water hammer in viscoelastic pipes have been developed applying the viscoelastic model, neglecting fluid-structure interaction (FSI) and unsteady friction (UF) ...
Mansoura University Faculty of Engineering Mechanical Power Eng. Dept.
NUMERICAL AND EXPERIMENTAL STUDY OF WATER HAMMER IN VISCOELASTIC PIPES A THESIS Submitted in Partial Fulfillment for the Degree of Master of Science in Mechanical Power Engineering
By Eng. Mohamed Mostafa Hassan Tawfik B.Sc. of Mechanical Power Engineering Faculty of Engineering, Mansoura Univeristy
Supervisors Prof. Dr. Hassan Mansour El-Saadany
Prof. Dr. Mohamed Safwat Saad El-Din
Mechanical Power Engineering Department Faculty of Engineering Mansoura University
Mechanical Power Engineering Department Faculty of Engineering Mansoura University
Prof. Dr. Berge Ohanness Djebedjian Mechanical Power Engineering Department Faculty of Engineering Mansoura University
2014
Mansoura University Faculty of Engineering Mech. Power Eng. Dept.
Supervisors Research Title: Numerical and Experimental Study of Water Hammer in Viscoelastic Pipes Researcher Name: Mohamed Mostafa Hassan Tawfik Scientific Degree: M.Sc.
Supervision Committee Name Prof. Dr. Hassan Mansour El-Saadany Prof. Dr. Mohamed Safwat Saad El-Din Prof. Dr. Berge Ohanness Djebedjian
Position Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University
Signature
Head of the Department
Vise Dean for Post Graduate Studies and Researches
Prof. Mohamed Ghasoub Saafan
Prof. Dr. Kassem Salah El-Alfy
Dean of the Faculty Prof. Dr. Zaki M. Zeidan
Mansoura University Faculty of Engineering Mech. Power Eng. Dept.
Examination Committee
Research Title: Numerical and Experimental Study of Water Hammer in Viscoelastic Pipes Researcher Name: Mohamed Mostafa Hassan Tawfik Scientific Degree: M.Sc.
Supervisor Committee Name Prof. Dr. Hassan Mansour El-Saadany Prof. Dr. Mohamed Safwat Saad El-Din Prof. Dr. Berge Ohanness Djebedjian
Position Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University
Signature
Examination Committee Name Prof. Dr. Nabil Ibrahim Ibrahim Hewedy Prof. Dr. Lotfy Hassan Rabie Sakr Prof. Dr. Hassan Mansour El-Saadany Prof. Dr. Mohamed Safwat Saad El-Din
Position Mechanical Power Engineering Dept., Faculty of Engineering, Minoufia University Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University Mechanical Power Engineering Dept., Faculty of Engineering, Mansoura University
Signature
Head of the Department
Vise Dean for Post Graduate Studies and Researches
Prof. Mohamed Ghasoub Saafan
Prof. Dr. Kassem Salah El-Alfy Dean of the Faculty Prof. Dr. Zaki M. Zeidan
Acknowledgments
ACKNOWLEDGMENTS First and foremost, I thank ALLAH for endowing me with health, patience, and knowledge to complete this work. I would like to express my sincere gratitude and indebtedness to Prof. Dr. Hassan M. El-Saadany, for his helpful comments and valuable directive. I owe my deepest gratitude to Prof. Dr. Mohamed S. Saad El-Din, for his great support, kind guidance and supervision of this thesis. I cannot find words to express my deep thanks and appreciation to Prof. Dr. Berge O. Djebedjian, for his sincerely advices, constructive cooperation, enthusiastic assistance and encouragement to finish this work. It gives me great pleasure in acknowledging the support and help of Dr. Alireza Keramat, lecturer at Jundi-Shapur University of Technology, Dezful, Iran, for his kind cooperation to understand the viscoelastic model. I would not forget to remember Eng. Mohamed Mansour, Comatrol Inc., Cairo, Egypt, who donated me a new pressure transducer (DSU110F). This thesis is dedicated to my parents, wife and son who have always stood beside me and supported me with patience, encouragement and dealt with my stress with lovely smiles.
i
Abstract
ABSTRACT Analysis of unsteady flow helps in designing pipe systems to withstand additional loads resulting from the water hammer phenomenon, as it can damage pipes severely and causes fittings rupture. Performing this analysis requires, firstly, performing a steady-state analysis to get the initial values to start the unsteady-state analysis. The present study introduces the development of a model, using the Gradient Method, and a FORTRAN code to perform steady-state network analysis. The developed code is a development of the code presented by Larock et al. (2000), which used the Newton-Raphson method. The code is validated by comparing its results with Bryan et al. (2006) case study results. The comparison showed that the maximum error was 0.125%; therefore the validation of code is verified. The mechanical behavior of the pipe material affects the pressure response of a fluid system during water hammer. Traditionally, the unsteady-state analysis to simulate water hammer focused on applying the elastic model, as elastic materials were commonly used as fluid conduits. But in recent years, the application of plastic pipes, such as polyethylene (PE), polyvinyl chloride (PVC) and polypropylene (PP) have been increasingly used in piping systems due to their excellent mechanical, chemical and thermal properties. This led to
ii
Abstract
the evolution of a new research trend in which the viscoelastic behavior of pipe-walls was taken into account. In the present work, a mathematical model and a FORTRAN code to simulate water hammer in viscoelastic pipes have been developed applying the viscoelastic model, neglecting fluid-structure interaction (FSI) and unsteady friction (UF) effects. A simple tank-pipe-valve test rig was designed and carried out using a pipe made of PP, which is characterized by a creep function that is determined experimentally. The developed code results were tested against both of Covas et al. (2004) results and the carried out experimental test rig facility results. The code results showed a good agreement with the experimental results in both cases, therefore it is considered valid. The parameters affecting the viscoelastic model were studied, including material, location, Courant number and wave speed effects. Materials of higher creep curve slope demonstrated more viscoelastic behavior represented by higher damping rate for pressure fluctuations caused by water hammer. For locations closer to the downstream end of the pipe, the maximum rise in pressure head, ΔHmax, increases while the damping rate and frequency of fluctuating pressure waves decrease. Increasing Courant number decreases the pressure wave frequency and it is recommended to apply Courant number of unity to achieve the best matching between numerical and experimental results. Wave speed has a great effect on numerical results. Increasing its value increases the pressure wave frequency approximately in a linear form. iii
Table of Contents
TABLE OF CONTENTS Subject
Page
Acknowledgments ......................................................................................... i Abstract ......................................................................................................... ii Table of Contents ......................................................................................... iv List of Tables ................................................................................................. xii List of Figures ............................................................................................... xiv Nomenclature ................................................................................................ xvii Abbreviations ................................................................................................ xxii
CHAPTER 1. INTRODUCTION ............................................................... 1 1.1 Introduction ........................................................................................ 1 1.2 Water Hammer Definition ................................................................. 1 1.3 Water Hammer Phenomena ............................................................... 2
iv
Table of Contents
1.3.1 Classic Water Hammer Phenomenon ....................................... 2 1.3.2 Fluid-Structure Interaction Phenomenon ................................. 6 1.4 Viscoelastic Pipes ................................................................................ 7 1.4.1 Advantages of Viscoelastic Pipes ............................................. 7 1.4.2 Viscoelastic Materials Characteristics ...................................... 8 1.5 Network Analysis Methods ................................................................. 8 1.5.1 Hardy Cross Method ................................................................. 9 1.5.2 Newton-Raphson Method ......................................................... 10 1.5.3 Gradient Method ....................................................................... 11 1.6 Thesis Structure ................................................................................... 13
CHAPTER 2. LITERATURE REVIEW ................................................... 15 2.1 Introduction ...................................................................................... 15 2.2 Brief Summary of Historical Development ..................................... 15 2.3 Experimental Work .......................................................................... 17
v
Table of Contents
2.3.1 Experimental Work on Elastic Pipes ........................................ 17 2.3.2 Experimental Work on Viscoelastic Pipes ............................... 21 2.4 Numerical Work .................................................................................. 26 2.4.1 Numerical Work Related to Elastic Pipes ................................ 26 2.4.2 Numerical Work Related to Viscoelastic Pipes ....................... 35 2.5 Review of Network Analysis Methods ............................................... 41 2.6 Aim of the Present Work ..................................................................... 46
CHAPTER 3. Theoretical Approach .......................................................... 49 3.1 Introduction ......................................................................................... 49 3.2 Steady State Network Analysis ........................................................... 49 3.2.1 Governing Equations ................................................................ 50 3.2.2 Gradient Method ....................................................................... 51 3.2.2.1 Mathematical Model ....................................................... 52 3.2.2.1.1 Networks with Known Pipe Resistances ............... 52
vi
Table of Contents
3.2.2.1.2 Networks with Pumps ............................................ 54 3.2.2.2 Code Development .......................................................... 55 3.2.2.3 Code Validation ............................................................... 57 3.3 Unsteady State Analysis ...................................................................... 58 3.3.1 Viscoelasticity .......................................................................... 58 3.3.2 Governing Equations ................................................................ 62 3.3.3 Method of Characteristics Implementation .............................. 64 3.3.3.1 Development of the Characteristic Equations ................. 64 3.3.3.2 The Finite Difference Equations Representation ............ 66 3.3.4 Code Development ................................................................... 69 3.3.5 Model Verification ................................................................... 71 3.3.6 Comparison between Elastic and Viscoelastic Models - Theory and Results ............................................................................... 74
CHAPTER 4. Experimental Work ............................................................. 77
vii
Table of Contents
4.1 Introduction ........................................................................................ 77 4.2 Water Hammer Experiment Setup ..................................................... 77 4.3 Available Instruments ........................................................................ 80 4.3.1 Solenoid Valve ......................................................................... 80 4.3.2 Flow Meter ................................................................................ 81 4.3.3 Pressure Transducers ................................................................ 83 4.3.3.1 H0 Pressure Transducer ................................................... 83 4.3.3.2 Water Hammer Pressure Transducer .............................. 85 4.3.4 Data Logger .............................................................................. 86 4.4 Test Rig Sizing ................................................................................... 87 4.4.1 Pipe Material Selection ............................................................. 87 4.4.2 Pipe Diameter Selection ........................................................... 88 4.4.3 Wave Speed Solution ................................................................ 90 4.4.4 Selecting Pipe Length ............................................................... 92 4.4.5 Selecting Minimum Reservoir Head ........................................ 93 viii
Table of Contents
4.4.6 Selecting Main Pump Operating Point ..................................... 97 4.5 Water Hammer Experimental Procedure ........................................... 101 4.5.1 Test Preparation ........................................................................ 101 4.5.2 Run under Steady-State Condition ........................................... 102 4.5.3 Run under Unsteady-State Condition ....................................... 102 4.6 Creep Experiment .............................................................................. 103 4.6.1 Experimental Setup ................................................................... 103 4.6.2 Experimental Procedure ........................................................... 105 4.6.3 Creep Experimental Results ..................................................... 107
CHAPTER 5. RESULTS AND DISCUSSION .......................................... 109 5.1 Introduction ........................................................................................ 109 5.2 Experimental Results ......................................................................... 109 5.2.1 Water Hammer Experimental Results ...................................... 109 5.2.2 Comparison with Code Results ................................................ 115
ix
Table of Contents
5.3 Further Investigations of Code Parameters ........................................ 117 5.3.1 Material Effect .......................................................................... 117 5.3.1.1 Creep Curve Comparison ................................................ 117 5.3.1.2 Damping Rate Comparison ............................................. 118 5.3.2 Location Effect ......................................................................... 119 5.3.2.1 Effect on Damping Rate .................................................. 119 5.3.2.2 Effect on Maximum Amplitude ...................................... 120 5.3.2.3 Effect on Fluctuating Head Frequency ........................... 120 5.3.3 Time Step Effect ....................................................................... 122 5.3.4 Wave Speed Effect ................................................................... 126
CHAPTER 6. CONCLUSIONS AND FUTURE WORK ......................... 131 6.1 Introduction ........................................................................................ 131 6.2 Conclusions ........................................................................................ 132 6.3 Future Work Recommendations ........................................................ 134
x
Table of Contents
REFERENCES ............................................................................................. 135 APPENDIX A. GRADIENT METHOD SUBROUTINE CODE ............ 153 APPENDIX B. VISCOELASTIC EQUATIONS DERIVATION ........... 165 ARABIC ABSTRACT ................................................................................. 179
xi
List of Tables
LIST OF TABLES Table No.
Title
Page
1.1
Comparison of common network analysis methods ......................... 12
2.1
Experimental systems information and parameters (Elastic Pipes) ................................................................................................ 18
2.2
Experimental
systems
information
and
parameters
(Viscoelastic Pipes) .......................................................................... 22 2.3
Numerical studies related to elastic pipes ........................................ 27
2.4
Numerical studies related to viscoelastic pipes ................................ 36
2.5
Summary of the scientific contributions to develop the common network analysis methods .................................................. 41
3.1
Comparison between developed code and Bryan et al. (2006) four-pipe network case study results .................................... 57
3.2
Specifications of the reservoir-pipeline-valve experiment performed by Covas et al. (2004) ..................................................... 71
3.3
Calibrated creep coefficients τk, Jk for the Imperial College test with Q0 = 1.01 l/s and a = 395 m/s, neglecting unsteady friction [Covas et al. (2005)] ............................................................ 72
4.1
Solenoid valve specifications ........................................................... 80
4.2
Flow meter specifications ................................................................. 81
4.3
H0 Pressure transducer specifications ............................................... 83
xii
List of Tables
4.4
Water hammer pressure transducer specifications ........................... 85
4.5
Data logger specifications ................................................................ 87
4.6
Specifications of selected aquatherm® green pipe ............................ 89
4.7
Wave speed values at different pipe support situations ................... 91
4.8
(Lmin) corresponding to different pipe support situations ................. 93
4.9
(H0,min) corresponding to different pipe support situations .............. 93
4.10
Main pump manufacturer specifications .......................................... 97
4.11
Auxiliary pump manufacturer specifications ................................... 97
4.12
Water hammer test rig specifications ............................................... 101
4.13
Digital caliper specifications ............................................................ 105
4.14
Kelvin–Voigt model coefficients ...................................................... 107
xiii
List of Figures
LIST OF FIGURES Figure Title Page No. 1.1 Steady flow from a reservoir in the absence of friction ................... 3 1.2
Evolution of a transient pressure wave in the pipe in Figure 1.1 .......................................................................................... 4
3.1
Gradient method subroutine flow chart ............................................ 56
3.2
Schematic diagram of Bryan et al. (2006) four-pipe network case study .......................................................................................... 57
3.3(a) Stress and strain for an instantaneous constant load ........................ 59 3.3(b) Boltzmann superposition principle for two stresses applied sequentially ....................................................................................... 59 3.4
Generalized Kelvin–Voigt model ..................................................... 60
3.5
The method of characteristics grid ................................................... 66
3.6
Viscoelastic code flow chart ............................................................. 70
3.7
Comparison between mathematical model results and experimental data of Covas et al. (2004) at three locations ............. 73
3.8
Summary of differences between water hammer models ................ 76
4.1(a) Water hammer test rig schematic diagram ....................................... 78 4.1(b) Water hammer test rig isometric detailed view ................................ 79 4.2
Solenoid valve schematic diagram ................................................... 80
4.3
Spring loaded variable area flow meter schematic diagram ............. 81
4.4
Flow meter calibration test rig .......................................................... 82 xiv
List of Figures
4.5
Flow meter calibration curve ............................................................ 82
4.6
H0 pressure transducer schematic diagram ....................................... 83
4.7
Pressure transducer calibration test rig ............................................. 84
4.8
H0 pressure transducer calibration curve .......................................... 84
4.9
Water hammer pressure transducer schematic diagram ................... 85
4.10
Water hammer pressure transducer calibration curve ...................... 86
4.11
Operating point change during transient event ................................. 94
4.12
Upstream reservoir simulation circuit .............................................. 95
4.13
Operation of upstream reservoir simulation circuit .......................... 96
4.14
Pump performance test rig ................................................................ 98
4.15
Comparison between new, used main, and auxiliary pumps performance curves ........................................................................... 98
4.16
Operating point selection .................................................................. 100
4.17
Tensile Testing Equipment ............................................................... 104
4.18
Tensile Testing Equipment schematic diagram ................................ 104
4.19
Creep test experimental results ......................................................... 108
5.1
Water hammer pressure transducer output data at three locations ............................................................................................ 111
5.2
Water hammer pressure heads at three locations ............................. 114
5.3
Comparison between code results and experimental data at three locations ................................................................................... 116
5.4
Comparison between PP-R and HDPE creep functions ................... 117
xv
List of Figures
5.5
Comparison between average (ΔH/ΔHmax %) for PP-R and HDPE . ............................................................................................... 118
5.6
Comparison between (ΔH/ΔHmax%) for Covas et al. (2004) case and present case ........................................................................ 119
5.7
Location effect on maximum amplitude (ΔHmax) value for Covas et al. (2004) case and present case ......................................... 120
5.8
Comparison between frequency of fluctuating heads for Covas et al. (2004) case and present case ......................................... 121
5.9
Time-step and Courant number effects on head fluctuations for Covas et al. (2004) case and present case ................................... 123
5.10
Time-step and Courant number effects on average head value for Covas et al. (2004) case and present case ................................... 124
5.11
Time-step and Courant number effects on average frequency of fluctuating head for Covas et al. (2004) case and present case .................................................................................................... 125
5.12
Wave speed effect on head fluctuations for Covas et al. (2004) case and present case ............................................................ 127
5.13
Wave speed effect on frequency for Covas et al. (2004) case and present case ................................................................................ 129
5.14
Wave speed effect on average wave frequency for Covas et al. (2004) case and present case ....................................................... 129
B.1
The method of characteristics grid ................................................... 171
B.2
Pump performance curve .................................................................. 178 xvi
Nomenclature
NOMENCLATURE Symbol
Description
A
Pipe cross sectional area, m2
AP
Constant in the pump performance curve equation, m-5s2
a
Wave speed, m/s
BP
Constant in the pump performance curve equation, m-2s
C0
Dimensionless parameter that describes the effect of pipe constraint condition on the wave speed
CP
Constant in the pump performance curve equation, m
D
Pipe internal diameter, m
di
Inner diameter of the creep test sample, m
do
Outer diameter of the creep test sample, m
E
Modulus of elasticity, Pa
Ek
Modulus of elasticity of the Kelvin–Voigt k-element spring, Pa
e
Pipe thickness, m
f
Friction coefficient
g
Gravitational acceleration, m/s2
H
Head, m
H0
Head at steady state condition, m
xvii
Nomenclature
t H0i
H0,min
Known or assumed head at node (i), at tth iteration, m Minimum head at steady state condition without column separation, m
H0s
Sump level at pumped source node (s), m
H1
Head of the auxiliary pump circuit, m
Hmax
Maximum pressure head during the transient event, m
Hmin
Minimum pressure head during the transient event, m
HP
Nodal head at general point P, m
HP,max Pump maximum delivery head, m Hsump
Sump level, m
H
Dynamic head (HH0), m
hP
Head increase across a pump, m
hv
Vapor pressure head, m
hx
Friction head loss in pipe (x), m
IH
Function representing the retarded response to water hammer
J
Creep compliance function, Pa-1
J0
Creep compliance of the first spring of the Kelvin–Voigt model, Pa-1
Jk
Creep compliance of the spring of the Kelvin–Voigt k-element, Pa-1
K
Fluid bulk modulus, Pa
k
Element number of the Kelvin–Voigt model, or polynomial order
xviii
Nomenclature
L
Pipe length, m
L0
Initial length of the creep test sample, m
Lmin
Minimum design pipe length, m
NKV
Number of Kelvin-Voigt elements
p p(x)
Pressure, Pa Value of the smoothed sample as a k-order polynomial
Q
Discharge, m3/s
Q0
Discharge at steady state condition, m3/s
t Q0x
Known or assumed discharge through a pipe (x) at tth iteration, m3/s
QP,max Maximum pump discharge, m3/s QP
Discharge of a pump, m3/s
q
Nodal flow rate, m3/s
q0
Known nodal flow rate, m3/s
R
Pipe resistance constant
R0
Known or assumed pipe resistance constant
S
Axial coordinate
t
Time, s, or iteration number
tc
Valve closure time, s
tcritical
Critical valve closure time, s
xix
Nomenclature
V
Fluid velocity, m/s
V0
Fluid velocity at steady state condition, m/s
VP
Nodal fluid velocity at general point P, m/s
X
Number of pipes in a network
x
Polynomial independent parameter
y(x) z
Samples before smoothing Elevation, m
Dimensionless Numbers C
Courant number, C = [Δt × max| a + V | ]/ΔS
Re
Reynolds number, Re VD
Greek Symbols
Difference
δ
Change in value
Strain
e
Instantaneous-elastic strain
r
Retarded strain
xx
Nomenclature
Circumferential strain
Pipe inclination angle, degree
Lagrange multiplier
μ
Viscosity, kg.m-1.s-1
μk
Viscosity of the dashpots of the Kelvin–Voigt k-element, kg.m-1.s-1
Poisson’s ratio
Fluid density, kg/m3
Stress, Pa
Instantaneous stress, Pa
Circumferential stress, Pa
Retardation time, s
k
Retardation time of the dashpot of the Kelvin–Voigt k-element, s
xxi
Abbreviations
ABBREVIATIONS
CFD
Computational Fluid Dynamics
C.V.
Control Valve
FDM
Finite Difference Method
FEM
Finite Element Method
FSI
Fluid-Structure Interaction
HDPE
High Density Polyethylene
IDQM
Incremental Differential Quadrature Method
IMOC
Implicit Method Of Characteristics
ITA
Inverse Transient Analysis
MDPE
Medium Density Polyethylene
MOC
Method Of Characteristics
NPT
National Pipe Thread
PE
Polyethylene
PIMOC
Point-Implicit Method of Characteristics
PP
Polypropylene
xxii
Abbreviations
PP-R
Polypropylene Random Copolymer
PVC
Polyvinylchloride
SF
Steady Friction model
SIMPLE
Semi-Implicit Method for Pressure Linked Equations
UF
Unsteady Friction model
WCM
Wave Characteristics Method
xxiii
Chapter One Introduction
Chapter 1
Introduction
CHAPTER 1 INTRODUCTION
1.1
Introduction Pipelines play a great role in everyone's life and are also essential to
industries. An extensive network of underground pipelines exists in every city, state, and nation to transport water, sewage, crude oil, petroleum products, natural gas, and many other liquids and gases. There are many problems associated with fluids transported through pipelines, which must be taken into consideration in piping networks design, maintenance, and operation. However, one of these problems, that can cause pipe and fittings rupture, is water hammer which is a form of transient flow. Transient flow (or unsteady flow) is the flow whose conditions at a point may change with time. A transition is caused by a disturbance to the flow, for example, when the pumps are shut off or valves are closed.
1.2
Water Hammer Definition Water hammer is a pressure surge or wave that occurs in any liquid
conductor like pipe or open channel when the momentum of flowing liquid is 1
Chapter 1
Introduction
disturbed. It commonly occurs due to the sudden change in the flow conditions, such as sudden closing or opening of a valve at the end of a pipe; pump failure; and valve malfunction. Water hammer can occur in pumping systems or gravity systems. However, since the disturbance in the flow in pumping systems is more intense, yet, it is more frequent in pumping systems than in gravity systems. This phenomenon may prove to be detrimental if such pressures created exceed the pressure for which the pipeline is designed. Any positive pressure that exceeds the pipeline design pressure can result in the bursting of the pipeline. Any negative pressure not only can cause crumbling of the pipeline if it is less than the design negative pressure, but also cavitations in pipes may occur if the pressure drops to the liquid vapor pressure and the one-phase flow is transformed to two-phase flow.
