow resistance constant. Further, because of the non-constant flow rate Q 3 , the pressure head on the bottom of the surge tank is not equal to the height h 3 . The motion of the water in the surge tank is then described by ρ A3 h 3

dc3 = ρg(h m − R3 |Q 3 |Q 3 )A3 − ρ A3 h 3 g. dt

(5.4)

With Q 3 = A3 c3 and dQ 3 /dt = dQ 1 /dt − dQ 2 /dt, the above equation is further written as dQ 2 g A3 dQ 1 − = (h m − h 3 − R3 |Q 3 |Q 3 ). dt dt h3

(5.5)

This is the second equation of motion in the considered hydraulic system. Like for Eq. (5.3), the changing rate of the flow rate Q 3 , as given on the l.h.s. of equation, can more reasonably be ignored. Now, the equation of motion in the upstream branch from the lake to the surge tank should be established. The flow resistance constant is assumed to be R1 . By further assuming a constant pipe cross-sectional area, the static pressure difference between the two ends of the branch can be expressed by the difference z 3 − h m , when neglecting the dynamic pressure for approximation. The motion equation applied to this upstream branch is then given by1 ρ A1 L 1

1 When

dc1 = ρg(z 3 − h m )A1 − ρg A1 R1 |Q 1 |Q 1 . dt

(5.6)

the inlet is not found at the same height as the node point m, then the gravitational force in the branch has to be added, in order to obtain pressure head h at the node (m) under the surge tank. Consequently, the same equation of motion is again obtained.

108

5 Surge Tank Functionality and System Stability

By using the volume flow rate, this equation is further written as g A1 dQ 1 = (z 3 − h m − R1 |Q 1 |Q 1 ). dt L1

(5.7)

This is the third equation of motion in the considered hydraulic system. If this upstream branch consists of pipes of different diameters, an equivalent pipe crosssectional area according to Eq. (3.18) should be used. Equations (5.3), (5.5) and (5.7) together form an ordinary differential equation system for computing time-dependent flows in the considered hydraulic system. By comparing these non-linear equations with Eq. (3.7) for one-strand flow system, it is obvious that the computations become much complex. One must simultaneously solve all three equations which are tightly connected to each other. Basically, only a numerical method is employed, for which quasi-stationary or linear flows within each finite time increment have been commonly assumed. For solving the established system of differential equations, first, the derivative dQ 1 /dt in Eq. (5.7) is inserted into Eq. (5.5). The pressure head h m is then resolved and further inserted into Eq. (5.3). One obtains, respectively   1 1 h 3 dQ 2 T2 − , hm = T1 g A3 dt     1 L2 1 1 h 3 dQ 2 T2 2 + + Q − + z 2 = 0, g A2 T1 A3 dt T1 2gϕ 2 A2D0 2

(5.8) (5.9)

in which the terms T1 and T2 are defined by T1 = 1 + T2 =

A1 h 3 , A3 L 1

A1 h 3 (z 3 − R1 |Q 1 |Q 1 ) + (h 3 + R3 |Q 3 |Q 3 ), A3 L 1

(5.10) (5.11)

For numerical solutions, Eq. (5.9) needs to be written in the finite difference form. Within each time increment, quasi-stationary flows are assumed. This includes the quasi-constant pressure head h3 and all three volume flow rates. Such an assumption is permissible, because in the current consideration only the low-frequency flow oscillations are in focus. The high-frequency pressure fluctuations remain unconcerned. Thus, one obtains     1 1 h 3 Q 2,i − Q 2,i−1 T2 1 L2 2 (5.12) + + Q − + z 2 = 0, g A2 T1 A3 t T1 2gϕ 2 A2D0 2,i or in short Q 22,i + TB Q 2,i − TC = 0,

(5.13)

5.2 Momentum Equations and Numerical Solutions

109

with  L2 1 h3 , + A2 T1 A3      1 L2 1 h3 T2 TC = 2ϕ 2 A2D0 Q 2,i−1 + g + + z2 . t A2 T1 A3 T1 TB =

2ϕ 2 A2D0 t



(5.14) (5.15)

Equation (5.13) represents a quadratic polynomial, from which the flow rate Q 2 at each instant in the numerical time series can immediately be determined. Then, the changing rate dQ 2 /dt in the passage 2 is calculated from Eq. (5.9). Moreover, both the pressure head h m at “m” and the changing rate dQ 1 /dt of the volume flow rate in passage 1 are obtainable from Eqs. (5.8) and (5.7), respectively. The volume flow rate in passage 1 is numerically computed as Q 1,i = Q 1,i−1 +

dQ 1 t. dt

(5.16)

The flow rate in the surge tank is then simply given as Q 3 = Q 1 − Q 2 . Correspondingly, the water height in the surge tank is obtained as h 3,i = h 3,i−1 +

Q3 t. A3

(5.17)

In these last two equations, the respective last terms can be computed more accurately by applying respective mean values within a finite time increment. This means that, forinstance, Q 3 in the last equation can be replaced by the mean value  Q 3,i + Q 3,i−1 /2. The pressure head at the injector is determined from Eq. (5.1) as h2 =

Q 22 . 2gϕ 2 A2D0

(5.18)

It is worth mentioning that this pressure head is not equal to h m + z 2 but h 2 < h m + z 2 in the case of positive flow acceleration dc2 /dt > 0, see Eq. (5.2). With the pressure head h2 and the flow rate Q2 , the hydraulic power through the injector is calculated as Phyd = ρgh 2 Q 2 .

(5.19)

The computations shown above are accurate and can be performed without any difficulties. By reaching sufficient computational accuracies, however, computations can be simplified for direct applications. This will be shown in the next section.

110

5 Surge Tank Functionality and System Stability

5.2.2 Simplifications of Computations The basic computations shown in the last section can be simplified by approximations. First, the inertial force in branch 2 in Fig. 5.3 can be neglected. From Eq. (5.3), one then obtains 1 Q 2 − z 2 − h m = 0. 2gϕ 2 A2D0 2

(5.20)

As expected, this reduced equation agrees with Eq. (2.29) because of h m +z 2 = h tot when the dynamic pressure is emitted. The neglect of the inertial force in branch 2 leads to a remarkable computation error which, however, is only restricted to the period of flow regulation including the start and stop of the machines. This is because within this time the changing rate of the volume flow rate dQ 2 /dt is usually of significant magnitude. It is only negligible after completing the flow regulation. Second, the inertial force exerted on the water in the surge tank, proportional to dQ 3 /dt, can be neglected in any case. This is mainly because the surge tank usually has a large diameter compared to branches 1 and 2 and thus the velocity in it is very small. From Eq. (5.5) with its left side terms equal to dQ 3 /dt, one then obtains h m − h 3 − R3 |Q 3 |Q 3 = 0.

(5.21)

For further computations, Eq. (5.20) is combined with Eq. (5.21) for eliminating the pressure head h m . This leads to  Q 2 = ϕ AD0 2g(z 2 + h 3 + R3 |Q 3 |Q 3 ).

(5.22)

The volume flow rate Q 2 can thus be directly computed. In the form of numerical computations and with regards to the quasi-stationary flow within each finite time interval, the above equation is also written as 



 Q 2,i = ϕi AD0 2g z 2 + h 3,i−1 + R3 Q 3,i−1 Q 3,i−1 .

(5.23)

After the volume flow rate Q 2,i has been computed, the pressure head h m, i is obtainable, for instance, from Eq. (5.20). The pressure head at the injector can be computed either from Eq. (5.18) or directly as h 2 ≈ h m + z 2 , both are based on the neglect of the inertial force which is associated with the flow acceleration or deceleration in branch 2. To compute the volume flow rate Q 1 , Eq. (5.7) in the last section should be used. In finite form, the equation is rewritten as



 Q 1,i − Q 1,i−1 g A1  z 3 − h m, i − R1 Q 1,i−1 Q 1,i−1 . = t L1

(5.24)

5.2 Momentum Equations and Numerical Solutions

111

As stated, the volume flow rate Q 1,i−1 in the last time step is used for the friction force term in the parentheses. Furthermore, the flow rate in the surge tank is simply given as Q3 = Q1 − Q2.

(5.25)

The height of water in the surge tank is then computed from Eq. (5.17). While doing this, it should continuously be checked, whether overflow takes place on the upper edge of the surge tank. Finally, the available hydraulic power through the injector is calculated from Eq. (5.19). Corresponding computation examples will be shown below in Sects. 5.2.3 and 5.2.4 for both the start and the shutdown of a Pelton turbine, respectively. The applicability of the above simplified computations will be confirmed. Further simplifications Basically, for only revealing low-frequency system oscillations between the lake and the surge tank, it is often sufficient for computations to directly define a function Q 2 = f(t) in place of Eqs. (5.22) and (5.23). This will, furthermore, contribute to the simplification of the entire computation. Especially, one can further assume L 2 = 0. In this way, one does not necessarily have to know the characteristics of the injector. The computation begins with the combination of Eqs. (5.5) and (5.7), which yields 

 A1 A3 A1 A3 1 dQ 2 hm = . + (z 3 − R1 |Q 1 |Q 1 ) + (h 3 + R3 |Q 3 |Q 3 ) − L1 h3 L1 h3 g dt (5.26)

This is the equation for directly determining the pressure head h m with given Q 2 = f(t). Both flow rates Q 1 and Q 3 as well as height h 3 should refer to the last time step (subscript i − 1). After the pressure head h m has been computed, then Eq. (5.24) can be further applied for determining the flow rate Q 1 . All other parameters can be subsequently determined, as described above. An application example of using this approximation will be shown in Sect. 5.2.4.

5.2.3 Reaction of the Surge Tank on the Turbine Start During the start of a turbine, the surge tank serves as a reservoir and provisionally provides the water flow for the turbine. The process lasts until the flow in the passage upstream of the surge tank is progressively accelerated to a stable value which is equal to the discharge at the turbine. The same hydraulic transient at the surge tank occurs when the load of the turbine increases by further opening the regulation devices. As

112

5 Surge Tank Functionality and System Stability

a result, the height of the water in the surge tank falls. For the operational safety of the considered hydraulic system, the maximum drop of water in the surge tank has to be limited. This means first that the capacity of the surge tank must be sufficiently high. If necessary, the design form of the surge tank with a lower chamber (Fig. 5.2b) can be used. Second, the start or the operation regulation of the turbine should be carefully accomplished. Generally, for a machine house with several turbines, the turbines should be started one after the other. To the considered hydraulic transient in association with the surge tank, the computational background and the concept have been presented above in Sects. 5.2.1 and 5.2.2. The computation will be implemented here for the start of a turbine in the given hydraulic system in Fig. 5.3. The initiation of the flow is simulated by the opening of an injector nozzle of the Pelton turbine. It is about an equivalent injector (diameter and needle stroke) to ensure the desired normal flow rate of Q N = 64.2 m3 /s. The system components and parameters are given in Table 5.1. The injector is assumed to be linearly opened within topen = 40 s with a needle-stroke speed equal to ds/dt = 17.5 mm/s, see Fig. 2.6. Significant steps of the computation, based on simplified computations (Sect. 5.2.2), are presented in Table 5.2, which can be completed, for instance, by means of MS Excel. The background of computations is Eq. (5.13). Computational results are shown in Fig. 5.4, with comparisons with accurate computations based on the method described in Sect. 5.2.1. The parameters which Table 5.1 A simplified hydraulic system with an equivalent injector nozzle of the Pelton turbine (referred to Fig. 5.3) Parameter

Symbol

Unit

Value

Available hydraulic head

H0

m

540

Diameter of pipeline 1

d1

m

6.8

Length of pipeline 1

L1

m

3700

Friction coefficient

λ1

–

0.0143

Resistance constant

R1

s2 /m5

0.0003

Diameter of pipeline 2

d2

m

3.8

Length of pipeline 2

L2

m

1400

Diameter of the surge tank

d ST (d 3 )

m

13

Resistance constant

R3

s2 /m5

0.001

Normal flow rate (discharge)

QN

m3 /s

64.2

Injector nozzle diametera

D0

mm

1120

Injector nozzle constant, Eq. (2.30)

a1

–

1.536

Injector nozzle constant, Eq. (2.30)

a2

–

−0.857

Normal needle position(open)a

sN

mm

700

a The

nozzle diameter and the normal needle stroke given in the table are related to an equivalent injector

0.078

0.094

0.109

0.125

5

6

7

8

:

0.063

4

:

0.047

3

0.141

0.031

2

0.156

0.016

1

10

0.00

0

9

s/D0

t(s)

:

0.22

0.20

0.18

0.16

0.14

0.11

0.09

0.07

0.047

0.024

0.000

ϕ

:

22.2

20.2

18.1

16.0

13.9

11.6

9.4

7.1

4.8

2.4

0.0

Q2

:

81.9

82.1

82.3

82.5

82.6

82.8

82.9

82.9

83.0

83.0

83.0

hm

:

538.9

539.1

539.3

539.5

539.6

539.8

539.9

539.9

540.0

540.0

540.0

h2

:

0.11

0.08

0.07

0.05

0.03

0.02

0.01

0.01

0.00

0.00

0.00

dQ1 /dt

Table 5.2 Numerical computations of an opening flow process with ds/dt = 17.5 mm/s

:

0.33

0.23

0.16

0.10

0.06

0.03

0.01

0.00

0.00

0.00

0.00

Q1

83.0 82.9 82.9 82.8 82.7 82.6 82.4 82.3 82.1

−4.8 −7.1 −9.4 −11.6 −13.8 −15.9 −18.0 −20.0 −21.9

:

83.0

:

83.0

0.0

h3

−2.4

Q3

0.0

:

117.6

106.8

95.9

84.7

73.3

61.7

49.8

37.7

25.3

12.8

P

5.2 Momentum Equations and Numerical Solutions 113

114

5 Surge Tank Functionality and System Stability

Fig. 5.4 Flow oscillations in the hydraulic system (Fig. 5.3) during and after opening the flow, computed based on the rigid water column theory; Comparison between accurate computations according to Sect. 5.2.1 and simplifications according to Sect. 5.2.2; Results obtained by simplified computations (dashed lines) are hardly visible. a Volume flow rate, b height of water in surge tank, c pressure head at injector, d Hydraulic power associated with flow rate Q2

5.2 Momentum Equations and Numerical Solutions

115

have been considered are volume flow rates in three passages, the height of the water in the surge tank (h3 ), the pressure head at the injector (h2 ), and the hydraulic power associated with the volume flow rate Q 2 . Between accurate and approximated computations there are almost no differences visible. All presented quantities are subject to damping effects. While carrying out these computations, it has been verified that under frictionless conditions all oscillations are sinusoidal functions. For the currently considered hydraulic system (Fig. 5.3), the period of low-frequency flow oscillations is computed, according to Eq. (4.44), as  T = 2π

L 1 A3 = 233 s. g A1

(5.27)

This value has been well confirmed by computations. The opening process persists 40 s. To meet the continuous growth of discharge Q 2 through passage 2, on the one hand, the flow in passage 1 undergoes a strong acceleration. The maximum reachable volume flow rate is Q 1,max = 106 m3 /s which is about 65% larger than the stabilized reference flow rate Q 1,R = Q 2,R = 64.2 m3 /s. On the other hand, the surge tank continuously provides the water (with Q 3 < 0) up to a time of about t = 75 s, with a maximum drop of the water level down to h 3,min = 67 m. Later, as the result of an increased volume flow rate Q1 , the maximum height of the water in the surge tank reaches h 3,max = 91 m. From Fig. 5.4, one also confirms that between oscillations of the volume flow rate Q3 and the height of the water h3 in the surge tank there is a phase shift of about 58 s, which corresponds to 1/4 of the oscillation period T. This, actually, has been theoretically expected from Eqs. (4.41) and (4.40). The flow rate Q 2 , after the opening of the flow (t > 40 s), is almost constant. Its weak oscillation, visible by enlarged scale (Fig. 5.4a), is synchronous with the oscillation of the height (h3 ) of water in the surge tank and thus with the oscillation of pressure head h 2 at the injector. In reality, the flow rate Q 2 is directly determined by the pressure head h 2 according to the characteristic of the used injector nozzle, see Eq. (2.33). For the weak oscillation of the volume flow rate Q 2 , a further interesting phenomenon can be confirmed from Fig. 5.4. While oscillations of flow rates Q1 and Q3 are almost synchronous, there is a phase shift of exactly T /4 between oscillations of Q1 and Q2 . This circumstance reveals that between the oscillations of the flow rates Q1 and Q3 a small phase shift exists. If, for instance, Q1 is expressed as Q 1 = Q 0 + sin(ωt + α1 ) and Q3 as Q 3 = A sin(ωt + α3 ) with α3 = α1 + α, then it follows from Q 2 = Q 1 − Q 3 with respect to α  1 by performing the linearization (linear term in Tailor series) Q 2 = Q 0 − A cos(ωt + α1 )α.

(5.28)

A phase shift of exactly T /4 = 58 s between oscillations of the volume flow rates Q2 (a cosine function) and Q3 (a sinusoidal function) has thus been proved. Against the amplitude A of the oscillation of the volume flow rate Q1 , for instance,

116

5 Surge Tank Functionality and System Stability

the amplitude of Q2 -oscillation is reduced by the factor α. In the computation example shown in Fig. 5.4, we obtained A = 42.0 in the Q3 -oscillation, measured at t = 127 s, and Aα = 0.54 in the Q 2 -oscillation, measured at t = 185 s. The phase shift between oscillations of volume flow rates Q1 and Q3 is then computed to be α = 0.0129 or t = α/ω = 0.5 s. Because this is a very small time, oscillations of Q1 and Q3 are almost synchronous. The oscillation of the height of the water in the surge tank leads to a synchronous oscillation of the pressure head h2 at the injector. Together with the weak oscillation of flow rate Q2 , the oscillation of the available hydraulic power is obtained as P ≈ ρgh 2 Q 2 , with the approximation h 2 ≈ h 2,tot . The simplified computations are based on the neglect of the inertial force in branch 2, as outlined in Sect. 5.2.2. Because during the time of opening the valve (t < 40 s), the inertial force (proportional to dQ 2 /dt) is significant, its neglect leads to remarkable computational inaccuracy. This can be well confirmed in Fig. 5.4c for the pressure head h2 . The present computations also help to check the capacity of the surge tank which is given in the simple form with constant cross-sectional area. The most reasonable quantity is the amplitude of oscillations of the water in the surge tank. In the present case, it takes about 16 m at the beginning of the event. As a matter of fact, the computed amplitude also depends on the season-dependent altitude of the water level in the lake. The most critical minimum height of the water in the surge tank occurs when the lake level is at its minimum. Correspondingly, the most critical maximum height of the water in the surge tank is given when the lake level is at its maximum. Because of viscous friction in passage 2, the real oscillation of the flow in the system will be a bit more damped than here computed. The easy way to achieve more damping effect on the oscillations of the flow is to enhance the resistance at the throttle of the surge tank. This will be shown below in Sect. 5.2.5. Computations shown in Fig. 5.4 can be well verified by comparison with accurate computations based on the elastic water column theory. This will be presented later in Sect. 7.15.4.

5.2.4 Reaction of the Surge Tank on the Turbine Shutdown Either for the load regulation in the power grid or for an emergency case, water turbines in a hydro power plant often need to be shut down. The surge tank shown in Fig. 5.3 then behaves as a storage device to pick up the flow in the upstream passage. As a consequence, the level of the water in the surge tank rises. It can be expected that after shutting down the turbine, similar system oscillations as those in the last section (Fig. 5.4) will occur between the lake and the surge tank. Corresponding computations can be accomplished. Against the computations in the last section, further simplifications will be applied by predefining the changing rate of the flow rate in the passage 2 to be dQ 2 /dt = const

5.2 Momentum Equations and Numerical Solutions

117

and by applying Eq. (5.26) for direct computations. The computation step is similar to that in Table 5.2, however, without considering the injector characteristic. Figure 5.5 shows the computation results for the case of the shut-down of the turbine in the same hydraulic system as used in the last section. For comparison reasons, computations without approximation (Sect. 5.2.1) have also been shown for the height of the water in surge tank h3 . The initial flow rate is Q 2,0 = 64.2 m3 /s, which corresponds to the case when all turbines in the considered hydropower station are in operation. A linear closing of the flow within 40 s leads to dQ 2 /dt = −1.606 m3 /s2 . Obviously, similar flow oscillations have been confirmed as for opening the flow (Fig. 5.4). The maximum reachable height of the water in the surge tank is about h 3,max = 97.8 m for the given height of the upper water in the lake, as denoted in Fig. 5.3. For other cases, it changes correspondingly. It especially depends on the season-dependent maximum height of water in the upper lake. The flow states at the moment of fully closing the turbine will now be considered according to Fig. 5.5. When resetting the time to be zero, both the height of the water level in the surge tank and the flow velocity are measured to h 3,t=0 = 90 m and c1,t=0 = 1.377 m/s, respectively. The latter is computed from Q 1,t=0 = 50.0 m3 /s.

Fig. 5.5 Flow oscillations in the hydraulic system (Fig. 5.3) during and after closing the flow; comparison of h3 between accurate computation according to Sect. 5.2.1 and approximations with a linear decrease of the volume flow rate Q2

118

5 Surge Tank Functionality and System Stability

According to Eq. (4.34) with respective parameter definitions,2 the maximum height is obtained as   2 L 1 A1 2 c = 98.6 m. (5.29) h 3,max = z 3 + h 3,t=0 − z 3 + g A3 1,t=0 From Fig. 5.5, one reads out a maximum height of h 3,max = 97.8 m. It is smaller than the maximum, mainly because of the viscous friction effect in flow passage 1 and the flow resistance at the throttle of the surge tank.

5.2.5 Damping Effect of the Surge Tank Throttle Area From the above computation examples, as shown in Figs. 5.4 and 5.5, considerable flow oscillations between the lake and the surge tank have been confirmed. This occurrence determines the necessary size and thus the capacity of the surge tank. To prevent the over-dimensioning of the surge tank in some cases, setting small entrance areas at the surge tank throttle is often very effective. This simply implies the enhancement of the flow resistance there. For the simple design of the surge tank throttle, the flow resistance depends on the ratio between the throttle area ( Athrottle ) and the cross-sectional area of the surge tank ( AST ). Basically, the flow by entering into the surge tank suffers from sudden expansions. The related shock loss can be approximated by the Borda-Carnot equation according to Eq. (2.22), given as RST

  Athrottle 2 1 1− = . AST 2g A2throttle

(5.30)

In almost all cases there is Athrottle < AST . In the computation example shown in Fig. 5.4, the surge tank diameter is dST = 13 m, giving AST = 132.7 m2 . Then, a throttle area of Athrottle = 6 m2 , for instance, will result in a resistance constant RST = 0.0013. In the computation, RST = 0.001 has been applied, see Table 5.1 and Fig. 5.3. If the throttle area is reduced to Athrottle = 1.3 m2 , a resistance constant of RST = 0.03 in value can be reached. The effect of reducing the throttle area of the surge tank on the damping of flow oscillations in the considered hydraulic system can be well demonstrated by repeating previous computations. One only needs to assume different resistance constants RST . Corresponding computation results regarding the height of the water level in the surge tank are shown in Fig. 5.6 for the case of opening the flow. When compared with Fig. 5.4, obviously, the damping effect of reducing the throttle area of the surge tank is significant. height z in Eq. (4.34) and Fig. 4.5 is the same parameter as the height h3 − z3 in Eq. (5.29) and Fig. 5.3, with z3 = 83 m.

2 The

5.2 Momentum Equations and Numerical Solutions

119

Fig. 5.6 Opening of the flow in Fig. 5.3 and damping of the flows through the throttle section of the surge tank with different resistance constants

Another alternative method to enhance the flow resistance at the entrance of a surge tank is the use of grids or perforated plate screens.

5.3 System Stability Performance and the Thoma Criterion 5.3.1 System Instability Owing to External Stimulations In the computational examples presented in foregoing sections (see Figs. 5.4 and 5.5), all flow oscillations in connection with the turbine start and stop are damped oscillations. This can be considered to be of the natural feature of the phenomenon, simply because of the non-vanished viscous friction and other reasons causing the

120

5 Surge Tank Functionality and System Stability

energy losses in the system. In actual reality, this natural feature of flow oscillations with the damping effects applies to all hydraulic and hydropower systems. In other words, all hydropower systems are damped and thus self-stabilized systems. A traditional more or less significant aspect in dimensioning and evaluating a hydropower system is related to the system instability. This aspect appears to be against the natural law of all independent hydraulic systems with damping effects. However, a natural stable system can become unstable, if it is subjected to certain external stimulations. The related phenomenon, i.e., the system instability is generally referred to as resonance. In hydropower stations, at least at earlier times, the system instability could happen in a hydraulic system, if an external stimulus is added to it for the purpose of flow regulations. One application example with regulation purpose is to keep the constant power output of a turbine during the system oscillation. According to Eq. (5.19) as well as to Fig. 5.4, the hydraulic power or power output at a turbine enters into oscillations after each flow regulation or turbine start because of oscillations of both the discharge and the effective pressure head. Such a power oscillation has, for instance, already been confirmed in Fig. 5.4. One method to keep the power output constant would be to simultaneously regulate the discharge through the injector in accordance with the system oscillation. This can be achieved, for instance, by synchronously regulating the opening of the injector nozzle. The action, i.e., the regulation itself behaves as a synchronized external stimulation to the flow oscillation in the system between the lake and the surge tank. The mechanism could thus lead to the system instability and the hydro-mechanical resonance if the damping effect of the system is insufficient. The sensibility of the system to fall into the system instability depends on the system design. This topic has been treated by Thoma (1910). The so-called Thoma criterion has been developed to evaluate the system stability performance and further to estimate the minimum necessary diameter of the surge tank in a given hydraulic system.

5.3.2 Thoma Criterion The basic principle in Thoma’s concern is the concept of keeping the power output at the turbine constant during the low-frequency flow oscillation. According to Fig. 5.7, this could be achieved by synchronously regulating the volume flow rate Q 2 in the pressure shaft through the synchronous regulation of the injector nozzle. To suppress the possible occurrence of flow instability in the system between the lake and the surge tank, the Thoma’s criterion regarding the size of the surge tank must be fulfilled. To derive the Thoma criterion, the momentum equation in penstock 1 of constant cross-sectional area should be established by applying Newton’s second law of motion. First, one needs to determine the static pressure at the end of the penstock, i.e., locally beneath the surge tank. Because the rise or drop of the water level in the surge tank (ST) usually represents a rather slow process, the acceleration and the related forces in the surge tank can be neglected. Except for the special design of

5.3 System Stability Performance and the Thoma Criterion

dST

s s

Surge tank

121

Resistance constant R1

Q3

Penstock: L1, d1

hST,R

Q1 Pressure shaft

hST

H0

Lake

Q2

Injector nozzle Fig. 5.7 Flow oscillation between the lake and the surge tank (diameter d ST )

the throttle cross-section, the flow resistance at the entrance of the surge tank can be further neglected against the friction resistance in the long penstock.3 Under these conditions, the height of water in the surge tank basically represents the static pressure in the penstock locally beneath the surge tank. The difference in the static pressure between both ends of the penstock is then given by H0 − h ST , if the dynamic pressure at the penstock inlet, leading to the reduction of the static pressure, is neglected. Besides the effect of the pressure force, the mass flow in the penstock is additionally subject to the friction resistance which is given as R1 Q 21 . Here, only the positive flow with Q 1 > 0 is considered (see Fig. 5.4a). The momentum equation for the flow in the penstock is then obtained as L 1 d Q1 = H0 − h ST − R1 Q 21 . A1 g dt

(5.31)

Besides, the continuity equation of the flow in the surge tank is simply expressed, with notation Q 3 = Q ST , as AST

dh ST = Q3. dt

(5.32)

In Thoma’s concern, the constant power output is referred to the stable flow state after the flow oscillation has been completely damped. Before reaching this stable state, the constant power output is obtained by regulating the flow rate, i.e., by forced regulation of the opening of the injector nozzle in accordance with the oscillating pressure head. With h2 as the available pressure head at the injector nozzle, the 3 This assumption is only conditionally correct. In many cases, the throttle resistance is intentionally

enlarged to effectively dampen the flow oscillations, see Sect. 5.2.5. The system under Thoma’s concern is thus usually always stable. In the current context of deriving the Thoma criterion, the neglect of the throttle resistance is primarily because of the small volume flow rate Q 3 (surge tank) against the volume flow rate Q 1 in the penstock, see Fig. 5.4, for instance.

122

5 Surge Tank Functionality and System Stability

constant hydraulic power is expressed in the proportional form as Q 2 h 2 = Q 2,R h 2,R = const.

(5.33)

The reference flow state under stable flow conditions is denoted by Q 2,R and h 2,R . The length of the pressure shaft (Fig. 5.7) is usually much shorter than the length of the penstock. By neglecting the associated inertial force exerted on the mass flow, the approximation h 2 ≈ h ST can be made. From Eq. (5.33), one then obtains Q 2 dh ST dQ 2 =− . dt h ST dt

(5.34)

The oscillation of the flow rate Q2 under the regulation is around the mean value of Q 2,R . The same is so far the oscillation of the pressure head h ST . Thus, the approximation Q 2 /h ST ≈ Q 2,R /h ST,R in the above equation can be applied. Because of Q 2 = Q 1 − Q 3 and in view of Eq. (5.32), it further follows from the above equation d2 h ST Q 2,R dh ST dQ 1 = AST . − 2 dt dt h ST, R dt

(5.35)

Then, Eq. (5.31) becomes AST

 A1 g  d2 h ST Q 2,R dh ST + h ST − H0 + R1 Q 21 = 0 − 2 dt h ST, R dt L1

(5.36)

When substituting s = h ST − h ST,R according to Fig. 5.7, one obtains with H0 − R1 Q 21,R = h ST,R AST

 A1 g d2 s A1 g  2 Q 2,R ds + − s + R1 Q 1 − Q 21,R = 0. 2 dt h ST, R dt L1 L1

(5.37)

On the other hand, the discharge Q 2 , which is regulated by an external enforced mechanism, varies around Q 2,R . Since it represents a small variation (Q 2 − Q 2,R  Q 2,R , i.e., Q 2 ≈ Q 2,R , see Fig. 5.4), the volume flow rate Q 1 can be expressed as Q 1 ≈ Q 3 + Q 2,R . Based on Eq. (5.32) for Q 3 and s = h ST − h ST,R , its square leads to Q 21 ≈ Q 23 + 2Q 2,R AST

ds + Q 22,R . dt

(5.38)

This relation is inserted into Eq. (5.37). One obtains with Q 2,R = Q 1,R   ds g A1 d2 s A1 1 g A1 + + Q 2,R 2g R1 − s+ R1 Q 23 = 0. (5.39) dt 2 L1 AST h ST,R dt AST L 1 AST L 1

5.3 System Stability Performance and the Thoma Criterion

123

The last term on the l.h.s. of this equation is usually negligibly small, as stated by Thoma (1910). Thus, the above equation is also written as   ds g A1 A1 1 d2 s + + Q 2,R 2g R1 − s = 0. 2 dt L1 AST h ST,R dt AST L 1

(5.40)

Obviously, this equation is a type of vibration equation and describes a lowfrequency oscillation of the flow in terms of the deviation s of the water height from the stable state in the surge tank. The oscillation is damped, only when the coefficient of ds/dt is positive. Thus, one should have AST >

L1 . 2g R1 A1 h ST,R

(5.41)

In view of Eq. (2.10) for the definition of R1 , one further obtains AST >

π d13 . 4 λh ST,R

(5.42)

This equation represents the Thoma criterion for the minimum cross-sectional area of the surge tank. Although this criterion has sometimes been mentioned in system designs and specifications (Giesecke and Mosonyi 2009; Chaudhry 2014), it has practically not found any applications. This is so because the described regulation mechanism finds no longer applications in any hydropower stations. The power oscillation, as confirmed in Fig. 5.4 after each start or load regulation, is commonly of small scale and thus insignificant. The low-frequency oscillations of both the flow rate and the pressure head will be attenuated as a result of the flow resistance and the viscous frictions in the system. Moreover, the Thoma criterion has been derived by neglecting the flow resistance at the throttle of the surge tank. In many cases, such a resistance might take a decisive role in damping out the system oscillation, especially, if it is intentionally enlarged. At least, the method of enhancing the throttle resistance represents a much easier and more reliable alternative to the method of enlarging the surge tank. The period of the considered flow oscillation is mainly determined by the coefficient of the third term on the l.h.s. in Eq. (5.40), so that by neglecting the damping effect it follows  AST L 1 . (5.43) T ≈ 2π g A1 As has been expected, the computed period of oscillations is equal to the natural period of the system, see Eq. (4.44). A computational verification of the Thoma criterion for the performance of system stabilities has been carried out by Zhang (2012). There, for the intended extension

124

5 Surge Tank Functionality and System Stability

Fig. 5.8 Computations of stability in a surge tank to demonstrate the Thoma criterion

of an existing pumped storage system (see Fig. 1.5), a new surge tank on the lowpressure side of the machine house 2 has been considered under the condition of the Thoma criterion. From Eq. (5.42), the minimum diameter of the new surge tank has been computed to be 7 m. Then, flow oscillations after opening the flow have been simulated in relying on the same regulation concept as in Thoma’s consideration and by supposing the surge tank to be 6 and 8 m in diameters, respectively. Figure 5.8 shows the simulated flow oscillations in the surge tank. The Thoma criterion has been well validated from these simulations. The flow oscillation in the surge tank would diffuse if the diameter of the surge tank would be 6 m, which is less than the Thoma’s minimum diameter in the current case. In the computation example shown in Fig. 5.8, the expected damping of the flow oscillations for D = 8 m appears to be rather insignificant. It is even less significant than the case of natural damping without any external stimulation. This fact represents another reason, in addition to those mentioned above, that the Thoma criterion and the related design concept regarding the surge tank size have practically not been further applied.

References Chaudhry, M. H. (2014). Applied hydraulic transients (3rd ed.). New York Inc: Springer. Giesecke, J. & Mosonyi, E. (2009) Wasserkraftanlagen (5 Auflage). Springer. Thoma, D. (1910). Theorie des Wasserschlosses bei selbsttätig geregelten Turbinenanlagen, Dissertation, Kgl. Technische Hochschule zu München, Oldenburg in München, Germany. Zhang, Z. (2012). Hydrodynamisches Verhalten von Wasserschlössern im Triebwassersystems Gr2 und Gr3, technical report, Oberhasli Hydroelectric Power Company (KWO), Innertkirchen, Switzerland, Nr. A000246540.

Chapter 6

Elastic Water Column Theory and Fundamentals

Methods of transient computations have been progressed from the earlier graphical to the nowadays numerical method based on computer technologies. Especially, the method of characteristics (MOC) has dominated applications in engineering hydraulics. This method, however, requires both time and space discretizations. The use of “characteristic grids”, as stated in Sect. 1.3, inevitably increases the computational complexity, so that only large and complex commercial software programs are available for transient computations. In many cases, in fact, the space discretization does not appear to be necessary if only the local pressure rises, for instance, at the regulation valve or significant nodes along the pipeline are interested for the operational safety. In contrast, the method of directly tracking shock waves only in time series looks to be much more convenient and applicable, primarily, because of its distinct and simple computational algorithms. In reality, both the method of characteristics (MOC) and the wave tracking method (WTM) are based on the same fundamentals of fluid mechanics. These are the momentum and continuity equations as well as the laws of wave propagation, as already presented in Sect. 3.3. In this chapter, such fundamental equations are further considered for approaching computational algorithms which are applied in both the method of characteristics (MOC) and the wave tracking method (WTM).

6.1 Transient Flow Mechanics and Differential Equations The basic equations in the elastic water column theory are the differential momentum and the continuity equations. Both have been derived and given in Eqs. (3.41) and (3.53), respectively. For further applications, they are rewritten here as ∂c 1 ∂p dz λ ∂c +c + +g + c|c| = 0, ∂t ∂x ρ ∂x dx 2d © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_6

(6.1)

125

126

6 Elastic Water Column Theory and Fundamentals

  1 ∂p ∂p ∂c + 2 +c = 0. ∂x ρa ∂t ∂x

(6.2)

In these two equations, a positive velocity is assumed, according to Fig. 3.14, to be in the positive direction of the coordinate x. The viscous friction effect, causing the pressure drop in the flow, is considered in the momentum equation by considering the friction coefficient λ. Equation (6.2) is multiplied by the wave propagation speeds a and −a, respectively. Then each resultant equation is added to Eq. (6.1). One obtains   ∂c 1 ∂p ∂p dz λ ∂c + (c ± a) ± + (c ± a) +g + c|c| = 0. ∂t ∂x ρa ∂t ∂x dx 2d

(6.3)

Since c  a, this equation is also written as   ∂c ∂c 1 ∂p ∂p dz λ ±a ± ±a +g + c|c| = 0. ∂t ∂x ρa ∂t ∂x dx 2d

(6.4)

Before a transient process begins, the flow is found under stationary conditions. Both the flow velocity and the static pressure at a given location in the flow are expressed by c0 and p0 , respectively. For flows in a pipeline of constant cross-sectional area, which simply means ∂c0 /∂ x = 0, one obtains from Eq. (6.1) immediately dz λ 1 ∂ p0 +g + c0 |c0 | = 0. ρ ∂x dx 2d

(6.5)

This equation is then subtracted from Eq. (6.4) for eliminating the term dz/dx. One obtains, after a rearrangement     ∂ λ ∂ p − p0 p − p0 c − c0 ± ±a c − c0 ± + (c|c| − c0 |c0 |) = 0. ∂t ρa ∂x ρa 2d (6.6) The static pressures in this equation are replaced by respective static pressure heads h = p/ρg and h 0 = p0 /ρg. It then follows   ∂ g ∂  g (c − c0 ) ± (h − h 0 ) ± a (c − c0 ) ± (h − h 0 ) ∂t a ∂x a λ + (c|c| − c0 |c0 |) = 0. 2d

(6.7)

If the viscous friction effect is first neglected, then with dx/dt = ±a, the above equation represents a vanishing total derivative

6.1 Transient Flow Mechanics and Differential Equations

 d  g (c − c0 ) ± (h − h 0 ) = 0, dx a

127

(6.8)

which is further converted, by multiplying it with the constant ±a/g; the result is   d a h − h 0 ± (c − c0 ) = 0. dx g

(6.9)

The vanishing total derivative signifies that respective quantities, i.e., the shock wave parameters h − h 0 ± a/g(c − c0 ) remain constant while propagating along the pipeline at the speed dx/dt = ±a in both positive and negative x-directions. In the terminology of solving the wave equation by the method of characteristics, the conditions dx/dt = ±a in the x−t domain are denoted as characteristic lines, see Sect. 8.1. For general frictional flows, it follows from Eq. (6.7) in a similar way   a λ d h − h 0 ± (c − c0 ) = − (c|c| − c0 |c0 |). dx g 2gd

(6.10)

The assumption of deriving Eqs. (6.9) and (6.10) is the constant cross-sectional area of the pipe so that the initial velocity c0 is constant along the pipeline. The initial static pressure head h 0 , however, is usually a variable along the pipeline axis. This is not only because of the viscous friction effect, but especially because of the installation form of the pipeline (horizontal, inclined or vertical). Worth mentioning is that in almost all applications up to now, the initial flow parameters c0 and h 0 are not included in the above equations. Instead, one commonly uses the so-called piezometric head in form of h = p/ρg + z, in which a reference altitude for z has to be given.1 This would lead to a great confusion in computations because the symbol h is commonly used to represent the static pressure head. As a matter of fact, Eq. (6.10) represents the fundamental background for both the method of characteristics and the wave tracking method. The former has found its wide applications up to now. It, however, suffers from great disadvantages that characteristic grids in the x−t domain must always be accounted for. The latter method shows its great convenience and applicability by only tracking the wave propagation in the time series. Such an advantage of the wave tracking method has recently convincingly been demonstrated by Zhang (2016, 2018). For completeness and the reference purpose, both the wave tracking method (WTM) and the method of characteristics (MOC) will be separately presented in Chaps. 7 and 8. Because of its advanced applicability, the wave tracking method will be described in all details with application examples.

1 In fluid mechanics, basically, the expression h = p/ρg + z has no common physical meaning. Only in the field of geology in relation with groundwater, the terminology “piezometric head” has found its applications.

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6 Elastic Water Column Theory and Fundamentals

6.2 Wave Equation and Wave Parameters For frictionless flow (λ = 0) in a pipeline of constant cross-sectional area, Eq. (6.7) becomes     ∂ a a ∂ (6.11) (h − h 0 ) ± (c − c0 ) ± a (h − h 0 ) ± (c − c0 ) = 0, ∂t g ∂x g The respective shock wave parameter (h − h 0 ) ± a/g(c − c0 ) retains its value while traveling along the pipeline at the wave speed a. Further differentiations of the above equation lead to     ∂2 a a ∂2 (h − h 0 ) ± (c − c0 ) ± a 2 (h − h 0 ) ± (c − c0 ) = 0, ∂ x∂t g ∂x g   2  2  ∂ ∂ a a (h − h 0 ) ± (c − c0 ) ± a (h − h 0 ) ± (c − c0 ) = 0. ∂t 2 g ∂ x∂t g

(6.12) (6.13)

By eliminating the mixed partial derivatives, one obtains    2  ∂2 a a 2 ∂ (h − h 0 ) ± (c − c0 ) − a (h − h 0 ) ± (c − c0 ) = 0. ∂t 2 g ∂x2 g

(6.14)

This is a hyperbolic wave equation of the parameter (h − h 0 ) ± a/g(c − c0 ). It describes the one-dimensional wave propagation in pipe flows under a constant speed equal to a. According to d’Alembert, its solution is a (c − c0 ) = 2 f (x − at), g a h − h 0 − (c − c0 ) = 2F(x + at). g h − h0 +

(6.15) (6.16)

F and f represent two wave parameters which travel against and along the xdirection, respectively. Equations (6.15) and (6.16) are called water hammer equations. The significance of the d’Alembert solution of the wave equation can be explained with the aid of Fig. 6.1. While the valve at x = 0 is regulated, the wave parameter F 1 Fig. 6.1 Generation of an upstream and a downstream pressure shock wave on both sides of a valve

c1 f1

c2

Valve F1

f2

Upstream

F2 Downstream

x

6.2 Wave Equation and Wave Parameters

129

in the upstream flow is immediately generated which propagates upstream. Its value depends on dynamical regulations of the valve. At the very beginning of a performed flow regulation, the wave parameter f 1 in the upstream domain is equal to zero. In the flow downstream of the regulation valve, correspondingly, the generated wave travels downstream and is therefore denoted by f 2 . For frictionless flow in a pipe of constant cross-sectional area, both the downstream and the upstream wave parameters (f and F) remain constant while traveling in the pipe. This exactly agrees with Eq. (6.9). At x = 0 and as long as the wave parameter f 1 in the upstream flow is equal to zero, the pressure head at the entrance of the valve is computed from Eq. (6.15) as a h − h 0 = − (c − c0 ). g

(6.17)

This is exactly the Joukowsky equation for direct shock pressures, see Eq. (3.21). Equations (6.15) and (6.16) form the basic equations for the Wave Tracking Method (WTM) which will be described in details in the next chapter.

References Zhang, Z. (2016). Transient flows in pipe system with pump shut-down and the simultaneous closing of the spherical valve. In IAHR, 28th Symposium on Hydraulic Machinery and Systems. Grenoble, France, see also IOP conference series: Earth and Environmental Science (Vol. 49, p. 052001). https://doi.org/10.1088/1755-1315/49/5/052001. Zhang, Z. (2018, August). Wave tracking method of hydraulic transients in pipe systems with pump shut-off under simultaneous closing of spherical valves. Journal of Renewable Energy, 132, 157–166.

