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Modular calculation and optimization toolkit for power plant
applications
Matias Nyman
Master’s Thesis in Process and Systems Engineering
Supervisor: Frank Pettersson
Faculty of Science and Engineering
Åbo Akademi University
31-Aug-2021
1
Abstract
Automation is an ever-increasing phenomenon in today’s world, almost exponentially
so. If a task is expected to be repeated in a predictable and quantifiable way, it will at
some point be subject to automation. Naturally, engineering is not exempt from this
phenomenon, as a part of calculation and design work involves data collection and
munging, trivial descriptive and explanatory tasks and so forth. This type of work
generally fits the description of tasks subject to automation. This thesis work for Citec
Oy Ab aims to reduce repetitive tasks associated with process and power plant
engineering, while allowing for calculation and optimization of different power plant
processes yielding higher overall efficiency, thereby reducing costs and emissions.
This will be accomplished by providing a dependable well-documented calculation
toolkit for the process engineering team at Citec. The toolkit will be constructed by
introducing and further developing a comprehensive Python library named pcalcs. The
library consists of methods for calculating a substantial number of common process
engineering challenges and creating calculation reports automatically. The goal is to
make a well-documented and robust calculation library with flexible interfaces. The
missing functionality that is a core focal point of this thesis is the development of the
hydraulic network solver capabilities needed to calculate the hydraulic steady state
condition of large piping networks.
Powerplant design can be conducted in a discrete modular fashion where systems
functionally unrelated to each other can be designed separately. The calculation
modules will be created in a similar modular fashion.
2
Acknowledgements
Considerable gratitude is directed towards William Laurén for countless hours of help
and guidance with programming in Python, as well as being the key driving force and
enabling factor behind this thesis. I would also like to thank Tom Thorn and Frank
Pettersson for process engineering guidance and administrative help. And finally, a
big thanks’ to the Citec process engineering team for comments and process
engineering expertise.
3
1 CONTENTS
Abstract .................................................................................................................................... 1
Acknowledgements .................................................................................................................. 2
Symbols and abbreviations ...................................................................................................... 5
2 Introduction ...................................................................................................................... 7
3 Fundamental theory .......................................................................................................... 9
3.1 Power plant structure................................................................................................ 9
3.2 Pressure loss ............................................................................................................. 9
3.3 Determining fluid properties .................................................................................. 11
3.4 Prime movers ......................................................................................................... 14
3.4.1 Positive displacement pumps ......................................................................... 14
3.4.2 Centrifugal pumps .......................................................................................... 16
4 Hydraulic network solver development ......................................................................... 19
4.1 Hardy Cross Method .............................................................................................. 22
4.2 Vertex-Loop Method .............................................................................................. 23
4.3 Global Gradient Algorithm .................................................................................... 26
4.4 Challenges in network solver creation ................................................................... 28
4.5 EPANET ................................................................................................................ 29
4.5.1 Extended period analysis ................................................................................ 30
4.5.2 Hydraulic network logic control .................................................................... 31
4.6 Accounting for compressibility .............................................................................. 32
4.7 Network temperature distribution .......................................................................... 33
5 Calculation library .......................................................................................................... 36
5.1 Physical quantity control ........................................................................................ 36
5.2 Python source code documentation ........................................................................ 36
5.3 Library structure ..................................................................................................... 37
4
5.4 Graphical user interface ......................................................................................... 37
5.5 Testing and version control .................................................................................... 39
6 Automatic calculation report creation ............................................................................ 41
6.1 System specific report creation and structuring ..................................................... 41
6.2 Master project report creation ................................................................................ 47
7 Results ............................................................................................................................ 50
7.1 Fuel gas distribution ............................................................................................... 50
7.1.1 Fuel gas distribution ....................................................................................... 52
7.2 Lube oil system ...................................................................................................... 56
7.2.1 PRV operational pressure setting ................................................................... 60
7.3 Fuel oil distribution ................................................................................................ 69
7.4 Cooling water system ............................................................................................. 73
7.4.1 Cooling water system header sizing ............................................................... 76
7.4.2 Recirculation temperature simulation ............................................................ 78
7.5 Drainage systems ................................................................................................... 81
7.6 Instrument air system ............................................................................................. 82
8 Conclusion and future development ............................................................................... 85
9 Svensk sammanfattning ................................................................................................. 86
10 References .................................................................................................................. 90
5
Symbols and abbreviations
�̇� Volume flow
�̇� Mass flow
𝑝 Pressure
𝜌 Density
H Head
∆𝑝 Change in pressure
P Power
𝜂 Efficiency
ω Rotational frequency
𝑐𝑝 Specific heat capacity
T Temperature
𝑁𝑃𝑆𝐻𝑎 Available net pressure suction head
𝑁𝑃𝑆𝐻𝑟 Required net pressure suction head
𝑓𝐷 Darcy friction factor
L Length
𝐷𝑖 Inner pipe diameter
𝑣 Flow velocity
𝐾 Hydraulic resistance coefficient
𝑉𝐿𝑀 Vertex loop method
𝐻𝐶𝑀 Hardy cross method
6
𝐺𝐺𝐴 Global gradient algorithm
PRV Pressure regulating valve
PSV Pressure sustaining valve
PBV Pressure breaker valve
FCV Flow control valve
TCV Throttle control valve
GPV General purpose valve
PDA Pressure driven analysis
DDA Demand driven analysis
7
2 Introduction
The Python programming language was used as the primary platform for developing
a tool capable of creating modularized calculation reports detailing plant engineering
calculations. The Python programming language together with the scientific
computing open-source community is well suited for this task. The extensive and well-
documented scientific and engineering libraries that are maintained by developers
from both the professional industry and academia, provide a capable foundation used
for this thesis. One key calculation capability requirement was the ability to solve the
steady state of complex hydraulic networks. These networks are almost exclusively
solved computationally due to the iterative or probabilistic nature of the algorithms
developed for solving the hydraulic states of the circuit. [1]
In this thesis work, an extended period analysis solver for hydraulic networks is
implemented and further developed to suit the specific needs of Citec. The hydraulic
network solver developed uses a modification and development of the EPANET 2.2
solver, developed by the United States Environmental Protection Agency and open
sourced under a MIT license. [2]
In order to achieve accurate results in any hydraulic calculation, the fluid properties
are of considerable importance. In this toolkit, fluid property calculation is automated
with the Python interface to the CoolProp library and used as input for the hydraulic
network calculations. For fluids that are not well defined or missing from the CoolProp
library, the applicable industry standards are used. The common input parameters to
define the fluid properties and state are temperature and pressure, but any of the
available CoolProp inputs can be used.
All calculations are performed and visualized in Jupyter Notebooks, while the code
functionality is defined in modules (Python files), from which class and function
definitions can be imported. This allows faster development of new calculation
modules.
The notebooks are first converted to LaTeX code and then to PDFs. The created
calculation report modules are normally segmented into four main sections: theory,
system overview, detailed results and conclusion. The theory section definitions
8
include but are not limited to: fluid properties, assumptions, pressure loss formulas and
friction factor formulas depending on flow type. In the system overview section, all
network data such as pipe length, size, roughness and minor losses are detailed for all
pipes in the network, while the connections between vertices are detailed in tabular
formats and block diagrams. This section can also include pump curves, tank volumes,
control philosophies or other relevant input data. The detailed results section generates
result data that is presented in tabular formats or in suitable visualizations such as line
graphs, scatter graphs, heat maps or distribution graphs. In the conclusion section, the
user can draw project specific conclusions based on the results obtained in the previous
section. If applicable, some of the conclusions can be automatically generated from
analysing the result data from the detailed results section.
Furthermore, the reports are constructed to allow for editing of input variables, such
as pipe lengths, sizes, inlet conditions to the networks and more. This allows for swiftly
creating calculation reports in different design stages with updated input data.
9
3 Fundamental theory
In this chapter, the most central underlying theory needed for the implementation of
the calculation toolkit is presented. This mainly consists of thermodynamic, hydraulic
and pneumatic theory. The theory presented in this section will be limited to common
process technologies in the context of power plant engineering.
3.1 Power plant structure
Power plants have many different types of process structures, dependent on the core
electricity-generating mechanism. Common process systems are for example process
water cooling or heating networks, fuel distribution networks, instrument air networks,
steam distribution systems and exhaust gas systems. These systems are set up to serve
the needs of the core electricity-generating mechanism. Generally, the limitations and
points of interest are among others delivery pressures to consumer vertices, pump
sizing, actual flow rate, pipe sizing, flow velocities, pressure losses and piping network
temperature distributions. These calculations need to be conducted for different design
conditions and varying operating conditions to ensure the power plant is functional in
the entire operating range.
Steam distribution and cooling water systems are common in most power plants due
to the utilization of the Rankine cycle for electricity generation. Instrument air systems
are commonly used for electro-pneumatic valve actuation and utility purposes. In the
case of some combustion power plants, fuel gas and fuel oil distribution systems are
needed.
3.2 Pressure loss
Fluid flow in enclosed or open-channel pipes is always accompanied by a certain
reduction in the static pressure of the fluid in the direction of the flow. This reduction
is referred to as pressure loss and is caused by internal friction in the fluid due to
viscous forces and turbulence also friction between the fluid and the wall.
In its most fundamental form, the relationship between the pressure loss and flow rate
can be described by equation 3.2.1.
10
∆p = k1 + k2�̇�2 (3.2.1)
where 𝑘1 is pressure loss independent of flow rate, such as height difference at pipe
inlet and outlet. The 𝑘2 is the coefficient relating the volume flow to the pressure loss.
This can also be a pressure increase in case the height difference is negative in respect
to the flow direction. This equation is empirically determined and there are a couple
of derivations that are used, of these the Darcy-Weisbach equation is most commonly
used. Other derivations in common use are the Hazen-Williams equation, used
exclusively for water networks and the Chézy-Manning equation, used for calculating
pressure loss resulting from turbulent flow in open channels. The 2 last equations are
commonly referred to as Reynolds independent, as they do not include the Reynolds
number. [3] [4]
To calculate hydraulic and pneumatic behaviours in process engineering scenarios, the
most appropriate derivation is the expanded Darcy-Weisbach equation. This allows for
minor losses in network elements such as valves, bends and orifices while also
including pressure differences due to height differences between the pipe inlet and
outlet.
∆𝑝 = 𝑔𝜌(ℎ1−ℎ2) + (∑𝐾 + 𝑓𝐷𝐿
𝐷𝑖)𝑣2
2𝑔 (3.2.2)
where 𝐾 is the resistance coefficient for a minor loss element and 𝑓𝐷 is the Darcy
friction factor. The other formulas mentioned are also implemented through the
EPANET solver if necessary.
