APPLICATION OF THE GRADIENT METHOD FOR THE ANALYSIS
OF UNSTEADY FLOW IN WATER NETWORKS
by
Iftekhar Ahmed
A Thesis Submitted to the Faculty of the
DEPARTMENT OF CIVIL ENGINEERING
In Partial Fulfillment of the RequirementsFor the Degree of
MASTER OF SCIENCE
In the Graduate College
THE UNIVERSITY OF ARIZONA
1997
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for anadvanced degree at The University of Arizona and is deposited in the University Libraryto be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission,provided that accurate acknowledgment of source is made. Requests for permission forextended quotation from or reproduction of this manuscript in whole or in part may begranted by the head of the major department or the Dean of the Graduate College when inhis or her judgment the proposed use of the material is in the interests of scholarship. Inall other instances, however, permission must be obtained from the author.
SIGNED:
2
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
DateL?
Kevin E. L, eyProfessor of Civil ngineering
3
ACKNOWLEDGMENTS
The author would like to express his appreciation to his major advisor, Professor
Kevin E. Lansey for his support, guidance and patience during the length of this work.
Acknowledgments are also due to Dr. Dinshaw N. Contractor and Dr. Avi Ostfeld for
serving as members of the author's committee and for their valuable advice and moral
support.
Dr. Vincent F. Bralts and Dr. Rabi H. Mohtar of the Department of Agricultural
and Biological Engineering at Purdue University, and Dr. Don J. Wood of the
Department of Civil Engineering at the University of Kentucky were instrumental in
providing information on water distribution network modeling.
Special thanks are also extended to Farhana Ahmed for sharing valuable ideas on
computer modeling, and to Alicia L. Urban for editing a significant portion of this thesis.
The author would like to recognize his parents, Nazim U. Ahmed and Mumtaj
Nazim, who encouraged him to follow an engineering career and instilled in him the
work ethic and integrity essential for undertaking and completing the tasks required of
this work.
4
DEDICATION
The author would like to dedicate this work to his grandparents who, through their
stalwart struggle in the face of insurmountable odds taught him the meaning of
perseverance and fortitude.
5
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS 7
LIST OF TABLES. 9
ABSTRACT 10
1. INTRODUCTION 11
2. LITERATURE REVIEW 14
3. BASIC EQUATIONS AND SOLUTION TECHNIQUESFOR STEADY-STATE FLOW CONDITIONS 18
3.1 Introduction 183.2 Basic Equations for Steady-State Flow Conditions in Pipe Networks. 19
3.2.1 The Flow (Q) Equations 213.2.2 The Node (h) Equations 263.2.3 The Loop (AQ) Equations 293.2.4 Comparison of the Solution Techniques 313.2.5 Gradient Method 323.2.6 Comparison of the Gradient Method with Traditional Methods 383.2.7 Governing Equations in Dynamic Pipe Network Analysis 41
3.3 Conclusions 51
4. GRADIENT ALGORITHM FOR UNSTEADY FLOW CONDITIONS 52
7.1 Introduction 817.2 Model Verification 817.3 Model Sensitivity 1157.4 Conclusions 121
8. CONCLUSIONS 122
APPENDIX A: LIST OF SYMBOLS 124
APPENDIX B: LIST OF VARIABLES IN SOURCE CODE 126
APPENDIX C: SOURCE CODE 133
APPENDIX D: EXAMPLE INPUT DATA 165
APPENDIX E: NETWORK SPECIFICATION 173
APPENDIX F: TABLES 175
REFERENCES 185
LIST OF ILLUSTRATIONS
7
Figure 3.1
Figure 3.2
Figure 4.1
Figure 5.1
Figure 5.2
Figure 6.1
Figure 6.2
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7
Figure 7.8
Figure 7.9
Figure 7.10
Figure 7.11
Figure 7.12
Figure 7.13
Figure 7.14a
Figure 7.14b
Figure 7.15
Figure 7.16
Figure 7.17
Figure 7.18
Network Analyzed by Todini for Examples A, B, C and D 40
Free Body Diagram with Notation for Dynamic Equations 43
Typical Setup of Water Distribution Network 55
Variable Level Tank Setup 67
Pump Operation Characteristic Curve 69
Flowchart (Steady-State) 79
Flowchart (Unsteady-State) 80
Network A (taken from Holloway, 1985) 83
Network B 84
Demand Factor for Networks A and B 85
Comparison of Calculated Heads for Network A 86
Comparison of Calculated Flows for Network A 87
Head Profile for Varied Time Step in Network A 88
Flow Profile for Varied Time Step in Network A 89
Calculated Head (node 2) for Network B 91
Calculated Head (node 7) for Network B 92
Calculated Flows for Network B 93
Network C (taken from Onizuka, 1986) 95
Head Profile for Network C 96
Flow Profile for Network C 97
Network D 99
Applied Demand Function 99
Head Profile (node 2) for Network D 100
Head Profile (node 5) for Network D 101
Flow Profile (pipe 2) for Network D 102
Flow Profile (pipe 7) for Network D 103
LIST OF ILLUSTRATIONS --- Continued
8
Figure 7.19a
Figure 7.19b
Figure 7.20
Figure 7.21
Figure 7.22
Figure 7.23
Figure 7.24
Figure 7.25
Figure 7.26
Figure 7.27
Figure 7.28
Figure 7.29
Figure 7.30
Figure 7.31
Figure D- 1 a
Figure D- lb
Network E 105
Applied Demand Function 105
WSEL Profile (D-10m) for Network E 106
Head Profile (node 4) at D=10m for Network E 107
Flow Profile (pipe 5) at D=1 0m for Network E 108
WSEL Profile (D=12m) for Network E 109
Head Profile (node 4) at D-12m for Network E 110
Flow Profile (pipe 5) at D=12m for Network E 111
Flow Profile (pipe 1) for Network D with Pump in Pipe 1 113
Pump Head Profile for Network D with Pump in Pipe 1 114
Sensitivity Analysis (± %D) for Network A 117
Sensitivity Analysis (± %L) for Network A 118
Sensitivity Analysis (± %f) for Network A 119
Sensitivity Analysis (± %q ) for Network A 120
Setup for Example Input Data 172
Example Demand Function 172
9
LIST OF TABLES
Table 3.1 Comparison of the Gradient Method with Traditional Methods. 39
Table E-1 Specification for Networks C, D and E 174
Table F-1 Hourly Variation of Flow (cfs) and Head (ft) for Network A 176
Table F-2 Hourly Variation of Flow (cfs) and Head (ft) for Network A 177
Table F-3 Hourly Variation of Flow (cfs) and Head (ft) for Network B 178
Table F-4 Variation of Flow and Head for Network C 179
Table F-5 Hourly Variation of Head (m) at Node 2 of Network D 180
Table F-6 Hourly Variation of Head (m) at Node 5 of Network D 181
Table F-7 Hourly Variation of Flow (cms) in Pipe 2 of Network D 182
Table F-8 Hourly Variation of Flow (cms) in Pipe 7 of Network D 183
Table F-9 Sensitivity Analysis on Network A 184
10
ABSTRACT
Gradually varied unsteady flow in water distribution networks is studied using the
Gradient Method and the Newton-Raphson Solution Technique. The Gradient Algorithm
has shown robust behavior when run with examples simulating different operational
situations. In the attempt to obtain solutions rapidly, a computer program has been
developed in FORTRAN 77. The computer program developed herein is general, easy to
use, and allows a dynamic analysis of complex looped and open water distribution
networks. Piezometric heads at nodes and flow rates in pipes are computed for water
distribution networks. Results obtained from the Gradient Method are compared with an
existing dynamic model. The benefits of using the Gradient Algorithm appear to be
flexibility in coding, versatility in operational conditions, and its application to
complicated water distribution networks.
1 1
CHAPTER 1
INTRODUCTION
Water is vital to man's existence. Apart from the day to day potable
requirements, water is needed for irrigation, power generation, recreation, industrial
production, and receiving wastewater. According to the United Nations Population
Reference Bureau, by the end of the century the world population may reach 6 billion.
Therefore, the total water requirement will continue to increase. In addition, per capita
water consumption is also on the rise. In 1965, the per capita consumption in the USA
for domestic, public, and commercial purposes was 460 liters per day (lpd), and the
industrial requirement was 137 lpd; by 1985 it increased to approximately 465 and 152
lpd respectively. It is now predicted that usage will increase to 474 lpcpd and 163 lpcpd,
respectively, or an overall increase of 0.3 percent per year by the year 2000.
As demands increase, building of state-of-the-art water distribution systems and
rehabilitation of the existing ones to satisfy the demand are major concerns for the water
industry. The two primary objectives in design and operation of water distribution
systems are to (1) minimize the cost of supplying water and (2) maximize the reliability
of water supply. However, these two objectives are in conflict because increased
reliability leads to increased cost.
The problem of design and operation of water distribution networks is multi-
objective in nature. One of the major issues facing engineers and governments involved
12
in the design, operation, and maintenance of water distribution system is to reconcile cost
and reliability. Significant capital investments are required to improve the current level
of service with emphasis on correct establishment of priorities and development of
adequate engineering decision support tools. One such tool for engineers is a well
documented computer model that is reliable.
A hydraulic model will provide information on local water consumption,
geographical distribution of design flows, and pressure gradients for the system. The
simplest hydraulic model considers a steady-state analysis where the flow parameters are
time invariant. The more realistic representation is to incorporate the time variant or
unsteady nature of the network demands. Flow conditions within a water supply system
vary throughout the year and each day, but these changes generally occur slowly. Typical
conditions within a system range from the small winter demands to the large summer
discharges. Flow conditions vary not only seasonally but also hourly within a typical
day. For instance, demand is low at night when most people have gone to rest and
factories sit idle. Demand is high during the day with several peak flow rates that occur
in the morning and the evening, when people take showers, wash dishes, and do laundry.