Chapter 7

Wave Tracking Method

The method of tracking waves in the fluid flows goes back to the suggestion of Allievi (1902) in the earlier time. The method, in effect, relies on the d’Alembert solution of the one-dimensional hyperbolic wave equation and represents a Lagrangian approach in fluid mechanics. The method is simply based on tracking the shock wave propagation in the time series. This indicates that both the shock pressure and the flow velocity at any interested position and the components in a hydraulic system can be directly computed. The latter may for instance be hydraulic machines, regulation valves or surge tanks. The approach clearly represents a significant advantage against the method of characteristics, which always requires the predefined characteristic grids in the time-space domain. Although some researchers, like Wood et al. (1966), had confirmed the distinct advantage and the related significant reduction of the computation time, the method, unfortunately, had neither been further nor systematically been developed. Only recently, a systematic contribution to the related computational method has been made by Zhang (2016, 2018). It has been shown quite evidently that the advanced wave tracking method (WTM) is able to directly compute all hydraulic transients in all possible complex hydraulic systems. This especially includes the programming for numerical computations by the user himself, simply based on the use of MS Excel, for instance. In this chapter, the fundamental background of the wave tracking method (WTM) will be outlined in detail, with the hope that this method could become the standard method for transient computations. Then, the generation and transmission performances as well as conservation and traveling laws of shock waves will be presented. Different boundary conditions, as listed in Sect. 7.2, will be considered. Some simple application examples of using the wave tracking method are presented in this chapter; they demonstrate the computational algorithms. Application examples for more complex hydraulic systems with pumps and water turbines will be shown in Chap. 11.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_7

131

132

7 Wave Tracking Method

7.1 Fundamental Equations of the Wave Tracking Method At first, Eq. (6.10) will be integrated along the path of wave propagations. This will be done, according to Fig. 7.1, in both the positive x-direction from xA to x and the negative direction from xB to x. With L A = x − xA and L B = −(x − xB ), both are positive, one obtains   a a h − h 0 + (c − c0 ) = h − h 0 + (c − c0 ) − g g A   a a h − h 0 − (c − c0 ) = h − h 0 − (c − c0 ) + g g B

λL A ¯ c| ¯ − c0 |c0 |), (c| 2gd

(7.1)

λL B ¯ c| ¯ − c0 |c0 |). (c| 2gd

(7.2)

While performing the integration of the friction terms, the first mean value theorem for integrals has been applied. Moreover, for simplicity, equal mean velocities in both equations are used, because they are found in the same pipe and at the same time. In almost all engineering applications with continuous regulation processes, except for sudden closing of the valve, the use of the mean value of velocities is sufficiently accurate. These two equations indicate that from known flow states at A and B, the flow state at any arbitrary location x between A and B and at a later time can be determined. This includes the solutions of both the flow velocity c and the shock pressure h. The time delays between the given location and the initial wave positions are tA = L A /a and tB = L B /a, respectively. For further computations, the following water hammer equations for expressing the shock waves are introduced a (c − c0 ), g a 2F(x + at) = h − h 0 − (c − c0 ), g   a 2 f A = h − h 0 + (c − c0 ) , g A 2 f (x − at) = h − h 0 +

(7.3) (7.4) (7.5)

x Lake

A

B

x f

LA

F

c

LB

Fig. 7.1 Illustrating propagation of pressure shock waves in a simple pipeline system

7.1 Fundamental Equations of the Wave Tracking Method

  a 2FB = h − h 0 − (c − c0 ) . g B

133

(7.6)

Equations (7.3) and (7.4) are equal to Eqs. (6.15) and (6.16), respectively. The background of using wave parameters F and f for expressing shock waves is that both F and f are solutions of the wave equation, as explained in Sect. 6.2. In actual fact, the method has been suggested by Allievi (1902), see Sect. 1.4. As stated in Sect. 6.2, the pressure shock wave f (x − at), for instance, is considered to propagate in the positive x-direction at a speed a. For clarity and within the frame of this book, this pressure shock wave is arranged to coincide with the positive direction of flows, without explicitly specifying the x-coordinate each time. Thus, f (x − at) can be simply written as f and is denoted as downstream wave. On the other hand, the pressure shock wave F(x + at), i.e. F, is agreed to always propagate in the negative direction of the flow; it is thus called the upstream wave. In many articles dealing with hydraulic transients in pipelines, the notation F is used to represent the primary or direct shock waves, which are directly generated at regulation organs found at the downstream end of the pipeline. Correspondingly, f is used to denote the reflected waves, which are commonly reflected from the upper reservoir or lake and travel back to the regulation organs. These notations, however, are only appropriate in simple pipeline systems like that in Fig. 7.1, where only the flow upstream of the valve is considered. In other cases, for instance, if the downstream flow in Fig. 6.1 is considered, the primary or direct shock waves are denoted by f. The same is given when considering the flow on the pressure side of a pump, see Sect. 7.3.5. Throughout this book, therefore, F and f are used to plainly represent upstream and downstream shock waves, respectively. Both may be primary waves, depending on applications like that in Fig. 6.1. Based upon these notations and in using the flow rate Q = c A, Eqs. (7.1) and (7.2), respectively, are now written as fx = fA −

1 RA (Q|Q| − Q 0 |Q 0 |), 2

(7.7)

Fx = FB +

1 RB (Q|Q| − Q 0 |Q 0 |), 2

(7.8)

in which the viscous resistance constants are given by RA =

λL A λL B and RB = . 2 2gd A 2gd A2

(7.9)

By neglecting the viscous friction effect, both shock waves F and f remain constant (in terms of values and shape) while traveling along the axis of the pipeline. The influence of two viscous resistances on shock waves will be accounted for in upcoming sections in connection with numerical solutions of transient flows (Sect. 7.10). As can be confirmed from Eq. (7.7), the shock wave f x at coordinate x can be determined from the shock wave f A at point x A with a time delay. The same is for

134

7 Wave Tracking Method

shock wave F x in Eq. (7.8), which is related to FB at point x B . The time delay between f x and f A is tA = L A /a. Correspondingly, there is tB = L B /a for the time delay between F x and F B . For meanings of L A and L B , see Fig. 7.1. Note that tA and tB may be different from one another and do not have to be equal to the time interval in numerical time series, when performing numerical computations. In further considerations, f x and F x are simply denoted by f and F, respectively. Both the shock pressure and the flow velocity at pipe section x are computed by combining Eqs. (7.3) and (7.4) as follows h − h 0 = F + f, c − c0 =

g ( f − F). a

(7.10) (7.11)

These two equations represent the fundamentals of the so-called wave tracking method. As already specified, f describes downstream and F upstream shock waves. The changes of both the shock pressure and the flow velocity against their initial values have been expressed as simple functions of F and f, respectively. The possible way of computing transient flows is thus to track shock waves F and f in the flow with respect to viscous friction effects. When viscous friction effects are negligible, then we have simply f x = f A and Fx = FB from Eqs. (7.7) to (7.8). If compared with the method of characteristics (MOC, Chap. 8), the current method with two shock wave parameters (F and f ) is highly flexible. This is especially so because of the use of individual paths L A and L B in the respective friction terms. In addition, the use of two wave parameters F and f also contributes to the simplification of related computations at discontinuous pipe cross-sections or T-junctions. According to Fig. 7.1, for instance, there is simply f = −F at the entrance of the pipeline on the upper side to the lake. This is mathematically obtainable from Eq. (7.10) because of h = h 0 = const. It physically represents the full reflection of shock waves. Contrary to the method of characteristics, it is clearly no longer necessary to compute the transient of inlet flows that usually has no engineering significance.

7.1.1 Joukowsky’s Equation in Upstream Flows In Sect. 3.3.1, the Joukowsky equation for transient flows upstream of a regulation organ has been given by Eq. (3.21), which is only applicable within a short time up to T2L . For its extension, Eqs. (7.10) and (7.11) are now applied to such an upstream flow according to Fig. 7.1. At the fixed location close to the valve, the respective derivatives are ∂F ∂f ∂h = + , ∂t ∂t ∂t

(7.12)

7.1 Fundamental Equations of the Wave Tracking Method

  g ∂f ∂F ∂c = − . ∂t a ∂t ∂t

135

(7.13)

By eliminating the derivative ∂ F/∂t and in finite difference form, one obtains a h = − c + 2 f. g

(7.14)

This equation, which actually can also be directly obtained from Eq. (7.3), has been already used in Sect. 3.3.4 at Eq. (3.30) for explaining the “self-stabilization effect” of transient flows through a regulation valve. In deriving the above equations from Eqs. (7.10) to (7.11) or directly from Eq. (7.3), it has not been distinguished between the flows up- and downstream of a regulation organ, respectively. Therefore, the equation is applicable for both flow areas. However, only in the flow area upstream of the regulation organ (Fig. 6.1) one has f = 0 provided that t ≤ T2L (F will never be zero). One then obtains Joukowsky’s equation in its initial form in the time domain. This is the reason of eliminating ∂ F/∂t for achieving Eq. (7.14). The case of eliminating ∂ f /∂t will be shown in the following section. The condition for f = 0 has been given here for t ≤ T2L . It is actually restricted to the interested location at the regulation organ (Fig. 7.1), because it takes T2L = 2(L A + L B )/a seconds at most, until f = 0 appears there first. If the location x in Fig. 7.1, for instance, is in focus, then the time for f = 0 is accordingly shorter.

7.1.2 Joukowsky’s Equation in Downstream Flows The Joukowsky equation, given by Eq. (3.21), initially applies to the flow upstream of a regulation organ. In many flow processes, one also needs to compute hydraulic transients in the flow area downstream of a regulation valve or a machine (Fig. 6.1). Similar computations can be completed as in the last section, for instance, by eliminating ∂ f /∂t from Eqs. (7.12) to (7.13). Or, one obtains directly from Eq. (7.4) for the flow at a fixed location a ∂c ∂F ∂h = +2 . ∂t g ∂t ∂t

(7.15)

In finite difference expression, this equation is also written as h =

a c + 2F. g

(7.16)

As Eq. (7.14), this equation is also applicable for flow areas both up- and downstream of the regulation organ. In other words, it is equivalent to Eq. (7.14). However, F = 0 under the condition t ≤ T2L is only obtained in the flow area downstream of

136

7 Wave Tracking Method

the regulation organ, as given by h =

a c. g

(7.17)

This is the Joukowsky equation in the time scale (h and c within the time t) and in the flow area downstream of the regulation organ. It states that a positive shock pressure will be created by increasing the flow rate which, for instance, is realized by opening a valve. It differs from the initial form of Joukowsky formula in Eq. (3.21) by a minus sign.

7.1.3 Joukowsky’s Equation in Spatial Scale In above considerations regarding Eqs. (7.14) and (7.17), the extended Joukowsky equation in time scale has been presented. The corresponding Joukowsky equation in spatial scale can be obtained by considering Eq. (7.3), for instance. Along the pipeline of constant cross-sectional area and at a fixed time, the derivative of Eq. (7.3) leads to ∂(h − h 0 ) ∂f a ∂c =2 − . ∂x ∂x g ∂x

(7.18)

This is applicable for general pipe installations. Only at a horizontal pipe of constant cross-sectional area and for flows with negligible viscous frictions there is ∂h 0 /∂ x = 0. Equation (7.18) is again generally applicable to flow areas both up- and downstream of the regulation valve. In the former case and under the condition t ≤ T2L , we have f = 0, so that it follows from the above equation a ∂c ∂(h − h 0 ) =− . ∂x g ∂x

(7.19)

This equation can be considered as the Joukowsky equation in spatial scale. It is restricted in the flow upstream of the regulation valve and conditionally within a maximum time of t = T2L . In analogy to Eq. (7.18), one obtains from Eq. (7.4) ∂F a ∂c ∂(h − h 0 ) =2 + . ∂x ∂x g ∂x

(7.20)

This equation is equivalent to Eq. (7.18). However, only in application to the flow area downstream of the regulation organ and conditionally within t ≤ T2L , there is F = 0. Then, it follows

7.1 Fundamental Equations of the Wave Tracking Method

∂(h − h 0 ) a ∂c = . ∂x g ∂x

137

(7.21)

It differs from Eq. (7.19) by a minus sign.

7.2 Multiple Initial and Boundary Conditions In practical applications, each computation of hydraulic transients requires detailed specifications of initial and boundary conditions. These generally include – – – – – –

Generation of shock waves at regulation organs or machines, Viscous friction effects on the traveling behavior of pressure shock waves, Local resistance effects, Stepped pipe section, nozzle and diffuser effects, Wave travelling performances at T-junctions, Full reflection of shock waves at the pipe section connected to a reservoir of constant level, – Reflection of shock waves at the varying free surface of water in a surge tank, – Total reflection of shock waves at a closed valve, – Two regulation organs and thus two sources of generating shock waves. All these topics represent the fundamentals of hydraulic transients and will be treated in details in the following sections.

7.3 Generation of Primary Shock Waves Hydraulic transients in a hydraulic system may be caused by flow regulations, startup or emergency shut-off of one or more operating machines. A regulation organ or a hydraulic machine, while getting started or stopped, thus, behaves as a mechanical source of shock waves in the associated pipeline network. In Sect. 7.1, regarding Fig. 7.1, the related primary pressure shock wave has been specified by F which, according to the definition there, is actually an upstream shock wave and propagates against the flow direction. In the downstream flow of either a regulation organ or a hydraulic machine, the initiated primary pressure shock wave is correspondingly the downstream shock wave f. For general applications, the generation mechanism and the computational algorithms of both up- and downstream shock waves have first to be accounted for.

138

7 Wave Tracking Method

7.3.1 Regulation Mechanism Flow regulations in hydraulic systems often occur by installed regulation devices like injector nozzles, butterfly valves, gate valves etc. The exact consideration of hydraulic characteristics of respective regulation organs would make transient computations rather complicated. For this reason, transient computations have often been conducted by assuming a regulation mechanism which is much simplified by considering an “abstract valve”. Such an “abstract valve” is found, for instance, uniquely at the downstream end of a pipe. The flow rate or discharge through the “valve” has, in the past, been commonly assumed to be proportional to the opening of the valve (τ ) as follows:  H Q =τ . (7.22) Q0 H0 The reference discharge (Q 0 ) through the valve is related to the reference pressure head H0 at the valve. The problem with above equation is, on the one hand, that the opening process of the valve τ is often not clearly defined. On the other hand, at almost all regulation devices, like injector of Pelton turbines for instance, the discharge is not proportional to the opening degree, i.e., the spear-needle stroke in the injector nozzle. Especially at spherical, i.e., ball valves or butterfly-disc valves, the opening and thus the discharge is far from being proportional to the position angle of the ball or disc, see Sect. 2.3. For this reason, Eq. (7.22) is not used to simulate the flow regulation in the current book. Instead, the injector nozzle of Pelton turbines represents a real regulation device which is ideally always found at the downstream end of a pipeline. Because of its simple characteristic, the injector nozzle will throughout this book be frequently used as a regulation device at the downstream end of a pipe for simulating flow regulations. For easy applications, Eq. (2.32) for computing the discharge through the injector is rewritten here as Q=

ϕ AD0 1 − (ϕ AD0 /A)2



2gh.

(7.23)

The discharge coefficient is computed from Eq. (2.30). The difference between Eqs. (7.23) and (7.22) is only restricted to the regulation dynamics. While in Eq. (7.22) linear regulation means linear variation of τ and Q with the time, it means in Eq. (7.23) plainly the linear relation of the spear-needle stroke s(t); the discharge coefficient ϕ(t) changes as a quadratic function of time. For more details about the injector and its characteristics, the readers are kindly referred to Sect. 10.2.

7.3 Generation of Primary Shock Waves

139

7.3.2 Upstream Shock Waves F of Primary Order A pipeline system with a regulation apparatus at the pipeline end can be found, for instance, in a hydraulic system of the Pelton turbine according to Fig. 7.2 (see also Fig. 4.7). The pressure shock wave, primarily arising from the regulation device, is the upstream shock wave F. Its generation mechanism and the computational algorithm will be presented here according to Fig. 7.2. For computational simplicity, the surge tank is provisionally not included in computations. The regulation apparatus is simply the injector nozzle. Its characteristic is given in Eq. (7.23). Equation (7.23) is inserted into Eq. (7.3). With Q = c A one obtains1 aϕ AD0 h+ g A



√  2g h − h ∗0 + 2 f = 0. 2 1 − (ϕ AD0 /A)

(7.24)

In this equation, the initial stable flow condition is expressed as h ∗0 = h 0 +

a Q0. gA

(7.25)

Equation (7.24) is the determination equation in form of a quadratic polynomial √ for the square root of the shock pressure h at the injector as a function of the discharge coefficient ϕ. Obviously, this equation is valid only for h > 0. In some applications, for instance, at low head systems and by rapidly closing or opening the valve, the pressure head h could fall into areas of negative values (h < 0). Because of the associated phase change in the flow, Eq. (7.24) becomes non-defined and computations cannot be further performed. More about this occurrence can be found in Sect. 7.16. The discharge coefficient in the above equation is given by Eq. (2.30) as a function of the needle stroke in the injector nozzle. The downstream shock wave f in Eq. (7.24) can be obtained simply by tracking the primary, i.e., the upstream shock wave. For a pipeline system as shown in Fig. 7.1, the stated downstream shock wave is zero (f = 0) within the time t ≤ 2L/a. The time T2L = 2L/a is the travel duration of the upstream shock wave F towards the lake and then, as reflected and denoted by f, back to the injector. Therefore, the downstream shock wave f in Eq. (7.24) basically behaves as a known quantity.

1 In

combining two equations for eliminating 1 − (ϕ AD0 /A)2 2g(ϕ AD0 )

2

√

h, one obtains alternatively

Q2 +

 a Q − h ∗0 + 2 f = 0. gA

(7.26)

This is the determination equation in form of a quadratic polynomial for discharge. The fact to be mentioned is that this equation is undefined, as soon as ϕ = 0, i.e., Q = 0 is reached by closing the injector. The pressure head is then directly computed from Eq. (7.24) to h = h ∗0 + 2 f . In addition, the approximation 1 − (φ AD0 /A)2 ≈ 1 can be made in both Eqs. (7.24) and (7.26).

7 Wave Tracking Method

Lake

Surge tank

Butterfly valve

H0

he

140

f c

F Injector nozzle

Fig. 7.2 A simple pipeline system with an injector nozzle for flow regulations and a butterfly valve for closing the flow

After the shock pressure has been computed from Eq. (7.24), the discharge through the injector nozzle is further computed from Eq. (7.3) with respect to Eq. (7.25) Q=

gA ∗ h0 − h + 2 f . a

(7.27)

From Eq. (7.10), the upstream shock wave F primarily arising from the injector is obtained as F = h − h 0 − f.

(7.28)

As mentioned before, this primary shock wave travels against the positive flow to the lake and will be reflected there. During this propagation, a great variety of disturbances may occur, like the viscous friction effect, change of shock waves at the transitions of pipes or T-junctions, etc. All about these aspects of wave propagations will be described below in respective sections. The reflection time T2L = 2L/a is considered again. Within this time, there is f = 0, so that it follows from Eq. (7.3) a h − h 0 = − (c − c0 ). g

(7.29)

This is exactly the Joukowsky equation, see Eq. (3.21). It has also been denoted as the Joukowsky equation in time scale at Eq. (7.14).

7.3.3 Downstream Shock Waves f of Primary Order The case of flow regulation at the entrance of a pipeline is now considered. As shown in Fig. 7.2, a butterfly valve is usually available for closing the flow. Following a similar route as in the last section, the downstream shock wave f, primarily generated by the valve, can be determined.

7.3 Generation of Primary Shock Waves

141

The characteristic of the butterfly valve is given, for instance, by Eq. (2.38). Because the pressure head he on the high-pressure side is constant and known, the flow rate through the butterfly valve is simply a function of the opening of the valve and the pressure head h on the low-pressure side (exit of the valve). With h butterfly = h e − c2 /2g − h, one obtains from Eq. (2.37) c2  . h e − h = 1 + cp 2g

(7.30)

The flow velocity is then resolved as  c=

2g(h e − h) . 1 + cp

(7.31)

This equation is inserted into Eq. (7.4). One obtains a (h e − h) + g



 2g  h e − h + h ∗∗ 0 − h e + 2F = 0, 1 + cp

(7.32)

with h ∗∗ 0 = h0 −

a c0 . g

(7.33)

√ This is the determining equation in form of a quadratic polynomial for h e − h at the exit of the butterfly valve as a function of the opening degree β because of Eqs. (2.38) and (2.39). It is generally an equation for the head difference h e − h. In the current case, it is also for the head h because the pressure head h e is known. The upstream shock wave F in Eq. (7.32) is the approaching wave and therefore a known parameter. It is equal to zero (F = 0), conditionally, within the time t ≤ T2L . After the shock pressure has been computed from Eq. (7.32), the volume flow rate through the butterfly valve can be computed from Eqs. (2.37) to (2.38). As another option, it is also computed from Eq. (7.4) using Eq. (7.33). This yields Q=

gA h − h ∗∗ 0 − 2F . a

(7.34)

From Eq. (7.10), the downstream shock wave f departing from the butterfly valve is finally obtained as f = h − h 0 − F. It is a primary wave.

(7.35)

142

7 Wave Tracking Method

In case of a butterfly valve, which is found in the middle of the pipe, as in Fig. 6.1, the above computations can be applied to only determine the head difference h in place of h e − h. For determining both the flow rate and the primary wave f, the transient flow upstream of the butterfly valve must be additionally accounted for. Below in Sect. 7.3.5, another similar case of using a pump in place of the butterfly valve will be considered. Corresponding detailed computational algorithms will be shown in Sect. 7.15.5.

7.3.4 Shock Waves by Predefined Flow-Rate Regulation In the above two sections, respective flow rates through the injector nozzle and the butterfly valve have been determined employing their characteristics. If no such characteristics are available or if it sometimes appears to be unnecessary to make accurate computations, the flow rate through a valve can be directly given as a function of the time. Depending on the flow area of observation, one distinguishes between the flows up- and downstream of the regulation valve. For the flow upstream of the regulation valve, as in Fig. 7.1, the primary upstream shock wave is obtained from Eq. (7.11) F= f −

a (Q − Q 0 ). gA

(7.36)

For the flow downstream of the regulation valve, as in Fig. 7.2 downstream of the butterfly valve, the primary downstream shock wave is computed from Eq. (7.11) as f =F+

a (Q − Q 0 ). gA

(7.37)

It should, however, be mentioned that the condition and computations with predefined flow rate regulation Q = f(t) are not exact. This is simply because the so-called self-stabilization effect, as described in Sect. 3.3.4, is not included.

7.3.5 Connection of Shock Waves on Both Sides of a Hydraulic Machine A regulation valve or a hydraulic machine, which is found within a pipeline and possesses regulation functions, also behaves as a mechanical source of generating both up- and downstream shock waves (F and f ). This has been explained in Sect. 6.2, see Fig. 6.1. Another concrete case is the creation of both up- and downstream shock waves on both sides of a pump. For this example, as shown in Fig. 7.3, the initiated, i.e., primary up- and downstream shock waves are again denoted by F1 and

7.3 Generation of Primary Shock Waves c1

143 h2

h1 F1 f1

Unified pump & valve

c2 F2 f2

Fig. 7.3 Scheme to compute the up- and downstream shock waves on both sides of a unified pump unit (pump and spherical valve)

f 2 , respectively. Their computations indispensably require knowledge of the pump characteristic or the unified characteristic of the pump and the valve (see Sect. 10.1) in the general form h 2 − h 1 = f(Q, μ). Here, the regulation parameter is denoted by μ which may be the opening degree of the valve and/or the rotational speed of the pump. On the one hand, the shock waves on both the low- and the high-pressure side of the pump unit (pump and valve) need to be separately considered. From Eqs. (7.3) to (7.4) one obtains a Q + h ∗1,0 , g A1 a h 2 = 2F2 + Q + h ∗∗ 2,0 . g A2 h 1 = 2 f1 −

(7.38) (7.39)

The initial parameters h ∗1,0 and h ∗∗ 2,0 are defined as follows a Q0, g A1 a = h 2,0 − Q0. g A2

h ∗1,0 = h 1,0 +

(7.40)

h ∗∗ 2,0

(7.41)

They are actually the same as Eqs. (7.25) and (7.33). In addition, Eq. (7.38) is the same as Eq. (7.27). From Eqs. (7.38) to (7.39), one obtains h2 − h1 =

a g



  1 1 ∗ Q + h ∗∗ + 2,0 − h 1,0 + 2(F2 − f 1 ). A1 A2

(7.42)

On the other hand, the unified characteristic of the pump plus the valve is assumed to be given as h 2 − h 1 = f(Q, μ). Then, Eq. (7.42) can further be written as a f(Q, μ) − g



  1 1 ∗ Q − h ∗∗ + 2,0 − h 1,0 − 2(F2 − f 1 ) = 0. A1 A2

(7.43)

144

7 Wave Tracking Method

This is the equation for finally resolving the flow rate at the pump as dependent on the flow regulation which is specified by μ = f(t). Both f 1 and F2 are the approaching shock waves towards the pump unit and, therefore, behave as known quantities. At the very beginning within the respective reflection times T2L,1 and T2L,2 , one simply has f 1 = 0 and F2 = 0. Moreover, from Eq. (7.11), two primary shock waves on both sides of the pump unit are computed as2 a (Q − Q 0 ), g A1 a f 2 = F2 + (Q − Q 0 ). g A2

F1 = f 1 −

(7.44) (7.45)

The pressure heads on both sides of the pump can be computed, for instance, by Eqs. (7.38) and (7.39), or simply by Eq. (7.10) of directly using wave parameters F and f. Concrete computation examples will be shown below in Sect. 7.3.6.

7.3.6 Application Cases with a Pump and a Francis Turbine 7.3.6.1

Pump Flow Regulation Over a Regulation Device

An application example for determining the up- and downstream shock waves at a pump will be shown in this section. For simplicity, the flow regulation takes place under the condition of constant rotational speed simply by regulating the opening of a spherical valve, which is installed on the high-pressure side of the pump (Fig. 7.3). The pump characteristic for a given rotational speed can be approximated, at least piece by piece, by the following polynomial Hpu = k2 Q 2 + k1 Q + k0 .

(7.48)

On the other hand, the characteristic of the spherical valve has been known according to Eq. (2.34). The unified characteristic of the pump and the spherical valve is obtained from the pressure balance h 2 − h 1 = Hpu − h sph as

2 If

the pressure heads h1 and h2 are computed from Eqs. (7.38) to (7.39), respectively, then two primary waves can also be computed from Eq. (7.10) as follows F1 = h 1 − h 1,0 − f 1 ,

(7.46)

f 2 = h 2 − h 2,0 − F2 .

(7.47)

7.3 Generation of Primary Shock Waves

h 2 − h 1 = k2 Q 2 + k1 Q + k0 − cp

145

Q2 . 2g A2sph

(7.49)

The general rule of computing the unified characteristic of the pump and the spherical valve with respect to the changeable rotational speed can be found in Sect. 10.1, see also the next subsection. Equating the last equation to Eq. (7.42) yields, after a rearrangement

k2 −



 

2+ k − a − a ∗∗ − h ∗ Q + k − 2(F2 − f 1 ) = 0. − h Q 1 0 2,0 1,0 g A1 g A2 2g A2sph cp

(7.50) This equation represents a concrete case of Eq. (7.43). To each given cp -value, i.e., from Eq. (2.35) as a function of the opening degree β of the considered spherical valve, the flow rate can be easily determined. Then the shock pressures on the low- and the high-pressure sides (h 1 and h 2 ) can be computed from Eqs. (7.38) to (7.39). Finally, one obtains two primary shock waves from Eqs. (7.44) to (7.45), respectively. In this example, the spherical valve has been considered as a “regulation apparatus”. Usually, a spherical valve as a closure organ is used to only close the flow in a pipeline rather than to regulate the flow. In hydropower stations, the start of a pump commonly takes place at the closed spherical valve. After or shortly before the rated rotational speed has been reached, the spherical valve begins to open. During the time of opening, the flow has to be considered to be “regulated” by the spherical valve. The condition of constant rotational speed of the pump (n = const), however, is not always fulfilled. The related dynamics needs to be considered in the above computations. A comparable computation can be found in Zhang (2016, 2018) as well as in Sects. 11.2 and 11.3. If a butterfly valve is used in place of a spherical valve, then, one only needs to replace the resistance coefficient cp in all above equations by the resistance coefficient cp according to Eq. (2.38) with respect to Eq. (2.39).

7.3.6.2

Pump Flow Regulation Over the Rotational Speed

Flow regulations at a pump can also be achieved by changing the pump rotational speed via a static frequency converter. In this case, hydraulic characteristics of the pump are basically expressed by using dimensionless pressure and discharge coefficients, see Eqs. (10.1) and (10.2) in Sect. 10.1. Since such a type of pump-load regulations is usually restricted to a relatively small variation range of two coefficients, the used head characteristic can be well expressed by the following polynomial ψ = m2ϕ2 + m1ϕ + m0.

(7.51)

146

7 Wave Tracking Method

With u 2 = π d2D n, this equation is further written in an explicit form as3 h 2 − h 1 = Hpu

  1 m2 2 m1π n 2 = Q + Q + m 0 (π d2D n) . 4 2g d2D d2D

(7.52)

Equalizing this equation to Eq. (7.42) yields    m2 m 0 (π d2D n)2  ∗∗ m1π n a 1 1 2 Q + − h 2,0 − h ∗1,0 Q + − + 4 2gd g A A 2g 2gd2D 2D 1 2 − 2(F2 − f 1 ) = 0.

(7.53)

From this equation, the flow rate can be easily solved as a function of the rotational speed of the pump and two known wave parameters ( f 1 and F2 ). Then, both primary shock waves (F1 and f 2 ) can be computed from Eq. (7.44) to (7.45), respectively. One finally obtains both the pressure head h 1 and the pressure head h 2 from Eq. (7.10). Actually, the rotational speed in the above equation behaves as a variable. Its determination basically depends on the dynamic behavior of the rotor system, as given by J

d(2π n) = −Mshaft + Mmotor , dt

(7.54)

with J as the moment of inertia of the rotor system. The active torque exerted on the rotor system mainly consists of the shaft torque Mshaft and the torque Mmotor on the motor shaft. The former is related with the characteristics of the pump and behaves as a known function of rotational speed and discharge, see Sect. 10.1. The active driving torque of motor Mmotor is basically also a function of the rotational speed and even of the gradient dn/dt. If this functional relation is not accessible, the rotational speed in Eq. (7.53) can be simply assumed to be predefined, for instance, as a function of time. However, both at the beginning and the end of the speed regulation, the changing rate of the speed, i.e., dn/dt should be zero, because otherwise the motor power would undergo a sudden change. More reasonable is to assume the motor power as a linear function of time or the rotational speed. In the case of emergency shut-down of the pump, the motor power simply disappears. Corresponding computational examples will be exclusively shown in Sects. 11.2 and 11.3. In both examples, Eq. (7.54) is applied.

3 The

impeller diameter is denoted by d2D with subscript 2 for impeller exit and D for impeller as a rotating disc. This designation is meaningful because another designation d2 is often used for the pipe diameter of the pump exit or at the exit of a unified pump and spherical valve.

7.3 Generation of Primary Shock Waves

7.3.6.3

147

Start, Stop and Load Regulation of a Francis Turbine

Similar computations can be applied to the case of the start, stop and load regulation of a Francis turbine. As the first requirement, however, one needs to apply corresponding turbine characteristics which involve variations of both the rotational speed and the guide vane angle. According to Sect. 10.3, the characteristics of a Francis turbine are basically derived from the reconstructed master equation, which is given as m 2 Q 211 + m 1 (Q 11 n 11 ) + m 0 n 211 − 1 = 0.

(7.55)

With Htu as the reference pressure head, the explicit form of the characteristic of a Francis turbine is obtained as   1 m2 2 m1π n 2 (7.56) Q + Q + m 0 (π d1D n) . Htu = 4 2g d1D d1D For reference, the impeller diameter d1D is used, with subscript 1 to denote the impeller inlet and subscript D the impeller as a “rotating disc”. This is necessary, because designation d1 is used throughout this book to denote the pipe diameter directly ahead of the turbine unit. For a given Francis turbine, m 0 and m 1 are two known constants. The coefficient m 2 is additionally a function of the guide vane angle. The method of determining all these three coefficients is described in Sect. 10.3. When the turbine is combined with a spherical valve, for instance, then, because of H12 = h 1 − h 2 = Htu + h sph according to Fig. 7.4, the unified characteristic of the Francis turbine and the valve is given as   1 m2 2 m1π n 2 Q + Q + m 0 (π d1D n) + h sph . h1 − h2 = 4 2g d1D d1D

(7.57)

Except for a minus sign, this equation is equal to Eq. (7.42). By equalizing them and with regard to Eq. (2.34) for h sph , one finally obtains

Valve & turbine h1

c1 F1 f1

c2

h2 Unified

F2 f2

Fig. 7.4 Scheme to compute the up- and downstream shock waves on both sides of a unified Francis turbine

148

7 Wave Tracking Method



    cp m2 m1π n a 1 1 2 Q + + + Q + 4 2gd1D g A1 A2 2gd1D 2g A2sph

+

m 0 (π d1D n)2  ∗∗ + h 2,0 − h ∗1,0 + 2(F2 − f 1 ) = 0. 2g

(7.58)

This equation applies to all cases of simultaneously changing the rotational speed, the guide vane angle and the opening degree of the installed spherical valve. The rotational speed covers a range from n = 0 to the runaway speed n R . For full-opening of the spherical valve, one has cp = 0. The rotational speed in the above equation again behaves as a variable. Its determination basically depends on the dynamic behavior of the rotor system, as given by J

d(2π n) = Mshaft − Mgenerator , dt

(7.59)

with J as the moment of inertia of the rotor system. The active torque exerted on the rotor system consists of the shaft torque Mshaft on the side of the turbine wheel and the counter torque Mgenerator on the generator side. The shaft torque is related to the characteristics of the turbine and, thus, is basically a known function of the rotational speed and the discharge, see Sect. 10.3. In contrast, the counter torque on the generator shaft Mgenerator is usually an unknown parameter. Therefore, for estimation purposes, the rotational speed in Eq. (7.58) can be predefined as a known function of time. The discharge can then be straightforwardly solved. Thereafter, both primary shock waves (F1 and f 2 ) can be computed from Eqs. (7.44) to (7.45), respectively. One finally obtains both the pressure head h 1 and the pressure head h 2 from Eq. (7.10). As a special case, the resistance torque Mgenerator from the generator vanishes during the start-up stage of the Francis turbine, at which the turbine speeds up only under the hydraulic torque in association with the opening of the guide vanes. Another special case with vanished resistance torque Mgenerator = 0 is the load rejection or emergency shut-down of the turbine. In such a case, Eq. (7.59) becomes completely known. From its finite form, the rotational speed can be resolved and applied to Eq. (7.58) to accurately compute the discharge at the turbine. Because the load rejection leads to the speed-up of the turbine towards the runaway speed and thus represents a serious incident, the spherical valve is usually arranged to be automatically closed for mechanical safety.

7.4 Conservation and Traveling Laws of Shock Waves at Series …

149

7.4 Conservation and Traveling Laws of Shock Waves at Series Junctions of Pipes In almost all hydraulic systems, a pipeline has often been constructed by serial connection of pipes of different cross-sectional areas. Usually, smooth transitions of pipes are configured for ensuring minimum disturbances in the flow. According to Fig. 7.5, for instance, one has an expansion of flow area at the positive flow and a contraction of flow area at the negative flow. In an extreme case, it may be of sudden expansion or sudden contraction in flow areas (Fig. 2.3). For considering hydraulic transients and especially for their accurate computations, entire hydraulic features of shock waves, when traveling through such series junctions of pipes, needs to be accurately determined. Basically, each flow under the change in flow area is characterized by two effects: energy loss and exchange between static and dynamic pressures according to the Bernoulli equation. For computing hydraulic transients, these two effects are negligibly small against the shock pressure itself. Therefore, according to Fig. 7.5, the following approximation can be generally applied h 2 − h 2,0 ≈ h 1 − h 1,0 .

(7.60)

From this equation, both the conservation and the traveling laws of shock waves can be derived, as shown below. These two laws, in effect, behave as the key and main contents of the wave tracking method and thus form the essential rules for traveling shock waves in any complex hydraulic system.

7.4.1 Conservation Laws of Shock Waves First, Eqs. (7.10) and (7.11) are applied to the flow at the cross-sections 1 and 2 in Fig. 7.5. One obtains F1 + f 1 = h 1 − h 1,0 ,

(7.61)

x A 1, L 1 Lake

c 1 (Q1)

1

A 2, L 2

2

F1

F2

f1

f2

c 2 (Q2) Valve

Fig. 7.5 Traveling performance of shock waves in the pipe transition of a series junction

150

7 Wave Tracking Method

f 1 − F1 =

a c1 − c1,0 , g

F2 + f 2 = h 2 − h 2,0 , f 2 − F2 =

a c2 − c2,0 . g

(7.62) (7.63) (7.64)

  Because of equal volume flow rates, one has c1 − c1,0 A1 = c2 − c2,0 A2 . Then, it follows from Eqs. (7.62) to (7.64) ( f 1 − F1 )A1 = ( f 2 − F2 )A2 .

(7.65)

Moreover, from Eqs. (7.61) to (7.63) and under the assumption of h 2 − h 2,0 ≈ h 1 − h 1,0 , as mentioned above, one obtains F1 + f 1 = F2 + f 2 .

(7.66)

These last two equations are called conservation laws of shock waves. They describe both the mass and the momentum conservations. Both equations are independent of flow direction and play essential roles in computing hydraulic performances of shock waves while passing through a pipe transition in form of series junctions. They were first applied by Zhang (2016, 2018). In Sect. 7.5, they will be generally extended to the case of shock waves through a T-junction.

7.4.2 Traveling Laws of Shock Waves From the conservation laws of shock waves, as just derived above, their traveling laws can be derived. According to Fig. 7.5, shock waves, which approach the pipe transition from both sides, are F2 and f 1 . Both are considered to be given. They may be, for instance, the upstream shock wave coming from the regulation valve and the downstream wave coming from the lake, respectively. The departure waves are correspondingly F1 and f 2 . They are determined from Eqs. (7.65) to (7.66) as F1 =

A1 − A2 2 A2 f1 + F2 , A1 + A2 A1 + A2

(7.67)

f2 =

2 A1 A2 − A1 f1 + F2 . A1 + A2 A1 + A2

(7.68)

These two relations are called the traveling laws of shock waves while passing through any pipe transitions with A2 = A1 . The physical meaning of these two equations can be revealed. The departure shock wave F1 , for instance, consists of the

7.4 Conservation and Traveling Laws of Shock Waves at Series …

151

partial reflection of wave f 1 and the partial transmission of wave F2 . The reflection part is always proportional to the area difference A1 − A2 , while the transmission part is directly proportional to the own flow area of the respective approaching shock wave. Special cases of applying the traveling laws are the wave reflections at the lake with A1 = ∞ and at the dead end of a pipe (closed valve) with Q = 0. Both are treated below in Sects. 7.7 and 7.8, respectively. After two departure shock waves F1 and f 2 have been computed by Eqs. (7.67) and (7.68), respectively, the shock pressure h 1 −h 1,0 at cross-section 1 of the pipe, for instance, can then be computed by Eq. (7.61). The same shock pressure is obtained at cross-section 2 from Eq. (7.63). The flow velocity in pipe 1 is obtainable from Eq. (7.62). The traveling laws of shock waves represent the fundamentals of the wave tracking method. They describe how shock waves will travel from one into another pipe of different cross-sectional areas. Their background is the conservation laws of shock waves, as described in Sect. 7.4.1. It is also independent of the form of pipe transitions. In other words, one does not need to distinguish between smooth and sudden changes of the flow areas at series junctions.

7.4.3 Traveling Laws of Using Volume Flow Rate The traveling laws of shock waves, which are derived and given by Eqs. (7.67) and (7.68), are directly related to the shock wave F2 . Both the generation of F2 at the downstream end of branch 2 and its computational algorithms have already been presented in Sect. 7.3. In place of shock wave F 2 , the flow rate in branch 2, Q2 , can be directly used as an alternative way, which sometimes shows some application advantages. For this purpose, the relation F2 = f 2 − a/g c2 − c2,0 , according to Eq. (7.64), is first combined with Eq. (7.68) for eliminating the shock wave parameter f 2 ; one obtains F2 = f 1 −

A1 + A2 a  Q 2 − Q 2,0 . 2 A1 g A2

(7.69)

Inserting this expression into Eq. (7.67) yields F1 = f 1 −

a  Q 2 − Q 2,0 . g A1

(7.70)

This is the traveling law of shock waves while traveling through a series junction of a pipeline, based on the direct use of the volume flow rate in branch 2. The targeted replacement of shock wave F 2 through volume flow rate Q2 has the consequence that all other quantities related to branch 2 ( f 2 and F2 as well as A2 and L 2 ) will no longer be used for computations. This is comparable to the situation which would be given, if the regulation organ were directly installed at cross-section 2 in Fig. 7.5.

152

7 Wave Tracking Method

One would have confirmed that the above equation agrees with Eq. (7.62), because of Q 1 = Q 2 and Q 1,0 = Q 2,0 between pipe sections 1 and 2. If, however, Eq. (7.70) alone is used to determine the shock wave F 1 , then the flow rate in branch 2, Q 2 , has to be known. This just provides a possibility to compute shock waves in branch 1 by directly predefining a time-dependent discharge in branch 2, Q 2 = f(t). The shock wave F 1 is then directly obtainable from Eq. (7.70). This appears sometimes to be useful for rapid estimation of shock waves in the system and thus for computational simplicity. According to Eq. (7.70), the departure shock wave F1 , see Fig. 7.5, seems to be the reflection of the approaching wave f 1 , however, with an additive term. Especially, for Q 2 = 0 in the case of a closed valve, one then obtains, with Q 2,0 = Q 1,0 = A1 c0 F= f +

a c0 . g

(7.71)

It represents the total reflection of shock waves at the dead end of a pipeline. For more details, see Sect. 7.8 below. Equation (7.70) is accurate, if the volume flow rate Q 2 is achieved and applied based on transient computations. However, the use of a predefined discharge Q 2 = f(t) will cause some inaccuracies in computing high-frequency shock waves. In Sect. 3.3.4, the self-stabilization mechanism of flows at regulation organs or valves has been demonstrated to be able to simultaneously regulate the discharge. As a consequence, a smoothened shock pressure rise will be achieved, see Fig. 3.10 as well as Fig. 4.9. Such a natural smoothening effect on the pressure rise, however, cannot be expected from computations by applying predefined Q 2 = f(t) which does not exactly represent the reality.

7.5 Conservation and Traveling Laws of Shock Waves at T-Junctions 7.5.1 T-Junction of Diverging Flows A T-junction in a hydraulic system may take different forms and be subjected to different flow conditions. In hydropower stations, a T-junction is found, for instance, at the section of connecting a surge tank according to Fig. 7.6. With defined positive flow in branch 3, the T-junction is confirmed to be a section of diverging flows. The volume flow rate in branch 1, Q 1 , is divided into flows Q 2 and Q 3 with Q 1 = Q 2 +Q 3 . The traveling performance of shock waves, while passing through such a T-junction, is a necessary information that should be revealed. In Fig. 7.6, corresponding shock waves (F and f ) have been shown, as appointed, with f as the downstream wave in accordance with the positive flow direction. The regulation organ is found at the end of branch 2 which, for instance, can be thought

7.5 Conservation and Traveling Laws of Shock Waves at T-Junctions

153

Surge tank (for instance) Q3 f3

F3

3

1 Q1

2 F1

h-h0

f1

F2

Q2

f2

F2 f2

Fig. 7.6 Traveling performance of shock waves in a T-junction of diverging flows

to be connected with a group of turbines. From there, the primary shock wave, F 2 , is generated by regulating the flow rate Q 2 . For computing related hydraulic transients and their attributes at a T-junction, Eq. (7.11) is applied to respective flow sections of three branches of the T-junction. This leads to a c1 − c1,0 , g a f 2 − F2 = c2 − c2,0 , g a f 3 − F3 = c3 − c3,0 . g f 1 − F1 =

(7.72) (7.73) (7.74)

With respect to the conservation of volume flows, which is given by c1 A1 = c2 A2 + c3 A3 , it follows from the above equations immediately ( f 1 − F1 )A1 = ( f 2 − F2 )A2 + ( f 3 − F3 )A3 .

(7.75)

On the other hand, the shock pressures at the connecting sections of all three branches can be assumed to be equal to that in the zone of T-junction, h − h 0 . From Eq. (7.10), one immediately obtains F1 + f 1 = F2 + f 2 = F3 + f 3 .