1.3
Water Hammer Phenomena
1.3.1 Classic Water Hammer Phenomenon In this section, a reservoir is attached to a simple frictionless pipeline with a valve at its end is assumed to observe how water hammer waves evolve in time, Figure 1.1. The valve is assumed to be closed completely instantaneously. At this instant (t = 0), the fluid nearest the valve is brought to rest and compressed. As a consequence, the head at the valve abruptly 2
Chapter 1
Introduction
increases by an amount ΔH. The amount of this increase is just sufficient to reduce the momentum of the moving water to zero. The increased head enlarges the pipe slightly and also increases the density of the fluid. The amount of the stretching of the pipe depends on the diameter and thickness of the pipe and on the compressibility of the pipe material and the liquid, but it normally changes by less than one-half percent. In Figure 1.2, the amount of the deformation is exaggerated.
H0 V0 L Figure 1.1. Steady flow from a reservoir in the absence of friction
The rise in pressure head causes a pressure wave to propagate upstream at the speed of sound (a). When the wave front reaches the reservoir (t = L/a), the velocity is zero throughout the pipe, all the fluid is under extra head ΔH, all momentum has been lost, and the pipe is enlarged, see Figure 1.2(a).
3
Chapter 1
Introduction
Under these conditions the fluid in the pipe near the reservoir connection is locally not in equilibrium, since the reservoir pressure head is only H0. Hence, fluid begins to flow toward the region of lower head (the reservoir) as the distended pipeline forces flow in that direction. Now, the source of the liquid for this flow is the compressed liquid that is stored in the enlarged pipe cross section under the increased pressure head.
ΔH
a (a)
a
t = L/a
t = 2L/a
40 V=
V =4 V0
(a)
(b)
t = 3L/a
t = 4L/a
a
4
ΔH 4
a
(a) V = V0
V=0 (b)
(d)
(c)
Figure 1.2. Evolution of a transient pressure wave in the pipe in Figure 1.1
4
ΔH
ΔH
Chapter 1
Introduction
The process continues to evolve with time. At (t = 2L/a), Figure 1.2(b), the pressure throughout the pipe has returned to its original value, but with the velocity reversed from its original direction. At this instant the store of compressed liquid is exhausted, and the pressure wave appears to undergo a reflection. That is, the pressure head drops an amount ΔH below the original steady head H0, and this pressure drop and the closed valve cause the velocity behind the wave front to return to zero. Behind this negative wave the pipe cross section shrinks and the liquid expands. At the instant the negative wave has reached the reservoir (t = 3L/a), Figure 1.2(c), the fluid velocity is everywhere zero, but uniformly at head ΔH less than before closure. However, the pressure head at the reservoir is again not in equilibrium with the reservoir head, so fluid is drawn from the reservoir into the pipe at velocity (V0(. Behind the new advancing wave the head is in equilibrium with the reservoir head. When the wave has reached the valve at (t = 4L/a), Figure 1.2(d), all variables have returned to the original steady state that existed before the valve was closed. This process is repeated every (4L/a) seconds and would, in the absence of friction, continue without abating.
5
Chapter 1
Introduction
1.3.2 Fluid-Structure Interaction Phenomenon In this phenomenon, describing the complete cycle of pressure wave resulting from hammering wave, which caused by sudden valve closure, requires taking many parameters into consideration. The most important parameters should be considered: a. Poisson coupling; fluid pressure waves cause elastic radial deformations (expansion, contraction) in the pipe wall, which induce an axial stress wave in the wall. This stress wave propagates along the wall with sonic velocity and produces a secondary fluid pressure wave (precursor wave), in which the sonic velocity in the wall material is greater than in the fluid-moves ahead of the primary fluid pressure front. In addition, the stress wave causes axial motions of the affected pipe. b. Friction coupling; this effect accounts for the mutual friction between fluid and wall. c. Junction coupling; this effect is caused by pressure difference between specific points of the pipe system such as bends, tees, changes of cross section, valves or pipe ends. The resultant hydraulic forces induce pipe motions (oscillations in elastic systems) which influence the fluid pressure and velocity (secondary pressure waves).
6
Chapter 1
1.4
Introduction
Viscoelastic Pipes These are pipes made of viscoelastic (plastic) materials, such as
Polyvinylchloride (PVC), Polyethylene (PE) and Polypropylene (PP). These materials have both viscous and elastic characteristics when undergoing deformation. They exhibit time-dependent strain under a certain stress.
1.4.1 Advantages of Viscoelastic Pipes The application of viscoelastic pipes has been increasingly used recently in pipe systems for both hot and cold water in residential, commercial and industrial water supply systems due to their light weight, easy and fast installation, and excellent chemical and mechanical characteristics. In addition, viscoelastic pipes resist corrosion, accommodate temperature fluctuations, and energy efficient as they have a high degree of self-insulation, so that they can maintain fluid temperatures. Some types of viscoelastic pipes, such as food grade thermoplastic PP pipes, allow transporting water faster, cleaner and quieter with no annoying flow noise. Other types, such as PVC pipes, can be used in transporting wastewater. Viscoelastic pipes not only used to transport water, but they can also convey aggressive fluids like acids for industrial purposes.
7
Chapter 1
Introduction
1.4.2 Viscoelastic Materials Characteristics Viscoelastic materials are polymers which exhibit a viscoelastic mechanical behavior [Ferry (1970), Aklonis et al. (1972) and Riande et al. (2000)] in which strain under a constant stress is a function of time. The relation between strain and time under a constant stress is known as creep phenomenon. The actual creep behavior of a viscoelastic material depends on the molecular structure, the temperature stress, the age, and the loading history of the material as a result of the manufacturing process. One of the most important properties which describes the mechanical behavior of the viscoelastic materials under stress is known as creep-compliance function or creep function which describes the time-variation of strain for a constant stress. It can be defined as the ratio of strain to the constant applied stress. This function is not given so it can be determined experimentally using a mechanical test.
1.5
Network Analysis Methods Many methods have been used to compute flows in a network of pipes.
Such methods range from graphical methods to the use of physical analogies and finally to the use of mathematical models [Ormsbee (2006)]. This section focuses on the following well-known methods: - Hardy Cross Method 8
Chapter 1
Introduction
- Newton-Raphson Method - Gradient Method
1.5.1 Hardy Cross Method Hardy Cross method was developed by Cross (1936) to solve head-loss equations in a looped network and it can be solved iteratively. This method requires a flow balance before the first iteration and initial guessed flow. Within a network with multiple loops, the Hardy Cross method determines a loop equation for each loop and solves one loop at a time [Newbold (2009)]. This method was based on loop-flow correction (Q) equations, where these corrections are calculated for each loop. Later, Cornish (1939) applied the principle of the Hardy Cross method to nodal-head correction (H) equations, where these corrections are calculated for each node. Both approaches proposed by Cross (1936) and Cornish (1939) are presently included under the Hardy Cross method [Bhave and Gupta (2006)]. The Hardy Cross method can be classified in the category of successive substitution methods, as in this method each equation is expressed explicitly to calculate one variable and this calculated variable is then substituted into the other equations to calculate another variable. The advantages of the Hardy Cross method can be summarized in its simplicity to program and its low code overhead. Also, it is a powerful tool for calculation of looped distribution network without limitation factors, such as: number of 9
Chapter 1
Introduction
loops, number of nodes or number of input nodes. One of the disadvantages of the Hardy Cross method is that it may encounter convergence difficulties for large circuits or even fail to provide a solution in some cases because this method does not solve the system of equations simultaneously. Another reason which causes a convergence problem; as reported by Wood and Rayes (1981); is the presence of some lines with very high and very low head losses included in the same energy equations which lead to ill-conditioning.
1.5.2 Newton-Raphson Method In order to improve the convergence of the Hardy Cross method, especially for large networks, Martin and Peter (1963) described a procedure to compute the flow adjustments for closed loop systems in which all the governing equations are solved simultaneously. Their procedure is known as Newton-Raphson method [Waheed (1992)]. This method achieves a solution through an iterative guess and correction procedure. Correction equations are constructed for all of the variables, instead of solving for the variables directly, to eliminate the error in each equation. The correction equations are solved by the Gaussian elimination method, then applied to each variable, which completes one iteration cycle. These iterative cycles of calculations are repeated until the residual error in all of the equations is reduced to a specified limit [Majumdar et al. (2007)].
10
Chapter 1
Introduction
Wood and Rayes (1981) concluded excellent convergence characteristics for the Newton-Raphson method and only few failures were reported. Although the Newton-Raphson method is numerically more stable and reliable than the Hardy Cross method, it requires a large amount of computer memory. Another disadvantage of this method is its sensitivity to the initial guess values as if these values are not close to the final solution, the procedure may not be converged. This problem is magnified in the network with pump or pressure reducing valve.
1.5.3 Gradient Method Todini and Pilati (1987) presented an efficient solution approach for steady-state flow conditions. Their algorithm is known as the gradient method. Their method operates simultaneously on the field of piezometric heads and flows. It allows a modeler to analyze large networks by solving a system of partly linear and non-linear equations that express the balance of mass and energy. It requires an initial guess for the pipe flows or nodal heads, but unlike the Hardy-Cross solution, a flow balance is not required. In this method, simple head-flow relationship for each pipe and continuity at the nodes are sufficient to solve any network problem. Although the number of equations in this method is larger than the other methods, Salgado et al. (1987) reported that the gradient method has lower number of iterations and lower convergence time because of solving all pipe 11
Chapter 1
Introduction
flows and nodal heads in each iteration. The composed system of equations to be solved in each iteration is a sparse, symmetric and positive-definite, which allows using highly efficient sparse matrix techniques for their solution. This method does not require defining loop equations, which may be a time consuming task. Another advantage of this method is that it can easily handle pumps and valves without having to change the structure of the equation matrix when the status of these components changes. Table 1.1 represents a comparison between the Hardy Cross, Newton-Raphson and Gradient methods. Table 1.1. Comparison of common network analysis methods
Unknown parameters Equations used Initialization Convergence problem Convergence speed Mode of analysis Hand calculation level Number of iterations Effect of network size on number of iterations Time per iteration Effect of network size on time per iteration Computer storage capacity requirements
Hardy Cross Newton-Raphson Gradient Method Method Method Q and H Q and H LHL* & NFC PHL & NFC Necessary Not necessary May be present Absent (as reported) Slow Fast Solved in parts Solved simultaneously Easy Hard Usually large Small Increases with the Independent of the network size network size Small Large Increases gradually Increases rapidly with the network size with the network size Small & increases Large & increases rapidly with the gradually with the network size network size
* LHL: Loop Head Loss equation; NFC: Node Flow Continuity equation; PHL: Pipe Head Loss equation
12
Chapter 1
1.6
Introduction
Thesis Structure This thesis is divided into six chapters as follows:
Chapter 1: Introduction, which illustrates the outlines of the thesis topics. Chapter 2: Literature Review, which covers the previous work in the field of the thesis topic and presents the aim of the study. Chapter 3: Theoretical Approach, which deals with the numerical analysis of both steady and unsteady states applying gradient method for steady analysis and viscoelastic model for unsteady state analysis. It also includes code validation by comparing with reference case studies. Chapter 4: Experimental Work, in which the water hammer experiment and the creep experiment were carried out on a viscoelastic pipe made of polypropylene. Chapter 5: Results and Discussion, which includes the experimental results and their comparison with viscoelastic model results. It also includes investigations of parameters affecting the viscoelastic model results. Chapter 6: Conclusions and Future Work, which includes the conclusions of the present study and recommendations for further work and studies.
13
Chapter 1
Introduction
14
Chapter Two Literature Review
Chapter 2
Literature Review
CHAPTER 2 LITERATURE REVIEW
2.1
Introduction Although hydraulic transients in closed conduits have been a subject of
both theoretical study and intense practical interest for more than one hundred years, it has not been fully understood. It is important to analyze transient flow in pipes to withstand additional loads resulting from the water hammer phenomenon. Analysis of water hammer in piping systems may be performed in different ways. It can be performed either by mathematical or numerical models. This chapter presents the development of previous studies in the field of hydraulic transients which has been used as reference sources, support and background for this research.
2.2
Brief Summary of Historical Development The problem of water hammer was first studied in 1858 by Menabrea,
although Michaud is generally accorded that distinction. In 1878, Michaud studied the problem of water hammer, and the design and use of air chambers and safety valves for controlling water hammer [Ghidaoui et al. (2005)]. 51
Chapter 2
Literature Review
By the year 1883, the friction losses were included in the analysis of water hammer, for the first time, by Gromeka. He assumed that the liquid was incompressible and the friction losses were directly proportional to the flow velocity. In 1898, Frizell attempted to develop a theoretical relationship for the velocity reduction in a pipe and the corresponding pressure rise. He was successful in this endeavor. He developed expressions for the velocity of water hammer waves and for the pressure rise due to an instantaneous reduction of the flow. He also discussed the effects of branch lines, wave reflections, and successive waves on turbine speed regulation [Chaudhry (1979)]. However, similar work by Joukowsky attracted greater attention. Joukowsky (1900) attempted to develop expressions relating pressure and velocity changes in a pipe. He developed a formula for the wave velocity, taking into consideration the elasticity of both the water and the pipe walls. He also developed the relationship between the velocity reduction and the resulting pressure rise. The propagation of a pressure wave along the pipe and the reflection of the pressure waves from the open end of a branch was the aim of the study. The effects of the variation of the closing rates of a valve were investigated also. In 1903, Allievi developed a general theory of water hammer from first principles and showed that the convective term in the momentum equation was negligible. He introduced two important dimensionless parameters that are widely used to characterize pipelines and valve behavior. He also produced 51
Chapter 2
Literature Review
charts for pressure rise at a valve due to uniform valve closure [Chaudhry (1979)]. After Joukowsky's and Allievi's publications, in the first two decades of the twentieth century, further refinements to the governing equations of water hammer appeared in books by Rich (1951), Parmakian (1963), and Streeter and Wylie (1967) and numerous papers were published on the analysis of water hammer. Because of their large number, they are not listed herein.
2.3
Experimental Work To validate the numerical methods, the computed data, including
pressure and/or velocity profiles, have to be compared with the experimental data. So, a review of previous experimental works is presented.
2.3.1 Experimental Work on Elastic Pipes Table 2.1 represents a summary of the previous experimental systems information and parameters. In these systems, pipes of elastic materials were used.
51
Chapter 2
Literature Review
Table 2.1. Experimental systems information and parameters (Elastic Pipes) Test Rig Data Authors
Year
Facility
Pipe D Material (mm)
L (m)
Wave Initial Fluid Speed Velocity (m/s) (m/s)
Water 1350 Tank-pipeCopper 25.4 36.09 valve Oil 1324
Holmboe and Rouleau
1967
Bergant et al.
2 Tank2001 pipe-valve
Lambert et al.
2001
2 Tankpipe-valve
6166
Turb.
0.128
82
Lam.
Copper 22.1 37.23 Water 1319
0.1 0.2 0.3
1870 3750 5600
Lam. Turb. Turb.
Copper 22.1 37.23 Water 1319
0.3
5600
Turb.
---
---
---
0.408
1570
Lam.
0.336
5700
Turb.
0.931
15800
Turb.
---
---
---
2 TankCopper 22.1 37.23 Water 1319 pipe-valve Tank-pipe2005 Steel 42.0 72.00 Water 1245 valve
Marcinkiewicz et al.
2008
Ismaier and Schlücker
Tank-pipecentrifugal 2009 pumpvalve
2 Tankpipe-valve
Flow Pipe Range Design
0.244
Vítkovský et al. 2007 Szymkiewicz and Mitosek
Re No
Copper 16.0 98.00 Water 1309
---
0.45 75.00 Water 1300 (D ) (max.) o
Findings
Spiral Considering the viscous coiled shear as frequency-dependent pipe provides good results. Using unsteady friction Straight model provides more pipe accurate results. Boundary-layer growth Straight model produces more pipe realistic results. Straight Using head measurements pipe reduces sensitivity. Straight Their method produces better pipe results. Spiral Using unsteady friction coiled model provides more pipe accurate results. Pulsating centrifugal pumps Closed can damp pressure surges loop during fluid transients.
Lam. = Laminar flow Turb. = Turbulent flow The flow range of this case is transient flow. It can be considered as low Reynolds number turbulent flow according to Bergant et al. (2001).
51
Chapter 2
Literature Review
Holmboe and Rouleau (1967) made their tests on a reservoir-pipe-valve system. They presented two test results, one represented turbulent flow case, while the second represented laminar flow one. In the first test, the operating fluid was water. In the second test, the operating fluid was oil. Their results were credible, as they used as reference data for comparison in many researches over more than four decades. They considered the viscous shear to be frequency-dependent and found good agreement between theoretical and experimental results. Bergant et al. (2001) investigated water hammer wave forms using a flexible laboratory apparatus for investigating water hammer and column separation events in pipelines which has been designed and constructed by Bergant and Simpson (1995). Bergant et al. (2001) performed three experimental runs with water as an operating fluid. Their results showed that applying the unsteady friction model provides more accurate results than using quasi-steady friction model. The experimental apparatus which has been designed and constructed by Bergant and Simpson (1995) is used also by Lambert et al. (2001) and Vítkovský et al. (2007). The experimental test performed by Lambert et al. (2001) to test their new unsteady friction model for turbulent flows which based on the growth and destruction of the boundary layer during a transient event. They found that the boundary-layer growth model is producing a more realistic unsteady friction
19
Chapter 2
Literature Review
effect compared to the standard quasi-steady friction model. It produced more damping results. Vítkovský et al. (2007) presented further development of inverse transient analysis (ITA) and experimental observations for leak detection. They explored that using head measurements as boundary conditions for ITA reduces significantly sensitivity, making both detection and quantification problematic. Szymkiewicz and Mitosek (2005) proposed a modified finite element method to solve the unsteady pipe flow equations. They carried out a set of experiments to check the validity of their proposed method. They focused attention on the simplest case of their work. Their results showed that the required damping and smoothing of a pressure wave can be obtained when numerical diffusion is produced by their applied method. Marcinkiewicz et al. (2008) carried out experiments to verify the ability of Relap5, Flowmaster2™ and Drako® programs to calculate the water hammer loadings using the unsteady friction model. They used a test stand specially erected at the laboratory of IMP PAN in Gdansk (Poland), which was described in details by Adamkowski et al. (2010). The experimental test consisted in several runs of unsteady flow after sudden closing of the valve from different initial flow conditions. Their analyses showed quite differences between damping of pressure waves and forces calculated with Relap5, Flowmaster2™ and Drako®. They showed that the differences can be reduced by using unsteady friction models in calculations. 20
Chapter 2
Literature Review
The dynamic interaction between water hammer and pressure pulsations, generated by centrifugal pumps, was investigated experimentally by Ismaier and Schlücker (2009). Their experimental investigations were performed at a piping system, built at the Institute for Process Technology and Machinery. They found that interaction can cause damping and amplifying effects and that pressure surges pass centrifugal pumps almost unhindered, because they are hydraulically open. Their measurements at that testing facility showed that pulsating centrifugal pumps can damp pressure surges generated by fast valve closing. So, pressure pulsation should be considered in critical applications right from the planning and computation stage. Their results also showed that one dimensional fluid codes can be used to calculate this phenomenon.
2.3.2 Experimental Work on Viscoelastic Pipes Table 2.2 summarizes the previous experimental systems information and parameters, in which viscoelastic pipes were used.
21
Chapter 2
Literature Review
Table 2.2. Experimental systems information and parameters (Viscoelastic Pipes) Test Rig Data Authors
Year
Facility Pipe Material
2 Tank-pumpPolyethylene pipe-valve
Brunone et al.
2000
Covas et al.
2004
Tank-pipevalve
Mitosek and Chorzelski
2003
Kodura and Weinerowska
Diameter Length Thickness Fluid (m) (m) (mm)
Unsteady loss model has more ability to Water simulate the pressure peak decay after the first cycle. - Applying elastic model leads to Water inaccurate results. - Creep test has variable results. Pressure disturbance velocity depends Water on the pipe length.
0.0938
352.0
8.1
Polyethylene (HDPE)
0.0506
277.0
6.2
Tank-pipevalve
Polyethylene (MDPE)
0.0408
36.00
4.6
2005
Tank-pipevalve
Polyethylene (MDPE)
0.0408
36.00
4.6
Water
Bergant et al.
2008
2 Tank-pipevalve
Polyethylene (MDPE)
0.0220
37.20
1.63
Water
Bergant et al.
2011
Tank-pipevalve
0.2354
261.2
7.3
Water
Bergant et al.
2013
Tank-pipevalve
0.2354
261.2
7.3
Water
Himr
2013
0.029
58.28
5.0
Water
Polyvinyl chloride (PVC) Polyvinyl chloride (PVC)
2 Tank-pipePolyethylene valve
22
Findings
Pressure disturbance velocity increases when the length of the pipe decreases. - Unsteady friction model is unable to simulate pressure fluctuations alone. - Better results can be achieved by applying UF model with viscoelastic model. Applying unsteady friction model with viscoelastic model leads to favorable simulation of hydraulic transient events. The influence of unsteady friction and FSI are important to be considered with viscoelastic model. Using Matlab-Simulink-SimHydraulics or HYDRA softwares can simulate water hammer well.
Chapter 2
Literature Review
Brunone et al. (2000) used an experimental apparatus to measure the velocity profile and pressure decay in a pipe system during two transient events caused by valve operation. He compared the experimental pressure readings to a water hammer model using a conventional quasi-steady representation of head loss and one with an improved unsteady loss model. They demonstrated that the unsteady loss model has a superior ability to track the decay in pressure peak after the first cycle. Covas et al. (2004) studied the effect of the pipe material on the pressure response of a fluid system during the occurrence of transient events. In order to study this effect in the case of using viscoelastic pipes, they performed their experimental work on a high density polyethylene (HDPE) pipe-rig at Imperial College (London, UK). They found that overpressures estimated assuming a linear elastic mechanical behavior of the pipe can be underestimated in 5–25% in polyethylene pipes. They determined the creep function experimentally and observed variability in the results, due to pipe constraints and stress-time history, which cannot be accounted for in these tests. Another experimental work has been performed to study the effect of using viscoelastic pipes on pressure response of fluid systems under transient conditions. This work was performed by Mitosek and Chorzelski (2003) and Kodura and Weinerowska (2005), who carried out measurements on a water hammer test rig with a medium density polyethylene (MDPE) pipe. The carried out experiments showed that the pressure disturbance velocity in MDPE pipe 23
Chapter 2
Literature Review
strongly depends on the length of the pipeline while the velocity increases noticeably when the length of the pipe decreases. Bergant et al. (2008b) presented a number of case studies with numerical results showing how a number of important parameters, such as cavitation and gas pocket, leakage and blockage and the viscoelasticity of the pipe wall, affect water hammer in a simple reservoir-pipeline-valve system. They used the test system to study the effect of applying the viscoelastic model. They concluded that applying the unsteady friction model alone cannot simulate the total damping observed in pressure fluctuations as the case of applying the elastic model for a plastic pipe. However, results obtained by combining unsteady friction and viscoelastic models have a slightly higher dampening than the actual observed data. Bergant et al. (2011) used a large-scale pipeline apparatus in order to validate their numerical and mathematical models which study the viscoelastic effect. Their apparatus consisted of a reservoir-pipe-valve system. They observed a favorable fitting between numerical results and observed data when both unsteady friction and viscoelastic pipe wall mechanical behavior are incorporated together in the hydraulic transient model. Bergant et al. (2013) investigated experimentally the effects of pipe-wall viscoelasticity on water hammer pressures using a large-scale pipeline apparatus made of polyvinyl chloride (PVC), which is described in details by Bergant et al. (2011). Their results revealed that beside the direct influence of 24
Chapter 2
Literature Review
the rheological behavior of the pipe wall, some effects should be taken into account for a precise comparison with numerical calculations such as FSI and unsteady friction. While other effects can partially contribute to the transient responses such as, nonlinear viscoelastic effects, pipe-wall vibrations, pipewall thickness, convective terms and free small gas pockets. Himr (2013) performed experimental measurements to compare between two programs; Matlab-Simulink-SimHydraulics, a commercial software to solve hydraulic systems, and HYDRA software, a program developed in Matlab by Brno University of Technology. He carried out his measurements on a simple gravitational pipeline test rig, which consists of a polyethylene pipe with a length of 58.28 m and a diameter of 29 mm connecting the two tanks and with a fast closing valve to generate the transient event. He found that both simulations were in very good agreement with experimental data and differences in comparison were minimal.