Other network parameters that change over time are the water levels in system storage
tanks, the cycle of pumps, and the effect of pressure reducing valves. To model this
variation, extended period simulation (EPS) is commonly applied to a network hydraulic
model. Extended period simulation is performed assuming steady flow conditions and
adjusting tank water levels over the simulation period at discrete time intervals.
13
Steady-state analysis, even with the "extended simulation" option, may not truly
represent the flow and pressure variations within a system. In addition, water quality
changes may be sensitive to the time variant flow conditions. Some work has been done
on dynamic network modeling where the flow parameters are time variant. The water
hammer distribution analysis program by Streeter (1967) is a good example of such a
model. Unfortunately, the program has very limited "real world" use. Simple network
modifications require extensive input data changes, and the program is limited in the size
network that can be analyzed. Such models are designed for rapid changes and small
time step in order to effectively study the water hammer phenomenon. Rapid changes
may occur in a network due to unavoidable circumstances. As the shock wave travels
through the network, it rapidly decreases in severity as it passes through each node in the
system. Since change in pressure is proportional to change in velocity, the pressure
changes become smaller as the wave passes through each node. Thus, rapid changes are
local problems, and they do not normally necessitate the use of water hammer analysis
over the entire network. This practice is simply not cost effective.
A gap exists in modeling unsteady flow conditions, between very slow changes
to the rapid changes resulting in water hammer. In this study, an efficient hydraulic water
distribution model is developed using the Gradient Algorithm to model gradually varied
unsteady flow in every link (pipe, pump, valve etc.) and the piezometric heads throughout
a system.
14
CHAPTER 2
LITERATURE REVIEW
A large number of papers have been published on numerical methods for
determining the solution of unsteady flow problems in water distribution networks. But,
among these publications only a handful investigate the problem of gradually varied
unsteady flow in hydraulic networks. Due to the fact that rapid pressure changes are
actually local problems in water distribution networks, a literature review on gradually
varied unsteady flow analyses is of utmost importance.
Sevuk (1979) attempted to simulate gradually varied unsteady flows by a single
mathematical model. His formulation of such model is based on the fact that at any
particular time t, the mathematical model of flow consists of:
(i) two link equations expressing the principles of conservation of mass and energy
for each element (e.g., pipes)
(ii) a node equation expressing the principle of conservation of mass at the junction
of elements
(iii) an external boundary condition expressing either the inflow to the system, or the
piezometric head at each node.
The primary variables in Sevuk's mathematical model are:
15
(i) inflow to and outflow from each link, i e, link flow
(ii) inflow to or outflow from each node, i.e., node flows
(iii) piezometric head at each node.
An Implicit Finite Difference Scheme was used to solve the following one-dimensional
mass and momentum conservation equations:
aQ/0x + (gA/a2) ah/at = 0 (2.1)
aQ/at + (gA) ah/ax + f Q1Q1/(2DA) = 0 (2.2)
where
g = gravitational acceleration
D = diameter of pipe
f = friction factor (Darcy-Weisbach)
a = (K/01/2 / (1 + KD/Ee)/2
where
K = bulk modulus of the fluid
E = modulus of elasticity of pipe walls
p = fluid density
e = wall thickness of the pipe
16
Holloway (1985) performed surge analysis to simulate gradually varied flow in
water distribution networks. The surge analysis differs from the water-hammer analysis
in that the flow change is felt throughout the pipe instantaneously. Holloway obtained
the solution by solving one non-linear ordinary differential equation per pipe line instead
of solving numerous non-linear partial differential equations per pipeline. Due to slow
rates of flow variation in a water distribution network during its normal operating cycle,
surge analysis gives satisfactory results with far less computational effort than that
required by water-hammer analysis. A detailed discussion of Holloway's work is
presented in this study to appreciate the flexibility of the Gradient Method.
Onizuka (1986) presented a method of analysis based on rigid water column
theory for slow transients in pipe networks. Onizuka formulated a system of first order
ordinary differential equations that describes the dynamic response of the network. The
time integration is performed directly by using, for example, the Runge-Kutta method
without involving any iterative procedure. A network example from Onizuka's paper has
been presented in this study with minor modifications.
Todini and Pilati (1987) proposed a gradient algorithm that operates on the field
of piezometric heads and flows simultaneously. Salgado (1987) extended the original
method to incorporate pumps into the system, and showed some of the advantages of the
method when compared with the methods recommended by Wood (1981). Todini
provided a unified mathematical framework for the most widely used existing gradient
based methods. This study extends the application of the Gradient Method for
17
simultaneous solution of piezometric heads and flows in distribution networks under the
slow-transient conditions.
Reference to further literature study is embedded throughout this work in order to
appreciate the efficiency and robustness of the Gradient Method for solving water
distribution network problems.
18
CHAPTER 3
BASIC EQUATIONS AND SOLUTION TECHNIQUES
FOR STEADY-STATE FLOW CONDITIONS
3.1 Introduction
The equations describing steady and gradually varied unsteady fluid flow in
closed conduits are derived in this chapter. The determination of steady-state pressure
and flow in a complex pipe network has long been of great importance in hydraulic
engineering. Steady-state network analysis is not only an important problem when
studied by itself, but is also essential in giving the transient analysis a starting point.
Existing basic equations such as the flow (Q), the node (h), and the loop (AQ) equations
are derived. Numerical solution techniques corresponding to these equations are also
investigated. A full derivation of the gradient method for the steady-state analysis and
comparison with existing models is presented. The chapter concludes with a detailed
derivation of existing basic equations as presented by Holloway (1985) describing the
physical behavior of fluid in a pipeline undergoing gradual changes in the flow
conditions. The assumptions made in the derivations, and the limitations that these
assumptions impose, are briefly discussed.
Following definitions are relevant to the derivation of the governing equations:
19
a. Steady-State Flow: Steady-state flow condition is that in which the velocity (of
the fluid in motion) at any fixed point does not change over time.
b. Dynamic (Unsteady) Flow: Dynamic or unsteady flow condition is that in
which the velocity (of the fluid in motion) at any fixed point changes from instant to
instant.
c. Rigid Wall (gradually varied unsteady flow): Under the assumption of rigid wall
conduit flow, a flow change is felt throughout a pipeline instantaneously and the fluid
in the pipe moves like a solid slug. In this approach, the inertia of the fluid in a
particular pipe is treated as lumped instead of continuously distributed.
d. Extended Period Simulation (EPS): Adjustment of water surface elevation in a
tank over a time period. Extended period simulation is performed by adjusting tank
water levels over the simulation period at discrete time intervals.
3.2 Basic Equations for Steady-State Flow Conditions in a Pipe Network
Fluid flow in a network is in steady-state when at any given point the velocity
does not vary in magnitude or direction with time. The steady-state network problem is
solved for pressure at nodes and flow distributions in pipes. The following two
fundamental physical laws result in a system of equations that are solved to determine the
unknowns:
1. Conservation of Mass --- the algebraic sum of the flows (known or unknown) at a
junction must equal zero (Kirchhoff' s 1" law), and
20
2. Conservation of Energy --- the net algebraic change in energy around any closed loop
must equal zero (Kirchhoff's 2' law).
In practice, for closed conduit flow, the "energy per unit weight" is given in terms
of the pressure head (ply), the elevation head (Z), and the velocity head (v2/2g), where p is
the pressure in pipes, y is the unit weight of the fluid, v is the fluid velocity and g is the
gravitational acceleration. It is common practice to lump all these head or energy terms
together as total hydraulic head with dimensions of length. In order for the pressure
throughout the system to be uniquely determined, at least one location must have a fixed
hydraulic head or a head that is a known function of the external flow. Every nodal
boundary (i.e., pump, reservoir, valve) will introduce an additional unknown (external
flow) and an additional equation (head/flow relationship). Therefore, these nodes do not
influence the balance between the number of equations and the number of unknowns.
The relationship between pipe flow and the change in head across that pipe is non-
linear. Friction losses are usually modeled in the following form:
hf = K Q IQI m
(3.2.1)
where
hf = the head loss due to friction in the pipeline
Q = the pipeline flow rate
K & m = constant parameters that depend on the units used and the particular
equation chosen (e.g., Hazen-Williams or Darcy-Weisbach)
21
Likewise, the head-flow relationships introduced by pumps, and valves are also non-
linear functions of flow. These relationships are usually approximated by piecewise
polynomials, oftentime, using a least-squares routine, over the operating range
(Holloway, 1985).
The two conservation laws can be posed in several ways to determine the desired
unknowns. Three classes of approaches have been investigated for steady-state
conditions. One such approach is to use the pipe flow rates as the unknown and set up a
total of p equations where p is the number of pipes. There will be one energy equation
for each independent loop and one continuity equation for each node. These equations
are called the flow (or Q) equations since the flows are the unknowns. The next section
deals with this formulation. Two other formulations, the node (h) and the loop (AQ)
equations are discussed in the following sections.
3.2.1 The Flow (Q) Equations
The flow equations are given below for a system with L loops including psuedo-
loops and NN nodes:
PiE - E hpk, = dh, (/ = 1,2, .L) (3.2.1.1)
i=1 i=1
Pn
1=1 Q11 = q
(n = 1,2, NN) (3.2.1.2)
22
where
= head loss in pipe i in loop Ihpki = head supplied by pump k in loop 1
dh, = change in head between the nodes at the beginning and end of loop Ip, = number of pipes in loop IP, = number of pumps in loop IQin = flow into node n from pipe i connected to the nodeqn = consumptive use at node n, and
p„ = number of pipes into node n
The 1 loops consist of the number of independent closed circuits in which the head
difference around the loop, dh, is zero. Also within this set are so called pseudo-loops 1
which are paths connecting two fixed-grade nodes. The dh for these loops is the
difference in energy between the fixed grade nodes.
