Technical Reference WaterCAD V8 XM

7/27/2019 Technical Reference WaterCAD V8 XM

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Technical Reference

Pressure Network Hydraulics

In practice, pipe networks consist not only of pipes but of miscellaneous fittings, services,

storage tanks and reservoirs, meters, regulating valves, pumps, and electronic and

mechanical controls.

Network Hydraulics Theory

For modeling purposes, these system elements are organized into the following

categories:

PipesTransport water from one location (or node) to another. Junctions/NodesSpecific points, or nodes, in the system at which an event of

interest is occurring. This includes points where pipes intersect, where there are

major demands on the system such as a large industry, a cluster of houses, or a

fire hydrant, or critical points in the system where pressures are important foranalysis purposes.

Reservoirs and TanksBoundary nodes with a known hydraulic grade thatdefine the initial hydraulic grades for any computational cycle. They form thebaseline hydraulic constraints used to determine the condition of all other nodes

during system operation. Boundary nodes are elements such as tanks, reservoirs,

and pressure sources. PumpsRepresented as nodes. Their purpose is to provide energy to the system

and raise the water pressure.

ValvesMechanical devices used to stop or control the flow through a pipe, or tocontrol the pressure in the pipe upstream or downstream of the valve. They resultin a loss of energy in the system.

An event or condition at one point in the system can affect all other parts of the system.

While this complicates the approach that the engineer must take to find a solution, there

are some governing principles that drive the behavior of the network, including theConservation of Mass and Energy Principle, and the Energy Principle.

The two modes of analysis are Steady-State Network Hydraulics and Extended Period

Simulation. This program solves for the distributions of flows and hydraulic grades using

the Gradient Algorithm.


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The Energy Principle

The first law of thermodynamics states that for any given system, the change in energy is

equal to the difference between the heat transferred to the system and the work done bythe system on its surroundings during a given time interval.

The energy referred to in this principle represents the total energy of the system minus

the sum of the potential, kinetic, and internal (molecular) forms of energy, such as

electrical and chemical energy. The internal energy changes are commonly disregarded inwater distribution analysis because of their relatively small magnitude.

In hydraulic applications, energy is often represented as energy per unit weight, resulting

in units of length. Using these length equivalents gives engineers a better feel for the

resulting behavior of the system. When using these length equivalents, the state of thesystem is expressed in terms of head. The energy at any point within a hydraulic system

is often represented in three parts:

Pressure Head: p/

Elevation Head: z

Velocity Head: V2/2g

Where: p = Pressure (N/m2, lb./ft.

2)

= Specific weight (N/m3, lb./ft.

3)

z = Elevation (m, ft.)

V = Velocity (m/s, ft./sec.)

g = Gravitational acceleration constant (m/s2, ft./sec.

2)

These quantities can be used to express the headloss or head gain between two locations

using the energy equation.

The Energy Equation

In addition to pressure head, elevation head, and velocity head, there may also be head

added to the system, by a pump for instance, and head removed from the system due to

friction. These changes in head are referred to as head gains and headlosses, respectively.Balancing the energy across two points in the system, you then obtain the energy

equation:

Where: p = Pressure (N/m , lb./ft. )


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g = Specific weight (N/m3, lb./ft.

3)

z = Elevation at the centroid (m, ft.)

V = Velocity (m/s, ft./sec.)


2)

hp = Head gain from a pump (m, ft.)hL = Combined headloss (m, ft.)

The components of the energy equation can be combined to express two useful quantities,

which are the hydraulic grade and the energy grade.

Hydraulic and Energy Grades

Hydraulic Grade

The hydraulic grade is the sum of the pressure head (p/g) and elevation head (z). Thehydraulic head represents the height to which a water column would rise in a piezometer.

The plot of the hydraulic grade in a profile is often referred to as the hydraulic grade line,

or HGL.

Energy Grade

The energy grade is the sum of the hydraulic grade and the velocity head (V2/2g). This is

the height to which a column of water would rise in a pitot tube. The plot of the hydraulicgrade in a profile is often referred to as the energy grade line, or EGL. At a lake or

reservoir, where the velocity is essentially zero, the EGL is equal to the HGL, as can be

seen in the following diagram.


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Conservation of Mass and Energy

Conservation of Mass

At any node in a system containing incompressible fluid, the total volumetric or massflows in must equal the flows out, less the change in storage. Separating these into flows

from connecting pipes, demands, and storage, you obtain:

Where: QIN = Total flow into the node (m3/s, cfs)

QOUT = Total demand at the node (m3/s, cfs)

VS = Change in storage volume (m3, ft.

3)

t = Change in time (s)

Conservation of Energy

The conservation of energy principle states that the headlosses through the system must

balance at each point. For pressure networks, this means that the total headloss between

any two nodes in the system must be the same regardless of what path is taken betweenthe two points. The headloss must be sign consistent with the assumed flow direction

(i.e., gain head when proceeding opposite the flow direction and lose head when

proceeding in the flow direction).

The same basic principle can be applied to any path between two points. As shown in the

figure above, the combined headloss around a loop must equal zero in order to achieve

the same hydraulic grade as at the beginning.