25
Chapter 2
2.4
Literature Review
Numerical Work Many researchers addressed the mathematical and numerical modeling
of the water hammer phenomenon to simulate this hydraulic problem. The mechanical behavior of the pipe material was found to be an effective parameter which affects the response of the hydraulic systems, especially under transient loads. However, this section reviews some of the numerical studies carried out to model the water hammer phenomenon. Some of these studies deal with pipes made of elastic materials, e.g. steel and copper, while the others deal with pipes manufactured from viscoelastic materials, e.g. PVC and polyethylene.
2.4.1 Numerical Work Related to Elastic Pipes Table 2.3 represents the summary of the previous numerical work which related to elastic pipes. It also shows whether the numerical model was validated by an experimental work or not.
26
Chapter 2
Literature Review Table 2.3. Numerical studies related to elastic pipes
Authors
Year
Mathematical Model
Vardy and Hwang
1991 Method of characteristics
Sun and Wang
1995
Šavar et al.
1998 Method of characteristics
Brunone et al.
2000
Guinot
2000
Kochupillai et al.
2005
Szymkiewicz and Mitosek
2005
Wood et al.
2005
Taïeb
2006
Saikia and Sarma
2006 Method of characteristics
Hashemi and Abedini 2007 Barten et al.
- Eigenfunction method - COMMIX code
Experimental Work (used for validation)
Theoretical work only Holmboe and Rouleau (1967) Holmboe and Rouleau (1967)
Findings Quasi-steady relationships were inaccurate in transient laminar or turbulent flows. Their results fitted the experimental results.
Steady friction model was suitable for high Re number flow, but not for low Re numbers. - A quasi-steady loss model Unsteady loss model simulates pressure peaks Brunone et al. (2000) - An unsteady friction in a better manner. - The 1st order Godunov-type scheme gives Finite volume scheme faster, inaccurate predictions. Theoretical work only (Godunov-type scheme) - The 2nd order Godunov-type scheme gives slower, very accurate predictions. In some cases, the structural vibration response Finite element method Theoretical work only increases with FSIs. Szymkiewicz and The numerical factors influences on the quality Modified finite element method Mitosek (2005) of the results. - Method of characteristics - MOC and WCM have the same accuracy. Theoretical work only - Wave characteristic method - WCM is more efficient for large systems. Holmboe and Rouleau Lax-Wendroff method Model results fitted with reference results. (1967) Incremental Differential Quadrature Method
2008 - TRACE and Relap5 codes
Theoretical work only Using a constant friction is not recommended. IDQM has unconditional stability and accuracy, as an advantage. For low pressure and temperature, the TRACE Theoretical work only code showed better results. Theoretical work only
27
Chapter 2 Kwon and Lee
Marcinkiewicz et al.
Literature Review
- Method of characteristics 2008 - The axisymmetrical model - The implicit scheme model - A quasi-steady loss model - An unsteady friction model, 2008 using Relap5, Flowmaster2™, and Drako® codes
Pothof
2008 New unsteady friction model
Rohani and Afshar
2008
Wahba
2008 2-D water hammer model
Alireza et al.
2009
- Method of characteristics - k-ω turbulence model
Riasi et al.
2009
- 4th order Runge-Kutta scheme - k-ω turbulence model
Wahba
Baldwin–Lomax turbulence 2009 model
Li et al.
2010
Rohani and Afshar Hou et al.
Implicit method of characteristics
- SIMPLE algorithm - CFD Point-implicit method of 2010 characteristics 2012
Corrective smoothed method
Kwon and Lee (2008)
Head loss coefficient for 1-D models should be bigger than the Darcy-Weisbach frictional coefficient.
Marcinkiewicz et al. (2008)
Unsteady friction has a strong damping effect on dynamic forces acting on pipe during the transient events.
- Quasi-steady friction models are accurate for fluid transients without flow reversal. - Unsteady friction model is accurate for fluid transients with flow reversal. The IMOC requires more computational effort Theoretical work only than the MOC. Holmboe and Rouleau Model results fitted with reference results. (1967) Holmboe and Rouleau (1967), Brunone et al. Model results fitted with reference results. (2000) Bergant and Simpson (1995), Brunone et al. (2000)
Theoretical work only Model results fitted with reference results. - For low Re number, model results agreed well with experimental data. - For high Re number, model over predicted the attenuation of the transient. SIMPLE algorithm for compressible flow can Theoretical work only simulate water hammer effectively.
Bergant et al. (2001) and Marcinkiewicz et al. (2008)
Theoretical work only PIMOC more efficient than IMOC. particle
CSPM has potential for water hammer Theoretical work only problems with free surfaces, as column separation and slug impact. 28
Chapter 2
Literature Review
Vardy and Hwang (1991) developed a quasi two-dimensional model of transient flows in pipes of circular cross-section, which applied to laminar and turbulent flows. They found that quasi-steady relationships were highly inaccurate in transient laminar or turbulent flows. Sun and Wang (1995) analyzed water hammer by an eigenfunction method and used the general-purpose computer code COMMIX. Their results compared with Holmboe and Rouleau (1967) experimental data and comparison showed good agreement. Šavar et al. (1998) performed a numerical analysis in a simple pipe and applied the steady friction model. They compared their results with Holmboe and Rouleau (1967) experimental results. The comparison showed that such friction factor model was suitable for high Reynolds number flow, but a better model was needed for low Reynolds numbers. Brunone et al. (2000) carried out a set of experiments relating the evolving fluid structure to a transient event. They compared the experimental pressures with a water hammer model using a conventional quasi-steady representation of head loss and one with an improved unsteady loss model. They found that the decay in pressure peak after the first cycle was simulated better by applying the unsteady loss model. Guinot (2000) used a Godunov-type scheme to model water hammer problem with the first and second order schemes based on Taylor series expansions of
29
Chapter 2
Literature Review
Riemann invariants. He found that the first-order scheme is almost 2000 times faster than the exact one, but gives inaccurate predictions at low densities. While the second-order approximate solver gives very accurate solutions and is 300 times faster than the exact, iterative one. Kochupillai et al. (2005) developed a simulation model for the FSIs that occur in pipeline systems. A new FEM formulation, based on flow velocity, has been developed to deal with the valve closure transient excitation problems. Their results showed that there are situations where the structural vibration response increases with FSIs. This result is in contrast to what is normally accepted in the literature as FSI reduces the structural displacements. Szymkiewicz and Mitosek (2005) proposed a modified finite element method to solve the unsteady pipe flow equations. They compared their numerical results with experimental data. This comparison showed an essential influence of the numerical factors on the quality of the results and the nature of this influence was explained by the accuracy analysis carried out. Wood et al. (2005) compared the formulation and computational performance of the MOC and Wave Characteristics Method (WCM). Results indicated that, for the same modeling accuracy, the WCM was more computationally efficient for analysis of large water distribution systems. Taïeb and Taïeb (2006) presented a model, inspired by the model presented in Prado and Larreteguy (2002), for transient laminar flow in pipes. The new
30
Chapter 2
Literature Review
model was tested against mathematical model presented in Bird et al. (1960), numerical simulations worked out by Vardy and Hwang (1991), and experimental results published by Holmboe and Rouleau (1967). These tests showed that the new model was able to obtain comparable results. Saikia and Sarma (2006) presented a numerical model for solution of water hammer problems. The stability and accuracy of the method were tested by comparing the results with the solutions obtained by Lax Diffusive Method, which developed by Streeter (1969) and found that their presented numerical model was well applicable for all flow conditions. The results indicated that use of constant friction is not recommended. Hashemi and Abedini (2007) applied Incremental Differential Quadrature Method (IDQM) to model unsteady flow in pipeline systems. They found that the unconditional stability and accuracy of the proposed method were the main advantages of IDQM method compared with classical numerical method. Barten et al. (2008) analyzed codes to model cavitation water hammers and discussed time-dependent pressure, flow behavior, and the generation and collapse of vapor bubbles at the valve and the first bridge. They showed that the codes were able to model the flow behavior of the water hammer for the high pressure and temperature case, but for the lower pressure and temperature case, the TRACE code provided a good approximation of the propagation of the pressure wave.
31
Chapter 2
Literature Review
Kwon and Lee (2008) studied the transient flow in a pipe using both experimental and computer models. They utilized and compared the method of characteristics model, axisymmetrical model, and implicit scheme model. They found that the head loss coefficient for the one dimensional models should be much bigger than the Darcy-Weisbach frictional coefficient. Marcinkiewicz et al. (2008) verified the ability of Relap5, Flowmaster2™ and Drako® programs to calculate the water hammer loadings by comparing calculations with experimental results. They used a quasi-steady friction model and unsteady friction model. Their analysis showed quite small differences between the pressures and the forces calculated with Relap5, Flowmaster2 and Drako and that the unsteady friction has a very strong damping effect on dynamic forces acting on pipe segments during the unsteady flow conditions. Pothof (2008) proposed a new formulation for the unsteady shear stress and validated it. He used the apparatus constructed by Bergant and Simpson (1995) to validate his model. He observed that quasi-steady friction models have a reasonable accuracy of the pressure damping during a high Reynolds number (over 100,000) fluid transient without flow reversal, such as pump trip or start-up scenarios. While during fluid transients with flow reversal, such as valve closure scenarios, the unsteady friction is dominant. Rohani and Afshar (2008) proposed an Implicit Method of Characteristics (IMOC) to alleviate the shortcomings and limitations of the conventional MOC. The proposed method was used to solve two example problems of
32
Chapter 2
Literature Review
transient flow and the results presented and compared with those of the explicit MOC. The results showed that the IMOC requires more computational effort than the conventional explicit version, as it requires the solution of a nonlinear system of equations at each time step. Wahba (2008) studied the damping of laminar fluid transients in piping systems numerically using a two-dimensional water hammer model solved using the numerical scheme developed by Wahba (2006). He compared his theoretical results with Holmboe and Rouleau (1967) experimental results and comparison showed good agreement between numerics and experimentation. Alireza et al. (2009) investigated the behavior of unsteady velocity profiles in laminar and turbulent water hammer flows numerically and applied a k-ω turbulence model. They compared the numerical results for both steady and unsteady turbulent pipe flows with the experimental results presented in Holmboe and Rouleau (1967) and Brunone et al. (2000) and they found that the numerical results were in good agreement with the experimental data. Riasi et al. (2009) studied the behavior of unsteady turbulent pipe flow resulting from water hammer numerically; using the fourth order Runge-Kutta scheme and an accurate k-ω turbulence model. The numerical results found to be in good agreement with test data obtained. Two-dimensional simulations of turbulent water hammer flows are carried out by Wahba (2009). The implemented turbulence model was the Baldwin–
33
Chapter 2
Literature Review
Lomax model. He compared the numerical results with the experimental results of Bergant et al. (2001), for low Reynolds number case and found that numerical simulations using both forms agreed well with experimental data, while he compared his results with Marcinkiewicz et al. (2008), for high Reynolds number case and showed that the frozen form over predicted the attenuation of the transient. Li et al. (2010) derived the SIMPLE series numerical algorithm for weakly compressible water flow to simulate the water hammer. They compared their results with CFD numerical calculations of water hammer. The results showed that the SIMPLE algorithm for compressible flow can effectively simulate water hammer and the numerical dissipation can smooth the wave front of water hammer. A Point-Implicit Method of Characteristics (PIMOC) was presented by Rohani and Afshar (2010). They used their proposed method to simulate some numerical examples and the results were compared with those of original IMOC and basic pump element formulation. The comparison indicated higher efficiency of the proposed model compared to the original one. Hou et al. (2012) simulated water hammer by applying the corrective smoothed particle method (CSPM) which was used in order to solve the classical water hammer equations. In this method, the time derivatives were approximated by Euler-forward time integration algorithm and the spatial derivatives by the corrective kernel estimate. The CSPM results were 34
Chapter 2
Literature Review
compared with conventional MOC solutions and they were in good agreement. Their results showed that the cubic spline kernel is the most efficient and accurate to be used. Although CSPM does not beat MOC in classical water hammer, it has potential for water hammer problems with free surfaces such as column separation and slug impact. 2.4.2 Numerical Work Related to Viscoelastic Pipes Table 2.4 summarizes the previous numerical work which related to viscoelastic pipes. It also shows whether the numerical model was validated by experimental work or not.
35
Chapter 2
Literature Review Table 2.4. Numerical studies related to viscoelastic pipes
Authors
Year
Mathematical Model
Experimental Work Friction (used for validation) Model
Covas et al.
2005 Method of characteristics Covas et al. (2004)
Weinerowska
2006
Bergant et al.
2008 Method of characteristics Bergant et al. (2008b)
UF
Duan
2009
Quasi-2D water hammer Brunone et al. (2000), model Covas et al. (2004)
UF
Bergant et al.
2011 Method of characteristics Bergant et al. (2011)
Keramat and Tijsseling
2012 Method of characteristics
Landry et al.
2012 Method of characteristics Bergant et al. (2001)
UF
Achouyab and Bahrar
2013 Method of characteristics Güney (1977)
SF
Himr
2013
---
UF
Kodura and Weinerowska (2005)
SF
UF
Güney (1983), Covas et al. (2004)
UF
- ode15s method Himr (2013) - Lax-Wendroff method
SF
36
Findings - Applying the viscoelastic model is important. - Viscoelasticity caused highly damped and attenuated pressure wave. - Applying the viscoelastic model gives satisfactory results. - Applying the elastic model is not accurate. - At the lower phase velocity, higher attenuation and dispersion of the wave form occurs. - The viscoelastic behavior effect of the pipe wall is dominant over unsteady friction. - The unsteady friction has a slight effect on the damping and dispersion of the pressure wave. - During the initial transient stage, the effect of the unsteady friction to damp the pressure peak was comparable to the viscoelasticity effect. - As time increases, the viscoelasticity effect becomes dominant both in terms of damping and phase shift. Incorporation of both unsteady friction and viscoelasticity leads to more accurate results. The viscoelasticity effect was much larger than the unsteady friction effect. The unsteady friction damping contribution is important for small-scale systems, not for large-scale transmission lines. The viscoelastic behavior of the pipe wall is even more pronounced as the temperature is high. Frequency and amplitude were satisfying for about 15% of the study time period, then simulation became less accurate.
Chapter 2
Literature Review
Covas et al. (2005) developed a mathematical model, which incorporates additional terms to take into account unsteady friction and pipe-wall viscoelasticity, to calculate hydraulic transients in pressurized polyethylene pipe systems. The developed model was calibrated and tested using with experimental data collected by Covas et al. (2004). They found that the retarded deformation of the pipe-wall caused the mechanical damping of the transient pressure wave and the pressure wave was highly attenuated and dispersed in time. Their results also showed that taking the viscoelastic behavior into account is important. Weinerowska (2006) presented and analyzed a viscoelastic model of water hammer in a single polymer pipeline. To avoid model complications, he calculated the friction factor like a steady flow, neglecting its dependence on time and space derivatives of the velocity and other factors. He used measurements collected by Kodura and Weinerowska (2005) in order to validate his model. He found that applying the viscoelastic model of water hammer enables obtaining of a satisfactory solution, compared with the classical elastic model which is not sufficiently accurate in the case of polymer pipe. Bergant et al. (2008a) investigated key parameters that may affect the pressure wave form predicted by the classical theory of water hammer. They presented one-dimensional mathematical models which describe unsteady friction, cavitation, fluid–structure interaction, pipe-wall viscoelasticity, and leakages
37
Chapter 2
Literature Review
and blockages in the transient pipe flow. These models were based on the MOC, such that they can easily be implemented in standard water hammer codes. They used Bergant et al. (2008b) experimental results in order to study the effect of applying the viscoelastic model. Their results showed that higher attenuation and dispersion of the wave form occurs at the lower phase velocity. They also found that the effect of the viscoelastic behavior of the pipe wall in the transient event is dominant over unsteady friction, as the addition of unsteady friction to the viscoelastic model has hardly any effect on the pressure response other than a slightly higher damping and dispersion of the pressure wave. Duan (2009) investigated the relative importance of unsteady friction and viscoelasticity effects in pipe fluid transients on the wave propagation, including peak damping and phase shift and pointed out new interpretations to the viscoelastic term in the water hammer model. He carried out the investigation using a quasi-2D water hammer model that incorporates both unsteady friction and viscoelasticity effects. He used the experimental results of the first test run, which was performed by Brunone et al. (2000) and the experimental results presented by Covas et al. (2005) to verify their proposed model. The results showed that the contribution of the unsteady friction effect to the peak damping of pressure head was comparable to the viscoelasticity effect during the initial transient stage while, as time increases, the viscoelasticity effect becomes dominant both in terms of damping and phase shift. 38
Chapter 2
Literature Review
Bergant et al. (2011) studied the mathematical modeling of hydraulic transients in PVC pipes by adding the retarded strain in the transient elastic pipe flow equations. Their mathematical model used to predict water-hammer responses in viscoelastic pipes taking into account unsteady friction losses. To validate their numerical and mathematical models, a large-scale pipeline apparatus has been used. They found a favorable fitting between numerical results and observed data, due to the incorporation of both unsteady skin friction and viscoelastic pipe wall mechanical behavior in the hydraulic transient model. Keramat and Tijsseling (2012) investigated water hammer in a reservoirpipe-valve system with taking all of unsteady friction, column separation, FSI and viscoelasticity effects into account. They applied MOC for numerical simulation. They compared their model results with two experiments on polyethylene pipes carried out by Güney (1983) and Covas et al. (2004). They found that both of unsteady friction and viscoelasticity caused attenuation of the transient pressures, where the viscoelasticity effect was much larger than of the unsteady friction effect. Landry et al. (2012) developed a theoretical formulation of the viscoelastic damping in piping systems without cavitation. Their new viscoelastic model implemented in SIMSEN software was described by adding a diffusion term corresponding to the expansion viscosity and a new inductance corresponding to significant unsteady shear at the pipe-wall. The unsteady friction model has
39
Chapter 2
Literature Review
been incorporated into the method of characteristic algorithm (MOC). Their numerical results were compared with the experimental results presented by Bergant et al. (2001). The results revealed that the unsteady friction damping effect on numerical results is important for small-scale, such as laboratory systems, but not for large-scale water supply and transmission lines. However, they concluded that it is very difficult to make the distinction between the frictional and viscoelastic behavior. Achouyab and Bahrar (2013) presented a numerical code to analyze water hammer phenomena in plastic pipes by applying the steady friction model with taking FSI into consideration. Digital processing was achieved using the method of characteristics. They attempted to couple the dynamic behavior of the pipe wall with the inclusion of the Poisson coupling and pressure wave effects. They compared their numerical results with those experimentally obtained by Güney (1977). Their study showed that the viscoelastic behavior of the pipe wall is even more pronounced as the temperature is high. Himr (2013) investigated two programs; Matlab-Simulink-SimHydraulics, commercial software that uses ode15s numerical method, and HYDRA software, which is based on the Lax-Wendroff numerical method to solve the momentum and continuity equations. He considered the physical properties of liquid and pipe elasticity parameters. Numerical simulation with both programs showed that the frequency and amplitude were satisfying for the first three seconds; about 15% of the study time period; then simulation became less
40
Chapter 2
Literature Review
accurate, but still sufficient. He attributed these results to the unknown behavior of the pipe damping, as it is not constant. He also suggested replacing the Kelvin-Voight model of the pipe wall with some more complicated one.
2.5
Review of Network Analysis Methods Table 2.5 summarizes the scientific contributions made to develop the
three methods.
Table 2.5. Summary of the scientific contributions to develop the common network analysis methods
Hardy Cross Method
Method
Author Cross Cornish Hoag and Weinberg Adams Voyles and Wilke Dillingham McCormick and Bellamy Barlow et al. Williams
Year
Contribution
1936 Developing the method based on Q equations. 1939 Applying the method principle to H equations. Suggesting a factor applied to the magnitude of the 1957 corrections to speed up the method convergence. 1961 Developing a program based on the method. Improving the method convergence by selecting the loops 1962 with minimum sum of the common flow resistance factors. Applying the computed head correction to both the end 1967 nodes of the pipe of a large resistance value to improve convergence. Suggesting special measures to maximize individual 1968 correction to improve convergence of the method. Suggesting the use of a quadratic correction factor to 1969 overcome the convergence problem. Using distributional factors for initializing pipe flows and 1973 suggested the use of a convergence acceleration factor.
Kootattep and 1985 Suggesting the use of buffered successive over relaxation. Aya
41
Chapter 2
Literature Review
Gradient Method
Newton-Raphson Method
Martin and Peter Shamir and Howard Epp and Fowler Lam and Wolla Larock et al.
1963 1968 1970 1972 2000
Djebedjian et al.
2005
Yaseen Mondy
2007 2008
Vaseti
2011
Hamam and Brameller
1971
Osiadacz
1987
Todini and Pilati Salgado et al. Ahmed and Lansey Elhay and Simpson Simpson and Elhay
1987 1987 1999 2011 2011
Describing the method which computes the flow adjustments for closed loop systems by solving all equations simultaneously. Introducing a program based on node-oriented equations using the method to linearize the nonlinear equations. Introducing a program based on loop-oriented equations using the method to linearize the nonlinear equations. Suggesting a check at the end of each iteration to assume different initial solutions if the method failed to converge. Developing a program which applied the method. Linking the network solver, that uses the method, to transient analyzer and genetic algorithm as an optimization tool. Developing a software which used the method. Developing a program which applied the method. Presenting a methodology to remove the effects of initial guess on the convergence of the method. Suggesting the "Hybrid Method", which is similar to the gradient method. Suggesting the "Newton Loop-Node Method", which is similar to the gradient method. Developing the method to simultaneously obtain (Q, H) values. Extending the method to incorporate pumps into the system. Formulating the pipe-node equations in an integral form to analyze larger systems easily. Proposing a regularization technique to overcome computation failure caused by zero flows. Presenting the correct Jacobian matrix formulas when the method is applied with the Darcy-Weisbach head-loss formula.
In order to speed up the convergence of the Hardy Cross method, many techniques were improved by Hoag and Weinberg (1957), Voyles and Wilke (1962), Dillingham (1967), McCormick and Bellamy (1968), Barlow et al. (1969), Williams (1973) and Kootattep and Aya (1985).
42
Chapter 2
Literature Review
Hoag and Weinberg (1957) suggested a factor of 0.5 applied to the magnitude of the corrections. Voyles and Wilke (1962) have shown that the convergence is improved when the loops are selected such that the sum of the common flow resistance factors is minimum. Dillingham (1967) treated a pipe with a large resistance value as an elongated joint and applied the computed head correction to both the end nodes of the pipe. McCormick and Bellamy (1968) suggested special measures to maximize individual correction and thereby improve convergence, while Barlow et al. (1969) suggested the use of a quadratic correction factor. Williams (1973) used distributional factors for initializing pipe flows in his studies on convergence problems in network analysis. He also suggested the use of a convergence acceleration factor. Kootattep and Aya (1985) suggested the use of buffered successive over relaxation, i.e., the use of an accelerator and decelerator. The Hardy Cross method was also used in developing programs applied to network analysis. Adams (1961) developed one of the first mainframe programs for pipe-network analysis, which based on the Hardy Cross method. Many researches carried out to improve the Newton-Raphson method. Shamir and Howard (1968) introduced a program based on node-oriented equations and solves for pressure, demand, and the parameters of pipes and nodes. Epp and Fowler (1970) developed another program which based on loop-oriented equations and solves only for pressures and flow rates. Both programs used the Newton-Raphson method to linearize the nonlinear mass and energy equations.