To state the above equations, the appropriate head loss formula (Darcy-Weisbach
or Hazen-Williams) must be used to determine the head loss (h), and a pump head curve
must be used to determine the head supplied by the pump. The flow (Q) equations result
in a larger number of equations than the next two approaches, but the continuity
equations are linear and the energy equations can be linearized. Therefore, efficient
numerical solution methods are available to solve this formulation's equations. One such
numerical tool is discussed next.
Psuedo-loops are non-existent loops that have to be created by introducing an imaginarypipeline between the fixed-grade nodes (e.g., reservoirs). These fictitious pipes have alarge frictional factor, thereby assuring that the flow through this pipeline will be small.
23
Linear Theory Method
In the linear theory method, the continuity equations are obviously linear.
However, the energy equations are non-linear since they cannot be written in the form:
PIE K, Q = dh
i=1(3.2.1.3)
where
Q = flow in pipe i in loop
= constant
dh = difference in water elevation in tanks (zero for loops, L), and
Pi = number of pipes in loop.
The energy equations are linearized by setting Qm as a constant for a given
iteration. By doing this, a system of linear equations are created that can be solved by
some linear technique (e.g., Gaussian elimination, Jacobi iterative method). The a, terms
are then given by:
= KiQm (3.2.1.4)
The a, terms can also contain the coefficients for the pump head characteristic
equations. Thus, L + NN - 1 linear equations will be created with the same number of
unknowns. For this purpose, the a's must be double subscripted: the first subscript (1)
refers to the loop number, while the second (i) refers to the number of the pipe. The
entire system of equations becomes
24
anQi + a12Q2 + + a., ;Q; = dh l
L equations (3.2.1.5)
aLl au _2 + + Q atiQ; = dhi
bilQi + b12Q2 + + bliQi qi
NN-1 equations (3.2.1.6)
bn -1,1Q1 + bn -1,2Q2 + + bn q -1
where
Q. = flow in pipe i in a loop
, if pipe i is in loop 1
0, if pipe i is not in loop 1
+1 if flow in pipe i is positive into node n
bni = -1 if flow in pipe i is negative into node n
0 if pipe i is not connected to node n
To solve the network problem, it is necessary to solve the linear set of equations,
recalculate the a terms, and resolve the linear equations iteratively until the solution
converges. The linear-theory method tends to overcorrect the Q's, so a weighted average
is usually taken of the old and new Q's. The major advantage of the linear-theory method
is that it does not require an initial solution. The Q's can be set to some arbitrary value,
and it will still converge correctly.
25
It has been reported (Holloway 1985) that Wood (1972, 1980) uses the linear-
theory method in his KYPIPE program. Wood's technique is best described as a
modified-linear-theory-newton (Holloway, 1985) method. The equation for a given
pipeline within a network system can be written as
F=KQIQr+MQ2±hp (3.2.1.7)
where,
M = algebraic sum of the minor losses in the pipe, and
= head supplied by a pump (if one is located in the pipeline considered) and
can be represented as:
hp = C l Q2 + C2Q + C3 (3.2.1.8)
This quadratic pump head-flow relationship is obtained by using a least-squares best fit of
three head-flow points on the pump discharge curve. The derivative of function F is:
G = (m+1)KT + 2MQ + 2C,Q + C2 (3.2.1.9)
Wood then uses the following gradient relationship to solve for the unknown Q:
GoQ = [GoQo - Fo]
(3.2.1.10)
where the subscript (0) indicates an initial guess or previous iteration. This equation is
summed around loops and solved simultaneously with the continuity equations to obtain
Qk , +„ or the flow at the following iteration step. This step will be repeated until Q 0 and Q
are within some desired tolerance.
26
3.2.2 The Node (h) Equations
By combining the energy equation for each pipe with the continuity equation, the
number of equations to be solved can be reduced from 4 + NN - 1 to NN, where NN is
the total number of nodes in the network. The unknowns are the head at each node.
Once the heads are known, the flow rates in each pipe can be obtained. The head loss
equation for a single pipe which has previously been written as hf = KT can now be
rewritten as
- h = 1%1 ti) gn) Qi; (3.2.2.1)
where
= head at node i
hi = head at node j
= head loss coefficient for pipe from node i to node j
= flow in pipe from node i to node j
= exponent in head loss equation for pipe from i to j
Since the head loss is positive in the flow direction, (sgn) Q ii = (sgn) (hi - hi), it is
possible to rewrite equation (3.2.2.1) to give
Q (sgn) (hi - h;) (lhi - (3.2.2.2)
The continuity equation at node i can be written as
Pi
E qi (3.2.2.3)k =1
where
27
Qki = flow into node i from node k
q 1 = consumptive use at node i, andp i = number of pipes at node i
The energy and continuity equations can be combined by substituting equation (3.2.2.2)
for each flow into equation (3.2.2.3) to give
Pi
k=1 (sgn) (hk (Ihk hiliKki) 1mki ) = ch
(3.2.2.4)
One such equation can be written for each node, and the set of node equations can be
solved for the unknown head (h i). Since the equations are non-linear, several solution
techniques are available for solving them, such as the Newton-Raphson method.
Newton-Raphson Method
The Newton-Raphson method is a powerful numerical equation solver. It is suited for
problems that can be expressed as F(x) 0, where the solution is the value of x for which
F is zero. The derivative of F can be approximated by:
F' = [F(x+Ax) - F(x)]/(Ax) (3.2.2.5)
With an initial estimate of x, the solution to the problem is simply the value of x + Ax
that forces F to zero. Therefore,
Ax = -F(x)/F'(x) (3.2.2.6)
The new value of x + Ax becomes x for the next iteration, and the process is repeated
until F converges toward zero within a specified tolerance.
28
The Newton-Raphson method can also be applied to a system of equations. The
remainder of this section will discuss its application to the NN - 1 nodal equations
(application to loop equations is also possible).
For each node (1 through k), the following head equation can be written
piE (sgn) (h - h) h - - q i = 0 (i = 1,2,..., NN-1)
J=1(3.2.2.7)
where
p i = number of pipes at node i, and
q i = consumptive use at node i
The difference in the value of the function at iteration k' and (k' + 1) is
dF = F(k' + 1) - F(k') (3.2.2.8)
This can also be approximated by the total derivative:
dF = (aF/ah,)Ah l + (ap5h2).612 + + (apahjAnk (3.2.2.9)
where
Ah = change in head between iteration k' and (k' + 1)
Thus the problem is finding values of Ah that force F(k'+1) to zero. This is accomplished
by setting equation (3.2.2.7) equal to equation (3.2.2.9), resulting in a system of k linear
equations with k unknowns (Ah) that can be solved by one of the linear methods
mentioned earlier. hi is then found by tracking through paths.
29
The solution procedure consists of picking some initial values of Ah, calculating
the partial derivatives of each F with respect to each h, solving the system of linear
equations to find h, and repeating until all of the F's are sufficiently close to zero.
3.2.3 The Loop (AQ) Equations
The third approach for solving for the flow and pressure distributions is to set up
the looped system so that the energy equations, as an initial condition, satisfy the
continuity equation. Thus, the problem becomes a matter of correcting the flows in each
loop in such a way that the continuity equations are not violated. To do so, flow in every
pipe in the loop can be corrected by adding flow (i.e., if there is too little head loss) or
subtracting flow (i.e., if there is too much head loss). Thus, the problem is reduced to
finding the correction factor, AQ, such that each loop energy equation is satisfied. These
loop energy equations may be written as:
PI
E Ki (Q0 + AQ) IQ0 + AQ 1 m = = (3.2.3.1)i =1
where
Q ic, = initial estimate of flow in the pipe i
AQ I = correction to flow in loop 1
Pi = number of pipes in lth loop, and
= number of loops
30
The AQ terms refer to the loop in which the pipe falls. If a pipe lies only in one loop,
then the flow in the pipe is Q . AQ. For a pipe that is part of several loops (i.e., a, and
b), the flow might be
Q, + AQa - AQb
(3.2.3.2)
The negative sign in front of the b term is included merely to show that a given pipe may
be so situated such that it is in the positive direction in one loop and in the negative
direction in another. When the loop approach is used, a total of 4 equations are required
since there are 4 unknowns, one for each loop. The Hardy-Cross method, which dates
back to 1936, is usually applied to the AQ equations (Walsh, 1984).
Hardy-Cross Method
When applied to the AQ equations, the method requires an initial solution which
satisfies the continuity equation. For loop l in a pipe network the AQ equation can be
written as:
PIF(AQ I) = E K [sgn (Qio AQ)] I Qio AQI r - dh, = 0 (3.2.3.3)
j=1
where
AQ, = correction to loop / to achieve convergence
Q0 = initial estimate of flow in pipe i (satisfies continuity)
= number of pipes in loop I
Applying the Newton-Raphson method for a single equation gives:
31
PiKi(Qi0 AQ) I Qi0 AQ, r
i=1
AQ(k' + 1) = AQ (3.2.3.4)
Pi
E Koni I Qio AQ1 r1=1
where the (k'+1) refers to the values of AQ in iteration (k' + 1). All other values refer to
iterations k', but they will be omitted from the equation for ease of reading. Equation
(3.2.3.4) should be recognized as being equivalent to:
AQ (k' + 1) = AQ (k') - F(k')/F'(k')
The signs on the Qi terms depend on how that pipe is situated in the loop under
consideration. It should be emphasized that the same pipe may have different signs in
different loops.
3.2.4 Comparison of the Solution Techniques
According to Jeppson (1976), the linear-theory method as used by Wood (1972,
1980) has two distinct advantages over the Newton-Raphson method:
1. it requires no initialization, and
2. according to Wood and Charles (1972), it always converges quickly.
Jeppson (1976) suggests that the linear-theory method should not be used with the head
oriented equations or the corrective loop equations. Linear-theory is reportedly best
suited for the flow equations, which is the system of equations chosen by Wood.