(7.76)

It is comparable to Eq. (7.66). These last two equations are denoted as conservation laws of shock waves at a Tjunction. While Eq. (7.76) is independent of specifications of positive flow directions and propagation directions of shock waves, the first Eq. (7.75) clearly depends on such specifications. It basically represents the relation Q 1 = Q 2 + Q 3 . In Fig. 7.6, the approaching shock waves are f 1 , F2 and F3 , which are considered to be known, as from the regulation organ or, as reflected, from the lake and the

154

7 Wave Tracking Method

surge tank. The departure shock waves to be determined are F1 , f 2 and f 3 . From Eqs. (7.75) to (7.76), one finally obtains F1 =

A1 − A2 − A3 2 A2 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.77)

f2 =

2 A1 A2 − A1 − A3 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.78)

f3 =

2 A1 2 A2 A3 − A1 − A2 f1 + F2 + F3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.79)

These three equations are called traveling laws of shock waves at a T-junction, as derived and presented by Zhang (2016, 2018). They represent the fundamental equations in the wave tracking method for computing hydraulic transients in any complex hydraulic system. At a series junction of two pipes, simplifications can be made with A3 = 0. One then obtains Eqs. (7.67) and (7.68) for this special case. One again confirms in the above equations that each term on the right-hand side represents either the reflection or the transmission part of the respective approaching shock waves. The magnitude of shock wave F1 , for instance, comprises the reflection part of f 1 and the transmission parts of F2 and F3 . In many other text books, the concept about the transmission and reflection of shock waves can also be found. The overall relation, however, is never formulated in the additive forms, as in Eqs. (7.77)–(7.79). In addition, three individual wave speeds (a1 , a2 and a3 ) are often taken into account. This is actually not necessary at all, as long as the media in all three branches are the same. First, the wave speed in a given hydraulic system is almost constant, so that a = const can be commonly assumed in all engineering applications. Second, in the zone of a T-junction, the wave propagation speed can anyhow be assumed to be constant. The reasons for using a unique constant wave speed in a given hydraulic system have already been discussed and proved in Sect. 3.3.3, see Eq. (3.25) as well as Eq. (3.27) with related discussions. In derived traveling laws of shock waves, F2 is considered to be the primary shock wave generated in branch 2, i.e., downstream of a T-junction. For the same reason, as 2, Eqs. in Sect. 7.4.3 for direct use of the flow rate Q2 in branch  (7.77)–(7.79) can be modified. For this reason, relation F2 = f 2 − a/g c2 − c2,0 according to Eq. (7.73) is combined with Eq. (7.78) for eliminating the wave parameter f 2 . This leads to F2 =

A3 A1 a A1 + A2 + A3  Q 2,0 − Q 2 + f1 + F3 . A1 + A3 g A2 2(A1 + A3 ) A1 + A3

(7.80)

By inserting this expression, respectively, into Eqs. (7.77) and (7.79), one obtains F1 =

 1 2 A3 A1 − A3 a Q 2,0 − Q 2 + f1 + F3 , A1 + A3 g A1 + A3 A1 + A3

(7.81)

7.5 Conservation and Traveling Laws of Shock Waves at T-Junctions

f3 =

 A3 − A1 1 2 A1 a Q 2,0 − Q 2 + f1 + F3 . A1 + A3 g A1 + A3 A1 + A3

155

(7.82)

These two equations are also called traveling laws of shock waves while crossing a T-junction, based on direct use of the volume flow rate in branch 2. The direct use of this volume flow rate has the consequence that all other quantities related to branch 2 ( f 2 and F2 as well as A2 and L 2 ) do not need to be further considered. This is comparable to a situation which would be given, if the regulation organ in branch 2 were directly installed close to the T-junction. Both equations shown above are valid in the current context for T-junctions which are configured for diverging flows. If the volume flow rate in branch 2 is predefined by Q 2 = f(t), two equations are applicable for computational simplicity and thus for rapid estimation of shock waves in the flow system. The computations, however, suffer from inaccuracies, because the so-called self-stabilization effect of flows at the regulation organ is not included in the computations. For more details about the self-stabilization mechanism, see Sect. 3.3.4. A special case of A1 = A3 = A is considered. From the last two equations given above, one obtains F1 = F3 +

a 1  Q 2,0 − Q 2 , g 2A

(7.83)

f3 = f1 +

a 1  Q 2,0 − Q 2 . g 2A

(7.84)

The term with flow rate Q 2 in each of these two equations does not drop out, even if the regulation valve is closed and the flow condition Q 2 = 0 is reached. The flow between branch 1 and branch 3 is simply a through-flow (A1 = A3 ), however, with F1 = F3 and f 3 = f 1 in general.

7.5.2 T-Junction of Converging Flows A T-junction is also found when two flows are merged into one. This may be encountered for instance at a parallel connection of two pumps, as illustrated in Fig. 7.7. Following the same computational procedure as in the foregoing sub-section, one obtains the traveling laws of shock waves at the T-junction for converging flows as follows F1 =

A1 − A2 − A3 2 A2 2 A3 f1 + f2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.85)

F2 =

2 A1 A2 − A1 − A3 2 A3 f1 + f2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.86)

156

7 Wave Tracking Method c3

f3 c1

F3

f1

f2

F1

c2

F2

Fig. 7.7 Traveling performance of shock waves in a T-junction of converging flows

(a)

(b)

Q2

Q2 Second T

Q4

Q1

Q

Q1

Q4

First T Q3

Q3

Fig. 7.8 X-junction and its equivalence consisting of two T-junctions

f3 =

2 A1 2 A2 A3 − A1 − A2 f1 + f2 + F3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.87)

Obviously, the same transmission and reflection laws as for Eqs. (7.77)–(7.79) are applicable. In effect, the above three equations can be analogously and thus directly written out. Corresponding conservation laws of shock waves like those given at Eqs. (7.75) and (7.76) are automatically fulfilled. At the end of this sub-section, computational algorithms for both the transmission and the reflection of shock waves at an X-junction should be indicated. An X-junction can be considered to consist of two T-junctions which are closely connected in series, as shown in Fig. 7.8. Between two T-junctions, the time delay is zero.

7.6 Traveling of Shock Waves Through an Orifice As an application example, the traveling law of shock waves is applied here to the flow through an orifice, which is installed in a pipeline, as shown in Fig. 7.9. The flow through an orifice causes, on the one hand, local pressure drops according to Eq. (2.27). In terms of a local flow resistance, the influence of an orifice on the traveling performance of a shock wave will be revealed in Sect. 7.11. On the other hand, as shown in Fig. 7.9, an orifice can be considered to consist of two cross-sections

7.6 Traveling of Shock Waves Through an Orifice 1

f1

2 F2

Fd fd

c

d

D

F1

157

f2

Fig. 7.9 Propagation of shock waves through an orifice

with sudden contraction and sudden expansion in series. The traveling performance of shock waves through such an orifice can be determined by applying conservation or traveling laws of shock waves to two representative sections. According to Fig. 7.9, the flow through an orifice is considered to be given from cross-Sect. 1 to cross-Sect. 2. The conservation laws from Eqs. (7.65) to (7.66) are then applied to the flow between cross-Sect. 1 and the inlet of the orifice, for one time, and further to the flow between the orifice exit and cross-Sect. 2, for another time. This directly provides ( f 1 − F1 )A1 = ( f d − Fd )Ad = ( f 2 − F2 )A2 ,

(7.88)

F1 + f 1 = Fd + f d = F2 + f 2 .

(7.89)

Because of A1 = A2 one immediately obtains F1 = F2 and f 2 = f 1 .

(7.90)

It looks like in a through-flow without any disturbance. This is due to the use of Eq. (7.66), at which the energy loss in the flow has been neglected. For orifices of small area ratios d/D, the related energy loss can become significant. If this energy loss should not or cannot be neglected, the related local pressure drop must be separately considered. Corresponding computations will be presented in Sect. 7.11. From this example, one simply confirms that an orifice basically behaves only as an obstacle in the flow. As a matter of fact, Eq. (7.90) can also be derived from the traveling laws of shock waves according to Eqs. (7.67) and (7.68). An orifice is basically considered to be a serial connection of a sudden contraction and a sudden expansion of the pipe. One then only needs to apply the traveling laws twice.

158

7 Wave Tracking Method

7.7 Full Reflection of Shock Waves at the Reservoir 7.7.1 Entrance Velocity Effect An important aspect in tracking shock waves in the flow along a pipeline is the general behaviour of shock waves at the section of the connection of a pipe to a lake or reservoir, which has a constant altitude of the water surface (Fig. 7.10). The case represents a special boundary condition for wave propagations. Because the static pressure at the connecting section is essentially constant, the approaching shock wave F must be instantaneously balanced by a coexisting shock wave f. The related hydro-mechanics is known as the full reflection of the approaching shock wave at a flow section of constant pressure, here because of the constant altitude of the water surface of the lake. Actually, the pressure at the pipe entrance, i.e., at the connecting section to the lake is not exactly constant. According to Fig. 7.10, the inlet flow from the lake into the pipeline can be considered to be free of energy loss. Because the static pressure at the pipe entrance depends on the flow rate and is equal to h = h e − c2 /2g, it follows with respect to the initial flow state (h0 and c0 ) h − h0 =

1  2 c − c2 . 2g 0

(7.91)

On the other hand, one obtains from comparison with the fundamental equation, i.e., Eq. (7.10) F+ f =

1  2 c0 − c2 . 2g

(7.92)

After a full reflection the wave is then accurately computed as f = −F +

1  2 c0 − c2 . 2g

(7.93)

Fig. 7.10 To the full reflection of shock waves at the connection section to a lake

he

The full reflection of the approaching shock wave is obviously affected by the flow velocity at the pipe entrance. This is called the entrance velocity effect. In all

F e f Lake

c

7.7 Full Reflection of Shock Waves at the Reservoir

159

transient computations found in the literature up to now, this effect has simply been ignored. The case considered assumes positive flow, i.e., the flow from the lake into the pipeline. For reverse flows, the static pressure at the connecting section to the lake is independent of the flow rate, as given by h = h e = const. This means that in this case f = −F, is exactly fulfilled. Later in Sect. 7.15.5, the difference between computations with and without considering the entrance velocity effect will be shown.

7.7.2 Quantification of the Entrance Velocity Effect The velocity term in Eq. (7.93) can be written as c02 − c2 = (c0 + c)(c0 − c). The velocity difference can be again replaced by Eq. (7.11). With c¯ = (c0 + c)/2 as a mean velocity, Eq. (7.93) is further written as f =−

a − c¯ F. a + c¯

(7.94)

Commonly, as in almost all cases with a wave propagation speed a > 1000 m/s, the condition c/a ¯  1 can be applied. It then follows f ≈ −F.

(7.95)

This is the relation which has been used up to now in transient computations. The neglect of the entrance velocity effect simply implies the neglect of dynamic pressure heads in Eq. (7.93). In general, it applies to c/a ¯ < 0.005.

7.8 Total Reflection of Shock Waves at Closed Valve 7.8.1 Valve at the Downstream End of a Pipe Another special boundary which determines the propagation of shock waves is the dead end of a pipeline. This is encountered, for instance, where the regulation organ is fully closed, as shown in Fig. 7.11. The boundary condition in this case is then formulated as Q = 0. This also means that no additional shock waves F will further be generated. The shock wave towards the closed valve will be reflected, however, by a mechanism other than that at the connecting section to a lake. From Eq. (7.11), one obtains F= f +

a c0 . g

(7.96)

160

7 Wave Tracking Method

Fig. 7.11 Total reflection of shock waves at the closed valve at the downstream end of the pipe

h-h 0 t

f +c F Closed Q=0

This equation has already been briefly presented in Eq. (7.71), which is directly obtained from the traveling laws of shock waves. It is called total reflection of shock waves at closed valves. The shock wave F clearly depends on the initial flow velocity. This is obvious because both F and f are process parameters since the transient flow begins with the initial flow velocity c0 at the time t = 0. The pressure head at the dead end of a pipeline is then computed from Eq. (7.10) as h = h0 +

a Q 0 + 2 f. gA

(7.97)

With the definition of h ∗0 in Eq. (7.25), one, thus, obtains h = h ∗0 + 2 f.

(7.98)

In reality, this relation is the consequential result of computations from Eq. (7.27) or Eq. (7.38) for the case of Q = 0 at the end of a closing action. A special case prevails, if the initial flow velocity is zero, which means that at t = 0 the pipeline is initially found in the closed state. This situation concerns a process of opening and subsequently again closing the flow towards Q = 0. From Eq. (7.96), it then follows F = f.

(7.99)

This result has often been found in the science and engineering literature dealing with hydraulic transients. It must be considered there only as a very special case of Eq. (7.96).

7.8.2 Valve at Upstream of a Pipe For comparison reasons, the case of a closed valve, which is found upstream, as illustrated in Fig. 7.12, is now considered. From Eq. (7.11), one obtains for c = 0

7.8 Total Reflection of Shock Waves at Closed Valve Fig. 7.12 Total reflection of shock waves at the closed valve at an upstream position

161

h-h 0

t

F +c f Closed Q=0

f =F−

a c0 . g

(7.100)

As in Eq. (7.96), the reflection of an approaching shock wave depends on the initial flow state. Only for a very special process of opening and subsequently again closing the flow towards Q = 0, there is again F = f . The pressure head at the upstream dead end of a pipeline is then computed from Eq. (7.10) as h = h0 −

a c0 + 2F = h ∗∗ 0 + 2F. g

(7.101)

The definition of the parameter h ∗∗ 0 is given in Eq. (7.41). In fact, one directly obtains the above equation also as a consequential result of computations from Eq. (7.39) with Q = 0.

7.9 Reflection of Shock Waves on the Moving Surface of Water in the Surge Tank A real hydraulic system in a hydropower station commonly contains at least one surge tank. The functionality of the surge tank has been outlined already in Sect. 5.1. For general construction of the system, see Fig. 5.7 for instance. During the occurrence of transient flows, the height of the water surface in the surge tank changes because of flow oscillations in the system. The phenomenon has a significant impact on the growing of the shock pressures in the system. In Sect. 7.5 dealing with flows at a T-junction (Fig. 7.6), the surge tank has been considered as the third branch of the T-junction. In reality, both shock waves F 3 and f 3 depend on the boundary condition whether and how rapidly the height of the water in the surge tank changes. This circumstance, therefore, represents quite a common condition of computing hydraulic transients in hydropower stations.

162

7 Wave Tracking Method

Fig. 7.13 Effect of upsurge of flows in the surge tank on the reflection performance of shock waves

c

f

F

h ST

F

h0

F= f +(h h0)

f

Q3

For computational purposes, a surge tank of constant cross-sectional area according to Fig. 7.13 is first considered. The initial height of the water surface in the surge tank is denoted by h0 . At this height, the approaching and the departing shock waves are denoted, respectively, by the downstream shock wave f (in the same direction as the positive velocity c) and the upstream shock wave F. This specification agrees with that in Fig. 7.6, so that all relations found there for F3 and f 3 can be applied. In the terminology in the current context, the shock wave F can be considered as the reflection of the approaching wave f on the free surface of the water in the surge tank. According to the full reflection rule in Sect. 7.7, see Eq. (7.95), there should be F = − f in the current case, if the water height in the surge tank would remain constant h ST = h 0 . The upsurge or down-surge of the water in the surge tank, however, acts as a non-stationary source of generating an additional shock wave, which has to be added to the reflected shock wave F. For simplicity of the analysis, it is assumed that with the beginning of the flow oscillation the upsurge of the water in the surge tank is induced, which means c > 0. If referred to the initial height h 0 of the water in the surge tank, the change of this height for t > 0 is given by h = h ST − h 0 . According to Eq. (7.10), we must have F + f = h ST − h 0 which is written here again as F = − f + (h ST − h 0 ).

(7.102)

By neglecting the traveling time of shock waves in the height h, the departure wave F comprises both the reflection of the approaching wave f and h = h ST − h 0 . The latter must be considered as a new part of additionally generated shock waves. It changes with time.

7.9 Reflection of Shock Waves on the Moving Surface of Water …

163

The above equation also applies to the case of down-surge which, at t = 0, is characterized by the fall of the height of water in the surge tank from the initial height h 0 . The new part of the shock wave h = h ST −h 0 is thus called upsurge with h > 0 and down-surge with h < 0. The upsurge, for instance, leads to the pressure rise directly below the surge tank at the T-junction and thus to the pressure rise in the entire system. The treatment of the throttle effect at the surge tank entrance is comparable to the treatment of a local resistance. Corresponding computational algorithms will be presented in Sects. 7.11 and 7.12.

7.10 Friction Effect on the Propagation of Shock Waves In Sect. 1.1, it has been reviewed that transient friction differs from the friction in stationary pipe flows. The difference roots in the fast transients caused by sudden closing of the valve rather than in common transients caused by flow regulations. Because even in fast transients the resultant difference in shock pressures, in the viewpoint of engineering applications, is negligibly small and insignificant, the basic difference between the transient friction and the friction in stationary flows can be neglected. This means that in transient computations, the Darcy-Weisbach friction coefficient for stationary flow can be applied. The transient flows are simply considered to be quasi-stationary.

7.10.1 General Friction Effect and Computations Viscous friction effect on the traveling performance of shock waves in transient flows can be determined and accounted for in transient computations. According to Eqs. (7.7) and (7.8), such an effect causes linear changes in shock waves F and f, while propagating along the pipeline axis (Fig. 7.1). As usual, its determination is based on numerical solutions of shock waves as a function of time. To the flow rate in friction terms in Eqs. (7.7) and (7.8), two aspects should be concerned. First, viscous friction behaves as an influencing factor, however, of only a lower order of magnitude in affecting the shock pressure, if compared with the shock pressure itself. Second, the flow rate does not significantly change within a time increment in the time series because of the inertial force exerted on the mass flow. For these two reasons, when accounting for the viscous friction effect, quasi-stationary flows can be assumed. For the flow rate in both friction terms in Eqs. (7.7) and (7.8), thus, the flow rate from the last time step (ti−1 ) can directly be used. It follows then fi = fA −

1 RA (Q i−1 |Q i−1 | − Q 0 |Q 0 |), 2

(7.103)

164

7 Wave Tracking Method

Fi = FB +

1 RB (Q i−1 |Q i−1 | − Q 0 |Q 0 |). 2

(7.104)

If the initial flow direction agrees with the positive flow direction, then Q 20 in both equations can be used. The expression Q i−1 |Q i−1 | has to stay, because possible reverse flow in a pipeline must be accounted for. The time delay between f i and f A is tA = L A /a. Correspondingly, there is tB = L B /a for the time delay between Fi and FB . For L A and L B see Fig. 7.1. As denoted in Sect. 7.1, tA and tB may be largely different from one another; furthermore, they do not have to be equal to the time increment in the time series.

7.10.2 Overall Friction Effect in a Round Trip of a Shock Wave 7.10.2.1

Flow Upstream of a Control Valve

A simple pipeline of constant cross-sectional area, as shown in Fig. 7.14, connects the reservoir (lake) at higher altitude and a regulation valve at lower altitude. While the flow is regulated, the primary pressure shock wave F propagates upstream towards the lake and gets there reflected. The reflected shock wave f then travels back to the valve. During such a round trip of shock waves, denoted by 2t = T2L = 2L/a, both wave parameters, F and f, are subjected to the viscous friction effect and the entrance velocity effect. The latter has been outline in Sect. 7.7. For the sake of later computational simplicity, the total effect of both physical phenomena should be analyzed. For this purpose, Eqs. (7.103) and (7.104) are considered again. According to Fig. 7.14, the primary shock wave is denoted by F B . It travels upstream towards the lake and changes to F i which is computed by Eq. (7.104). With respect to the entrance velocity effect at the lake, which is given by Eq. (7.93), the reflected shock wave is obtained as f A = −Fi +

1  2 c0 − c2 2g

Fig. 7.14 On the overall effect of viscous friction on shock waves after a round trip (B → A → B)

Lake

A

B fA

c

FB Valve L

7.10 Friction Effect on the Propagation of Shock Waves

= −FB −

165

1 1  2 RB (Q i−1 |Q i−1 | − Q 0 |Q 0 |) + Q 0 − Q 2i−1 . 2 2 2g A

(7.105)

This shock wave further travels downstream to the regulation valve. It is denoted as f t in further considerations and computed by Eq. (7.103) for accounting for the viscous friction effect as follows ft = fA −

1 RA (Q i−1 |Q i−1 | − Q 0 |Q 0 |). 2

(7.106)

Because of the low order of the friction effect in influencing the system dynamics, for simplicity, the same volume flow in the viscous friction term is used as in Eq. (7.105). Between the shock waves F B and f t there is a time delay 2t. Thus, the considered primary shock wave is denoted as Ft−2t . From Eqs. (7.105) to (7.106) with RA = RB = R, one then obtains the shock wave at the regulation valve f t = −Ft−2t − R(Q i−1 |Q i−1 | − Q 0 |Q 0 |) +

1  2 Q − Q 2i−1 . 2g A2 0

(7.107)

This equation represents the overall friction effect during a round trip of a shock wave in the flow upstream of a control valve. As the volume flow rate Q i−1 from the last time step, the value Q t - t at the mean time between t and t − 2t can be used. In the above equation, the last term, which accounts for the entrance velocity effect, can often be neglected, without causing any remarkable computational inaccuracy, as explained in Sect. 7.7. Then one obtains f t ≈ −Ft−2t − R(Q t−t |Q t−t | − Q 0 |Q 0 |).

(7.108)

In almost all hydraulic systems in hydro power stations, a surge tank is installed between the lake at high altitude and hydraulic machines at low altitude, see Chap. 5 and Fig. 5.1. For positive flows from the lake towards the surge tank, as in the turbine mode of system operation, Eq. (7.108) is applicable for traveling the shock waves within a round trip between the lake and the surge tank in the pipeline of constant cross-sectional area.

7.10.2.2

Flow Downstream of a Control Valve

The overall friction effect in the flow downstream of a control valve can be obtained in a similar way as above. The case corresponds to the flow with the primary shock wave f 2 in Figs. 7.3 or 7.4. By ignoring the formation of reverse flows, the entrance velocity effect automatically disappears. After a round trip, the shock wave comes back to the regulation valve in form of the upstream shock wave F t . Then, the following similar computation is obtained

166

7 Wave Tracking Method

Ft = − f t−2t + R(Q t−t |Q t−t | − Q 0 |Q 0 |).

(7.109)

Note that this relation is not simply obtained from Eq. (7.108). The application conditions of Eqs. (7.108) and (7.109) are different.

7.11 Local Resistance Effect on the Propagation of Shock Waves Another similar significant aspect to be concerned is the local flow resistance and its effect on the traveling performance of shock waves through the location. For local resistances, one accounts for disturbances in the flow at devices such as a spherical valve or an orifice, as illustrated in Fig. 7.15. The outcomes of local flow resistances are commonly pressure drops in the flow, as exactly comparable to the viscous friction effect. For this reason and in analogy analogous to Eqs. (7.103) and (7.104), the traveling performance of shock waves is given as follows: f2 = f1 −

1 Rlocal (Q i−1 |Q i−1 | − Q 0 |Q 0 |), 2

(7.110)

F1 = F2 +

1 Rlocal (Q i−1 |Q i−1 | − Q 0 |Q 0 |). 2

(7.111)

The local resistance constant Rlocal depends on the type of devices which are installed in the pipeline. For spherical and butterfly valves, corresponding resistance constants have been presented in Sect. 2.3. For orifices, Eqs. (2.26) and (2.27) should be used. As long as local pressure head drops are small compared to the shock pressures, they can be neglected. One obtains then

Fig. 7.15 Hydraulic devices affecting the traveling performances of shock waves

f2 ≈ f1,

(7.112)

F1 ≈ F2 .

(7.113)

x

Sph. valve

c

Orifice

2

1 F1

F2

f1

f2

Δh

c

7.11 Local Resistance Effect on the Propagation of Shock Waves

167

The background of these simplifications is simply h 2 ≈ h 1 . It should be noted that the case treated here is only for devices which are installed in a pipeline of constant cross-sectional area. It does not include the local resistances which are caused, for instance, by discontinuous cross-sections like a series junction. At the flow with sudden expansion or contraction, for instance, the traveling performance of shock waves behaves totally differently. While this traveling performance can be computed by equations in Sect. 7.4 by neglecting the resistance effect, the related resistance can be separately accounted for, for instance, by serially connecting an “artificial obstacle” like an orifice. In almost all engineering applications regarding hydraulic transients, however, the resistance at cross-sections like series junctions can be neglected. The resistance at the entrance throttle of a surge tank behaves differently, as the throttle resistance is often intentionally enlarged.

7.12 Throttle Resistance at the Entrance of the Surge Tank Transient flows at surge tanks represent quite complex flow processes which are related to different features of shock waves. First, as shown in Sect. 7.9, the moving surface of water in the surge tank generates an additional shock wave which leads to the pressure rise in the entire system. Second, the surge tank is often constructed with an enlarged throttle resistance at its entrance (Fig. 7.16), which causes the local energy loss and thus local changes in shock waves. The background of enlarging the throttle resistance is to effectively damp the flow oscillation in the system and thus to enhance the capacity of the surge tank. Because it commonly deals with a

Surge tank

Q3

f3,ST

hST

F

h0

f

3

F 3,ST

neglected

1 f3

Q1

F1

f1

2

F3

h-h0

F2 f2

Q2

Fig. 7.16 Throttle resistance at the entrance of the surge tank and its effect on the traveling performance of shock waves

168

7 Wave Tracking Method

significant resistance, its effect on the travelling performance of shock waves cannot be neglected. For application convenience, the related computational algorithm for transient computations under such complex boundary conditions will be outlined in this section. The throttle resistance at the entrance of the surge tank can be estimated based on a simple flow model. Corresponding computations have already been given in Sect. 5.2.5. The main point of the current computations is to combine the damped traveling performance of shock waves at the entrance of the surge tank with the effect of the moving surface, i.e., changeable height of water in the surge tank. According to Fig. 7.16, the throttle resistance can be considered to be found in branch 3 beyond the T-junction. This implies that computations of traveling performances of shock waves at the T-junction, as outlined in Sect. 7.5.1, remain unchanged. While passing through the resistance throttle of the surge tank, all shock waves are subjected to local changes, which are given by Eqs. (7.110) and (7.111) with the initial value of the volume flow rate being Q 3,0 = 0: f 3,ST = f 3 −

1 Rthrottle Q 3 |Q 3 |, 2

(7.114)

F3 = F3,ST +

1 Rthrottle Q 3 |Q 3 |. 2

(7.115)

In these two equations, the subscript ST denotes the flow in the surge tank. The resistance constant Rthrottle has been assumed to be independent of the flow direction. Shock waves f 3,ST and F3,ST are given by Eq. (7.102) as F3,ST = − f 3,ST + (h ST − h 0 ).

(7.116)

Inserting f 3,ST and F3,ST from Eqs. (7.114) to (7.115) in the above equation yields F3 = − f 3 + Rthrottle Q 3 |Q 3 | + (h ST − h 0 ).

(7.117)

According to Fig. 7.16, this equation represents the wave status after a round trip of the shock wave f in the surge tank and back to the T-junction ( f 3 → f 3,ST → f → F → F3,ST → F3 ). It involves both the effect of the changeable height of the water in the surge tank and the effect of the throttle resistance at the entrance of the surge tank. Between the two wave parameters f 3 and F3 , the phase displacement is equal to t3 ≈ 2h 0 /a that has to be used in numerical computations.

7.13 Two Regulation Organs and Origins of Wave Generations

169

7.13 Two Regulation Organs and Origins of Wave Generations Hydraulic transients occur, for instance, when the flow undergoes regulations. In all above sections, flow regulations have been assumed to be performed only at one hydraulic apparatus which may be a regulation organ or a hydraulic machine like a pump or a turbine. All presented transient computations by means of the wave tracking method appear to be quite convenient and applicable, even for flows in highly complex hydraulic networks. In some applications, a hydraulic system may contain two or more regulation organs and thus with two or more sources of generating shock waves. Figure 1.5, for instance, showed such a typical hydraulic system, in which two machines (Ma1 and Ma2) can be separately operated. Another example with two sources of generating shock waves is found at a hydraulic ram according to Fig. 7.17. A hydraulic ram is a cyclic water pump which operates on the water hammer effect to lift the water from a lower altitude to a higher destination without using any electrical power. Two valves in a hydraulic ram behave as two sources of periodically generating shock waves. Basically, computations of transient flows in hydraulic systems with two or more regulation organs and thus with two or more sources of generating shock waves are equally convenient and applicable as in the case with only one active valve. The background of this reality is that, regarding the generation of shock waves, two sources do not simultaneously affect each other. The related computational algorithms can be well demonstrated according to Fig. 7.18, where two injector nozzles are assumed to be used to independently regulate the flows. In pipeline 1 with injector 1, an arbitrary resistance is also assumed to exist.

Delivery pipe

Pond or stream

check valve Drive pipe

Air chamber

Waste water Waste valve

Fig. 7.17 A hydraulic ram used to lift the water from a lower altitude to a higher destination without using any electrical power

170

7 Wave Tracking Method

Fig. 7.18 Traveling performance of shock waves in a pipeline system with two injectors and hence two sources of generating primary shock waves

c3

f3 c1

f1V

Injector 1

F1V

R

F3

f1

f2

c2

F1

F2

Injector 2

Like in common cases, the generation process of shock waves must be first computed. It is sufficient to show this procedure only by considering injector 1. From Eq. (7.24), the shock pressure at the injector is computed as aϕ AD0 h+ g A



√  2g h − h ∗0 + 2 f 1V = 0. 2 1 − (ϕ AD0 /A)

(7.118)

The initial stable flow state is represented by h ∗0 according to Eq. (7.25). The discharge coefficient of the injector nozzle is referred to Eq. (2.30). In this equation, the approaching downstream shock wave f 1V behaves as known. At the beginning of the flow regulation, it takes zero. At the later time, it is known generally from tracking shock waves, for instance, from the T-junction, over each series junction and resistance R in the traveling path, and downstream to the injector nozzle. After the shock pressure h at the injector nozzle has been resolved from the above equation, the primary upstream shock wave F can then be computed from Eq. (7.28) as F1V = h − h 0 − f 1V .

(7.119)

The traveling of this upstream shock wave towards the T-junction has to be further tracked in the same way as before for the downstream shock wave f 1V . In the same way, one obtains the upstream shock wave F 2 at injector 2. Afterwards, one only needs to compute the transmission performance of all shock waves at the T-junction. This can be basically performed by applying the conservation laws of shock waves, like Eqs. (7.75) and (7.76), as follows: ( f 3 − F3 )A3 = ( f 1 − F1 )A1 + ( f 2 − F2 )A2 .

(7.120)

F1 + f 1 = F2 + f 2 = F3 + f 3 .

(7.121)

At the T-junction, the approaching shock waves are F1 , F2 and f 3 which are all known from computations in previous time steps. Departure shock waves are f 1 , f 2 and F3 which need to be computed for further tracking them. From the above two equations, or directly from Eqs. (7.77) to (7.79) for the T-junction of diverging flow,

7.13 Two Regulation Organs and Origins of Wave Generations

171

one obtains f1 =

A1 − A2 − A3 2 A2 2 A3 F1 + F2 + f3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.122)

f2 =

2 A1 A2 − A1 − A3 2 A3 F1 + F2 + f3, A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.123)

F3 =

2 A1 2 A2 A3 − A1 − A2 F1 + F2 + f3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.124)

This group of equations is called traveling laws of shock waves at the T-junction according to Fig. 7.18. In effect, these last three equations can be directly written out by taking into account respective reflection and transmission parts of all three approaching wave parameters, see Sect. 7.5.1. From the above computations, one would be again firmly convinced by the high applicability of the wave tracking method (WTM).

7.14 Shapes of Shock Waves In Sect. 3.3.4 accounting for the significant self-stabilization effect of transient flows through a regulation valve, it has been demonstrated that under different conditions of computations quite different shapes of shock waves will be obtained. In general, the condition of the predefined volume flow rate does not fulfill the self-stabilization mechanism of transient flows. It therefore does not represent the reality and hence will lead to significant computational inaccuracies. Especially, under the condition of a constant changing rate of the volume flow rate (dQ/dt = const), the shape of computed shock waves is evidently like that of regular triangular waves, even during the flow regulation (Fig. 3.8d). The shape of high-frequency shock waves actually depends on the configuration of each hydraulic system and the regulation dynamics. Its accurate determination requires computations under real flow conditions, as in Fig. 3.12b. There, and also in many other computations in foregoing chapters and sections, see Fig. 4.9b for instance, the shape of shock waves after closing the valve highly approaches to that of a regular triangular wave. In fact, this is only found in a simple hydraulic system like Fig. 7.14 and by slowly closing the regulation valve. By rapidly closing the valve, not only the shock pressure but also the shape of shock waves change. Figure 7.19 shows the comparison between two computations with different closing time for closing an injector nozzle. The presented computations refer to the pipeline system given in Table 4.2. The fact to be mentioned is that most hydraulic systems in hydropower stations are complex because of the presence of surge tanks, series and T-junctions, and other components. Because of multiple interactions between waves which are subjected

7 Wave Tracking Method

H0

172

L Injector

approx. regular triangular wave

non-regular triangular wave

Fig. 7.19 Shock-pressure response on the closing time of the flow in a simple hydraulic system; computations refer to Table 4.2

to reflections and transmissions, the shapes of shock waves are highly non-uniform. This can be confirmed for a stepped pipeline, as already encountered in Fig. 4.11, see also Sect. 7.15.3 below. Other examples can be found in Chap. 11.

7.15 Computational Examples and Algorithms In all above sections, the wave tracking method paired with generation mechanisms and travelling performances of shock waves has been outlined in details. Some computation examples in Chap. 4, see Fig. 4.9 to Fig. 4.10, were only made to compare

7.15 Computational Examples and Algorithms

173

the difference between the rigid and the elastic water column theory without showing any computation details. For engineering application purposes, therefore, the capability and applicability of the wave tracking method should be demonstrated by showing detailed computations and computational algorithms. For the sake of simplicity, computational algorithms of some basic hydraulic concepts will be presented below. Computation examples for more complex hydraulic systems with pumps and turbines as well as with real surge tanks will be given in Chap. 11.

7.15.1 Closing of the Injector Nozzle For the sake of simplicity, a pipeline system of constant diameter and with a Pelton injector as the regulation organ is considered according to Fig. 7.20. The existence of the surge tank is ignored. To simulate the shut-down of the turbine and thus the cut-off of the flow in the pipeline, the initial steady flow state is specified by a flow rate Q0 . The injector is assumed to be closed within a limited time. Relevant transient quantities to be computed are the changeable discharge and the pressure head at the injector. Other significant system parameters including the linear closing of the injector are listed in Table 7.1. The initial flow rate is computed, according to Eq. (2.46) including the viscous friction, as Q0 = 

ϕ0 AD0 1 + 2g R(ϕ0 AD0 )

2

 2g H0 .

(7.125)

Fig. 7.20 A simple pipeline system with an injector nozzle for flow regulations

Surge tank

Lake

H0

The point to be mentioned is that for self-programming, for instance, by means of MS Excel, all important equations should be written out each time prior to making their implementations. This is so far important to avoid any programming error and to clearly document computations for later understanding. In the present example, only the pressure head at the injector is interested and hence will be computed. Because a round trip of the primary shock waves takes a time equal to T2L = 2L/a, the time step t = T2L = 2 s can be applied. Otherwise, if, for instance, the pressure response at the middle section of the pipeline (L/2) is also interested, then the time step has to be set to t = L/2a = 0.5 s. As in all transient computations, the first attempt is to compute primary shock waves at the injector. To this end, Eq. (7.24) is directly used, as rewritten here:

c d=1.0, L=1250, Resistance constant R

Injector

174

7 Wave Tracking Method

Table 7.1 Pelton turbine system and operational parameters Parameter

Symbol

Unit

Value

Available hydraulic head

H0

m

300

Pipe length

L

m

1250

Pipe diameter

d

m

1.0

Viscous friction coefficient

λ

–

0.005

Resistance constant

R

s2 /m5

0.54

Injector nozzle diameter

D0

m

0.3

Initial needle stroke

s0

mm

250

Initial discharge coefficient

ϕ0

–

0.71

Injector constant

a1

–

1.536

Injector constant

a2

–

−0.857

Initial flow rate

Q0

m3 /s

3.81

Initial flow velocity c0 = Q 0 /A

c0

m/s

4.85

Initial pressure head at injector

h0

m

291.0

Wave propagation speed

a

m/s

1250

Linear closure time

tc

s

40

aϕ AD0 h+ g A



√  2g h − h ∗0 + 2 f = 0, 2 1 − (ϕ AD0 /A)

(7.126)

with h ∗0 = h 0 +

a Q0. gA

(7.127)

In the present example, a linear closing law s = s0 (1 − t/tc ) has been assumed. The discharge coefficient ϕ as the injector characteristic is obtainable from Eq. (2.30). At the very beginning of closing the injector, i.e., within the time t = T2L for the first round trip of the first shock wave, the downstream shock wave f in Eq. (7.126) is zero. The time-dependent pressure head at the injector can then immediately be solved from the above equation. The primary shock wave at the injector is then computed from Eq. (7.10) as F = h − h 0 − f.

(7.128)

Correspondingly, the flow velocity or the discharge is obtained from Eq. (7.11) as Q = Q0 +

g A( f − F). a

In fact, the same discharge can also be obtained from Eq. (7.23).

(7.129)

7.15 Computational Examples and Algorithms

175

In the considered simple pipeline of constant cross-sectional area, one needs to track the shock wave from the injector towards the upper reservoir and, after the reflection, back to the injector. During such a process which is related to a round trip of each shock wave, only the entrance velocity effect and the viscous friction effect need to be considered, as given in Eq. (7.107). When neglecting the entrance velocity effect, Eq. (7.108) can be directly applied, as rewritten here:  f t = −Ft−2t − R Q|Q| − Q 0 |Q 0 | .

(7.130)

Numerical computational steps of solving the current transients are shown in Table 7.2. In the tabulated computations, the parameters kB and kC , respectively, represent the coefficient of the second term and the third quasi-constant in Eq. (7.126). The downstream shock wave f is set to be zero at t = 0. To other times, it has been tracked and computed according to Eq. (7.130) with a time delay of T2L = 2 s. The computed discharge and shock pressure head at the injector are shown in Fig. 7.21 as functions of time. First, the self-stabilization effect of the flow during the closing time has been again verified, see Sect. 3.3.4. Second, as expected, the highest pressure head is confirmed at the end of the closing time, i.e., while reaching Q = 0. The resultant pressure head rise is about 30 m which can be again verified by Eq. (3.7) with Q = 0 and dQ/dt = −0.183 in the current case. Obviously, the maximum shock pressure at the injector is simply determined by the maximum gradient of flow velocity in the pipeline. While the injector nozzle is closed, the viscous friction effect disappears because of Q = 0. This simply means that the pressure rise h = 30 m in Fig. 7.21b is independent of viscous frictions. Therefore, if only the maximum shock pressure in the flow is interested, the entire computation can be simplified by setting λ = 0.

7.15.2 Opening of the Injector Nozzle Similar computations can be performed for the process of opening the injector nozzle. In this case, one computes from Eq. (7.127) the initial quantity h ∗0 = 300 because of Q = 0 and h 0 = H0 . Figure 7.22 shows corresponding computation results for the same hydraulic system as that considered in Fig. 7.21. From computations, the final stable state has been reached at Q = 3.814 m3 /s and h = 292.2 m. The expected value with the opening of the injector at s = 250 mm, however, is obtainable from Eq. (2.46) and h = H0 − c2 /2g − R Q 2 , leading to Q = 3.807 m3 /s and h = 291.0 m, respectively. The difference between computed and expected values is due to the entrance velocity effect (see Sect. 7.7.1), which has been ignored in the current transient computation. Another computation example which clearly shows the entrance velocity effect is found in Sect. 7.15.5.

Injector

s

250

238

225

213

200

188

175

163

150

:

Time

t

0

2

4

6

8

10

12

14

16

:

:

0.57

0.59

0.62

0.64

0.66

0.68

0.69

0.70

0.71

ϕ

Table 7.2 Numerical computational steps

31.5 30.2 28.8

−29.2 −38.9 −49.8 :

32.7

−20.9

:

34.5 33.7

−13.8

35.2

−3.4 −7.9

35.8

36.2

kB

0.0

0.0

f

Computations

17.2

−909.2

:

−809.6

−831.5

−850.8

−867.5

−881.7

−893.3

:

17.5

17.4

17.4

17.3

17.3

17.3

17.2

17.1

−909.2 −902.5

h0.5

kC

:

306

304

303

301

299

298

296

295

291

h

:

64.6

51.9

40.5

30.4

21.7

14.3

8.2

3.4

0.0

F

:

3.10

3.25

3.38

3.49

3.59

3.67

3.74

3.79

3.81

Q

176 7 Wave Tracking Method

7.15 Computational Examples and Algorithms

177

Fig. 7.21 Discharge and shock pressure response to the closing of the injector nozzle (referred to Fig. 7.20 without considering the surge tank)

Fig. 7.22 Discharge and shock pressure response to the opening of the injector nozzle (referred to Fig. 7.20 without considering the surge tank)

178

7 Wave Tracking Method

7.15.3 Stepped Pipeline A pipeline which consists of three pipes of different diameters and is referred to Fig. 4.11 should be considered. In fact, computation results have already been applied there for comparison with computations based on the rigid water column theory. Computations begin with the generation of the primary shock waves at the injector. To this end, both Eqs. (7.24) and (7.26) can be applied. The former seems to be rather more convenient than the latter, because it is also applicable to the closed injector with ϕ = 0. After the pressure head has been computed from Eq. (7.24), both the flow rate Q and the primary shock wave F can be computed from Eqs. (7.27) to (7.28), respectively. Afterwards, one only needs to track all shock waves along the pipeline. This includes both wave reflections and transmissions while passing through sections A and B. Corresponding computation steps are outlined in Table 7.3 for the case of closing the injector nozzle. The time increment has been set to be t = 1/3 s. This agrees with the shortest traveling time of a shock wave within a pipe of length L = 430 m, when the wave speed is assumed to be a = 1290 m/s. In Table 7.3, the parameter k B represents the coefficient of the second term in Eq. (7.24), which is then written as √  h + kB h − h ∗0 + 2 f = 0.

(7.131)

At sections A and B, one only needs to calculate the reflections and the transmissions of shock waves by using Eqs. (7.67) and (7.68), as rewritten here for direct use F1 =

A1 − A2 2 A2 f1 + F2 , A1 + A2 A1 + A2

(7.132)

f2 =

2 A1 A2 − A1 f1 + F2 . A1 + A2 A1 + A2

(7.133)

Both the wave parameter f 1 and F 2 are approaching parameters. Therefore, they behave as known quantities which are obtainable by simply tracking them. One confirms that it is really no longer necessary to compute the flow rates and the shock pressures at the two pipe sections A and B. This is so, if only the shock pressures at the pipe section ahead of the injector nozzle are interested. Figure 7.23 shows the shock pressure response at the injector which is closed within 30 s. This computation result has been already used in Fig. 4.11 for reference. The non-regular fluctuations of shock pressures after closing the injector arises from multiple interactions between shock waves coming from sections A and B, where all shock waves are subjected to both reflections and transmissions. Similar computations can be completed for the case of opening the injector nozzle. For computation results, one is referred to Fig. 4.11.