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The Gradient Algorithm

The gradient algorithm for the solution of pipe networks is formulated upon the full set of

system equations that model both heads and flows. Since both continuity and energy arebalanced and solved with each iteration, the method is theoretically guaranteed to deliver

the same level of accuracy observed and expected in other well-known algorithms suchas the Simultaneous Path Adjustment Method (Fowler) and the Linear Theory Method(Wood).

In addition, there are a number of other advantages that this method has over other

algorithms for the solution of pipe network systems:

The method can directly solve both looped and partly branched networks. Thisgives it a computational advantage over some loop-based algorithms, such asSimultaneous Path, which require the reformulation of the network into

equivalent looped networks or pseudo-loops.

Using the method avoids the post-computation step of loop and path definition,which adds significantly to the overhead of system computation.

The method is numerically stable when the system becomes disconnected bycheck valves, pressure regulating valves, or modeler's error. The loop and pathmethods fail in these situations.

The structure of the generated system of equations allows the use of extremelyfast and reliable sparse matrix solvers.

The derivation of the Gradient Algorithm starts with two matrices and ends as a workingsystem of equations.

Derivation of the Gradient Algorithm

Given a network defined by N unknown head nodes, P links of unknown flow, and Bboundary or fixed head nodes, the network topology can be expressed in two incidence

matrices:

A12 = A21 (P x N) Unknown head nodes incidence matrix

and

A10 = A01T

(P x B) Fixed head nodes incidence matrix

The following convention is used to assign matrix values:


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A12(i,j)

= 1, 0, or -1 (PxN) Unknown head nodes incidence matrix

Assigned nodal demands are given by:

qT

= [q1, q2,..., qN] (1 x N) Nodal demand vector

Assigned boundary nodal heads are given by:

Hf = [Hf1, Hf2,..., HfB] (1 x B) Fixed nodal head vector

The headloss or gain transform is expressed in the matrix:

FT(Q) = [f1, f2..., fp] (1 x P) Non-linear laws expressing headlosses in links

These matrix elements that define known or iterative network state can be used to

compute the final steady-state network represented by the matrix quantities for unknownflow and unknown nodal head.

Unknown link flow quantities are defined by:

QT

= [Q1,Q2..., Qp] (1 x P) Unknown link flow rate vector

Unknown nodal heads are defined by:

H = [H1, H2 ..., HN] (1 x N) Unknown nodal head vector

These topology and quantity matrices can be formulated into the generalized matrix

expression using the laws of energy and mass conservation:

A second diagonal matrix that implements the vectorized head change coefficients is

introduced. It is generalized for Hazen-Williams friction losses in this case:


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This yields the full expression of the network response in matrix form:

To solve the system of non-linear equations, the Newton-Raphson iterative scheme canbe obtained by differentiating both sides of the equation with respect to Q and H to get:

with

The final recursive form of the Newton-Raphson algorithm can now be derived after

matrix inversion and various algebraic manipulations and substitutions (not presentedhere). The working system of equations for each solution iteration, k, is given by:

The solution for each unknown nodal head for each time iteration is computationally

intensive. This high-speed solution utilizes a highly optimized sparse matrix solver that is

specifically tailored to the structure of this matrix system of equations.

Sources:


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Todini, E. and S. Pilati, "A gradient Algorithm for the Analysis of Pipe Networks,"

Computer Applications in Water Supply, Vol. 1Systems Analysis and Simulation, ed.

By Bryan Callback and Chin-Hour Or, Research Studies Press LTD, Watchword,Hertfordshire, England.

The Linear System Equation Solver

The Conjugate Gradient method is one method that, in theory, converges to an exact

solution in a limited number of steps. The Gradient working equation can be expressedfor the pressure network system of equations.

The use of this approach over more general sparse matrix solvers that implement

traditional Gaussian elimination methods without consideration to matrix symmetry is

preferred since performance gains are considerable. The algorithm utilized in this

software solves the system of equations using a variant of Cholesky's method which hasbeen optimized to reduce fill-in of the factorization matrix, thus minimizing storage and

reducing overall computational effort.

Pump Theory

Pumps are an integral part of many pressure systems. Pumps add energy, or head gains,

to the flow to counteract headlosses and hydraulic grade differences within the system.

A pump is defined by its characteristic curve, which relates the pump head, or the headadded to the system, to the flow rate. This curve is indicative of the ability of the pump to

add head at different flow rates. To model behavior of the pump system, additional

information is needed to ascertain the actual point at which the pump will be operating.

The system operating point is based on the point at which the pump curve crosses thesystem curve representing the static lift and headlosses due to friction and minor losses.

When these curves are superimposed, the operating point can easily be found. This is

shown in the figure below.


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System Operating Point

As water surface elevations and demands throughout the system change, the static head(Hs) and headlosses (HL) vary. This changes the location of the system curve, while the

pump characteristic curve remains constant. These shifts in the system curve result in a

shifting operating point over time.