32
Wood and Charles (1972) did a comparison between Hardy Cross, Newton-
Raphson, and Linear-Theory methods. Pipe flows in a 19 pipe network were computed
using the three different analysis techniques. Only the Linear-Theory analysis technique
was analyzed by Wood; the other data was taken from previous literature. The results of
this comparison were (Holloway, 1985)
I. Hardy Cross --- 635 iterations,
2. Newton-Raphson --- 24 iterations, and
3. Linear-Theory --- 4 iterations.
Holloway and Chaudhry (1983) did a comparative study between Newton-Raphson
models (NETWK by Jeppson of the University of Utah (1980) and FLOW by Epp and
Fowler of the University of British Columbia (1970)), and the Linear-Theory model
(KYPIPE) by Wood of the University of Kentucky (1980). These studies indicated that
the rate of convergence of the two methods for a large 200 pipe network were close or
equal (Holloway, 1985). NETWK (Newton-Raphson) and KYPIPE (Linear-Theory) both
converged in eight iterations, and FLOW (Newton-Raphson) converged in nine iterations
(Holloway, 1985). However, FLOW required least amount of computation time,
followed by KYPIPE and then NETWK.
3.2.5 Gradient Method
A gradient algorithm is derived for flow analysis of pipe networks. The Newton-
Raphson technique is applied both in terms of nodal heads (h) and pipe flows (Q) to
33
obtain a simultaneous solution of the system of equations expressing mass and energy
balance. The problem is then solved through an iterative solution of a system of linear
equations, the size of which equals the number of unknown heads.
A pipe network is defined by its topology, pipe characteristics and system
constraints. System constraints include water demands (q) or fixed heads (ho) for each
node and head loss laws for each pipe [Ki(Q,)]. The problem is to determine all of the
flow rates (Q) in the pipes and all of the unknown heads (h) at the nodes under the steady-
state assumption.
In matrix form the problem can be formulated as follows (Todini, 1987):
Al2 h + K(Q) = -A10 ho
A21 Q
(3.2.5.1)
Al2 A21 T
(ne,nn) unknown head nodes incidence matrix
= A 01T
(ne,no) fixed head nodes incidence matrix
QT = EQI,Q2,....,Q.el (1 ,ne) flowrates in each pipe
T= (1 ,nn) nodal demands
hT = (1 ,mi) unknown nodal heads
hoT = (1,no) fixed nodal heads
KT(Q) = (1,ne) laws expressing head losses in pipes (K, = KI(Q))
where, nn = number of nodes with unknown head
34
no = number of nodes with fixed head
ne = number of pipes with unknown flow
1 if flow of pipe i enters node j
0 if pipe i and node j are not connected
-1 if flow of pipe i leaves node j
and with A 10 defined similarly to A l2 for fixed head nodes.
The system represented by equation (3.2.5.1) may have more than one solution
depending upon the form of [K,(Q,)]. If all [KI(Q)] are monotonically increasing
functions, it can be proved that the solution of the system exists and is unique (Todini and
Pilati, 1987). Note the first set of equations (3.2.5.1) are head loss equations for each
pipe while the second are the set of nodal balance relationship.
Derivation
Todini and Pilati (1987) showed that the necessary conditions for the steady-state
flow are simply the simultaneous fulfillment of nodal balance and the head loss-flow
relationship. Both conditions can be expressed in the following compact system of
equations:
Al210Q1 F— Alohol
q(3.2.5.2)
where
An = matrix (ne,ne) dependent on the particular head loss-flow relationship
Al2 = topological (incidence) matrix (ne,nn)
A21 [A-12]
T
A 21 0 dh dq (3.2.5.6)NA, A l2 1 [dQi [dE]
35
A0-10 = known vector (ne, 1)
The upper part of the system represents the head loss-flow relationship and the lower part
corresponds to the nodal flow (mass) balance. Since A 11 clearly depends on the flow, the
upper part of the above system is a set of non-linear equations in Q. A generalized
relation between flow and head may be written as
= Qim (i = 1 ,2,..., ne) (3.2.5.3)
Using the link to node topological matrix (A l2), one can express the head loss (or head
gain in the presence of pumps) of each link connecting two different nodes as
A„ Q + Au h = - ADD ho (3.2.5.4)
where
O
K 21Q2I m (3.2.5.5)
o ne IsrL e
Because the system represented by equation (3.2.5.2) is non-linear, a direct solution is not
possible, but an iterative gradient approach is indeed feasible. Differentiating the system
of equation (3.2.5.2) gives
where
36
N = diagonal matrix (ne,ne) of the exponents m of the head loss-flow relationship.
At iteration k', where convergence has not yet been achieved, a (ne, 1) vector dE
and a (nn, 1) vector dq are defined as
dE = A,, Qi k' Al2 Aloho (3.2.5.7)
and dq A21 Q ik' - q (3.2.5.8)
Equation (3.2.5.7) represents the energy imbalance at each link while (3.2.5.8) represents
the flow imbalance at each node. The solution of (3.2.5.6) is
[NA,,
Ldh A21
Al dE1 [1dgi (3.2.5.9)
Following a similar approach to Todini and Pilati (1987), the inverse of the block-
triangular matrix in (3.2.5.9) can be computed as another block matrix:
rNA ll Al2 -I B11 B 12
[ A21 0(3.2.5.10)
B21 B 22
Assuming
NA,, = (3.2.5.11)
(and therefore D A il = N-1 ; N, A,, and D being diagonal), and computing the blocks of the
inverse in equation (3.2.5.10), one gets:
13 1 , = D - D Al2 (Ai2D Al2) -1 AND
B12 = D Al2 (A21 D Al2)
B21 = (A21 D A l2) -1 A2 1 D
B 22 = — (A21 D A l2 )4
The solution of equation (3.2.5.2) is thus obtained from the fact that
dQ = B 11 dE + B 12 dq
dh = B21 dB + B22 dq
(3.2.5.12)
(3.2.5.13)
37
By substituting set of equations (3.2.5.12) into equations (3.2.5.13), the following can be
obtained:
dh (A21 D A IX A21 D (An + Al2 hic' A10 h0) — (A21 D A l2) 1 (A2I Qk' C1)
= hk' (A21 D Al2) - ' {A21 D (An Qk' Alo ho) ± - A2I Q1(')}
(3.2.5.14)
dQ = {D - D Al2 (A21 D A l2) 1 A21 D) (A ll Qk' + Al2 hie + A10 hip) + D A l2 (A2I D Al2) -1
s, (A21 Qk'
= D (A,, Qk ' + Alo ho) - D Al2 (A21 D A l2)- ' {A2 , D (An Qle + A lo ho) + (ci - A21 Qk)}
(3.2.5.15)
Using the definition of D as presented earlier:
dQ = Q ic - Qk '+1
dh = hic - (3.2.5.16)
A recursive Newton-Raphson algorithm is obtained as follows:
38
hk'-1 -(A211\1-1 A ll -1 A l2) -1 {A l2 N I (Q1(' A 11 -1 A10 h0) ± (C1 - A21 Qk.)}
(3.2.5.17)
Q" (1 - N-1 ) - A„' (A, 2 h" + A, o ho)
(3.2.5.18)
where A, is computed using Q . k' and (k'+1) are consecutive iteration steps.
3.2.6 Comparison of the Gradient Method with Traditional Methods
Salgado and Todini (1987) compared the gradient algorithm for the analysis of
water distribution networks with other traditional gradient algorithms (Simultaneous Path
(or Loop) and Linear Theory). The gradient algorithm has shown robust behavior when
run with some examples simulating different operational situations. A set of examples as
presented by Todini and Salgado (1987) is investigated in this section. A common
network is used to simulate different hydraulic conditions.
The behavior is presented in a tabular form.
1. Example A
The network shown in Figure 3.1 incorporates small diameter pipes connected to larger
diameter pipes and contains 66 pipes, 41 nodes and 2 reservoirs (nodes 1 and 2). Nodal
demands are controlled to maintain fluid velocities under 0.2 m/s to simulate the behavior
of the network during the night. This example network includes 2 gravitational sources
and does not include pumps.
2. Example B
39
This example resembles Example A with the addition of 6 pumps. Also, a branched
connection has been added at node 25 to give a total of 74 pipes, 48 nodes and 8 sources.
3. Example C
This is the same basic network as presented in Example B, but nodal demands have been
increased tenfold to simulate extremely high daytime consumption.
4. Example D
The previous network has been modified to include 2 valves (in links 67 and 70) that are
completely closed in order to simulate the case when a particular link ceases to carry
flow. When the valves are fully closed, the original network becomes two independent
systems with their own sources. It is clear from Table 3.1 that the gradient method
worked extremely well for steady-state conditions. Thus, it is extended in chapter 4 to
consider gradually varied flow conditions. The equations describing Dynamic flow are
presented in the next section.
Table 3.1. Comparison of the Gradient Method with Traditional Methods (Todini, 1987)
hoT = [h01 ,h02 ,....,h00] (1,no) fixed nodal heads
KT(Q) = (1 ,ne) laws expressing head losses in pipes (K.; = Ki(Q))
where, nn = number of nodes with unknown head
no = number of nodes with fixed head
ne = number of pipes with unknown flow
1 if flow of pipe i enters node j
= 0 if pipe i and node j are not connected
-1 if flow of pipe i leaves node j
and with A 10 defined similarly to A l 2 for fixed head nodes.
60
The following compact system of equations was introduced earlier in Chapter 3
for the of nodal balance and the head loss-flow relationships:
[A ll A l2 1 [Q] Awhol
A21 0 h q(4.3.2)
where,
A1 1 = matrix (ne,ne) dependent on the particular head loss-flow relationship
A l2 = topological (incidence) matrix (ne,nn)
A21 = [AIX
Awho = known right hand side vector (ne,l) obtained from the dynamic head-flowequations
The upper part of the system represents the head-flow relationship and the lower one
corresponds to the nodal flow (mass) balance. Since A11 clearly depends on the flows, the
upper part of the above system is a set of non-linear equations in Q.