43
Chapter 2
Literature Review
In order to remove some of the shortcomings of the Newton-Raphson method, Lam and Wolla (1972a,b) suggested making a check at the end of each iteration to verify the convergence of the method, so that different initial solutions can be assumed if the method had a very slow rate of convergence or failed to converge. Vaseti (2011) presented a methodology to improve the weaknesses of the Newton-Raphson method and remove the effects of initial guess on the convergence of the method. He revised the components of the Jacobean Matrix and made some modifications to reduce the variable difference between two iterations in order to solve the oscillation problem around the final solution. Larock et al. (2000), Djebedjian et al. (2005), Yaseen (2007) and Mondy (2008) developed FORTRAN codes which applied the Newton-Raphson method in the steady-state analysis of networks. After presenting the Gradient method by Todini and Pilati (1987), other researchers contributed to their efforts to improve this method. Salgado et al. (1987) extended the original gradient method to incorporate pumps into the system and compared the gradient algorithm for the analysis of water distribution networks with other traditional algorithms and showed some of the advantages of the method. Hamam and Brameller (1971) suggested the "Hybrid Method", whereas Osiadacz (1987) presented the "Newton LoopNode Method". These methods are similar to the gradient method. The difference between these methods is the way in which flows are updated after a
44
Chapter 2
Literature Review
new trial solution for nodal heads has been found. But the gradient method was found to be simpler, so it was chosen for use in EPANET. Ahmed and Lansey (1999) formulated the pipe-node equations in an integral form to analyze larger systems easily. They developed an approach similar to the gradient method. Elhay and Simpson (2011) proposed a regularization technique to overcome computation failure caused by zero flows when the gradient algorithm is used to solve for the steady state of a system in which the head loss is modeled by the Hazen-Williams formula or the Darcy-Weisbach formulation. They suggested a new convergence stopping criterion for the iterative process based on the infinity-norm of the vector of nodal head differences between successive iterations. They recommended checking the infinity norms of the residuals once iteration has been stopped, as this test ensures that inaccurate solutions are not accepted. In order to overcome the gradient method shortcomings, which appear when the Darcy-Weisbach head-loss formula is used, Simpson and Elhay (2011) presented the correct Jacobian matrix formulas which take into account the friction factor’s dependence on flow when the gradient method is applied with the Darcy-Weisbach head-loss formula.
45
Chapter 2
2.6
Literature Review
Aim of the Present Work Studying the water hammer phenomenon is very important to predict the
pressure fluctuations during transient events. In order to get information about a network during transient event numerically, it is required firstly to perform a steady-state analysis for the network. According to previous review, the gradient method was chosen to perform this analysis as it has no reported convergence problems and requires a lower number of iterations and lower convergence time. Using polypropylene in Egypt in a wide range increased the importance of studying flow phenomena related to this type of pipes, especially the water hammer phenomenon. In addition, there is no experimental work carried out to study water hammer in polypropylene pipes, according to the author's knowledge, as it is clear in Table 2.2. So, pipes made of polypropylene are selected in the present work to be studied. According to literature, many researches applied both of unsteady friction and viscoelastic models together to improve numerical results, while some authors reported that the viscoelastic effect is dominant over unsteady friction effect. Therefore, in order to simplify the unsteady-state analysis, the viscoelastic model only is selected to perform this analysis, with neglecting the unsteady friction effect.
46
Chapter 2
Literature Review
This study aims to simulate the water hammer phenomenon in viscoelastic pipes. The specific objectives of the study are: 1- Develop a FORTRAN code for steady flow analysis based on the gradient method. 2- Compare code solution, for steady flow analysis, with a reference case study solution. 3- Develop a FORTRAN code to simulate hydraulic transient, in which the viscoelastic model is applied. 4- Validate the code by comparing its results with a reference case study results. 5- Set up an experimental test rig to validate the numerical work. 6- Compare code solutions and experimental measurements.
47
Chapter 2
Literature Review
48
Chapter Three Theoretical Approach
Chapter 3
Theoretical Approach
CHAPTER 3 THEORETICAL APPROACH
3.1
Introduction The simulation of hydraulic transients in closed conduits can be
performed in two steps; the first is to perform a network analysis in steady state, then water hammer model can be applied to simulate the flow in the unsteady state, using the data obtained from the first step. This chapter presents the development of a steady-state model to analyze networks, using the Gradient Method. It, also, presents a mathematical model that is used to simulate water hammer in viscoelastic pipes.
3.2
Steady State Network Analysis The analysis of a pipe network requires determining all the unknown
parameters with the help of the known ones and the available interrelating equations. There are four main parameters governing the analysis of a network. These parameters are: pipe discharge, Q, nodal head, H, nodal flow, q, and the pipe resistance constant, R. The pipe discharges are almost unknown and obtained from analysis [Bhave and Gupta (2006)].
94
Chapter 3
Theoretical Approach
3.2.1 Governing Equations The governing equations applied to such a network are: the continuity and energy equations. For a number of pipes equal to (X) connected to a junction (j), the continuity equation can be written in the Discharge-Form as follows:
X Q x q j 0 x 1
(3.1)
If the node (j) is connected to the node (i) through the pipe (x), which has a pipe resistance constant (Rx), then, the friction head loss through the pipe (x) can be determined from
H i H j Rx Qxn
(3.2)
where n is an exponent. Therefore, the continuity equation can be rewritten in the Head-Form as follows: X Hi H j x 1 R x
1/ n
qj 0
(3.3)
In Equation (3.1), the unknown pipe flows are taken as the basic unknown parameters, while the unknown nodal heads are taken as the basic unknown parameters in Equation (3.3).
05
Chapter 3
Theoretical Approach
The loop head loss equation (energy equation) for a loop can be written in the following form:
Rx Qxn 0
(3.4)
Loop
In order to determine unknown parameters in a pipe network, the continuity equation is applied to all junctions, while the energy equation is applied to available loops in the network. This generates a system of linear and nonlinear equations, which have to be solved simultaneously. This system of equations can be solved iteratively, as in Hardy Cross Method and Newton-Raphson Method. But these methods may encounter convergence difficulties, as Hardy Cross Method may fail to converge for large circuits while Newton-Raphson Method also may not be converged if the initial guess values are not close to the final solution. That is why another procedure was developed. It is called Gradient Method.
3.2.2 Gradient Method This method was presented by Todini and Pilati (1987). In this method, the generated nonlinear equations, i.e. the pipe head loss relationship, are linearized, and then a system of linear equations is ready for solving. The application of the gradient method is described herein for different situations. In the following sections, a mathematical model is developed. Then, the
05
Chapter 3
Theoretical Approach
developed code used to solve the model is discussed and verified through different case studies.
3.2.2.1 Mathematical Model In this section, the mathematical model of the gradient method for two different cases is developed.
3.2.2.1.1 Networks with Known Pipe Resistances In order to linearize the pipe head loss relationship (Equation 3.2), initial values of heads and flow rates at all nodes and pipes at the tth iteration can be assumed to be known, as an initial guess for nodal heads or the pipe flows supposed to be set. Then, the corrected nodal head at the node (i) can be determined from the relation: t 1 H i t H 0i t H i
where
t H i
(3.5)
is the correction of the assumed
t H 0i
value for the tth
iteration. Using Taylor's series expansion neglecting the residue after two terms, then Equation (3.2) can be written as
t H 0i t H i t H 0 j t H j R0 x t Q0nx nR0 x t Q0 x
05
n 1
t Qx
(3.6)
Chapter 3
Theoretical Approach
t H 0 j
where
t H j
t Q0 x
values for the tth iteration. For fixed nodal heads, no correction is
and
t Qx
are the correction of the assumed
and
necessary. Equation (3.6) can be rewritten, using Equation (3.5), as follows
t1 H i t 1H j nR0 x t Q0 x n1 t Qx R0 x t Q0nx Subtracting
n R
n 0 x t Q0 x
from both sides and replacing
(3.7)
t Q0 x t Qx
by
t 1 Q x get:
t 1 H i t 1 H j nR0 x t Q0 x n1 t 1 Qx 1 nR0 x t Q0nx
(3.8)
This equation provides (X) number of linearized equations involving corrected values of pipe discharges and nodal heads as unknowns. The continuity equation; Equation (3.1); is linear and can be rewritten for corrected discharge values as: X t 1 Qx q0 j x 1
(3.9)
It is applied for (N) number of nodes of unknown nodal heads. Therefore, Equation (3.9) provides (N) number of linear equations which can be solved simultaneously with Equation (3.8) to get the corrected values of X-pipe discharges and N-unknown nodal heads. For the first iteration, the pipe
05
Chapter 3
discharge,
Theoretical Approach
t Q0 x , can be assumed to be unity or can be taken as some other
arbitrarily chosen value.
3.2.2.1.2 Networks with Pumps When a pump is located in a pipe, x, then the head supplied by this pump, hP, can be expressed in terms of the discharge passing through it, Qx, as follows: hP AP Qx2 BP Qx CP
(3.10)
where AP, BP and CP are constants determined by fitting Equation (3.10) to three points taken from the expected working range of the pump performance curve. The linearized pipe head loss equation for this case was presented by Bhave and Gupta (2006) as follows: n AP R0 x t Q0 x
n 1
t 1 Qx
BP
t 1 Qx t 1 H j
H 0 s C P 1 n AP R0 x t Q0nx
(3.11)
where H 0 s is the sump level at pumped source node (s). When Equation (3.11) is solved simultaneously with Equation (3.9), all unknown pipe discharges and nodal heads are got.
09
Chapter 3
Theoretical Approach
3.2.2.2 Code Development From the previous section, it is clear that Equation (3.11) is a generalized form of the linearized pipe head loss equation, therefore it is used in the code used in order to solve for unknown pipe discharges and nodal heads. This code is a development of the FORTRAN code presented by Larock et al. (2000), which used the Newton-Raphson method to solve for unknown pipe discharges and nodal heads. Therefore, it was easier to develop a FORTRAN subroutine code to solve the steady state network problems using the gradient method. This code is attached in Appendix A. In this code, Equations (3.9) and (3.11) were expressed in the matrix form:
A X B
(3.12)
where for (n) equations, i.e. (n) unknowns, [A] is a square sparse (nn) matrix containing the coefficients of the system, while [X] and [B] are the unknowns and the known constants column vectors of n-rows, respectively. The developed subroutine code firstly scans all pipes and nodes and generates an expanded (mm)-[A] matrix and m-rows [X] and [B] vectors, where (m) is the sum of the numbers of all pipes and nodes. Then, the columns and rows corresponding to the known nodal heads were eliminated to reduce the [A] matrix size to (nn) and to get n-rows of [X] and [B] vectors. In order to solve this system of linear equations, the following form is used:
00
Chapter 3
Theoretical Approach
X A1 B
(3.13)
where [A–1] is the inverse of the coefficient matrix. This procedure is shown in Figure 3.1, which represents a flow diagram of the code.
Start
Read all pipes, nodes and pumps data
Checking of known heads nodes
Filling the expanded [A] matrix
Filling the expanded [B] vector
Elimination of columns & rows corresponding to known nodal heads, in matrix [A] & vector [B]
Iterations start
Initialization of unknown discharges in pipes and nodal heads
Updating discharges & nodal heads values
Solving the set of linear equations No
Error ≤ Max. Error Yes
No
Iteration No. ≤ Max. Iteration No. Yes
Printing solution
End
Figure 3.1. Gradient method subroutine flow chart
05
Chapter 3
Theoretical Approach
3.2.2.3 Code Validation In
order
to
validate
the 50 m
developed code, a reference case study
1
was investigated. It represents a four-
[1]
pipe, shown in Figure 3.2, which was
2
presented by Bryan et al. (2006). For [2]
all pipes, the resistance constant was
[3]
assumed to be 0.935 and Hazen3
Williams equation was applied with an exponent, n = 1.852. The comparison between the developed code and
3
4
[4]
2.5 m3/s
2.0 m /s
Figure 3.2. Schematic diagram of Bryan et al. (2006) four-pipe network case study
Bryan et al. (2006) results were presented in Table 3.1. The comparison shows that the gradient method has been modeled and implemented herein correctly. Table 3.1. Comparison between developed code and Bryan et al. (2006) four-pipe network case study results
Unknown Parameters Pipe Discharges (m3/s)
Nodal Heads (m)
Code Results
Bryan et al. (2006) Results
Q1
4.5000
4.50
Q2
2.2403
2.24
Q3
2.2596
2.26
Q4
0.2403
0.24
H2
34.8448
34.85
H3
30.6800
30.69
H4
30.6133
30.62
05
Chapter 3
3.3
Theoretical Approach
Unsteady State Analysis Unsteady state analysis for flow in pipes can be carried out by applying
either elastic or viscoelastic model. An appropriate model should be selected according to the pipe material type. However, elastic model can be applied in order to predict the water hammer pressure fluctuations in elastic pipes, i.e. steel, copper or concrete pipes, whereas the viscoelastic model is valid to be applied to investigate water hammer effects for both elastic or viscoelastic pipes. The reason of this fact is explained in section 3.3.6. For pipes made of plastic such as PE, PVC and PP, viscoelasticity is a crucial mechanical property which changes the hydraulic and structural transient responses [Keramat et al. (2012)]. In this section, the mathematical representation of viscoelasticity is introduced. It was concluded by Keramat et al. (2012) that is the FSI has no significant effects when a pipe is fixed rigidly while Soares et al. (2008) noted that unsteady friction effects are negligible when compared to pipe-wall viscoelasticity. So, a mathematical model to simulate water hammer in viscoelastic pipes is developed taking into account the viscoelastic behavior of pipe walls and neglecting FSI and UF effects. 3.3.1 Viscoelasticity Viscoelastic materials exhibit both viscous and elastic characteristics when undergoing deformation due to its molecular nature. When viscoelastic materials subjected to a certain instantaneous stress σ0, they do not respond 05
Chapter 3
Theoretical Approach
according to Hooke’s law. The viscosity of the viscoelastic material gives the substance a
σ(t) σ(t)
an
instantaneous
response and a retarded viscous response,
as
shown
t
ε(t) ε(t)
(t) εεrr(t)
εε00
Figure 3.3(a), so that the total strain ε can be decomposed into an instantaneous-elastic strain,
t1
t ε1 (t)+ ε22(t) (t)
ε0,2
ε11(t) (t)
ε0,1 0,1 ε0,2 0,2
εεee tt
in
σσ22
ε(t)
Loading Unloading Unloading Loading Phase Phase Phase Phase
elastic
(b) (b)
σσ11+ +σ σ22 σ11
σσ00
strain rate which is dependent on time. These materials have
σ(t) σ(t)
(a) (a)
ε22(t) (t) tt11
tt
Figure 3.3 (a) Stress and strain for an instantaneous constant load (b) Boltzmann superposition principle for two stresses applied sequentially
εe, and a retarded strain, εr as:
e r (t )
(3.14)
Basically, the relation between stress σ and strain ε for linear viscoelastic materials involves higher-order time derivatives of both stress and strain. But for small strains, applying “Boltzmann superposition principle”, an alternative representation of the relation between stress and strain can be presented. It is a combination of stresses that act independently resulting in strains that can be added linearly. This is shown in Figure 3.3(b) for the particular case of two stresses. So, the total strain generated by a continuous application of stress σ(t) is [Bergant et al. (2008a)]:
04
Chapter 3
Theoretical Approach
J (t )
(t ) J 0 (t ) * t
(3.15)
where J0 is the instantaneous creep-compliance and "*" denotes convolution, while the term [(t)J0] represents the immediate response. For linearly elastic materials, the constant creep-compliance J0 is equal to the inverse of Young’s modulus of elasticity, i.e., J0 = 1/E0. The creep-compliance function J(t), which describes the viscoelastic behavior of the pipe material, can be determined experimentally using a mechanical test or calibrated on collected transient data [Covas et al. (2004), Covas et al. (2005) and Bergant et al. (2011)]. In order to determine the stress-strain interactions and temporal dependencies of viscoelastic materials, many
1
NKV
E1
E2
E NKV
viscoelastic models have been developed. For small deformations,
E0
it is usually applicable to apply linear viscoelastic models, for
Figure 3.4. Generalized Kelvin–Voigt model
example a generalized Kelvin– Voigt model consisting of NKV parallel spring and dashpot elements in series with one additional spring, as shown in Figure 3.4. In this mechanical model, the elastic behavior of viscoelastic materials is modeled using springs, while dashpots are used to model the viscous behavior. It is used to describe the creep function [Aklonis et al. (1972)] as follows:
55
Chapter 3
J (t ) J 0
Theoretical Approach
J k 1 e t /
N KV k 1
k
(3.16)
where Jk is defined by Jk = 1/Ek, Jk and Ek are the creep-compliance and the modulus of elasticity of the spring of the Kelvin–Voigt k-element, and τk = μk/Ek. τk and μk are the retardation time and the viscosity of the dashpots of k-element. The parameters Jk and τk of the viscoelastic mechanical model are adjusted to the creep-compliance experimental data. The creep-compliance function for a material is dependent on temperature, stress, age, and orientation as a result of the manufacturing process [Lai and Bakker (1995)]. These effects are not included in Equation (3.16). The total strain indicated by Equation (3.15) consists of an elastic and a viscoelastic part. The viscoelastic part is a function of the whole loading history. In the case of water hammer, this loading comes from the fluid pressure head within the pipe. The steady head H0 accounts for the static situation as one can assume that a long time has been passed after the establishment of H0. It fully determines the static response of the structure. As the dynamic response is of interest here, the dynamic head H , which represents the difference between the fluid pressure head H and static head H0, as defined by Equation (3.17), is used in the coming formulations: H H H0
(3.17)
55
Chapter 3
Theoretical Approach
If the dynamic head is substituted for stress σ in the integrand of Equation (3.15), a function which represents up to a constant factor the retarded response to water hammer is given by [Keramat et al. (2010)]: I H (t )
N KV
Jk
t
H (t S )e k 1 k
0
s / k
N KV dS I H k (t ) k 1
(3.18)
3.3.2 Governing Equations In this section, the governing equations for water hammer in a viscoelastic pipe are presented. In the presented model, the relevant assumptions are: the piping system consists of thin-walled (i.e. the ratio of thickness to diameter e/D ≤ 0.1 1,2), linearly viscoelastic pipes with no buckling and no large deformations and neglecting FSI, UF and convective terms. Another important assumption made in this derivation is that the pipe is totally restrained from axial movements. Applying the Kelvin-Voigt model to simulate viscoelasticity, then the continuity and momentum equations can be written in the following forms [Keramat et al. (2010)]:
1 2
g H V 2 2 0 S a t t
(3.19)
f V 1 p dz g V | V | 0 t S dS 2 D
(3.20)
http://courses.washington.edu/me354a/Thin%20Walled%20Pressure%20vessels.pdf http://www.freestudy.co.uk/statics/torsion/torsion2.pdf
55
Chapter 3
Theoretical Approach
where,
1 2 * dJ
(3.21)
where V is the fluid velocity in m/s, S the axial coordinate, g the gravitational acceleration in m/s2, a the wave speed in m/s, H the head in m, t the time in s,
the circumferential strain, p the pressure in Pa, z the elevation in m, f the friction coefficient, D the pipe internal diameter in m, ν Poisson’s ratio, circumferential stress in Pa, and J stands for creep compliance function in Pa-1. The thin-walled pipe assumption also allows the dynamic circumferential hoop stress to be calculated according to:
g DH
(3.22)
2e
where is the fluid density in kg/m3, and e is the pipe wall thickness in m. Using Equations (3.20) and (3.22) in (3.19), and eliminating the convolution operator, Equation (3.19) can be rewritten in the pressure form as follows: a2
V 1 p C 01 0 S t
(3.23)
where
55
Chapter 3
Theoretical Approach
I H s z 2 2 gD C 01 g a 1 e t t S
A comparison between the theories of elastic and viscoelastic models is summarized in section 3.3.6.
3.3.3 Method of Characteristics Implementation From the previous section, the continuity and momentum equations, which govern unsteady fluid flow in pipelines, are found as partial differential equations. The standard procedure to solve these equations is the MOC. This procedure yields water-hammer compatibility equations that are valid along characteristic lines in the distance (S)–time (t) plane.
3.3.3.1 Development of the Characteristic Equations Multiplying Equation (3.20) by Lagrange multiplier, a constant linear scale factor (), and adding the result to Equation (3.23), get: f V 1 p p dz V a2 V | V | C 01 0 (3.24) g S t S dS 2 D t
The first group can be replaced by (.dV/dt), then (a2 = .dS/dt), while the second group can be replaced by (1/.dp/dt), then ( = dS/dt). To satisfy these two requirements for (dS/dt), then (2 = a2). This leads to:
59
Chapter 3
Theoretical Approach
a
(3.25)
Then, Equation (3.24) after replacing groups can be rewritten as follows: f dz dV 1 dp V | V | C 01 0 g dS 2 D dt dt
(3.26)
Replacing Lagrange multiplier with its values from Equation (3.25), then Equation (3.26) can be rewritten in the head form, as it is easier to visualize the propagation of pressure waves, as follows: C dV g dH f V | V | 01 0 only when dt a dt 2 D a
dS dt a (3.27)
C dV g dH f V | V | 01 0 only when dt a dt 2 D a
dS dt a (3.28)
Equations (3.27) and (3.28) are called the characteristic equations. These equations describe a family of straight lines of slope (1/a) on the S-t plane. For the lines associated with the characteristic equation (3.27), which has a positive slope, they are referred to as C+ characteristics. Similarly, the lines associated with the characteristic Equation (3.28), which has a negative slope, they are referred to as C– characteristics. Figure 3.5 shows the C+ and C– characteristic lines on the S-t plane for a reservoir-pipe-valve system.
50
Chapter 3
Theoretical Approach
S
t t
P
t
C
C+
t-t
Le
Ri
S L i 1
i
i+1
Figure 3.5. The method of characteristics grid
3.3.3.2 The Finite Difference Equations Representation In order to get the numerical solution of Equations (3.27) and (3.28), they have to be written in finite difference form. They become after multiplying them by (t) as follows:
VP VLe g H P H Le
C t ft VLe | VLe | 01 0 2D a
(3.29)
VP VRi g H P H Ri
C t ft VRi | VRi | 01 0 2D a
(3.30)
a
a
55
Chapter 3
Theoretical Approach
C t The term 01 can be rewritten in the following form: a I C 01t z gt C 07 H a S t
I C 01t z gt C 07 H a S t
where, C07 1 2 t. a
gD e
dS only when a dt
(3.31)
dS only when a dt
(3.32)
.
By substitution in Equations (3.29) and (3.30), get:
VP VLe g H P H Le a
I ft VLe | VLe | gt. sin C07 H 2D t
0
(3.33)
VP VRi g H P H Ri a
I ft VRi | VRi | gt. sin C07 H 2D t
0
(3.34) where [sin θ = (z/S)] is positive for pipes sloping upward in the downstream
direction. Keramat et al. (2010) evaluated the term I H / t using the following linear relation for the unknown head at the current time step:
55
Chapter 3
Theoretical Approach
I H N KV I H k a1 H P (t ) a2 t k 1 t
(3.35)
where, N KV
J k a1 1 e k k 1 t
t
Sum 01
(3.36)
a2 a3 H P t t H 0 a1 H P t t a4 N KV
t J k k a3 e k 1 k
Sum 02
(3.37)
(3.38)
t N KV e k a4 I H k t t Sum03 k 1 k
(3.39)
Then, Equations (3.33) and (3.34) can be rewritten as follows: ft VLe | VLe | 2D gt.sin C09 0
(3.40)
ft VRi | VRi | 2D gt.sin C09 0
(3.41)
VP VLe g C08 H P g H Le a
a
VP VRi g C08 H P g H Ri a
a
55
Chapter 3
Theoretical Approach
where C08 C07 * a1 and C09 C07 * a2 . Equations (3.40) and (3.41) represent the C+ and C– characteristic lines equations.