Variable Speed Pumps

A pump's characteristic

curve is fixed for a given motor speed and impeller diameter, but can be determined forany speed and any diameter by applying the affinity laws. For variable speed pumps,

these affinity laws are presented as:


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Constant Horsepower Pumps

During preliminary studies, the exact characteristics of the constant horsepower pumpmay not be known. In these cases, the assumption is often made that the pump is adding

energy to the water at a constant rate. Based on power-head-flow rate relationships for

pumps, the operating point of the pump can then be determined. Although this

assumption is useful for some applications, a constant horsepower pump should only beused for preliminary studies.

Note: It is not necessary to place a check valve on the pipe immediately

downstream of a pump because pumps have built in check valves that

prevent reverse flow.

This software currently models six different types of pumps:

Note: Whenever possible, avoid using constant power or design point pumps.

They are often enticing because they require less work on behalf of theengineer, but they are much less accurate than a pump curve based on

several representative points.

Constant PowerThese pumps may be useful for preliminary designs andestimating pump size, but should not be used for any analysis for which more

accurate results are desired.


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Design Point (One-Point)A pump can be defined by a single design point (Hd@ Qd). From this point, the curve's interception with the head and discharge axes

is computed as Ho = 1.33Hd and Qo = 2.00Qd. This type of pump is useful forpreliminary designs but should not be used for final analysis.

Standard (Three-Point)This pump curve is defined by three pointstheshutoff head (pump head at zero discharge), the design point (as with the single-point pump), and the maximum operating point (the highest discharge at which

the pump performs predictably).

Standard ExtendedThe same as the standard three-point pump but with anextended point at the zero pump head point. This is automatically calculated bythe program.

Custom ExtendedThe custom extended pump is similar to the standardextended pump, but allows you to enter the discharge at zero pump head.

Multiple PointThis option allows you to define a custom rating curve for apump. The pump curve is defined by entering points for discharge rates at various

heads. Since the general pump equation, shown below, is used to simulate the

pump during the network computations, the user-defined pump curve points areused to solve for coefficients in the general pump equation:

Where: Y = Head (m, ft.)

Q = Discharge (m3/s, cfs)

A,B,C = Pump curve coefficients

The Levenberg-Marquardt Method is used to solve for A, B and C based on the given

multiple-point rating curve.

Valve Theory

There are several types of valves that may be present in a pressurized system. These

valves have different behaviors and different responsibilities, but all valves are used for

automatically controlling parts of the system. They can be opened, closed, or throttled toachieve the desired result.

Check Valves (CVs)


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Check valves are used to maintain flow in only one direction by closing when the flow

begins to reverse. When the flow is in the specified direction of the check valve, it is

considered to be fully open. Check valves are added to the network on a pipe element.

Flow Control Valves (FCVs)

FCVs are used to limit the maximum flow rate through the valve from upstream to

downstream. FCVs do not limit the minimum flow rate or negative flow rate (flow from

the To Pipe to the From Pipe). These valves are commonly found in areas where a waterdistrict has contracted with another district or a private developer to limit the maximum

demand to a value that will not adversely affect the provider's system.

Pressure Reducing Valves (PRVs)

Pressure reducing valves are often used for separate pressure zones in water distribution

networks. These valves prevent the pressure downstream from exceeding a specifiedlevel in order to avoid pressures that could have damaging effects on the system.

Pressure Sustaining Valves (PSVs)

A Pressure Sustaining Valve (PSV) is used to maintain a set pressure at a specific point inthe pipe network. The valve can be in one of three states:

Partially opened (i.e., active) to maintain its pressure setting on its upstream sidewhen the downstream pressure is below this value.

Fully open if the downstream pressure is above the setting. Closed if the pressure on the downstream side exceeds that on the upstream side

(i.e., reverse flow is not allowed).

Pressure Breaker Valves (PBVs)

Pressure breaker valves create a specified headloss across the valve and are often used tomodel components that cannot be easily modeled using standard minor loss elements.

Throttle Control Valves (TCVs)

Throttle control valves simulate minor loss elements whose headloss characteristicschange over time.

General Purpose Valves (GPVs)

GPVs are used to model situations and devices where you specify the flow-to-headloss

relationship, rather than using standard hydraulic formulas. GPVs can be used to


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represent reduced pressure backflow prevention valves, well draw-down behavior, and

turbines.

Friction and Minor Loss Methods

Chezy's Equation

Chezy's equation is rarely used directly, but it is the basis for several other methods,

including Manning's equation. Chezy's equation is:

Where: Q = Discharge in the section (m3/s, cfs)

C = Chezy's roughness coefficient (m1/2

/s, ft.1/2

/sec.)

A = Flow area (m2, ft.

2)

R = Hydraulic radius (m, ft.)

S = Friction slope (m/m, ft./ft.)

Colebrook-White Equation

The Colebrook-White equation is used to iteratively calculate for the Darcy-Weisbach

friction factor:

Free Surface:

Full Flow (Closed Conduit):

Where: f = Friction factor (unitless)

k = Darcy-Weisbach roughness height (m, ft.)

Re = Reynolds Number (unitless)



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D = Pipe diameter (m, ft.)