A generalized relation between flow and head may be written as
h=KQm
(i = 1,2,..., ne) (4.3.3)
Using the link-to-node topological matrix (A l2), one can express the head loss (or head
gain in presence of pumps) of each link (i e, pipes, pumps, etc.) connecting two different
nodes as
A„ Q + h = - Aio ho (4.3.4)
where, for the network in Figure 4.1,
-
K i 1Q i tm-1 L,
At — 'gA i
61
i 1 m-1 L2 K21Q2 tl At —
gA2
-
0
0 K51 m 1 L5 Q 5 t 1At —
gA5
(4.3.5)
and
A11
Al2 = (4.3.6)
Combining all the terms at the known time step, matrix A ieho is writen in the following
compact form:
L1QltgA i 1 —1 0 0 0- h0
1t
± h ol t+At
L 2 0 1 —1 0 0 hitQ2t
gA 2 + 0 0 1 —1 0 * h2t */2
0 1 0 —1 0 h3t•
L s t_ 0 0 —1 0 1 h 02 t + h 02t+At
_Q 5 .gA 5
-
A lchi, = -
(4.3.7)
which, upon algebraic manipulation, is a (ne,l) matrix for the system.
62
Because the system represented by equation (4.3.2) is non-linear, a direct solution
is not possible, but an iterative gradient approach has been introduced in the last chapter
that solves the problem.
The resulting recursive Newton-Raphson algorithm is:
hk '+' = - (A21 N1 A il -
A l2) -1 {A 12 (Qk ' + A,0 ho) + (q - A21 Qie)}
(4.3.8)
= _ N-1 ) Qie _(An h" ADD ho)
(4.3.9)
where A ll is computed using Q k'. k' and (k'+1) are consecutive iteration steps. The
above set of equations are solved at every time step with a specified number of iterations.
A computer program has been developed in this study that can perform this simulation.
4.4 Conclusions
This chapter introduced the gradient algorithm for simulation of gradually varied
unsteady flow problems in water distribution networks. Governing equations have been
derived in reference to a typical water distribution network (Figure 4.1). The gradient
algorithm effectively simplifies the analysis whereas the existing dynamic algorithms
remain fairly complicated with loop and flow equations that can be time consuming to
solve. Furthermore, this chapter represented the application of approximation for the
non-linear friction loss term in the head loss-flow equations. Direct application of the
friction loss equation (3.2.7.17) allows equation (4.3.4) to be solved directly rather than
applying the gradient method. Any future work with friction loss equations (3.2.7.18)
63
and (3.2.7.19) can be successfully implemented with the application of the gradient
method. In Chapter 7, results obtained from these approximations are in good agreement
with the existing models. Appropriate boundary conditions implemented in this model
are discussed in the next chapter.
64
CHAPTER 5
BOUNDARY CONDITIONS AND INTERNAL COMPONENTS
5.1 Introduction
In the previous chapter, the gradient algorithm was derived. For a complete
solution of these equations, the initial values of pipe flows, nodal heads, and the boundary
conditions (if any) throughout the transient analysis must be specified. More importantly
the boundary conditions of the dynamic model must be defined.
Dynamic boundary conditions are similar to steady-state boundary conditions ---
the difference being that they may vary with time. At the end of each time step during the
simulation, proper subroutines in the developed code update the possible changes in the
boundary conditions. Such changes may be the variation of water surface elevations in
tanks, the variations in the energy gained by internal components like pumps, etc. The
following subsections describe alternative boundary conditions and internal components
used in the dynamic model.
5.2 Boundary Conditions
5.2.1 Constant Nodal Demand
A constant nodal demand is the simplest of all the boundaries in a water
distribution network. The analyst is required to input the demand for each node in a
65
network. Thus, the proper execution of the mass balance equation in the model is
performed. These nodal demands are time invariant.
5.2.2 Variable Nodal Demand
It is often the case that nodal demands vary over a typical day depending on the
water usage. These cases can be studied effectively with the introduction of a functional
change in demands. The functional relationship with respect to time can be introduced
within a subroutine. Data input is thus not required for nodal demands described as
functions; the demands at different time steps are simply generated by the program. It is
up to the analyst to decide whether the demand function be linear, step-wise or of any
other mathematical form.
5.2.3 Constant-level Reservoir (FGN)
A constant-level reservoir is treated as in any steady-state model. Such a
reservoir, popularly known as the fixed grade node, is used as a starting point in the
computation of the pressures throughout a system. Pseudo-loops are used in the existing
loop equations for more than one fixed grade node. This study investigates head loss-
flow equations for each pipe element in a network and therefore, no matter how many
fixed grade nodes are present, pseudo-loops are never introduced in the dynamic
equations. The energy or hydraulic head at such reservoir locations remain time invariant
during a simulation period.
66
5.2.4 Variable-level Tank
More common in networks are tanks in which water level varies over time.
Variable-level tanks are treated similar to the constant-level reservoir boundary with a
few additional features. As depicted in Figure 5.1, the minimum and maximum
allowable head, and the tank diameter are physical data necessary to describe the tank.
The initial head is treated as a fixed grade node. The model will subtract or add volume,
and thus head, to the tank as dictated by the calculated flow rate through the connecting
pipe. The flow in the pipeline to the tank will be shut off if the tank empties or becomes
totally full. Whenever flow conditions allow, the tank will go back on-line. A subroutine
is introduced in the model that simulates water surface level in such tanks at the end of
each time step.
5.3 Internal Components
5.3.1 Minor Losses
Minor losses occur when fluid flow through appurtenances in pipelines. A minor
loss coefficient for components, typically valves, can be input for any pipe. The common
form for the relation representing minor losses is:
dhi = 1(„; Qm (5.3.1)
where
dhi = minor head loss in pipe i
K, = minor loss coefficient for valve(s) in pipe i
Q = flow rate in pipe i
m = minor loss exponent (typically 2)
67
D
y
Hmax
Hmin
D = Tank diameterHmax = Maximum allowable water surface elevationHmin = Minimum allowable water surface elevation
Figure 5.1. Variable Level Tank Setup
68
In this study, the minor loss constant is simply added to the friction-loss term in the head
loss-flow equations. The minor loss equation is of the same form as the friction loss
formula discussed in Chapter 3.
5.3.2 Check Valves
Check valves act in the same manner as in the existing steady-state analyses and
EPS. They allow pipe flow to occur in only one direction. These valves may switch
between open and closed condition during a temporal simulation. Valve shut-off has not
been coded in this study.
5.3.3 Pumps
A pump adds energy to water and can be included in any line of the pipe system.
The characteristic of a pump can be described from pump curve data (Wood 1980). An
exponential curve can be fit to this data to obtain a pump characteristic curve of the form:
Ep = h, - CV (5.3.2)
Following Wood (1980), the developed computer program determines the coefficient C
and exponent n for this curve from the pump shut-off head, h1, and two additional points
of operating data (head-discharge) input for this purpose. Figure 5.2 depicts this
representation. The data points are shown along with the curve of the form of Equation
(5.3.2) which passes through these data points. The procedure for determining C and n is
illustrated on the next page with an example. Pump shut-off has not been coded in this
study.
Ep = A - SQConstant Slope
QQI
Shut-off Head (hl)
t
i
t
' Normal Operating Range 1g1
g
1
t1
Figure 5.2. Pump Operation Characteristic Curve
69
70
EXAMPLE OF OBTAINING A CHARACTERISTIC
PUMP CURVE FROM PERFORMANCE DATA (taken from Wood, 1980)
A pump head (Ed, flow rate (Q) relation is used of the form:
Ep = h, - CQ" (a)
where C and n are determined by passing the curve through (13.2 , Q2) and (n3 , Q3).
The constants are computed as
n = log [(h 1 -h3)/(h 1 -h2)] / log (Q 3/Q2) (b)
and C= (h, -112) / Q211 (C )
This expression holds in range 0 Q Q3. Above Q3 the characteristic is extended at aconstant slope (as shown in Fig. 4.2) equal to the slope of Equation (a) evaluated at Q =Q3*
For Q> Q3 the characteristic has the form:
Ep =A+SQ (d)
where
S= - nCQ,' (e)
A = h3 - S Q3 (f)
An example of this characteristic follows for the data below:
Shut-off head, 11 1 = 200 ft.
Second point: h, = 180 ft. Q2 = 2 cfs
Third point: h3 = 150 ft. Q3 = 4 cfs
From Equation (b), n = 1.322 and from Equation (c ), C = 8.000
Therefore,
= 200 - 8 Q' 322 for 0 � Q � 4 cfs
For the range Q > 4 cfs, the following holds. From Equation (e),
S = - 16.53 and from Equation (f), A = 216
Thus, ;= 216- 16.53 Q for Q >4 cfs
71
The exponential relation given in Equation (5.3.2) represents a pump very well
between zero flow and the third data point (h3 , Q3) but is not suitable for negative flows
and may not be suitable for flows in excess of Q3. In order to satisfactorily handle this,
the computer program is designed to do the following:
1. A normal pump design is to include a check valve downstream of a pump. The check
valve closes if pressure conditions cause flow reversal through the pipe. In these
conditions the pump is not adequate to overcome the system grade caused by other
factors, and the pump as described will be unable to operate.
2. If the solution indicates that the pump is operating at a flow rate above the third
operating point (Q,) then it operates on a straight line described by Ep = A + S Q
where the slope is the same as given by Equation (5.8.1) at Q = Q3 . The computer
program determines the values of A and S as described on the previous page. This
representation gives a characteristic which is more realistic than Equation (5.8.1) for
Q > Q3*
5.4 Conclusions
In summary, all of the network boundary conditions and internal components in
this dynamic model are in the same form as those in the existing steady-state models.