Injector

s/D0

0.75

0.74

0.73

0.73

0.72

0.71

0.70

0.69

0.68

0.68

0.67

0.66

:

Time

t (s)

0.00

0.33

0.67

1.00

1.33

1.67

2.00

2.33

2.67

3.00

3.33

3.67

:

:

0.64

0.64

0.65

0.65

0.65

0.66

0.66

0.66

0.66

0.67

0.67

0.67

ϕ

15.4

15.3

15.3

15.2

15.1

15.1

15.0

14.9

−0.4

−0.6

−0.9

−1.9

−2.9

−3.6

−4.4

−5.0

:

15.5

−0.2

:

15.5

15.6

15.6

kB

0.0

0.0

0.0

f

:

410

410

410

410

410

411

410

410

409

409

408

407

h

:

8.0

7.2

6.4

5.7

5.2

4.6

3.8

3.0

2.2

1.5

0.7

0.0

F

:

1.80

1.81

1.82

1.83

1.84

1.85

1.85

1.86

1.87

1.87

1.88

1.88

Q

:

−3.2

−2.8

−2.4

−1.9

−1.4

−0.7

−0.1

−0.1

0.0

0.0

0.0

0.0

f1

Section B

Table 7.3 Numerical computations of transient flows at the injector nozzle being closed

:

7.1

6.4

5.7

5.1

4.6

3.8

3.0

2.2

1.5

0.7

0.0

0.0

F2

:

4.5

4.0

3.6

3.3

3.0

2.6

2.2

1.6

1.1

0.5

0.0

0.0

F1

:

−5.8

−5.1

−4.4

−3.6

−2.9

−1.9

−0.9

−0.7

:

−2.5

−2.3

−2.0

−1.7

−1.3

−0.9

−0.5

0.0

0.0

0.0

−0.2 −0.4

0.0

0.0

f1

Section A

0.0

0.0

f2

:

4.0

3.6

3.3

3.0

2.6

2.2

1.6

1.1

0.5

0.0

0.0

0.0

F2

:

3.1

2.7

2.5

2.3

2.0

1.7

1.3

0.9

0.5

0.0

0.0

0.0

F1

:

−3.5

−3.2

−2.8

−2.4

−1.9

−1.4

−0.7

−0.1

−0.1

0.0

0.0

0.0

f2

7.15 Computational Examples and Algorithms 179

7 Wave Tracking Method

L=430, d=1.5

L=430, d=1.3

1

L=430, d=1.0 B

A F1 f1

H0

180

2

1 F2 f2

F1 f1

2

Injector F2 f2

Fig. 7.23 Stepped pipeline and shock pressure at the injector nozzle being closed

7.15.4 Flow Oscillation in Surge Tanks The surge tank is a very important component in almost all hydro-power stations. Its functionality has been outlined in Chap. 5, where the design and specification of surge tanks have also been provided based on the rigid water column theory. As indicated there, the application of the rigid water column theory can be considerably complicated at hydraulic networks which, for instance, comprise pipes of different diameters, T-junction, and other hydraulic components. It has also indicated that at such hydraulic systems, the elastic water column theory can be favorably applied. In order to demonstrate this circumstance, the wave tracking method will be applied to a hydraulic system that, according to Fig. 5.3 and Table 5.1, has already been computed and presented in Figs. 5.4 and 5.5 (application of the rigid water column theory). As usual, the first step is to compute the primary shock waves generated at the injector nozzle. For reference purposes of direct applications, as this is always recommended, Eqs. (7.24), (7.27) and (7.28) are rewritten here aϕ AD0 h+ g A



√  2g h − h ∗0 + 2 f = 0, 2 1 − (ϕ AD0 /A)

(7.134)

7.15 Computational Examples and Algorithms

181

h3

Q3

z3

f3

Q1

F3

F2

F1 m

m f2

Q2

f1

Fig. 7.24 Transmission and reflection of shock waves at the connection point m below the surge tank, referred to Fig. 5.3

Q=

gA ∗ h0 − h + 2 f , a

F = h − h 0 − f.

(7.135) (7.136)

At the junction point m of the surge tank, all wave parameters (f and F) are shown in Fig. 7.24, in accordance with defined positive flow directions. As appointed in Sect. 6.2 and Sect. 7.1, the wave f is denoted as the downstream and F as the upstream waves, respectively. In the current case, the traveling laws of shock waves at the junction point m are given by Eqs. (7.77), (7.78) and (7.79), as rewritten here F1 =

A1 − A2 − A3 2 A2 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.137)

f2 =

2 A1 A2 − A1 − A3 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.138)

f3 =

2 A1 2 A2 A3 − A1 − A2 f1 + F2 + F3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(7.139)

Equations (7.134)–(7.139) form the basis for computations in the considered hydraulic system. They can be solved in time series of tabulated forms. Against the method based on the rigid water column theory, which basically requires simultaneous solutions of all three momentum equations in Sect. 5.2.1, the wave tracking method shows great advantages. It is also against the use of Eq. (5.13) which is obtained after complicated computations. Under equal regulation laws as those leading to Fig. 5.4 with an opening time of the injector nozzle equal to t = 40 s, numerical steps for computing the shock pressures and the flow rates at the junction point m are outlined in Table 7.4. The time increment has been selected to be t = 0.064 s. This time increment has been set with respect to the initial height (z3 = 83 m) of water in the surge tank (Fig. 5.3) and a wave speed equal to a = 1250 m/s. The propagation of waves in the surge tank can thus be accurately tracked. The time delay between shock waves at the injector and the surge tank (L 2 = 1400 m) is t = 1.12 s. Between the surge tank and the

Injector

s/D0

0.000

0.001

0.002

0.003

0.004

0.005

:

0.078

0.079

0.080

0.081

0.082

0.083

:

Time

t (s)

0

0.064

0.128

0.192

0.256

0.32

:

4.992

5.056

5.120

5.184

5.248

5.312

:

ϕ

:

0.12

0.12

0.12

0.12

0.12

0.11

:

0.01

0.01

0.00

0.00

0.00

0.00

f

:

31.6

30.0

28.4

26.7

25.1

23.5

:

0.0

0.0

0.0

0.0

0.0

0.0

kB

:

6.0

5.9

5.8

5.8

5.7

5.6

:

0.4

0.3

0.2

0.2

0.1

0.0

h

:

473

472

470

469

467

466

:

531

533

535

537

538

540

Q2

:

11.6

11.4

11.3

11.1

11.0

10.8

:

0.8

0.6

0.5

0.3

0.2

0.0

0.0 0.0 0.0

−98.1 −98.2 −98.3 :

0.0

−98.1

:

0.0

−98.0

0.0

−8.7 0.0

0.0

−6.9

−97.9

0.0

−5.2

:

0.0

−3.5

:

0.0

−1.7

f1 0.0

0.0

F2

:

−98.0

−97.8

−96.5

−95.2

−93.8

−92.5

:

0.0

0.0

0.0

0.0

0.0

0.0

F2

Surge tank

Table 7.4 Numerical computations of transient flows at the injector nozzle being opened F3

:

7.9

7.8

7.7

7.6

7.5

7.4

:

0.0

0.0

0.0

0.0

0.0

0.0

F1

:

−0.6

−0.8

−0.8

−0.7

−0.7

−0.7

:

0.0

0.0

0.0

0.0

0.0

0.0

f2

:

97.3

97.1

95.8

94.4

93.1

91.8

:

0.0

0.0

0.0

0.0

0.0

0.0

f3

:

−8.6

−8.6

−8.5

−8.4

−8.3

−8.1

:

0.0

0.0

0.0

0.0

0.0

0.0

Q1

:

0.2

0.2

0.2

0.2

0.2

0.2

:

0.0

0.0

0.0

0.0

0.0

0.0

Q2

:

17.4

17.3

17.1

16.9

16.6

16.4

:

0.0

0.0

0.0

0.0

0.0

0.0

Q3

:

−17.2

−17.1

−16.9

−16.7

−16.4

−16.2

:

0.0

0.0

0.0

0.0

0.0

0.0

h3

:

82.7

82.7

82.7

82.7

82.7

82.8

:

83.0

83.0

83.0

83.0

83.0

83.0

182 7 Wave Tracking Method

7.15 Computational Examples and Algorithms

183

lake, the distance L 1 = 3700 m makes a time delay of t = 2.96 s. All these time delays must be considered in numerical computations presented in Table 7.4. In Table 7.4, the parameter k B represents the coefficient of the second term in Eq. (7.134). Computational algorithms shown in Table 7.4 demonstrate that computations can be well accomplished by taking into account Eq. (7.134)–(7.139). Figure 7.25 shows the computational results according to Table 7.4 for an opening process of the hydraulic system shown in Fig. 5.3. For comparison reasons, computational results from Fig. 5.4 based on the use of the rigid water column theory have also been shown. Obviously, low-frequency flow oscillations between the surge tank and the lake, as in Fig. 5.3, can be well computed from both the rigid and elastic water column theories. As shown in Table 7.4, the shock pressure at the injector is also computed. Figure 7.26 shows its fluctuation, which is obviously impressed by the oscillation of the water level in the surge tank.

Fig. 7.25 Application of the wave tracking method to compute flow oscillations in the hydraulic system (Fig. 5.3) during and after opening the flow; comparison with computation results from Fig. 5.4 based on rigid water column theory

184

7 Wave Tracking Method

Fig. 7.26 Pressure repose at the injector nozzle during and after opening the flow, from the same computations as in Fig. 7.25

7.15.5 Flow Regulation Between Two Reservoirs 7.15.5.1

Background and Equations

f1 F1

Q

Resistance constant R1 L1

h1

h2

f2 F2

Resistance constant R2 L2

Fig. 7.27 Flow regulation between two reservoirs by using a butterfly valve

z2

z1

H0

A flow process between two reservoirs can be controlled, for instance, by a butterfly valve which is found in the middle of a pipeline, as illustrated in Fig. 7.27. The hydraulic transients which occur in association with the flow regulation in such a hydraulic system can be accurately computed by the wave tracking method. For the purpose of demonstrating computations, the pipeline is assumed to be of constant cross-sectional area. The significant knowledge for computing the current transient flow is the characteristic of the butterfly valve. This has been introduced by Eq. (2.38) with respect to Eq. (2.39) with sufficient accuracy. In place of the butterfly valve, other types of valves may be available. From Eq. (2.37) and with h butterfly = h 1 − h 2 , one obtains

7.15 Computational Examples and Algorithms

185

h 1 − h 2 = cp

c2 . 2g

(7.140)

On the other hand, the pressure head drop h 1 − h 2 should be computed by the method of hydraulic transients. First, on the high-pressure side of the butterfly valve, one obtains from Eq. (7.3) with the aid of Eq. (7.25) h 1 = h ∗1,0 −

a c1 + 2 f 1 . g

(7.141)

On the low-pressure side of the butterfly valve, it follows then from Eqs. (7.4) and (7.33) h 2 = h ∗∗ 2,0 +

a c2 + 2F2 . g

(7.142)

Combing Eqs. (7.141) and (7.142) with c1 = c2 = c yields a h 1 − h 2 = h ∗1,0 − h ∗∗ 2,0 − 2 c + 2( f 1 − F2 ). g

(7.143)

This equation for shock waves is further combined with the characteristic of the butterfly valve given by Eq. (7.140). One obtains with Q = Ac  cp 2a ∗ Q + h ∗∗ Q2 + 2,0 − h 1,0 + 2(F2 − f 1 ) = 0. 2 2g A gA

(7.144)

This is the determination equation for the volume flow rate Q through the butterfly valve. At the very beginning of the flow regulation, one has f 1 = 0 and F2 = 0. At other times, both f 1 and F2 are obtainable by tracking them. After the flow rate has been computed from Eq. (7.144), both the pressure head h1 and h2 can be computed from Eqs. (7.141) and (7.142), respectively. It should be indicated that the above equation does not apply to the case Q = 0 under the condition of a closed butterfly valve, because then cp would be infinite: cp = ∞. In this case, both pressure heads h1 and h2 will be directly obtained, respectively, from Eqs. (7.141) and (7.142) with c = 0. The primary shock waves on both sides of the butterfly valve are denoted by F 1 and f 2 , respectively. They can be determined from the fundamental equation given by Eq. (7.10), as follows F1 = h 1 − h 1,0 − f 1 ,

(7.145)

f 2 = h 2 − h 2,0 − F2 .

(7.146)

As an alternative, the fundamental equation given by Eq. (7.11) can also be applied.

186

7 Wave Tracking Method

Further computations all rely on tracking two primary waves. In the flow upstream of the butterfly valve, for instance, the primary shock wave F 1 travels upstream towards the upper-side reservoir. There, it will be reflected under the entrance velocity effect and, in terms of the downstream wave f 1 , will further travel back to the butterfly valve. For such a round trip which takes 2t = 2L 1 /a, the returning downstream wave is computed, according to Eq. (7.107), as  f 1 = −F1,−2t − R1 Q 2 − Q 20 +

1  2 Q0 − Q2 . 2 2g A

(7.147)

In the following computations, the entrance velocity effect (last term in above equation) will be, respectively, taken into account and ignored (simplification). In the same way, one obtains the returning upstream wave on the low-pressure side of the butterfly valve from Eq. (7.109) as  F2 = − f 2,−2t + R2 Q 2 − Q 20 .

(7.148)

The time delay between f 2 and F 2 is equal to 2t = 2L 2 /a. The flow rate Q in the above two equations should be the respective mean value during the round trip of the respective shock waves. Because the associated flow resistances are usually of small order of magnitude in influencing the shock pressure in the system, the volume flow rates in the last time step can be used. Especially, equal volume flow rates in Eqs. (7.147) and (7.148) can be applied. The resistance constants R1 and R2 are of general character and may contain viscous frictions and local resistances. In numerical computations, the time increment has to be set, by which the two different round trip times T1,2L and T2,2L are divisible. The pipeline system shown in Fig. 7.27 can be extended by including one or two surge tanks, pipes of different diameters, etc. For use of the wave tracking method, this does not mean any significant increase of the computational complexity.

7.15.5.2

Computational Examples

For the flow system illustrated in Fig. 7.27, the design and operation parameters are listed in Table 7.5. Some parameters have been computed for the help that interesting readers could well follow the computation examples by themselves, for instance, for the training purpose. Example 1: Closing of the Butterfly Valve The starting point of computations is the calculation of the initial stationary flow. For general purposes, the stationary flow for each given opening of the butterfly valve is obtainable by balancing the resistance and pressure forces in the flow. The total flow resistance between two reservoirs (or tanks) is determined by R = R1 + R2 + Rvalve with Rvalve from Eq. (2.41). In stationary flow, the pressure force exerted on the flow

7.15 Computational Examples and Algorithms

187

Table 7.5 Hydraulic system (Fig. 7.27) with a butterfly valve for regulating the flow Upper reservoir

z1

m

60

Lower reservoir

z2

m

30

Total hydraulic head

H0

m

30

Pipeline diameter

d

m

1

Pipeline length

L1

m

150

Pipeline length

L2

m

300

friction coefficient

λ

–

0.01

Resistance constant (computed)

R1

s2 /m5

0.124

Resistance constant (computed)

R2

s2 /m5

0.248

Butterfly valve closed

β0

°

10

Butterfly valve opening at operation

βN

°

60

Valve resistance coefficient (computed)

cp,N

–

2.32

Valve resistance constant (computed)

RN

s2 /m5

0.192

Flow rate (computed)

QN

m3 /s

6.81

Pressure drop at the valve (computed)

ΔhN

m

8.9

Upper pressure head (computed)

h1,N

m

50.4

Lower pressure head (computed)

h2,N

m

41.5

Valve closing/opening time

t valve

s

20

Wave propagation speed

a

m/s

1250

is related to the total pressure head and given as H0 − c2 /2g, in which the effect of the dynamic on the static pressure head at the entrance of the pipeline is taken into account. Thus, one obtains H0 −

1 2 c = R Q2. 2g

(7.149)

With Q = c A, the flow rate under the condition of stationary flow is then computed as  Q=A

2g H0 . 1 + 2g R A2

(7.150)

With the initial resistance constant R0 in a transient process, one obtains the initial flow rate Q0 as the starting point. For approximation, the above equation can also be applied to the period during the flow regulation, if the flow can be assumed to be quasi-stationary, for instance, by slowly opening or closing the valve. Computations have been carried out by the numerical algorithm according to Table 7.6. The parameters k A and k C refer to the quadratic Eq. (7.144) and represent,

188

7 Wave Tracking Method

respectively, the coefficient of Q2 and the sum of the last two terms on the l.h.s. of the equation. The coefficient of Q is a constant. In this computational example, the employed butterfly valve has its closed position at β 0 = 10°, at which cp approaches infinity. In the conducted computations, the closing of the butterfly valve occurs from β = 60° linearly to β = 10.2° within 20 s. The selection of β = 10.2° is only for the reason that Eq. (7.144) with cp < ∞ can be directly applied. Transient computations are carried out under the following three different conditions: (1) with respect to the entrance velocity effect according to Eq. (7.147), (2) by neglecting the entrance velocity effect in Eq. (7.147), and (3) under the condition of quasi-stationary flows according to Eq. (7.150). The computational steps under condition (1) are shown in Table 7.6. For the case (2) by neglecting the entrance velocity effect, only the computations of the wave parameter f 1 need to be updated. Computations under the condition (3) for quasi-stationary flow simply follow Eq. (7.150). Figure 7.28 shows computational results involving the flow rate and the pressure heads on both sides of the butterfly valve. One also compares the computations under all three conditions listed above. Once the flow regulation is completed (t = 20 s), the flow tends to be rapidly stabilized (Fig. 7.28b). Regarding this flow performance, no difference has been confirmed between computations under the condition (1) and (3). In addition, it can be easily demonstrated that by very slowly closing the butterfly valve (t > 100 s), the difference between two computations disappear almost completely, already from the beginning of the flow regulation (t = 0). In Fig. 7.28c, computations without considering the entrance velocity effect have been shown. A deviation of the computed pressure head h1 from the expected stabilized value has been confirmed to be given by about 4 meters. Basically, this deviation depends differently on all parameters in both the system design and flow regulations. It would not be recognized if no comparable computations were made. The computations again evidently demonstrate that hydraulic transients can be accurately computed, as long as all related initial and boundary conditions are well specified and correctly accounted for. A very important phenomenon regarding the pressure head h2 has to be concerned. Computations are only representative, if the pressure head is greater than the saturation pressure. By rapidly closing the flow, for instance, the pressure head h2 may drop down to the saturation pressure, so that cavitation and high-level noise on the low-pressure side of the valve will take place. On the other hand, if computations show significant negative pressures, then, cavitation in the flow has definitely happened. Local cavitations even happened earlier. For this reason, current transient computations are much helpful for both designing the hydraulic system and setting the regulation dynamics of the butterfly valve.

β

60.0

59.4

58.8

58.2

57.6

57.0

56.4

:

t

0.00

0.24

0.48

0.72

0.96

1.20

1.44

:

:

0.56

0.55

0.54

0.53

0.53

0.52

0.51

σ

:

3.37

3.17

2.97

2.79

2.63

2.47

2.32

cp

Transient computations

Time F2

0.26 0.82 1.39 2.25

−0.56 −1.13 −1.99 −2.87 :

−0.01

−0.28

:

0.00

0.00

0.00

0.00

f1

:

0.28

0.26

0.25

0.23

0.22

0.20

0.19

kA

:

−2210

−2214

−2216

−2219

:

6.77

6.78

6.79

6.80

6.81

6.81

−2220 −2220

Q 6.81

kC −2220

Table 7.6 Numerical solution of transient flows related with the closing of the butterfly valve h1

:

51.7

51.4

51.3

51.0

50.7

50.7

50.4

h2

:

38.9

39.3

40.0

40.3

40.6

41.2

41.5

F1

:

4.19

2.96

2.04

1.16

0.57

0.28

0.00

f2

:

−4.81

−3.56

−2.35

−1.45

−0.86

−0.28

0.00

7.15 Computational Examples and Algorithms 189

190

7 Wave Tracking Method

Fig. 7.28 Transient flows between two reservoirs by closing the butterfly valve. a Volume flow rate; b Pressure heads h1 and h2 , computed with respect to the entrance velocity effect; c Pressure heads h1 and h2 , computed by neglecting the entrance velocity effect

7.15 Computational Examples and Algorithms

191

Example 2: Opening of the Butterfly Valve By following the same computational steps as in example 1, transient computations for the case of opening the butterfly valve have also been completed, as shown in Fig. 7.29. This time, the neglect of the entrance velocity effect in computations

Fig. 7.29 Transient flows between two reservoirs by opening the butterfly valve. a Volume flow rate; b Pressure heads h1 and h2 , computed with respect to the entrance velocity effect; c Pressure heads h1 and h2 , computed by neglecting the entrance velocity effect

192

7 Wave Tracking Method

(Fig. 7.29c) clearly leads to computational errors in both the pressure head h1 and h2 . Both pressure heads tend to be stabilized, however, with deviations from respective values which are expected under stationary flow conditions.

7.16 Evaporation of Flows and Restrictions of Computations When considering Figs. 7.19 and 7.21 for closing the flow, one confirms that after fullclosing the pressure fluctuations take place around the initial available pressure head H 0 . The amplitude of the high-frequency pressure fluctuations primarily depends on the closing time. These circumstances of transient flows will be exactly revealed in Chap. 9. At the present moment, a special phenomenon has to be indicated which could occur in transient computations. The phenomenon is associated with the case when the amplitude of pressure fluctuations is greater than the mean value (H 0 ), and hence negative pressures will be obtained. The same or comparable negative pressures could also be encountered, when the flow is rapidly opened and the local pressure considerably drops. The phenomenon often happens where the initial pressure head H 0 is relatively low. The occurrence of negative pressures has both physical and mathematical consequences. As for the former, the reverse flow at the injector in Fig. 7.19, for instance, would take place, which is physically impossible. Negative pressures always lead to evaporation of water and thus to the change of the assumed one-phase flow model. Because of repeated rise and fall of pressures, water is subjected to rapid change between evaporation and condensation. The phenomenon is often confirmed by huge cavitation noises. For this reason, computations have to be stopped, as soon as the first negative pressure takes place, see Fig. 7.30.

Fig. 7.30 Occurrence of negative pressures and limitation of computations

7.16 Evaporation of Flows and Restrictions of Computations

193

As for the mathematical consequence of negative pressures, Eq. (7.24), for instance, becomes non-applicable. Any further computations will lead to errors. Similar mathematical problems will be given, if Eq. (7.26) is used for the same transient computations. As soon as Q = 0 is reached, computations must be stopped.

References Allievi, L. (1902). General theory of the variable flow of water in pressure conducts, see “Theory of Water Hammer”, (E. E. Halmos, Trans.). Typography Riccardo Garoni, Rome, 1925. Wood, D. J., Dorsch, R. & Lightner, C. (1966). Wave analysis of unsteady flow in conduits. Journal of the Hydraulics Division ASCE, 92(HY2), 83–110. Zhang, Z. (2016). Transient flows in pipe system with pump shut-down and the simultaneous closing of the spherical valve. In IAHR, 28th Symposium on Hydraulic Machinery and Systems. Grenoble, France, see also IOP Conference Series: Earth and Environmental Science (Vol. 49, p. 052001). https://doi.org/10.1088/1755-1315/49/5/052001. Zhang, Z. (2018, August). Wave tracking method of hydraulic transients in pipe systems with pump shut-off under simultaneous closing of spherical valves, Journal of Renewable Energy, 132, 157–166.

Chapter 8

Method of Characteristics

The method of characteristics has played an essential role in computing hydraulic transients up to now. Almost all applicable software tools are based on this method. In principle, the method of characteristics is a mathematical technique for solving so-called hyperbolic partial differential equations. Unlike the wave tracking method which is a Lagrangian approach in fluid mechanics (Chap. 7), the method of characteristics relies on the Eulerian description of fluid flows. It basically makes use of the invariance of wave propagation along the characteristic lines and requires, therefore, computations at equidistant nodes along a pipeline. Although the wave tracking method has been confirmed in the last chapter to be much applicable, for the completeness, the method of characteristics and the computational algorithms will be presented in this chapter.

8.1 Characteristic Lines The basic equation in the method of characteristics is the partial differential Eq. (6.7) or equivalently the total differential Eq. (6.10) under the condition dx/dt = ±a. For frictionless flows, for instance, Eq. (6.10) simply means c ± (g/a)h = const with h = f(x, t) and c = f(x, t). In Chap. 7 when describing the wave tracking method, the expression c ± (g/a)h or, equivalently, h ± (a/g)c have been expressed as wave parameters F and f. The condition dx/dt = ±a, which connects two independent variables x and t by the wave speed, represents mathematically two characteristic lines, as illustrated in Fig. 8.1. The principle of the method of characteristics can be well explained with the aids of Fig. 8.1. For simplicity, the frictionless flows are again considered. At the location xA and xB , two wave quantities (c−c0 )A +g/a(h −h 0 )A and (c−c0 )B −g/a(h −h 0 )B are assumed to be, respectively, known at the times tA and tB . They, in fact, behave as two constants and change their positions in opposite directions along the pipeline axis. In x-t plane in Fig. 8.1, this implies that two wave quantities maintain their constant values along their respective characteristic lines which are given by dx/dt = ±a. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_8

195

196

8 Method of Characteristics

Fig. 8.1 Propagations of shock waves and characteristic lines

t P

tP

tA

A B

tB a

a

xA

c

xP

xB

x

At the time tP , both shock waves reach the location xP and are characterized by the same pressure and the same flow velocity. These two flow parameters can then be straightforwardly solved for from the following two equations: g (h − h 0 )P = (c − c0 )A + a g (c − c0 )P − (h − h 0 )P = (c − c0 )B − a

(c − c0 )P +

g (h − h 0 )A , a g (h − h 0 )B . a

(8.1) (8.2)

The characteristic lines of shock waves represent conditions which must be complied with during the wave propagation. The quantities to be considered are (c − c0 ) ± g/a(h − h 0 ). For real flows with viscous friction effects, the term on the r.h.s of Eq. (6.10) has to be accounted for in the computations. In practical implementations of the method of characteristics, both time and space are discretized into finite lengths, commonly with equidistant steps x along the pipeline. This requirement, basically, relies on equidistant time steps t in each numerical solution of transient computations with respect to the constant wave speed. For this reason, the so-called characteristic grids, as illustrated in Fig. 8.2, must always be predefined when performing transient computations. Quite often, this Fig. 8.2 Characteristic grids in the method of characteristics

t t

t P t B

A x

x

x

x

x x

8.1 Characteristic Lines

197

seems to be rather inconvenient, because one must always compute the flows also at grids and nodes which are almost irrelevant or non-interesting. This just represents the shortcoming of the method of characteristics against the wave tracking method, as already pointed out many times.

8.2 Computational Algorithms First, Eq. (6.10) should be integrated along the path of wave propagation. According to Fig. 8.1, the integration is completed simply along the pipeline from x A to x P as well as from x B to x P , respectively. One then obtains     xP a a λ (h − h 0 ) + (c − c0 ) − (h − h 0 ) + (c − c0 ) = − (c|c| − c0 |c0 |)dx, g g 2gd P A  (h − h 0 ) −

a (c − c0 ) g



  a λ − (h − h 0 ) − (c − c0 ) = − g 2gd P B

(8.3)

xA xP

(c|c| − c0 |c0 |)dx.

(8.4)

xB

For frictionless flows (λ = 0), these two equations reduce to Eqs. (8.1) and (8.2), respectively. The integration of two friction terms in the above equations can be approximated and hence simplified. As known, the viscous friction effect in hydraulic transients basically behaves as an influence factor which, however, is of a lower order in magnitude, when compared with the shock pressure itself. Therefore, quasi-stationary flows along each characteristic line can be assumed, each time when accounting for the viscous effect. In practical applications, one sets x = xP − xA = xB − xP for generating characteristic grids along the pipeline, as shown in Fig. 8.2. The friction coefficient λ, as in the above equations, is generally assumed to be constant. Under conditions of quasi-stationary flows for the friction terms and by applying the velocities cA and cB , respectively, one obtains     a a λx (h − h 0 ) + (c − c0 ) − (h − h 0 ) + (c − c0 ) = − (cA |cA | − cA0 |cA0 |), g g 2gd P A

(8.5)

    a a λx (h − h 0 ) − (c − c0 ) − (h − h 0 ) − (c − c0 ) = (cB |cB | − cB0 |cB0 |). g g 2gd P B (8.6) These two equations have similar structures as Eqs. (7.1) and (7.2). The method of characteristics (MOC), however, follows the computational algorithms which are different from the wave tracking method (WTM). It should be indicated that in almost all applications of the method of characteristics up to now the initial flow parameters c0 and h 0 are not included in the above

198

8 Method of Characteristics

equations. For computations, one exclusively uses the so-called piezometric head in the form h = p/ρg + z by setting a reference altitude with z = 0 each time. Actually, such a terminology is commonly used only in geology, but not in hydraulic pipeline networks. Yet, the use of the symbol h for representing the piezometric head instead of the static pressure head would lead to a great confusion in complex transient computations. From the above two equations, both the velocity change (c − c0 )P and the change of the pressure head (h − h 0 )P at the point x P can be determined. In using Q = Ac for the volume flow rate and with the initial flow rate Q A0 = Q B0 = Q 0 , one obtains (h − h 0 )P = (Q − Q 0 )P =

1 1 (CA + CB ) − R(Q A |Q A | − Q B |Q B |), 2 2

(8.7)

1 gA 1 gA (CA − CB ) − R(Q A |Q A | + Q B |Q B | − 2Q 0 |Q 0 |), 2 a 2 a (8.8)

in which the parameterized variables CA and CB , as well as the resistance constant R, are defined as a (Q − Q 0 )A , gA a (Q − Q 0 )B , CB = (h − h 0 )B − gA

CA = (h − h 0 )A +

R=

λx . 2gd A2

(8.9) (8.10) (8.11)

This group of equations represents in this context the fundamentals of the method of characteristics (MOC). Both parameters CA and CB are, respectively, referred to the times tA and tB which are equal because of the choice of the uniform characteristic grids. Regarding the friction terms in the above equations, further simplification of computations can be made. In view of the lower order significance of the viscous friction effect and thus for the same reason as that leading to Eqs. (8.5) and (8.6), approximation Q A ≈ Q B at each time can be applied. Then, the last term in Eq. (8.7) vanishes. Furthermore, in performing numerical computations, the respective mean flow rate within the grid length x and within the time interval t = ti − ti−1 can be considered to be equal to the flow rate at the location x P and to the time ti−1 , i.e., the time one time-step before. This simply means that Q A ≈ Q B ≈ Q P,i−1 can be applied in Eq. (8.8). One then obtains in this approximation (h − h 0 )P =

1 (CA + CB ), 2

(8.12)

8.2 Computational Algorithms

(Q − Q 0 )P =

199

      g A CA − CB − R Q P,i−1  Q P,i−1  − Q 0 |Q 0 | . a 2

(8.13)

Obviously, these two equations are comparable to Eqs. (7.10) and (7.11), respectively, which are obtained by tracking the shock waves. One would confirm and realize that the method of characteristics by using CA and CB is in fact an Eulerian method, while the wave tracking method of using wave parameters F and f is simply a Lagrangian approach. The obvious shortcoming of the method of characteristics is the requirement of equidistant grid size x. In addition, one has always to compute CA and CB instead of tracking them. This means that at all grid-nodes both the pressure heads (h) as well as the flow rates (Q) must be computed. In engineering applications, however, knowledge of these is in most cases not required. The method presented here also slightly differs from some other applications that employ flow rates in the friction terms. The applied simplification, leading to Eqs. (8.12) and (8.13), as can be fully verified, has no remarkable influence on the computational accuracies. It is thus well justified. Since each hydraulic system is configured with a great variety of substantial components such as the reservoir, pipelines, surge tanks as well as hydraulic machines and regulation organs, the computations with Eqs. (8.12) and (8.13) strongly depend, in each application, on the initial and boundary conditions, like those listed in Sect. 7.2. This will be described below in the respective sections. At the end of this section, it should be mentioned that the presented computational algorithms with CA and CB are a common concept in applications of the method of characteristics (Giesecke and Mosonyi 2009, Chaudhry 2014, Horlacher and Lüdecke 2012). These two parameters, however, are used sometimes in the dimension “pressure head m”, e.g., in Giesecke and Mosonyi (2009), and sometimes in the dimension “volume flow rate m3 /s”, e.g., in Chaudhry (2014).

8.3 Generation of Shock Waves and C B at an Injector Nozzle To simulate the occurrence of hydraulic transients in a hydraulic system, it appears to be convenient to simulate the load regulation at a water turbine. For this reason, the injector nozzle of the Pelton turbine type is preferred and can be well used (Fig. 8.3). This is because the flow after the injector and thus under the constant atmospheric pressure does not backwards affect the flow in the pipeline network. The hydraulic system is thus mechanically disconnected with the turbine wheel (Fig. 2.5). The discharge through the injector nozzle is computed by Eq. (2.32), as rewritten here for direct application: Q=



ϕ AD0 1 − (ϕ AD0 /A)

2

2gh.

(8.14)

200

8 Method of Characteristics

x

A

P

+QA

x Fig. 8.3 Injector nozzle of Pelton turbines, used to simulate the flow regulation in a pipeline system

It is performed by regulation of the discharge coefficient ϕ according to Eq. (2.30). On the other hand, the shock pressure at the section upstream of and close to the injector is obtainable from Eqs. (8.12) and (8.13) by eliminating the variable CB . This is necessary, because the flow downstream of the injector, i.e., at the point B which does not exist, is irrelevant (Fig. 8.3). In form of finite element computations, one obtains (h i − h 0 )P = CA,i−1 −

  a (Q i − Q 0 )P − R Q 2P,i−1 − Q 20 . gA

(8.15)

In this equation, Q 2P,i−1 is used, because in the considered pipeline no reverse flow will occur. In the above equation, the variable CA,i−1 and the resistance constant R are computed, respectively, by Eqs. (8.9) and (8.11). The variable CA,i−1 refers to the first grid-node at x A upstream of the regulation injector and to the time ti−1 . At the very beginning of computations (t0 = 0) as well as at the first time node t1 one has CA = 0. Equations (8.14) and (8.15) are basic when computing both the pressure head at the injector and the discharge through the injector nozzle. By inserting Eq. (8.15) into (8.14), for instance, one obtains with removal of subscript P Q 2i

    2 a a 2 Qi − ψ h0 + Q 0 + CA,i−1 − R Q i−1 − Q 0 = 0. +ψ gA gA

(8.16)

The scale parameter ψ represents the opening degree of the injector nozzle and is defined as ψ=

2g(ϕ AD0 )2 ≈ 2g(ϕ AD0 )2 . 1 − (ϕ AD0 /A)2

(8.17)

The approximation made here is in effect h tot ≈ h, which means that the total pressure head is approximated by the static pressure head. It is followed by directly neglecting the dynamic pressure head at the pipe section immediately ahead of the injector, see Eqs. (2.32) and (2.33). Correspondingly, Eq. (8.14) is simplified to Eq. (2.33). Such an approximation does not lead to any significant computational

8.3 Generation of Shock Waves and C B at an Injector Nozzle

201

error because the dynamic pressure head is usually less than 1 m and certainly negligible as compared to the static pressure head h. Equation (8.16) represents a quadratic polynomial for the unknown discharge. One immediately obtains

   a 2 ψ a 1 2 ψ + 4ψ h ∗0 + CA,i−1 − R Q 2i−1 − Q 20 , Qi = − + 2 gA 2 gA

(8.18)

with h ∗0 = h 0 +

a Q0. gA

(8.19)

The shock pressure head (h) at the injector can then be computed from Eq. (8.15). Regarding the upstream grid-nodes, the grid-node at the injector (section P) behaves as the downstream node B (Fig. 8.1). The corresponding variable CB can be computed from Eq. (8.10) with the obtained volume flow rate Q and the pressure head h. It will then be further applied to computations at other upstream grid-nodes according to Eqs. (8.12) and (8.13). As it can be confirmed, Eq. (8.16) agrees with Eq. (7.26) in Chap. 7, which was obtained with the wave tracking method (WTM). This implies that both methods lead to identical computational results. The viscous friction effect in Eq. (7.26) is accounted for in the wave parameter f, see Eq. (7.103).

8.4 Flow State at a Reservoir of Constant Height The upper-side end of a pipeline system is usually connected to a lake or reservoir of constant height of water in it, as shown in Fig. 8.4. The static pressure head at the pipe entrance (x = 0) is then basically determined by the constant height of water and the currently existing flow rate. For c > 0, it is computed as

he

Fig. 8.4 Scheme to determine the pressure head at the entrance of a pipeline

Lake p x

B

202

8 Method of Characteristics

hP = he −

Q 2P . 2g A2

(8.20)

The second term on the r.h.s. of the equation represents the dynamic pressure head. In Sect. 7.7.1, it has been denoted as the entrance velocity effect. Its quantitative influence on computational results was demonstrated in Sect. 7.15.5 based on an illustrative application example, see Figs. 7.27 and 7.28. In almost all applications up to now, it was simply neglected. On the other hand, one obtains from Eqs. (8.12) and (8.13), this time by eliminating the variable CA , (Q i − Q 0 )P =

    gA (h i − h 0 )P − CB,i−1 − R Q P,i−1  Q P,i−1  − Q 0 |Q 0 | . (8.21) a

These last two equations, Eqs. (8.20) and (8.21), are capable to compute the pressure head h P and the flow rate Q P at the pipe entrance. By neglecting the entrance velocity effect, the pressure head difference (h − h 0 )P in the above equation disappears. It follows then by removing the subscript P Qi ≈ Q0 −

gA CB,i−1 + R(Q i−1 |Q i−1 | − Q 0 |Q 0 |) . a

(8.22)

The variable CB,i−1 in the above equation refers to the flow state at the first grid-node x B downstream of the pipe entrance and to the time ti−1 . From an engineering viewpoint, the transient flow state at the pipe entrance is commonly not interesting at all. This simply means that computations of the volume flow rate from Eqs. (8.21) or (8.22) are only of minimum value. They are, unfortunately, indispensible because the computed volume flow rate will be further used for computations in the downstream flow at all other grid-nodes; one sees the role played by the variable CA,i−1 in Eq. (8.15). Obviously, the necessity of computing the detailed flow states at the pipe entrance repeatedly reflects the shortcomings of the method of characteristics for certain applications. On the contrary, by using the wave tracking method, one only needs to apply Eq. (7.95).

8.5 Pressure Head at the Closed Valve When the flow in a pipeline system is stopped by completely closing the valve at the downstream end of the pipeline, the flow at the valve completely vanishes. Because of the compressibility of the fluid, however, the fluctuations of both the shock pressure and flow velocity in the pipeline continue. The shock pressure fluctuation appears in a similar form as previously demonstrated in Fig. 4.9b. In all real cases, all fluctuations are damped. According to Fig. 8.5, the boundary condition at the closed valve is simply

8.5 Pressure Head at the Closed Valve

203

hP-h0 t A

P +QA Closed Q=0

LA

Fig. 8.5 Scheme to determine the boundary condition at the closed valve at the downstream end of the pipe

Q P = 0.

(8.23)

From Eq. (8.5), and in view of the negligible viscous friction effect, one obtains the shock pressure head at the closed valve: (h − h 0 )P = (h − h 0 )A +

a QA. gA

(8.24)

The positive flow is defined in the direction towards the closed valve. The pressure head at the closed valve is simply determined by the pressure head and the flow rate, which are found at the cross-section A upstream of the valve, however, with a time delay equal to L A /a. In fact, Eq. (8.24) can also be obtained from Eqs. (8.7) and (8.8) by eliminating the wave parameter CB and setting Q P,0 = Q A,0 for the initially stationary flow. For comparison, the corresponding condition in using the wave tracking method is simply expressed as F = f + (a/g)c0 , which is called total reflection of shock waves at closed valves, see Eq. (7.96).

8.6 Traveling Laws of Shock Waves at Series Junctions of Pipes From Eqs. (8.3) and (8.4), it can be confirmed that for frictionless flows both variables (h − h 0 )+a/g(c −c0 ) and (h − h 0 )−a/g(c −c0 ) are wave parameters which remain constant while traveling along a pipeline of constant cross-sectional area. They are represented in Eqs. (8.9) and (8.10) as CA and CB , respectively. When they travel through a series junction of two pipes with different cross-sectional areas, they are subjected to both transmission and reflection. In order to work out related traveling performances of such waves, the computational algorithms will be applied, which

204

8 Method of Characteristics x

Fig. 8.6 Exchange of pressure waves at a discontinuous pipe section A1 c1

CA1

CA2

CB1

CB2

A2 c2

are comparable to those in the previous section (Sect. 7.4) for applying the wave tracking method. First, a pipeline connection with an abrupt change in the cross-sectional area is considered in Fig. 8.6. A stationary flow through such a discontinuous section is characterized by two effects, which are related, respectively, to the energy loss and the conversion between static and dynamic pressures. Compared with the shock pressures in transient flows, both effects can be neglected. This approximation especially applies to practical cases at which smooth series junctions of pipes are always preferred. For this reason, equal static pressure heads on both sides of the stepped crosssection can be assumed, i.e., h 1 ≈ h 2 and h 1,0 ≈ h 2,0 . From Eqs. (8.9) and (8.10), one first obtains the following relations a (c1 − c1,0 ), CA2 = (h 2 − h 2,0 ) + g a = (h 1 − h 1,0 ) − (c1 − c1,0 ), CB2 = (h 2 − h 2,0 ) − g

CA1 = (h 1 − h 1,0 ) + CB1

a (c2 − c2,0 ), g a (c2 − c2,0 ). g

(8.25) (8.26)

With the additional relations (c1 − c1,0 )A1 = (c2 − c2,0 )A2 and h 1 − h 1,0 ≈ h 2 − h 2,0 , one further obtains (CA1 − CB1 )A1 = (CA2 − CB2 )A2 ,

(8.27)

CA1 + CB1 = CA2 + CB2 .

(8.28)

and

These two equations, respectively, correspond to Eqs. (7.65) and (7.66) and can, thus, be denoted as the conservation laws of shock waves. They will be extended in the next section to the transient flows passing through a T-junction. According to Fig. 8.6, the approaching waves are CA1 and CB2 from both sides of the considered cross-section. The departure waves are denoted by CB1 and CA2 . Both can be computed from Eqs. (8.27) and (8.28). One obtains CB1 =

A1 − A2 2 A2 CA1 + CB2 , A1 + A2 A1 + A2

(8.29)

8.6 Traveling Laws of Shock Waves at Series Junctions of Pipes

CA2 =

2 A1 A2 − A1 CA1 + CB2 . A1 + A2 A1 + A2

205

(8.30)

These two equations have the same forms as Eqs. (7.67) and (7.68). Therefore, for the same reason, they are called the traveling laws of shock waves when passing through a series junction. Their physical meanings are related to the transmission and the reflection of shock waves at the section and can be found there as Eqs. (7.67) and (7.68). In the above considerations, the consequence of energy loss in the flow, while passing through a discontinuous section, has been ignored. The mechanism of this type of the energy loss is the turbulent vortex mixing and hence the dissipation in the flow downstream of the discontinuous section. In case of a sudden expansion of the pipe section, for instance, the related energy loss is computed from the Borda-Carnot formula according to Eq. (2.22). If necessary, like the treatment of viscous friction effects, the energy loss can be accounted for in the resistance constant R in Eqs. (8.7) and (8.8). One does not have to make any corrections in computations of CA and CB . In place of Eq. (8.11), the following resistance constant should be applied R = Rexpans +

λx . 2gd A2

(8.31)

In real pipeline systems, except for cases of using orifices or at the throttle section below the surge tank (Figs. 3.1 and 4.5), sudden expansions or contractions in pipe cross-sectional areas with large area ratio are quite rare. Thus, the energy loss arising from a series junction of pipes in all possible forms can be neglected in applied transient computations, without leading to any remarkable computational inaccuracy.

8.7 Traveling Laws of Shock Waves at T-Junction In almost all hydraulic systems of hydropower stations, T-junctions of pipelines have often been found in common applications. One confirms in Fig. 5.3, for instance, a T-junction at the node m. In fact, each surge tank connection forms a T-junction. The surge tank itself behaves as a side branch with a considerably large diameter. Figure 8.7 illustrates a general T-junction with given flow directions. The traveling performance of shock waves at such a T-junction needs to be clarified. The main computational steps are similar to those which have been presented in the last section dealing with shock waves at a series junction of pipes. Under the assumption of equal static pressures at all three branches, the conservation laws of shock waves in the flow at a T-junction can be derived, as given by CA1 + CB1 = CA2 + CB2 = CA3 + CB3 , and

(8.32)

206

8 Method of Characteristics

Fig. 8.7 Exchange of pressure waves at a T-Junction

A3 c3

CA3 CB3

A1 c1

CA1 CB1

CA2

A2 c2

CB2

(CA1 − CB1 )A1 = (CA2 − CB2 )A2 = (CA3 − CB3 )A3 .

(8.33)

Correspondingly, the traveling performance of waves at a T-junction can be derived. According to Fig. 8.7, the approaching and hence the known wave parameters are CA1 , CB2 and CB3 . The departure quantities to be determined are CB1 , CA2 and CA3 . From Eqs. (8.32) and (8.33), and by following similar computational procedures as in Sect. 8.6, one obtains the wave traveling laws at a T-junction as follows:

CB1 =

A1 − A2 − A3 2 A2 2 A3 CA1 + CB2 + CB3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(8.34)

CA2 =

2 A1 A2 − A1 − A3 2 A3 CA1 + CB2 + CB3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(8.35)

CA3 =

2 A1 2 A2 A3 − A1 − A2 CA1 + CB2 + CB3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(8.36)

In fact, these relations can also be obtained by directly taken over the traveling laws of shock waves from Eqs. (7.77) to (7.79) which are derived when applying the wave tracking method. Obviously, Eqs. (8.29) and (8.30), which are applicable to series junctions of pipes, are comprised in above equations. One only needs to set A3 = 0.

References Chaudhry, M. H. (2014). Applied hydraulic transients (3rd ed.). New York Inc: Springer. Giesecke, J., & Mosonyi, E. (2009), Wasserkraftanlagen, Vol. 5 (Auflage, Springer). Horlacher, H., & Lüdecke, H. (2012). Strömungsberechnung für Rohrsystem (Expert Verlag).