3.3.4 Code Development The viscoelastic model is programmed by the development of a FORTRAN code. It is a development of the FORTRAN code presented by Larock et al. (2000), which applied the elastic model only for the unsteady state analysis. The user has the ability to apply either the elastic model, if MODEL_SW equals one, or the viscoelastic one, if MODEL_SW equals two. The code can be applied to analyze a network of (NP) pipes during a time interval of (tmax) seconds with a fixed time step (Δt). When the viscoelastic model is applied, the code solves for the elastic model also. This allows the user to compare viscoelastic model against the elastic model results. Figure 3.6 represents a flow diagram of the code.
54
Chapter 3
Theoretical Approach
Steady State Analysis Analyze given network Get heads & velocities (H0,V0)
Read all pipes, nodes and pumps data
Start
Perform Elastic Model Analysis
1
Read MODEL_SW Print error message
Larock et al. (2000) code
Yes No
MODEL_SW = 1 Printing Elastic Model results
No MODEL_SW = 2
MODEL_SW = 2
Yes
No
Yes t = Δt
Printing Viscoelastic Model results
Start a new time step
End Calculate (C07, C08, C10, Sum01, Sum02) for the pipe (NP)
Start new pipe calculations
NP = 1 No
Calculate (C02, C03, C04, C06, C09, I Hk, Sum03) for the pipe (NP) and all junctions
1
Yes
t > tmax
No Compute heads & velocities (H,V) for all interior and boundary junctions NP = NP +1
NP > No of pipes
Yes
Figure 3.6. Viscoelastic code flow chart 55
t = t +Δt
Chapter 3
Theoretical Approach
3.3.5 Model Verification In order to verify the proposed mathematical model and its solutions, a case study presented experimentally by Covas et al. (2004) was investigated. They performed an experiment on a HDPE pipe-rig, which is rigidly fixed to a wall, at the Department of Civil and Environmental Engineering, Imperial College, London, with the specifications given in Table 3.2. It is a reservoirpipeline-valve system where the length between the vessel and the downstream globe valve is 277 m.
Table 3.2. Specifications of the reservoir-pipeline-valve experiment performed by Covas et al. (2004) Inner Wall Steady State Reservoir Valve Closure Wave Length Poisson Diameter Thickness Discharge Head Time Speed (m) Ratio (mm) (mm) (l/s) (m) (s) (m/s) 277
50.6
6.3
0.46
1.01
45
0.09
385
Covas et al. (2005) performed creep tests to determine the creep function of HDPE. They also estimated the order of magnitude of this creep function based on data collected from the pipe-rig and in the calibration of a mathematical model developed by Covas et al. (2004). The creep function coefficients provided by Covas et al. (2005) is mentioned in Table 3.3. The developed mathematical model results for the heads locations 1, 5 and 8 corresponding to distances from the upstream end of 271 m, 197 m and 55
Chapter 3
Theoretical Approach
116.5 m, respectively, are compared with the experimental results of Covas et al. (2004) in Figure 3.7. Table 3.3. Calibrated creep coefficients τk, Jk for the Imperial College test with Q0 = 1.01 l/s and a = 395 m/s, neglecting unsteady friction [Covas et al. (2005)] Sample Size
20.0 s
Retardation Times τk [seconds] and Creep Coefficients Jk (10-10Pa-1)
Number of Elements 5
K τk Jk
1 0.05 1.057
2 3 4 5 0.50 1.50 5.0 10.0 1.054 0.9051 0.2617 0.7456
It can be observed that viscoelasticity has been modeled and implemented herein correctly, as the average amplitude and frequency of the numerical solution are 96% and 95.1% of their values of the experimental results, respectively. The effect of FSI was not significant in the experimental results because the test-rig pipe sections were rigidly fixed and assumed to be constrained from any axial movement. The viscoelastic effect becomes more dominant with respect to unsteady friction, as time progresses. So, slight discrepancy between the numerical and experimental results with time progress can be observed, due to the neglect of the unsteady friction effect in the present model. The figure also shows that for viscoelastic model, the pressure wave damps faster than its damping in the elastic model. This agrees with the experimental results. Therefore, in case of studying viscoelastic pipes the viscoelastic model should be applied to predict pressure head fluctuations better. 55
Chapter 3
Theoretical Approach
x
L x/L= 0.42
Location #
8
0.71
0.972
5
1
Figure 3.7. Comparison between mathematical model results and experimental data of Covas et al. (2004) at three locations 55
Chapter 3
Theoretical Approach
3.3.6 Comparison between Elastic and Viscoelastic Models - Theory and Results According to this chapter, a comparison between elastic and viscoelastic models can be held, as shown in Figure 3.8. This comparison deals with models application, governing equations and effect on curve shape, as it explained below. a. Models Application Applying the elastic model is limited to investigate water hammer effects for elastic pipes only, while the viscoelastic model is more general as it is valid to be applied for studying transient events in both elastic or viscoelastic pipes. This is because of the fact that when both of elastic or viscoelastic materials are subjected to a certain instantaneous stress σ0, they have an instantaneous elastic response and a retarded viscous response, as shown in Equation (3.14), so that the total strain ε can be decomposed into an instantaneous-elastic strain, εe, and a retarded strain, εr. For elastic materials, the retarded strain term is zero. b. Models Governing Equations Another reason for the generality of the viscoelastic model lies in its governing equations. The momentum equation is a common one between elastic and viscoelastic models, while the continuity equation in
59
Chapter 3
Theoretical Approach
the viscoelastic model, Equation (3.19), includes an additional term (∂ευ/∂t) which can be assumed to equal zero for elastic materials. c. Models Effect on Curve Shape The selected model affects the pressure fluctuations during transient events. When the elastic model is applied, pressure waves appear likesquare peaks with high amplitude and low frequency with low damping rate. Conversely, applying the viscoelastic model makes pressure fluctuations appear in sharpened peaks with low amplitude and high frequency with high damping rate. The reason of these differences can be attributed to the dependence of the viscoelastic model on creep function, while for elastic model, it is creep function independent.
50
Chapter 3
Theoretical Approach
Water Hammer Models
Elastic Model (a)
Applied for
Viscoelastic Model Under fixed stress
Elastic Pipes
Viscoelastic/Elastic Pipes
Eq. (3.14) can be applied e r (t )
(b)
Continuity Eq.
Momentum Eq. [Eq. (3.20)]
Governed by Continuity Eq.
[Eq. (3.19)]
[Eq. (3.19)]
with
with
V g H 2 2 0 s a t t
Applied for
(b)
0 → for elastic pipes
Governed by
(a)
V g H 2 2 0 s a t t (c) Effect
Effect on Curve Shape (c)
on Curve Shape 1. Higher amplitude. 2. Lower frequency. 3. Low damping rate. 4. Like-square peaks. 5. Creep function independent.
1. Lower amplitude. 2. Higher frequency. 3. High damping rate. 4. More sharpened peaks. 5. Creep function dependent.
Figure 3.8. Summary of differences between water hammer models 55
Chapter Four Experimental Work
Chapter 4
Experimental Work
CHAPTER 4 EXPERIMENTAL WORK 4.1
Introduction This chapter aims to study and analyze the characterization of the
viscoelastic behavior during transient events experimentally. In order to perform this analysis, a simple tank-pipe-valve test rig was designed and carried out. The viscoelastic behavior is characterized by a creep function, which was determined by a creep experimental test.
4.2
Water Hammer Experiment Setup The test rig used to carry out the water hammer experiment is shown in
Figure 4.1. Figure 4.1(a) shows a schematic diagram of the test rig. It mainly represents a reservoir, pipe and valve system. The reservoir is replaced with the dotted block, including parts from (1) to (9) due to some design considerations which are explained in details in section 4.4. The test rig consists of a suction small-head reservoir (1), which is used as fixed-head water supply. The upstream pressure is measured using a pressure transducer, called H0 Pressure Transducer (3), while another pressure transducer is used to measure pressure fluctuations during the transient event, called Water Hammer Pressure Transducer (12). The pressure transducers output data are collected at a high 77
Chapter 4
Experimental Work
sampling rate by the data logger (13) which records these data on a computer (14). The solenoid valve (15) is used to generate the transient event. The flow rate is measured using a variable area flow meter (16). Flow rate at different parts of the test rig is controlled using three control valves (4), (7) and (17). The test section (10) is a polypropylene pipe with 20.0 m length and an internal diameter of 0.0212 m and wall thickness of 5.4 mm (e/D=0.25). It is supported with unequally spaced, 2.5 m height test section supports (11). Figure 4.1(b) represents an isometric detailed view of the test rig which developed using the SolidWorks Premium™ 2012 edition. In Figure 4.1(b), the test section is drawn to a scale of (1:10) and other parts are not to scale. Specifications of the used instruments are listed in detail in the following section. 14 13
To atmosphere
S
12
3 2
17
16
1- Reservoir 2- Main pump 3- H0 Pressure transducer 4- Control valve (C.V.2) 5- Non-return valve 6- Pressure gauge (H1) 7- Control valve (C.V.1) 8- Auxiliary pump 9- Relief valve
15
11
10
10- Test section 11- Test section supports 12- Water hammer pressure transducers 13- Data logger 14- Computer 15- Solenoid valve 16- Flow meter 17- Control valve (C.V.3)
(a) Schematic diagram 77
9
4 5 6 7
8
1
14
13
77 (b) Isometric detailed view
16
10
11
6
Figure 4.1. Water hammer test rig
17
15
12
3
8
4
2
5
1
7 9
Chapter 4 Experimental Work
Chapter 4
4.3
Experimental Work
Available Instruments The specifications of instruments used to carry out the water hammer
experiment are listed in this section in order to ease designing the test rig. These instruments include a solenoid valve, a flow meter, pressure transducers and a data logger.
4.3.1 Solenoid Valve The transient event in the water hammer experiment is
1
generated using a fast-closing, normally closed solenoid valve.
2
1- Closing spring 2- Armature
Figure 4.2 shows a schematic
3- Valve plate
diagram of the used solenoid
4- Valve orifice
5
3
5- Coil
valve and its specifications are
4
listed in Table 4.1. Figure 4.2. Solenoid valve schematic diagram Table 4.1. Solenoid valve specifications Media Permissible Valve Orifice Pressure Temperature Differential Closing Brand Type Code Function Diameter Range Pressure Time, Min. Max. (mm) (bar) (bar) tc (ms) (°C) (°C) Normally Danfoss EV210B 032U3624 – 10 + 100 25 0.25 70 0 - 30 Closed 78
Chapter 4
Experimental Work
4.3.2 Flow Meter One of the basic steps in the experiment is to measure the steady-state flow rate and velocity before the transient event
Figure 4.3. Spring loaded variable area flow meter schematic diagram
is generated. Therefore, a spring loaded variable area flow meter is used. In this flow meter, a spring and piston mechanism within a polysulfone metering tube is utilized as shown schematically in Figure 4.3. This mechanism is used as a balancing force, in order to make the meter independent of gravity, allowing it to be used in any plane. The specifications of the flow meter are listed in Table 4.2.
Table 4.2. Flow meter specifications
Brand
Code
Type
OMEGA
FL-9016
Variable Area
Measuring Max. Max. Range Connection Pressure Temperature Accuracy (LPM) (bar) (°C) Min. Max. 1" NPT male
3.8
78
60.6
22.4
+ 121
±5% Full Scale
Chapter 4
Experimental Work
The flow meter was calibrated experimentally
using
the
5
1
volumetric method, in which the amount of liquid collected
4
in a tank within a certain time interval
is
determined.
2
1- Reservoir 4- Control valve 5- Known-volume tank 2- Pump 3- Flow meter
A
schematic diagram of the test
3
Figure 4.4. Flow meter calibration test rig
rig used for this purpose is shown in Figure 4.4. In this calibration test, water is pumped through the flow meter
Qa & Qm are in LPM
filling a known-volume tank. The time required to fill a certain volume in the tank is measured using a stopwatch. The control valve is used to repeat the test at different flow rates. The relation between the actual
flow
rate,
Qa,
and
measured flow rate, Qm, is
Figure 4.5. Flow meter calibration curve
shown in Figure 4.5. The shown curve can be linearly fitted with the correlation shown in Figure 4.5.
78
Chapter 4
Experimental Work
4.3.3 Pressure Transducers 4.3.3.1 H0 Pressure Transducer It
is
a
mechanically-based
6
5
transducer as it senses the pressure by
4
1- Pressure fitting
means of a bourdon tube which
2- Housing
produces a force on a conversion
3- Bourdon tube
3
4- Conversion spring
spring. The resultant movement is
5- Inductive distance
converted into an electrical signal by
5- sensor 6- Electrical connectors
an inductive distance sensor, as shown in Figure 4.6. The output signal rises in proportion
to
the
pressure.
2
1
1 Figure 4.6. H0 pressure transducer schematic diagram
Its
specifications are listed in Table 4.3. Table 4.3. H0 Pressure transducer specifications
Brand
Code
SAUTER
DSU 110 F001
Measuring Max. Max. Range Output Connection Pressure Temperature Accuracy (bar) Signal (bar) (°C) Min. Max. 0-10 V DC
1" NPT male
0.0
78
10.0
16.0
+ 70
±1% Full Scale
Chapter 4
Experimental Work
The calibration of the transducer P
3
is achieved by the test rig shown
1
in Figure 4.7. In this experiment, water is pumped under a certain
5
pressure head, which is measured
2
1- Reservoir 4- Pressure gage 2- Pump 5- Control valve 3- Pressure transducer
using a calibrated Bourdon tube pressure gage (4). The transducer
4
Figure 4.7. Pressure transducer calibration test rig
(3) output signal is measured using a voltmeter. The control valve (5) allows repeating the test
P is in (bar) & Vt is in (Volt)
at different pressure head values. The
relationship
measured
between
pressure
and
the the
transducer output signal is shown in Figure 4.8, with the linear fitting correlation relating these parameters.
Figure 4.8. H0 pressure transducer calibration curve
78
Chapter 4
Experimental Work
4.3.3.2 Water Hammer Pressure Transducer During a transient event, rapid changes in pressure heads occur. So, an electronically-based,
fast
5
response,
pressure transducer is used. It uses a high-accuracy
silicon
strain
gages
1- Pressure fitting 2- Stainless steel
molecularly bonded to a stainless steel
2- diaphragm
diaphragm, as shown in Figure 4.9. Its
3- Silicon strain gages
4
4- Housing
specifications are listed in Table 4.4. It
1
5- Electrical connectors
was also calibrated, in a steady-state condition, using the test rig shown in
3
Figure 4.7, with the same procedure used
1
2
1
in calibrating the H0 pressure transducer as mentioned before. Figure 4.10 shows the relationship between the measured
1
1 Figure 4.9. Water hammer pressure transducer schematic diagram
pressure and the transducer output signal and the linear fitting correlation relating these parameters. Table 4.4. Water hammer pressure transducer specifications
Brand
Code
OMEGA
PX309300GV
Measuring Response Max. Range Output Connection Time Temperature Accuracy (bar) Signal (ms) (°C) Min. Max. 0-100 mV DC
1/2" NPT male
0.0 78
20.7
1.0
+ 50
±0.25% Full Scale
Chapter 4
Experimental Work
P is in (bar) & Vt is in (mV)
Figure 4.10. Water hammer pressure transducer calibration curve
4.3.4 Data Logger The study of the fast pressure fluctuations during the transient event requires saving the data collected from pressure transducers at a relatively high sampling rate. Therefore, a high-sampling rate data logger is used for this purpose. It is a USB 2.0 full speed voltage input data acquisition module with user programmable voltage inputs ranging from ±30 mV to ±10 V, full scale. All analog input channels can be measured sequentially at about 1 ms/channel. A total of 1000 samples/second can be taken, divided across all active channels. More specifications are listed in Table 4.5. 78
Chapter 4
Experimental Work
Table 4.5. Data logger specifications
Brand
Code Input Channels
Max. Sampling Rate (sample/second)
OM8 Differential or DAQOMEGA 16 Single-ended USBInputs 2401
4.4
1000.0
Input Range (V) -10 -5.0 -2.5 -2.0 -1.0
to to to to to
(mV) + 10 +5.0 +2.5 +2.0 +1.0
-500 -250 -125 -75 -30
to + 500 to + 250 to + 125 to + 75 to + 30
Max. Temperature (°C)
+ 50
Test Rig Sizing In this section, the sizing of the test rig and its design parameters,
including: pipe material, diameter, length and minimum reservoir head to avoid occurring column separation, are studied. All the available instruments' specifications listed in the previous section are considered.
4.4.1 Pipe Material Selection In recent years, the use of polypropylene material in piping systems for public and industrial water supply transport has been gradually increasing throughout the world, particularly in Egypt. Pipes made from Fusiolen Polypropylene Random Copolymer (shorted to Fusiolen PP-R) material are best known worldwide due to their good physical and chemical properties which are well-suited to the transport of potable water and to the heating field. The PP-R material is also an environmental friendly material, as it is fully 77
Chapter 4
Experimental Work
recyclable/PVC and toxin free. Pipes made of this material are available in the Egyptian market and they are manufactured by aquatherm® brand under the name of Fusiotherm® pipes (or aquatherm green pipes®, as a new brand name).
4.4.2 Pipe Diameter Selection Pipes used in water distribution and house connection in Egypt have commonly nominal diameters ranging from one-half inch to one inch. Therefore, the pipe diameter is selected within this range. The criterion used to select the appropriate diameter is the minimum steadystate head (H0,min) allowed to avoid column separation, to protect used pressure transducers against vacuum pressure. This critical head can be determined by applying Joukowsky equation, proved by Joukowsky (1900), Wylie and Streeter (1993) and Larock et al. (2000), which describes the change in pressure head (ΔH), as function of wave speed (a) and flow velocity change (ΔV) as follows: a H V g
(4.1)
4 Q0 and consequently If the flow is completely stopped, then V V0 2 D
Equation (4.1) can be rewritten as:
77
Chapter 4
Experimental Work
H
a 4Q0 g D 2
(4.2)
Since column separation occurs when pressure head decreases below fluid's vapor pressure head (hv), then, to avoid column separation, the value of (H0,min) can be determined as follows: a 4 Q0 H 0,min hv H hv 2 g D
(4.3)
From Equation (4.3), it is clear that the larger pipe diameter gives the lower (H0,min) value. The lower value of (H0,min) represents the minimum requirement of the steady state head without column separation. Therefore, it is preferable to choose this case, i.e. choosing the larger pipe diameter of the range mentioned previously. So, the pipe diameter was selected to be one-inch nominal diameter. The actual dimensions of aquatherm® green pipe SDR6, one inch in nominal diameter is shown in Table 4.6. Table 4.6. Specifications of selected aquatherm® green pipe
Pipe Series
Material
Art.No.
SDR 6/ S
fusiolen PP-R
10012
Permissible Working Pressure (bar) Internal Wall Diameter, Thickness, at Temperature at Temperature of 20°C (1-50of 30°C (1-50D (mm) e (mm) year service time) year service time) 21.2
5.4
77
30 - 25.7
25.5 - 21.8
Chapter 4
Experimental Work
4.4.3 Wave Speed Solution The type of pipe support affects the wave speed value, as derived by Wylie and Streeter (1993). They developed formulas to determine the wave speed for three different support situations: Case (a): pipe anchored at its upstream end only; Case (b): pipe anchored throughout against axial movement. Case (c): pipe anchored with expansion joints along the pipeline. They showed that the equation for wave speed can be conveniently expressed in the general form: a
K/ K D 1 C0 E e
(4.4)
where, For Case (a) restraint:
C0 5 / 4
(4.5.a)
For Case (b) restraint:
C0 1 2
(4.5.b)
For Case (c) restraint:
C0 1.0
(4.5.c)
78
Chapter 4
Experimental Work
In a practical sense, the actual pipe restraint situation probably is not conform precisely to any of these cases, but lies somewhere in this range of possibilities [Larock et al. (2000)]. In order to calculate the wave speed in the pipe selected in the previous section, it is required to collect the values of parameters in Equations (4.4) and (4.5). For water at a temperature of 20○C, the density1, , bulk modulus2, K, and vapor pressure head3, hv, are 998.2 kg/m3, 2.15 GPa and 0.23826 m, respectively. Whereas for PP-R material, Poisson ratio, ν, and modulus of elasticity, E, are 0.45 and 1,100 MPa4, respectively. From these data, the wave speed can take three different values according to pipe support situations. These values are listed in Table 4.7. Table 4.7. Wave speed values at different pipe support situations
1 2 3 4
Case
Pipe Support Situations
Wave Speed, a (m/s)
(a)
Pipe anchored at its upstream end only
549.2882
(b)
Pipe anchored throughout against axial movement
550.0277
(c)
Pipe anchored with expansion joints along the pipeline
498.3285
http://www.engineeringtoolbox.com/water-density-specific-weight-d_595.html http://www.engineeringtoolbox.com/bulk-modulus-elasticity-d_585.html http://eweb.chemeng.ed.ac.uk/jack/newWork/Chemeng/Chemeng/water.html http://www.ineos.com/Global/Olefins%20and%20Polymers%20USA/Products/Technical%20information/En gineering%20Properties%20of%20PP.pdf
78
Chapter 4
Experimental Work
4.4.4 Selecting Pipe Length The study of the water hammer phenomenon caused by sudden valve closure requires that the valve close quickly before the pressure wave reaches the valve. This means that valve closing time is required to be less than the critical time of closure which can be calculated from the following relation: t critical
2L a
(4.6)
However, the water hammer is generated using a fast-closing solenoid valve with data listed in Table 4.1. From the table, it is clear that the closing time (tc) is specified with 70 ms, which must be less than the critical closing time described in Equation (4.6). Therefore, the minimum pipe length (Lmin) at which the valve closing time (tc) is equal to the critical closing time (tcritical) is determined as: Lmin
tc a 2
(4.7)
Substituting with wave speed values mentioned in Table 4.4 in Equation (4.7) leads to get three values of (Lmin) shown in Table 4.8.
78
Chapter 4
Experimental Work
Table 4.8. (Lmin) corresponding to different pipe support situations Pipe Support Situation Cases
a (m/s)
Lmin (m)
Case (a)
549.2882
19.225
Case (b)
550.0277
19.251
Case (c)
498.3285
17.441
From Table 4.8, there are three different available lengths according to support situation. The length of the test rig pipe should be higher than the maximum value of these values. Thus, a pipe of 20 m length was selected.
4.4.5 Selecting Minimum Reservoir Head In this section, Equation (4.3) is applied to calculate (H0,min) at different support situations, assuming a flow rate of 3.0 LPM and using hv value mentioned before. The (H0,min) values are illustrated in Table 4.9.
Table 4.9. (H0,min) corresponding to different pipe support situations Pipe Support Situation Cases
a (m/s)
H0,min (m)
Case (a)
549.2882
8.1695
Case (b)
550.0277
8.1802
Case (c)
498.3285
7.4337
78
Chapter 4
Experimental Work
From Table 4.9, it is clear that there was a problem in determining the reservoir dimensions, as the Hydraulic Machines Laboratory, Faculty of Engineering, Mansoura University, is approximately 5 m height only. So, it is required to use a pump to get an appropriate head simulating the required reservoir head. Reservoir head supposed to remain constant during the transient event and equal to steady state head (H0). When a pump is used instead of the reservoir, the (H0) value changes during the transient event, as a result of changing operating point. This fact is shown in Figure 4.11. Therefore, a hydraulic circuit is designed to maintain the main pump operating point and head at (H0).