The basic difference is the possible time variation of demands, pump operation, and water
surface elevations in variable-level tanks. The boundary conditions are introduced in
subroutine forms in the developed computer program. This provides flexibility in
72
possible future updates of the model. Moreover, it is easy to understand the basic theory
behind such special conditions in hydraulic water distribution networks.
73
CHAPTER 6
DEVELOPMENT OF THE NETWORK MODEL
6.1 Introduction
The objective of this study is the development of an efficient, reliable algorithm
for the analysis of complex water distribution networks in the gradually varied transient
state. This goal is accomplished in three stages:
1. Derivation of mathematical model,
2. Development of a procedure to solve the adopted mathematical model, and
3. Selection of a convenient representation for the input data needed by the
model.
Chapters 3 through 5 developed the steady-state and dynamic gradient algorithm
for fluid flow in closed conduits, and the equations describing the typical network
boundary conditions. Thus, these three chapters have satisfied the first stage of the
computer program development with the introduction of a full mathematical description
of a hydraulic network. In addition, a recursive Newton-Raphson algorithm presented in
Chapter 3 fulfills the requirement for the second stage of the development.
To the author's knowledge, the use of the gradient algorithm for dynamic analysis
of water distribution networks has not been previously completed. Thus, the main
74
concern of the third stage in development is the formulation of procedures and data input
structure that will allow the solutions developed in the first two stages to be efficiently
implemented.
The program must be general --- with the ability to solve a wide variety of
systems. The organization of the input data should allow the calculation of the final
solution and still keep the input requirement of the program at a bare minimum. The
implementation of the solution procedures previously presented is described in this
chapter.
6.2 Network Representation
After reliability, the most important function of computer representation of water
distribution networks is for it to accurately describe the hydraulic system while requiring
a minimal amount of input data. Unlike network water-hammer analysis, the model
developed in this study requires no more input data than that required by the available
steady-state models. To simulate the system, the following information must be supplied
to the model:
1. System data:
a. Option for dynamic or steady state analysis
b. Flow units (SI or English), determined by the gravitational constant used
c. Number of pipes
75
d. Number of nodes
e. Option to print out all calculations or just the summary of nodal heads and
pipe flows
f. Option to change the maximum number of iterations allowed
g. Option to change the relative accuracy for convergence (default value is
0.0001)
h. Option to use Darcy-Weisbach or Hazen-Williams friction formula
i. Option to allow non-consecutive nodal numbering
2. Pipe line data:
a. Check valve
b. Upstream node number
c. Downstream node number
d. Line length
e. Inside pipe diameter
f. Friction coefficient (Hazen-Williams or Darcy-Weisbach)
g. Total grade head for pipes connecting to a fixed grade node
h. Pipe number as required by non-consecutive numbering
3. Node (junction) data:
a. External demand (+ outflow, - inflow)
b. Elevation
c. Node number
4. Dynamic data:
a. Total analysis time (sec), and
b. Time period for calculations (At)
5. Pump data (if any):
a. Number of pumps in the network
b. Pipe number in which pump is located
c. Shut-off head
d. Two points (h,Q) on operating curve
6. Variable tank data (if any):
a. Number of tanks in the network
b. Pipe number connecting to variable level tank
c. Maximum allowable water surface elevation
d. Minimum allowable water surface elevation
e. Tank diameter
76
77
7. Valve data:
a. Number of valves in the network
b. Pipe number in which valve is located
c. Valve coefficient (KJ
8. Demand function (if applicable):
Supplied in the program subroutine.
6.3 Model Formulation
The gradient algorithm developed by Todini (1987) was adopted for the initial
steady-state analysis. Ease of model set up (pipe connections), and smaller number of
equations with apparent stability (Todini 1987) for steady-state were the two reasons for
adopting the algorithm. Redundant work was saved by coupling the dynamic model with
this steady-state model. The coding for the steady-state analysis was also programmed
(in FORTRAN 77) by the author. The input data is read into the program via the normal
input statements. After the initial values of nodal heads (h) and flow rates in pipe (Q) are
obtained, the analysis is transferred to the dynamic algorithm. After performing a
specified number of iterations per time step the results are sent to an output file. A
stopping criteria based on desired tolerance is taken into account
The stopping criteria and program FLOWCHARTS are presented in following
pages.
78
6.4 Stopping Criteria
The following convergence scheme is used as a stopping criteria:
t-1-itQ
t+t _ t-(4.4-11 ÷ Qtk+,At > CLOSR (6.4.1)
where
Q t = Flow at previous iteration step at time t+ At
rit+At = Flow at current iteration step at time t+ AtV k '+1
CLOSR ---- desired tolerance (default is 0.0001)
Iterations will be performed as long as the above relation is true. Otherwise, the program
will terminate with the indication of convergence.
• This program simulates water distribution systems using the Gradient• Algorithm. The program setup is analogous to the governing Newton-* Raphson Recursive Algorithm presented in Chapter 3.
• The program reads in data from an Input File (free format) and writes• results in an Output File (formatted). Both Input and Output file• names are to be supplied by the user at the execution of the program.
Major computations for HEAD values are performed within programSubroutine MULTMX. Calculated Head values are used to obtain the
• FLOWS in Subroutine FLOWCALC. Intermediate subroutines perform minormatrix calculations. FLOWCHART is included in Chapter 5.
While the program has been succesfully compiled and run on MicrosoftFORTRAN Powerstation, the author accepts no responsibility for itscorrectness.
PROGRAM NETTINTEGER 1(48), J(48), X, Y, W, NE, NN, NR, NC, CNR, CNC, DNR, DNCINTEGER ENR, ENC, FNR, FNCREAL D(48),L(48),C(48),K0(48),H(30),Q(48),AR(48),Z(30),QD(30)REAL HFGN(48),M,GR,PI,FR(48)DIMENSION A(48,48),B(48,48),FLOW(48,1),DEM(30,1),ICON(48,30)DIMENSION AOH0(48,1),ICONT(30,48)PARAMETER (GR=32.2, PI=3.142)CHARACTER*20 FNAMEIN, FNAMOUTWRITE(*, 1 (AW) 'ENTER THE NAME OF YOUR INPUT FILE :'READ(*,! (A)') FNAMEINOPEN(5,FILE=FNAMEIN, STATUS='OLD')WRITE(*, 1 (AW) 1 ENTER NAME OF YOUR OUTPUT FILE :'READ(*,'(A)') FNAMOUTOPEN(6,FILE=FNAMOUT, STATUS='NEW',FORM='FORMATTED')CALL INPUT(I,J,D,L,C,KO,H,Q,Z,QD,AR,HEGN,NE,NN,GR,M,FR,PI)INITIALIZE VARIABLES FOR THE FOLLOWING SUBROUTINESFNC = 1NR = NENC = NNCNR=NECNC=NNDNR=NNDNC=NEENR=NNENC=NNFNR=NECALL SETUPF(I,J,X,Y,W,NE,NN,QD,HFGN,B,DEM,ICON,ICONT,
1317 FORMAT(///10X,'NUMBER OF ELEMENTS :',I3,/,10X,*'NUMBER OF NODES : 1 ,I3,/,10X,'NUMBER OF FGN :',I3,/,10X,*'EXPONENT FOR FRICTION FORMULA, M :',F7.5,////)
READ DATA OF ELEMENTSREAD (5,*) (I(II), II=1,NE)READ (5,*) (J(II), II=1,NE)READ (5,*) (D(II), II=1,NE)READ (5,*) (L(II), II=1,NE)READ (5,*) (C(II), II=1,NE)READ (5,*) (FR(II), II=1,NE)DO 1890 II = 1, NE
AR(II) = (PI/4.)*(D(II)**2.)1890 CONTINUE
WRITE (6,1891)1891 FORMAT(//16X,'ELEMENT I J D(FT) L(FT) AR(FT^2) FR',/,15X,
*46(1H-))DO 1885 II=1,NEWRITE(6,1893) II, I(II), J(II), D(II), L(II), AR(II),FR(II)
2221 CONTINUE2220 FORMAT (22X,I3,3X,F8.2,5X,F8.2,5X,F7.3)
READ Q, HFGN & SET UP 'KO'=CONSTANT IN D-W OR H-W FRICTION FORMULA]READ (5,*) (Q(II), II=1,NE)READ (5,*) (HFGN(II), II=1,NE)DO 2222 II = 1, NEIF (M .EQ. 1.852) THEN
WRITE (6, *)WRITE (6, *) 'MATRIX [A(12)] OR [ICON] :'DO 50 II = 1, NEWRITE (6,1826) (ICON(II,Y), Y = 1,NN)
1826 FORMAT(/1X,12(I2,X))50 CONTINUE
DEVELOP TRANSPOSE MATRIX [A(21)] OR [ICONT(NN*NE)]DO 46 X = 1, NNW=XDO 47 Y = 1, NEII = Y
136
ICONT (X,Y) = ICON (II,W)47 CONTINUE46 CONTINUE
WRITE OUT MATRIX A(21)WRITE (6, *)WRITE (6, *) 'MATRIX [A(21)] OR [ICONT]DO 55 X = 1,NNWRITE (6,1828) (ICONT(X,Y), Y = 1,NE)
1828 FORMAT(/1X,12(I2,X))55 CONTINUE
DEVELOP MATRIX A(10)*H(0): NE*1& WRITE OUT
WRITE (6, *)WRITE (6, *) 'MATRIX [A(10)H(0)] OR [AOHO]Y= 1DO 65 II = 1, NE
AOHO(II,Y) = -HFGN(II)WRITE (6,1830) AOHO(II,Y)
1830 FORMAT(/1X,(F8.2))65 CONTINUE
RETURNEND
************************************************************************• Subroutine SOLVE• This subprogram sets up the iteration# and checks convergence• criteria by calling on Subroutine SETALL.************************************************************************
INTEGER NE,NN,FNR,FNC,I,J,X,Y,ENR,ENCDIMENSION F(FNR,FNC),FLOW(NE,FNC),G(NE,FNC),E(ENR,ENC),B(NE,NE)DIMENSION A(NE,NE),ICON(NE,NN),AOHO(NE,FNC),QRES(NE,FNC)J= 1DO 85 I = 1,NE
G(I,J) = F(I,J) + FLOW(I,J)85 CONTINUE
WRITE OUT RESULT OF [F(FNR,FNC) + FLOW(NE,1)]WRITE (6, *)WRITE (6, *) 'RESULT OF [F(FNR,FNC) + FLOW(NE,FNC)]Y= 1DO 90 X = 1, NEWRITE (6, 1004) G(X,Y)
1004 FORMAT (/1X, (F9.5))90 CONTINUE
RETURNEND
***********************************************************************• Subroutine SUBMX• Intermediate subprogram to perform minor matrix algebra.***********************************************************************
24 continuewrite out the result of inverted matrix 'e'
write (6, *)write (6, *) 'result of inverted matrix Edo 25 x = 1, enrwrite (6,1009) (e(x,y), y = 1, enc)
1009 format (/x,7(f10.4,2x))25 continue
write out the result of [H(k+1)] stored in matrix 'r'write (6, *)write (6, *) 'result of [H(k+1)] stored in matrix (r) by GAUSSJy= 1do 26 x = 1, nnwrite (6, 1010) r(x,y)
1010 format (./lx, (f12.5))26 continue
multiply the result of [H(k+1)] by -1 to take care of sign(s)in the original formulation(s)
j = 1do 27 i = 1, nn
RESH(i,j) = S*r(i,j)27 continue
write out the result of [H(k+1)] with proper sign, stored inmatrix RESH(nn,fnc)
write (6, *)write (6, *)'result of [H(k+1)] with proper sign; RESH(nn,fnc):y= 1do 28 x = 1, nnwrite (6, 1011) RESH(x,y)
CALCULATE [N^-1*A11^-1]*[Al2*H(K+1)+A(10)H(0)] & STORE IN [AHB(NE,FNC)]
IF TEST TO DO THE ABOVE MULTIPLICATIONIF (NE .NE. NE) ERROR = .TRUE.IF (NE .NE. NE) ERROR = .TRUE.IF (FNC .NE. FNC) ERROR = .TRUE.IF (.NOT. ERROR) THENDO 235 I = 1, NE
*****************************************************************************• Program NETT (Unsteady-State)
• This program simulates water distribution systems using the Gradient• Algorithm. The program setup is analogous to the governing Newton-• Raphson Recursive Algorithm presented in Chapter 4.