The friction factor 𝑓𝐷 is calculated based on the flow regime (as determined by the
Reynolds number limits for internal flow in pipes). For laminar flow (Re < 2000), the
friction factor is calculated with 𝑓𝐷 = 64
Re. For transitional flow (2000 < Re < 4000), a
cubic interpolation of the Moody chart is used. For turbulent flow (Re > 4000) the
Swamee and Jain approximation of the Colebrook-White equation is used. This value
depends on the absolute roughness 𝜀 of the inner pipe wall. [2]
11
3.3 Determining fluid properties
Accurately determining fluid properties is a critical part of achieving accurate pressure
loss calculations. For power plant applications, the number of different fluids can be
vast.
Where possible, the established CoolProp library has been used as a back-end. The
CoolProp library is an open-source library that defines the thermophysical properties
of many pure fluids and common mixtures. The core underlying equations in the
CoolProp library, which determine the thermodynamic properties of fluids are the
Helmholtz-energy-explicit equations of state, which are used in many similar high
accuracy fluid software libraries [5]. Specifically, the CoolProp library was used for
determining the properties of the following fluids:
▪ Air
▪ Water
▪ Water and Ethylene glycol mixtures at varying relative mass proportions
▪ Water and Propylene glycol mixtures at varying relative mass proportions
▪ Methane
▪ Natural gas from different sources
▪ Exhaust gas
For power-plant applications, some of the required fluid properties are not available in
the CoolProp library and must be implemented as custom solutions. Such fluids are
for example: varying grades of fuel oils and lubrication oils. Both lube oils and fuel
oils are long-chain hydrocarbons, and as such have viscosities that increase
logarithmically with decreasing fluid temperature. [6]
Lubrication oils are differentiated by their viscosity at different temperatures and the
temperature's effect on the oil viscosity. The lube oil grades predominantly used in
power plant applications due to their low costs and appropriate viscosities are
monograde oils such as SAE30 and SAE40. The SAE oil classifications define the
viscosity of the oil. SAE30 is defined according to [7], as having a kinematic viscosity
between 9.3 cSt and 12.5 cSt at 100°C. Conversely, the SAE40 lube oil viscosity is
defined according to [7] as having a kinematic viscosity between 12.5 cSt and 16.3 cSt
at 100°C. For lubrication oils there are currently two separate ways of accurately
12
determining the viscosity in the entire operational temperature interval implemented
in this toolkit. The first is using interpolated data of measurement values. The second
is obtaining 2 reference sets of reference points for viscosity and temperature and then
use the industry-standard formulas provided in [6] to extrapolate the data for any
arbitrary temperature. The density of lubrication oils is defined by similar
measurement data interpolation or alternatively according to the ASTM D1250
standard [8]. Figure 1 shows the calculated viscosity and density of SAE30 and SAE40
dependent on temperature.
Figure 1: Lube oil viscosities and densities
Fuel oil viscosity calculation provides the additional challenge of having hydraulic
properties that are variable with their source. As such, project-specific measurement
data is needed to establish fluid properties with reasonable accuracy. To accurately
determine the fuel oils properties for the entire temperature range, two reference point
sets for viscosity and temperature are needed. Standard [6] is also used for fuel oils
13
when extrapolating the data for any arbitrary temperature. In Figure 2 the fuel oil
properties can be found for an example fuel with a viscosity of 15 cSt at 100°C.
Figure 2: Example fuel oil viscosity and density
The input for any given oil must be the 2 reference point sets needed for calculating
the viscosity-temperature relationship according to [6]. The fuel oils most commonly
used have one reference point defined according to the ISO8216 [9] or ASTM D396
[10] standards. The ISO8216 standard defines the maximum viscosity of a certain fuel
oil grade at a temperature of 50°C. The ASTM D396 standard gives the interval in
which a fuel oil viscosity can be at 40°C for lighter fuel oil fractions and at 100°C for
heavier fuel oil fractions. For any given grade, the upper limit will be used as the
reference point as this will result in the worst-case viscosity at all temperatures when
extrapolated with [6].
14
3.4 Prime movers
In piping networks, prime movers (pumps, fans, compressors and similar machinery)
are chosen based on a multitude of criteria. The most essential criteria are the network
fluid properties, as these will impact the efficiency and overall viability of using any
type of prime mover. The underlying theory describing how these devices function
and how they impact the hydraulics of the network which they are part of is presented
in this section.
For compressible fluids i.e. gases, compressors or fans are usually chosen. The theory
describing gas compression will not be detailed as it is not essential for this thesis. For
liquids, the primary selection criteria are the kinematic viscosity and the density of the
fluid. For low viscosity fluids, centrifugal pumps are the most common as the available
pressure increase is not impacted by the viscosity of the fluid. The popularity of
centrifugal pumps and well-understood affinity laws also contribute to their wide-
spread use. For higher viscosity fluids, the most common choice of prime mover is
some type of positive displacement pump, normally a screw pump or diaphragm pump.
[11]
Implementing positive displacement pumps into the environment of the hydraulic
network solver presented some challenges, as they have a widely varying model-
dependent behaviour and were not included in the solver back-end discussed in section
4.5.
3.4.1 Positive displacement pumps
Positive displacement pumps work on the principle of trapping a fixed amount of fluid
in the suction side and forcing the fluid to flow out to the discharge side of the pump.
This mechanism results in ideal positive displacement pumps behaving as constant
flow machines, i.e. providing a constant flow independent of pressure increase, at a
constant rotational speed. In practice, the flow provided by the pump will decrease
marginally when the pump is operating at higher pressure increases, due to internal
leakage. Figure 3 shows an example construction of a single-rotor screw pump.
15
Figure 3: Screw pump adapted from [12]
A crucial safety device for the operation of positive displacement pumps is the safety
bypass valve which connects the pump discharge side to the suction side. The safety
bypass valve is of type pressure regulating valve (PRV). This valve opens in case the
pressure at the inlet side of the valve exceeds a set value. The flow provided by the
pump will then normally be recirculated to the suction side or discharged to the
atmosphere in case recirculation is undesirable. The pressure regulating valves used
are commonly classified as pressure compensated and non-pressure compensated. The
former type opens only in case the differential pressure across the inlet and outlet side
of the valve exceeds the set value, the second type opens based on the valve inlet
pressure regardless of the outlet side pressure (the valve inlet is connected to the
discharge side of the pump). This distinction is important when considering simulation
of positive displacement pumps in pressurized systems. [12] [13] [14]
Simulating positive displacement pumps is often more challenging than centrifugal
pumps since the safety valve opening and closing action is associated with a
phenomenon known as pressure accumulation. This will cause the pressure to increase
above the setpoint of the PRV after the valve has opened due to hydraulic resistance
in the valve and the valve outlet piping. The hydraulic resistance in the valve will also
vary depending on fluid properties and the valve position mechanism in sometimes
unpredictable ways. The exact behaviour of the pressure accumulation is usually not
provided by pump manufacturers in case that the pump has an internal PRV. Normally
only a maximum pressure accumulation value is provided. If the pressure regulating
valve is externally mounted, data from the valve manufacturers needs to be used. [14]
16
To be able to simulate the pressure accumulation of a screw pump, a screw pump
object which has a customizable pressure accumulation curve was implemented to suit
the needs of the specific calculation performed. The pressure accumulation behaviour
is shown for an ideal screw pump in Figure 4 and is system and pump dependent.
Figure 4: Example of pressure accumulation
In order to calculate the pressure accumulation behaviour for all cases, a pump
abstraction was programmatically implemented where the polynomial relationship
between reduction in flow and pressure accumulation could be modified to suit the
pump used. If no data is used and the recirculation piping has reasonably low hydraulic
resistance, zero pressure accumulation will be the default assumption.
3.4.2 Centrifugal pumps
Centrifugal pumps operate as prime movers by converting rotational kinetic energy to
hydrodynamic energy. The theory governing the modelling of these devices is detailed
in extent in [11]. The structure of a centrifugal pump is shown in Figure 5.
17
Figure 5: Centrifugal pump [15]
The centrifugal pump curve is calculated in accordance with equation 3.4.1, because
the EPANET solver (detailed in section 4.5) is used as back-end.
∆𝑝𝑖𝑗 = −𝜔2(𝑝𝑚𝑎𝑥 − 𝑟(𝑄𝑖𝑗/𝜔)
𝑛) (3.4.1)
where 𝑝𝑚𝑎𝑥 is the shutoff pressure of the pump, while 𝑟 and 𝑛 are pump coefficients.
𝑄𝑖𝑗 and ∆𝑝𝑖𝑗 are the flow rate and pressure drop between inlet i and outlet j. In case
equation 3.4.1 has inadequate accuracy when transforming pump characteristics at
different rotational frequencies, an external implementation of the standard affinity
laws is also included in this toolkit.
This type of formulation is needed to describe pump behaviour at different rotational
speeds as there currently is no way of implementing the affinity laws detailed in [11]
into the hydraulic network solving algorithms, without significantly affecting the
solver algorithm convergence times. Equation 3.4.1 will always have a negative value
as in the context of a network, as pumps are defined as edges with a negative pressure
loss to flow rate relationship. Negative pressure loss is exactly analogous to the
pressure increase shown in Figure 6. [13] [11] [2]
18
Figure 6: Centrifugal pump comparison
In Figure 6, pump data from a real pump is compared to the results when fitting the
data points to equation 3.4.1. The equation can be observed fitting the example pump
data in Figure 6 to a satisfactory accuracy.
19
4 Hydraulic network solver development
Piping networks are multiple pipes connected in series and parallel. Water distribution
networks are common examples of piping networks. An example of a water
distribution network is shown in Figure 7.
Figure 7: Water distribution network
These networks present more challenges than linear or simple parallel systems where
calculating the flow distribution is less challenging. Normally piping networks also
consist of pipes, valves, pumps, tanks, fittings, flanges, nozzles etc., which all provide
their own sets of challenges in modelling properly in the context of the hydraulic
network. In order to calculate the hydraulic steady state of the system, it is abstracted
as a discrete graph. In graph theory, a discrete graph is defined as a set of objects
having some form of relation to each other. Normally, a discrete graph is thought of
as a set of vertices connected by edges. More specifically, the piping network needs to
be defined as a connected discrete graph, meaning that sub-graphs of exactly one
vertex are not allowed. [16]
20
The example network shown in Figure 8 is used to more easily demonstrate
these definitions.