Hazen-Williams Equation

The Hazen-Williams Formula is frequently used in the analysis of pressure pipe systems

(such as water distribution networks and sewer force mains). The formula is as follows:

Where: Q = Discharge in the section (m3/s, cfs)

C = Hazen-Williams roughness coefficient (unitless)


2)



k = Constant (0.85 for SI units, 1.32 for US units).

Darcy-Weisbach Equation

Because of non-empirical origins, the Darcy-Weisbach equation is viewed by many

engineers as the most accurate method for modeling friction losses. It most commonlytakes the following form:

Where: hL = Headloss (m, ft.)

f = Darcy-Weisbach friction factor (unitless)

D = Pipe diameter (m, ft.)

L = Pipe length (m, ft.)

V = Flow velocity (m/s, ft./sec.)


2)

For section geometries that are not circular, this equation is adapted by relating a circular

section's full-flow hydraulic radius to its diameter:

D = 4R

Where: R = Hydraulic radius (m, ft.)

D = Diameter (m, ft.)

This can then be rearranged to the form:


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Where: Q = Discharge (m

3/s, cfs)


2)



f = Darcy-Weisbach friction factor (unitless)


2)

The Swamee and Jain equation can then be used to calculate the friction factor.

Swamee and Jain Equation

Note: The Kinematic Viscosity is used in determining the friction coefficient

in the Darcy-Weisbach Friction Method. The default units are initiallyset by Bentley Systems.

Where: f = Friction factor (unitless)

= Roughness height (m, ft.)

D = Pipe diameter (m, ft.)Re = Reynolds Number (unitless)

The friction factor is dependent on the Reynolds number of the flow, which is dependent

on the flow velocity, which is dependent on the discharge. As you can see, this processrequires the iterative selection of a friction factor until the calculated discharge agrees

with the chosen friction factor.

Manning's Equation

Note: Manning's roughness coefficients are the same as the roughnesscoefficients used in Kutter's equation.

Manning's equation, which is based on Chezy's equation, is one of the most popular

methods in use today for free surface flow. For Manning's equation, the roughnesscoefficient in Chezy's equation is calculated as:


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Where: C = Chezy's roughness coefficient (m

1/2/s, ft.

1/2/sec.)


n = Manning's roughness (s/m1/3

)

k = Constant (1.00 m /m , 1.49 ft. /ft. )

Substituting this roughness into Chezy's equation, you obtain the well-known Manning's

equation:

Where: Q = Discharge (m3/s, cfs)

k = Constant (1.00 m1/3

/s, 1.49 ft.1/3

/sec.)

n = Manning's roughness (unitless)

A = Flow area (m , ft. )



Minor Losses

Minor losses in pressure pipes are caused by localized areas of increased turbulence that

create a drop in the energy and hydraulic grades at that point in the system. The

magnitude of these losses is dependent primarily upon the shape of the fitting, which

directly affects the flow lines in the pipe.

Flow Lines at Entrance

The equation most commonly used for determining the loss in a fitting, valve, meter, or

other localized component is:


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Where: hm = Loss due to the minor loss element (m, ft.)

K = Loss coefficient for the specific fitting

V = Velocity (m/s, ft./sec.)g = Gravitational acceleration constant (m/s

2, ft./sec.

2)

Typical values for fitting loss coefficients are included in the Fittings Table.

Generally speaking, more gradual transitions create smoother flow lines and smaller

headlosses. For example, the figure below shows the effects of entrance configuration ontypical pipe entrance flow lines.

Water Quality Theory

The governing equations for Bentley WaterCAD V8 XM Edition water quality solver are

based on the principles of conservation of mass coupled with reaction kinetics.

1. Advective Transport in Pipes2. Mixing at Pipe Junctions3. Mixing in Storage Facilities4.

Bulk Flow Reactions5. Pipe Wall Reactions

6. System of Equations7. Lagrangian Transport Algorithm

Engineer's Reference

This section provides you with tables of commonly used roughness values and fitting loss

coefficients.

Roughness ValuesManning's Equation

Commonly used roughness values for different materials are:

Manning's Coefficient (n) for Closed Metal Conduits Flowing Partly Full


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Channel Type and Description Minimum Normal Maximum

a. Brass, smooth 0.009 0.010 0.013

b. Steel

1. Lockbar and welded 0.010 0.012 0.014

2. Riveted and spiral 0.013 0.016 0.017c. Cast iron

1. Coated 0.010 0.013 0.014

2. Uncoated 0.011 0.014 0.016

d. Wrought iron

1. Black 0.012 0.014 0.015

2. Galvanized 0.013 0.016 0.017

e. Corrugated metal

1. Subdrain 0.017 0.019 0.021

2. Storm drain 0.021 0.024 0.030

Roughness ValuesDarcy-Weisbach Equation

(Colebrook-White)