• The program reads in data form an Input File (free fromat) and writes *• results in an Output File (formatted). Both Input and Output file• names are to be supplied by the user at the execution of the program. **• Major computations for HEAD values are performed within program• Subroutine MULTMX. Calculated Head values are used to obtain the• FLOWS in Subroutine FLOWCALC. Intermediate subroutines perform minor *• matrix calculations. FLOWCHART is included in Chapter 5.
• While the program has been succesfully compiled and run on Microsoft *• FORTRAN Powerstation, the author accepts no responsibility for its• correctness.
PROGRAM NETTINTEGER I(12), J(12),X,Y,W,NE,NN,NR,NC,CNR,CNC,DNR,DNCINTEGER ENR,ENC,FNR,FNC,DT,I0(12),JI(12),TMAX,TS,INCR,NNNINTEGER NEL,ADJN(1),SLN(1),FGNN(1),NVL,PLN,PSLN(1)REAL D(12),L(12),C(12),K0(12),H(7),Q(12),Z(7),QD(7),HFGN(9),M,GRREAL AR(12),FR,PI,HMAX(1),HMIN(1),TD(1),TA(1),WSEL(1),KV(12)REAL H1(1),H2(1),H3(1),Q2(1),Q3(1),N1(1),C1(1),A1(1),S1(1),EP(1)DIMENSION A(12,12),B(12,12),FLOW(12,1),DEM(7,1),ICON(12,7)DIMENSION AOH0(12,1),ICONT(7,12),HED(9,1),INC(12,9),ICONI(12,7)DIMENSION ICONTI(7,12),HP(7,1),QAG(12,1),QRES(12,1),RESH(7,1)PARAMETER (GR=9.81, PI=3.142)CHARACTER*20 FNAMEIN, FNAMEOUTWRITE(*,'(AW) 'ENTER THE NAME OF YOUR INPUT FILE :'READ(*,'(A) 1 ) FNAMEINOPEN(5,FILE=FNAMEIN, STATUS='OLD')WRITE(*, 1 (AW)'ENTER NAME OF YOUR OUTPUT'FILE .16READ(*,'(A)') FNAMEOUTOPEN(6,FILE=FNAMEOUT, STACUS :='NEW,FORM='FORMATTED')
'IO' & 'JI' ARE USE6 TO SET UP R.H.S. INCIDENCE MATRIX [INC(NE,NE)]CALL INPUT(I,J,D,L,C,KO,H,Q,Z,QD,HFGN,NE,NN,DT,AR,I0,JI,TMAX,PI,M,
1317 FORMAT(///10X,'NUMBER OF ELEMENTS :',I3,/,10X,*'NUMBER OF NODES :',I3,/,10X,'NUMBER OF FGN :',I3,/,10X,*'DT :',I4,/,10X,'TMAX :',I6,/,10X,'EXPONENT FOR FRICTION FORMULA,* M :',F7.5,/,10X,'DARCY-WEISBACH FRICTION FACTOR, FR :',F7.5,/,* lox, 'NUMBER OF ELEMENTS CONNECTED TO VARIABLE HEAD TANKS, NEL :',* I3,/,10X,'NUMBER OF VALVES IN THE NETWORK, NVL :',I3,/,*10X,'NUMBER OF PUMPS IN THE NETWORK, PLN :',I3,////)
READ DATA OF ELEMENTSREAD (5,*) (I(II), II=1,NE)READ (5„*) (J(II), II=1,NE)READ (5,*) (D(II), II=1,NE)READ (5,*) (L(II), II=1,NE)READ (5,*) (C(II), II=1,NE)DO 1890 II = 1, NE
AR(II) = (PI/4.)*(D(II)**2)1890 CONTINUE
WRITE (6,1891)1891 FORMAT(//16X,'ELEMENT I J D(FT) L(FT) C AR(FTA2)',/,15X,
READ PARAMETERS FOR VALVES IN THE NETWORKREAD "ONLY IF" THERE ARE VALVES IN THE NETWORK PIPES
IF (NVL .NE. 0) THEN
149
READ (5,*) (KV(II), II = 1, NE)WRITE OUT PARAMETERS FOR VALVES IN THE NETWORK PIPES
WRITE (6,130)130 FORMAT (//10X, 'ELEM# KV',/,8X,20(1H-))
DO 16 II = 1, NEWRITE (6,140) II,KV(II)
16 CONTINUE140 FORMAT (12X,I3,3X,F5.2)
END IFREAD PARAMETERS FOR VALVES IN THE NETWORKREAD "ONLY IF" THERE ARE VALVES IN THE NETWORK PIPES
IF (PLN .NE. 0) THENREAD (5,*) (PSLN(II), II = 1, PLN)READ (5,*) (H1(II), II = 1, PLN)READ (5,*) (H2(II), II = 1, PLN)READ (5,*) (H3(II), II = 1, PLN)READ (5,*) (Q2(II), II = 1, PLN)READ (5,*) (Q3(II), II = 1, PLN)
WRITE OUT PARAMETERS FOR PUMPS IN THE NETWORK PIPESWRITE (6,150)
150 FORMAT (//10X,'PUMP# PSLN H1 H2 H3 Q2 Q3',/,8X,46(1H-))DO 18 II = 1, PLNWRITE (6,160) II, PSLN(II), 51(II), H2(II), H3(II), Q2(II), Q3(II)
18 CONTINUE160 FORMAT ( 12X, 13,1X,13,1X,F8.2,1X,F8.2,1X,F8.2,1X,F8.2,1X,F8.2)
END IFRETURNEND
************************************************************************Subroutine SETUPFThis subprogarm sets up the matrices that remain fixed over time.*
74 CONTINUEDO 75 X = 1, NNW=XDO 76 Y = 1, NEII = YICONTI (X,Y) = ICONI (II,W)
76 CONTINUE75 CONTINUE
RETURNEND
***********************************************************************• Subroutine DEMF• This subprogram sets up the user-defined Demand Function.• A sample Step Demand Function is presented here.*************************************************************** * *******
• Subroutine MULTMX• This subprogram performs necessary computations to obtain HEAD *• values. It calls on Subroutine GAUSSJ for matrix inversion.************************************************************************
WRITE OUT RESULT OF [A(11)^-1 * A(12)] OR [A * ICON]WRITE (6, *)WRITE (6, *) 'RESULT OF [A(11) ^- 1 * A(12)] OR [A*ICON]DO 20 X = 1,CNRWRITE (6, 1000) (C(X,Y), Y = 1, CNC)
1000 FORMAT (/1X,12(F6.4,2X))20 CONTINUE
IF TEST TO MULTIPLY [ICONT(NN,NE)*B(NE,NE)]IF (NR .NE. NE) ERROR = .TRUE.IF (NC .NE. DNR) ERROR = .TRUE.IF (NE .NE. DNC) ERROR = .TRUE.IF (.NOT. ERROR) THENDO 25 I - 1, DNR
***********************************************************************• Subroutine ADDMX• Intermediate subprogram to perform minor matrix algebra.***********************************************************************
INTEGER NE,NN,FNR,FNC,I,J,X,Y,ENR,ENCDIMENSION F(FNR,FNC),FLOW(NE,FNC),G(NE,FNC),E(ENR,ENC),B(NE,NE)DIMENSION A(NE,NE),ICON(NE,NN),AOHO(NE,FNC),QRES(NE,FNC)DIMENSION RESH(NN,FNC)J= 1DO 85 I = 1,NE
G(I,J) = F(I,J) + FLOW(I,J)85 CONTINUE
WRITE OUT RESULT OF [F(FNR,FNC) + FLOW(NE,1)]WRITE (6, *)WRITE (6, *) 'RESULT OF [F(FNR,FNC) + FLOW(NE,FNC)]Y= 1DO 90 X = 1, NEWRITE (6, 1004) G(X,Y)
1004 FORMAT (/1X,(F9.5))90 CONTINUE
RETURNEND
***********************************************************************• Subroutine SUBMX• Intermediate subprogram to perform minor matrix algebra.***********************************************************************
INTEGER NN,FNC,I,J,X,Y,ENR,ENC,NEDIMENSION DEM(NN,FNC), P(NN,FNC), PP(NN,FNC),H(NN,FNC),E(ENR,ENC)DIMENSION B(NE,NE),FLOW(NE,FNC),A(NE,NE),ICON(NE,NN),AOHO(NE,FNC)DIMENSION QRES(NE,FNC),RESH(NN,FNC)J= 1DO 135 I = 1, NN
PP(I,J) = DEM(I,J) - P(I,J)135 CONTINUE
WRITE OUT RESULT OF [q - (A(21)*Q(K)] OR [DEM(NN,FNC)-P(NN,FNC)]WRITE (6, *)WRITE (6, *) 'RESULT OF [DEM(NN,FNC)-P(NN,FNC)]Y= 1DO 140 X = 1, NNWRITE (6, 1007) PP(X,Y)
1007 FORMAT (/1X, (F7.3)140 CONTINUE***********************************************************************
***********************************************************************• Subroutine ADDMXII• Intermediate subprogram to perform minor matrix algebra.***********************************************************************
24 continuewrite out the result of inverted matrix 'e'
write (6, *)write (6, *) 'result of inverted matrix Edo 25 x = 1, enrwrite (6,1009) (e(x,y), y = 1, enc)