Figure 8: Example network
The most common categories used for distinguishing graphs are: directed graphs or
undirected graphs and simple graphs or multigraphs. Directed graphs have vertices
that contain an intrinsic direction to the connection between the vertices of the graph,
while undirected graphs represent undirected connections between vertices. Simple
graphs are commonly defined as graphs that only allow vertices to be connected by
one edge. Conversely, multigraphs can have many vertices connecting vertices to each
other, the definition normally also allows for edges connecting a vertex to itself. [16]
For most methods of solving piping networks, the network is abstracted as a directed
multigraph, i.e. vertices can be connected by multiple edges that have an associated
direction. Pipes, valves and pumps are often abstracted as directed edges while
junctions, tanks and reservoirs are abstracted as vertices. [2]
One other central definition for most piping network solving methods are loops, which
are a special case of paths. In graph theory, paths are defined as a sequence of edges
and vertices joining two vertices. Loops are special cases of paths where the first vertex
in the path is also the last vertex. It is important to note, even though the piping network
21
is abstracted as a directed multigraph. For the purpose of network solver creation, the
loops must be undirected. [17]
For the network in Figure 8, there are three undirected loops that can be identified as
the following sets of vertices and edges:
𝐿1: {N1, N2, N4, N6, N5, N3}, {E1, E3, E7, E6, E5, E2}
𝐿2: {N1, N2, N4, N3}, {E1, E3, E4, E2}
𝐿3: {N3, N4, N6, N5}, {E4, E7, E6, E5}
Most deterministic models of solving piping systems are developed formulations of
Kirchhoff's 1st &2nd law. The added complexity arises from the pressure loss not being
linear with respect to the flow rate, as the potential difference is to the current in the
analogous simple electrical circuits. In most solvers, this is overcome by applying
some derivation of the Newton-Rhapson method to each edge, which entails
computing the pressure loss in the edge as well as the derivative of the pressure loss.
This approach is applied in parallel to each edge in each loop to achieve iterative
convergence to a solved state where pressure losses and flow rates are approximately
constant over each iteration.
What differentiates each method and their popularity in hydraulic solvers is how
quickly the newton iterations are performed and the needed pre-processing to achieve
executability of the algorithm and post-processing of network characteristics. In the
following sections, the advantages and disadvantages of three methods are compared.
These methods are: the Hardy Cross Method (HCM), the Vertex-Loop Method (VLM)
and the Global Gradient Algorithm (GGA). The goal of this comparison is to obtain
an overview of the evolution of these algorithms and why a specific algorithm is
chosen for an application. Each algorithm is set up for the network in Figure 8.
The VLM is an improved formulation of the HCM, where some of the pre- processing
and post-processing is circumvented algebraically. However, the algorithms are
separated by a classification where the VLM and the GGA are referred to as
Simultaneous while the HCM is referred to as Local. [18] This is mainly due to the use
of matrix implementations of the vertex and loop equations, causing each iteration to
adjust the flow in all edges simultaneously. The main difference is the use of incidence
22
matrices and other modified topology matrices to define and solve the network. The
classifications used are shown in Figure 9.
Figure 9: Network solver algorithm classifications [18]
4.1 Hardy Cross Method
The HCM is the first iterative method developed for solving the hydraulic state of
piping networks and is an adaptation of the moment distribution method also
developed by Hardy Cross. [19] The method is an application of two fundamental
principles, the first of which is the conservation of mass. For the vertices in the network,
this is analogous to Kirchhoff's 1st law for electric circuits: the sum of all flows entering
the vertex must be equal to the sum of all flows exiting the vertex. The second
fundamental principle is the conservation of energy. For a loop in the piping network,
this requires the sum of all pressure losses (accounting for the edge direction) around
the loop to be equal to zero. [19] This, in turn, is analogous to Kirchhoff's 2nd law for
electric circuits. The most outstanding advantages of the HCM are the simple
mathematics and the clearly outlined solution procedure. The main disadvantages are
the below detailed substantial pre-processing and post-processing related to
algorithmically automating all the solution procedure steps listed below for any
arbitrary hydraulic circuit.
The following variable definitions are used for the HCM: 𝑄 ̇ is the flow rate of an edge,
𝑛 is the flow exponent (the flow rate exponent in the pressure loss formula) and 𝑟 is
the flow relation (all terms in the pressure loss formula that are not 𝑄 ̇ or 𝑛).
Method procedure:
23
1. Determine initial flow state guess. This initial guess must satisfy the
conservation of mass constraint such that the sum of all flows entering any
vertex must be equal to the sum of all flows exiting the vertex.
2. Find all closed loops in the system.
3. Determine the pressure losses in pipes flowing in the clockwise and counter-
clockwise direction, accounting for the defined loop direction while using the
chosen volume flow to pressure loss correlation.
4. Calculate sum of pressure losses for all loops ∑𝑟𝑄�̇� by subtracting the
pressure losses in the counter clockwise direction from the pressure losses in
the clockwise direction.
5. Find ∑n𝑟�̇�(𝑛−1) for each loop irrespective to the direction, this term is
identical to the absolute value of the sum of the derivative of the pressure losses
in the pipes in each loop.
6. The change in the flow in each pipe of the loop is equal to ∑𝑟𝑄�̇�
∑n𝑟�̇�(𝑛−1) .
7. If the flow change term is negative it applies to the pipes in the clockwise
direction, if the term is positive it applies to the pipes in the counter clockwise
direction.
8. Repeat using updated flows, starting from step 3, until the flow change is
within the specified tolerance.
The HCM does not require any specific relation between pressure loss and flow,
although the Darcy-Weisbach equation is commonly used. The HCM was the first of
its kind and is therefore mentioned here. However, many improvements to this method
have been developed, and are therefore preferred for implementing in a hydraulic
network solver.
4.2 Vertex-Loop Method
The first step in the VLM solution procedure is constructing vertex equations in
accordance with Kirchhoff’s 1st law. These state that the sum of flows entering the
vertex must be equal to the sum of flows exiting the vertex. For piping networks, some
vertices are input vertices where the flow is entering the network and others are output
vertices where the flow is exiting the network. Assuming the network does not have
storage ability, the input must also be equal to the output. The vertex equations can be
24
represented in matrix form. The matrix is the equivalent of an incidence matrix for a
directed graph in graph theory. This matrix will be modified so that one row
(representing the connections for one vertex) is removed in order to achieve linear
independency among the vertex equations, in this example vertex N6 is omitted. [20]
These examples showcase some of the formulations needed to solve the example
network in Figure 8.
𝑁 =
[ −1 −1 0 0 0 0 01 0 −1 0 0 0 00 1 0 −1 −1 0 00 0 1 1 0 −1 00 0 0 0 1 0 −1]
𝑁1𝑁2𝑁3𝑁4𝑁5
When starting the calculation procedure, the initial guess of flow rates in all edges
must satisfy the conservation of mass in every vertex in the network. As the solution
procedure is iterated, the sum of the pressure loss in all loops approaches zero. This,
in turn, means that Kirchhoff's 2nd law must be satisfied within the required tolerance
when the solution is obtained. The 𝐿 matrix defines the edge directions for each loop.
𝐿 = [1 1 −1 −1 0 0 00 0 0 1 1 −1 −1
] 𝐿2𝐿3
The next step is defining the V matrix. This matrix is a column matrix where the first
terms are the inflows to the vertices in the loop amount of vertex except for the omitted
node, in this case node N6.
𝑉 =
[ −|�̇�𝑁1|
|�̇�𝑁2|
|�̇�𝑁3|
|�̇�𝑁4|
|�̇�𝑁5|
𝐿2𝐿3 ]
where �̇�𝑁𝑥 is the inlet or outlet flow rate from the network in that vertex 𝑁𝑥. Sign -1
denotes inlet of fluid to the network and sign +1 denotes outlet of fluid from the
network. In case there is no outlet the �̇�𝑁𝑥 term will be zero. 𝐿2 and 𝐿3 are defined
according to the L matrix with the following equations:
25
𝐿2 = −∑∆𝑝𝑒 ∀𝑒 ∈ 𝑆𝐿2 +∑sgn(𝑒)∆𝑝𝑒 |∆𝑝𝑒′ | ∀ 𝑒 ∈ 𝑆𝐿2 (4.2.1)
𝐿3 = −∑∆𝑝𝑒 ∀𝑒 ∈ 𝑆𝐿3 +∑sgn(𝑒)∆𝑝𝑒 |∆𝑝𝑒′ | ∀ 𝑒 ∈ 𝑆𝐿3 (4.2.2)
The 𝑆𝐿2 and 𝑆𝐿3 are the sets of edges that belong to that specific loop, while 𝑒 denotes
an edge and sgn(𝑒) denotes the direction of the edge compared to the loop direction.
The sgn(𝑒)∆𝑝𝑒 term is the pressure loss in edge e (accounting for direction), while
∆𝑝𝑒′ is the first order derivative of the pressure loss in edge e. [17]
The last matrix that needs to be constructed is the NL matrix, this matrix must have
the same number of rows as the previously defined V matrix, the columns in this matrix
represent the edges between vertices. This matrix is constructed from the vertex matrix
and the loop matrix, where the first rows consist of the vertex matrix and the last rows
are the loop matrix where each term is multiplied by the absolute value of the
derivative of the pressure loss function at the previously defined flow in that edge. For
the example network in Figure 8, the NL matrix is defined as:
𝑁𝐿 =
[
−1 −1 0 0 0 0 01 0 −1 0 0 0 00 1 0 −1 −1 0 00 0 1 1 0 −1 00 0 0 0 1 0 −1
1 ∙ |∆𝑝𝑒1′ | 1 ∙ |∆𝑝𝑒2
′ | −1 ∙ |∆𝑝𝑒3′ | −1 ∙ |∆𝑝𝑒4
′ | 0 0 0
0 0 0 1 ∙ |∆𝑝𝑒4′ | 1 ∙ |∆𝑝𝑒5
′ | −1 ∙ |∆𝑝𝑒6′ | −1 ∙ |∆𝑝𝑒7
′ |]
Finally, the new vector of flows for all the edges in the system are calculated according
to equation 6.2.3.
�̇�𝑛+1 = 𝑁𝐿−1 × 𝑉 (4.2.3)
In case some of the flows in the resulting vector are negative, this means that the flow
in that edge will be reversed in respect to the previously assumed flow direction. The
calculated flows in the result vector are used in the next iteration to initiate the solution
procedure again. This procedure is then repeated until the stopping criteria are met.
Usually the stopping criterion will be the satisfaction of Kirchhoff's 2nd law at the
required tolerance. The disadvantages of the Vertex-Loop method include the critical
pre-processing stage where the correct loops are identified. [21] [20]
26
The number of loops needs to be:
L = E − V − 1 (4.2.6)
where 𝐿 is the number of loops, 𝐸 is the number of edges and 𝑉 is the number of
vertices.
However, selecting loops to exclude from the calculation is not trivial. The loop that
is excluded cannot contain any edges or vertices that are not included in at least one of
the included loops. I.e. the complete set of all edges and all vertices in the network
must be possible to construct only from the vertices and loops included in the circuit.
One additional post processing stage is algorithmically calculating the pressures in
each node based on the starting node pressure and the calculated pressure drops in each
node.