Darcy-Weisbach Roughness Heights e for Closed Conduits

Pipe Material (mm) (ft.)Glass, drawn brass, copper (new) 0.0015 0.000005

Seamless commercial steel (new) 0.004 0.000013

Commercial steel (enamel coated) 0.0048 0.000016

Commercial steel (new) 0.045 0.00015

Wrought iron (new) 0.045 0.00015

Asphalted cast iron (new) 0.12 0.0004

Galvanized iron 0.15 0.0005

Cast iron (new) 0.26 0.00085

Concrete (steel forms, smooth) 0.18 0.0006Concrete (good joints, average) 0.36 0.0012

Concrete (rough, visible, form marks) 0.60 0.002

Riveted steel (new) 0.9 ~ 9.0 0.003 - 0.03

Corrugated metal 45 0.15


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Roughness ValuesHazen-Williams Equation


Hazen-Williams Roughness Coefficients (C)

Pipe Material C

Asbestos Cement 140

Brass 130-140

Brick sewer 100

Cast-iron

New, unlined 130

10 yr. Old 107-113

20 yr. Old 89-100

30 yr. Old 75-90

40 yr. Old 64-83

Concrete or concrete lined

Steel forms 140

Wooden forms 120

Centrifugally spun 135

Copper 130-140

Galvanized iron 120

Glass 140

Lead 130-140

Plastic 140-150

Steel

Coal-tar enamel, lined 145-150

New unlined 140-150

Riveted 110

Tin 130

Vitrified clay (good condition) 110-140

Wood stave (average condition) 120

Typical Roughness Values for Pressure Pipes

Typical pipe roughness values are shown below. These values may vary depending on the

manufacturer, workmanship, age, and many other factors.

Comparative Pipe Roughness Values


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Material

Manning's

Coefficient

n

Hazen-

Williams

C

Darcy-Weisbach Roughness

Height

k (mm) k (0.001 ft.)

Asbestos cement 0.011 140 0.0015 0.005

Brass 0.011 135 0.0015 0.005

Brick 0.015 100 0.6 2

Cast-iron, new 0.012 130 0.26 0.85

Concrete:

Steel forms 0.011 140 0.18 0.6

Wooden forms 0.015 120 0.6 2

Centrifugally

spun

0.013 135 0.36 1.2

Copper 0.011 135 0.0015 0.005

Corrugated metal 0.022 45 150Galvanized iron 0.016 120 0.15 0.5

Glass 0.011 140 0.0015 0.005

Lead 0.011 135 0.0015 0.005

Plastic 0.009 150 0.0015 0.005

Steel

Coal-tar enamel 0.010 148 0.0048 0.016

New unlined 0.011 145 0.045 0.15

Riveted 0.019 110 0.9 3

Wood stave 0.012 120 0.18 0.6

Fitting Loss Coefficients

For similar fittings, the K-value is highly dependent on things such as bend radius and

contraction ratios.

Typical Fitting K Coefficients

Fitting K Value Fitting K Value

Pipe Entrance 90 Smooth Bend

Bellmouth 0.03-0.05 Bend Radius / D = 4 0.16-0.18

Rounded 0.12-0.25 Bend Radius / D = 2 0.19-0.25

Sharp-Edged 0.50 Bend Radius / D = 1 0.35-0.40

Projecting 0.80 Mitered Bend

ContractionSudden = 15 0.05


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D2/D1 = 0.80 0.18 = 30 0.10

D2/D1 = 0.50 0.37 = 45 0.20

D2/D1 = 0.20 0.49 = 60 0.35

ContractionConical = 90 0.80

D2/D1 = 0.80 0.05 TeeD2/D1 = 0.50 0.07 Line Flow 0.30-0.40

D2/D1 = 0.20 0.08 Branch Flow 0.75-1.80

ExpansionSudden Cross

D2/D1 = 0.80 0.16 Line Flow 0.50

D2/D1 = 0.50 0.57 Branch Flow 0.75

D2/D1 = 0.20 0.92 45 Wye

ExpansionConical Line Flow 0.30

D2/D1 = 0.80 0.03 Branch Flow 0.50

D2/D1 = 0.50 0.08D2/D1 = 0.20 0.13

Variable Speed Pump Theory

The variable speed pump (VSP) model within Bentley WaterCAD V8 XM Edition lets

you model the performance of pumps equipped with variable frequency drives. Variable

frequency drives continually adjust the pump drive shaft rotational speed in order to

maintain pressure and flow requirements in a network while improving energy efficiencyand other operating characteristics as summarized by Lingireddy and Wood (1998);

Minimization of excess pressures and energy usage, Leakage control through more precise pressure regulation, Flexible pump scheduling, improving off peak energy utilization, Control of tank drain and fill cycles, Improved system performance during emergency water usage events such as fires

and main breaks,

Reduction of transients produced when pumps start and stop, Simplification of flow control procedures.

Bentley WaterCAD V8 XM Edition variable speed pumping feature will allow designersto make better decisions by empowering them to fully evaluate the advantages and

disadvantages associated with VSPs for their unique application.

Within Bentley WaterCAD V8 XM Edition there are two different ways to model VSPs

depending on the data available to describe pump operations. The relative speed factor isa unitless number that quantifies the rotational speed of the pump drive shaft. 1) If the


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relative speed factor (or for EPS simulations a series of factors) is known, a pattern based

VSP can be used. 2) If the relative speed factor is unknown, it can be estimated using the

VSP with Bentley WaterCAD V8 XM Edition new Automatic Parameter EstimationeXtension (APEX).