1009 format (/x,7(f10.4,2x))25 continue
write out the result of [H(k+1)] stored in matrix 'r'write (6, *)write (6, *) 'result of [H(k+1)] stored in matrix (r) by GAUSSJy= 1do 26 x = 1, nnwrite (6, 1010) r(x,y)
1010 format (/lx, (f12.5))26 continue
multiply the result of [H(k+1)] by -1 to take care of sign(s)in the original formulation(s)
j = 1do 27 i = 1, nn
RESH(i,j) = S*r(i,j)27 continue
write out the result of [H(k+1)] with proper sign, stored in
matrix RESH(nn,fnc)write (6, *)write (6, *)'result of [H(k+1)] with proper sign; RESH(nn,fnc):y = 1do 28 x = 1, nnwrite (6, 1011) RESH(x.Y)
' STORED IN [AHB(NE,FNC)]Y= 1DO 250 X = 1, NEWRITE (6, 1017) AHB(X,Y)
1017 FORMAT (/1X, (F10.5)250 CONTINUE
CALCULATE [Q(K+1)] = [BFQ(NE,FNC) - AHB(NE,FNC)] & STORE IN[QRES(NE,FNC)]
J= 1DO 255 I = 1, NE
QRES(I,J) = BFQ(I,J) - AHB(I,J)255 CONTINUE
WRITE OUT THE RESULTWRITE (6, *)WRITE (6, *) 'RESULT OF [Q(K+1)]Y= 1DO 260 X = 1, NEWRITE (6, 1018) QRES(X,Y)
1018 FORMAT (/1X, (F10.5)260 CONTINUE
RETURNEND
***********************************************************************• Subroutine TLEVEL• This subprogram updates WSEL in tank at the end of every time• step.***********************************************************************
100 FORMAT (//10X,'ELEM# WSEL',/,8X,10(1H-))DO 12 II = 1,NELWRITE (6,110) II, WSEL(II)
12 CONTINUE110 FORMAT (10X,I3,3X,F8.2)
RETURNEND
***********************************************************************• Subroutine PUMP• This subprogram calculates Pump Head at the end of each time• step.***********************************************************************
+10% D h 1493.457 1330 1278.7hL 6.543 170.014 221.302
-10% D h 1493.457 1237.23 1193hL 6.543 262.77 307.008
+50% L h 1493.457 1253.85 1198.11hL 6.543 246.15 301.891
-50% L h 1493.457 1340.39 1304.28hL 6.543 159.613 195.718
+25% f h 1493.457 1276.63 1230hL 6.543 223.372 269.911
-25% f h 1493.457 1318 1264.62hL 6.543 182.3 235.373
+25%q h 1493.457 1188.95 1141.82hL 6.543 311.053 358.176
-25% q h 1493.457 1362.02 1356.73hL 6.543 137.978 143.27
Dimension in feetResults obtained at t = 1 hourh = piezometric headhL = head Loss = FGNh - h = 1500 - h
184
REFERENCES
Al-Layla, M. A., et. al., Water Supply Engineering Design, Ann Arbor Science, AnnArbor, Michigan, 1977.
Cabrera, E., and Vela, A. F., ed., Improving Efficiency and Reliability in WaterDistribution Systems, Kluwer Academic Publishers, London, 1994.
Chaudhry, M. H., Applied Hydraulic Transients, Van Nostrand Rienhold, New York,1979.
Epp, R., and Fowler, A. G., "Efficient Code for Steady-State Flows in Networks,"Journal of the Hydraulics Division, ASCE, Vol. 96, No. HY1, January,1970, pp. 43-56.
Etter, D. M., Structured FORTRAN 77 for Engineers and Scientists,Benjamin/Cummings, New York, 1993.
Fox, J. A., Hydraulic Analysis of Unsteady Flow in Pipe Networks, MacMillan Press,London, 1977.
Gerald, C. F., and Wheatley, P. 0., Applied Numerical Analysis, Addison-Wesley, NewYork, 1994.
Holloway, M. B., "Dynamic Pipe Network Computer Model," Ph.D. dissertation(Civil Engineering), Washington State University, Pullman, Washington,1985.
Holloway, M. B., Chaudhry, M. H., "Comparison of Pipe-Network-AnalysisComputer Programs," Proceedings Volume 2, 6th Canadian Hydro-technical Conference, the Canadian Society for Civil Engineering, June,1983, Ottawa, Ontario, Canada, pp. 543-558.
Jeppson, R. W., Analysis of Flow in Pipe Networks, Ann Arbor Science, Ann Arbor,Michigan, 1976.
Jeppson, R. W., "User's Manual---Pipe Network Simulation Analysis ComputerProgram---NETWK," Civil and Environmental Engineering, Utah Water
185
186
Research Laboratory, Utah State University, Logan, Utah, 1980.
Koffman, E. B., and Friedman, F. L., Problem Solving and Structured Programming inFORTRAN 77, Addison-Wesley, New York, 1990.
Mohtar, R. H., Bralts, V. F., and Shayya, W. H., "A Finite Element Model forthe Analysis and Optimization of Pipe Networks," Soil and WaterDiv. of ASAE, Vol. 34, No. 2, March-April, 1991, pp. 393-401.
Onizuka, K., "System Dynamics Approach to Pipe Network Analysis," Journalof Hydraulic Engineering, ASCE, Vol. 112, No. 8, August, 1986.
Prasuhn, A. L., Fundamentals of Hydraulic Engineering, Saunders HBJ, New York,1987.
Press, W. H., et. al., Numerical Recipes (FORTRAN), Cambridge University Press,Cambridge, 1989.
Roberson, J. A., and Crowe, C. T., Engineering Fluid Mechanics, Houghton Mifflin,New Jersey, 1993.
Salgado, R., Todini, E., and O'Connel, P. E., "Comparison of the Gradient Method withsome Traditional Methods for the Analysis of Water Supply DistributionNetworks," International Conference on Computer Applications for Water Supplyand Distribution 1987, Leicester Poly-technic, U.K., 8-10 September, 1987.
Sevuk, S., "Steady and Unsteady flow Simulation in Pipe Networks," EngineeringSoftware Conference 1979, Southampton, U.K., 4-6 September, 1979.
Streeter, V. L., "Water Hammer Analysis of Distribution Systems," Journal of theHydraulics Division, ASCE, Vol. 93, No. HY5, September, 1967, pp. 185-201.
Todini, E., Pilati, S., "A Gradient Method for the Analysis of Pipe Networks,"International Conference on Computer Applications for Water Supply andDistribution 1987, Leicester Polytechnic, U.K., 8-10 September, 1987.
Walski, T. M., Analysis of Water Distribution Systems, Van Nostrand Reinhold, NewYork, 1984.
Wood, D. J., Funk, J. E., and Boulos, P. F., "Pipe Network Transients---Distributed andLumped Parameter Modeling," Pressure Surges—Proceedings of the 6thInternational Conference 1990, Bedford, U.K., 1990, pp. 131-142.
Wood, D. J., "Algorithms for Pipe Network Analysis and Their Reliability,"Research Report No. 127, Water Resources Research Institute, Universityof Kentucky, Lexington, Kentucky, 1981.
Wood, D. J., "User's Manual---Computer Analysis of Flow in Pipe Networks IncludingExtended Period Simulations," Department of Civil Engineering, University ofKentucky, Lexington, Kentucky, 1980.
Wood, D. J., Charles, C. O. A., "Hydraulic Network Analysis Using Linear Theory,"Journal of the Hydraulics Division, ASCE, Vol. 98, No. HY7,July, 1972, pp. 1157-1170.
Wylie, E. B., "The Microcomputer and Pipeline Transients," Journal of theHydraulics Division, ASCE, Vol. 109, No. HY12, December, 1983, pp.1723-1739.