4.3 Global Gradient Algorithm
The GGA is a popular foundation of commercial and open-source hydraulic network
solving software. The GGA that will be used was developed by Enzio Todini and
bypasses some of the problems associated with hydraulic network algorithms. [17]
The hydraulic network solving procedure for the GGA starts with categorizing vertices
in the network as known-pressure and unknown-pressure vertices. The known-
pressure vertices are either input or output flow rates to or from the network.
Unknown-pressure vertices are arbitrary connections or junctions in the network
where a solution is of interest. All vertices are connected by edges with initially
unknown flow rates.
The vector of nodal inlet/outlet demands 𝑞 (1 by p) is defined by equation 4.3.1 while
the vector of network pressure losses in each edge ℎ𝐿 is defined by equation 4.3.2.
Where 𝐴 is the network incidence matrix (𝑛 by 𝑝), and F is the vector of flowrates in
all edges (1 by p). The variable n is the number of unknown pressure nodes while p is
the number of edges in the network.
𝑞 = 𝐴𝐹 (4.3.1)
ℎ𝐿 = 𝑅𝐹2 (4.3.2)
27
R is the diagonal pipe resistance matrix where each individual element 𝑟𝑖𝑗 is defined
according to equation 4.3.3.
𝑟𝑖𝑗 = { 0 ↔ 𝑖 ≠ 𝑗
𝛼1,𝑖|𝐹𝑖| + 𝛼2,𝑖|𝐹𝑖| −𝑎𝑖+𝑏𝑖|𝐹𝑖|
𝑐𝑖
|𝐹𝑖| ↔ 𝑖 = 𝑗
(4.3.3)
where 𝛼1,𝑖 is the major pressure loss coefficient, 𝛼2,𝑖 is the minor pressure loss
coefficient and 𝑎𝑖, 𝑏𝑖 and 𝑐𝑖 are the pump coefficients.
The energy equation will take the form:
𝑅𝐹 + 𝐴𝑇𝐻 = −𝐵𝐻𝑓 (4.3.4)
where 𝐻𝑓 is the vector of vertex pressures in all k known head vertices (k by 1) and 𝐻
is the vector of vertex pressures in all n unknown head vertices (n by 1). Lastly, B is
a topology matrix describing the connections of the k known head vertices to the rest
of the network.
Equation 4.3.1 and 4.3.4 can then be combined as in equation 4.3.5 describing the
entire network.
[0 𝐴𝐴𝑇 𝑅
] [𝐻𝐹] = [
𝑞−𝐵𝐻𝑓
] (4.3.5)
Equation 4.3.5 can be reshaped to solve for the nodal pressure estimations in the next
iteration in equation 4.3.6 and the edge flowrates in equation 4.3.7
𝐻𝑖+1 = −[𝐴𝑀−1𝑅−1𝐴𝑇]−1{𝐴𝑀−1[𝐹−1 + 𝑅−1𝐵𝐻𝑓] + [𝑞 − 𝐴𝐹𝑖]} (4.3.6)
𝐹𝑖+1 = [𝐼 − 𝑀−1]𝐹𝑖 −𝑀−1𝑅−1[𝐴𝑇𝐻𝑖+1 + 𝐵𝐻𝑓] (4.3.7)
where I is the identity matrix (p by p) and M is the identity matrix (p by p) where each
element is the flow exponent for the pressure loss formula used (this number is 2 for
the Darcy–Weisbach equation). Convergence has been reached when 𝐹𝑖+1 ≈ 𝐹𝑖 and
𝐻𝑖+1 ≈ 𝐻𝑖 within the specified tolerance. A full derivation of the GGA is provided in
[22].
There are numerous advantages of the GGA. Fast convergence due to the superior
speed at which the Newton iterations are performed is a desirable property often
28
mentioned. The most interesting advantage however is that it is not needed to have an
initial solution which satisfies the mass balance equations for all vertices in the
network, as opposed to the VLM or the HCM which both require an initial solution
which satisfies the vertex mass balances. This advantage enables easier
implementation of extended period analysis solver capabilities. Extended period
analysis is detailed further in section 4.5. The most notable disadvantage is the
increased complexity of the mathematical formulations as compared to the HCM and
VLM. [2] [22]
4.4 Challenges in network solver creation
The main challenges in creating a circuit solver programmatically arise not from
implementing the solver algorithm itself, but from the required pre-processing of the
piping network structure. One of the main challenges in using the network-solving
algorithms based on the balancing of loops, is the identification of the correct number
of minimal loops for matrix-based algorithms such as the VLM. For very large
networks there can be a substantial number of loops that consist of edges and vertices
that can be composed by other loops. As such, including the loop in the calculation is
unnecessary and for very large networks it can have a negative impact on solution
times. For some algorithms, for example, the VLM, the addition of unnecessary loops
will also turn the matrix into a non-square matrix, and thereby non-invertible. As a
result, VLM will not be executable or alternatively, diverge as Moore-Penrose inverses
are not guaranteed to provide convergence. [21] [16] [18]
The usual approach to minimal loop extraction is some variant of Depth-First Search
(DFS) or Breadth-First Search (BFS) coupled with a heuristic that stores vertex
connections and identifies loops. Another challenge is loop initialization, for the
presented Hardy-Cross method and Vertex-Loop method, the initial flows in the
network must for all vertices satisfy the conservation of mass constraint. This can
cause problems as the initialization of flows in edges belonging to loops will be
initialized many times, and the flows are not guaranteed to satisfy the conservation of
mass for all dependent loops as the flow in the independent loop is initialized. This
problem requires robust heuristics for initializing flows that are quite cumbersome to
develop. [17] [16] [23]
29
4.5 EPANET
The EPANET solver uses the GGA for solving the hydraulic steady state. However, a
lot of useful auxiliary formulations are used to add functionality to the solver.
For closing pipes in the network, the EPANET solver applies a linear pressure loss
formula where a high hydraulic resistance value is applied to the pipe in question.
∆𝑝𝑖 = 𝑅ℎ𝑖𝑔ℎ𝐹𝑖 (4.5.1)
Fully open valves are modelled in the same fashion where a very low linear pressure
loss formula where a high hydraulic resistance value is applied to the pipe in question.
∆𝑝𝑖 = 𝑅𝑙𝑜𝑤𝐹𝑖 (4.5.2)
When a pipe is open, the minor loss caused by the valve will still occur. The default
values for 𝑅𝑙𝑜𝑤 and 𝑅ℎ𝑖𝑔ℎ are 10−6 and 108.
The following types of valves are also implemented in the EPANET solver. Pressure
reducing valves (PRV), which act to limit the pressure in the downstream vertex of the
valve to a specified setting 𝑃𝑠𝑒𝑡. Pressure sustaining valves (PSV) which act to limit
the pressure in the upstream vertex of the valve to a specified setting 𝑃𝑠𝑒𝑡. Flow control
valves (FCV), which limit the flow to the user specified flow setting 𝑄𝑠𝑒𝑡 through the
valve by splitting the edge and imposing negative and positive nodal demands at the
outlet and inlet vertices. Throttle control valves (TCV), which have a hydraulic
resistance coefficient that is specified by the user. Pressure breaker valves (PBV),
which specify a pressure loss across the valve which should occur regardless of flow.
Finally, General purpose valves (GPV), which use a piecewise linear curve to relate
pressure loss to flow rate, these are very useful for defining prime movers which have
pressure increase to flow rate relations which are not implemented in EPANET, for
example, positive displacement pumps. A curve can be defined for GPVs with a
negative pressure loss to flow rate relation, which is exactly analogous to pressure
increase to flow rate. The slight modifications to the matrices in the GGA that the
valve implementations entail are detailed in [2].
One other major advantage is the implementation of pressure-driven analysis as an
alternative in EPANET 2.2. This type of network analysis entails a consumption flow
30
rate behaviour in the consumer vertices that is dependent on the pressure in the vertex.
The alternative, demand-driven analysis assumes a flow demand that is independent
of the pressure in the vertex. This can result in unrealistic network solutions where
solutions with negative pressures are accepted as valid. When performing a pressure-
driven network analysis, the demand for each vertex is calculated according to
Wagner’s Equation:
𝑞𝐷𝑖 =
{
𝐷𝑖 , 𝑝𝑖 ≥ 𝑃𝑓
𝐷𝑖 (𝑝𝑖− 𝑃0
𝑃𝑓−𝑃0)
1
𝑒 , 𝑃0 < 𝑝𝑖 < 𝑃𝑓
0 , 𝑝𝑖 ≤ 𝑃0
(4.5.3)
where 𝑞𝐷𝑖 is the actual flow demand in the node, 𝐷𝑖 is the full nodal demand when the
pressure in the consumption node is greater than the maximum consumption pressure
limit 𝑃𝑓. 𝑃0 is the zero-consumption pressure limit and 𝑝𝑖 is the actual consumption
node pressure. Finally, 𝑒 is the pressure exponent, which has a default value of 0.5.
The pressure exponent in equation 4.5.3 models the exact relationship between vertex
pressure and nodal demand. This is useful for calculating actual consumption
behaviour for different types of consumers. [2]
4.5.1 Extended period analysis
Analyzing the behaviour of a hydraulic network over an extended time period is known
as Extended Period Analysis (EPA). This is a pseudo-dynamic approximation of
changes in the hydraulic state of the network. EPA analysis is mainly used to update
the network state when known pressure vertices such as tanks have changing fluid
levels due to in/outflow and thereby updating the static pressure in the tank.
The volume contained in a known head vertex is determined according to
𝑑𝑉𝑠
𝑑𝑡= 𝑄𝑠 (4.5.4)
where 𝑉𝑠 is the liquid volume contained in the tank and 𝑄𝑠 is the flowrate to or from
the tank vertex. In systems with vertically uniform tanks, where the fluid level
31
decreases linearly, the level and therefore the static pressure is updated according to
the flow rate out of the tank with equation 4.5.5.
𝑑ℎ𝑡+1 =𝑑𝑉
𝐴𝑐 (4.5.5)
where ℎ𝑡 is the fluid level in the tank at timestep 𝑡, and 𝐴𝑐 is the cross-sectional area
of the tank.
In tanks or containers with vertically variable geometries, the updated tank volume is
used to calculate the static pressure in the next timestep according to
𝑃𝑠 = 𝐸𝑠 + 𝑌(𝑉𝑠) (4.5.6)
where 𝑃𝑠 is the static tank pressure, 𝐸𝑠 is the minimum static pressure in the tank and
𝑌(𝑉𝑠) is a user defined function which details the relationship between volume and
level. The 𝑌(𝑉𝑠) term allows the user to define tanks of variable geometries. These
equations coupled with equation 4.3.6 & 4.3.7 are used to implement the network
states at user specified time intervals of ∆𝑡. [2] [24]
4.5.2 Hydraulic network logic control
The implementation of timesteps with EPA detailed in chapter 4.5.1 allows for
building conditional logic into the network, where rules closing pipes, opening valves,
changing pump speeds, and so forth can be implemented based on user-defined rules.