Pattern Based VSPsThe variable speed pumping model lets you adjust pumpperformance using the relative speed factor. A single relative speed setting or a

pattern of time varying relative speed factors can be applied to the pump. This isespecially useful when modeling the operation of existing VSPs in your system.

The Affinity Laws are used to adjust pump performance according to the relative

speed factor setting.

SeePump Theoryfor more information about pump curves.

VSPs with APEXAPEX can be used in conjunction with the VSP model toestimate an unknown relative speed setting sufficient to maintain an operatingobjective. APEX uses an explicit algorithm to solve for unknown parameters

directly (Boulos and Wood, 1990). This technique has proven to be powerful,robust, and computationally efficient for estimation of network parameters and

has been improved to allow use for steady state and extended period simulations.

To use APEX for estimating relative speed factors, the control node and control

level setting for the pump must be selected and the pump curve and operatingrange for the pump must be defined. The following paragraphs provide guidelines

for performing these tasks.

Control Node LocationThe location of the control node is an importantconsideration that affects pump operating efficiency, pressure maintenanceperformance, and, in rare instances, the stability of the parameter estimation

calculation. The algorithm has been designed to allow multiple VSPs to operate

within one pressure zone of a network; however, the pump and control node pairsshould be decoupled from one another. In other words, a control node should be

located such that only the pump it controls influences it. If the pressure zone of

the model contains a tank or reservoir (hydraulic boundary conditions), consider

making the boundary condition the control node as opposed to selecting apressure junction near the boundary. This will eliminate the possibility of

specifying a set of hydraulic conditions that are impossible to maintain and thus

reduce the possibility of computational failure. Setting the Target HeadThe control node target head is the constant elevation

of the hydraulic grade line (HGL) that the VSP will attempt to maintain. The

target head at the control node must be within the physical limitations of the VSPas it has been defined (pump curve and maximum speed setting). If the target

head is greater then the maximum head, the pump can generate at the demanded

flow rate the pump will automatically revert to fixed speed operation at themaximum relative speed setting, and the target head will not be maintained.


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Note: Navigating to the target head settingsThe VSP target head forjunction nodes can be set on the VSP tab of the Pump dialog box and

for tanks on the Section tab of the Tank dialog box by adjusting the

initial level.

Setting the Maximum Relative Speed FactorFor flexible operation, avariable speed drive and pump should be configured such that it can efficientlyoperate over a range of speeds to satisfy the pressure and flow requirements it will

be subject. The value selected for the maximum relative speed factor depends on

the normal operating range of the drive motor. To set the proper maximum value,you must determine the drive motor's normal operating speed and maximum

operating speed (the maximum speed at which the drive motor normally operates,

not the speed at which the drive catastrophically fails). The relative speed factor isdefined as the quotient of the current operating speed and the normal operating

speed. Thus the maximum relative speed factor is the maximum operating speed

of the drive divided by the normal operating speed. For example, a maximum

relative speed factor of 2.0 means that the maximum speed is two times thenormal operating speed, and a maximum relative speed factor of 1.0 means that

the maximum operating speed is equal to the normal operating speed.

Defining the Pump CurveIn order to determine the relative speed factor usingAPEX, the pump curve must be smooth and continuously differentiable; thus a

one point or three point power function curve definition must be used. For best

results, the curve should be defined for the normal operating speed of the pump(corresponding to a relative speed factor equal to 1.0, regardless of the maximum

speed setting).

Variable speed pump theory includes:

VSP Interactions with Simple and Logical Controls

The VSP model and APEX have been designed to fully integrate with the simple and rulebased control framework within Bentley WaterCAD V8 XM Edition. You must keep in

mind that the definition of controls requires that the state (On, Off, Fixed Speed

Override) and speed setting of a VSP be properly managed during the simulation.Therefore, the interactions between VSPs and controls can be rather complex. We have

tried to the extent possible to simplify these interactions while maintaining the power and

flexibility to model real world behaviors. The paragraphs that follow describe guidelines

for defining simple and logical controls with VSPs.

Pattern based VSPsThe pattern of relative speed factors specified for a VSPtakes precedence over all simple and logical control commands. Therefore, the

use of controls with pattern based VSPs is not recommended. Rather, the patternof relative speed factors should be defined such that control objectives are

implicitly met.


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VSPs with APEXA VSP can be switched into any one of three different states.When the VSP is On, the APEX will estimate the relative speed sufficient to

maintain a constant pressure head at the control node. When the VSP is Off, therelative speed factor and flow through the pump are set to zero, and the pressure

head at the control node is a function of the prevailing network boundary and

demand conditions. When the control state of a VSP is Fixed Speed Override, thepump will operate at the maximum speed setting and the target head will no

longer be maintained. The Temporarily Closed state for a VSP indicates that the

check valve (CV) within the pump has closed in response to prevailing hydraulic

conditions, and that the target head cannot be maintained. The VSP control nodecan be specified at any junction node or tank in a network model. As described

below, however, the behavior of simple and logical controls depends on the type

of control node selected.