Chapter 9

Method of Direct Computations and Transient Conformity

For computing hydraulic transients in a hydraulic system, basically, two computational models or theories are available: rigid and elastic water column theories. In applications, they are also called rigid and elastic water column methods. Both theories (methods) are based on quite different hydro-mechanical principles. More precisely, the rigid water column theory cannot be simply considered as an approximation of the elastic water column theory. The former assumes incompressibility of the fluid and thus infinity of the wave speed (a = ∞) in the flow. Obviously, it is in no case an approximation, if compared with the real wave speed of about a = 1200–1400 m/s in the elastic water column theory. The computational background of the rigid water column theory is the use of the momentum equation, which is directly applied to the water column in a pipeline. Because of the assumption of incompressibility of the flow, each change of the flow state, for instance, caused by a regulation valve, will simultaneously lead to some changes of flow state in the entire hydraulic system. At a pipeline system including a T-junction, one must simultaneously solve three momentum equations which are related to all three branches of that T-junction. This often means great time consumption and the high complexity in computations. Therefore, the computational applicability of this method is mainly restricted to simple flow systems. Quite contrary, such a computational trouble does not occur when applying the elastic water column theory to any complex system. This is simply so because the flow at each given position in the system does not simultaneously depend on flows elsewhere. As already shown in Chap. 7 describing the wave tracking method, one only needs to track shock waves in the time series. Transient computations are, therefore, quasi one-dimensional. The method based on the elastic water column theory clearly represents an accurate and highly applicable method for transient computations. For short pipes, the method provides the same computational results as the method based on the rigid water column theory. This circumstance straightforwardly implies that an intrinsic relation between the elastic and the rigid water column theories must exist, although

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_9

207

208

9 Method of Direct Computations and Transient Conformity

both theories are based on different hydro-mechanical principles. This fact can be well demonstrated by considering a simple pipeline system, as will be outlined in this chapter. As an associated result, the method for directly computing the shock pressures at the regulation organ will be derived.1

9.1 Method of Direct Computations (MDC) According to Chap. 7, the wave tracking method is characterized by following two shock waves, F and f, which are computed from their values in previous time steps. Thus, both parameters apparently behave as two process parameters which are superimposed to one another during the entire computations. This feature of computations essentially differs from that based on the rigid water column theory. Although the wave tracking method represents quite a convenient method and does not provide computational complexity, it seems to be interesting to reveal and work out the supposed intrinsic relation between this method and the method based on the rigid water column theory. This consideration arises from the statement in Sect. 4.2.3, in view of Fig. 4.9, that the wave propagation speed does not influence the pressure response in the flow while being closed, see also Fig. 3.10. As a result of this fact, an explicit direct method without using two wave parameters (F and f ) seems to be possible and hence highly interesting. The existence of such a direct method will be demonstrated in this section in association with the demonstration of the intrinsic relation between the rigid and the elastic water column theory. The two shock waves, F and f, will be proved to be actually two state parameters. For this purpose as well as for the general case, viscous frictional flows are considered. A simple pipeline system is considered that merely consists of a lake, a pipeline of constant cross-sectional area, and a regulation organ, as shown in Fig. 9.1. For a pipeline with stepped pipes, an equivalent cross-sectional area according to Sect. 3.2.4 should be applied in computations. As the regulation organ, an injector nozzle of Pelton turbines is taken into account, whose characteristic is known according to Eq. (2.30). The equation for the determination of the pressure head at the injector nozzle has been derived in Sect. 7.3.2, see Eq. (7.24). With respect to (ϕ AD0 /A)2  1, the equation is simplified to h+

1 The

  a AD0  ϕ 2gh − h ∗0 + 2 f = 0, g A

(9.1)

author of this book would like to claim that this so-called “Method of Direct Computations” is novel for computing hydraulic transients in simple pipeline systems.

9.1 Method of Direct Computations (MDC)

209

1

F f

H0

Lake c F L, A

f 2

Fig. 9.1 Transient flows and propagations of pressure shock waves in a simple hydraulic system with an injector nozzle

in which the initial flow constant h ∗0 is computed from Eq. (7.25). The shock wave f in the above equation currently behaves as a process parameter, which, according to Chap. 7, has been tracked through all previous time steps. The objective of further computations is to replace the process parameter f by other state parameters like h and Q. For this computational purpose, Eq. (9.1) for the determination of the shock pressure at the injector section is differentiated with respect to time, as given by a AD0 dϕ  df AD0 1 dh dh + −2 = 0. 2gh + aϕ √ dt g A dt A dt 2gh dt

(9.2)

It is known from previous chapters that the wave tracking method is commonly based on the numerical solution in form of time series. This numerical method initially applies to Eq. (9.1) for the pressure head at the injector section. It, thus, is applicable to Eq. (9.2) as well. Usually, the time increment used in numerical computations is determined by the shortest spatial resolution L and is given by t = L/a. In the current case, according to Fig. 9.1, the flow state at the entrance of the pipeline (cross-section 1) is actually not in focus and thus does not need to be computed. The reflected shock wave f simply travels downstream back to the injector. This allows a maximum time step t = 2L/a to be used, which simply corresponds to the time for a round trip of a pressure shock wave. Thus, the finite time increment can be specified as t = ti − ti−1 =

2L = T2L , a

(9.3)

with L as the length of the pipeline between the upper lake and the injector section. It has been shown in Sect. 7.7 that the downstream wave f is the reflection of the upstream wave F by reaching the upper lake. In the first approximation, the entrance velocity effect according to Sect. 7.7 can be neglected. Then from Eq. (7.108), the downstream wave after a round trip and subject to the viscous friction effect is given by

210

9 Method of Direct Computations and Transient Conformity

f i = −Fi−1 − R(Q i−1 |Q i−1 | − Q 0 |Q 0 |).

(9.4)

The resistance constant R primarily considers the viscous friction effect along the pipeline. The time delay between two waves, f i and Fi−1 , is equal to the specified time interval t = T2L . On the other hand, there is Fi−1 = h i−1 − h 0 − f i−1 according to Eq. (7.10) for the shock pressure at the injector and the time ti - 1 . Thus, it follows from Eq. (9.4)   dFt=0 dQ 2t=0 dh t=0 d f t=0 dQ 2t=0 d f1 =− −R =− − −R . dt dt dt dt dt dt

(9.5)

Here, dh t=0 /dt simply means the changing rate of the pressure head at time t = 0, i.e. (dh/dt)t=0 . The same holds for the derivative dQ 2t=0 /dt. Moreover, it should be distinguished here between Q t=0 and Q 0 . The latter is a constant, while the former is the value of the variable Q at the time t = 0. For d f /dt at other time nodes, one immediately obtains the following series (exemplarily only for three time nodes):   dQ 2i−4 dh i−4 d f i−4 d f i−3 =− − −R , dt dt dt dt   dQ 2i−3 d f i−2 dh i−3 d f i−3 =− − −R , dt dt dt dt   dQ 2i−2 d f i−1 dh i−2 d f i−2 =− − −R . dt dt dt dt

(9.6) (9.7) (9.8)

For the flow rate, Q|Q| should be used in place of Q 2 , if negative flows are also expected. In the current case, the flow through the injector nozzle (Fig. 9.1), however, is always positive. Numerical computations start at t = 0 so chosen as the start of a transient process. Within the first time step (t ≤ t), there is f 1 = f 0 = 0 and further d f t=0 /dt = 0. The above equations are then combined one by one by substituting the respective d f /dt; one obtains     dQ 2i−2 dQ 2i−3 dQ 2t=0 d f i−1 dh i−2 dh i−3 dh t=0 =− + + ··· + −R + + ··· + . dt dt dt dt dt dt dt

(9.9) Both sides of this equation are multiplied by the time increment t. In view of the rule of derivatives, e.g., (dh i−2 /dt)t = h i−1 − h i−2 , then, the sum of all terms in the first parentheses on the r.h.s. of the equation simply represents the pressure head rise h i−1 − h 0 , with h 0 as the initial available pressure head in the flow. The

9.1 Method of Direct Computations (MDC)

211

  expression in the second parentheses depicts the difference R Q 2i−1 − Q 20 for the viscous friction effect, with Q 0 as the initial flow rate. Thus, one obtains   d f i−1 t = −(h i - 1 − h 0 ) − R Q 2i−1 − Q 20 . dt

(9.10)

The dynamic pressure head in each transient flow is usually very small and can thus be neglected. The initial pressure head in the flow at the injector nozzle is then computed as h 0 = H0 − R Q 20 . With this relation and by removing the subscript, the above equation is generally written as   df t = H0 − h + R Q 2 . dt

(9.11)

This equation implies that the computation of the downstream wave f as an apparent process parameter has been transferred through its derivative into the computation of two state parameters h and Q. For frictionless flows or by neglecting the viscous friction effect, the derivative d f /dt simply becomes a function of the static pressure head h (shock pressure) only. Equation (9.11) is inserted into Eq. (9.2). With the selection of a = 2L/T2L and t = T2L , both from Eq. (9.3), one obtains   √ 1 dh t, h = H0 − R Q 2 − B h − C + 2 dt

(9.12)

with B=L

AD0 L 2 AD0 dϕ ϕ and C = . √ g A dt A T2L 2gh

(9.13)

The stable flow state is given by B = 0 and dh/dt = 0, from which one obtains h = H0 − R Q 2 , as expected with the neglect of the dynamic pressure head. Both parameters, B and C, are equal to those in Eq. (4.54), when considering T2L = t in the current analysis. In addition, Eq. (9.12) is comparable with or approximately equal to Eq. (4.53), because the last term in Eq. (4.53) has been confirmed to be negligible within the period of flow regulation (Fig. 4.8) and so is the last term in Eq. (9.12). When closing the flow by shutting the injector nozzle, there is B < 0 because of dϕ/dt < 0. The derived equation is an analytical formula for computing shock pressures in the flow. It also explicitly takes into account the viscous friction effect. If compared with Eq. (9.1), one confirms that this analytical equation represents a computational method without use of the process wave parameter f and even without explicit use of the wave propagation speed. The method presented above is, thus, called here Method of Direct Computations (MDC) or shortly direct method. It technically differs from both the wave tracking method (WTM) and the method of characteristics

212

9 Method of Direct Computations and Transient Conformity

(MOC). As it is derived from WTM and thus from the elastic water column theory, it must fundamentally differ from the method based on the rigid water column theory (Sect. 3.2). In addition, as will be shown below, Eq. (9.12) is also able to determine high-frequency pressure fluctuations in transient flows. This is clearly a significant feature against Eq. (4.53), which was derived from the rigid water column theory. On the other hand, non-consideration of the compressibility of the fluid flow, i.e., the wave propagation speed evidently represents a common feature of two different water column theories. In this form, the existence of an intrinsic relation between two theories has been thus confirmed. The corresponding verification will be demonstrated below in Sect. 9.5.

9.1.1 Remarks on the Method of Direct Computations From Eq. (9.12), some special features of transient flows which are related to the computational method can be revealed as follows: 1 For continuous variations of both B and C, the shock pressure must also be a continuous function of the time h = f(t). When closing the injector nozzle, for instance, this continuous behavior exactly reflects the smoothening, i.e., the selfstabilization effect of the flow, as already outlined in Sect. 3.3.4, see Fig. 3.10 as well as Fig. 4.8. 2 At the moment of reaching the full-closing of the flow (Q = 0), for instance, a sudden change in the shock pressure takes always place. This circumstance is represented by a jump of the parameter B from a certain value towards B = 0. This plainly implies that the related pressure jump in the flow can be determined from Eq. (9.12), see Sect. 9.3 below. 3 In fact, Eq. (9.12) is also able to effectively identify significant high-frequency pressure fluctuations in the flow, for instance, after closing the regulation valve (Fig. 4.9), or in the first few seconds after opening the valve (Fig. 4.10). For more details, see Sect. 9.4 below.

9.1.2 Numerical Computational Algorithms For numerical computations, Eq. (9.12) needs to be written in finite difference form. The derivative dh/dt can be replaced by h/t with h = h i − h i−1 and t = T2L . The latter is required because of Eq. (9.3) and the definition of parameter C in Eq. (9.13). It then follows    1 h. h i−1 = H0 − R Q 2i−1 − B h i−1 − C + 2

(9.14)

9.1 Method of Direct Computations (MDC)

213

At this equation, basically, respective mean values of B and C in each finite time interval t = ti − ti - 1 should be applied. For simplicity, both Bi and Ci can be applied in accordance with the condition of quasi-stationary flow within each finite time interval t. With h i−1 = h i − h to replace the term on the left hand side of Eq. (9.14), one obtains    1 2 h. (9.15) h i = H0 − R Q i−1 − Bi h i−1 − Ci − 2 The last term in this equation represents the pressure fluctuation in the flow. As will be demonstrated below (Sect. 9.4), it is of dominant influence, e.g., after the injector nozzle, according to Fig. 9.1, has been completely closed. From Eq. (9.15) with h = h i − h i−1 , the pressure head at the time node ti is obtained as √ H0 − h i−1 − R Q 2i−1 − B h i−1 . (9.16) h i = h i−1 + C + 1/2 These last two equations represent the computational algorithm of the method of direct computations (MDC). For computing the discharge Q which is included in the friction term, Eq. (2.33) can be applied. The method of direct computations generally applies to flow regulations from one to another discharge. This especially includes both the closing and the opening operations, for instance, by closing and opening the injector nozzle. The related hydraulic characteristic ϕ = f(s) has been included in the parameters B and C. Corresponding computation examples will be shown in the next section.

9.2 Validation of the Direct Method and Computation Examples In order to demonstrate the reliability and applicability of the method, computations by employing the method of direct computations (MDC) should be compared with computations performed with the wave tracking method (WTM). This simply means that Eq. (9.16) should be compared with Eq. (9.1). For this purpose, the hydraulic pipeline according to Fig. 4.7 (viz. Fig. 9.1) is again taken into account. The pipeline system with the injector nozzle is assumed to be given, as already listed in Table 4.2. Both the closing and the opening of the injector nozzle should be considered, respectively. For simplicity, linear regulation of the needle stroke ds/dt = const is assumed. According to Eq. (2.30), this leads to a non-linear change of the discharge coefficient as follows

214

9 Method of Direct Computations and Transient Conformity

dϕ = dt



 a1 a2 s ds = f(s). +2 D0 D0 D0 dt

(9.17)

Because of a2 < 0, the maximum changing rate of the discharge coefficient is found at s = 0. Calculations using the wave tracking method according to Eq. (9.1) follow the same computational algorithms, as already depicted in Sect. 7.15, see Figs. 7.20 and 7.21.

9.2.1 Closing of the Injector Nozzle As first, the method of direct computations (MDC) will be applied to the case of closing the injector nozzle in Fig. 9.1. The corresponding computational algorithm when using Eq. (9.16) is outlined in Table 9.1. Because of the use of the time increment t = T2L , the computations are able to resolve the high-frequency flow fluctuations after closing the injector. The computational algorithm is comparable to that in Table 4.3 which, however, is not capable for determining that type of fluctuations. Table 9.1 Computational algorithm for the case of partially closing the injector nozzle within 30 s Injector nozzle

MDC: Eq. (9.16)

t (s)

s/D0

ϕ

dϕ/dt

B

C

h

Q

0.0

0.750

0.670

−0.005

−0.123

0.188

404.3

1.87

2.0

0.707

0.657

−0.007

−0.159

0.185

409.4

1.85

4.0

0.663

0.642

−0.009

−0.195

0.179

408.3

1.80

6.0

0.620

0.623

−0.010

−0.232

0.174

410.3

1.76

8.0

0.577

0.601

−0.012

−0.268

0.167

410.8

1.69

10.0

0.533

0.575

−0.013

−0.304

0.160

412.2

1.63

12.0 .. .

0.490 .. .

0.547 .. .

−0.015 .. .

−0.341 .. .

0.152 .. .

413.2 .. .

1.55 .. .

26.0

0.187

0.257

−0.026

−0.595

0.071

421.5

0.73

28.0

0.143

0.203

−0.028

−0.631

0.056

422.6

0.58

30.0

0.100

0.145

−0.030

−0.668

0.040

423.7

0.42

32.0

0.100

0.145

0.000

0.000

0.040

397.8

0.40

34.0

0.100

0.145

0.000

0.000

0.041

419.9

0.41

36.0

0.100

0.145

0.000

0.000

0.040

401.1

0.40

38.0

0.100

0.145

0.000

0.000

0.041

417.1

0.41

40.0 .. .

0.100 .. .

0.145 .. .

0.000

0.000

0.040 .. .

403.5 .. .

0.41 .. .

.. .

.. .

9.2 Validation of the Direct Method and Computation Examples

215

Fig. 9.2 Shock pressure at the injector while being partially closed (s0 = 150, send = 20); Comparison between the method of direct computations (MDC) and the wave tracking method (WTM)

Figure 9.2 shows comparison between computations employing the methods MDC and WTM, when applied to closing the injector nozzle from the initial needle stroke s0 = 150 to send = 20. The excellent agreement between the two computations, clearly, demonstrates the high accuracy as well as the high applicability of the method of direct computations. Even the pressure fluctuation in the flow after completing the flow regulation could be well resolved. Its period is equal to T4L = 2T2L = 4 s, as expected in the current example. Because of the use of the time increment t = T2L , the plotted pressure fluctuation is exactly of triangular wave form. For this reason, strictly speaking, both computations are approximations. The exact wave form basically depends on the pressure rise within the period of closing the flow and significantly differs from the regular triangular wave form, if the closing time is excessively short, see Fig. 7.19 in Sect. 7.14. Obviously, the maximum pressure rise in the flow is found at the end of the closing period, where, according to Eq. (9.17), the regulation parameter dϕ/dt reaches the maximum. After completing the partial closing of the flow, the pressure fluctuation tends to be significantly attenuated. Both the fluid-dynamic background and the quantitative prediction of such a form of flow attenuation will be outlined below in Sect. 9.4. When completely closing the injector nozzle, the corresponding comparison between computations by the two methods is shown in Fig. 9.3. The pressure fluctuation, after closing the injector nozzle, is characterized by an amplitude which is almost constant and corresponds to the pressure excess above H 0 at the end of closing the injector. The abrupt pressure change both at the beginning and the end of the closing period is related to the abrupt change of the parameter B in both Eqs. (9.12) and (9.16). At

216

9 Method of Direct Computations and Transient Conformity

Fig. 9.3 Shock pressure at the injector while being completely closed (s0 = 150, send = 0); Comparison between the method of direct computations (MDC) and the wave tracking method (WTM)

the end of the closing period, for instance, the parameter B abruptly changes from a certain value towards B = 0. As will be shown in Sect. 9.3 below, the method of direct computations is also able to directly compute such pressure jumps (h B and h E ). The fact to be mentioned here in advance is that within the period of closing the flow the computation according to Eq. (9.16) can further be considerably simplified; this will be done in Sect. 9.6. The applicability of the method of direct computations to transient flows caused by two-step closings of the injector nozzle and subjected to some special appearances will be shown in Sect. 9.7.

9.2.2 Opening of the Injector Nozzle For comparison purposes, Fig. 9.4 shows the computational results for the opening of the injector nozzle from s0 = 0 towards send = 150. As can again be confirmed in this case, excellent agreement between two computations has been achieved. At the very beginning of opening the injector nozzle, the abrupt pressure drop reads h B = 33 m. At the end of the opening period, a positive jump of the pressure head is detected. Such pressure jumps arise because of the same mechanism as in Figs. 9.2 and 9.3, namely due to the jump of the parameter B. In general, when opening the flow, for instance, by opening the injector nozzle, no any critical over-pressure in the flow will occur. All shock pressures are found below the initial static pressure head at t = 0.

9.2 Validation of the Direct Method and Computation Examples

217

Fig. 9.4 Shock pressure at the injector while being opened (s0 = 0, send = 150); Comparison between the method of direct computations (MDC) and the wave tracking method (WTM)

9.2.3 Stepped Pipeline The method of direct computations represents a very effective method for computing hydraulic transients in a simple pipeline of constant cross-sectional area, as shown in the foregoing two sub-sections. In practical applications, stepped pipes are often encountered. In Sect. 3.2.4 of applying the rigid water column theory, a stepped pipeline has been approximated to be a pipeline of constant cross-sectional area by using an equivalent pipe diameter. The same approximation can be approached in the current case when applying the method of direct computations to the stepped pipeline. For this reason, a stepped pipeline according to Fig. 9.5a is considered. The equivalent pipe diameter is computed from Eq. (3.16) to deq = 1.214 m. The total resistance constant of the flow is obtained by considering the series connection of three viscous resistances based on Eq. (2.12), leading to R = 0.747. For completeness, both the closing and the opening of the injector nozzle should be considered. The corresponding closing and opening laws (ds/dt = const and treg = 30 s) are supposed to be equal to those applied in Figs. 9.2 and 9.4. Respective computational results, obtained by the method of direct computations, are shown in panels (b) and (c) of Fig. 9.5 for the purpose of comparison with accurate computations by using the wave tracking method. For accurate reference computations, see Sect. 7.15.3 and Fig. 7.23 (for full closing). Except for the obvious difference between two computations of subsequent pressure fluctuations, the shock pressures in the flow have been well predicted by the simple direct method. In practical applications, such a prediction is often sufficiently accurate because, when closing the injector nozzle, only the maximum pressure rise in the flow is relevant for the operational safety of the pipeline system.

218

9 Method of Direct Computations and Transient Conformity

L=430, d=1.5

L=430, d=1.3

H0

(a)

L=430, d=1.0

(b)

(c)

Fig. 9.5 Shock pressure at the injector in a stepped pipeline; Comparison between the method of direct computations (MDC) and the wave tracking method (WTM)

9.3 Pressure Jumps at the Beginning and End of Each Flow Regulation From Figs. 9.2 and 9.4, pressure jumps at both the beginning and the end of each flow regulation have always been observed. They can be well confirmed by considering Eq. (9.15). At the instant of ending the flow regulation (t = treg ), e.g., the parameter

9.3 Pressure Jumps at the Beginning and End of Each Flow Regulation

219

B in Eq. (9.15) abruptly changes from a certain value, which is determined by dϕ/dt, towards B = 0. The opposite jump of parameter B is found at the beginning of each flow regulation. Such pressure jumps will now be directly computed.

9.3.1 Pressure Jumps at the Beginning of Flow Regulations When considering the starting phase of a flow regulation, one obtains from Eq. (9.15) with B = 0 and h = 0 at t < 0 as well as with B > 0 at t ≥ 0, respectively, h 0 = H0 − R Q 20 , h 1 = H0 −

R Q 20

− B1



  1 h 0 − C1 − (h 1 − h 0 ). 2

(9.18) (9.19)

The second equation refers to the first time node for t > 0. Combining these two equations by eliminating the pressure head H0 , it follows with h B = h 1 − h 0 as the expected pressure jump at the beginning of a flow regulation (closing or opening) h B = −

 B1 h0. C1 + 1/2

(9.20)

This equation allows the pressure jump within the first time increment at the beginning of each flow regulation to be directly determined. One only needs to use the initial values of B and C. As a matter of fact, the computation provides the same result as from Eq. (9.1) with f = 0. By opening the flow from Q 0 = 0, i.e., ϕ0 = 0, one immediately obtains from Eq. (9.20) with C = 0  dϕ 2L AD0  2gh 0 . h B = −2B1 h 0 = − g A dt

(9.21)

The pressure jump is directly proportional to the opening speed of the injector nozzle (dϕ/dt). Because of Eq. (2.33), the above equation can also be written as h B = −

2L dc . g dt

(9.22)

With t = T2L as in Eq. (9.3) and thus 2L/t = a, this last equation agrees well with the Joukowsky equation, see Eq. (3.21). When linearly opening the injector nozzle (ds/dt = const) from the closed state (s = 0), one has C1 ≈ 0, and B1 is a maximum because dϕ/dt is a maximum at the

220

9 Method of Direct Computations and Transient Conformity

beginning of the injector opening (Fig. 2.7). Therefore, an abrupt significant pressure drop has been detected in Fig. 9.4. By the same principle, a small pressure rise at the beginning of closing the injector nozzle (Figs. 9.2 and 9.3) can be explained. As stated in Sect. 9.1.2 in connection with Eq. (9.14), the respective mean values of B and C in each finite time interval should be applied. This is also so for Eq. (9.20). In the engineering viewpoint and for simplicity of applications, both B1 and C1 in the above equation can be replaced by B0 and C0 , respectively. This is generally based on the condition of quasi-stationary flows within a finite time interval, without having to be concerned with higher accuracy. In the current computational example according to Fig. 9.4, it leads to h B = −35.1 m by using the lower values B0 and C0 , while h B = −31.8 m is obtained by using the upper values B1 and C1 . At least in this example, the difference between the two computations is absolutely irrelevant.

9.3.2 Pressure Jumps at the End of Flow Regulations With the same principles, the pressure jumps at the end of each flow regulation can be well explained. From Eq. (9.15) with Bi = BE = 0 at ti = treg and BE+1 = 0 at ti+1 , respectively, one obtains    1 h E = H0 − R Q 2E−1 − BE h E−1 − CE − (h E − h E−1 ), 2   1 h E+1 = H0 − R Q 2E − CE+1 − (h E+1 − h E ). 2

(9.23) (9.24)

These two equations are combined for eliminating the pressure head H0 . With Q E−1 ≈ Q E and CE+1 ≈ CE as well as with h E = h E+1 − h E as the expected pressure head jump at the end of each flow regulation, one obtains      1 1 CE + h E = BE h E−1 + CE − (h E − h E−1 ). 2 2

(9.25)

In this equation, the last term refers to the last time step before the end of flow regulation. It can thus be considered to be related to an infinitesimal time step. Correspondingly, it represents an infinitesimal change in the pressure head and can thus be neglected. The pressure head h E−1 is then considered to be available at the moment of ending the flow regulation. As a result, Eq. (9.25) becomes h E =

 BE hE. CE + 1/2

(9.26)

9.3 Pressure Jumps at the Beginning and End of Each Flow Regulation

221

This formula generally represents the pressure jump in the flow at the end of each flow regulation. It refers to the flow at the section of regulation valves. In the case of closing the flow (BE < 0), the formula determines an abrupt and significant pressure drop (Figs. 9.2 and 9.3). Otherwise, when opening the valve for the flow (BE > 0), it determines an abrupt but relatively small pressure rise (Fig. 9.4). Such a small pressure rise occurs simply because of the small value of B and the large value of C at the end of each opening of the injector. This is confirmed in Fig. 2.7 in view of the discharge coefficient ϕ for parameter C and the gradient dϕ/dt = (dϕ/ds) · ds/dt for parameter B, respectively for large values of s/D0 . By computations it has been found h E = −25.5 m in Fig. 9.2 and h E = −36.5 m in Fig. 9.3, respectively. The latter corresponds to the case of C = 0 in Eq. (9.26). In Fig. 9.4, a small value of h E = 3.5 m has been computed. Concerning the first pressure jump, the first approaching minimum/maximum of the pressure head is obtained from Eq. (9.24) with the rise of Eq. (9.26) and CE+1 ≈ CE as   CE − 1/2  BE h E . h E+1 = H0 − R Q 2E − CE + 1/2

(9.27)

It represents the beginning of pressure fluctuations in the flow. For the illustrated case of Fig. 9.2, the minimum pressure head after the pressure jump is computed from the above equation to be h E+1 = 398.0 m. In Fig. 9.4, the corresponding maximum pressure head is found at h E+1 = 405.9 m. At full-closing of the flow, one obtains the simplification with C = 0 and dc/dt < 0 h E+1 = H0 + BE



  L AD0 dϕ  L dc h E = H0 + 2gh E = H0 + . g A dt g dt E

(9.28)

Correspondingly, it follows from Eq. (9.26)    2L dc h E = 2BE h E = . g dt E

(9.29)

In view of Eq. (9.28), it is a mirrored pressure jump from H0 − h E /2 above to H0 + h E /2 below the mean pressure head H 0 , with h E < 0. As can be confirmed at Eq. (9.26), except for a minus sign, the equation takes the same form as Eq. (9.20). The same reverse expression is given by Eq. (9.29) and Eq. (9.22). Obviously, it is about an opposite phenomenon of hydraulic transients between the start and the end of each flow regulation.

222

9 Method of Direct Computations and Transient Conformity

9.4 Pressure Fluctuations 9.4.1 Pressure Fluctuations After Flow Regulations The ending of a flow regulation is specified by B = 0 which arises from dϕ/dt = 0, see Eq. (9.13). The pressure fluctuation in the flow afterwards is then determined by the last term in Eq. (9.15). The parameter C represents the remaining constant opening (ϕ = const) of the injector nozzle and hence satisfies C ≈ const (see Table 9.1 for t > 30 s). From Eq. (9.16) and with Ci+1 ≈ Ci , one obtains h i+1 =

H0 − R Q 2i Ci − 0.5 + hi. Ci + 0.5 Ci + 0.5

(9.30)

This equation applies to cases of both opening and closing the injector nozzle. It is actually automatically included in Eq. (9.16) while performing numerical computations. From this equation, one confirms two performances of observed pressure fluctuations. First, the pressure fluctuation, as computed and shown in Figs. 9.2, 9.3 and 9.4, takes place around a constant value, which is approached after a long time and determined from Eq. (9.30) by setting h i+1 = h i = h 0 . One obtains h 0 = H0 − R Q 2 .

(9.31)

This value has been expected. It is equal to that in the stationary flow without considering the dynamic pressure head, which always behaves as a tiny value and has been neglected while deriving Eq. (9.11) at the very beginning of the current chapter. Second, with respect to Eq. (9.31), one rewrites Eq. (9.30) to h i+1 − h 0 =

Ci − 0.5 (h i − h 0 ). Ci + 0.5

(9.32)

Because of C < 0.5 (see Table 9.1), a positive deviation h i − h 0 > 0 will always be followed by a negative deviation h i+1 − h 0 < 0 and vice versa. This feature of Eq. (9.32) mathematically represents a consecutive change of h i+1 and thus its periodic fluctuation. Because of |h i+1 − h 0 | ≤ |h i − h 0 |, this is generally an attenuated pressure head fluctuation, as illustrated in Fig. 9.6. The attenuation effect is quantitatively determined by the non-vanishing parameter C = 0 and thus by nonvanishing flow rate, see Table 9.1 for computations. The fluid dynamic background of such an attenuation effect is the self-stabilization of transient flows through a regulation valve, as already described in Sect. 3.3.4. The viscous friction effect only plays a negligible role.

Fig. 9.6 Pressure jump at the end of a flow regulation and the pressure fluctuation after that (redrawn of Fig. 9.2)

223 h (m)

9.4 Pressure Fluctuations

hE

hi

hi+2

h0 hi+1

hi+3

t (s)

The pressure fluctuation of constant amplitude is given by C = 0 which corresponds to a full-closing of the flow. It follows from the above equation with h 0 = H0 immediately h i+1 − H0 = −(h i − H0 ).

(9.33)

Such a pressure fluctuation of constant amplitude and around the mean value h = H0 has already been observed in Fig. 9.3. It should be mentioned that the condition C = 0 is only restricted to the flow close to the regulation organ, i.e., the injector nozzle. The flow in the upstream pipeline, actually, oscillates with nonvanishing velocities. Because these velocities are very small and close to zero, the viscous effect on damping the flow oscillation is negligible. The main damping effect is found at the entrance of the pipeline at the lake (Fig. 9.1), where each reverse flow with c < 0 leads to dissipation of the related kinetic energy which can never be recovered again. Both Eqs. (9.30) and (9.33) represent pressure fluctuations in the form of regular triangular waves with a constant period of T4L = 2T2L , as explained in Sect. 9.2 and confirmed in Fig. 9.2. The precondition for obtaining this period is the use of the time interval t = T2L in numerical solutions, as predetermined by Eq. (9.3).

9.4.2 Pressure Fluctuations During Flow Regulations The start of each flow regulation is specified by a jump of the parameter B from B = 0 to a value B = 0 which is related, for instance, to the injector opening with dϕ/dt > 0. An example of computations has already been shown in Fig. 9.4. The pressure fluctuation in the flow was directly computed from Eq. (9.16) with non-vanishing parameters B and C. At Eq. (9.15) it was pointed out in advance, that the last term in the equation represents the pressure fluctuation in the flow. This explicitly implies that the sum of the other three terms on the r.h.s. of the equation represents the time-dependent mean value h¯ i , around which the pressure fluctuation takes place. The fluctuation term can then be expressed as

224

9 Method of Direct Computations and Transient Conformity

  1 h i = − Ci − (h i − h i−1 ). 2

(9.34)

Because the time-dependent mean value of the pressures is a slow process against the pressure fluctuation, the expression h i − h i−1 in the above equation can be approximated as h i − h i−1 . From Eq. (9.34), one then obtains h i =

Ci − 0.5

h . Ci + 0.5 i−1

(9.35)

It takes the same form as in Eq. (9.32). In the current case, the centered mean pressure h¯ i of the pressure fluctuations changes with time. Finally, Eq. (9.15) is written as Ci − 0.5

h , h i = h¯ i + Ci + 0.5 i−1

(9.36)

 h¯ i = H0 − R Q 2i−1 − Bi h i−1 .

(9.37)

with

Fig. 9.7 Pressure jump and fluctuations during the period of a flow regulation (redrawn of Fig. 9.4)

h (m)

Referring to Fig. 9.4, the computed pressure fluctuations together with the timedependent mean pressure rise during the period of opening the injector nozzle is again shown here in Fig. 9.7. The parameter C has values only of order Ci < 0.2 (see Table 9.1 for C of about the same order of magnitude by closing the injector nozzle). It is especially small, because at the very beginning of opening the injector nozzle, there is Q ≈ 0 and thus C ≈ 0. This means, first, that the pressure h i in

fluctuation







Eq. (9.35) will continuously change its sign. Second, there is h i < h i−1 for C = 0. The pressure fluctuation thus will be attenuated since C = 0. The background of such an attenuation effect is again the self-stabilization effect of the flow through a valve, see Sect. 3.3.4. The viscous friction effect only plays a negligible role.

H0

h2

h4

h6 h7 h5

h3 h1 t (s)

9.5 Transient Conformity of Rigid and Elastic Fluid Flows

225

9.5 Transient Conformity of Rigid and Elastic Fluid Flows From the elastic water column theory, the method of direct computations has been derived; the result is given by Eq. (9.12). Because of the use of the time T2L in parameter C and the time interval t = T2L , Eq. (9.12) is actually a finite element equation. Its numerical solution basically requires the assumption of quasi-stationary flows within each finite time interval t. This principle has already been applied while obtaining Eq. (9.14) and deriving Eq. (9.15). For accurate computations by using t = ti − ti−1 , all flow parameters in Eq. (9.12) should be related to respective mean values within the time interval, especially for the most sensitive variable h on the left hand side of Eq. (9.12). For this reason, Eq. (9.12) is explicitly written as    1 h i + h i−1 = H0 − R Q 2i−1 − Bi h i − Ci + (h i − h i−1 ). 2 2

(9.38)

The pressure head h i is then resolved as  h i = H0 − R Q 2i−1 − Bi h i − Ci h.

(9.39)

This equation exactly agrees with Eq. (4.53) which has been derived in Sect. 4.2.2 based on the rigid water column theory. This agreement obviously reveals an intrinsic relation between rigid and elastic fluid flows. This relation is called Transient Conformity. It exactly explains the background of excellent agreement between computations based on the two different models, as already shown in Fig. 4.9 within the period of closing the injector nozzle up to tc = 30 s. The corresponding computational algorithm was given in Table 4.4; alternatively, see Table 4.3. Against Eq. (9.15), however, Eq. (9.39) is unable to resolve pressure fluctuations with a period of T4L , neither at the beginning of flow regulation nor after the regulation (Figs. 9.2, 9.3 and 9.4). This property of Eq. (9.39), i.e., Equation (4.53) has already been revealed and is documented in Figs. 4.8 and 4.9. If the pressure fluctuations are in focus, one has to apply either the wave tracking method (WTM) for flows in all complex hydraulic networks or the method of direct computations (MDC) according to Sect. 9.1 for flows in simple pipelines. For certain applications, both Eqs. (9.39) and (9.15) can be further simplified, as shown below.

9.6 Simplifications of Computations In connection with Eq. (9.15), it was indicated that the last term in the equation represents the pressure fluctuations in the flow. Corresponding computations have been given by Eqs. (9.35) and (9.32) for transient flows, respectively, during and

226

9 Method of Direct Computations and Transient Conformity

after a flow regulation. Observations of related pressure fluctuations in those flows can be found in respective figures in the foregoing sections. When considering the monotonous pressure rise in Figs. 9.2 and 9.3, for instance, within the period of flow regulations (t ≤ treg ), it can be shown that the last term in Eq. (9.15) is negligibly small against the sum of all other terms on the r.h.s. of the equation. As a matter of fact, the neglect of the last term in Eq. (9.15) is based on the same reason as the neglect of the last term in Eq. (9.39). The latter has already been described in Sect. 4.2.2 and proved to be reliable (Fig. 4.8). With the neglect of the last term in both Eq. (9.15) and Eq. (9.39), computations √ h i−1 is of can be simplified. Against the pressure head h i in Eq. (9.39), √ √ the term a lower order of magnitude and thus the approximation h i−1 ≈ h i is valid for quasi-stationary flow. With this approximation, Eqs. (9.15) and (9.39) become equal. One obtains  (9.40) h i = H0 − R Q 2i−1 − Bi h i , or simply √   h + B h − H0 − R Q 2 = 0.

(9.41)

Like Eq. (9.12), this equation again represents an analytical formula for directly computing the pressure response at the injector nozzle during the flow regulation. The regulation law is included in the parameter B in form of B = f(t) with given s = f(t) and thus ϕ = f(t). Because the time increment t = T2L is neither included in the equation, nor in all derived equations below, it can be arbitrarily refined for accurate numerical computations. For this reason, Eq. (9.41) is identical to Eq. (4.56). When compared with Eq. (9.37) for an application example, one confirms that Eq. (9.41) simply represents the time-dependent mean pressure rise like that in Fig. 9.7 for the case of opening the flow. √ The above equation represents a quadratic polynomial for the unknown parameter h. Correspondingly, the shock pressure at the injector nozzle is straightforwardly computed as 

 2  1 2 2 −B + B + 4 H0 − R Q h= . 4

(9.42)

The functional dependence h = f(t) is usually represented in tabular form. For the flow rate Q in the above equation, the value Q i−1 in the last time node can be used. The flow rate itself is computed from the characteristic of the injector nozzle, e.g., by Eq. (2.33). Figure 9.8 shows, for the case of closing the valve, comparison computations between Eqs. (9.42) and (9.15). It is about the same hydraulic configuration and the same closing process of the injector nozzle as applied in Fig. 9.2. Obviously, the

9.6 Simplifications of Computations

227

Fig. 9.8 Comparison between computations with and without the fluctuation term in Eq. (9.15) for the case of closing the flow

fluctuation term in Eq. (9.15) can be well neglected, if only the maximum pressure rise in the flow by closing the regulation organ is in focus. One then simply uses Eqs. (9.41) or (9.42). For the case of opening the flow, Eq. (9.42), as stated before, represents the time-dependent mean pressure rise. Comparison computations have been shown in Fig. 9.9. Different from Eq. (9.15) for accurate computations, Eq. (9.42) is conditionally applicable. It, however, can be applied, if only the maximum pressure rise in the flow is of interest. For frictionless flows or by neglecting the viscous friction effect, it follows from the above equation straightforwardly

Fig. 9.9 Comparison between computations with and without the fluctuation term in Eq. (9.15) for the case of opening the flow

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9 Method of Direct Computations and Transient Conformity

h=

2  1 −B + B 2 + 4H0 . 4

(9.43)

For the case of closing the valve, the computation is valid up to the end of the closing period, as demanded by the application condition B = 0. For the case of opening the flow, the computation provides a satisfactory approach to the timedependent mean pressure rise like that in Fig. 9.7. Furthermore, the computation of Eq. (9.41) can also be performed in the following direct way:    h i = H0 − R Q 2i−1 − Bi h i−1 .

(9.44)

This computational algorithm has already been given in Eq. (4.57) when representing the “approximation” method of using the rigid water column theory. It is now clear that that equation is not about the approximation or inaccuracy of the used theory but about the approximation or simplification of the employed computational algorithms. Equation (9.43) is further considered. In a hydraulic system with high pressure head H 0 , one usually has B 2  H0 with B being less than unity in magnitude, see Table 9.1 for instance. Then, Eq. (9.43) is further simplified as  h − H0 ≈ −B H0 .

(9.45)

√ √ If this is compared with Eq. (9.41), one confirms that h is simply replaced by H0 . Therefore, only this last equation has to be considered as a real approximation. Inserting the parameter B from Eq. (9.13) and accounting Eq. (2.33) yield h − H0 ≈ −

L dc . g dt

(9.46)

This is the equation which has often been encountered in the rigid water column theory, see e.g., Equations (3.6) and (3.57). The fact to be mentioned is that this last equation has been known to be applicable to “short pipes”. This means, in the case of opening flows according to Fig. 9.9 for instance, that it is applicable for times t > 6T2L . For closing flows, it is restricted to t > T2L . Corresponding statements can be found in Sect. 3.2.1 regarding application restrictions of the rigid water column theory.

9.7 Application to the Two-Step Closing of an Injector Nozzle In Sect. 9.2.1, the case of injector closing in one-step, i.e., ds/dt = const, has been considered. The MDC can actually also be applied to the case of two-step closing of the injector. Such a closing concept has often been found in large Pelton turbines,

9.7 Application to the Two-Step Closing of an Injector Nozzle

229

Fig. 9.10 One- and two-step closing of the injector nozzle under the condition of equal changing rate of the discharge coefficient at t 1 = 20 s (two-step) and t = 30 (one-step)

where the maximum pressure rise in the flow system has to be limited. For the pipeline system which has been considered in Sect. 9.2.1 with T2L = 2 s for a round trip of waves, a two-step closing of the injector nozzle has been defined, as shown in Fig. 9.10. with a total closing time equal to 30 s. The first step of the closing process is so specified that at its end (t = 20 s) the changing rate of the discharge coefficient is equal to that at the end of one-step closing of the injector nozzle. The purpose of computations is to reveal different phenomena which are related to the two-step closing of the injector and to show the applicability of MDC. These include the explicit explanations of the viscous friction effect and the overlap effect of shock waves.

9.7.1 Comparison Between MDC and WTM For the closing of the injector nozzle in two steps, as specified in Fig. 9.10, transient computations have been carried out by both the wave tracking method (WTM) and the method of direct computations (MDC). Because of T2L = 2 s, the time increment for MDC computations has been set to be t = 2 s. For WTM computations, a time increment of t = 0.5 s has been used. Figure 9.11 shows the comparison between two computations. As can be confirmed, the MDC is highly accurate even for the two-step closing of the injector nozzle. The high-frequency pressure fluctuations begin at the end of the first closing step as a result of the jump of the parameter B because of the jump of the changing rate of the discharge coefficient.

230

9 Method of Direct Computations and Transient Conformity

Fig. 9.11 Shock pressure response in the flow by two-step closing of the injector; Comparison between computations, respectively, by the wave tracking method (WTM) and the method of direct computations (MDC)

For the hydraulic effect of the two-step closing of the injector nozzle against the one-step closing, the readers are kindly referred to Sect. 10.2.3 for general concepts or below in Sect. 9.7.4 for brief information.

9.7.2 Time Increment Effect and Computational Bias The method of direct computations requires the use of the time increment t to be equal to T2L . This means, for T2L = 2 s in the current case, that the time nods in the time series have to be an integer multiple of 2. This requirement may cause some computation inaccuracies. If, for instance, the first step of a two-step closing process of the injector ends at t = 19 s rather than at t = 20 s, then a shift of the time series of computed pressure fluctuations after the first step of closing will take place. Figure 9.12 demonstrates such a shift which can be considered as a computational bias. For evaluating the hydraulic performance of the pipeline system, such a computational bias, however, has practically no meaning and can be ignored. If compared with Fig. 9.11, however, one confirms that the amplitude of the pressure fluctuations after the full-closing of the injector has changed a lot. This is simply because of the different overlap of the first pressure fluctuation after t 1 = 20 and 19 s, respectively, with the second pressure fluctuation after t = 30 s. Such a real flow phenomenon, which also depends on the used regulation organ and the regulation dynamics, is commonly automatically accounted for while performing computations.