An example is provided below written in the .inp file syntax used by the EPANET
solver.
RULE X
IF SYSTEM CLOCKTIME >= 6
AND SYSTEM CLOCKTIME < 10
AND TANK 1 LEVEL BELOW 15
THEN PUMP 335 SPEED IS 0.9
AND PIPE 58 STATUS IS OPEN
AND NODE 23 STATUS IS OPEN
This allows for calculating many networks with complex conditional automation and
behaviours. For example: filling many tanks from one large storage tank if the fluid
level is above the minimum level or varying pump speeds based on the pressure in
some vertex. [2]
32
4.6 Accounting for compressibility
The EPANET solver cannot handle compressible fluids as is, and this entails some
challenges in accounting for compressibility with reasonable accuracy.
Incompressible pressure loss calculations have built-in thermodynamic assumptions.
In order to achieve incompressibility and thereby regarding the density of the fluid as
constant, the system must be considered as adiabatic and isothermal, as temperature
fluctuations will cause a varying fluid density and viscosity. Furthermore, liquids have
densities that vary only marginally with the pressure of the liquid.
For gases, however, the density will vary substantially with pressure. For pressure loss
calculations in piping networks, this entails an increased volumetric flow rate in the
direction of decreasing pressure due to the expansion of the gas. This will in turn
increase the pressure loss in the system as the flow velocity increases.
For systems with very low total pressure losses, this effect is negligible as the decrease
in density is only marginal. The limit on the total network pressure losses dictating
when accounting for compressibility is necessary to achieve the desired accuracy is
dependent on how closely the gas behaves as an ideal gas. Standard practice limits for
when compressibility needs to be accounted for vary between 5-20% total pressure
loss. In case that the downstream pressure is not greater than approximately 40% of
the inlet pressure in the pipe, the standard practice way of accounting for
compressibility is to calculate fluid properties at the average of the incompressible
pressure loss inlet and outlet conditions [13]. This enables an approach of
approximating compressibility in piping networks that have bridges i.e. edges that if
removed would separate the graph into two disconnected sub-graphs. The approach
implemented is structured as follows.
1. Identify bridges in the network
2. Solve the network with an incompressible fluid.
3. Identify network vertices where the pressure has dropped to a certain pressure
limit 𝑃𝐿, check if the edges leading out or in of the vertex are bridges.
4. If either the vertices inlet or outlet edges are bridges, split the network into 2
subnetworks. Calculate the pressure losses intended for use in the first network
first, use the conditions at the outlet vertex to recalculate the fluid properties
33
for using when solving the second network. If the pressure loss in any vertex
in the second network decreases more than the limit 𝑃𝐿, split the sub-network
further until the total pressure loss in the last sub-circuit no longer exceeds 𝑃𝐿.
5. If neither the vertices inlet nor outlet edges are bridges, go to a previous vertex
in the topologically ordered network until a bridge is found and go to step 4. If
no bridges are found connected to any of the previous vertices, raise an error.
Each subnetwork will first be calculated as incompressible and then the solution will
be iterated with fluid properties recalculated at the mean pressure in the subnetwork.
Using this approach to approximate compressible pressure losses might require some
trial and error, as the user-defined limit will determine the needed number of sub-
networks. For very low limits, appropriate subnetworks might be impossible to locate,
especially for very parallel or circular piping networks.
4.7 Network temperature distribution
Heating and cooling of fluid streams are an integral part of power plant process
technology. To get an idea of the temperature in the circuit caused by heating and
cooling, the water quality modeling capabilities of the EPANET 2.2 were modified to
represent temperature distribution. The key capability used was the EPANET 2.2
solvers capability to model a non-reactive tracer substance distribution throughout the
network. Heat load source vertices and known head vertices are modelled as chemical
mass booster vertices, where the mass of the chemical injection is taken to be exactly
analogous to the temperature at an appropriate scaling factor. The disadvantage of
using this method is the fluid properties are not impacted by the varying temperature.
As such it is the users responsibility to determine a suitable mean temperature to be
used for the fluid properties. If the temperature will impact the fluid properties
drastically, as is the case for many longer-chain hydrocarbons, the network can be split
after the heat load source and the fluid properties recalculated at the heat load vertex
outlet temperature. The mass booster vertices can have negative or positive mass
injections, modelling cooling or heating respectively.
The normal approach to mixing at network junctions where pipes with different fluid
temperatures join, is coupling mass and energy balances with known inflow conditions
34
to calculate the outflow condition. However, this approach is infeasible for some
network types where a heat load is added to the circuit during recirculation. As with
each timestep, the temperature will increase in case no regard is taken to model the
behaviour of individual volume segments of fluid in the pipes, an example is provided
in Figure 10.
Figure 10: Recirculation in a vertex
To solve this problem the Lagrangian transport algorithm for tracer concentration
splits at junctions is used by the EPANET hydraulic network solver which is detailed
in section 4.5. The Lagrangian transport algorithm found in Figure 11 uses a
distribution model at junctions, which is based on the interaction between individual
volume segments in the system, this will ensure that the concentration will not
approach infinity when a tracer is added during recirculation. The timesteps at which
the new concentration segment is added will be smaller than the timesteps used in
Extended Period Analysis by the EPANET solver. [2]
35
Figure 11: Lagrangian transport algorithm [2]
This formulation is central to the temperature calculations in the network as they will
be implemented by adding an analogous concentration of a tracer substance to each
starting vertex and heat load vertex in the network. The tracer substance does not
impact the hydraulic properties of the specified fluid and is as such an exact analogy
of how the temperature will distribute itself in the network. [2]
36
5 Calculation library
The piping network solver developed in this thesis work is built-in a Python library
capable of unit handling, result visualization, fluid property calculation, compressible
fluid handling, network structure defining, report creation, and handling pumps & heat
exchangers. The library is named Pcalcs. Result and input data visualizations are
implemented by the Matplotlib and Seaborn libraries. Tensor manipulations, data
formatting, and mathematical operations are performed with the Numpy, SciPy, and
Pandas libraries.
5.1 Physical quantity control
For this toolkit, the Pint library was chosen [25] for implementing physical quantity
handling and control of units. The Pint library enables the rigorous control of units, as
each variable is converted to a data types that have 2 attributes, magnitude and
dimensionality. The dimensionality represents the components of the unit, for example
the unit feet/second and meter/minute both have the dimensionality length/time, and
conversion of the magnitude in feet/second to meter/minute is thus possible. If the user
tries to convert a unit to a dimensionality that is different from the source unit
dimensionality, an error will be raised by the Pint library. This functionality
automatically ensures that unit conversion will be robust. Normal arithmetic
operations also have built-in automatic conversion, such that the resulting unit can then
be converted to an appropriate unit.
The Pint library also has built-in functionality that allows the user to define custom
units for any dimensionality if needed. The Pint library also enables some dimensional
analysis through inbuilt Buckingham Pi theorem calculation capabilities, if necessary.
[25]
5.2 Python source code documentation
The network solver is built-in an existing Python library. The library was created as
an internal development project at Citec to provide a collection of engineering
calculation capabilities with intuitive and flexible interfaces. The Pcalcs code library
is based on well-documented and established packages. Documentation of the key
classes and functions in the code library is provided by documentation comments
37
which are written in reStructuredText (RST) [26] within each function and class. The
RST documentation strings are then combined with general documentation text and
converted to appropriately structured HTML documentation pages. The
documentation home page can be found in Appendix A.
5.3 Library structure
The library is split into separate packages. Class and function implementations are
split into modules which are separate .py-files based on subject. The central modules
in the main package are:
▪ pcalcs main package
▪ pcalcs.circuit contains the network solver
▪ pcalcs.circuit.gui GUI for swiftly defining a piping network including all
relevant metadata
▪ pcalcs.tests package which contains tests for the library
▪ pcalcs.medium contains fluids properties and fluid handling functions
▪ pcalcs.pumps contains pump curves and related functions
▪ pcalcs.heat contains heat transfer functions
▪ pcalcs.units contains functions and classes to handle units
5.4 Graphical user interface
The user can load an existing piping network from a .json file or create a new file with
the option boxes in Figure 12.
38
Figure 12: Loading screen of the GUI
The networks (Figure 13) are built and edited using the graphical user interface (GUI)
application and stored as JSON files. It is also possible to generate JSON files from
Excel inputs or a script.
Figure 13: Example network in the GUI
39
Each vertex and edge have the following fields available:
• Width, Height: Defines the size of the vertices. These fields are not applicable
for edges.
• Name
• Description: Text input that describes the vertex or edge for documentation
purposes.
• Input: Text input that is parsed into the Properties table. The syntax
is Key: value, where each key is on a new line. The keys are used to define
inputs relating to the piping system.
The GUI can also solve the circuit, define prime movers and export results.
5.5 Testing and version control
Version control for the entire toolkit and calculation modules is implemented with the
Git version control system. The Pcalcs code base is located in a remote server. The
convention for developing or maintaining the code base is making all changes on a wip
(work in progress) branch. After work is completed (including developing new tests)
the entire test sequence is run to ensure that every test will pass before committing
changes and pushing to the remote server. Once changes are made and documented in
commits, a pull request is issued to the dev branch. The dev branch is then merged into
the master branch and published under a new version when the maintenance or
development is completed.
The calculation modules are located in another Git repository. The version control
convention for the calculation modules is developing standard calculation report
module templates in a system-modules branch. When initializing project work, a new
branch will be created based on the system-modules branch. When work is completed
in the project, the branch will be committed and pushed to the server for future
reference.
To ensure the correctness and accuracy of the calculations and dependable deployment
of the solver to all end users adequate testing of the written code is needed. The Pytest
library is used to create and structure test frameworks. The test suite was constructed
mainly for the code produced for this thesis, and will not test any of the dependencies
by default.
40
The most central aspect is to ensure correctness of the pressure loss calculations. The
pressure loss was tested against an existing pressure loss template, a separate
implementation using [27] and a completely from scratch implementation of a pressure
loss calculation. All methods used the Swaimee Jain friction factor for a pipe with the
internal diameter of 54.5 mm, a pipe roughness of 0.05 mm, and a length of 100 m.
The test passed with an assertion that the pressure losses calculated by the EPANET
implementation need to be within 1 % for all comparisons.