Junction NodesWhen the VSP control node type selected is a junction node,the VSP will behave according to some automatic behaviors in addition to the

controls defined for the pump. If the head at the control node is above the target

head, the pump state will automatically switch to Off. If the head at the controlnode is less then the target head, the pump state will automatically switch to On.

The VSP will automatically switch into and out of the Fixed Speed Override and

Temporarily Closed states in order to maintain the fixed head at the control nodeand prevent reverse flow through the pump. Additional controls can be added to

model more complex use cases.

TanksWhen the VSP control node is a tank, you must manage the state of thepump through control definitions, allowing for flexible modeling of the complex

control behaviors that may be desired for tanks. If a VSP has a state of On, the

pump will maintain the current level of the tank. For example, at the beginning ofa simulation, if a VSP has status of on it will maintain the initial level of the tank.

As the simulation progresses and the pump happens to turn off, temporarily close,

or go into fixed speed override, the level in the tank will be determined inresponse to the hydraulic conditions prevailing in the network. When the VSP

turns on again, it will maintain the current level of the tank, not the initial level.

Thus control statements must be written that dictate what state the pump should

switch to depending on the level in the tank. A pump station with a VSP and afixed-speed pump operating in a coordinated fashion can be used to model tank

drain and fill operations.

Performing Advanced Analyses

The VSP model is fully integrated with the Energy Cost Manager for easy estimation ofpump operating costs. When comparing the energy efficiency of fixed speed and variable

speed pumps, however, it is important to bear in mind that the pumps are not maintaining

the same pressures in the network. The performance of the pumps should be compared insuch a way that takes this difference into account; otherwise the comparison is of little

value. For example, consider a comparison between a VSP and a fixed-speed pump is

prepared, but the target head at the control node is greater than the head maintained there


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by the fixed speed pump. The VSP energy efficiency numbers will be disappointing

because the VSP is maintaining higher pressures.

The concept of a minimum acceptable head (or pressure) can be useful when evaluating

the performance of fixed speed and variable speed pumps. Both pumps should be sized

and operated such that the pressure is equal to or greater than the minimum acceptablehead. In this way, the heads maintained by the respective pumps can be used to define

equivalency between the respective designs. When the comparison is thoughtfullydesigned and conducted, it is likely that the energy efficiency improvements possible

with VSPs will come to light more clearly.

Hydraulic Equivalency Theory

This section outlines the rules that Skelebrator uses for creating equivalent pipes from

parallel or series pipes.

These equations can be solved for equivalent diameter or roughness (C, n or k). With the

Darcy-Weisbach equation, the equations are solved only for D because there aresituations where the roughness can be negative. Both solutions are presented. In general,

there will be one pipe that is the dominant pipe, and the properties of that pipe will be

used when a decision must be made. There will be some default rule for picking thedominant pipe, but you will be able to override it.

You will not use equivalent lengths because you want to preserve the system geometry.For pipes in series, you will add the lengths of the two pipes while for pipes in parallel.

You will use the length of the dominant pipe as follows:

Lr= L1 + L2

Principles

The equations derived below are based on the following principles. The equations below

are for two pipes but can be extended to n pipes.

For pipes in series:

Qr= Q1 + Q2

where Q = flow, r refers to the resulting pipe, and 1 and 2 refer to the pipes being

removed.


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hr= h1 + h2

For pipes in parallel:

Qr= Q1 + Q2

and

hr= h1 + h2

As long as the units are consistent, then any appropriate units can be used. For example,if the diameters are in feet, then the resulting diameter will be in feet.

Thiessen Polygon Generation Theory

Nave Method

A Thiessen polygon of a site, also called a Voronoi region, is the set of points that arecloser to the site than to any of the other sites.

Let P = {p1,p2,...pn} be the set of sites andV= {v(p1), v(p2),...v(pn)} represent the

Voronoi regions or Thiessen polygons forPi,which is the intersection of all of the half

planes defined by the perpendicular bisectors ofpi and the other sites. Thus, a nave

method for constructing Thiessen Polygons can be formulated as follows:

Step 1 For each i such that i = 1, 2,..., n, generate n - 1 half planesH(pi,pj), 1


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(1986; 1987) has developed a sweepline algorithm for constructing Thiessen polygons.

This algorithm has been implemented in the WaterGEMS Thiessen Polygon Generator.

The detailed working algorithm is given as follows:

1. Q


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In order to deal with a recoverable calamity, the concept of water supply is introduced to

quantify the supply capacity of a water distribution system. It is defined as a percentage

of the supplied demand over the normal demand. Water companies are required tocomply the minimum water supply level under a calamity of one element outage, which

is expected to be fully repaired within 24 hours. The modeling approach for evaluating

water supply level for the use cases as follows.

Use Cases

In 1994, the Dutch water authority posted the guideline for water companies to evaluate

the level of water supply while coping with calamity events. A tentative guideline

requirement is that a water system must meet 75% of the original demand for themajority of customers and no large group of customers (2000 resident addresses) should

receive less than 75% of their original demand.