9.7 Application to the Two-Step Closing of an Injector Nozzle

231

Fig. 9.12 Time increment effect and computational bias

9.7.3 Explicit Explanation of the Viscous Friction Effect Conceptually, the difference between one- and two-step closings of an injector in a pipeline system is a hydraulic nature instead of the consequence of computations. However, some related hydraulic phenomena can be well explained from transient computations. To this end, the method of direct computations (MDC), because of its explicit form given by Eq. (9.15), behaves as more applicable than the wave tracking method (WTM). For demonstration purposes, the WTM is first applied to the one- and the two-step closings of the injector nozzle according to the specification in Fig. 9.10. The computation results are shown in Fig. 9.13. Although the same changing rate dϕ/dt = −0.0384 at t 1 = 20 s (for two-step closing) and t = 30 (for one-step

Fig. 9.13 Difference in the pressure rise at t 1 = 20 s (for two-step closing) and at t 1 = 30 s (for one-step closing) despite equal changing rate of the discharge coefficient

232

9 Method of Direct Computations and Transient Conformity

closing) is assumed, different pressure rises have been detected. This difference and its origin could not be simply explained by the applied computational method, i.e., WTM. If, however, Eq. (9.15) is considered, then, it is clear that at the time t 1 = 20 s (end of the first step of the two-step closing), the viscous friction term R Q 2 is still active because of Q = 0. At the end of one-step closing (t = 30), the friction term in the equation disappears because of Q = 0. The reachable pressure rise is then proportional to the parameter B and thus to the changing rate dϕ/dt. The last term in the equation does not have any influence on the continuous growth of pressures. This example significantly demonstrates that the method of direct computations can be well used to explain different hydraulic phenomena in hydraulic transients. Its other applicabilities like those to explain both the pressure head jumps at the end of each flow regulation and the attenuations of flow fluctuations can be found in the foregoing sections. Incidentally, it has been confirmed that the two-step closing process of the injector nozzle is effective in suppressing the pressure rise and pressure fluctuations in the flow system. More about this aspect can be found in Sects. 10.2.3 and 10.2.4.

9.7.4 Explicit Explanation of Overlapping Pressure Shock Waves In Sect. 9.3, it was demonstrated that each abrupt change in the discharge coefficient leads to a jump of the shock pressure in the flow. This occurs both at the beginning and the ending of each flow regulation. As a consequence, the shock pressure after its jump begins to fluctuate. For a two-step flow regulation, the first pressure jump and the related shock wave will be overlapped by the second pressure jump, as already shown in Fig. 9.11. Basically, the overlap of the two shock waves may occur in a superimposed or a suppressed form. They lead to enlarged wave fluctuations and a reduction of the fluctuation amplitude, respectively. The decisive factor is the time delay between the two shock waves, i.e., the two pressure jumps. This can be well explained based on the knowledge in Sect. 9.3. According to Fig. 9.14a which shows an application example, the first downward jump of the pressure occurs at t 1 = 20 s. The subsequent pressure fluctuation has a period of T 4L which is equal to 4 s in this example. After 10 s, which is a non-integer multiple of T 4L , the pressure tends to jump upwards. Because the second downward jump of the pressure just occurs at the moment (end of closing time), the overlap of two pressure jumps is a type of “wave suppression”. The suppressed overlap of shock waves thus leads to a decreased amplitude of wave fluctuations. In contrast, a superimposed overlap of shock waves will be given, if the time between two pressure jumps is an integer multiple of T 4L . In Fig. 9.14b, the second pressure jump is set to occur at t 2 = 32 s. At this time, the pressure in the first shock wave tends to jump downwards. Because of the coincidence of two downward pressure jumps, their overlap shows an enlarged amplitude.

9.7 Application to the Two-Step Closing of an Injector Nozzle

233

Fig. 9.14 Overlap effect of shock waves in a two-step regulation. a Suppressed overlap of pressure waves. b Superimposed overlap of pressure waves

In fact, both cases shown in Fig. 9.14 have a common background in mechanics. The second pressure jump by reaching Q = 0 (complete closing of the flow) always occurs from the current value of pressure to its mirrored value below the mean pressure head H 0 , as described in Sect. 9.3.2, see Eq. (9.29). For the reason of the second case in Fig. 9.14b, if possible, the time interval t 2 − t 1 between two jumps of pressures should be set to be a non-integer multiple of T 4L , in order to prevent the superimposed overlap of shock waves.

9.8 Remarks to the Method of Direct Computations The method of direct computations is derived from the wave tracking method and is thus a solution of the elastic water column theory. It is therefore accurate and also able to resolve pressure fluctuations both during and after flow regulations. Although it is only applicable to relatively simple pipeline systems, the method evidently revealed

234

9 Method of Direct Computations and Transient Conformity

the intrinsic relation between the rigid and the elastic water column theories, which is called Transient Conformity. It has been again demonstrated that the wave propagation speed does not affect any critical shock pressure rise in transient flows. The wave speed is only used to determine the time T2L which is found in the parameter C for flow fluctuations, but is not included in the parameter B for maximum pressure rise in the flow. It, thus, only determines the period and, more or less, the attenuation of shock pressure fluctuations in a given hydraulic system. In addition, the Joukowsky equation generally lost its significance, because it basically only applies to a short time t < T2L at the very beginning of flow regulations. As a significant hydraulic feature, it has also been clearly demonstrated that the pressure rise in a transient flow is mainly determined by the parameter B and hence by the dynamical procedure of flow regulations. In case of using injector nozzles, it is determined by the changing rate of the discharge coefficient dϕ/dt and therefore by the changing rate of the needle stroke ds/dt. From this viewpoint, an optimized regulation rule will be presented in Sect. 10.2.4. The method of direct computations with its explicit expression is able to exactly explain diverse pressure jumps in the flow, attenuation of pressure fluctuations after each flow regulation, viscous friction effect, overlap of shock pressure waves and many other hydraulic phenomena. After fully closing the flow, for instance, the pressure fluctuation appears to be highly dominant against that after partially closing the flow. The method of direct computations can be well applied to transient computations in the case of closing the injector nozzle in two linear steps. It is certainly equally applicable to any other two-step flow regulations.

Chapter 10

Hydraulic Characteristics of Pumps and Turbines

Almost all hydraulic systems in hydropower stations are constructed for turbine and pump operations. Except for Pelton turbines, at which only the injector nozzles are included in the hydraulic system, all other types of turbines and pumps are found within the hydraulic network and the pipeline system. This fact determines that to compute each hydraulic transient the hydraulic characteristics of the respective turbines and pumps must be taken into account. In case of emergency shutdowns of turbines or pumps, for instance, the extended characteristics of these machines with changeable rotational speed must be known. Furthermore, the moment of inertia of the entire rotor system plays an essential role in affecting the changing rate of rotor rotations. For this reason, in this chapter, the characteristics of different types of turbines and pumps will be described. The characteristics should include both the variation of the rotational speed and the reverse flow; the latter is particularly necessary for pump flows. Thus, the characteristics in form of four-quadrant diagrams are often used. For reasons of application, respective four-quadrant diagrams usually need to be approximated by appropriate quadratic polynomials.

10.1 Hydraulic Characteristics of the Pump Pump characteristics have frequently been given for constant rotational speeds and stationary operations. They are primarily represented by the created pressure head, the shaft power, the hydraulic efficiency, and the cavitation performance as functions of the volume flow rate. For computing hydraulic transients in a system with pumps, however, only the head and the shaft power need to be accounted for. The related characteristics are commonly represented in the general form of Hpu = f(Q, n) and Pshaft = f(Q, n), where the rotational speed n is included as an independent

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_10

235

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10 Hydraulic Characteristics of Pumps and Turbines

parameter. This is necessary because it changes when the flow through the pump is regulated by the rotational speed through a static frequency converter (Merino and Lopez 1996; Terens and Schäfer 1993; Käling and Schütte 1994) or when the pump is started or shut down. The pump characteristics with variable rotational speed have been treated commonly in two different manners. First, the coefficients of the discharge and the head in dimensionless form have often been used to characterize the stable operation of the pump. Second, the so-called four-quadrant diagrams of a pump have been applied to represent the pump characteristics which also cover the undesired conditions with reverse flow and inverse rotation. In fact, this second form of pump characteristics is often used in computations of hydraulic transients which, for instance, arise from emergency shutdown of the pump. In the following two sub-sections, both forms of pump characteristics will be outlined.

10.1.1 Characteristics in Terms of Coefficients of Discharge and Head The pump is considered, as desired, to operate in the pumping mode. This means that both the rotation and the flow direction are positive. The rotational speed, however, behaves as a variable. This corresponds to the flow regulation by varying the rotational speed with the frequency converter technique. The complete hydraulic characteristics of a given pump with variable rotational speed are described in terms of coefficients of discharge, head, and shaft power, which are defined as ϕ=

Q Q , = 2 2 d2D u2 d2D (nπ d2D )

(10.1)

2g Hpu 2g Hpu = , u 22 (nπ d2D )2

(10.2)

2Pshaft 2Pshaft = . 2 3 5 ρd2D u 2 ρπ 3 n 3 d2D

(10.3)

ψ= λ=

Here, the impeller diameter is denoted by d2D (Fig. 10.1), with the subscript 2 for the impeller exit and D for the impeller as a rotating disc. This notation is meaningful because another notation d2 has often been used for the pipe diameter at the pump exit or at the exit of unified pump and spherical valve, see Sect. 10.1.3 below. In the above definitions, the pump-impeller diameter d2D has been used as reference diameter. The reference speed is the related circumferential speed u 2 = π d2D n. Further, the pump efficiency is simply computed as ηpu =

ρg Hpu Q 1 = ϕψ. Pshaft λ

(10.4)

10.1 Hydraulic Characteristics of the Pump

(a)

237

(b)

d2D

(c)

Fig. 10.1 Characteristics of an example pump (d 2D = 2.258 m, nN = 750 rpm, QN = 20.4 m3 /s, H N = 400 m, PN = 90 MW), obtained from the 4-quadrant diagram, as shown latter in Fig. 10.5, BEP: best efficiency point

The hydraulic characteristics of a pump are then generally expressed by ψ = f(ϕ), λ = f(ϕ), ηpu = f(ϕ)

(10.5)

They are primarily independent of the rotational speed and, therefore, can be applied to general flow regulations including the use of varspeed motors (variable speed). The hydraulic background is simply the high similarity of flows in a pump under different rotational speeds. Corresponding characteristics, for a special pump as an example, have been shown in Fig. 10.1. The nominal operation point or the best efficiency point (BEP) has been marked in each diagram. It is obtained from the design data H N = 400 m, nN = 750 rpm and QN = 20.4 m3 /s. At closed flow (Q = 0), the so-called shut-off pressure coefficient in the current case reads ψshut-off = 1.29. For a given shut-off pressure equal to H shut-off = 400 m, for instance, the shut-off rotational speed is computed from Eq. (10.2) as n shut-off

60 = π d2D



2g Hshut-off = 660 rpm. 1.29

(10.6)

As a matter of fact, the shut-off speed of a large pump in a hydropower plant, for instance, represents a very significant parameter, which determines both the start

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10 Hydraulic Characteristics of Pumps and Turbines

and the stop processes of the pump. It is, therefore, a very important parameter in transient computations. As shown in Fig. 10.2, the pump is considered to transport the water from a reservoir at a lower altitude to a reservoir at a higher altitude; therefore, it always operates against an almost constant pressure head. For reasons of safely closing the flow, a spherical valve is usually installed downstream of the pump. While the pump is started and the rotational speed progressively increases, the shut-off pressure continuously increases in proportion to the square of the rotational speed (Fig. 10.2b). In order to avoid a reverse flow, the spherical valve must remain closed, before the shut-off speed is reached and the shut-off pressure head is equal or comparable to the altitude difference between the upper and the lower reservoirs. In the current case of the pump, which is shown in Fig. 10.1, the applied altitude difference between the lower and the upper reservoirs is supposed to be 400 m (design data). The corresponding shut-off speed, according to Eq. (10.6), is computed as n shut-off = 660 rpm. Thus, when starting the pump, only after this speed has been reached, the spherical valve downstream of the pump can be opened. The same shut-off speed of the pump should be taken for closing the valve. While the pump is shut down, reverse flow could take place if the pump speed falls below the shut-off speed. For safety reasons, therefore, the spherical valve should be sequentially closed as soon as the pump is shut down. The shut-off speed is determined for Q = 0. From Fig. 10.1b, one confirms that the corresponding power coefficient falls to a minimum. As will be demonstrated

(b)

H0

(a)

Sph. valve

Pump

Fig. 10.2 Pump unit with a spherical valve in a hydropower station. a Spherical valve in the closed position; b Shut-off pressure head as a function of the rotational speed of the pump shown in Fig. 10.1

10.1 Hydraulic Characteristics of the Pump

239

below in connection with the four-quadrant diagram of the pump, this minimum corresponds to the minimum of the torque of the shaft and is always given at the shut-off speed. At each stationary operation point, the power applied to the pump shaft is used to create the hydraulic power and to overcome all relevant mechanical resistances. The former is measured in the flow at the pump exit. Under shut-off conditions (Q = 0), the shaft power then only represents the power, which is required to overcome all mechanical resistances. These mechanical resistances include those arising from the flow circulations at the impeller inlet, within the impeller and at the impeller exit. They also include mechanical resistances due to viscous friction on both the front as well as the rear sides of the impeller. Further mechanical resistances included are found at the bearings and sealings of the rotor system. With λshut-off = 0.0235, as read out from Fig. 10.1b and for n shut-off = 660 rpm in the current example (Hshut-off = 400 m), one deduces from Eq. (10.3) the total mechanical power loss at the shut-off speed Pmech, shut-off = 28.5 MW.

(10.7)

If compared with the rated, i.e., nominated power of about PN = 91 MW, the total mechanical loss at the shut-off speed appears to be very large. The greater part in Pmech, shut-off is consumed to maintain internal flow circulations, as outlined above. At the nominated flow rate, it disappears because of blade-congruent and thus sound flows in the pump. This can be confirmed from the efficiency diagram that has been shown in Fig. 10.1c. At the nominated flow rate, the mechanical loss is only of about 10% (ηN = 90%). Equation (10.7) represents the mechanical power loss at the shut-off point as a special case. Because for Q = 0 the mechanical power loss is physically proportional to the cubic of the rotational speed, it can be written in general form as  n 3 MW. Pmech = 28.5 660

(10.8)

This is the resistance power which must be overcome when starting and speeding up the pump up to the shut-off speed. The corresponding relation is shown in Fig. 10.3. It ends at n = 660 rpm for the head H = 400 m according to Eq. (10.6), and at n = 699.6 rpm for a head H = 450 m from an analogous computation. Such a relation must always be taken into account in related transient computations. After the shut-off speed has been reached, the water in the pump begins to move. The mechanical loss then changes with another mechanical law. While the hydraulic power is computed by Phyd = ρg Hpu Q, the shaft power is computed from Eq. (10.3) by λ. The difference is simply the mechanical loss Pmech = Pshaft − Phyd . With the use of Eq. (10.4) there is obviously Pmech

    1 1 Pshaft = 1 − ϕψ Pshaft . = 1− ηpu λ

(10.9)

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10 Hydraulic Characteristics of Pumps and Turbines

Pmech

28.5

n 660

3

Fig. 10.3 Mechanical power loss of the pump as a function of the rotational speed under the condition of given heads H pu = 400 and 450 m, respectively

Regarding its computation and graphical representation as a function of the rotational speed, one goes back to the tabulated pump characteristics for ϕ, ψ and λ, and follows the scheme in Fig. 10.4. For a given head H pu , one obtains from Eq. (10.2) to each head coefficient ψ a corresponding rotation speed n, i.e., n = f(ψ). Further, from Eq. (10.3), the shaft power can be computed. Finally, the mechanical power loss is obtained from Eq. (10.9). One thus obtains the relation Pmech = f(n) in tabular form, which is valid for n ≥ n shut-off , i.e., Q > 0. For the considered pump, the computational results are shown in Fig. 10.3 for H pu = 400 m and 450 m, respectively. It is evident that the maximum mechanical power loss is found at the shut-off speed. For transient computations, the shaft torque or the moment of force is often used to balance the rotor dynamics. As long as the rotational speed is found below the shutoff speed, Eq. (10.8) also represents the shaft power and thus can be used to compute the mechanical resistance torque by Mmech = Pmech /ω, with ω as the angular speed of the pump. After the shut-off speed has been reached, however, the shaft torque also comprises the hydraulic torque and is computed with the aid of Eq. (10.3) as

Eq. (10.2): n

0.068 0.367 0.055 0.059 0.731 0.056 0.055 0.812 0.055 : : :

Hpu=400

Eq. (10.3): Pshaft

Eq. (10.9): Pmech Fig. 10.4 Scheme to compute the mechanical power loss

Pmech=f(n)

10.1 Hydraulic Characteristics of the Pump

Mshaft =

π2 2 5 Pshaft = ρn d2D λ. 2π n 4

241

(10.10)

The use of discharge, head and shaft power coefficients is based on the law of similarity of flows in the pump under variable rotational speed. Specifically, proportions Q ∝ n, Hpu ∝ n 2 and Pshaft ∝ n 3 exist. They are, however, restricted within a range of the rotational speed, usually down to about 0.3n N . Below this value, remarkable flow separations in pumps will occur; corresponding characteristics are commonly not accessible. For this reason, pump characteristics in terms of head and discharge coefficients are often insufficient for computing hydraulic transients in hydraulic systems with pumps, where reverse flows and inverse rotation of the pump may take place. For large flow variation, therefore, as e.g., when starting and turning down the pump, its characteristics in terms of so-called four-quadrant diagrams must always be applied. This will be presented in the next section.

10.1.2 Four-Quadrant Diagrams and Operation Map Pump characteristics, which also account for the cases of inverse rotation and reverse flows, are given by so-called four-quadrant diagrams. The diagrams basically relate the discharge and the mechanical torque to both the rotational speed and the pumping head. They are commonly constructed with rotational speed and flow rate as two coordinates, so that they describe the full characteristics of pumps, also under conditions with reverse flow and inverse rotation. Under the assumption of flow similarities regarding flow rate, rotational speed, head and hydraulic power, a four-quadrant diagram can be shown in advanced form by using dimensionless parameters. Against the basic form of using n and Q as coordinates and H pu = const as curve parameter, a four-quadrant diagram for different pumping heads H pu can be unified to a single curve, which is necessary when used for transient computations. The essential quantities used in four-quadrant diagrams are therefore dimensionless unit parameters for rotational speed, discharge and shaft torque, as defined by π d2D n Q Mshaft , Q 11 = 2  , M11 = . n 11 =  3 ρd2D g Hpu d2D 2g Hpu 2g Hpu

(10.11)

In these definitions, n is in 1/s and Q in m3 /s. The impeller diameter d2D has been again used as the reference diameter. The author would like to justify the use of dimensionless unit parameters. In some other applications, respective unit parameters are defined without accounting for the gravity constant g. This might be sufficient, if it is only about simple applications. For applications with much more mathematical complexity, however, the dimensionless expressions will desirably lead to much simpler computational result, as shown in Zhang (2018a, b) and Zhang et al. (2018); this will be again demonstrated below.

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10 Hydraulic Characteristics of Pumps and Turbines

Commonly, the four-quadrant diagrams show the following relations: Q 11 = f(n 11 ),

(10.12)

M11 = f(n 11 ).

(10.13)

Each of these two relations represents a single curve, independent of the pumping head. This is exactly so because of the utilization of similarity laws for pump flows. Obviously, the relation between the shut-off pressure and the shut-off speed can be obtained from the unit rotational speed n 11,shut-off that is found from Eq. (10.12) by setting Q 11 = 0. In almost all cases, the four-quadrant diagrams can only be obtained from experimental measurements. Figure 10.5 shows, based on measurement, such a set of four-quadrant diagrams of the pump that has been accounted for in the last sub-section and is shown in Fig. 10.1. The pump operation in the pumping mode is commonly found in the first quadrant, in which both the rotational speed and the discharge are positive. The nominal operation point has been again denoted by BEP (best efficiency point). Equations (10.12) and (10.13) represent the basic equations of the complete hydraulic characteristics of a pump. Other parameters like discharge, head, and shaft power coefficients as well as the pump efficiency can be derived by applying Eqs. (10.1), (10.3) and (10.4) with respect to Pshaft = 2π n M. One obtains ψ=

Q 11 M11 Q 11 1 ,ϕ= , λ = 2 2 , ηpu = . 2 n 11 2n 11 M11 n 11 n 11

(10.14)

The pump efficiency, which is also shown in Fig. 10.5, is commonly only plotted for n 11 > 0 and Q 11 > 0, where the pump operates in the pumping mode. From Fig. 10.5a, the shut-off speed, which is given at Q = 0, is confirmed to be n 11,shut-off = 0.88. According to Eq. (10.14), the related shut-off pressure coefficient is computed as ψshut-off = 1.29, which is indicated in Fig. 10.1a. Correspondingly, one obtains from Fig. 10.5b the related unit torque M11,shut-off = 0.0091, which exactly represents the local minimum of the curve. From Eq. (10.14), the power coefficient is computed as λshut-off = 0.0235. It agrees with that in Fig. 10.1b. The corresponding mechanical power loss is given by Pmech = 2π n Mshaft . With respect to the definitions of n 11 and M11 in Eq. (10.11), it is further computed for Hpu = 400 m as  3/2 2 2g Hpu n 11,shut-off M11,shut-off = 28.5 MW. Pmech . shut-off = ρd2D It is equal to the former computational result in Eq. (10.7). Correspondingly, Eq. (10.8) can be used to compute the mechanical resistance torque Mmech = Pmech /ω when starting the pump and before the shut-off speed is reached.

10.1 Hydraulic Characteristics of the Pump

243

Fig. 10.5 Four-quadrant diagrams of the pump that has been given in Fig. 10.1. a Unit volume flow rate, b unit toque, c efficiency

Once the shut-off speed is exceeded, the unit speed n11 can be computed based on its definition in Eq. (10.11). The unit torque M 11 is then obtained from the pump characteristic (Fig. 10.5b). Again from Eq. (10.11), the shaft torque is further computed as 3 g Hpu M11 . Mshaft = ρd2D

(10.15)

At this moment, it should be mentioned that the Suter format of four-quadrant diagrams has sometimes been used in computations of hydraulic transients of pump systems (Suter 1966; Dörfler 2010). It is about a reconstruction of the common format of four-quadrant diagrams (using Q, n, H and M) by making the parameters dimensionless in special forms. As will be demonstrated in the next section dealing with the unification of the pump and the valve characteristics, the four-quadrant

244

10 Hydraulic Characteristics of Pumps and Turbines

diagrams of using unit parameters n 11 , Q 11 and M11 , as shown in Fig. 10.5, is highly significant and most applicable to all related hydraulic transients. For this reason, the Suter format of four-quadrant diagrams will not be considered. For the purpose of operation optimization and restriction, an operation map of the considered pump has been designed by the author from the used four-quadrant diagrams (Fig. 10.5), as presented in Fig. 10.6. In such an operation map, not only the operation limitation of the pump is indicated. From it, one can also directly read out the shut-off speed as a function of the pumping head (H), which is quasi equal to the altitude difference between the upper and the lower reservoirs. In reality, this function has been obtained from n 11,shut-off = 0.88 for the currently considered pump.

10.1.3 Unification of the Pump and the Valve Characteristics Large pumps, including pump-turbines in pumped storage power stations, are always installed in connection with a spherical valve close to the pump on its higher pressure side, as sketched in Fig. 10.7, see also Fig. 10.2. When e.g., the pump is shut down, the spherical valve should be simultaneously and progressively closed. Obviously, the resulting hydraulic transients in such a hydraulic system depend on hydraulic characteristics of both the pump and the spherical valve. For the characteristics of the pump, because of the possible reverse flow and inverse rotation, only those in form of four-quadrant diagrams should be used. For practical applications, it seems to be quite convenient to consider the pump and the spherical valve as a combined hydraulic unit and to exclusively use the unified characteristic. One expects that the unified characteristic explicitly shows the discharge as a function of both the rotational speed of the pump and the opening degree of the spherical valve. The latter functional relation is encountered, for instance, when the pump is shut down and the spherical valve is closed. It is desired that the unified characteristic is represented in a suitable mathematical form, in order to ensure easy application in transient computations. The significance of using such a unified characteristic in a quadratic mathematical form has already been shown in Sect. 7.3.6 for computing the generation of both the upstream and the downstream shock waves at a pump. While the start of a pump usually can be well controlled, the emergency shutdown of the pump mostly represents a highly serious case. The simultaneous closing of the spherical valve will additionally complicate the computations. The first computation of such a complex transient process with successful application of the unified characteristics has been performed by Zhang (2016b, 2018c). Besides the excellent agreement between computations and measurements, reverse flows through the pump during its shutdown could be accurately detected as well. To unify the pump and the valve characteristics, the positive flow direction is specified to agree with the pump flow, as shown in Fig. 10.7. The pressure head rise at the pump is denoted by Hpu , while the pressure head drop over the spherical valve is given by h sphV , which is positive for positive flows, see Eq. (2.34). The total rise of the pressure head over the combined pump unit is H12 . It is for stationary flow

10.1 Hydraulic Characteristics of the Pump

245

Fig. 10.6 Operation map of the considered pump under variable rotational speed, recalculated from the four-quadrant diagrams shown in Fig. 10.5

246

10 Hydraulic Characteristics of Pumps and Turbines

Fig. 10.7 Unification of the pump and the spherical valve

H0

L Unified Pump

Sph.valve

hsphV

Hpu H12

and by neglecting all possible losses in the pipeline, equal to the altitude difference between the lower and the upper reservoirs. It also equals the pressure head rise at the pump when the spherical valve is fully opened. Basically, the unified characteristic is simply given by H12 = Hpu − h sphV . For the case of varying rotational speed, the pump characteristic in terms of ψ = f(ϕ) or Q 11 = f(n 11 ) must be used. Correspondingly, there are two choices for applications.

10.1.3.1

Using the Characteristic ψ = f(ϕ)

If the characteristic of a pump is given by ψ = f(ϕ), e.g., in the form of tabulated data, then the following approximation by using the least-square curve fitting method can be achieved: ψ = m2ϕ2 + m1ϕ + m0,

(10.16)

in which m0 is equal to ψ0 at ϕ = 0 and reads m 0 = 1.29 from Fig. 10.1a. The two other coefficients m1 and m2 are fitting parameters. Equation (10.16) is then expressed in the explicit form as 2g Hpu =

m2 2 m1π n Q + Q + m 0 (π d2D n)2 . 4 d2D d2D

(10.17)

To obtain the unified characteristic, the characteristic of the spherical valve according to Eq. (2.34) should be taken into account. With H12 = Hpu − h sphV and with possibly reverse flow through the spherical valve, the unified characteristic is obtained as

10.1 Hydraulic Characteristics of the Pump

H12

247



|Q| cp 1 m2 m1π n m 0 (π d2D n)2 2 . = − + Q + Q 4 2g d2D Q A2sph 2gd2D 2g

(10.18)

The resistance coefficient cp = f(β) of the spherical valve can be computed with the aid of Eq. (2.35). Thus, the above equation is generally written as Q = f(H12 , n, β).

(10.19)

The unified characteristic is applicable where the flow regulation is performed by regulating the rotational speed (n) and/or the spherical valve (cp ). The latter is accounted for with Eq. (2.35) by changing the opening degree (β) of the given spherical valve, see Fig. 2.8. Depending on the flow direction, one has |Q|/Q = ±1. Furthermore, the flow area of the spherical valve is usually equal to the cross-sectional area of the pipe. Because the spherical valve is commonly used as a closure organ rather than a regulation valve, its regulation performance is basically restricted to the opening and the closing processes. In view of this issue, the characteristic of the pump in the form of ψ = f(ϕ) can be updated to the characteristic of the unified pump and valve. With Eq. (10.2) and Hpu = H12 + h sphV , the head coefficient is now expressed as ψ=

 2g H12 + h sphV (nπ d2D )

2

=

2g h sphV 2g H12 + . 2 (nπ d2D ) (nπ d2D )2

(10.20)

Because of Eq. (2.34), it follows further that ψ=

2g H12 Q2 1 + c . p (nπ d2D )2 (nπ d2D )2 A2sph

(10.21)

The first term on the r.h.s. of the equation can be considered to be the head coefficient of the unified pump and valve, denoted by ψU . The flow rate can be expressed by the discharge coefficient ϕ. Accordingly, with respect to Eq. (10.1), one rewrites the above equation and obtains ψU = ψ − cp

4 d2D ϕ2, A2sph

(10.22)

with ψU =

2g H12 . (nπ d2D )2

(10.23)

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10 Hydraulic Characteristics of Pumps and Turbines

Fig. 10.8 Pressure coefficient of the unified pump and the spherical valve

The basic head coefficient of the pump ψ = f(ϕ)  is updated to the head coefficient of the unified pump and valve in the form ψU = f ϕ, cp , i.e., ψU = f(ϕ, β) because of Eq. (2.35). The hydraulic performance of the spherical valve has been thus included, as shown in Fig. 10.8. By using the extended head coefficient of the unified pump and valve, the start-up process of the pump can be optimized, even under the condition of quasi-stationary flows. This will be shown in Appendix C. It should be noted that the pressure head H12 in Eq. (10.18) is a variable in hydraulic transients. This means that it always represents the instantaneous effective pressure head which is measured on both sides of a unified pump-valve component (Fig. 10.7). Only under conditions of stationary flows and by neglecting the flow friction in the pipeline, it is equal to the altitude difference H0 . For regulating pump flows by only varying the rotational speed of the pump under full-opening of the spherical valve, Eq. (10.18) reduces to Eq. (10.17). The latter has been already applied in Sect. 7.3.6, see Eq. (7.52). The quadratic form of the unified characteristic enables the computation there to be easily completed.

10.1.3.2

Using the Characteristic Q 11 = f(n11 )

If the characteristic of a pump is given in the form Q 11 = f(n 11 ), a regressed polynomial of second order could be computed in the same way as for Eq. (10.16). However, the explicit function Hpu = f(Q, n) would no longer be a quadratic polynomial of the discharge Q like that in Eq. (10.17). Therefore, Eq. (10.16) is further considered and directly reformulated with the aid of Eq. (10.14). One finally obtains the following quadratic equation: m 2 Q 211 + m 1 (n 11 Q 11 ) + m 0 n 211 − 1 = 0.

(10.24)

10.1 Hydraulic Characteristics of the Pump

249

Fig. 10.9 Validation of regression curves in term of m 2 Q 211 + m 1 (n 11 Q 11 ) + m 0 n 211 − 1 = 0

It is equivalent to Eq. (10.16). The three coefficients, which are the same as in Eq. (10.16), can be directly determined from the given characteristic Q 11 = f(n 11 ), for instance, in the form of tabulated data. The coefficient m0 represents the shut-off condition (Q = 0) and is given by m 0 = 1.29, as can be immediately obtained from the above equation with n 11,shut - off = 0.88 according to Fig. 10.5. For performing the least-square curve fitting, both n 11 Q 11 and n 211 behave as two independent variables.1 Because the computations may cover a range of unit speed from negative to positive values, the regression curve, as in the current case (Fig. 10.5a), must be approximated at least in two areas as for n 11 > 0.88 and n 11 < 0.88, respectively. The accuracy of regression computations is usually sufficiently high, as verified in Fig. 10.9 by comparing the regression computations with tabulated source data. Corresponding applications of using Eq. (10.24) can be found in Zhang (2016b, 2018c). From Eq. (10.24), the following explicit relation is obtained: 2g Hpu =

πn m2 2 Q + m1 Q + m 0 (π d2D n)2 . 4 d2D d2D

(10.25)

For the unified pump and valve, one has correspondingly H12 =

  |Q| cp 1 m2 π m1n m 0 (π d2D n)2 2 Q . − + Q + 4 2g d2D Q A22 2gd2D 2g

(10.26)

the known relation Q 11 = f(n 11 ), three data columns can be created: n 11 Q 11 and n 211 as independent variables and Q 211 as the dependent variable. Then, all three coefficients in Eq. (10.24) can be found out from the two-parameter regression model.

1 From

250

10 Hydraulic Characteristics of Pumps and Turbines

These last two equations are identical to Eqs. (10.17) and (10.18), respectively. Because all three coefficients m 0 , m 1 and m 2 are determined from the four-quadrantdiagram, the above two equations also apply to the cases with reverse flow and inverse rotational speed. The last equation is again applicable where the flow regulation is performed by regulating the rotational speed (n) of the pump and/or the opening of the spherical valve (cp ) with respect to Eq. (2.35). For applying the butterfly valve, for instance, the head drop coefficient cp in all above equations must be replaced by the head drop coefficient cp according to Eq. (2.38) with respect to Eq. (2.39). It should be noted that the mathematical advantage of expressing the unified characteristic in the quadratic form cannot be obtained by using the pump characteristic in the Suter format (Suter 1966). In other words, the Suter format is not applicable to the current unification method. Like in Eq. (10.18), it should be repeatedly noted that the pressure head H12 in Eq. (10.26) always applies to the instantaneous pressure head which is measured on both sides of the unified pump-valve. In addition, the volume flow rate used in the above equations refers to one pump. For two or more pumps in a pump house, the volume flow rate at the pump may differ from the volume flow rate in the pipeline system. The torque characteristic, M11 , in the combined unit (pump and valve) remains the same as for the pump alone. Figure 10.5b can then be directly used. Especially, Eq. (10.8) can be used to compute the mechanical torque Mmech = Pmech /ω while starting the pump and before the shut-off speed is reached. A complete computational example of a transient flow arising from the emergency shutdown of the pump and the simultaneously closing of the spherical valve will be shown in Sect. 11.2. For the same pump in the same hydraulic system, the transient computations in connection with the start-up of the pump will be demonstrated in Sect. 11.3.

10.2 Hydraulic Characteristics of the Pelton Turbine The Pelton turbine is a type of hydropower machines and has found the widest applications in mountain areas where water of high altitude is available. A Pelton turbine basically consists of one or more injectors and a Pelton wheel (Fig. 10.10). The injector is used to convert the pressure energy into the kinetic energy of the high-speed jet, on the one hand, and to regulate the discharge on the other hand. Because of the mechanical separation of the Pelton wheel from the injector, the characteristic of the Pelton turbine is simply represented by the regulation function, i.e., the characteristic of the injector. When hydraulic transients are concerned, a hydraulic system with Pelton turbines is often more preferable than others because of simpler computations. The background is simply the disconnection of the Pelton wheel from the injector. This simply means that the Pelton wheel and its dynamic behavior never affect the flow

djet

251

D0

10.2 Hydraulic Characteristics of the Pelton Turbine

s

Fig. 10.10 Pelton turbine and the injector

in the entire pipeline network. For this reason, hydraulic transients in an arbitrary pipeline system with other types of water turbines or regulation organs can be approximately simulated by using a “fictive” and equivalent Pelton injector for discharge regulation. This is especially applicable, if the hydraulic transient in a pipeline only needs to be roughly estimated. Such applications have been already demonstrated many times, e.g., in Chaps. 7, 8 and 9. For these reasons, the general characteristic of Pelton injectors and the appropriate regulation laws will be discussed in the following sections.

10.2.1 Characteristic of the Injector The speed of the jet out of the injector is a function of the total pressure head (h tot against the atmosphere) at the injector and is computed from the Torricelli equation as c0 =



2gh tot .

(10.27)

The flow through the injector practically causes no remarkable head loss and, thus, no drop in the jet speed. Except for the case of accurately determining the hydraulic efficiency of a given Pelton turbine, the jet speed in the above equation can be used for all other purposes, including the bucket design of Pelton turbines and operation optimizations.

252

10 Hydraulic Characteristics of Pumps and Turbines

Like at the pump or other hydraulic machines, the discharge through the injector can be represented by the discharge coefficient which is defined as ϕ=

Q . √ AD0 2gh tot

(10.28)

The reference area AD0 is the exit area of the injector nozzle with a diameter D0 (Fig. 10.10). In the terminology of the Pelton turbines, the discharge coefficient has often been denoted by ϕD0 (Zhang 2009, 2016a). The discharge through the injector is regulated by changing the position, i.e., the stroke of the movable spear needle in the injector. Because the discharge can also be expressed as the product of the jet speed and the jet section, i.e., Q = Ajet c0 = √ Ajet 2gh tot , the discharge coefficient in the above equation is also written as ϕ=

Ajet AD0

(10.29)

The discharge coefficient simply represents the ratio of the contracted jet section (jet waist) to the nozzle aperture. It is of geometrical character and hence also called the contraction factor of the flow out of the injector. The hydraulic characteristic of a Pelton injector can, thus, be understood as the discharge or discharge coefficient as a function of the jet contraction. Because the jet contraction, in turn, is determined by the needle stroke s of the injector, the discharge coefficient of a Pelton injector has often been approximated by the following law:  2 s s ϕ = a1 + a2 . D0 D0

(10.30)

This equation is called the hydraulic characteristic of the Pelton injector. Both the constant a1 and a2 , basically, depend on the geometrical design of the Pelton injector. They are mainly determined by both the nozzle contraction angle and the edge angle of the spear needle. In Fig. 10.11, hydraulic characteristics of three injectors of different designs are displayed (Zhang 2009, 2016a). For a injector design with αnozzle = 45◦ and αs = 25◦ , the two constants a1 and a2 can be set to be a1 = 1.536, a2 = −0.857.

(10.31)

These values have been used throughout this book. For other design of injectors, both constants (a1 and a2 ) can be computed by applying the similarity law for contraction flows out of a injector nozzle, as described in Zhang (2009, 2016a). It should be pointed out, however, that for computing hydraulic transients, the different characteristics shown in Fig. 10.11 do not lead to any remarkable difference in computational results. As a function of the discharge coefficient, the discharge through the injector can be computed. The injector is assumed to be connected to a pipe of a cross-sectional

10.2 Hydraulic Characteristics of the Pelton Turbine

(a)

253

(b) 1: 2: 3:

s=28° s=25°

1

s=22°

2

D0

nozzel

s

3

s

Fig. 10.11 Characteristics of Pelton turbine injectors according to Zhang (2009, 2016a)

area A. If both the static pressure and the flow velocity are related to this area, then, with respect to the total pressure head h tot = h + c2 /2g and Q = c A, it follows from Eq. (10.28)



Q = ϕ AD0 2gh +

Q A

2 .

(10.32)

2gh.

(10.33)

The discharge is then obtained as Q=



ϕ AD0 1 − (ϕ AD0 /A)

2

The volume flow rate as a function of the discharge coefficient in the given form can be well applied to transient computations. The relation represents an additional balance between the pressure head and the volume flow rate. For information, another relation between two flow parameters is found in the mechanism of water hammer. The pressure head in the above equation is then the instantaneous shock pressure found at the injector.

10.2.2 Power Output and Flow Regulations The hydraulic power included in the water jet is given by P = ρgh tot Q.

(10.34)

It is usually applied to stationary flows and represents, by multiplying the turbine efficiency, the power output of the Pelton turbine. The regulation of the power output

254

10 Hydraulic Characteristics of Pumps and Turbines

is simply achieved by regulating the flow rate and thus, according to Eq. (10.33), by regulating the discharge coefficient. The possibility of rapidly regulating the discharge and thus the power output of a Pelton turbine allows the Pelton turbine to be operated also for quickly balancing the load of the power grid. The related flow regulations, including the start and the shut-down of the turbine, might happen even within hours. Each time, shock pressures will be caused in association with water hammer in the pipeline system. For operational safety reasons, while shutting down the turbine, the injector has to be progressively closed by following a certain closing law. Such a closing law is basically validated by either transient analyses or measurements with regard to shock pressures, low-frequency flow oscillations, surge tank capacity etc. For closing the injector, changing the spear-needle stroke is often performed in two steps of different rhythms. Within each step, a linear regulation law s = f(t) can be applied. The stroke speed ds/dt in the second step is usually remarkably smaller than that in the first step. In Sect. 9.7, an application example has been shown to demonstrate the reliability of the method of direct computations. The most suitable regulation law, which effectively limits the maximum pressure rise in the flow, is the parabolic law that has been recommended by Zhang (2016a). Both the linear and parabolic closing laws and the related system responses will be presented in the following two sections.

10.2.3 Linear Closing Law of Injectors The linear closing of the injector is described by a constant needle stroke speed ds/dt = const. Because of the occurrence of high shock pressures in the respective hydraulic pipelines, the one-step linear closing of the injector is practically not recommended. The reason is, as shown in Fig. 10.12 by solid lines, that towards the full-closing of the injector, the changing rate of the discharge coefficient (dϕ/dt) continuously reaches its maximum. This will lead to high pressure rise in the flow, because, according to Eq. (9.45), the pressure rise in the flow is proportional to the parameter B and thus to the changing rate of the discharge coefficient dϕ/dt. A two-step linear closing of the injector can be arranged. An example of a special concept is shown in Fig. 10.12. The changing rate of the discharge coefficient at the end of the first step (t 1 = 24 s) is made to be equal to that at the end of the one-step closing (t 2 = 50 s). In the example, it reads (dϕ/dt)max = −0.021. For the two closing concepts in Fig. 10.12, corresponding transient computations have been conducted based on the wave tracking method (Chap. 7). The hydraulic system considered is the same as that shown in Table 4.2. Figure 10.13 shows the computational results and the comparison between two closing concepts. For this comparison, the following points regarding the two-step closing of the injector should be indicated:

10.2 Hydraulic Characteristics of the Pelton Turbine

255

Fig. 10.12 Linear closing of an injector nozzle with one- and two-step rhythms with t 1 = 24 s and s1 /D0 = 0.25 (comparable to Fig. 9.10)

Fig. 10.13 Pressure response in the flow and comparison between one- and two-step linear closing of the injector (comparable to Fig. 9.13)

(1) The two-step closing concept can be generally recommended. The shock pressure at the end of the first step reaches its maximum which is basically comparable to that at the end of the one-step regulation. A small difference between two maxima arises because of the viscous friction effect, as explained in Sect. 9.7.3, see also Fig. 9.13. The maximum pressure rise in the one-step closing case, however, only appears once. (2) The first downward jump of the pressure at the end of the first step (t 1 = 24 s) triggers the pressure fluctuation in the flow. The fast attenuation of this pressure fluctuation is again ascribed to the self-stabilization effect of the flow through

256

10 Hydraulic Characteristics of Pumps and Turbines

a valve, see Sect. 3.3.4. Because of this rapidity, the overlapping phenomenon, as explained in Sect. 9.7.4 and Fig. 9.14, is here not evident.

10.2.4 Parabolic Closing Law of Injectors In practice, however, non-linear closing laws are often utilized for achieving better pressure responses in pipeline systems with Pelton turbines. In Fig. 1.4, such a pressure response in a hydraulic system, both from measurements and computations has already been shown. First, the achieved excellent agreement between the measurement and the computation (Zhang 2018c) demonstrates high reliability of the used wave tracking method (Chap. 7). Second, within the closing period (t < 47 s), the rapid pressure rise at the injector could be successfully inhibited. This can be evidenced by comparison with the possible pressure response, which is given by linearly closing the injector, as presented in Fig. 10.14 based on similar computations. Obviously, the pressure response during the closing time strongly depends on the closing law of the injector. Following the above concept, the question arises how the closing law of an injector should be specified, in order to effectively inhibit the rapid pressure rise in the system. In general, a parabolic form of the closing law can be well proposed. For the nominal operation, i.e., the nominal opening (s0,N ) of the injector nozzle and for a given closing time t0,N , the closing law of a Pelton injector can be defined as (Zhang 2016a)   t 2 s s0,N 1− = . D0 D0 t0,N

(10.35)

Fig. 10.14 Response of the pressure head at the injector; comparison between the linear and the non-linear closing of the injector nozzle (hydraulic system according to Fig. 3.1)

10.2 Hydraulic Characteristics of the Pelton Turbine

257

Fig. 10.15 Parabolic closing law of the injector nozzle (Zhang 2016a)

This equation satisfies the condition that for t = t0,N one has both s = 0 and ds/dt = 0, as desired. For given values of s0,N and t0,N , the specified closing law is shown in Fig. 10.15. When the injector nozzle is intended to regulate the flow by changing the needle position from the full opening s0,N to a partial opening sP , then the closing time is calculated from the above equation to tP t0,N

 =1−

sP . s0,N

(10.36)

For complete closing of the injector nozzle (sP = 0), the closing time is again equal to t0,N . For closing an injector, which is found in a partial opening (s0 ), the same closing law should be followed, however, with a correspondingly shorter closing time as drawn in Fig. 10.15. The closing law 2 is exactly a shifting of the nominal closing law 1 towards the left. With s0 as the partial opening of an injector, the new closing time is calculated as  t0 s0 = . (10.37) t0,N s0,N Based on the coordinate transformation for shifting curve 1 towards the left, the new closing law is obtained as  s = s0,N

t0 − t t0,N

2 ,

(10.38)

258

10 Hydraulic Characteristics of Pumps and Turbines

or because of Eq. (10.37) to  s = s0,N

s0 t − s0,N t0,N

2 .