The temperature distribution approximation was also tested by creating and solving a
circuit with the specified heat load in a certain vertex and comparing the results to
negative and positive heat loads at different flows and comparing these to an energy
balance equation. Temperature distribution in mixing vertices with several inputs and
outputs was also tested by solving a network comparing to manually calculated energy
and mass balances. These tests pass with 0.01 % accuracy, the difference is assumed
to be due to numerical inaccuracy.
The custom implementation of a screw pump was also tested to ensure expected
behaviour when operating in pressure compensated and non-pressure compensated
mode.
41
6 Automatic calculation report creation
The report creation is differentiated by the actual calculations and the visualisations
presented in the reports.
6.1 System specific report creation and structuring
The reports detailing the calculation methods and results were created with the central
design criterion of reusability. Calculation templates are created for what is perceived
to be a standard design case. The standard design case consists of the network design
and parameters which are most commonly available for each system.
To simultaneously achieve flexibility and scalability, the calculation modules are
created from Jupyter Notebooks. The notebooks are server-based open-source
applications which among many other functions provide Python code execution
capabilities, data visualization, Markdown and LaTeX typesetting. To convert the
notebooks into pdfs with the desired appearance and functionality, LaTeX formatted
documents are created with the MiKTeX distribution.
Each module consists of text and calculation sections together with visualization of
parameters of the calculation results. One central functionality which was
implemented is the serialization of custom data types into the JSON format. In the case
of modules that include calculation sections that yield results that influence automatic
text generation in other notebooks, the resulting data must be shared with the other
modules. The produced data types are serialized into JSON files which are then
deserialized in the local variable space of the module in which the object is needed.
The JSON files can be stored in the generated directory of each calculation module,
found in Figure 14.
To generate a calculation report from the calculation modules, each module is included
in a master report notebook, where the structure and order of the final report is defined.
The master report notebook also creates a table of contents for the report and a title
page is generated as well. The report structure can be found in the tree structure
observed in Figure 14.
42
Figure 14: Calculation report tree structure
These tree structures also contain JSON, TEX and image files that are included in the
report creation. The JSON files are used to read in more substantial data such as pump
data, input variables or circuit properties (vertices, edges, connections and related
metadata for example pipe lengths and sizes). The TEX files are LaTeX templates
generated in the notebooks for the table of content generation, document typesetting
purposes and title page structure generation. The system-specific calculation reports
have a title page that includes the document metadata shown in Figure 15 and a table
of contents can be found in Figure 16.
43
Figure 15: Example document metadata
Figure 16: Example table of contents
Depending on the particularities of the system, sections can be added if appropriate. In
the Background and theory section the formatted equations, symbol definitions,
default assumption parameters and underlying methods used are detailed for report
reproducibility. The Medium properties details the equations, conventions, standards
and sources used to determine fluid properties. When displaying or delivering any type
of calculation report detailing pressure losses, temperature versus viscosity and density
plots such as Figure 1 & Figure 2 are included.
In the System Overview section, a high-level overview of the sub-systems to be
calculated are given, together with a flow diagram of the specific system.
44
Figure 17 shows a simplified flow diagram of a lube oil system included in the System
Overview section.
Figure 17: System overview P&ID diagram
In the following section Specific system parameters, a block diagram of the edges and
vertices in the system is presented together with the input used for each edge and vertex,
Figure 18 shows an example of the block diagram inputs used for an example circuit.
The input data table for vertices and edges of the network in Figure 18 will not be
included as these are found only in a tabular format of several hundred rows, to provide
reproducibility in an actual calculation report.
45
Figure 18: Example piping network
In the Calculation results section, the results are shown for the perceived area of
interest, alternatively the entire results can be shown in a tabular format. In Figure 19
the calculation results of the same example network in Figure 18 can be observed from
a single figure where edge colors signify flowrate and vertex colors represent the
pressures in vertices, the solution time for one timestep of this network was 1471 ms.
Tabular results are used for results where exact values for each result parameter is
available for each vertex and edge according to their respective colormaps. The
46
parameters include but are not limited to: velocity, pressure change, temperature
change, flowrate and pressure.
Figure 19: Example network result visualization
The last section in the report is the Conclusion section, certain system specific
conclusion texts can be automatically generated based on the relevant result data in the
Calculation results section. Custom conclusion text is written by the user. It is also
possible to include appendices with relevant information or used data-sheets.
47
6.2 Master project report creation
The directories are structured to allow for all system-specific reports to be combined
into a project report that includes all the system-specific calculation reports that are
needed for the project. The heading structure in the project-specific reports can be
offset to any degree specified by the author in the master notebook. Auxiliary
information such as site layouts, project-level assumptions and weather data or
customer specifications can be included at the beginning of this report.
The minimum and maximum ambient design temperatures of the power plant location
can be compared with historical temperature data to verify design temperatures or act
as supporting information in key design decisions. The weather data are obtained from
the Meteostat database through a python-based application programming interface
(API). The Meteostat database consists of open data from many different sources
detailed in [28].
The site coordinates are provided to the API which will return the weighted average
temperature data of multiple nearby weather stations for the specified time interval.
For most databases, more than 20 years of data have been registered, which will
include very extreme weather cases for referencing. Point data provide a more
complete dataset over the specified time, as data from multiple stations are included.
As mentioned, the data output is being interpolated based on the geographical distance
between the different weather stations and the reference point provided in the database
query, point data are also adjusted for differences in altitude. [28]
The city of Christchurch, New Zealand (latitude: -43.531656, longitude: 172.630584)
was randomly generated as an example site location. Three different weather stations
found in Table 1 were available to use for data interpolation.
48
Table 1: Example site weather data
Data 93780 93781 93773
name Christchurch Christchurch Aerodrome Aws
Timaru Aerodrome
Aws
country NZ NZ NZ
region CAN CAN CA
latitude -43.4833 -43.4833 -44.3
longitude 172.55 172.5167 171.2333
elevation 38 37 26
timezone
Pacific/Aucklan
d Pacific/Auckland Pacific/Auckland
distance 8642.05 10991.87 141804.50
The elevation and distance between the site and weather station in Table 1 are provided
in meters. The distance signifies the distance between the site and the weather station.
The resulting daily minimum, average and maximum temperatures can be plotted
against the design temperatures, as in Figure 20.
The Meteostat database can also provide humidity data, ambient pressure and wind
speeds for many site coordinates. This data can be useful for future development. [28]
49
Figure 20: Example site temperature variation
50
7 Results
Some of the calculation modules are presented included in this section.
Morphologically similar systems or systems that show confidential system design are
omitted. Systems, where hydrocarbons are included, are normally calculated for an
entire temperature interval, as these systems usually are extremely sensitive to
temperature variations. Other systems are calculated for the minimum and maximum
ambient temperatures.
7.1 Fuel gas distribution
Fuel gas distribution is a common system included in many combustion power plants
using natural gas or alternatively biogas as fuel. Fuel gas is delivered to power plants
commonly by pipeline and this is also one of the base assumptions underlying the fuel
gas system report. The first part included in the calculation module is the incoming
fuel gas pressure reduction to plant appropriate levels. This task is usually performed
by a control valve which is controlled based on the outlet pressure. As the pressure of
incoming fuel gas is reduced, a temperature decrease will occur due to the Joule-
Thompson effect [1]. As the fuel gas to be used usually will need to be kept at
appropriate temperatures, a temperature regulation system will need to be
implemented. A simplified example system is shown in Figure 21.This report will
assume a target temperature of 20 °C. As the enthalpy of the fuel gas after the pressure
reduction station is fixed due to constant target temperature and pressure, calculating
the cooling or heating need is done by the difference in the enthalpies of the incoming
and outgoing gas.
51
Figure 21: Simplified gas pressure reduction system
The results can be generated within the needed pressure and temperature interval by
iterating the solution with appropriately chosen intervals for the inlet temperature and
pressure. The visualization chosen was a heat map showcasing the bivariate data
generated. The results for a constant fuel gas mass flow rate with an outlet temperature
and pressure at 20 °C and 5 bar can be found in Figure 22. Negative values entail
cooling and positive values entail heating.
Figure 22: Inlet fuel gas heating demand
52
As the outlet flow rate, pressure and temperature are constant, the distribution within
the plant is based on the outlet setpoints for the pressure and temperature of the
pressure reduction.
7.1.1 Fuel gas distribution
The distribution of fuel gas to consumers around the plant is normally performed by
splitting branches from the incoming fuel gas header, an example network is shown in
Figure 23. The consumers are modelled to consume a set amount of fuel gas
independent of the consumer inlet pressure. The results of the calculation are
visualized by a network graph, if appropriate.
Figure 23: Network structure
The calculation results can be overviewed from Figure 24. The edge colors signify
flowrates while vertex colors signify pressure.
53
Figure 24: Vertex pressure and flow rate
A tabular format of the calculation results is always provided for non-trivial circuits as
large amounts of relational data can be presented in this way. For large circuits that
are morphologically symmetrical with regards to sizing, pipe lengths and minor losses,
results are presented for only the key vertices and edges (usually the topologically
sorted first and last vertices are included). Usually, the inlet pressure to the consumers
is of interest, as such a separate table is generated detailing the consumer inlet pressure
and the total pressure loss as compared to the pressure reduction outlet pressure. The
table constructed for showing consumer inlet pressure can be found in Table 2.
Table 2: Consumer inlet pressures
54
In fuel gas systems, a common limiter of pipe size dimensioning is the flow velocity.
In this example a commonly used limit of 20 m/s is imposed on the network. In order
to examine high flow velocities and thereby identify possible problem areas, the k:th
percentile of flow velocities can be automatically shown in a separate table. The 80:th
percentile of flow rates with the initial pipe sizes can be found in Table 3.
Table 3: 80:th percentile of fuel gas flow velocities with initial pipe sizing
The 20 m/s limit is exceeded in certain parts of the circuit. The pipe sizing is therefore
updated to DN300 for the pipes exceeding the velocity limit. The updated results for
the same edges can be found in Table 4. All the flow velocities are now below the
upper bound of 20 m/s.
Table 4: 80:th percentile of fuel gas flow velocities with updated pipe sizing
All flow rates are now below the specified upper bound of 20 m/s. The updated inlet
pressures and total pressure drop percentages are shown in Table 5.
55
Table 5: Updated consumer inlet pressures
This type of result enables swift validation of flow velocities in the entire fuel gas
distribution circuit.
56
7.2 Lube oil system
Many power plant types using roto-machinery for power generation or auxiliary tasks
need lube oil handling systems. The calculation module created here will assume that
the lube oil handling system is composed of the following parts:
▪ Unloading to storage tank from delivery vehicle
▪ Distribution to final lube oil consumer units from storage tank
▪ Removal of lube oil from consumer units to used lube oil storage tank
▪ Loading of lube oil from used lube oil storage tank to removal vehicle
The general structure of the example lube oil system can be found in Figure 25. As
lube oil becomes more viscous at lower temperatures, positive displacement pumps
are most commonly used in lube oil handling and pumping. As the viscosity increases
with decreasing temperature the pressure loss in the discharge line will increase,
depending on the hydraulic resistance in the discharge line, this can cause recirculation.