H Hmax H0 System Curve
Q=0
Performance Curve
Q0
Q
Figure 4.11. Operating point change during transient event
78
Chapter 4
Experimental Work
Figure 4.12 represents a schematic diagram of this hydraulic circuit. The idea of this circuit is based on controlling the bypass line of the main pump. This line was required to be fully closed at steady state condition. When the transient event occurred, it was required to open and allow the initial flow (Q0) to pass through it to maintain (H0) constant at the upstream end of the test section. This target was achieved using a non-return valve and an auxiliary pump. 3 2 10
9
4
5 6
7
1
8 1- Reservoir 2- Main pump 3- Pressure transducer (H0) 4- Control valve (C.V.2) 5- Non-return valve
6- Pressure gauge (H1) 7- Control valve (C.V.1) 8- Auxiliary pump 9- Relief valve 10- Test section
Figure 4.12. Upstream reservoir simulation circuit
At steady-state condition, the solenoid valve, main pump and auxiliary pump were energized. The pressure head (H1) is higher than (H0) to block the bypass line completely. This was achieved by controlling the control valve (C.V.1). This step is shown in Figure 4.13(a). 78
Chapter 4
Experimental Work
H0
H0
Q=Q0
Q=0
H1
H1
(a)
(b)
Figure 4.13. Operation of upstream reservoir simulation circuit (a) at steady state condition (b) at unsteady state condition
When transient event occurred, both of the solenoid valve and auxiliary pump were de-energized at the same moment. This leaded to drop the pressure head (H1) to a value less than (H0) and to open the bypass line. To control the flow rate passing through the bypass line, the second control valve (C.V.2) was used, so that the pressure head at the main pipe upstream end was remained constant at (H0). This step is shown in Figure 4.13(b). During the transient event, if pressure fluctuations caused increasing pressure at the upstream end of the test section to a value higher than (H0), then the relief valve works as a vent to drop the pressure to its original value. The relief valve is adjusted to open once its upstream pressure increased by a slight value higher than (H0).
78
Chapter 4
Experimental Work
4.4.6 Selecting Main Pump Operating Point There are two pumps used in test rig shown in Figure 4.1. The manufacturer specifications of the main pump are shown in Table 4.10, whereas specifications of the auxiliary pump are shown in Table 4.11. Table 4.10. Main pump manufacturer specifications Power
Brand Calpeda
Type
Code
Motor Speed (rpm) (kW) (hp)
Self-Priming NGM 4-60/A 0.75 Jet Pump
1.0
3450
Max. Flow (QPmax) (LPM)
Max. Head (HP,max) (m)
66.6
45.0
Table 4.11. Auxiliary pump manufacturer specifications
Brand
Type
Code
Pedrollo
Self-Priming Jet Pump
JSWm 10H
Power
Motor Speed (rpm) (kW) (hp) 0.75
1.0
2900
Max. Flow (QPmax) (LPM)
Max. Head (HP,max) (m)
50.0
56.0
To select an appropriate pump operating point, the pump performance curve experiment has to be performed on the main pump. The test rig used to carry out this experiment is shown in Figure 4.14, with a suction head of 1 m. In this experiment, water is pumped through the pipe and its pressure and flow rate were measured using the H0 pressure transducer and the flow meter which are calibrated before. The control valve was used to change the operating point. Figure 4.15 shows a comparison between the pump performance curve of a 77
Chapter 4
new
Experimental Work
pump,
according
to
P
3
manufacturer data, and the used pump, according to experimental data.
5
Now, there are several available
4
2
appropriate operating point should
1- Reservoir 2- Pump 3- Pressure transducer 4- Flow meter 5- Control valve
be selected within two boundary
Figure 4.14. Pump performance test rig
operating
points.
However,
the
1
conditions. The first is that the minimum pressure head during the transient event, Hmin, must exceed the fluid vapor pressure, hv, to avoid column
separation.
While
the
second one is that the maximum pressure head during the transient event, Hmax, must not exceed the safe limit, about 60 m, to avoid damaging
any
of
the
used
instruments. Hmin and Hmax can be determined
from
the
following
Figure 4.15. Comparison between new, used main, and auxiliary pumps performance curves
equations:
77
Chapter 4
Experimental Work
H min H 0
a 4 Q0 g D 2
(4.8)
H max H 0
a 4 Q0 g D 2
(4.9)
Applying Equations (4.8) and (4.9) for maximum and minimum allowable wave speed value listed in Table 4.9, the relation between Hmin and hv and the relation between Hmax and the maximum safe head can be plotted as shown in Figure 4.16. It is clear that there are three regions: (a) For (Q0 ≤ 8.36 LPM), there is no column separation and the maximum amplitude of the pressure wave occurs corresponding to any operating point within this range is less than the maximum safe head. So, this region is called Safe Region. (b) For (8.36 > Q0 > 13.8 LPM), there is no column separation, but the maximum pressure occurring within this range is higher than the maximum safe head, which may affect the used instruments negatively. So, this region is called Critical Region. (c) For (Q0 ≥ 13.8 LPM), column separation occurs and the maximum pressure occurring within this range may damage the used instruments. This region is called Unsafe Region.
77
Chapter 4
Experimental Work
Therefore, the operating point was selected within the safe region at (Q0 = 4.0 LPM) which corresponds to (H0 = 39.42547 m) as shown in Figure 4.16. Table 4.12 describes the carried out test rig specifications.
Figure 4.16. Operating point selection
888
Chapter 4
Experimental Work
Table 4.12. Water hammer test rig specifications Internal Wall Pipe Diameter, Thickness, Material D (mm) e (mm) fusiolen PP-R
4.5
21.2
5.4
Length, L (m)
Poisson Q0 Ratio, (LPM) ν
20.0
0.45
4.0
H0 (m)
Wave Speed, a (m/s) From To
39.42547 498.3285 550.0277
Water Hammer Experimental Procedure The aim of this experiment is to study the effect of viscoelasticity on
pressure head fluctuations during transient events. The test rig shown in Figure 4.1 is used for this purpose. The experiment is carried out in three consecutive steps: (1) Test preparation, (2) Run under steady-state condition, and (3) Run under unsteady-state condition. These steps are explained in the following sections.
4.5.1 Test Preparation In this step, the bypass line is adjusted to control the operating point during the experiment. First of all, the solenoid valve is fully closed and the auxiliary pump is de-energized. Then, the main pump is started and the control valve (C.V.2) is adjusted so that the H0 pressure transducer reading reaches a value of (3.8676 V) which corresponds a head of (39.42547 m). Now, the bypass line is ready for use and the main pump is de-energized. 888
Chapter 4
Experimental Work
4.5.2 Run under Steady-State Condition In this step, the solenoid valve is energized to make it open under a small differential pressure while the control valve (C.V.3) is fully opened. Then, the auxiliary pump is started and the control valve (C.V.1) is partially opened so that the pressure gauge (H1) reading exceeds (4.4 bar) which correspond a pressure head of (45 m), i.e. the maximum allowable head of the main pump. Now, the main pump is started and the flow rate is adjusted using the control valve (C.V.3) open so that the H0 pressure transducer reading reaches a value of (3.8676 V) which corresponds a head of (39.42547 m) and flow rate of (4.0 LPM).
4.5.3 Run under Unsteady-State Condition In this step, the transient event is generated. The solenoid valve is deenergized and closes rapidly, within 70 ms, and the auxiliary pump is deenergized also at the same moment. So, the bypass line is opened and H0 pressure transducer reading is fixed. The water hammer pressure sensor output signal is recorded in a computer file via the data logger, which is adjusted to record a sample each 3.0 ms (with a sampling rate of 333.3 sample/s). These results and their analysis are investigated in details in Chapter 5.
888
Chapter 4
4.6
Experimental Work
Creep Experiment Creep is the tendency of a solid material to slowly move or deform
permanently under constant stresses over a period of time at a certain temperature. It is important to describe the creep compliance function, J(t), represented by the generalized Kelvin–Voigt model in order to apply the viscoelastic transient code developed in Chapter three, as it requires this description as input data. This function, which characterizes the viscoelastic behavior of the pipe material, can be determined experimentally using a mechanical test with a sample of the used pipe. In this test, the strain response in a material subjected to uniaxial stress at a constant temperature is measured as a function of time. So, this experiment requires applying a constant stress and analyzing the strain response of the system.
4.6.1 Experimental Setup Creep experiment is performed at the Strength of Materials Laboratory, Faculty of Engineering, Mansoura University, Mansoura, Egypt. A Tensile Testing Equipment, shown in Figure 4.17, is used to apply a constant tensile stress on a sample of the PP-R pipe used in the water hammer experiment with a length of 25.15 cm. This experiment is carried out at room temperature (25C). A schematic diagram of this testing equipment is shown in Figure 4.18.
888
Chapter 4
Experimental Work
1
1 6 7 8 3
4
5 2 1- Frame 2- Base 3- Tensile grips 4- Sample
Figure 4.17. Tensile Testing Equipment
5- Power screw 6- Ram 7- Ram cylinder 8- Load indicator
Figure 4.18. Tensile Testing Equipment schematic diagram
In this equipment the sample (4) is clamped by tensile grips (3). The distance between these grips can be adjusted by the power screw (5). The load is generated due to a hydraulic force resulting from pushing the upper tensile grip up with the ram (6). This load can be measured by the indicator (8). Strain is measured by measuring the elongation using a digital caliper with the specifications shown in Table 4.13, while time is measured by a stopwatch.
888
Chapter 4
Experimental Work
Table 4.13. Digital caliper specifications
Brand
LUTRON
Model
Input
Output Signal
Measuring Range Resolution Accuracy (mm) (mm) Min. Max.
DC-515 1.5 V DC 0.19" LCD, 6 digits display 0.0 150.0
0.01
0.03
4.6.2 Experimental Procedure Firstly, the sample is clamped by the tensile grips (3) and a constant load of 7.5 tonf, which corresponds to a load of 73.575 kN, is applied for fifteen minutes. This load is the maximum allowable load to be applied to the sample without damaging it. Higher loads were applied to other samples and they reached the break point within in less than one minute. The stress is determined from the following relation:
Load
4
d
2 o
di
2
(4.10)
where do and di are the outer and inner diameters of the sample, respectively. The strain, ε, is calculated from:
L
(4.11)
L0 888
Chapter 4
Experimental Work
where δL and L0 are the elongation and original length of the sample, respectively. Elongation is measured each five minutes by the digital caliper. After that, the sample is replaced with another sample which would be subjected to a different constant load. Creep function is estimated as a function of time from the relation: J (t )
(t )
(4.12)
The creep function is used in the Kelvin–Voigt model form to specify the input parameters of the viscoelastic transient code developed before. The form of the model for NKV elements can be written as follows: t J (t ) J 0 J k 1 e k k 1 N KV
(4.13)
Specifying the values of (J0, Jk and τk) requires curve-fitting which can be carried out using the MATLAB function "lsqcurvefit" in which the objective function is a vector of the unknown values of (J0, Jk and τk). This function solves nonlinear curve-fitting (data-fitting) problems in least-squares sense by applying a large scale algorithm that uses linear algebra, which does not need to store, nor operate on, full matrices. This can be done internally by storing sparse matrices, and by using sparse linear algebra for computations whenever possible.
888
Chapter 4
Experimental Work
4.6.3 Creep Experimental Results In this experiment, the elongation of a sample is measured against time under a fixed stress of 15.68 MPa. The relation between elongation and time is shown in Figure 4.19(a). The relation between strain and time is obtained using Equation (4.11) for a length of the sample L0 = 25.15 cm, and shown in Figure 4.19(b).
The
creep
compliance
function
is
determined
from
Equation (4.12) and its relation with time is illustrated in Figure 4.19(c). The values of J0, Jk and τk in Equation (4.13) are specified using the "lsqcurvefit" MATLAB function which solves nonlinear curve-fitting (data-fitting) problems in least-squares sense. These values are shown in Table 4.14.
Table 4.14. Kelvin–Voigt model coefficients Loading Number Time of (minutes) Elements 15.0
7
Retardation Times τk (minutes) and Creep Coefficients Jk (Pa-1) k τk Jk
1 2 3 4 5 6 7 0.1 0.5 1.5 3.0 6.0 10.0 15.0 3.36e-14 2.59e-09 6.02e-10 2.22e-14 2.43e-14 7.36e-14 9.17e-09
The creep compliance function; Equation (4.13); is determined using the obtained values of J0, Jk and τk and compared with the experimental creep curve, as shown in Figure 4.19(d). It is evident that they fit well with the experimental results. Coefficients given in Table 4.14 are used as input data to the viscoelastic transient code.
887
Chapter 4
Experimental Work
(a)
(b)
(c)
(d)
Figure 4.19. Creep test experimental results (a) elongation vs. time (b) strain vs. time (c) creep vs. time (d) calculated creep vs. time 887
Chapter Five Results and Discussion
Chapter 5
Results and Discussion
CHAPTER 5 RESULTS AND DISCUSSION
5.1
Introduction This chapter aims to present the experimental data obtained from the
water hammer test facility, which carried out and explained previously in details in Chapter four. In addition, these results are compared against the viscoelastic transient code results. Finally, the viscoelastic code is analyzed through investigating four parameters of the model; material, location, time step (Courant number) and wave speed. The effects of these parameters on the viscoelastic model results are studied.
5.2
Experimental Results
5.2.1 Water Hammer Experimental Results In the water hammer experiment, the pressure heads are measured against time at locations 1, 2 and 3 which correspond to distances from the upstream end of 1 m, 10 m and 19 m, respectively. These measurements were received from the water hammer pressure transducer. Figure 5.1 shows the output raw data recorded by the data logger at the three locations. It shows that 109
Chapter 5
Results and Discussion
the pressure transducer output signals include noise. This noise is caused due to processing a sinusoidal analog input with a high sampling rate. This relationship between sampling rate and noise was observed by Pérez-Alcázar and Santos (2002) and Kester (1996). They remarked that during the evaluation of analog-to-digital converters, discrete frequencies can appear in the harmonics of the frequency of the input signal. Pérez-Alcázar and Santos (2002) found that the harmonics disappear gradually or their power spreads in the whole band defined by decreasing the sampling rate. That is why a lower sampling rate of 333.3 sample/s is used instead of 1000 sample/s. The selected sampling rate allows capturing about 25 samples per peak (as every peak lasts, theoretically, for "2L/a" seconds). Decreasing the sampling rate a lot more is not recommended as it may affect the output resolution badly.
110
Chapter 5
Results and Discussion
x
L
x/L= 0.05 Location # 1
0.5
0.95
2
3
Figure 5.1. Water hammer pressure transducer output data at three locations 111
Chapter 5
Results and Discussion
The reduction of the amount of noise reported by the transducer is achieved by a smoothing filter. Smoothing is a statistical process in which an approximate function is created and used to eliminate noise and short-term volatility data from a data set, allowing real trends and allowing important patterns to stand out. In smoothing, the data points of a signal are modified so that the individual points caused by noise are reduced, and points that are lower than the adjacent points are increased leading to a smoothed signal. Many different algorithms are used in smoothing. For processing analog-to-digital converted signals, it is recommended to use a low-pass filter, which passes low-frequency signals and attenuates signals with frequencies higher than the cutoff frequency. One of the most used low-pass filters is known as Savitzky-Golay Smoothing Filter, which is an algorithm presented by Savitzky and Golay (1964). The algorithm is discussed and explained by Madden (1978), Orfanidis (1999), Persson and Strang (2003), and Schafer (2011), and found to be well-adapted for data smoothing [William et al. (1992)]. Savitzky-Golay smoothing filter technique is based on replacing each sample value, y(x), with a k-order polynomial, p(x): k
p( x) a0 a1 x a 2 x 2 ... ak x k a j x j
(4.14)
j 0
The coefficients of the polynomial can be obtained by fitting an array of nequally-spaced data points (n = 2m + 1, with m a positive integer from 1 to 12) is based on the least-squares fitting technique. These n-points, known as 112
Chapter 5
Results and Discussion
approximation interval, are normally a subgroup of a larger group of points. Assuming that the approximation interval is centered at (x=0), then [p(0)=a0] which means that the coefficient a0 in Equation (4.14) gives the smoothed value of the data point at the middle of the approximation interval while the other coefficients al, a2, ... and ak correspond to the smoothed values of the 1st, 2nd, ..., and kth derivatives at the midpoint of the array, divided by 1!, 2!, ..., and k!, respectively. The output at the next sample is obtained by shifting the analysis approximation interval to the right by one sample, redefining the origin to be the position of the middle sample of the new block of (2m + 1) samples, and repeating the polynomial fitting and evaluation at the central location. This can be repeated at each sample of the input signal, each time producing a new polynomial, and a new value of the output sequence. Savitzky-Golay smoothing filter is applied using TableCurve 2D, Version 5.01, developed by SYSTAT© Software Inc. The pressure heads at locations 1, 2 and 3 are shown after smoothing and calibration in Figure 5.2.
113
Chapter 5
Results and Discussion
x
L
x/L= 0.05 Location # 1
0.5
0.95
2
3
Figure 5.2. Water hammer pressure heads at three locations
114
Chapter 5
Results and Discussion
5.2.2 Comparison with Code Results Herein, the viscoelastic transient code was run using the input data given in Tables 4.12 and 4.14. However, Figure 5.3 in this run, the wave speed was assumed to be 510 m/s for a reason which is explained in section 4.4.3 The code results are compared with the experimental results at locations 1, 2 and 3, as shown in Figure 5.3. It shows that there is a slight difference between the viscoelastic model results and experimental data, as the average amplitude and frequency of the numerical solution are overestimated by 100.3% and 100.2% of their values for the experimental results, respectively. This is because of neglecting FSI effect in the code while the experimental test rig pipe is not fully fixed, as it fixed diagonally at 2.5 m high with unequally spaced fixations, due to some practical considerations. From Figure 5.3, it can be observed that there is a slight discrepancy between the viscoelastic model and experimental results which increases with time. This discrepancy arises because the viscoelastic effect becomes more dominant with respect to unsteady friction, as time progresses. Figure 5.3 also shows that there is a large variation between the elastic model results and either the viscoelastic results or experimental data. This leads to the fact that is when the water hammer phenomenon in a viscoelastic pipe is studied, its behavior should be modeled using a viscoelastic model to get a more realistic prediction of pressure head fluctuations during the transient event. 115
Chapter 5
Results and Discussion
x
L
x/L= 0.05 Location # 1
0.5
0.95
2
3
Figure 5.3. Comparison between code results and experimental data at three locations
116
Chapter 5
5.3
Results and Discussion
Further Investigations of Code Parameters
5.3.1 Material Effect 5.3.1.1 Creep Curve Comparison A comparison between creep functions of both HDPE, Covas et al. (2005), and PP-R is shown in Figure 5.4. This comparison shows that the slope of PP-R creep curve is higher than the slope of HDPE creep curve. This can be analyzed as the PP-R has more viscoelastic properties than the HDPE.
Figure 5.4. Comparison between PP-R and HDPE creep functions
117
Chapter 5
Results and Discussion
5.3.1.2 Damping Rate Comparison In this section, the damping rate is indicted by the ratio of pressure head increase (ΔH) above the steady state head (H0) at each peak to the maximum pressure head increase occurred (ΔHmax). Figure 5.5 shows a comparison between average damping rates of both PP-R and HDPE during a time interval of 0.5 second, based on three-location data for each case. Such comparison shows that the damping rate of PP-R is much higher than it for HDPE. This means that PP-R behaves in a more viscoelasticity than HDPE, which confirms the previous analysis.
Figure 5.5. Comparison between average (ΔH/ΔHmax %) for PP-R and HDPE
118
Chapter 5
Results and Discussion
5.3.2 Location Effect 5.3.2.1 Effect on Damping Rate The damping rate indicated by (ΔH/ΔHmax %) is compared at three different locations for both Covas et al. (2004) and present cases. Figure 5.6(a) shows this comparison for Covas et al. (2004) case, while Figure 5.6(b) shows it for the present case. Comparison indicates that the damping rate increases near the upstream end of the pipe.
(a)
(b)
Figure 5.6. Comparison between (ΔH/ΔHmax%) (a) for Covas et al. (2004) case (b) for present case
119
Chapter 5
Results and Discussion
5.3.2.2 Effect on Maximum Amplitude The maximum amplitude (ΔHmax) is affected by location. Figure 5.7(a) shows this effect for Covas et al. (2004) case, while Figure 5.7(b) shows it for the present case. For both cases, it is clear that the maximum amplitude increases with getting closer to the downstream end of the pipe. The difference in curves concavity may be due to the material nature.
(a)
(b)
Figure 5.7. Location effect on maximum amplitude (ΔHmax) value (a) for Covas et al. (2004) case (b) for present case
5.3.2.3 Effect on Fluctuating Head Frequency Location of measuring point affects the fluctuating head frequency. Figure 5.8(a) shows this effect for Covas et al. (2004) case, while Figure 5.8(b) shows it for the present case. For both cases, it is clear that 120
Chapter 5
Results and Discussion
frequency decreases near the pipe downstream end. At a certain location, Figure 5.8(a) illustrates that frequency decreases with time for HDPE, while Figure 5.8(b) illustrates that frequency increases with time for PP-R. From Figures 5.6 and 5.8, it is clear that near the upstream end, both of damping rate and frequency increase. This means that the increasing damping rate is associated with increasing wave frequency. According to this conclusion, decreasing frequency with time for HDPE, in Figure 5.8(a), can be attributed to its low damping rate, as shown in Figure 5.5, while increasing frequency with time for PP-R, in Figure 5.8(b), may be attributed to its high damping rate.
(a)
(b)
Figure 5.8. Comparison between frequency of fluctuating heads (a) for Covas et al. (2004) case (b) for present case
121
Chapter 5
Results and Discussion
5.3.3 Time Step Effect In this section, the time step effect on the viscoelastic model is investigated by varying the time step value. The stability criterion for explicit time stepping developed by Courant et al. (1928) must be satisfied. This criterion requires that the Courant number, C, which is defined by Equation (5.1), must not exceed the unity.
C
t max a V s
≤1
(5.1)
Therefore, the time step effect as well as the Courant number effect on the viscoelastic model can be investigated by varying the time step value within the following limit: t
s max a V
(5.2)
Figure 5.9 shows the time step and Courant number effects on numerical results. Figure 5.9(a) shows these effects for Covas et al. (2004) case at a wave speed of 385 m/s at location 5, and Figure 5.9(b) shows them for the present case for a wave speed of 510 m/s at location 3. It can be observed that changing time step and Courant number cause amplitude and frequency distortions. The amplitude distortions are illustrated in Figure 5.10, whereas the frequency distortions are shown in Figure 5.11. 122
Chapter 5
Results and Discussion
(a)
(b) Figure 5.9. Time-step and Courant number effects on head fluctuations (a) for Covas et al. (2004) case (Location 5, a=385 m/s) (b) for present case (Location 3, a=510 m/s) 123
Chapter 5
Results and Discussion
A comparison between the average head of experimental data and those of the numerical results at different time steps is illustrated in Figure 5.10(a) and Figure 5.10(b) for Covas et al. (2004) and present cases, respectively. The average head ratio is the ratio of the numerical results average head to the experimental data average head obtained by Covas et al. (2004) at different time steps. The difference between curves trends may be related to the variation of the creep functions of HDPE and PP-R which affects the pipe wall response to fluid transients.
(b) (a) Figure 5.10. Time-step and Courant number effects on average head value (a) for Covas et al. (2004) case (Location 5, a=385 m/s) (b) for present case (Location 3, a=510 m/s)
A comparison between the frequency of experimental data and those of the numerical results at different time steps is shown in Figure 5.11(a) and Figure 5.11(b) for Covas et al. (2004) and present cases, respectively. The 124
Chapter 5
Results and Discussion
average frequency ratio is the ratio of the numerical results average frequency to the experimental data average frequency obtained by Covas et al. (2004) at different time steps.