The guideline is applicable to all the elements between the source and tap in a water

system and is required to find the effect of every element. In order to calculate the watersupply level under a calamity event, a hydraulic modeling approach is proposed:

1. Take one element at a time out of a model, copying the calamity event of elementoutage

2. Run the model for peak hours of all demand types and also the peak hours of tankfilling. The actual demand needs to be modeled as a function of pressure; the

supply is considered unaffected if the pressure is above the required pressure

threshold

3. Evaluate the water supply level for each demand node. If there is less than 2000resident customers receiving less than 75% of the normal demand, then the

requirement is met. Repeat Step 1 to simulate another calamity event. If therequirement is not met, continue with step 4.

4. Perform 24 hours pressure dependent demand simulation for the maximumdemand day under the calamity even

5. Sum up the actual demand for each node over 24 hours6. Check if there is any node where the totalized demand over 24 hours is less than

75% of the maximum day demand; if not, the guideline is met. Otherwise an

appropriate system improvement needs to be identified in order to meet theguideline.

UK water companies are required by law to provide water at a pressure that will, under

normal circumstances, enable it to reach the top floor of a house. In order to assess if thisrequirement is satisfied, companies are required to report against a service levelcorresponding to a pressure head of 10 meters at a flow of 9 liters per minute. In addition,

water companies are also required to report the supply reference for unplanned and

planned service interruptions.

Both use cases provide some generality for water utilities world wide to evaluate theperformance of water systems under emergency and low pressure conditions. An


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emergency event can be specified as one set of element outages. In order to quantify the

water supply level under such an event, the demand must be modeled as a function of

nodal pressure. Hydraulic model needs to be enhanced to perform pressure dependentdemand simulation and to compute the level of certainty/supply level.

Supply Level Evaluation

Assume Qi to be the normal demand at node i. Qis,j represents the actual supplied demand

at node i under calamity eventj, the supply level at node i for eventj is given as:

This gives the percentage of the demand that a system supplies to node i under calamity

eventj

. The key is to calculate the actual supply demandQ

i

s

under the outage that maycause lower than required junction pressure. The less the demand, the greater the impact

the calamity is on the system supplied capacity and the more critical the element is to the

system.

Pressure Dependent Demand

Whenever a calamity occurs, the systems pressures are affected. Some locations may not

have the required pressure. Nodal demand, water available at a location, is dependent on

the pressure at the node when the pressure is low. Unlike the conventional approach ofdemand driven analysis, demand is a function of pressure, Pressure Dependent Demand

(PDD). However, it is believed that a junction demand is not affected by pressure if the

pressure is above a threshold. The junction demand is reduced when the pressure isdropping below the pressure threshold and it is zero when the pressure is zero.

PDD can be defined as one of two pressure demand relationships including a power

function and a pressure demand piecewise linear curve (table). The power function isgiven as:


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Where:

Hi = calculated pressure at node i

Qri = requested demand or reference demand at node i

Qsi = calculated demand at node i

Hri = reference pressure that is deemed to supply full requested/reference demandHt = pressure threshold above which the demand is independent of nodal pressure

= exponent of pressure demand relationship.

A typical PDD power function is illustrated below. The actual demand increases to thefull requested demand (100%) as pressure increases but remains constant after the

pressure is greater than the pressure threshold, namely the percent of pressure threshold is

greater than 100%.

Pressure demand piecewise linear curve is specified as a table of pressure percentage vs.demand percentage. Pressure percentage is the ratio of actual pressure to a nodal

threshold pressure while demand percentage is the ratio of the calculated demand to the

reference demand.

Demand Deficit

When a calamity event is modeled, the total supplied demand may be less than the

normal required demand. The difference between the calculated demand and the normal

required demand is a demand deficit that is evaluated under a prescribed supply levelthreshold. The total system demand deficit under one possible calamity eventj:


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Where is the deficit demand at eventj andSt is the threshold of supply level. Thisformula provides the method for evaluating water supply level, element criticality, and

modeling pressure dependent demand.

Solution Methodology

The key solution methodology is how to solve for the pressure dependent demand.

Conventionally, nodal demand is a known value. Applying the mass conservation law to

each node and energy conservation law to each loop, the network hydraulics solution canbe obtained by iteratively solving a set of linear and non-linear equations. A unified

formulation for solving network hydraulics is given as a global gradient algorithm(GGA).

Where Q is the unknown pipe discharge andHis the unknown nodal head. q is the set of

nodal demand that is not dependent on the nodal headH.

For pressure dependent demand, the demand is no longer a known value but a function of

nodal pressure. The solution matrix becomes:

A new diagonal matrixA22 is added to the solution matrix. The non-zero diagonal

element is given as


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Direct GGA Solution

An alternative solution method is to directly apply GGA as derived but move the pressure

dependent demand term to the right

This method will require no matrix modification of original GGA, but the program will

update the nodal demand according to the pressure head of the left side of the matrix.

Technical Reference WaterCAD V8 XM

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