(10.39)

As happened quite often, the injector in its initial opening at s0 will be regulated to s1 . The time needed to reach this new value is computed as t1 t0,N

=

t0 t0,N

 −

s1 . s0,N

(10.40)

A Pelton injector nozzle is commonly designed with a ratio of the nominal opening of the injector s0,N to the nozzle diameter D0 at about s0,N /D0 ≈ 0.70. In the basic definition of the parabolic closing law in Eq. (10.35), then, the closing time t0,N should be determined. The criterion for this is obviously related to the permissible maximum rise of the shock pressure in the flow during the closing process. As known from Eq. (9.45), for instance, the shock pressure is directly proportional to the parameter B and thus to the changing rate of the discharge coefficient dϕ/dt because of Eq. (9.13) for the method of direct computations. One also confirms the same relation in Eq. (4.57) for computations based on the rigid water column theory. This circumstance requires computing the maximum dϕ/dt which is related to the parabolic closing law defined in Eq. (10.35). For this purpose, Eqs. (10.30) and (10.35) need to be combined. One first obtains     d s s dϕ = a1 + 2a2 dt D0 dt D0

(10.41)

In Fig. 10.16, both the discharge coefficient ϕ and its time rate of change are shown as functions of the related time t/t0,N . The maximum dϕ/dt is given, where (t m ) the condition d2 ϕ/dt 2 = 0 is fulfilled. The corresponding time is found as tm =1− t0,N

−

1 a1 D0 . 6 a2 s0,N

(10.42)

For a given injector nozzle, both constants a1 and a2 are known, with a2 < 0 according to Eq. (10.31). Because of s0,N /D0 ≈ 0.70 for commonly used injectors, the time ratio tm /t0,N is constant and approximately equal to tm /t0,N ≈ 0.35, as also observed in Fig. 10.16b. It can be expected that the maximum shock pressure will mostly occur at this time at the injector, which follows the parabolic regulation law. The maximum dϕ/dt, in absolute value, is then obtained from Eq. (10.41) as



dφ a 1 a1 s0,N 4 1

− = .

dt 3 t 6 a2 D0 0,N max

(10.43)

10.2 Hydraulic Characteristics of the Pelton Turbine

259

Fig. 10.16 Regulation behaviors of an injector nozzle subjected to the parabolic regulation law. a Discharge coefficient. b Changing rate of the discharge coefficient

For a given injector, this maximum is simply determined by the nominal closing time t0,N , which refers to the nominal opening of the injector at about s0,N /D0 ≈ 0.70, as shown in Fig. 10.17. It is now almost possible, to directly estimate the maximum pressure head, for instance, from Eq. (9.45) by directly specifying the closing time t0,N according to the following procedure: t0,N



dϕ →



→ B → h max . dt max

(10.44)

As can be confirmed from Fig. 10.17, the maximum changing rate of the discharge coefficient is very high, if the nominal closing time is less than 20 s. For the reason of suppressing the possible water hammer in the flow, the nominal closing time in usual applications should be greater than 30 s.

260

10 Hydraulic Characteristics of Pumps and Turbines

Fig. 10.17 Maximum changing rate of the discharge coefficient as a function of the nominal closing time at an injector following the parabolic closing law

In order to inhibit the rapid pressure rise in the flow during the period of closing the injector, a critical, i.e., maximum permissible changing rate of the discharge coefficient, |dϕ/dt|cr , should not be exceeded. This implies



dϕ 4 a1 1 a1 s0,N − <



. 3 t0,N 6 a2 D0 dt cr

(10.45)

The necessary closing time t0,N is then found to be t0,N

a1 4 1 a1 s0,N > . − 3 |dϕ/dt|cr 6 a2 D0

(10.46)

10.2.5 Unification of Characteristics of the Injector and the Spherical Valve The installation of each Pelton turbine in a hydropower station is usually accompanied by a spherical valve ahead of the injector. The spherical valve is basically employed as a closure rather than a regulation organ. It influences transient flows in a turbine system, however, by emergency shutdown of the turbine, at which the spherical valve will be closed together with the closing of the injector nozzle. In such an emergency case, it will be much helpful for transient computations, if the characteristics of both the injector nozzle and the spherical valve can be unified.

10.2 Hydraulic Characteristics of the Pelton Turbine

261

Fig. 10.18 Unification of characteristics of the injector and the spherical valve

Sph. valve Injector +c h

hsph

hs

This can be performed in the same way as in Sect. 10.1.3 for the case of the pump and a spherical valve. According to Fig. 10.18 with the pressure head hs at the injector, the discharge flow of the injector is obtained from Eq. (10.33) as  ϕ AD0 2gh s . Q= 2 1 − (ϕ AD0 /A)

(10.47)

With the pressure difference h − h s at the spherical valve, one obtains from Eq. (2.34) h − h s = cp

Q2 . 2g A2

(10.48)

Combining these two equations for eliminating the pressure head hs yields Q=



ϕ AD0 1 − (ϕ AD0 /A)2

 Q2 . 2g h − cp 2g A2 

(10.49)

The discharge is then obtained as Q=

√ ϕ AD0 2gh 1 + (cp − 1)(ϕ AD0 /A)2

.

(10.50)

This is the unified characteristic of the injector and the spherical valve. The resistance coefficient of the spherical valve (cp ) is a function of the opening degree of the valve and is given by Eq. (2.35), see also Fig. 2.8b.

10.3 Hydraulic Characteristics of the Francis Turbine The Francis turbine is another important type of water turbines and used in hydropower stations. For the purpose of regulating the power output which is basically given by P = ηρg H Q, only the discharge regulation is applicable. In almost

262

10 Hydraulic Characteristics of Pumps and Turbines

h1 GV

GV

d2

d1

h2

Fig. 10.19 Francis turbine and the guide vanes (GV)

all cases of Francis turbines, the discharge regulation is achieved by regulating the opening of the guide vanes, which are integrated in the turbine unit, see Fig. 10.19. Nowadays, this method still remains as the major method of regulating the turbine load. The way of regulating the rotational speed, as desirable sometimes, is basically a concept regarding the significant changes in the available hydraulic head, e.g., if the water level in the upper reservoir changes considerably (Heckelsmueller 2015). Each start and stop of a Francis turbine always leads to the occurrence of hydraulic transients in the system. In general, the extent of the resultant hydraulic transients depends not only on the predefined regulation process but also on the entire characteristics of the used Francis turbine. The latter, unlike for the Pelton turbine, behaves in a rather complex fashion, because the turbine machine itself is found within the considered hydraulic system. In emergency cases with, e.g., rejection of turbine loads, the turbine wheel will be speeded up towards the runaway speed, which represents a worst-case scenario for all machine components and should be assessed and possibly avoided. The speed-up of the machine itself is strongly correlated with the hydraulic transients in the hydraulic system. The hydraulic performance of a Francis turbine, when regarding the hydraulic transients in the related pipeline system, is also known as more complex than that of a pump due to the integration of the regulation device (guide-vane apparatus) into the turbine unit. The complete characteristics of a Francis turbine essentially consist of the discharge and the torque of the shaft as a function of both the guide-wane angle and the rotational speed. While the flow and thus the load regulation commonly takes place by regulating the guide-vane angle under constant rotational speed, the start and the stop of the Francis turbine always occur at variable speed, much often with

10.3 Hydraulic Characteristics of the Francis Turbine

263

simultaneous opening and closing of guide vanes. Sometimes, the process may also be accompanied, e.g., by opening or closing the installed spherical valve. In this case, the characteristics of the Francis turbine and the spherical valve should again be unified, like that for the pump in Sect. 10.1.3.

10.3.1 Characteristics in Terms of Unit Parameters Complete hydraulic characteristics of a Francis turbine are mainly obtained by experiments at which both the guide-vane angle and the rotation speed are changed. Figure 10.20 shows, for example, such a set of characteristics of a Francis turbine as functions of the unit speed for each given guide-vane angle. The unit rotational speed, the unit flow rate, and the unit hydraulic torque are defined as follows: Mhyd π d1D n Q , Q 11 = 2 √ , M11 = n 11 = √ . 3 ρd1D 2g H d1D 2g H (g H )

(10.51)

Fig. 10.20 Francis turbine characteristics from measurements. a Unit discharge with an example of data fitting (regression) by m 2 Q 211 + m 1 (Q 11 n 11 ) + m 0 n 211 − 1 = 0, b unit torque, c hydraulic efficiency

264

10 Hydraulic Characteristics of Pumps and Turbines

The hydraulic efficiency of the Francis turbine, is then computed as2 ηhyd = 2n 11

M11 . Q 11

(10.52)

All three definitions in Eq. (10.51) are comparable to Eq. (10.11) for the pump. The impeller diameter d1D is used here as reference diameter. The reference head H represents the effective available pressure head at the turbine. It varies and fluctuates, in the context of hydraulic transients, with the pressure shock in the flow. In general, a Francis turbine is a hydraulic unit which also involves the draft tube. Consequently, the pressure head on the lower pressure side should always refer to the water level in the lower reservoir. Diagrams like those in Fig. 10.20 can be used for computing hydraulic transients in pipeline systems with the Francis turbines. Basically, there are three varieties of regulations leading to hydraulic transients. These are • changing the rotational speed at constant guide-vane settings. This only represents a rare case and usually does not happen in practical applications for load regulations. However, it represents the runaway incident, at which the turbine speeds up at a constant guide-vane angle, as soon as the load on the side of the generator is rejected; • changing the opening of guide vanes under the constant rotational speed. This represents the common case of regulating the turbine load by regulating the discharge; • simultaneously changing the rotational speed and the guide-vane angle. This occurs when starting or stopping the turbine. In the former case, the turbine speeds up, while guide vanes are simultaneously opened. When computing hydraulic transients, which each time are related to one of these three cases, then the complete characteristics of the installed Francis turbine, like those in Fig. 10.20, must be expressed in different analytical forms. Curves shown in Fig. 10.20, for instance, are only directly applicable for the first case mentioned above. For general applications and recalculations of the characteristics, the master equation of the Francis turbine, which has been developed by Zhang (2018b), must be considered. 2 On the one hand, the hydraulic power transferred from the flow to the impeller of a Francis turbine

is computed as Phyd = ηhyd ρg H Q.

(10.53)

On the other hand, this power is also represented as the product of the hydraulic torque exerted on the impeller (Mhyd ) and the angular speed (2π n) of the shaft, as given by Phyd = 2π n Mhyd . By equalizing this equation to Eq. (10.53) and further by applying Eq. (10.51), then, Eq. (10.52) is obtained.

10.3 Hydraulic Characteristics of the Francis Turbine

265

10.3.2 Master Equation of the Francis Turbine Complete hydraulic characteristics of a given Francis turbine can be accurately computed by the master equation. This equation analytically relates the discharge through the turbine with the geometrical configuration of the turbine, the rotational speed, and the flow angle downstream of guide-vanes in the following form (Zhang 2018b): kA Q 211 + kB Q 11 + kC = 0,

(10.54)

with    2     1 d1D 2 16 d1D 4 1 d2 n 11,N 2 + μ + + 2 μ G , sh sw π 2 d2 tan β1b tan α1 πb d1D Q 11,N     d1D 1 − μsh μsh d22 n 11,N + μsw G(1 − 2μsw G) 2 , kB = 2n 11 − π b tan α1 tan β1b d1D Q 11,N   d2 kC = n 211 μsh + 2μsw G(μsw G − 1) 22 − 1. d1D

kA = ζQ

In further considerations, the guide-vane angle is used to represent the flow angle. As the variable operation parameters, both the rotational speed (n 11 ) and the guide-vane angle (α1 ) are included in the coefficients kA , kB and kC . The parameter G represents a geometrical structural constant which is related to the geometrical configuration at the impeller exit. Its computation can be found in Zhang (2018a, b). Other geometrical parameters are the design angle of impeller blades β1b and the channel width b at the impeller inlet, the impeller diameter d1D and the draft tube diameter d2 . The coefficient μsh accounts for the part of the theoretical maximum of the shock loss which occurs at the impeller inlet, see Eq. (10.57) below. For similar reasons, the coefficient μsw considers the deviation of the swirling flow at the impeller exit from the flow profile which fulfills the potential flow conditions. Both coefficients can be assumed to be equal to unity, when using the master equation only for computing hydraulic transients. By the way, ζQ represents the total energy loss arising from flow frictions in both the impeller and the draft tube. The subscript N denotes the nominal operation condition which can only be achieved at the nominal guide-vane settings (α1,N ). The influence of the hydraulic head is included in dimensionless unit parameters, as defined in Eq. (10.51). The master equation given by Eq. (10.54) is well justified by its functionality. For each given rotational speed and guide-vane angle, the discharge through a given Francis turbine can be straightforwardly computed. This enables the hydraulic efficiency of the turbine to be determined by the following energy equation: ηhyd = 1 − ηQ − ηshock − ηswirl .

(10.55)

266

10 Hydraulic Characteristics of Pumps and Turbines

It accounts for the friction loss of the flow, the shock loss at the impeller inlet, and the swirling loss at the impeller exit. The respective losses are computed as 4 16 d1D Q2 , π 2 d24 11   2 1 d1D 1 Q 11 − n 11 , + ηshock = μsh tan β1b tan α1 π b   2 n 11,N Q 11 2 2 2 2 d2 n 11 . ηswirl = 2μsw G 2 1 − n 11 Q 11,N d1D

ηQ = ζQ

(10.56) (10.57) (10.58)

The hydraulic power is given by Eq. (10.53). Finally, the hydraulic torque exerted on the impeller is computed from Eq. (10.52) as M11,hyd =

ηhyd Q 11 . 2 n 11

(10.59)

This equation, which is basically a definition equation, does not apply to the case of n = 0. An alternative and totally equivalent expression of the hydraulic torque has been presented in the same reference (Zhang 2018b) as follows: M11,hyd =

  n 11,N Q 11 d1D 1 d2 n 11 Q 11 . Q 211 − μsw G 22 1 − π b tanα1 n 11 Q 11,N d1D

(10.60)

It is directly derived from the law of conservation of angular momentum, which relies on the Euler equation for specific work of the Francis turbine (g H = u 1 c1u − u 2 c2u ). For M11,hyd = 0, it follows from the above equation the runaway condition of the Francis turbine. Detailed computations of the runaway speed can be found in Zhang (2018b). As demonstrated above, the master equation, given by Eq. (10.54), serves as the starting point for computing all relevant quantities and hydraulic characteristics of a given Francis turbine. According to the author, it represents the most important equation in fundamentals of the Francis turbine. The accuracy of the master equation primarily depends on the accuracy of the geometrical structural constant G, of which the detailed and exact computations have been presented in Zhang (2018a) in terms of the streamline similarity method. As demonstrated in a subsequent application (Zhang 2018b), both the concept of using the geometrical structural constant G and the master equation have been excellently validated by experiments.

10.3 Hydraulic Characteristics of the Francis Turbine

267

10.3.3 Reconstruction of the Master Equation of the Francis Turbine The master equation of the Francis turbine, as given is Eq. (10.54), is an implicit function of both the rotational speed and the guide-vane angel. For other applications, like transient computations which are subjected to load regulations, Eq. (10.54) is also written in the following reconstructed form m 2 Q 211 + m 1 (n 11 Q 11 ) + m 0 n 211 − 1 = 0,

(10.61)

with d22 , 2 d1D   μsh d22 n 11,N + μsw G(1 − 2μsw G) 2 , − tan β1b d1D Q 11,N

m 0 = μsh + 2μsw G(μsw G − 1)  d1D 1 − μsh m1 = 2 π b tanα1 

m 2 = kA . This equation formally looks like Eq. (10.24). Obviously, as a variable, only the guide-vane angle is included in the coefficients. For transient computations, both μsh = 1 and μsw = 1 can be applied without causing any remarkable computational inaccuracy. Thus, for a given Francis turbine, both coefficients m 0 and m 1 are two absolute constants that are independent of turbine operations. The variable guide-vane angle is only included in the constant m 2 . The coefficient m 2 , i.e., kA in Eq. (10.54), is rewritten in the following form  m 2 = m 2N +

1 1 + tan β1b tan α1

2 

d1D πb

2 ,

(10.62)

    16 d1D 4 d2 n 11,N 2 = ζQ 2 +2 G . π d2 d1D Q 11,N

(10.63)

with m 2N

Obviously, the parameter m 2N is also an absolute constant. It is directly related to the nominal operation point of a given Francis turbine. Its estimation by the regression computations, as shown below, also enables the geometrical structural constant G to be determined. The significance of reconstructing the master equation is as follows. Suppose that the characteristic of a given Francis turbine is known, like that in Fig. 10.20a and b from measurements. With the tabulated data of an arbitrary curve, better from that at the nominal guide-vane angle, regression computations can be performed, in order to determine all three coefficients in Eq. (10.61). This can be easily performed,

268

10 Hydraulic Characteristics of Pumps and Turbines

Fig. 10.21 Francis turbine characteristics, recalculated from Fig. 10.20

according to Table 10.1, by using “regenerated data 1”. It is about the same computation as in Sect. 10.1.3 for coefficients in Eq. (10.24). Finally, one obtains the best approach to the reconstructed master equation, i.e., Equation (10.61) with all known coefficients m 0 , m 1 and m 2 . Among them, m 0 and m 1 are absolute constant. Only the coefficient m 2 is a an explicit function of the guide-vane angle. In Fig. 10.20a, high reliability of regression computations has been demonstrated for a guide-vane angel α1 = 13.9◦ (dashed line). With known coefficient m 2 from regression computations for a given guide vane angle, one also obtains the absolute constant m 2N from Eq. (10.62) with known geometrical configuration of the turbine wheel (β1b and d1D /b). The direct computation of m 2N from Eq. (10.63) is no longer necessary. Once the coefficients m 0 , m 1 and m 2 have been computed, one is again able to compute the unit discharge from the master equation, i.e., Equation (10.61) for each given guide-vane angle and the rotational speed. Figure 10.21a, for instance, shows the recalculated unit discharge as a function of the guide-vane angle for different unit rotational speeds which are given by n 11 = const. This type of charts can be directly used for transient computations, for instance, if the discharge is regulated by changing the guide-vane angle under constant rotational speed (n 11 = const). The fact to be mentioned is that Fig. 10.21a is basically a regressed approximation of Fig. 10.20a from measurements. This is because all three coefficients m 0 , m 1 and m 2 in Eq. (10.61) are determined from Fig. 10.20a by least-square fitting regression. If no such measurements are available, then all three coefficients in Eq. (10.61) have to be computed from given formulas. As a premise, the geometrical structural constant G must be known. For its computation through numerical integrations, the reader is referred to Zhang (2018a, b). In a similar way, the unit torque that has been shown in Fig. 10.20b can be reconstructed by considering Eq. (10.60). To this equation, one in principle only needs to apply the relation Q 11 = f(α1 , n 11 ) which has just been obtainable from Eq. (10.61) and is shown in Fig. 10.21a. This would yield a functional relation M11,hyd = f(α1 , n 11 ). However, the geometrical structural constant G, in reality, the term μsw G has to be known. As said above, its determination basically relies on

10.3 Hydraulic Characteristics of the Francis Turbine

269

Table 10.1 Example of regression computations by using the data for α1 = 13.9° in Figs. 10.19b and 10.20a Basic data (measurements)

Regenerated data 1

Regenerated data 2

n11

Q11

103 M 11

Q 211

n11 Q11

n 211

x

y

0.000

0.085

88.3

0.007

0.000

0.000

0.007

0.084

0.181

0.085

82.3

0.007

0.015

0.033

0.007

0.085

0.319

0.084

76.4

0.007

0.027

0.102

0.007

0.085

0.402

0.084

72.2

0.007

0.034

0.161

0.007

0.084

0.492 .. .

0.082 .. .

66.7 .. .

0.007 .. .

0.041 .. .

0.242 .. .

0.007 .. .

0.083 .. .

Both x and y in “regenerated date 2” refer to Eq. (10.65)

numerical integrations. In the current case with available measurement data, however, it can also be determined through regression computations. For this reason, Eq. (10.60) is rewritten as   d2 n 11,N 2 d1D 1 Q 211 − M11,hyd = μsw G 22 n 11 Q 11 − Q 11 , π b tanα1 Q 11,N d1D

(10.64)

or simply y = (μsw G)x,

(10.65)

with known nominal rotational speed n 11,N and discharge Q 11,N . For a given guidevane angle, both x and y can be tabulated with corresponding computations, see “regenerated data 2” in Table 10.1. Then, the term μsw G can be easily computed by using the least-square regression method. In the current example and by using the data set of α1 = 13.9◦ , regression computation yields μsw G = 0.67. This value, by assuming μsw = 1, agrees well with G = 0.675 that is directly computed based on the geometrical design of turbine wheel and numerical integrations. Because of its geometrical property, the computed μsw G = 0.67 also applies to other guide-vane angles. As a matter of fact, there are also other ways to determine the constant G, for instance, from the coefficient m 0 in Eq. (10.61) after completing the regression computation shown in Table 10.1. From Eq. (10.60), then, the unit hydraulic torque as a function of the rotational speed and the guide-vane angle can be computed, as shown in Fig. 10.21b. The diagram can be directly applied for computing hydraulic transients which are caused, for instance, by changing the guide-vane angle under constant rotational speed. In general, the great advantage of using the master equation, which is given by Eq. (10.54) in basic form and Eq. (10.61) in reconstructed form, is confirmed by the fact that the master equation comprises all possible varieties of regulation forms for a

270

10 Hydraulic Characteristics of Pumps and Turbines

Francis turbine. For a constant guide-vane angle, for instance, it enables the transient flows up to the runaway speed (case of load rejection) to be computed based on the relation shown in Fig. 10.20. For practical transient computations, therefore, there is no restriction to regulate the guide-vane angle and the rotational speed. The quadratic form of Eq. (10.61) is of great applicability in transient computations. As mentioned before, a Francis turbine will directly cause pressure shocks in a hydraulic system, when the turbine is started, stopped or regulated. All these procedures are basically related to the variation of either the rotational speed or the guide-vane angle or the simultaneous variation of both, as clearly included in the master equation. If the effective pressure head at a Francis turbine is denoted as Htu , then, it follows from Eq. (10.61) with respect to Eq. (10.51) Htu =

  1 m2 2 m1π n 2 . Q + Q + m n) d (π 0 1D 4 2g d1D d1D

(10.66)

This equation has a similar structure as Eq. (10.18) for pump flows. It can thus be directly applied to compute primary shock waves which are generated at a Francis turbine. The explicit quadratic term of the discharge Q remains, so that it greatly contributes to the simplification of the entire computation. The related computational concept regarding the generation of the primary shock waves has been shown in Sect. 7.3.6 with Eq. (7.56). Equation (10.66) will be combined with the characteristic of a spherical valve, in order to compute transient flows which are associated with the stop of the turbine and the simultaneous closing of the spherical valve, see Sect. 10.3.4 below.

10.3.4 Unification of the Francis Turbine and a Spherical Valve The application area of the Francis turbine is marked with a high pressure head which may reach up to 600 m. For safety and insulation reasons, a safety valve, commonly a spherical valve, is installed in the upstream, i.e., on the pressure side of the turbine (Fig. 10.22). For normal operations of the turbine unit, including the start and the stop of the machine as well as the load regulation via guide vanes, the safety valve remains inactive, i.e., fully opened. At emergency cases, e.g., at the unexpected load rejection, however, the safety valve will often be automatically closed. If this occurs, one has to be concerned with a very complex mechanism of generating hydraulic transients in the related system. This may include a rapid change of the rotational speed of the turbine and simultaneous closing of both the guide vanes and the safety valve. Like for unifying the characteristics of the pump and the spherical valve in Sect. 10.1.3, the turbine unit and the safety valve in the current case can also be unified.

271

H0

10.3 Hydraulic Characteristics of the Francis Turbine

Turbine

Sph. valve

hsphV

Htu H12

Fig. 10.22 Unification of the Francis turbine and the spherical valve

According to Fig. 10.22 of using a spherical valve operating in the positive flow direction, it follows H12 = Htu + h sphV .

(10.67)

Combining this equation with Eq. (10.66), one obtains with respect to Eq. (2.34) for spherical valves H12



cp 1 m2 m1π n m 0 (π d1D n)2 . = + 2 Q+ Q2 + 4 2g d1D 2gd1D 2g Asph

(10.68)

This is the unified characteristic of the turbine and the spherical valve. It is of similar form as Eq. (10.18) for pump flows. For a given pressure head H12 , the discharge through the turbine is thus determined by the rotational speed, the guide-vane angle which is included in the coefficients m 1 and m 2 , and the resistance coefficient cp at the spherical valve. The last item is, in turn, a function of the opening degree of the used spherical valve and has been given by Eq. (2.35), see also Fig. 2.8.

References Dörfler, P. (2010, October–December). Improved Suter transform for pump-turbine characteristics. Journal of Fluid Machinery and Systems, 3(4).

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Heckelsmueller, G. (2015). Application of variable speed operation on Francis turbines. Ingeniería e Investigación, 35(1), 12–16. Käling, B. & Schütte, T. (1994). Adjustable speed for hydropower applications. In ASME Joint International Power Generation Conference, Phoenix, Arizona. Merino, J. M., & Lopez, A. (1996). ABB Varspeed generator boosts efficiency and operating flexibility of hydropower plant. Fuel and Energy Abstracts, 37(4), 272. Suter, P. (1966). Representation of pump characteristics for calculation of water hammer. Sulzer Technical Review ResearchIssue, 1966, 45–48. Terens, L. & Schäfer, R. (1993). Variable speed in hydropower generation utilizing static frequency converters. In Proceedings of the International Conference on Hydropower, Nashville, Tennessee, pp. 1860–1869. Zhang, Zh. (2009). Freistrahlturbinen. Springer-Verlag. Zhang, Zh. (2016a). Pelton turbines. Springer-Verlag. Zhang, Zh. (2016b). Transient flows in pipe system with pump shut-down and the simultaneous closing of the spherical valve. In IAHR, 28th Symposium on Hydraulic Machinery and Systems, Grenoble, France; See also IOP Conference Series: Earth and Environmental Science (Vol. 49) (2016) (052001) https://doi.org/10.1088/1755-1315/49/5/052001. Zhang, Zh., Li, Z., Wei, X,. & Qin, D. (2018, January). Structure constant G and streamline similarity method for flow distributions at the low pressure sides of the pump and the turbine impellers. Large Electric Machine and Hydraulic Turbine. Zhang, Zh. (2018a, February) Streamline similarity method for flow distributions and shock losses at the impeller inlet of the centrifugal pump. Journal of Hydrodynamics, 30 (1), 140–152. Zhang, Zh. (2018b, April). Master equation and runaway speed of the Francis turbine, Journal of Hydrodynamics, 2018, Vol. 30, Issue 2, pp. 203–217. Zhang, Zh. (2018c, August). Wave tracking method of hydraulic transients in pipe systems with pump shut-off under simultaneous closing of spherical valves. Journal of Renewable Energy, 132, 157–166.

Chapter 11

Application Examples of Complex Transient Computations

The wave tracking method basically represents a highly advanced method for computing hydraulic transients. Because of its simple and clearly structured computational algorithms, it can be easily applied to any complex hydraulic system. Both the applicability and reliability of the method will be demonstrated in this chapter by presenting diverse application examples. These include the shut-down of a Pelton turbine in a relatively simple pipeline system, the start-up of a pump, the emergency shut-down of a pump with simultaneous closing of a spherical valve. All computations are relying on the use of MS Excel, even for the transients originated by the rupture of pipelines. For the benefit of readers, it is always helpful, when computing hydraulic transients, each time to write down all relevant computational equations. All computational results in this chapter are selected from a series of case-studies that were carried out by the author at the Oberhasli Hydroelectric Power Company (KWO Ltd.). These include: • Start-up and shut-down of Pelton turbines, including emergency shut-down during the start-up process; • Start-up of the pump via the co-axis Francis-turbine; • Start-up of the pump with given rotational speed curve; • Start-up of the pump with given motor power curve; • Emergency shut-down of the pump with simultaneous closing of the spherical valve; • Burst of the pressurized pipeline ahead of a water turbine; • Dimensioning of surge tanks in the extended hydraulic system shown in Fig. 1.5. For performing all these transient computations, the wave tracking method was applied with the aid of MS Excel.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4_11

273

274

11 Application Examples of Complex Transient Computations

11.1 Shut-Down of a Pelton Turbine In Sect. 3.1, a transient measurement has been shown in Fig. 3.1. It was about a shut-down of a Pelton turbine by progressively closing the injector nozzle in a real hydraulic system. The related hydraulic system consists of one surge tank and three pipes of different diameters and lengths. It is somewhat more complex than that considered in Sect. 7.15.1, where the existence of the surge tank was ignored. The computational algorithm, however, is almost the same. In Sect. 7.15.4, the wave tracking method was applied to the computation of the flow oscillation between the lake and the surge tank. The basic concept of computations, as shown in Table 7.4, can be applied also to the current case. One only needs to additionally consider the traveling performance of shock waves at sections with changeable flow areas. The computations of transients in the hydraulic system shown in Fig. 3.1, as in general, include the following steps: – – – –

Computation of the injector characteristic; Generation of the primary shock waves; Creation of equations for discharge or shock pressures; Tracking the propagation of shock waves.

Because the computations basically rely on the construction of numerical solutions, it is recommended, for programing purposes, to always write down all necessary equations that are used for programming. Step 1: Injector Characteristic and Closing Law First, Eq. (2.32) or Eq. (10.33) is used to compute the discharge through the injector: √ ϕ AD0 2gh  Q=  2 ,  1 − ϕ AD0 A

(11.1)

with the discharge coefficient according to Eq. (10.30)  2 s s ϕ = a1 + a2 . D0 D0

(11.2)

The pressure head in Eq. (11.1) is related to the transient shock pressure at the injector. In order to compare the computations with the available measurements, the closing of the injector nozzle s = f(t) is defined as the same as that shown in Fig. 3.1. Step 2: Water Hammer Equation Another necessary equation to solve hydraulic transients is related to the water hammer. From Eq. (7.3), one obtains immediately h = h0 −

a (c − c0 ) + 2 f. g

(11.3)

11.1 Shut-Down of a Pelton Turbine

275

Within the time of a round trip of the first shock  wave in the first pipe segment (d = 1.4 m, L 1 = 167 m in Fig. 3.1), i.e., t = 2L 1 a, the downstream shock wave f in the above equation is zero. The next step is to relate the above equation to Eq. (11.1), in order to compute the primary shock wave. Step 3: Determination Equation and Primary Shock Waves As said, the transient computations are based on the solution of the valve characteristic and the water hammer equation. Combining Eqs. (11.1) and (11.3) and using Q = c A, one obtains aϕ AD0 h+ g A



√  ∗  2g   2 h − h 0 + 2 f = 0, 1 − ϕ AD0 A

(11.4)

with h ∗0 = h 0 +

a c0 . g

(11.5)

This equation is already known, see Eq. (7.24). At the very beginning of closing the valve there is f = 0. At other times, f is obtainable from tracking the propagation of the shock waves. Then, from Eq. (11.4), the pressure head h at the injector can be computed as a function of the specified flow regulation ϕ(t). The wave speed with value a = 1238 m/s can be applied, as it has been measured and is given in Eq. (3.56). The time increment can be set equal to t = 0.135 s. It corresponds to the traveling time of the pressure shock wave in the first pipe of length L 1 = 167 m. It can also approximately be used as the time for a round trip of shock waves in the surge tank. Once the pressure head h at the injector is determined from Eq. (11.4), the primary pressure shock is computed from Eq. (7.28) F = h − h 0 − f.

(11.6)

Within the first two time-intervals t ≤ 2t, one has f = 0. Then, the flow velocity is obtained, for instance, from Eq. (7.11) c − c0 =

g ( f − F). a

(11.7)

Step 4: Wave Traveling Performance The traveling performance of waves includes the wave transmission and reflection at pipe sections with changing cross-sectional area. In the current computations, this concerns, first, the T-junction at the surge tank and, second, the series junction of pipes with d = 1.65 m and d = 1.40 m, as shown in Fig. 11.1. With defined positive

276

11 Application Examples of Complex Transient Computations

c3 c1

f3

F3

f1

f2

F1

c2

F2

f1

f2

F1

F2

c1

c2

Fig. 11.1 Wave traveling performances at sections with changing cross-sectional areas (refers to Fig. 3.1)

flow velocities and thus the up- and downstream shock waves at such connections, the following traveling laws of pressure shock waves can be obtained. At the T-junction: F1 =

A1 − A2 − A3 2 A2 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(11.8)

f2 =

2 A1 A2 − A1 − A3 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(11.9)

f3 =

2 A1 2 A2 A3 − A1 − A2 f1 + F2 + F3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(11.10)

At the series junction of two pipes: F1 =

A1 − A2 2 A2 f1 + F2 , A1 + A2 A1 + A2

(11.11)

f2 =

2 A1 A2 − A1 f1 + F2 . A1 + A2 A1 + A2

(11.12)

This group of equations has been directly taken over from equations in Sects. 7.5.1 and 7.4.2. They help for easy programming of computations, for instance, by means of MS Excel. Step 5: Viscous Friction and Resistance Effects The numerical computations in the current case follow similar steps as in Tables 7.2 and 7.3 as well as in Table 7.4 showing the treatment of the T-junction (surge tank connection). When tracking the waves F and f, the viscous frictional resistances should be taken into account. According to Eqs. (7.103) and (7.104), this occurs in the following form fi = fA −

1 RA (Q i−1 |Q i−1 | − Q 0 |Q 0 |), 2

(11.13)

Fi = FB +

1 RB (Q i−1 |Q i−1 | − Q 0 |Q 0 |). 2

(11.14)

11.1 Shut-Down of a Pelton Turbine

277

For a round trip of shock waves between the surge tank and the reservoir, the viscous friction is considered according to Eq. (7.107) f t = −Ft−2t − R(Q i−1 |Q i−1 | − Q 0 |Q 0 |) +

 1  2 Q 0 − Q 2i−1 . 2 2g A

(11.15)

This equation also accounts for the entrance velocity effect by the last term which in most applications can be neglected. At the entrance of the surge tank, the throttle resistance behaves as a local resistance. Its effect on the status of pressure shock waves in the T-junction is considered together with the effect of the changeable height of the water in the surge tank. With the aid of Eq. (7.117), this is given as F3 = − f 3 + Rthrottle Q 3 |Q 3 | + (h ST − h 0 ).

(11.16)

Step 6: Programming and Computations Table 11.1 presents the numerical √ steps of the computations. The parameters kB and kC refer to the coefficient of h and the last quasi-constant term in Eq. (11.4), respectively. These six steps form the main and the general procedure of transient computations. They are necessary not only for the wave tracking method (WTM) but also for the method of characteristics (MOC), if this would be applied. Computational Results Computational results are shown in Fig. 11.2 for the shock pressure at the injector, together with measurements for comparison, see also Figs. 1.4 and 10.14. Obviously, excellent agreement between computations and measurements has been achieved. Further, in Fig. 11.3, computation results up to t = 400 s are shown, again for comparison with measurements. Some differences between the computations and the measurements occur because of the simplifications used in the computations. Both measurements and computations demonstrate that overflow of water at the surge tank has happened and took more than two hundred cubic meters. The surge tank has about the form of Fig. 5.2c with an upper chamber. While in the experiments this volume of water could flow back into the surge tank again, it has been ignored in computations. This also leads to a difference in the measured and computed periods of low-frequency flow oscillations. Because of back flow of water into the surge tank, the period of the real flow oscillation, as evidenced by the measurements, is significantly longer than that obtained from the computations. The initial purpose of performing the measurements and computations has been to estimate the maximum pressure rise in the system, as shown in Fig. 11.2. The low-frequency flow oscillations, as shown in Fig. 11.3, were not the topic at that time. The present computation again demonstrates that the wave tracking method is also able to compute the low-frequency flow oscillations in a hydraulic system. There is

s

145.4

145.4

144.7

143.9

143.2

142.4

141.7

141.0

140.2

139.5

138.8

138.1

137.3

136.6

135.9

135.2

134.4

133.7

133.0

132.3

t

0.00

0.13

0.26

0.39

0.52

0.65

0.78

0.91

1.04

1.17

1.30

1.43

1.56

1.69

1.82

1.95

2.08

2.21

2.34

2.47

Injector

0.52

0.52

0.52

0.53

0.53

0.53

0.53

0.53

0.53

0.54

0.54

0.54

0.54

0.54

0.55

0.55

0.55

0.55

0.55

0.55

ϕ

26.7

26.6

26.6

26.5

26.4

26.3

26.2

26.1

26.0

25.9

25.9

25.8

25.7

25.6

25.5

25.4

25.3

−0.2

−0.4

−0.6

−0.7

−0.9

−1.1

−1.3

−1.5

−1.7

−1.8

−2.0

−2.2

−2.4

−2.5

−2.7

−2.9

26.8

26.9

26.9

kb

0.0

0.0

0.0

0.0

f

510 511 513

−1118

−1118

−1118

−1112

−1113

−1113

−1113

−1114

−1114

−1114

−1115

−1115

−1115

−1116

−1116

−1116

−1117

−1117

530

528

527

526

525

524

523

522

521

520

519

518

517

516

515

514

510

−1118

−1118

h

kc

Table 11.1 Numerical steps of computations

22.1

20.8

19.5

18.3

17.0

15.7

14.5

13.2

12.0

10.8

9.6

8.4

7.1

6.0

4.8

3.6

2.4

1.2

0.0

0.0

F

4.62

4.63

4.64

4.65

4.66

4.67

4.68

4.70

4.71

4.72

4.73

4.74

4.75

4.76

4.77

4.78

4.79

4.80

4.81

4.81

c (m/s)

7.10

7.12

7.14

7.16

7.18

7.19

7.21

7.23

7.24

7.26

7.28

7.30

7.31

7.33

7.35

7.36

7.38

7.39

7.41

7.41

Q

0.2

0.2

0.1

0.1

0.1

0.1

0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

f1

20.7

19.4

18.2

16.9

15.7

14.4

13.2

12.0

10.7

9.5

8.3

7.1

5.9

4.7

3.6

2.4

1.2

0.0

0.0

0.0

F2

17.4

16.3

15.2

14.2

13.1

12.1

11.0

10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

0.0

0.0

F1

−3.2

−3.0

−2.8

−2.6

−2.4

−2.3

−2.1

−1.9

−1.7

−1.6

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.0

0.0

f2

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

f1

Pipe connection (ϕ1.4 − ϕ1.65)

7.8

6.8

5.8

4.8

3.9

2.9

1.9

0.9

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

F1

−2.6

−2.1

−1.6

−1.2

−0.7

−0.3

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

F2

1.0

1.0

1.0

0.9

0.9

0.8

0.7

0.3

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

F3

−6.8

−5.8

−4.8

−3.9

−2.9

−2.1

−1.2

−0.6

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

f1

T-junction (surge tank)

0.2

0.2

0.1

0.1

0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

Q3

(continued)

3.6

3.1

2.6

2.1

1.6

1.2

0.7

0.3

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

f3

278 11 Application Examples of Complex Transient Computations

s

131.6

130.9

130.2 .. .

t

2.60

2.73

2.86 .. .

Injector

0.51 .. .

0.52

0.52

ϕ

Table 11.1 (continued)

kb

25.2

25.1

25.1 .. .

f

−3.1

−3.2

−4.1 .. .

−1110 .. .

−1112

−1112

kc

532 .. .

532

531

h

25.8 .. .

24.7

23.4

F

4.58 .. .

4.59

4.60

c (m/s)

7.04 .. .

7.07

7.09

Q

−1.0 .. .

−0.4

0.2

f1

24.6 .. .

23.3

22.0

F2

20.4 .. .

19.4

18.5

F1

−5.1 .. .

−4.2

−3.3

f2

0.0 .. .

0.0

0.0

f1

Pipe connection (ϕ1.4 − ϕ1.65)

10.8 .. .

9.8

8.8

F1

−4.1 .. .

−3.6

−3.1

F2

1.1 .. .

1.0

1.0

F3

−9.7 .. .

−8.7

−7.8

f1

T-junction (surge tank)

5.1 .. .

4.6

4.1

f3

0.3 .. .

0.3

0.2

Q3

11.1 Shut-Down of a Pelton Turbine 279

280

11 Application Examples of Complex Transient Computations

Fig. 11.2 Comparison between measured and computed pressure response at injectors of a twoinjector Pelton turbine after the shut-down of the turbine under full-load (Q0 = 7.4 m3 /s)

no reasonable increase of computational expenditures. One only needs to extend the computations from Figs. 11.2 to 11.3 (up to 400 s) by enlarging the number of computational steps. This just represents a great advantage against the method based on the solid water column theory, at which many momentum equations must be iteratively solved, see Sect. 5.2.

11.2 Pump Emergency Stop with Simultaneous Closing of a Spherical Valve The computations in this section refer to the publications (Zhang 2016, 2018), but with more details. The considered hydraulic system in a hydropower station is shown in Fig. 11.4. It consists of three surge tanks and eight pipes of different diameters. Obviously, hydraulic transients in such a hydraulic system may be highly complex and cause highly complex computations. One must perform computations on both the high and low pressure sides of the pump. In addition, the emergency shut-down of the pump always leads to the simultaneous closing of the spherical valve. In such cases, it is recommended to apply the unified characteristics of the pump and the spherical valve.

11.2 Pump Emergency Stop with Simultaneous Closing …

281

Fig. 11.3 Comparison between the measured and computed pressure response and oscillations at the injector for the same case as in Fig. 11.2

282

11 Application Examples of Complex Transient Computations

Lake

Lake

Surge tank C

Tilted surge tank B

Surge tank A

c3 f3

Pump house f1

f2

F1

F2

F3

f1 c1

f2 c2

F1

F2

Fig. 11.4 Pumped storage power station (KWO GR2)

11.2.1 Unified Characteristics and Rotor Dynamics According to Fig. 11.4, a spherical valve is installed on the high pressure side of the pump. It remains, under normal operation conditions, fully opened and does not cause any flow resistance. In the case of emergency shutdown of the pump, however, the spherical valve will be automatically closed. This will additionally complicate the transient flow induced by the pump. In order to compute the associated complex hydraulic transients in the considered complex hydraulic system, as stated before, the unified characteristics of the pump and the spherical valve according to Eq. (10.26) should be applied. For the sake of clarity and further applications, this equation is rewritten here, viz.,

|Q| cp 1 m2 π m1n m 0 (π d2D n)2 . (11.17) − Q+ Q2 + H12 = 4 2 2g d2D Q AsphV 2gd2D 2g All three coefficients m 0 , m 1 and m 2 are determined from the four-quadrantdiagram of the pump, so that this equation is able to account for both the reverse flow and the inverse rotation of the pump. On the other hand, the discharge through the pump also depends on its rotational speed which varies during the shut-down of the pump. The slowing down of the rotational speed, in turn, is fundamentally determined by the dynamic balance of the rotor system. In case of pump shut-down, the active resisting torque exerted on the rotor shaft is simply the shaft torque which is related to the hydraulic characteristic of the pump given by Eq. (10.15). With respect to the moment of inertia (J) of the rotor system, the rotor dynamics is described by 2π J

dn = −Mshaft . dt

Because of Eq. (10.15), one further obtains

(11.18)

11.2 Pump Emergency Stop with Simultaneous Closing …

ρg 3 dn =− d Hpu M11 . dt 2π J 2D

283

(11.19)

For M11 one applies Eq. (10.13), i.e., the characteristic like that in Fig. 10.5b.