(b) (a) Figure 5.11. Time-step and Courant number effects on average frequency of fluctuating head (a) for Covas et al. (2004) case (Location 5, a=385 m/s) (b) for present case (Location 3, a=510 m/s)
From Figure 5.10, it is clear that the higher time step, and Courant number, gives better match. The best amplitude match is obtained at Courant number of unity and 0.8562 for Covas et al. (2004) and present cases, respectively. While Figure 5.11 illustrates that the best frequency match is achieved at Courant number of 0.983 (≈ 1.0) and unity for Covas et al. (2004) and present cases, respectively. However, frequency distortion is more important to be taken into account to avoid occurring phase shift between numerical and experimental
125
Chapter 5
Results and Discussion
results. Therefore, in order to achieve the optimum amplitude and frequency match, it is recommended to choose Courant number about unity to simulate the water hammer in a viscoelastic pipe. 5.3.4 Wave Speed Effect The type of pipe support affects the wave speed value as was mentioned before. Equation (4.4) expresses wave speed in its general form, for three different support situations. For Covas et al. (2004), despite the fixation type is known as case (b), but the value of the modulus of elasticity of HDPE varies from 0.8 to 1.43 GPa. Therefore, the wave speed value can vary from 345 m/s to 452 m/s. While for the present case, although the modulus of elasticity of PP-R is known at 1100 MPa, but the fixation type is unknown. Therefore, the wave speed can vary between 498.33 m/s and 550.03 m/s. In this section, the effect of varying the wave speed value within these ranges is studied. For all cases, the time step value was set so that the Courant number is unity. Figure 5.12 shows the wave speed effect on the viscoelastic model results.
126
Chapter 5
Results and Discussion
(a)
(b) Figure 5.12. Wave speed effect on head fluctuations (a) for Covas et al. (2004) case (Location 5, a=385 m/s) (b) for present case (Location 3, a=510 m/s) 127
Chapter 5
Results and Discussion
It can be observed that changing wave speed has no significant effect on pressure-head wave amplitude, while it affects the wave frequency. For a certain wave speed, it can be observed that the wave frequency changes with time. This effect can be observed from Figure 5.13. It shows the change in wave frequency as a function of time and the comparison against experimental results frequency. From Figure 5.13, it is clear that pressure wave frequency increases with wave speed. Figure 5.13(a) shows that for a certain wave speed, frequency decreases with time in Covas et al. (2004) case, while frequency increases with time in the present case, as shown in Figure 5.13(b). It can also be observed that the frequencies of the wave speeds of 385 m/s and 510 m/s are too close to the experimental data frequencies in Covas et al. (2004) and present cases, respectively. The difference between trends of curves in Figures 5.13(a) and 5.13(b) has been explained previously in section 5.3.2.3. Figure 5.14 represents the relation between average pressure-head wave frequency as a function of wave speed. It shows that the higher wave speed, the higher average pressure-head wave frequency. This relation tends to be linear. It can be observed that the best wave speed in which the best match between numerical and experimental results is about 388.7 m/s and 508 m/s for Covas et al. (2004) and present cases, respectively.
128
Chapter 5
Results and Discussion
(a)
(b)
Figure 5.13. Wave speed effect on frequency (a) for Covas et al. (2004) case (Location 5, a=385 m/s) (b) for present case (Location 3, a=510 m/s)
(a)
(b)
Figure 5.14. Wave speed effect on average wave frequency (a) for Covas et al. (2004) case (Location 5, a=385 m/s) (b) for present case (Location 3, a=510 m/s) 129
Chapter 5
Results and Discussion
130
Chapter Six Conclusions and Future Work
Chapter 6
Conclusions and Future Work
CHAPTER 6 CONCLUSIONS AND FUTURE WORK
6.1
Introduction In the present work, the water hammer phenomenon in viscoelastic pipes
is studied. In order to get information about a network during transient event numerically, a steady-state analysis for the network is performed using the gradient method. A FORTRAN code to perform this analysis has been developed. The code results were compared with the solution of a reference case study and it was found to be valid. The steady-state analysis solution is used as input data to the water hammer simulation model applying the viscoelastic model. The viscoelastic pipe wall behavior was simulated using the Kelvin-Voigt model. Governing equations for water hammer in a viscoelastic pipe were solved using the method of characteristics (MOC) neglecting the fluid-structure interaction (FSI) and unsteady friction (UF). A FORTRAN code to apply this model has been developed and its results were compared with experimental data obtained by Covas et al. (2004). Comparison demonstrated that the developed code was valid, as the average amplitude and frequency of the numerical solution were
131
Chapter 6
Conclusions and Future Work
underestimated and found to be 96% and 95.1% of the corresponding values of the experimental results, respectively. A simple tank-pipe-valve test rig was designed and carried out using a PP-R pipe. The viscoelastic behavior is characterized by a creep function, which was determined by a creep experimental test. The experimental data obtained from the test rig is compared with the numerical results of the water hammer developed code. The comparison indicated that the average amplitude and frequency of the numerical solution were overestimated and found to be 100.38% and 100.2% of the corresponding values of the experimental results, respectively. The effects of material, location, time-step and wave speed were investigated.
6.2
Conclusions From the previous results, one can conclude that:
1.
The fluctuating pressure waves damps faster in viscoelastic pipes than in elastic pipes as a result of Viscoelasticity.
2.
Viscoelastic Model should be applied for viscoelastic pipes, as it predicts pressure head fluctuations better than the Elastic Model.
3.
Materials with high creep function slope show more viscoelastic properties, i.e. more damped pressure waves.
131
Chapter 6
4.
Conclusions and Future Work
The maximum rise in pressure head, ΔHmax, increases near the downstream end of the pipe.
5.
Damping rate increases near the upstream end of the pipe.
6.
Fluctuating pressure waves frequency increases near the upstream end of the pipe.
7.
The unsteady friction effect arises with time progress. Therefore, a slight difference between the numerical and experimental results with time progress can be observed.
8.
Assuming thin pipe wall with a relatively high thickness-diameter ratio (e/D = 0.25) may affect results and cause a slight discrepancy between numerical and experimental results.
9.
For applying the developed code, there is an inverse relationship between Courant Number and wave frequency. In addition, choosing the time step size (Δt) so that the Courant Number equals unity gives best results matching.
10. There is a direct relationship between pressure wave frequency increases with wave speed increase. The wave speed value is found to have a great effect on the numerical results, so it has to be calculated precisely, or can be varied among a calculated range to get the best solution.
133
Chapter 6
6.3
Conclusions and Future Work
Future Work Recommendations The present work dealt with the viscoelastic model neglecting the fluid-
structure interaction (FSI) and the unsteady friction (UF) effects and assuming thin-walled, linearly viscoelastic pipes. Some recommendations on the possible direction for future research can be listed as follows: 1. Studying the effect of applying FSI on the viscoelastic model. 2. Applying UF model and studying its effect on the viscoelastic model results. 3. Investigating the material effect by running the experimental test with different pipe materials. Comparison between materials under the same conditions can be carried out. 4. Study of water hammer in a pipe network, including combination between elastic and viscoelastic pipes, or different viscoelastic pipes. 5. Study of non-Newtonian fluid flow in highly viscoelastic pipes using a reciprocating pump, as a simulation of blood in the arteries.
131
References
References
References
[1]
Achouyab, E. and Bahrar, B. (2013). Modeling of Transient Flow in Plastic Pipes. Contemporary Engineering Sciences, Vol. 6, pp. 35–47.
[2]
Adamkowski, A., Henclik, S., and Lewandowski, M. (2010). Experimental and Numerical Results of the Influence of Dynamic Poisson Effect on Transient Pipe Flow Parameters. 25th IAHR Symposium on Hydraulic Machinery and Systems, Timişoara, Romania, 2010. Published in IOP Conference Series: Earth and Environmental Science, Vol. 12, No. 1.
[3]
Adams, R.W. (1961). Distribution Analysis by Electronic Computer. Journal of the Institute of Water Engineers, Vol. 15, pp. 415–428.
[4]
Ahmed, I. and Lansey, K. (1999). Analysis of Unsteady Flow in Networks Using a Gradient Algorithm Based Method. ASCE Specialty Conference on Water Resources, Tempe, AZ, USA.
[5]
Aklonis, J., MacKnight, W., and Shen, M. (1972). Introduction to Polymer Viscoelasticity. New York: Wiley-Interscience, John Wiley & Sons.
[6]
Alireza, R., Ahmad, N., and Mehrdad, R. (2009). Unsteady Velocity Profiles in Laminar and Turbulent Water Hammer Flows. Journal of 535
References
Fluids Engineering, Transactions of the ASME, Vol. 131, pp. 121202.1– 121202.8. [7]
Barlow, J.F., White, C., Markland, E., and Cross, H. (1969). Computer Analysis of Pipe Networks. Proceedings of Institution of Civil Engineers (ICE), Vol. 43, pp. 249–259.
[8]
Barten, W., Jasiulevicius, A., Manera, A., Macian-Juan, R., and Zerkak, O. (2008). Analysis of the Capability of System Codes to Model Cavitation Water Hammers: Simulation of UMSICHT Water Hammer Experiments with TRACE and RELAP5. Nuclear Engineering and Design, Vol. 238, pp. 1129–1145.
[9]
Bergant, A. and Simpson, A.R. (1995). Water Hammer and Column Separation Measurements in an Experimental Apparatus. Report n. R128, Dept. of Civil and Environmental Engineering, University of Adelaide, Adelaide, Australia.
[10]
Bergant, A., Hou, Q., Keramat, A., and Tijsseling, A.S. (2011). Experimental and Numerical Analysis of Water Hammer in a Large-Scale PVC Pipeline Apparatus. Report 11-51, Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
[11]
Bergant, A., Hou, Q., Keramat, A., and Tijsseling, A.S. (2013). Waterhammer Tests in a Long PVC Pipeline with Short Steel End 536
References
Sections. Journal of Hydraulic Structures, Vol. 1, pp. 23–34. [12]
Bergant, A., Simpson, A., and Vítkovský, J. (2001). Developments in Unsteady Pipe Flow Friction Modelling. Journal of Hydraulic Research, Vol. 39, pp. 249–257.
[13]
Bergant, A., Tijsseling, A.S., Vitkovsky, J.P., Covas, D., Simpson, A.R., and Lambert, M.F. (2008a). Parameters Affecting Water-Hammer Wave Attenuation, Shape and Timing—Part 1: Mathematical Tools. Journal of Hydraulic Research, Vol. 46, pp. 373–381.
[14]
Bergant, A., Tijsseling, A.S., Vitkovsky, J.P., Covas, D., Simpson, A.R., and Lambert, M.F. (2008b). Parameters Affecting Water-Hammer Wave Attenuation, Shape and Timing—Part 2: Case Studies. Journal of Hydraulic Research, Vol. 46, pp. 382–391.
[15]
Bhave, P.R. and Gupta, R. (2006). Analysis of Water Distribution Networks. Oxford, UK: Alpha Science International Ltd.
[16]
Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960). Transport Phenomena. New York, USA: John Wiley & Sons.
[17]
Brunone, B., Karney, B.W., Mecarelli, M., and Ferrante, M. (2000). Velocity Profiles and Unsteady Pipe Friction in Transient Flow. Journal of Water Resources Planning and Management, Vol. 126, pp. 236–244.
[18]
Bryan, W., Kevin, E., and Paul, F. (2006). Comprehensive Water 537
References
Distribution Systems Analysis Handbook for Engineers and Planners. 2nd edition. Pasadena, CA: MWH Soft, Inc. [19]
Chaudhry,
M.H.
(1979).
Applied
Hydraulic
Transients.
Litton
Educational Publishing, Van Nostrand Reinhold Company. [20]
Cornish, R. (1939). The Analysis of Flow in Networks of Pipes. Journal of the Institution of Civil Engineers, Vol. 13, pp. 147–154.
[21]
Courant, R., Friedrichs, K., and Lewy, H. (1928). Über die Partiellen Differenzengleichungen der Mathematischen Physik. Mathematische Annalen (in German), Vol. 100, pp. 32–74.
[22]
Covas, D., Stoianov, I., Mano, J., Ramos, H., Graham, N., and Maksimovic, C. (2004). The Dynamic Effect of Pipe-Wall Viscoelasticity in Hydraulic Transients. Part I— Experimental Analysis and Creep Characterization. Journal of Hydraulic Research, Vol. 42, pp. 516–530.
[23]
Covas, D., Stoianov, I., Mano, J., Ramos, H., Graham, N., and Maksimovic, C. (2005). The Dynamic Effect of Pipe-Wall Viscoelasticity in Hydraulic Transients. Part II— Model Development, Calibration and Verification. Journal of Hydraulic Research, Vol. 43, pp. 56–70.
[24]
Cross, H. (1936). Analysis of Flow in Networks of Conduits or Conductors. Bulletin No. 286, Engineering Experiment Station, University of Illinois, Urbana, IL, USA.
538
References
[25]
Dillingham, J.H. (1967). Computer Analysis of Water Distribution Systems—Part II. Water and Sewage Works, Vol. 114, pp. 43–54.
[26]
Djebedjian, B., Mondy, A., Mohamed, M.S., and Rayan, M.A. (2005). Network Optimization for Steady Flow and Water Hammer Using Genetic Algorithms. Proceedings of the 9th International Water Technology Conference, IWTC, Sharm El-Sheikh, Egypt, pp. 1101–1115.
[27]
Duan, H.F. (2009). Relative Importance of Unsteady Friction and Viscoelasticity in Pipe Fluid Transients. 33rd IAHR Congress: Water Engineering for a Sustainable Environment, Vancouver, British Columbia, Canada.
[28]
ECOSSE (The Edinburgh Collection of Open Software for Simulation and Education), Water Vapour Pressure Calculator. http://eweb.chemeng.ed.ac.uk/jack/newWork/Chemeng/Chemeng/water.ht ml
[29]
Elhay, S. and Simpson, A. (2011). Dealing with Zero Flows in Solving the Nonlinear Equations for Water Distribution Systems. Journal of Hydraulic Engineering, Vol. 137, pp. 1216–1224.
[30]
Epp, R. and Fowler, A.G. (1970). Efficient Code for Steady State Flows in Networks. Journal of the Hydraulics Division, Proceedings of the American Society of Civil Engineers, Vol. 96, pp. 43–56.
539
References
[31]
Ferry, J. (1970). Viscoelastic Properties of Polymers. 2nd edition. New York, USA: John Wiley & Sons.
[32]
FREESTUDY, Free Tutorials on Engineering and Science. http://www.freestudy.co.uk/statics/torsion/torsion2.pdf
[33]
Ghidaoui, M.S., Zhao, M., Mclnnis, D.A., and Axworthy, D.H. (2005). A Review of Water Hammer Theory and Practice. Applied Mechanics Reviews, ASME, Vol. 58, pp. 49–76.
[34]
Guinot, V. (2000). Riemann Solvers for Water Hammer Simulations by Godunov Method. International Journal for Numerical Methods in Engineering, Vol. 49, pp. 851–870.
[35]
Güney, M. (1977). Contribution à l’étude du phénomène de coup de bélier en conduite viscoélastique. Thèse présentée à l’Université de Lyon.
[36]
Güney, M. (1983). Waterhammer in Viscoelastic Pipes where CrossSection Parameters are Time Dependent. Proceedings of 4th International Conference on Pressure Surges, BHRA, Bath, United Kingdom, pp. 189– 209.
[37]
Hamam, Y. and Brameller, A. (1971). Hybrid Method for the Solution of Piping Networks. Proceedings of the Institution of Electrical Engineers (IEE), Vol. 118, Issue 11, pp. 1607–1612.
[38]
Hashemi, M.R. and Abedini, M.J. (2007). Numerical Modelling of 541
References
Water Hammer Using Differential Quadrature Method. International Conference on Numerical Analysis and Applied Mathematics, Corfu, Greece. AIP Conference Proceedings, Vol. 936, pp. 263–266. [39]
Himr, D. (2013). Numerical Simulation of Water Hammer in Low Pressurized Pipe: Comparison of SimHydraulics and Lax-Wendroff Method with Experiment. EPJ Web of Conferences, Vol. 45, pp. 1–4.
[40]
Hoag, L.N. and Weinberg, G. (1957). Pipeline Network Analyses by Electronic Digital Computer. Journal of the American Water Works Association, Vol. 49, pp. 517–524.
[41]
Holmboe, E.L. and Rouleau, W.T. (1967). The Effect of Viscous Shear on Transients in Liquid Lines. Journal of Basic Engineering, Trans. ASME, Vol. 89, pp. 174–180.
[42]
Hou, Q., Kruisbrink, A.C.H., Tijsseling, A.S., and Keramat, A. (2012). Simulating Water Hammer with Corrective Smoothed Particle Method. Report 12-14, Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
[43]
INEOS, The World for Chemicals. http://www.ineos.com/Global/Olefins%20and%20Polymers%20USA/Pro ducts/Technical%20information/Engineering%20Properties%20of%20PP. pdf
545
References
[44]
Ismaier, A. and Schlücker, E. (2009). Fluid Dynamic Interaction between Water Hammer and Centrifugal Pumps. Nuclear Engineering and Design, Vol. 239, pp. 3151–3154.
[45]
Joukowsky,
N.
(1900).
Über
den
hydraulischen
Stoss
in
Wasserleitungsröhren. Mémoires de l'Académie Impériale des Sciences de St.-Petersburg, Series 8, Vol. 9. pp. 1–71. [46]
Keramat, A. and Tijsseling, A.S. (2012). Waterhammer with Column Separation, Fluid-Structure Interaction and Unsteady Friction in a Viscoelastic Pipe. Report 12-43, Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
[47]
Keramat, A., Tijsseling, A.S., and Ahmadi, A. (2010). Investigation of Transient Cavitating Flow in Viscoelastic Pipes. Report 10-39, Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
[48]
Keramat, A., Tijsseling, A.S., Hou, Q., and Ahmadi, A. (2012). Fluid– Structure Interaction with Pipe-Wall Viscoelasticity during Water Hammer. Journal of Fluids and Structures, Vol. 28, pp. 434–455.
[49]
Kester, W. (1996). High Speed Sampling and High Speed ADC. Section 4, in High Speed Design Techniques, Analog-Devices, Inc.
541
References
[50]
Kochupillai, J., Ganesan, N., and Padmanabhan, C. (2005). A New Finite Element Formulation Based on the Velocity of Flow for Water Hammer problems. International Journal of Pressure Vessels and Piping, Vol. 82, pp. 1–14.
[51]
Kodura, A. and Weinerowska, K. (2005). The Influence of the Local Pipe Leak on the Properties of the Water Hammer. Proc. of 2nd Congress of Environmental Engineering, Lublin, Poland, Vol. 1, pp. 399–407.
[52]
Kootattep, S. and Aya, H. (1985). Appropriate Method of Distribution Network Analysis for Developing Countries. Aqua, Vol. 6, pp. 311–315.
[53]
Kwon, H.J. and Lee, J. (2008). Computer and Experimental Models of Transient Flow in a Pipe Involving Backflow Preventers. Journal of Hydraulic Engineering, Vol. 134, pp. 426–434.
[54]
Lai, J. and Bakker, A. (1995). Analysis of the Non-Linear Creep of High-Density Polyethylene. Polymer, Vol. 36, pp. 93–99.
[55]
Lam, C.F. and Wolla, M.L. (1972a). Computer Analysis of Water Distribution Systems: Part I— Formulation of Equations. Journal of the Hydraulics Division, ASCE, Vol. 98, pp. 335–344.
[56]
Lam, C.F. and Wolla, M.L. (1972b). Computer Analysis of Water Distribution Systems: Part II— Numerical Solution. Journal of the Hydraulics Division, ASCE, Vol. 98, pp. 447–460.
543
References
[57]
Lambert, M.F., Vítkovský, J.P., Simpson, A.R., and Bergant, A. (2001). A Boundary Layer Growth Model for One-Dimensional Turbulent Unsteady Pipe Friction. 14th Australian Fluid Mechanics Conference, Adelaide University, Adelaide, Australia.
[58]
Landry, C., Nicolet, C., Bergant, A., Müller, A., and Avellan, F. (2012). Modeling of Unsteady Friction and Viscoelastic Damping in Piping Systems. Proceedings of the 26th IAHR Symposium on Hydraulic Machinery
and
Systems.
IOP
Conference
Series:
Earth
and
Environmental Science, Vol. 15, pp. 1–9. [59]
Larock, B.E., Jeppson, R.W., and Watters, G.Z. (2000). Hydraulics of Pipeline Systems. New York: CRC Press LLC.
[60]
Li, J., Wu, P., and Jiandong, Y. (2010). CFD Numerical Simulation of Water Hammer in Pipeline based on the Navier-Stokes Equation. Fifth European Conference on Computational Fluid Dynamics, Lisbon, Portugal.
[61]
Madden, H., (1978). Comments on the Savitzky-Golay Convolution Method for Least-Squares-Fit Smoothing and Differentiation of Digital Data. Analytical Chemistry, Vol. 50, pp. 1383–1386.
[62]
Majumdar, A., Steadman, T., and Moore, R. (2007). Generalized Fluid System Simulation Program (GFSSP) Version 5.0. Draft Report, Propulsion Department, George C. Marshall Space Flight Center. 544
References
[63]
Marcinkiewicz, J., Adamowski, A., and Lewandowski, M. (2008). Experimental Evaluation of Ability of Relap5, Drako ®, Flowmaster2TM and Program using Unsteady Wall Friction Model to Calculate Water Hammer Loadings on Pipelines. Nuclear Engineering and Design, Vol. 238, pp. 2084–2093.
[64]
Martin, D.W. and Peters, G. (1963). The Application of Newton's Method to Network Analysis by Digital Computer. Journal of the Institute of Water Engineers, Vol. 17, pp. 115–129.
[65]
McCormick, M. and Bellamy, C.J. (1968). A Computer Program for the Analysis of Networks of Pipes and Pumps. Journal of the Institute of Engineers, Vol. 38, pp. 51–58.
[66]
Mitosek, M. and Chorzelski, M. (2003). Influence of Visco-Elasticity on Pressure Wave Velocity in Polyethylene MDPE Pipe. Archives of HydroEngineering and Environmental Mechanics, Vol. 50, pp. 127–140.
[67]
Mondy, A. (2008). Optimization of Water Distribution Systems Subjected to Water Hammer using Genetic Algorithms. MSc Thesis, Faculty of Engineering, Mansoura University, Egypt.
[68]
Newbold, J.R. (2009). Comparison and Simulation of a Water Distribution Network in EPANET and a New Generic Graph Trace Analysis Based Model. MSc Thesis, Faculty of the Virginia Polytechnic Institute, State University. 545
References
[69]
Orfanidis, S. (1999). Introduction to Signal Processing. Prentice Hall.
[70]
Ormsbee, L.E. (2006). The History of Water Distribution Network Analysis: The Computer Age. 8th Annual Water Distribution Systems Analysis Symposium, Cincinnati, Ohio, USA.
[71]
Osiadacz, A.J. (1987). Simulation and Analysis of Gas Networks. Houston: Gulf Publishing Company, Book Division.
[72]
Parmakian, J. (1963). Water-Hammer Analysis. N.J.: Prentice-Hall Englewood Cliffs.
[73]
Pérez-Alcázar, P.R. and Santos, A. (2002). Relationship between Sampling Rate and Quantization Noise. Proceedings of the 14th International Conference on Digital Signal Processing, Vol. 2, pp. 807– 810.
[74]
Persson, P. and Strang, G. (2003). Smoothing by Savitzky-Golay and Legendre Filters. The IMA Volumes in Mathematics and its Applications, Vol. 134, pp. 301–316.