The lube oil can be emptied from the lube oil consumer units while still significantly
above the maximum ambient temperature, Figure 26 displays the results if the oil is
allowed too cool to ambient temperature. The lube oil calculation is performed for
each system at 1°C intervals between the minimum and maximum specified
temperatures. If the pump specific 𝑁𝑃𝑆𝐻𝑟 data is available, the calculated 𝑁𝑃𝑆𝐻𝑎 for
the pumps in each sub-system can be compared visually in Figure 27.
57
Figure 25: Example lube oil system
The flow rates and filling times can be shown independently for each sub-system, an
example from the same sub circuit can be found in Figure 26.
58
Figure 26: Flow rate and filling/emptying time versus lube oil temperature
59
To ensure that the 𝑁𝑃𝑆𝐻𝑎 is larger than the 𝑁𝑃𝑆𝐻𝑟, the results can be shown for each
sub-system as in Figure 27. At temperatures where the 𝑁𝑃𝑆𝐻𝑎 is smaller than the
𝑁𝑃𝑆𝐻𝑟 operation of the pump is considered infeasible as this would cause damage to
the pump. In this case, the 𝑁𝑃𝑆𝐻𝑟 is flow dependent and is shown for flow rates of 5
m³/h and 10 m³/h.
Figure 27: NPSH comparison
60
7.2.1 PRV operational pressure setting
The operational pressure of the PRV is the pressure at which it will open, allowing
fluid to be recirculated or expelled from the pipe, dependent on where the discharge
side is connected.
Figure 28: Pump PRV recirculation example
The PRVs operational pressure setting plays a central role in ensuring a sufficient
𝑁𝑃𝑆𝐻𝑟 and flow to all destination vertices. If the discharge piping has very high
hydraulic resistance a higher operational pressure setting is needed for the PRV. As
this will allow for a larger pressure loss from the pump discharge to the outlet of the
piping. However, a higher PRV operational pressure can cause problems in ensuring
sufficient 𝑁𝑃𝑆𝐻𝑎. Due to allowing a higher flow in the system before the recirculation
reduces the flow rate and therefore the pressure loss in the suction piping.
The importance of the PRVs operational pressure setting is demonstrated by
calculating the results of 2 different cases, one case where a transfer pump is used to
pump lube oil to the first consumer (Figure 29) and another case where a transfer pump
is used to pump lube oil to the last unit (Figure 30). For demonstration purposes a
constant 𝑁𝑃𝑆𝐻𝑟 of 60 kPa(a) will be assumed for the pump model. Assuming a
minimum temperature of 15 °C the PRV will first be adjusted such that the system can
provide the maximum flow at temperatures above 15 °C.
61
Figure 29: Transfer from a lube oil tank to the first unit
Figure 30: Transfer from a lube oil tank to the last unit
At a PRV operational pressure setting of 500 kPa, the 𝑁𝑃𝑆𝐻𝑎 can be observed in
Figure 31 to reach levels that are far below the specified 𝑁𝑃𝑆𝐻𝑟 . The flow rates
dependence on the lube oil temperature can be observed in Figure 32. The system will
allow for the nominal flow of 10 m³/h at lube oil temperatures above 6 °C for filling
the first unit and 10°C for filling the last unit, whereas the 𝑁𝑃𝑆𝐻𝑎 exceeds 𝑁𝑃𝑆𝐻𝑟 at
temperatures of 12 °C and above. As such pump operation is possible at temperatures
of 12 °C and above.
62
Figure 31: NPSH available at a PRV setting of 500 kPa(a)
63
Figure 32: Flow and filling time at a PRV setting of 500 kPa(a)
To ensure the 𝑁𝑃𝑆𝐻𝑎 will be sufficient for the entire temperature interval, the PRV
operational pressure setting was adjusted to 300 kPa. The results can be observed in
Figure 33 and Figure 34.
64
Figure 33: NPSH available at a PRV setting of 300 kPa(a)
The 𝑁𝑃𝑆𝐻𝑎 is now within the allowed region, however the flow rate is reduced due
to recirculation at temperatures below approximately 20 °C when pumping lube oil to
the last consumer, and 17 °C when pumping lube oil to the first consumer. The flow
and filling times dependency on the temperature can be found in Figure 34.
65
Figure 34: Flow and filling time at a PRV setting of 300 kPa
The most common solution to this problem is increasing the pipe sizes in the system
to reduce the pressure drops and therefore allow the nominal flow to be delivered for
the entire temperature interval. In case pipe sizes are fixed due to external reasons, the
pipe routing can be restructured according to Figure 35 and Figure 36. If possible, this
is the preferred circuit structure as it reduces the difference in hydraulic resistance of
the discharge piping when filling the first and last units.
66
Figure 35: Rerouted piping for last unit filling
Figure 36: Rerouted piping for first unit filling
The resulting flow rate, filling times and 𝑁𝑃𝑆𝐻𝑎 can be found in Figure 37 and Figure
38. The PRV operational setting was adjusted to 325 kPa in order to achieve the desired
results.
67
Figure 37: Restructured flow and filling times with a PRV setting of 325 kPa(a)
The flow rate can be observed to remain nominal for temperatures above 15 °C in
Figure 37. The 𝑁𝑃𝑆𝐻𝑎 can be found to be within the acceptable limits in Figure 38.
68
Figure 38: Restructured network NPSH with a PRV setting of 325 kPa(a)
For networks that have fixed pipe sizes and cannot be restructured, a good solution is
using a frequency converter to adjust the flow in the network based on the 𝑁𝑃𝑆𝐻𝑎 on
the suction side of the pump.
Another solution is using a heater to increase the lube oil temperature to decrease the
lube oil viscosity and thereby lowering the pressure losses. This solution might be
more expensive than using a frequency converter as there will be heating costs
associated with this approach, it will however allow for the full nominal flow of the
pump.
69
7.3 Fuel oil distribution
The calculations for the fuel oil distribution system detail the actual flow rates for the
network structure and the 𝑁𝑃𝑆𝐻𝑎 for all pumps included.
The fuel types available for use in the calculation templates are:
- LBF (Liquid Bio Fuel)
- LFO (Light Fuel Oil)
- HFO (Heavy Fuel Oil)
- CRO (Crude Oil)
The first parts of these systems consist of unloading pumps and one or more storage
tanks. It is important to consider the inlet nozzle elevation in the calculations, to
account for the needed static pressure increase. From the storage tanks, the fuel oil can
be pumped directly to the consumers with a fuel oil distribution pump or with a transfer
pump to an intermediate storage tank for treatment. The fuel oil calculation module
section also includes a separator circulation section between two tanks that is added to
the report in case a fuel oil separator is used.
An example fuel oil system structure can be found in Figure 39. The control for
sufficient 𝑁𝑃𝑆𝐻𝑟 for the transfer and unloading pumps is nearly identical to what is
presented for the lube oil system in section 7.2, and will therefore be omitted from this
section.
Depending on the fuel oil viscosity, both positive displacement and centrifugal pumps
can be used for fuel oil handling. This calculation module implementation uses screw
pumps.
The fuel oil distribution to consumers is in this calculation module performed with a
VFD-controlled pump. The pump maintains the pressure at the discharge side of the
pump to a specified setpoint below the shut-off head of the pump. The pump must also
be capable of providing a flow above the maximum consumption for all consumers.
The consumers are modelled with the pressure dependent analysis capability as having
a minimum pressure at which they can accept a flow demand while having a required
pressure at which they can accept the full flow demand.
70
Figure 39: Fuel oil system example P&ID
71
The inlet pressures at each consumer are distributed over an interval with the lowest
inlet pressure at the consumer furthest away from the pump and the highest pressure
at the consumer closest to the pump. An example network was created with 6
consumers, and a distribution pump outlet set point pressure of 700 kPa. The required
pressure is 600 kPa and the minimum pressure is 590 kPa. In Figure 40 an example
distribution of inlet pressures at the consumers can be found for temperatures between
5 ° and 40 °C.
Figure 40: Consumer inlet pressure distribution
For the distribution branch pipe sizing in Table 6 the flow provided to the last
consumers will be impacted at the lowest ambient temperature of 5 °C.
72
Table 6: Consumer inlet results
Possible solutions are increasing the pipe sizing or using a higher discharge pressure
setpoint, to allow for a higher pressure drop while still maintaining the consumer inlet
pressures above the required pressure. Table 7 shows the results when adjusting the
discharge pressure setpoint to 800 kPa.
Table 7: Consumer inlet results with adjusted discharge pressure setpoint
The results when increasing the consumer specific pipe sizes from DN20 to DN25
when maintaining the discharge pressure setpoint at 700 kPa can be found in Table 8.
Table 8: Consumer inlet results with increased consumer specific pipe sizes
73
7.4 Cooling water system
Cooling water systems are an integral part of many power plant types which require
removal of excess heat. In cases where the heat is too low grade for power generation,
radiators can be used to vent the thermal energy to the atmosphere. The cooling water
calculation module features a cooling water system which extracts heat from heat
exchangers and distributes the heated water to radiators, cooling towers or a central
heat exchanger. An example system featuring radiators is shown in Figure 41.
Figure 41: Example cooling water network
The cooling water network has different operating states. The hydraulic resistance in
the BYPASS2 and RECIRCULATION nodes can be adjusted to direct the flow in the
network as desired. The head-flow functions of the prime movers used in the example
cooling water network are shown in Figure 42.
74
Figure 42: Example cooling water network pump curves
The operating points calculated at different resistance values in the BYPASS2 and
RECIRCULATION nodes of the network can be found in Figure 41. The network was
solved with hydraulic resistances at approximately 0, 50 and 100 % for both the
BYPASS2 and RECIRCULATION vertices.
75
Figure 43: Operational points versus hydraulic resistance in key nodes
The impact on the flow rates in key parts of the network are shown in Figure 44 for all
the chosen combinations of hydraulic resistances in the BYPASS2 and
RECIRCULATION vertices.
76
Figure 44: Cooling water network flow rates
7.4.1 Cooling water system header sizing
A common calculation need is sizing the header piping of cooling water distribution
networks. In such applications, sizing of the cooling water header piping used for
distributing the cooling water to the radiators can be used.
Figure 45: Cooling water distribution network
77
The network in Figure 45 was constructed and solved with varying inlet pressure and
header sizes to determine the total available flow rate. The outlet pressure was assumed
to be constant at 200 kPa(a) while the network was solved with the inlet pressure in
the interval 500-800 kPa(a).
Calculating the system results with variable inlet conditions, allows for more accurate
pipe size selection, as the network behaviour can be compared for all cases. The
calculation results are shown in Figure 46.