11.2.2 Connection of Shock Pressures on Both Sides of the Pump Unit This is the second step of considering the water hammer equation. For the considered hydraulic system shown in Fig. 11.4, the hydraulic transients occur on both the suction (marked by 1) and the pressure side (marked by 2) of the pump unit. They are, however, always connected by equal flow rates Q. On the suction side of the pump, the pressure-head increase is computed from Eq. (7.3) as h 1 − h 1,0 = 2 f 1 −

a (Q − Q 0 ). g A1

(11.20)

Within the time before the first upstream shock wave gets reflected at the surge tank A and travels backwards towards the pump unit, there is f 1 = 0. On the pressure side, the pressure-head increase is deduced from Eq. (7.4) to be h 2 − h 2,0 = 2F2 +

a (Q − Q 0 ). g A2

(11.21)

For the same reason, there is F 2 = 0 within the time for a round trip of the first downstream pressure surge. At this time, the first reflection of the primary wave occurs at the first series junction of the pipes. The difference in the pressure heads on both sides of the unified pump and spherical valve is then obtained as   a h 2 − h 1 = h 2,0 − h 1,0 + 2(F2 − f 1 ) + g



 1 1 + (Q − Q 0 ). A1 A2

(11.22)

or with H12 = h 2 − h 1 and H12,0 = h 2,0 − h 1,0 according to Fig. 10.7 in short H12

a = H12,0 + 2(F2 − f 1 ) + g



 1 1 + (Q − Q 0 ). A1 A2

(11.23)

284

11 Application Examples of Complex Transient Computations

11.2.3 Determination Equation and Primary Shock Waves On both sides of the pump unit, the hydraulic transients are so connected that the flow rate and the pressure head at the pump unit must satisfy the unified characteristic. For this reason, Eq. (11.23) is combined with Eq. (11.17). One obtains the final determining equation as 

m2 4 d2D

−

+2g H0∗

|Q| cp Q A2sphV



Q2 +

πm 1 n d2D

− 2a



1 A1

+

1 A2

+ m 0 (π d2D n) − 4g(F2 − f 1 ) = 0 2

 Q

,

(11.24)

with H0∗

a = g



 1 1 Q 0 − H12,0 . + A1 A2

(11.25)

This equation has a structure of a quadratic form for the volume flow rate Q. From tracking the wave parameters f 1 and F 2 , the flow rate at each time can be immediately computed as functions of the rotational speed and the opening degree of the spherical valve (represented by cp ). At the very beginning of transient occurrences, i.e., within the times for a round trip of respective first primary waves, there are, respectively, f 1 = 0 and F2 = 0. The computations can thus be well started. They must be numerically performed with an appropriate finite time increment. The instant value  |Q| Q = ±1 in the equation can be set to be equal to that of the last finite time step. The rotational speed n is numerically obtained as  n i = n i−1 +

dn dt

 t,

(11.26)

i−1

 in which dn dt is from the last time step and is basically computed from Eq. (11.19). For its computation, the related computations of some other quantities have to be performed in advance, as shown below and described for more details in the next section of numerical computations. As soon as the flow rate is computed each time from the above quadratic equation, the primary shock waves generated on both the suction and the pressure sides, respectively, can be computed from Eq. (7.11) as a (Q − Q 0 ), g A1 a f 2 = F2 + (Q − Q 0 ). g A2

F1 = f 1 −

(11.27) (11.28)

In these two equations, both f 1 and F2 are incoming waves and, therefore, are known from previous time steps.

11.2 Pump Emergency Stop with Simultaneous Closing …

285

On the one hand, these two primary shock waves instantaneously travel in the lower and upper pipelines, respectively, away from the pump unit. One needs to track them along for further transient computations in the next time steps and at other interested positions in the system. On the other hand, the instant pressure heads on both sides of the pump unit are computed from Eq. (7.10) as h 1 = h 1,0 + F1 + f 1 ,

(11.29)

h 2 = h 2,0 + F2 + f 2 .

(11.30)

The effective head of the pump unit (pump and valve) is then obtained as H12 = h 2 − h 1 .

(11.31)

During computations, one can check that this pressure head difference is equal to that in Eq. (11.17). Correspondingly, the head over the pump alone is given by Hpu = H12 + h sphV = H12 +

|Q| cp Q2. Q 2g A2sphV

(11.32)

For h sphV and cp see Eqs. (2.34) and (2.35).  The computed pump  head Hpu will be used to compute dn dt as listed in Eq. (11.19). With dn dt at a given time, then the rotational speed n in the next time step can be straightforwardly computed from Eq. (11.26). Figure 11.5 shows the scheme of computations.

11.2.4 Tracking the Shock Waves In the last section, the primary shock waves (F1 and f 2 ) on both sides of the unified pump and valve are computed. These two shock waves propagate, respectively, in the lower and the upper pipelines towards the respective lakes, where the waves will be reflected and propagate back to the pump unit. During their propagations in each pipeline, the shock waves are subjected to viscous friction effects as well as to the disturbing (reflection and transmission) at each stepped pipe section and T-junction. The corresponding tracking paths are also illustrated in Fig. 11.5. If the viscous friction effect should be considered, then Eqs. (7.7) and (7.8) need to be applied; they take the form fx = fA −

1 RA (Q|Q| − Q 0 |Q 0 |), 2

(11.33)

286

11 Application Examples of Complex Transient Computations

i=0: n0, Q0, H12, 0, H0* f1=0, F2=0 f1

surge tanks B and C transitions of pipes lake

surge tank A transition of pipes lake

F2

tracking

i=1, 2, 3… Eq. (11.24): Q=f(n, cp)

n, β

Eq. (11.23): H12 Eq. (11.32): Hpu

i+1 Eq. (11.26): n

Eq. (10.11): n11, Q11 Fig. 10.5: M11=f(n11)

Eq. (11.19): dn/dt

f2 F1

Eq. (11.28): f2 Eq. (11.27): F1

Eq. (11.29): h1 Eq. (11.30): h2

Fig. 11.5 Scheme of computations

Fx = FB +

1 RB (Q|Q| − Q 0 |Q 0 |). 2

(11.34)

Usually, the entrance velocity effect, as described in Sect. 7.7.1, can be neglected in complex transient computations. The pumping system considered currently (Fig. 11.4) consists of three different surge tanks on both sides of the pump house and totally eight pipes of different diameters and lengths. For the purpose of demonstrating the computations, the performance of wave reflection and transmission at the T-junction of the surge tank C is considered. The representative wave parameters are shown in the figure. With three approaching waves ( f 1 , F2 and F3 ), which are known from computations in previous time steps, the three departure waves are computed by Eqs. (7.77)–(7.79), rewritten here as F1 =

A1 − A2 − A3 2 A2 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(11.35)

f2 =

2 A1 A2 − A1 − A3 2 A3 f1 + F2 + F3 , A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(11.36)

f3 =

2 A1 2 A2 A3 − A1 − A2 f1 + F2 + F3 . A1 + A2 + A3 A1 + A2 + A3 A1 + A2 + A3

(11.37)

11.2 Pump Emergency Stop with Simultaneous Closing …

287

This set of equations represents the traveling laws of pressure waves at a Tjunction. The flow area A3 refers to the sectional area of the surge tank behind the throttle. For computational purposes, for instance by self-programming, one should in principle always write down all the above relations at each T-junction and the simplified relations (with A3 = 0) at each series junction of two pipes. This is necessary, because for converging and diverging flows at a T-junction, the traveling laws of pressure waves have different forms, see Sect. 7.5. Further, as a rule throughout this book, the downstream wave f always propagates in the positive velocity direction, while the upstream F always propagates in the negative velocity direction. The total effect of both the throttle resistance and the moving free surface of the water in the surge tank on the traveling performance of the waves F 3 and f 3 should be considered according to Eq. (7.117) as follows: F3 = − f 3 + Rthrottle Q 3 |Q 3 | + (h ST − h 0 ).

(11.38)

When the pressure wave f 2 reaches the upper lake, then full reflection will occur, according to (7.95), in the form F2 = − f 2 .

(11.39)

This relation relies on the fact that the water level in the lake remains unchanged.

11.2.5 Numerical Computations The key equation for computing the current hydraulic transients is Eq. (11.24). This equation enables the flow rate at each instant to be computed in a simple way. For the time-dependent solution of the transient flow, only numerical methods can be applied. First, one usually specifies the maximum allowable time increment, in order to resolve the flow state in the shortest pipe in a pipeline network. In the current example, the shortest path is obviously found in the round trip of the wave in the surge tank (from the throttle to the free surface of the water and back to the throttle). However, one can only specify an average path, because the height of water in the surge tank changes as a result of transients. Such an approximation will hardly affect the computational results, as it has been confirmed many times by the author based on data simulations. By following the computational scheme in Fig. 11.5, the complete computations can be performed, for instance, simply by means of MS Excel. They are straightforward and therefore simple, because no syntax like “if-then-else” is used.

288

11 Application Examples of Complex Transient Computations

11.2.6 Computational Results At the considered pumping system, a test of emergency shut-down of the pump with simultaneous closing of the spherical valve was earlier conducted. A part of the measurement results will be used to verify the numerical computations, as outlined above and based on the use of MS Excel tools. Figures 11.6 and 11.7 show the pressure-head responses at the pump inlet and exit, respectively. The computations agree very well with measurements. The dominant pressure fluctuation at the pump exit, with a period of about 2.3 s, corresponds to the wave propagation between the pump house and the tilted surge tank B on the pressure side (T 4L = 2.3 s). Especially, as shown in Fig. 11.8, the time-dependent rotational speed after the pump shutdown could precisely be predicted by computations. It should be mentioned that the process of dropping off the rotational speed depends not only on the pump and the valve characteristics but also on the total moment of inertia (J) of the rotor system and the closing time of the spherical valve. In the considered case, the closing time of the valve took 17 s. Before the spherical valve is completely closed, the back flow is also confirmed, as shown in Fig. 11.9.

Fig. 11.6 Pressure response at the pump inlet

Fig. 11.7 Pressure response at the pump exit

11.2 Pump Emergency Stop with Simultaneous Closing …

289

Fig. 11.8 Rotational speed of the pump after the emergency shut-down

Fig. 11.9 Flow rate through the pump after the emergency shut-off

The computed system oscillation is shown in Fig. 11.10 for comparison with measurements. It comprises the main, i.e., the primary oscillation, between the lake

Fig. 11.10 Pressure head at the surge tank and system oscillations

290

11 Application Examples of Complex Transient Computations

(upper side) and the surge tank, and the secondary oscillation, between the vertical and the tilted surge tanks. Analytically, the primary oscillation period is computed according to Eq. (4.44) or Eq. (5.43) by  T = 2π

AST L pipe , g Apipe

(11.40)

with L pipe as the pipe length from the upper lake to the main surge tank which has a section equal to AST . In the current application, the primary oscillation period has been computed to be Tprimary = 234 s. It agrees very well with both measurements and transient computations. The high accuracy of computations, as presented above, relies on the applications of real operation conditions without any unreliable assumptions. The MS Excel tool has been demonstrated to be very powerful. Because of these properties, on-line investigations of all flow and operating parameters can be easily performed.

11.3 Pump Start Each pump start will cause a hydraulic transient in the related pipeline network. For demonstrating the associated phenomena, the hydraulic system with the known pump in Sect. 10.1 is again considered, as shown in Fig. 11.4 for a pumped storage power station. It is about a rather complex hydraulic system with three surge tanks and eight pipes of different diameters. On the pressure side of the pump, a spherical valve as a closure organ is installed, in order to completely block the flow from the upper to the lower reservoir. The opening of the spherical valve in connection with the pump start has to be performed with respect to the so-called shut-off speed of the pump. This represents a speed at which the water from the rest (Q = 0) tends to move. For the considered pump, the shut-off speed has been shown in Fig. 10.6. It obviously depends on the altitude difference between the upper and lower reservoirs. For an altitude difference H = 400 m, the shut-off speed takes n shut−off = 660 rpm, see Eq. (10.6). While starting the pump, the spherical valve should only be opened, after this shut-off speed has been reached, as described in Sect. 10.1 with the aid of Fig. 10.2. The start of the pump can be of different ways. This includes, for instance – predefined power or toque of the electric motor – speed-up of the pump by opening and starting the coaxial hydraulic turbine – predefined rotational speed of the pump n = f(t). In all cases, the angular moment of inertia of the pump is balanced by the electrical, hydraulic, and mechanical resistance torques. Before the shut-off speed is reached, the hydraulic torque is zero, so that no flow processes need to be considered. In view of the mechanical loss given by Eq. (10.8), the dynamic balance of the rotor system

11.3 Pump Start

291

is given by Pmotor = J

 n 3 dω . ω + 28.5 dt 660

(11.41)

Here, J represents the moment of inertia of the entire rotor system and ω is the angular velocity. The shut-off pressure, which also indicates the pressure in the pump casing, is directly determined by the shut-off speed and given by the shut-off pressure coefficient ψshut−off = 1.29, see Sect. 10.1.1. After the shut-off speed has been reached, the hydraulic torque becomes dominant in balancing the dynamics of the system. The equilibrium of the torques exerted on the rotor shaft is given by dω Pmotor =J + Mshaft . ω dt

(11.42)

To the shaft torque in this equation, the four-quadrant diagram (Fig. 10.5b) with respect to M 11 in Eq. (10.11) can be applied. Because for normal start of the pump, one always has n > 0 and Q > 0, the shaft torque can also be represented by Mshaft =  Pshaft ω, with Pshaft according to Eq. (10.3) and from Fig. 10.1b. In both cases, the unified characteristics, i.e., Eqs. (10.18) and (10.26), are identical. In using Eq. (10.3), one obtains with ω = 2π n dn 1 Pmotor 5 = 2π J + λρπ 2 n 2 d2D . ω dt 4

(11.43)

In the following computations, the motor power is assumed to be given as a linear function of time. The start-up of the pump by opening and starting the coaxial Francis turbine can be found in technical reports (Zhang 2011a, b).

11.3.1 Computational Specifications and Algorithms For the purpose of demonstrating computational procedures, only the start process of the pump is considered from the moment when the shut-off speed is reached and the spherical valve begins to open; This time is set to zero. Then, Eq. (11.41) does not need to be further considered. The motor power in Eq. (11.42) is predefined in the following linear form Pmotor = 35 +

91 − 35 t (MW). tmotor

Then, from Eq. (11.43), dn/dt is obtained as

(11.44)

292

11 Application Examples of Complex Transient Computations

1 dn = dt 2π J



 Pmotor 1 5 . − λρπ 2 n 2 d2D ω 4

(11.45)

According to Appendix C.1, the time for opening the spherical valve should be shorter than or comparable to the time tpower for the increase of the motor power. For simplicity, the linear opening of the valve is assumed with tsphV = tmotor = 30 s. The computational algorithm  is similar to that in Sect. 11.2 and Fig. 11.5. Especially, Eq. (11.24) with |Q| Q = 1 and Eq. (11.25) for H0∗ are also applicable to the current case, as rewritten here 

  cp m2 πm 1 n 1 1 2 Q Q + − 2a + 4 − 2 d2D A1 A2 d2D AsphV , (11.46) 2 ∗ +2g H0 + m 0 (π d2D n) − 4g(F2 − f 1 ) = 0 with H0∗

a = g



 1 1 Q 0 − H12,0 . + A1 A2

(11.47)

The discharge can then be computed. Further, Eqs. (11.23) and (11.32) are used to compute the pressure head H12 over the unified pump-valve and the pressure head Hpu over the pump alone. At this moment, the discharge coefficient ϕ can be computed from Eq. (10.1) and then the power coefficient λ from Eq. (10.5), i.e., the characteristic according to Fig. 10.1b. The corresponding computational algorithm is shown in Fig. 11.11. Next, from the given motor power in Eq. (11.44), the changing rate of the rotational speed is computed from Eq. (11.45). It behaves as the basis for the determination of the rotational speed in the next time step according to Eq. (11.26). In parallel, the two primary shock waves, f 2 on the pressure side and F 1 on the suction side of the unified pump unit, are computed from Eqs. (11.28) and (11.27), respectively. As shown in Fig. 11.11, one needs to further tracking both of them.

11.3.2 Computational Results Before showing the computational results, it should be indicated that for the current case of the start-up of the pump, the simplified computations by neglecting the water hammer have also been completed, as shown in Appendix C.2. The purpose of doing these additional computations is to confirm the outcomes of the applied simplifications. Figures 11.12, 11.13, 11.14 and 11.15 show the significant system reactions during the specified start-up of the pump with tsphV = tmotor = 30 s. At the end of the fullopening of the valve, i.e., by reaching the rated motor power, the rotation of the pump is speeded up to its rated value. Correspondingly, the rated discharge is also reached.

11.3 Pump Start

293

i=0, t=0 nshut-off, Q0=0, H12, 0, H0* f1=0, F2=0 f1

surge tank A transition of pipes lake

surge tanks B and C transitions of pipes lake

F2

tracking

f2

F1 Fig. 11.11 Scheme of computations Fig. 11.12 Rotational speed during the start-up process

Fig. 11.13 Discharge through the pump during the start-up process

i=1, 2, 3… Eq. (11.46): Q=f(n, cp)

β n

Eq. (11.23): H12 Eq. (11.32): Hpu

i+1 Eq. (11.26): n

Eq. (10.1): ϕ Fig. 10.1b: λ=f(ϕ)

Eq. (11.44): Pmotor Eq. (11.45): dn/dt

Eq. (11.28): f2 Eq. (11.27): F1

Eq. (11.29): h1 Eq. (11.30): h2

294

11 Application Examples of Complex Transient Computations

Fig. 11.14 Pressure head over the unified pump-valve during the start-up process

Fig. 11.15 Shock pressures on the suction (b) and the pressure (a) sides of the pump itself during the start-up process with tsphV = tmotor = 30 s

Both the rotational speed and the discharge are obviously subjected to the dominant system oscillation between the surge tank C (Fig. 11.4) and the upper lake. Such a dominant system oscillation can be well confirmed from Fig. 11.14, which shows the pressure head over the unified pump and valve. The same oscillation period as in Fig. 11.10 is obtained from computations. The secondary pressure fluctuation is the total effect of flow oscillations in all three surge tanks, which are connected by the pump and the pipelines. The shock pressure on the suction and the pressure side of the pump itself are shown in Fig. 11.15. The maximum pressure at the pump exit, i.e., prior to the spherical valve, reaches a level equal to 583 m. It is about 67 m above the static mean value of the pressure head (516 m) at the pump. This large pressure rise is mainly because of the partial opening of the spherical valve. This pressure rise has to be considered to be significantly high. It can only be reduced by slowly increasing the motor power, for instance, within a time period t motor = 60 s, while the opening of the spherical valve remains at t sphV = 30 s. Figure 11.16 shows the corresponding computational results. The maximum pressure head reaches 566 m. The pressure rise against the static value is this time about 50 m. When compared with computational results that are presented in Appendix C.2 by neglecting the water hammer, one confirms, as shown there, quite similar system

11.3 Pump Start

295

Fig. 11.16 Additional computations of shock pressures on the suction (b) and the pressure (a) sides of the pump during the start-up process with tsphV = 30 s and tmotor = 60 s

reactions on the start-up of the pump. However, the shock pressure at the pump exit, in view of Fig. 11.15a, cannot be estimated by those approximations. For this reason, transient computations for the start-up process of the pump are indispensible.

References Zhang, Z. (2011a). Betriebsverhalten von Pumpen mit variabler Drehzahl für den Fall Grimsel 2. KWO Report, No. A000219861. Zhang, Z. (2011b). Startvorgang der Pumpe in Grimsel 2 und Druckstossberechnungen. KWO Report, No. A000219862. Zhang, Z. (2016). Transient flows in pipe system with pump shut-down and the simultaneous closing of the spherical valve. In IAHR, 28th Symposium on Hydraulic Machinery and Systems. France: Grenoble. Zhang, Z. (2018). Wave tracking method of hydraulic transients in pipe systems with pump shut-off under simultaneous closing of spherical valves. Journal of Renewable Energy, 132.

Appendix A

Nomenclature

Symbol

Unit

Meaning

a

m/s

Wave propagation speed

a1 , a2

–

Injector nozzle constant

A

m2

Area

AD0

m2

Reference area of the injector nozzle

Aeq

m2

Equivalent pipe cross-sectional area

AST

m2

Cross-sectional area of the surge tank

B

m0.5

Parameter, see Eqs. (4.54) and (9.13)

c

m/s

Flow velocity

cf

–

Shear stress coefficient

cp

–

Resistance coefficient of a regulation valve

C

–

Parameter, see Eqs. (4.54) and (9.13)

CA, CB

m

Parameter used in Method of Characteristics (MOC)

d

m

Pipeline diameter

d 1D

m

Impeller diameter (turbine)

d 2D

m

Impeller diameter (pump)

D0

m

Aperture diameter of Pelton injector nozzle

EM

N/m2

Young’s modulus

f

m

Downstream shock wave

F

m

Upstream shock wave

F

N

Force

g

m/s2

Gravity constant

G

–

Geometrical structural constant

h

m

Static pressure head (continued)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4

297

298

Appendix A: Nomenclature

(continued) Symbol

Unit

Meaning

h0

m

Static pressure head in stationary flows (reference for transient flows)

he

m

Altitude head difference at the pipe entrance (Fig. 3.4)

htot

m

Total pressure head

H0

m

Available total pressure head

J

Nms2

Moment of inertia

k

m

Pipe roughness height

kA, kB, kC

–

Constants in master equation of the Francis turbine

KW

N/m2

Bulk modulus (volume modulus) of water according to Eq. (3.48)

L

m

Pipe length

m

–

Constant in regressed pump and turbine characteristics

M shaft

Nm

Shaft torque

n

1/s

Rotational speed

p

Pa

Static pressure

ptot

Pa

Total pressure

P

W

Power

Phyd

W

Hydraulic power

Pmech

W

Mechanical power loss

Pmotor

W

Motor power

Pshaft

W

Shaft power

Q

m3 /s

Flow rate

R

s2 /m5

Resistance constant

Re

–

Reynolds number

s

m

Spear-needle stroke of a Pelton injector

s

m

Pipe wall thickness

t

s

Time

tc

s

Closing time

T

s

Period of flow oscillation (fluctuation)

T2L

s

Time for a round trip of shock waves

T 4L

s

Period of pressure fluctuations in a simple pipeline system, according to Eq. (3.24)

x

m

Coordinate along the pipeline axis

z

m

Vertical coordinate

α

deg

Slope angle of a pipeline (Fig. 3.14)

α

deg

Guide vane angle or absolute flow angle at the impeller inlet of a Francis turbine

β

–

Diameter ratio of orifice

β

deg

Opening degree of a spherical valve

ε

–

Circumferential strain (continued)

Appendix A: Nomenclature (continued) Symbol

Unit

Meaning

ζ

–

Resistance coefficient

ζ

–

Shock loss coefficient (Borda-Carnot)

η

–

Efficiency

λ

–

Darcy-Weisbach friction coefficient

λ

–

Coefficient of shaft power (pump)

ν

m2 /s

kinematic viscosity

ρ

kg/m3

Specific density of water

σ

–

Relative opening angle of the disk of butterfly valves

σ

N/m2

Normal stress

τ

N/m2

Viscous wall shear stress

τ

–

Opening of the valve, see Eq. (7.22)

ϕ

–

Coefficient of discharge

ϕE

–

Ratio of kinetic to potential energies, see Eq. (4.9)

ψ

–

Pressure coefficient (pumps)

ψ

–

Scale parameter of the Pelton injector, see Eq. (8.17)

ω

1/s

Angular speed

MOC Method Of Characteristics WTM Wave Tracking Method MDC Method of Direct Computations

299

Appendix B

Characteristics of Regulation Organs

In hydropower stations, diverse regulation organs or control valves are used to control the flows in hydraulic systems. These are mainly spherical, butterfly and gate valves. While the flow is going to be regulated or the machines like pumps and turbines are started up or shut down, corresponding valves will be active. The resultant hydraulic transients can be only computed, if characteristics of respective control valves are known. For this reason, the hydraulic characteristics of spherical, butterfly and gate valves will be presented in this appendix. As a matter of fact, the hydraulic characteristics of the mentioned control valves are all obtainable from measurements and in terms of charts or tables. For transient computations, one basically applies the analytical expression of respective characteristics. One possibility is to approximate existing data by means of the regression method, as this has up to now been applied in all computations. The author of this book analytically derived respective hydraulic characteristics of control valves mentioned above, all by assuming frictionless flow and in relying on fluid mechanics. Because, for different reasons, all related research works have not yet been published, for convenience of readers and application engineers, significant results will be presented below. The accuracies of respective computations will be demonstrated by comparing computational results with known data from the literature.

B.1 Characteristics of Spherical Valves Spherical valves are also called ball valves. According to Fig. B.1a, a spherical valve is designed with a through-flow diameter equal to the pipe diameter. The various valve designs differ from each other only by the geometrical ratio of the throughflow diameter to the ball diameter. This ratio determines the beginning of the rotor position for just opening the flow. For optimum design of spherical valves used in hydropower stations, such a difference in geometrical design is negligible. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4

301

302

Appendix B: Characteristics of Regulation Organs

(a)

(b)

β

d

d

c

Δhsph

Fig. B.1 Spherical valve and the valve characteristic

The pressure-head drop by a spherical valve is determined by the following formula h sphV = cp

c2 Q2 = cp . 2g 2g A2

(B.1)

The reference flow velocity is c = Q/A. The resistance coefficient cp is of purely geometrical character and effectively only a function of the rotor position which is specified by the angle β. Based on flow analyses, the author suggested the following relation 106 cp =  2 − 1. aβ 2 + bβ

(B.2)

The formula is applicable for nearly all spherical valves that are used in hydropower stations. For a good approximation, both the constants a and b in the above equation can be set equal to a = 0.108, b = 1.35.

(B.3)

In Fig. B.1b, corresponding computations have been plotted against the position angle β of the ball. At the closed position (β = 0), the resistance coefficient tends to infinity and at full opening (β = 90◦ ) it vanishes. In dealing with regulation and control valves in other application fields, the flow coefficient or flow factor has often been used to represent the discharge capacity of valves. By using HsphV = h sphV + c2 /2g instead of the pressure drop p as reference, the flow coefficient of a spherical valve has been commonly defined as follows kV =

Q . A 2g HsphV 

(B.4)

Appendix B: Characteristics of Regulation Organs

303

This definition is significant for spherical valves if valves are used primarily for hydraulic control. The reason of including the dynamic head c2 /2g in the reference pressure head is simply because, otherwise, there would be HsphV = 0 at full opening of the valve and the flow coefficient kV would be infinity. Moreover, the flow coefficient defined above is dimensionless. This is particularly advantageous, because it is independent of the size of the valve. It is, however, against the convention in other application fields of valve technologies, where the flow factor is defined without accounting for the flow area A. In addition, the pressure drop p in bar and the flow rate Q in m3 /h in metric units have been commonly used in the field of valve control. In comparison with Eq. (B.1), and with c = Q/A, one immediately deduces the following relation kV = 

1 aβ 2 + bβ = . 103 1 + cp

(B.5)

This clear and simple relation evidently corroborates the meaningful definition of the flow coefficient in dimensionless form in Eq. (B.4). For transient computations, basically, only the resistance coefficient according to Eq. (B.2) is applied. This is because a spherical valve, during its opening or closing, behaves as a local resistance causing a pressure drop in the flow. In order to verify the formula given in Eq. (B.2), computational results from Eq. (B.5), as equivalent to Eq. (B.2), are compared with data from the literature, as shown in Fig. B.2. As the parameter for the valve rotor position, the complementary angle of β is applied. Obviously, Eq. (B.5) and thus Eq. (B.2) are satisfactorily validated.

Fig. B.2 Flow coefficient kV of spherical valve and comparison of Zhang’s equation with other measurements collected by Penninger and Benigni (2006)

304

Appendix B: Characteristics of Regulation Organs

B.2 Characteristics of Butterfly Valves The butterfly valve, as sketched in Fig. B.3, is a significant closure organ in hydropower stations. Depending on the application area, the rotary disc of the butterfly valve is sometimes designed with an eccentric axis, in order to ensure selfclosing of the valve in critical situations. In addition, the disc position in the closed state β0 noticeably influences the characteristics of butterfly valves. The hydraulic performance of a butterfly valve is interpreted by the pressure-head drop which is computed according to h butterfly = cp

c2 . 2g

(B.6)

The reference flow velocity is the mean flow velocity in the connected pipeline, c = Q/A. The resistance coefficient, cp , is again of purely geometrical character and essentially only a function of the rotor position which is specified by the angle β. It represents hydraulic characteristics of butterfly valves. Based on a flow analysis, the author derived the following accurate formula for the resistance coefficient of a butterfly valve with centered disc axis:  cp =

2 √ 1 + 0.42σ − 1 + 0.2. 1−σ

(B.7)

Fig. B.3 Butterfly valve and parameterization

β0 β d

c

Δhbutterfly

Appendix B: Characteristics of Regulation Organs

305

Fig. B.4 Resistance coefficient of butterfly valves computed by Eq. (B.7) and comparison with data from the literature

The parameter σ is computed by the ratio σ =

cos β . cos β0

(B.8)

The last term, i.e., the constant 0.2 in Eq. (B.7), corresponds to the residual resistance, when the valve is fully opened. It is basically determined by measurements and applications. In order to validate Eq. (B.7), corresponding computations have been compared with known data, for instance, from the VDI Heat Atlas (VDI Wärmeatlas) and Idelchik (2007), as shown in Fig. B.4 for valves with β0 = 0. The residual resistance is assumed to be given by cp = 0.2. Obviously, satisfactory agreement between computations from Eq. (B.7) and data from literatures has been achieved.

B.3 Characteristics of Gate Valves Another type of often applied regulation and closure organs in hydropower stations is the gate valve, which, according to Fig. B.5, may have a circular or rectangular gate plate. The hydraulic performance of a gate valve is again interpreted by the pressure-head drop which is computed as h gateV = cp

c2 . 2g

(B.9)

The reference flow velocity is the mean flow velocity in the connected pipeline, c = Q/A.

306

Appendix B: Characteristics of Regulation Organs

(a)

(b)

Rplate

h

d/2

h

d/2

Fig. B.5 Gate valves with different forms of gate plates

Hydraulic characteristics of gate valves are represented by a resistance coefficientcp , which is again of purely geometrical character and essentially a function of the gate-plate position. For circular form of a gate plate (Fig. B.5a), as from analyses by the author, the radius of the circular plate only has a negligible effect, provided the ratio of the radii or diameters dplate /dpipe does not exceed the limit value 1.3. This condition is practically always fulfilled. Under this condition and based on flow analyses, the author derived the following accurate formula of the resistance coefficient: ⎞2

⎛ ⎜ cp,circ = ⎝

1 ⎟ 2  − 1⎠ + 0.1.



h h 0.29 dpipe + 0.69 dpipe

(B.10)

For rectangular gate plates (Fig. B.5b), the following similar formula has been derived ⎞2

⎛ ⎜ cp,rect = ⎝

0.64

h dpipe

2

⎟  − 1⎠ + 0.1.

h + 0.36 dpipe

1

(B.11)

In both cases, the constant 0.1 in the above equations approximately or rather formally represents the residual resistance, when the valves are fully opened. It indeed depends on the configuration of the gate valve, that is to say whether the pipe is grooved for the gate plate or not. In Fig. 2.10, as found in Sect. 2.3.4, computational results, respectively, from Eqs. (B.10) and (B.11) have already been shown, for both forms of the gate valve.

Appendix B: Characteristics of Regulation Organs

307

Fig. B.6 Resistance coefficient of a gate valve with circular form of gate plate and comparison with data from Idelchik (2007)

In order to validate analyses, Eq. (B.10) is considered to be compared with known data from the literature. Fig. B.6 shows comparison of computations with data from Idelchik (2007). This time, excellent agreement between computations from Eq. (B.10) and data from the literature is achieved.

References

Idelchik, I. E. (2007). Handbook of hydraulic resistance, 4th Revised ed. USA: Begell House Publishers Inc. Penninger, G., & Benigni, H. (2006). Numerical simulation and design of spherical valves for modern pump storage power plants. In 14th International Seminar on Hydropower Plants (pp. 297–305). Vienna. VDI Heat Atlas. (2019). Springer Verlag.

Appendix C

Computation of the Pump Start by Neglecting the Water Hammer

At large pumps as those in hydropower stations, the pump operates always against an almost constant pressure, which is comparable to the altitude difference between the lower and upper reservoirs (Fig. 10.7). To prevent back flows, while the pump is found at the rest, the spherical valve is always closed. When starting the pump, the spherical valve should only be opened, once the shut-off speed of the pump is reached. In the showed example in Sect. 11.3, the shut-off speed takes n = 660 rpm for an altitude difference H 0 = 400 m between two reservoirs. Within the time, when the pump is further accelerated to its nominal speed nN = 750 rpm, the spherical valve should have completed its full-opening. This is a highly complex process, in which all parameters such as the motor power, the rotational speed, the head and discharge of the pump, and the pressure drop over the spherical valve, are involved. The first approximation of considering such a flow process is to assume the quasi-stationary flow by ignoring both the water hammer and the moment of inertia of the rotor shaft. This is under certain circumstances significant, if the time for fully opening of the spherical valve against the time for increasing the motor power towards its rated data should be evaluated. The second approximation is to only ignore the water hammer in the flow system. The resultant computational inaccuracy can be identified by comparing the computational results with those from the accurate computations (Sect. 11.3). Mechanically, the start-up process of the pump is balanced by all available torques exerted on the rotor system. As from Eq. (11.42), this is expressed as dω Pmotor =J + Mshaft . ω dt

(C.1)

The shaft torque is basically determined by the pump characteristics. By considering the start-up process of the pump from the time of reaching the shut-off speed, neither the reverse flow nor the inverse rotation occurs. Therefore, the characteristics of the pump according to Eq. (10.5) or Fig. 10.1 can be applied.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4

309

310

Appendix C: Computation of the Pump Start by Neglecting the Water Hammer

C.1 Quasi-Stationary Flows In this section, both the water hammer and the moment of inertia of the rotor system are neglected. The characteristic of the unified pump and valve has been shown in Sect. 10.1.3 and given by Eq. (10.22), rewritten here as ψU = ψ − cp

4 d2D ϕ2, A2sph

(C.2)

with ψU =

2g H12 . (nπ d2D )2

(C.3)

From these two equations, the rotational speed of the pump is calculated as  1  2g H12  n= .  4 π d2D ψ − cp d2D ϕ2 A2

(C.4)

sph

For a given head difference H 12 (altitude difference between two reservoirs) and an opening of the spherical valve, the rotational speed is obtained for each operation point along the initial characteristic ψ = f(ϕ) of the pump. Then, the shaft power of the pump is computed from Eq. (10.3) to

Pshaft

 3/2 2g H12 1 1 3 3 5 2 = λρπ n d2D = λρd2D . 4 2 2 ψ − cp ϕ 2 d2D /A2sph

(C.5)

Based on the above two equations, the shaft power can be shown as a function of the rotation speed of the pump for each given opening of the spherical valve (cp ). The computational algorithm is shown in Fig. C.1. For H 12 = 400 m (nominal altitude

ϕ ψ λ 0.068 0.367 0.055 0.059 0.731 0.056 0.055 0.812 0.055 : : :

H12, cp=f(βSphV)

Eq. (C.4): n Eq. (C.5): Pshaft Pshaft=f(H12, β, n) Hpu= H12+ΔhsphV

Fig. C.1 Computational algorithm of the starting process of the pump by neglecting the water hammer and the moment of inertia of the rotor system

Appendix C: Computation of the Pump Start by Neglecting the Water Hammer

311

Fig. C.2 Shaft power as functions of the rotational speed for different opening of the spherical valve

difference between two reservoirs)and three  different openings of the spherical valve, corresponding relations Pshaft = f n, cp , i.e., Pshaft = f(n, β) are plotted in Fig. C.2. From Fig. C.2, one confirms the following significant facts: (1) By reaching the full-opening of the spherical valve (β = 90◦ ) at the moment of reaching the nominal rotational speed (n = 750 rpm), the shaft power takes 91 MW, as expected. Because this is equal to the motor power, the rotational speed remains unchanged. The start-up of the pump is successful. (2) If the opening of the spherical vale is relatively slow and reaches, e.g., only β = 50◦ at the moment of reaching the nominal rotational speed (n = 750 rpm), then, the shaft power takes only about 83 MW. Because this shaft power is smaller than the motor power, the pump will be further accelerated to a rotational speed equal to 766 rpm. This clearly represents an undesired start-up process of the pump. (3) If the opening of the spherical vale is even slower, as given in Fig. C.2 with β = 40◦ at the moment of reaching the nominal rotational speed, then, the shaft power only measures 73 MW. Because it is significantly smaller than the motor power of 91 MW, the pump rotation will be further accelerated to a value equal to 785 rpm. This value, theoretically, represents a maximum possible speed, because during this process, the spherical valve is further opened towards β = 90◦ . From the above considerations, the opening of the valve should be completed at the latest when the nominal speed of the pump has been reached. The three remarks made above are based on the assumption of quasi-stationary operations, at which even the moment of inertia of the rotor system has been neglected. For this reason, the above consideration is rather qualitative. Below in Sect. C.2, a more quantitative case will be given, in which the moment of inertia is taken into account. The detailed computation, when also considering the water hammer, has already been presented in Sect 11.3.

312

Appendix C: Computation of the Pump Start by Neglecting the Water Hammer

C.2 Dynamic Start-Up of the Pump The dynamic starting process of the pump is considered here by further accounting for the moment of inertia of the rotor system. In order to simulate such a starting process, the motor power is assumed to be given by the following linear relation: Pmotor = 35 +

91 − 35 t. tmotor

(C.6)

As agreed above, the computation begins from the shut-off speed, which is denoted by n shut-off = 660 rpm and Pshut-off = 28.5 MW, see Eq. (10.7). In Eq. (C.6), the initial motor power is set to 35 MW, which must be greater than the shut-off power. With ω = 2π n and Mshaft = Pshaft /ω, one obtains from Eq. (C.1) Pmotor = 4π 2 J n

dn + Pshaft . dt

(C.7)

The shaft power is given by Eq. (C.5), so that Pmotor = 4π 2 J n

1 dn 5 + λρπ 3 n 3 d2D . dt 2

(C.8)

From this equation, the time rate of the speed change is computed as   dn 1 1 3 3 5 = Pmotor − λρπ n d2D . dt 4π 2 J n 2

(C.9)

The rotational speed in the next time step is computed as  n i+1 = n i +

dn dt

 t.

(C.10)

i

On the other hand, one obtains from Eqs. (C.2) and (C.3) ψ=

4 2g H12 d2D + c ϕ2. p A2sph (nπ d2D )2

(C.11)

This equation is basically the same as Eq. (10.22). Further, Eq. (10.16) is applied. The last equation above is then rewritten as m2ϕ2 + m1ϕ + m0 = or

d4 2 2g H12 + cp 2D ϕ , 2 A2sph (nπ d2D )

(C.12)

Appendix C: Computation of the Pump Start by Neglecting the Water Hammer

 4 d2D 2g H12 = 0. m 2 − cp 2 ϕ 2 + m 1 ϕ + m 0 − Asph (nπ d2D )2

313



(C.13)

This is a polynomial equation for the variable ϕ. At each instant with known cp = f(β) of the spherical valve and the computed rotational speed n, the discharge coefficient can be computed. The corresponding computational algorithm is shown in Fig. C.3. Since the water hammer in the current case is not considered, the pressure head H 12 can be approximated to be equal to the altitude difference between the two reservoirs, so that H 12 = H 0 with H 0 = 400 m as the nominal head in the current example. This also implies that the viscous friction is neglected. Figures C.4 and C.5, by solid lines, show the computed rotational speed of the pump and the discharge for a given opening time (t sphV = 30 s) of the spherical valve. The same time is set for the motor power increase according to Eq. (C.6), although, basically, both times can be different. In the current computational example, the pump has been continuously accelerated to its rated value.

Eq. (2.35): cp=f(β) Eq. (C.6): Pmotor=f(t)

t=0, i=0: n0=660, Q0=0

i+1 Eq. (C.10): n

Eq. (C.9): dn/dt

i=1, 2, 3… Eq. (C.13): ϕ=f(n, cp) Fig. 10.1b: λ=f(ϕ)

Eq. (10.1): Q Fig. 10.1a: ψ Eq. (10.2): Hpu

Fig. C.3 Computational algorithm of the starting process of the pump by neglecting the water hammer

Fig. C.4 Rotational speed of the pump during the start-up process

314

Appendix C: Computation of the Pump Start by Neglecting the Water Hammer

Fig. C.5 Discharge through the pump during the start-up process

For the purpose of comparisons, the computational results from accurate transient computations in Sect. 11.3 are also shown (dashed lines). Obviously, the approximation made by neglecting the water hammer does not lead to significant computational inaccuracy, if only the rotational speed and the discharge are in focus. Further, the time rate of the speed change and the pressure head over the pump are shown in Figs. C.6 and C.7, respectively. The comparison with accurate computations indicates again that the employed approximation does not lead to significant computational inaccuracy. Fig. C.6 Acceleration of the speed during the start-up process

Fig. C.7 Pressure head over the pump during the start-up process

Appendix C: Computation of the Pump Start by Neglecting the Water Hammer

315

However, it is no longer possible to estimate the shock pressure at the exit of the pump, which, according to Fig. 11.15a may be exceedingly high (e.g. hpu2 = 583 m). For this reason, if possible, hydraulic transients according to Sect. 11.3 should always be completed.

Index

A Abstract valve, 138 Analytical formula, 211, 226 Attenuation effect, 101, 222, 224 Attenuation of pressure fluctuations, 101, 255

B Borda-Carnot equation, 118 Borda-Carnot formula, 25, 205 Bulk modulus of water, 71, 72 Butterfly valve, 32, 33, 140–142, 184–191, 304, 305

C Characteristic grids, 13, 125, 127, 131, 196– 198 Characteristic line, 13, 127, 195–197 Characteristic of the injector, 28, 29, 250, 251, 260, 261 Characteristics of the Francis turbine, 261, 263 Characteristics of the Pelton turbine, 28, 29, 250 Characteristics of the pump, 235–244, 282 Closing law linear, 254–256 parabolic, 256–260 Computational bias, 230, 231 Condensation of water, 50, 192 Conformity of hydraulic transients, 9, 15, 16, 96, 207, 225 Conservation law, 149–153, 156, 157, 170, 204, 205

D Darcy-Weisbach friction coefficient, 20, 21, 163, 299 Discharge-related algorithm, 92, 93 Downstream shock wave, 133, 139–144, 147, 170, 276, 297 E Elastic modulus of water, 9 Elastic water column theory, 8–10, 41, 52, 53, 60, 73–75, 125 Emergency shut-down, 146, 148, 273, 280, 288, 289 Entrance velocity effect, 158, 159, 164, 165, 186, 188, 190, 191 Equivalent pipe diameter, 51, 52, 101, 217 Eulerian method, 199 Evaporation of water, 50, 192 F Fast transients, 6, 163 Four-quadrant diagram, 241–245 Full reflection of shock waves, 134, 137, 158, 287 G Gate valve, 34–36, 305–307 H Hydraulic ram, 2, 169 J Jump potential of the pressure, 63

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 Zh. Zhang, Hydraulic Transients and Computations, https://doi.org/10.1007/978-3-030-40233-4

317

318 L Lagrangian approach, 10, 131, 195, 199 Load rejection, 41, 77, 148, 270

M Method of direct computations, 207, 208, 211–218, 229–234 Mirrored pressure jump, 62, 63, 221, 233 Moving surface of water, 161, 167, 168, 287

O Operation map, 241, 244, 245 Opposite phenomenon, 221 Orifice, 27, 156, 157, 166 Overall friction effect, 164, 165 Overlap of shock pressures, 232, 256

P Period of oscillations, 89, 115, 123 Piezometric head, 42, 127, 198 Poisson’s ratio, 71 Pressure jump, 63, 212, 216, 218–221, 223, 224, 232–234 Pressure-related algorithm, 93

R Regular triangular wave, 171, 215, 223 Resistance constant, 22–27 Rigid water column theory, 8, 9, 42, 44, 46, 73–75, 77, 97–101 Round trip of waves, 45, 58, 59, 164, 165, 168, 209, 298

S Self-stabilization effect, 57, 61–63 Shapes of shock waves, 171, 172 Short pipe, 9, 44–46, 77, 99 Shut-off pressure, 237, 238, 242, 291 Shut-off speed, 237–240, 242, 244, 250, 290, 291, 309, 312

Index Spherical valve, 28, 31, 32, 166, 246–248, 260, 261, 270, 271, 301–303, 309– 311 Superimposed overlap, 232, 233 Suppressed overlap, 232, 233 Surge tank, 14, 85–92, 103–107, 109–112, 115–121, 123, 124, 161–163, 167, 168, 180, 181, 183 Surge tank size, 14, 90, 104

T Thoma criterion, 15, 104, 119, 120, 123, 124 Throttle resistance at surge tank, 167, 168, 277, 287 Time increment effect, 230, 231 Total reflection of shock waves, 152, 159– 161, 203 Transient conformity, 207, 225, 234 Transient friction models, 7 Traveling law, 11, 149–152, 154–157, 171, 181, 203, 205, 206, 276, 287 Two-step closing, 216, 228–232, 254, 255

U Unified characteristics, 143–145, 244, 260, 261, 271, 280, 282, 291 Unit parameters, 241, 244, 263, 265 Upstream shock wave, 134, 139–142, 170, 283, 297

V Viscous resistance constants, 133

W Water hammer equations, 128, 132, 274, 275, 283 Wave equation, 128 Wave propagation speed, 3, 9, 53, 57–60, 71–73

X X-junction, 156