[75]
Pothof, I. (2008). A Turbulent Approach to Unsteady Friction. Journal of Hydraulic Research, Vol. 46, pp. 679–690.
[76]
Prado, R.A. and Larreteguy, A.E. (2002). A Transient Shear Stress Model for the Analysis of Laminar Water-Hammer Problems. Journal of Hydraulic Research, Vol. 40, pp. 45–53. 546
References
[77]
Riande, E., Díaz-Calleja, R., Prolongo, M.G., Masegosa, R., and Salom, C. (2000). Polymer Viscoelasticity: Stress and Strain in Practice. New York, USA: Marcel Dekker, Inc.
[78]
Riasi, A., Nourbakhsh, A., and Raisee, M. (2009). Unsteady Turbulent Pipe Flow due to Water Hammer using k-ω Turbulence Model. Journal of Hydraulic Research, Vol. 47, pp. 429–437.
[79]
Rich, G. (1951). Hydraulic Transients. New York, USA: McGraw-Hill.
[80]
Rohani, M. and Afshar, M. (2008). Water Hammer Simulation by Implicit Method of Characteristic. International Journal of Pressure Vessels and Piping, Vol. 85, pp. 851–859.
[81]
Rohani, M. and Afshar, M. (2010). Simulation of Transient Flow Caused by Pump Failure: Point-Implicit Method of Characteristics. Annals of Nuclear Energy, Vol. 37, pp. 1742–1750.
[82]
Saikia, M.D. and Sarma, A.K. (2006). Numerical Modeling of Water Hammer with Variable Friction Factor. Journal of Engineering and Applied Sciences, Vol. 1, pp. 35–40.
[83]
Salgado, R., Todini, E., and O’Connel, P.E. (1987). Comparison of the Gradient Method with Some Traditional Methods for the Analysis of Water Supply Distribution Networks. International Conference on Computer Applications for Water Supply and Distribution, Leicester, UK.
547
References
[84]
Šavar, M., Korbar, R., and Virag, Z. (1998). The Comparison of Numerical and Experimental Results for the Simple Water Hammer Problem. Proceedings of the 5th International Design Conference, Dubrovnik, Croatia, pp. 689–694.
[85]
Savitzky, A. and Golay, M. (1964). Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry, Vol. 36, pp. 1627–1639.
[86]
Schafer, R. (2011). What is a Savitky-Golay Filter? Signal Processing Magazine, IEEE, Vol. 28, pp. 111–117.
[87]
Shamir, U. and Howard, C.D.D. (1968). Water Distribution Systems Analysis. Journal of the Hydraulics Division, ASCE, Vol. 94, pp. 219– 234.
[88]
Simpson, A. and Elhay, S. (2011). Jacobian Matrix for Solving Water Distribution System Equations with the Darcy-Weisbach Head-Loss Model. Journal of Hydraulic Engineering, Vol. 137, pp. 696–700.
[89]
Soares, A., Covas, D., and Reis, L. (2008). Analysis of PVC Pipe-Wall Viscoelasticity during Water Hammer. Journal of Hydraulic Engineering, ASCE, Vol. 134, pp. 1389-1394.
[90]
SolidWorks Premium™ 2012 edition. http://www.solidworks.com/sw/3d-cad-design-software.htm
548
References
[91]
Streeter, V.L. (1969). Water Hammer Analysis. Journal of the Hydraulics Division, ASCE, Vol. 95, pp. 1959-1972.
[92]
Streeter, V.L. and Wylie, E.B. (1967). Hydraulic Transients. New York, USA: McGraw-Hill.
[93]
Sun, J.G. and Wang, X.Q. (1995). Pressure Transient in Liquid Lines. ASME/JSME Pressure Vessels and Piping Conference, Honolulu, Hawaii, USA.
[94]
Szymkiewicz, R. and Mitosek, M. (2005). Analysis of Unsteady Pipe Flow using the Modified Finite Element Method. Communications in Numerical Methods in Engineering, Vol. 21, pp. 183–199.
[95]
Taïeb, L.H. and Taïeb, E.H. (2006). Transient Laminar Flow in Pipes and Water Hammer, Instantaneous Axial Velocity Profiles. La Houille Blanche, Vol. 2, pp. 100–105.
[96]
The Engineering ToolBox, Water Density and Specific Weight. http://www.engineeringtoolbox.com/water-density-specific-weightd_595.html
[97]
The Engineering ToolBox, Bulk Modulus and Fluid Elasticity. http://www.engineeringtoolbox.com/bulk-modulus-elasticity-d_585.html
[98]
Todini, E. and Pilati, S. (1987). A Gradient Algorithm for the Analysis of Pipe Networks. International Conference on Computer Applications for 549
References
Water Supply and Distribution, Leicester, UK. [99]
University of Washington, UW Courses Web Server. http://courses.washington.edu/me354a/Thin%20Walled%20Pressure%20v essels.pdf
[100] Vardy, A.E. and Hwang, K.L. (1991). A Characteristic Model of Transient Friction in Pipes. Journal of Hydraulic Research, Vol. 29, pp. 669–685. [101] Vaseti, M.M. (2011). Simulate the Water Distribution Networks by Improved Newton-Raphson Method. International Journal of Academic Research, Vol. 3, pp. 506–510. [102] Vítkovský, J.P., Lambert, M.F., Simpson, A.R., and Liggett, J.A. (2007). Experimental Observation and Analysis of Inverse Transients for Pipeline Leak Detection. Journal of Water Resources Planning and Management, Vol. 133, pp. 519–530. [103] Voyles, C.F. and Wilke, H.R. (1962). Selection of Circuit Arrangements for Distribution Network Analysis by the Hardy Cross Method. Journal of the American Water Works Association, Vol. 54, pp. 285–290. [104] Wahba, E. (2006). Runge–Kutta Time-Stepping Schemes with TVD Central Differencing for the Water Hammer Equations. International Journal for Numerical Methods in Fluids, Vol. 52, pp. 571–590.
551
References
[105] Wahba, E. (2008). Modeling the Attenuation of Laminar Fluid Transients in Piping Systems. Applied Mathematical Modelling, Vol. 32, pp. 2863– 2871. [106] Wahba, E. (2009). Turbulence Modeling for Two-Dimensional Water Hammer Simulations in the Low Reynolds Number Range. Computers & Fluids, Vol. 38, pp. 1763–1770. [107] Waheed, A. (1992). Computer Aided Design and Analysis of Closed Loop Piping Systems. MSc Thesis, Faculty of the Graduate College, Oklahoma State University. [108] Weinerowska, K. (2006). Viscoelastic Model of Waterhammer in Single Pipeline – Problems and Questions. Archives of Hydro-Engineering and Environmental Mechanics, Vol. 53, pp. 331–351. [109] William, H., Saul, A., William, T., and Brain, P. (1992). Numerical Recipes in C: The Art of Scientific Computing. 2nd edition. Cambridge, UK: Cambridge University Press. [110] Williams, G. (1973). Enhancement of Convergence of Pipe Network Solutions. Journal of the Hydraulics Division, ASCE, Vol. 99, pp. 1057– 1067. [111] Wood, D., Lindireddy, S., Boulos, P.F., Karney, B.W., and McPherson, D.L. (2005). Numerical Methods for Modeling Transient Flow. Journal of
555
References
the American Water Works Association, Vol. 97, pp. 104–115. [112] Wood, D.J. and Rayes, A.G. (1981). Reliability of Algorithms for Pipe Network Analysis. Journal of the Hydraulics Division, ASCE, Vol. 107, pp. 1145–1161. [113] Wylie, E. and Streeter, V. (1993). Fluid Transients in Systems. N.J.: Prentice-Hall Englewood Cliffs. [114] Yaseen, A.E. (2007). Reliability-Based Optimization of Potable Water Networks using Genetic Algorithms and Monte Carlo Simulation. MSc Thesis, Faculty of Engineering, Mansoura University, Egypt.
551
Appendix (A) Gradient Method Subroutine Code
Appendix A
APPENDIX A GRADIENT METHOD SUBROUTINE CODE C
This subroutine is used to solve a network using Gradient Method C developed by Todini and Plati(1987)
C C--------------------------------------------------------------------C INPUTS: C ======= C GENERAL DATA C -----------C NP = No. of pipes (links) C NJ = No. of all nodes (including nodes connected to tanks or C reservoir) C NPUMP = No. of pumps in the system C EXPONENT = Friction resistance exponent (i.e. N = 1.852 C for Hazen-Williams) C PIPES DATA C ---------C L1 & L2 = Vectors of upstream & downstream nodes of each pipe C K = Vector of pipes' resistances due to friction C NODES DATA C ---------C QJ = Vector of all nodes' demand C H = Vector of all (HGL's) of all nodes C NOTE: If the value of (QJ) or (H) of any node entered equal C to (ZERO), the code will consider it as unknown value. So, C if there is no demand @ any node (zero is an input value) C OR no head (i.e. a tank with zero level), PLEASE ENTER THIS C ZERO VALUE EQUAL TO (1E-100) NOT (0.0) C PUMPS DATA C ---------C II = Vector of No. of pipes on which the pumps are installed C AP,BP,CP = Vectors of pumps' constants C (Hp = AP*Qp^2 + BP*Qp + CP) C--------------------------------------------------------------------C SUBROUTINE GRADIENT(NP,NJ,NPUMP,EXPONENT,L1,L2,K,QJ,H,II,AP,BP, * CP,ERR,MAX) C C IMPLICIT NONE
351
Appendix A INTEGER NP,NJ,NPUMP,errorflag DOUBLEPRECISION Eqn,Soln,XSoln,Soln0,Q,QJ,K,N,H,L1,L2,EXPONENT, * EqnINV DIMENSION Eqn(1000,1000),Soln(1000),Soln0(1000),EqnINV(1000,1000) DIMENSION Q(1000),QJ(1000),K(1000),N(1000),H(1000),XSOLN(1000), * L1(1000),L2(1000),KHC(1000),TRANS(1000) DIMENSION AP(100),BP(100),CP(100),CONST1(1000),CONST2(1000) C C C
------PRINTING THE READ DATA IN (INPUT.DAT FILE)-----*
C C C C C
C C C C C
C C C C
OPEN(10,FILE= 'Input to GRADIENT SUBROUTINE.dat', STATUS='OLD', FORM='FORMATTED', ACCESS='SEQUENTIAL') WRITE(10,*) 'GENERAL INPUTS' WRITE(10,*) '==============' WRITE(10,*) ' NP|NJ|NPUMP|EXPONENT ' WRITE(10,*) NP,NJ,NPUMP,EXPONENT WRITE(10,*) ' ERR|MAX| ' WRITE(10,*) ERR,MAX WRITE(10,*) '********************************************'
------PRINTING PIPES DATA-----WRITE(10,*) 'PIPES DATA' WRITE(10,*) '===========' WRITE(10,*) WRITE(10,*) 'PIPE No.(I)|L1(I)|L2(I)|K(I)|N(I)' DO I = 1,NP N(I) = EXPONENT WRITE(10,*) I,L1(I),L2(I),K(I),N(I) ENDDO WRITE(10,*) '********************************************' ------PRINTING NODES DATA-----WRITE(10,*) 'NODES DATA' WRITE(10,*) '===========' WRITE(10,*) WRITE(10,*) 'NODE No.(I)|QJ(I)|H(I)' DO I = 1,NJ WRITE(10,*) I,QJ(I),H(I) ENDDO WRITE(10,*) '********************************************' ------PRINTING PUMPS DATA-----WRITE(10,*) 'PUMPS DATA'
351
Appendix A C
1003 C C C C C C
BEGINNING OF GRADIENT METHOD PROGRAMMING ======================================== NOE = Number of Equations (No. of Unknowns) NCP = Number of Connected Pipes Counter (No. of Pipes Connected C to Node II) KHC = Known Head Counter IKHC = Counter to check the number of known heads (to reduce NOE)
C C C 1002 7001 C C C
DO 7001 I = 1,NP Q(I) = 1.0 CONTINUE ------CHECKING THE KNOWN-HEAD NODES------
90 C C C C C C C C C C C C C C C C C C C
WRITE(10,*) '===========' IF(NPUMP.EQ.0) GOTO 1002 WRITE(10,*) 'PIPE No.(II)|AP(II)|BP(II)|CP(II)' DO 1003 I = 1,NPUMP WRITE(10,*) II,AP(II),BP(II),CP(II) CONTINUE
DO 90 I = 1,NJ IF(H(I).EQ.0.0) GOTO 90 KHC(NP+I) = H(I) IKHC = IKHC + 1 CONTINUE ------END OF KNOWN-HEAD NODES CHECK-----------FILING THE "Soln" VECTOR-----The "Soln" vector represents the (R.H.S) of Eq. (11.12), page(252) It will be defined in this program as follows: Soln(I) = CONST1(I) + CONST2(I)*(Q(I)**N(I)) Where: CONST1(I) = (H(L2(I)) - H(L1(I))) - CP(I) CONST2(I) = -(1.0-N(I))*(AP(I)-K(I)) NOTES: ====== 1) The 1st step is to calculate "CONST1(I) & CONST2(I)" 2) This step must be performed before starting iterations (before line #5001) 3) If one head, or both, is unknown (before starting iterations); its value will be (zero) and will not
355
Appendix A C C C C C C
affect the "constant" value 4) To determine the values of "CONST1(I) & CONST2(I)", the counter (I) will start from (I=1) to (I=NP), as these values will be used to determine "Soln(I)" corresponding to "Head-loss Equations" (from 1-NP Equations) ----------------------------------------------------------------DO I = 1,NP CONST1(I) = (H(L2(I)) - H(L1(I))) - CP(I) CONST2(I) = -(1.0 - N(I))*(AP(I) - K(I)) ENDDO
C C C
------ITERATIONS START-----5001
ITER = 0 NOE = NP + NJ
C ITER = ITER + 1 C DO I = 1,NOE DO J = 1,NOE Eqn(I,J) = 0.0 ENDDO ENDDO C DO 7000 I = 1,NOE IF(I.LE.NP) THEN DO 7500 J = 1,(NOE+1) IF(I.EQ.J) Eqn(I,J) = BP(I) + * (N(I)*(AP(I)-K(I))*Q(I)*ABS(Q(I))**(N(I)-2.0)) C CHECK THE ELIMINATED COLUMN (due to a known head) IF(KHC(J).NE.0) GOTO 7500 C END OF CHECK THE ELIMINATED COLUMN IF(J.EQ.(L1(I)+NP)) Eqn(I,J) = 1.0 IF(J.EQ.(L2(I)+NP)) Eqn(I,J) = -1.0 7500 CONTINUE ELSE C CHECK THE ELIMINATED ROW (due to a known head) IF(KHC(I).NE.0) GOTO 7000 C END OF CHECK THE ELIMINATED ROW II = I - NP DO 6500 NCP = 1,NP C IF(II.NE.L1(NCP).AND.II.NE.L2(NCP)) GOTO 6500 IF(II.EQ.L1(NCP)) Eqn(I,NCP) = -1.0 IF(II.EQ.L2(NCP)) Eqn(I,NCP) = 1.0 6500 CONTINUE ENDIF
351
Appendix A 7000
CONTINUE
C DO 4001 I = 1,NOE IF(I.LE.NP) THEN Soln(I)=CONST1(I)+CONST2(I)*Q(I)*ABS(Q(I))**(N(I)-1.0) ELSE II = I - NP Soln(I) = QJ(II) ENDIF CONTINUE
4001 C C ----TRANSFERRING ZERO COLUMNS & ROWS TO THE END OF THE MATRIX---C C (A) TRANSFERRING ZERO COLUMNS TO THE END OF THE MATRIX C C Determine the zero columns (from the # of known heads), i.e., H1&H2 C So the zero columns will be = NP+JJJ (i.e. for H1 column: JJJ=1, C then the zero column No.= 7+1=8, and for H2: the zero column C No.=7+2=9) C We will start transferring the furthest column (i.e. column #9), C then we transfer the next (the closest) zero column (i.e. column #8) C C The algorithm to transfer a zero column: C KKK = Number of column after the zero column C TRANS(:) = A zero vector C J = A counter to check all head columns (in descending order) C --> If the check condition is true, that means that the column (J) C is a zero column to be replaced with the next column (KKK) C DO I = 1,NOE TRANS(I) = 0.0 ENDDO C DO 1501 J = NOE,(NP+1),-1 C Check if the column is zero or not DO M = 1,NOE IF(Eqn(M,J).NE.0.0) GOTO 1501 ENDDO DO I=J,(NOE-1) KKK = I + 1 Eqn(:,I) = Eqn(:,KKK) Eqn(:,KKK) = TRANS(:) ENDDO 1501 CONTINUE C C (B) TRANSFERRING ZERO ROWS TO THE END OF THE MATRIX
351
Appendix A C C KKK = Number of row after the zero row C TRANS(:) = A zero vector C J = A counter to check all head rows (in descending order) C --> If the check condition is true, that means that the row (J) C is a zero row to be replaced with the next row (KKK) C C (B1) TRANSFERRING ZERO ROWS (OF "Eqn" MATRIX) TO THE END OF THE MATRIX C DO 1502 J = NOE,(NP+1),-1 C Check if the row is zero or not DO M = 1,NOE IF(Eqn(J,M).NE.0.0) GOTO 1502 ENDDO DO I = J,(NOE-1) KKK = I + 1 Eqn(I,:) = Eqn(KKK,:) Eqn(KKK,:) = TRANS(:) ENDDO 1502 CONTINUE C C (B2) TRANSFERRING ZERO ROWS (OF "Soln" MATRIX) TO THE END OF THE MATRIX C DO 1503 J = NOE,(NP+1),-1 C Check if the row is zero or not IF(Soln(J).NE.0.0) GOTO 1503 DO I=J,(NOE-1) KKK = I + 1 Soln(I) = Soln(KKK) Soln(KKK) = 0.0 ENDDO 1503 CONTINUE C C REDUCING THE MATRICES SIZE TO THE NEW (NOE) AFTER ELIMINATION OF C ZERO COLUMNS & ROWS C NOE = (NP + NJ) - IKHC DO I = 1,NOE DO J=1,NOE Eqn(I,J) = Eqn(I,J) ENDDO ENDDO C DO I = 1,NOE Soln(I) = Soln(I) ENDDO
351
Appendix A C C C
------PRINTING THE MATRICES------
100
OPEN(50, FILE='MATRICES.dat', STATUS='UNKNOWN') WRITE(50,*) 'THE MATRIX OF COEFFICIENTS' WRITE(50,*) ' ' WRITE(50,100) FORMAT( ' Eqn 1 2 3 * ' 5 6 7 * ' 9 10 11 ') WRITE(50,*) ' '
4', 8',
C 111
DO I = 1,NOE WRITE(50,111) I,(Eqn(I,J),J=1,NOE) FORMAT(1X,I3,11(F10.4)) ENDDO
C WRITE(50,*) WRITE(50,*) WRITE(50,*) WRITE(50,*)
' ' ' ' 'THE MATRIX OF KNOWNS' ' '
C 112 C C C
------INITIALIZATION OF VALUES------
6000 C C C C C C C C C C C C
DO I = 1,NOE WRITE(50,112) I,Soln(I) FORMAT(3X,'Soln('I3') =',F10.4) ENDDO
DO 6000 I = 1,NOE IF(I.LE.NP) Soln0(I) = Q(I) IF(I.GT.NP) Soln0(I) = H(I) CONTINUE ------SOLVING THE SYSTEM OF LINEAR EQUATIONS-----The system will be solved by getting the inverse matrix of "Eqn" and multiply it by the known matrix "Soln" The Inv(Eqn) = "EqnINV" will be got using the "FINDInv" SUBROUTINE NOTES ON THE "FINDInv" SUBROUTINE CALL FINDInv(Eqn, EqnINV, NOE, errorflag) ---> INPUT = Eqn(NOE,NOE) (THE MATRIX OF COEFFICIENTS) OUTPUT = EqnINV (THE INVERSE OF THE MATRIX "Eqn") CALL FINDInv(Eqn, EqnINV, NOE, errorflag) ------PRINTING THE INVERSE MATRIX "EqnINV------
351
Appendix A
114
WRITE(50,*) ' ' WRITE(50,*) ' ' WRITE(50,*) 'THE INVERSE OF THE COEFFICIENTS MATRIX' WRITE(50,*) ' ' WRITE(50,114) FORMAT( ' Eqn 1 2 3 4', * ' 5 6 7 8', * ' 9 10 11 ') WRITE(50,*) ' '
C
C
DO I = 1,NOE WRITE(50,101) I,(EqnINV(I,J),J=1,NOE) FORMAT(1X,I3,11(F10.4)) ENDDO ------END OF PRINTING THE INVERSE MATRIX-----DO I = 1,NOE SUM = 0.0 DO J = 1,NOE SUM = SUM + (EqnINV(I,J)*Soln(J)) XSOLN(I) = SUM ENDDO ENDDO ------PRINTING THE SOLUTION OF THE ITERATION "XSoln-----WRITE(50,*) ' ' WRITE(50,*) ' ' WRITE(50,*) 'THE SOLUTION OF ITERATION NO.',ITER WRITE(50,*) ' ' DO I = 1,NOE WRITE(50,102) I,XSoln(I) FORMAT(3X,'XSoln('I3') =',F10.4) ENDDO WRITE(50,*) ' ' WRITE(50,*) ' ' WRITE(50,*) '===================================================' WRITE(50,*) ' ' ------END OF PRINTING THE SOLUTION OF THE ITERATION "XSoln------
C
------CHECK THE ERROR VALUE & MAXIMUM No. of ITERATIONS------
101 C
C
102
DO 6001 I = 1,NOE ERROR1 = ABS(XSoln(I)-Soln0(I)) IF(ERROR1.LE.Err) GOTO 6001 Soln0(I) = XSoln(I) IF(I.LE.NP) Q(I) = XSoln(I) IF(I.GT.NP) H(I-NP) = XSoln(I) IF(ITER.GE.MAX)THEN
311
Appendix A WRITE (*,*) 'THE MAX. No. OF ITERATIONS IS REACHED' GOTO 6001 6001 C C C
ENDIF IF(I.EQ.NOE) GOTO 5001 CONTINUE ------PRINTING THE SOLUTION (SOLN(I)) -----OPEN(11, FILE='RESULTS.dat', STATUS='UNKNOWN')
C
4002 4003 4005
DO 4005 I = 1,NOE IF(I.LE.NP)THEN WRITE(11,4002) I, XSOLN(I) FORMAT(5X,'Q(',I3,') = ',F8.4) ELSE WRITE(11,4003) I-NP, XSOLN(I) FORMAT(5X,'H(',I3,') = ',F8.4) ENDIF CONTINUE
C END SUBROUTINE GRADIENT C C--------------------------------------------------------------------C SUBROUTINE FINDInv(matrix, inverse, n, errorflag) C IMPLICIT NONE INTEGER, INTENT(IN) :: n INTEGER, INTENT(OUT) :: errorflag C Return error status: -1 for error, 0 for normal DOUBLEPRECISION :: matrix (1000,1000) ! Input matrix DOUBLEPRECISION :: inverse (1000,1000) ! Inverted matrix LOGICAL :: FLAG = .TRUE. INTEGER :: i, j, k REAL :: m REAL, DIMENSION(n,2*n) :: augmatrix C augmented matrix C ------Augment input matrix with an identity matrix-----C WRITE(*,*) 'THE SEEN MATRIX OF COEFFICIENTS' C DO I = 1,n DO J = 1,n WRITE(*,117) I,J,matrix(I,J) 117 FORMAT(3X,'matrix(',I3,',',I3,')=',F10.4,\) ENDDO WRITE (*,113)
313
Appendix A 113
FORMAT(2X,//) ENDDO
C DO i = 1,n DO j = 1,2*n IF (j