Figure 46: Cooling water flow rate results
The most suitable pipe sizes can then be selected based on the results in Figure 46. To
calculate these types of results, the inputs to the network solver algorithm needs to be
iterated in nested for-loops, using updated input parameters. The worst-case time
complexity for nested for loops is 𝑛𝑘where k is the number of for-loops. This entails
substantial calculation times when the number of variables in the calculation is large.
The results in Figure 46 were calculated in 12.0 s.
78
7.4.2 Recirculation temperature simulation
The example cooling water system in Figure 41 features recirculation. To demonstrate
the temperature distribution simulation in the system, a simpler network featuring
recirculation was implemented in Figure 47. This calculation is possible due to the
theory presented in section 4.7.
Figure 47: Example recirculation network
The flow and pressure results in the network can be found in Figure 48 with the
following key input parameters:
▪ HL - Heat load: 100 kW
▪ N6 - Consumes: 10 l/s
▪ IN - Pressure: 5 bar, Temperature: 40 °C
▪ PUMP - Constant pressure increase: 40 kPa
79
Figure 48: Recirculation network results
The network was simulated with 100 timesteps of length 1 s. The temperature results
can be found for each node in Figure 49. As can be observed in Figure 49 the
temperatures in the network converge to steady state values. The stepwise behaviour
of the temperature in the HL-INLET and HL-OUTLET nodes is caused by the volume
segments of temperature which have circulated in the recirculation loop mixing with
the inlet from the rest of the network.
80
Figure 49: Recirculation network node temperatures [°C]
81
7.5 Drainage systems
Many power plant systems need to be drained for service. Such systems which utilize
static pressure differences to remove fluid from a source to a destination are referred
to as drainage systems. These systems are implemented according to the theory in
section 4.5.1. Two tanks with variable vertical geometries and one cylindrical tank
were constructed with the volume functions found in Figure 50. All tanks are drained
by identical pipes to a vertex of identical pressure, the resulting draining volume flow
rate as a function of time can be found in Figure 51.
Figure 50: Example volume functions
Figure 51: Drainage flow rates for all 100 m³ tanks
82
As is apparent from Figure 51 drainage times can be heavily dependent on the
geometry of the vessel. One interesting test of the correctness of these calculations is
summing up the flow rates for each timestep, which is equivalent to taking the integral
of the volume flow as a function of time, which will yield the tank volume.
7.6 Instrument air system
Instrument air systems are used for example for control actuation, filter cleaning or
utility purposes. This entails unpredictable instrument air consumption patterns, the
consumers in similar systems are inactive most of the time and as such the systems are
not normally dimensioned for all consumers to be active at once since this would be
exceedingly improbable.
Assuming a consumer can either be active or inactive with a fixed flowrate, entails that
the number of operational states is the binary combinations of all consumers 2𝑛. For a
relatively small network of 25 consumers this entails 225 = 33554432 operational
states. If allowing for multiple consumption rates the number of operational states will
be even larger.
Systems such as this are in the calculation modules performed on a case by case basis
of interesting consumer activation combinations which result in the worst-case
behaviour. In Figure 52, Figure 53, Figure 54 and Figure 55 different operational case
results are shown for an instrument air system with 25 consumers. In these calculations,
the flow consumption in some sections is deactivated or changed for different cases.
83
Figure 52: All consumers active
Figure 53: Operational consumers in case one
84
Figure 54: Operational consumers in case two
Figure 55: Operational consumers in case three
85
8 Conclusion and future development
In toto, the calculation and visualization capabilities, calculation report creation and
auxiliary calculation support functionalities of this toolkit are capable, well
documented and robust. As such, this toolkit will serve as a robust foundation, upon
which, process calculations and optimizations, which dictate process design in
different stages of a project can be conducted. This will serve to provide insights in
early design on piping morphology, equipment location, pipe sizing, pipe routing,
pump selection, inline component selection, energy requirements, the time needed to
conduct specific processes and more. Similarly, in later project stages where more
accurate input parameters are available, validation and final design decisions can be
based of the calculations provided by this toolkit.
Currently, standard templates are developed for many key networks, which are
commonly included in the design scope of Citec. These template projects can be used
to generate project-specific calculation reports by modifying the network structures in
the GUI presented in section 5.4 while providing the needed input data. To generate
custom calculation reports, modification of the python code and autogenerated text
fields is required.
For future development of this toolkit, the calculation capabilities of the Pcalcs library
could be expanded, more templates could be created for all common and often
occurring piping network configurations in the design scope of Citec. In the context of
hydraulic networks, economic optimization of pipe sizing could be a practical
development direction. Where a non-linear cost function describing the operating costs
of the prime movers in the network compared to the total procurement, installation and
transportation cost of the pipe sizing selected for each pipe in the network could be
optimized. The necessary boundary conditions could dictate minimal flow rates to all
consumers, maximum flow velocities, maximum pressures in all nodes, and so forth.
For this type of optimization to be possible, total initial cost functions need to be
established for the piping and included instrumentation and inline assets, such as
valves, filters and fittings.
86
9 Svensk sammanfattning
Denna avhandling ämnar att utveckla ett robust, väldokumenterat och snabbt verktyg
som ger användaren förmågan att utföra och presentera komplexa processtekniska
beräkningar. Till kärnan av arbetet hör implementeringen av en hydraulisk kretslösare
med tillräcklig funktionalitet för att analysera processtekniska förfaranden i alla
behövliga system. Algoritmer som används för att lösa flödes -och trycktillstånd i
komplexa hydrauliska nätverk är t.ex Hardy Cross metoden som var den första
explicita metoden för att lösa ut hydrauliska jämviktstillstånd i rörnätverk. Hardy
Cross metoden utvecklades 1936 och är därför ämnad för att utföras manuellt, därför
finns det många delsteg som är väldigt arbetsamma att generalisera programmatiskt
för ett hydrauliskt nätverk med en godtycklig morfologi. Som alternativ
implementerades en metod av Wood & Charles här benämnd Vertex-Loop metoden.
Vertex-Loop metoden är en utveckling av Hardy Cross metoden, med
omformuleringar baserade på koncept från grafteori. I sammanfattning generaliseras
nätverket som en specifik typ av graf, nätverkets egenskaper kan sedan generaliseras i
matris format för alla noder och bågar i nätverket. Utvecklingen av en hydraulisk
nätverkslösare i sig självt är ett väldigt omfattande arbete i beaktan på all
programmatisk för -och efterbehandling associerad med att lösa fullständigt
godtyckliga hydrauliska kretsar. Därför beslöts att vidareförädla EPANET-lösaren
som distribueras med öppen källkod. [19] [20] [2]
Figur 1: Olika typer av lösningsalgoritmer [18]
87
EPANET-lösaren baserar sig på en implementation av Global Gradient algoritmen,
vilken är en vidareförädling av koncepten i Hardy Cross metoden. Global Gradient
algoritmen har snabbare konvergens och är mera flexibel eftersom den undviker en del
extern bearbetning av indata och utdata. [17]
Behövliga funktioner byggdes vidare på EPANET-lösarens existerande funktionalitet.
Uppskattningar av tryckfall i fluider med densitet som varierar beroende på tryck
utförs med att beräkna icke-komprimerbart tryckfall i olika segment i nätverket. I
noder där trycket sjunkit under en gräns delas nätverket i olika segment vars
fluidegenskaper uppdateras i enlighet med trycket i noden. Detta kan enbart utföras för
noder som skulle separera nätverket fullständigt i två eller flera icke-sammankopplade
nätverk. [2]
Olika typer av pumpar inkluderas i många rörnätverk. Den vanligaste typen av
centrifugalpumpar finns färdigt inkluderade i EPANET. Strukturen av en
centrifugalpump hittas i Figure 5. I EPANET-lösaren beskrivs flöde-tryckhöjnings
förhållandet för en centrifugal pump med ekvationen:
∆𝑝𝑖𝑗 = −𝜔2(𝑝𝑚𝑎𝑥 − 𝑟(𝑄𝑖𝑗/𝜔)
𝑛) (3.4.1)
För att utföra tryckfallsberäkningarna med en noggrannhet som återspeglar
verkligheten måste de fluidegenskaper som påverkar tryckfall etableras med en
tillräcklig noggrannhet för hela temperatur-och tryckintervallen i vilka de används.
Fluidegenskaper som påverkar tryckfall är huvudsakligen densitet och viskositet.
CoolProp programbiblioteket har används för alla fluider som finns tillgängliga [5].
Oljors hydrauliska egenskaper (som vanligen inte är lika väl definierade) har
implementerats baserat på lämpliga standarder [10] [7] [6].
I rörnätverket där fluider med högre viskositet används ersätts centrifugalpumpar
vanligen av förträngningspumpar på grund av centrifugalpumparnas avtagande
verkningsgrad vid högre viskositet. Gränsen för viskositeten vid vilken
förträngningspumpar föredras över centrifugalpumpar beror på många faktorer som
t.ex. temperatur intervall, impellerns och pumpväggarnas konstruktion. [29].
Förträngningspumpar har ungefär samma flöde upp till en maximal tryckhöjning som
pumpen åstadkommer. I praktiken kommer internt läckage att orsaka mindre flöde vid
88
högre tryckhöjningar. Eftersom många typer av förträngningspumpar har en maximal
tryckhöjning som är mycket högre än designtrycket för pumpen, förses dessa med
övertrycksventiler som öppnar när trycket på utloppssidan överskrider ett börvärde.
Flödet recirkuleras vanligen till inlopps sidan. Dylika förfaranden associeras med
tryckackumulering på utloppssidan av pumpen. Fenomenet modelleras enligt Figure 4.
[14]
Funktionaliteten av beräkningsverktyget utvecklas i ett existerande Python
programbibliotek. Programbiblioteket har fått namnet Pcalcs och förser användaren
med funktionalitet som automatiserad kontroll av fysikaliska storheter, autogenererad
dokumentation, visualisering och konvertering av beräkningar till beräkningsrapporter.
Programbiblioteket är utvecklat med avsikt att automatisera grovjobbet associerat med
lösning av rörnätverk genom användningen av ett grafiskt användargränssnitt för att
rita, definiera och lösa hydrauliska jämviktsförfaranden i nätverken. Lösning av
nätverkens tillstånd kan också automatiseras för ändrade temperaturer, tryck,
rörstorlekar m.m. i beräkningsrapporterna.
Modulerade rapportstrukturer har utvecklats för många vanliga processer i kraftverk.
Rapporterna utvecklas med skalbarhet som ett centralt designkriterium. Därför
används Jupyter Notebooks som plattform till rapporterna. Jupyter Notebooks är
applikationer som förser användaren med bland annat exekvering av Python-kod och
visualisering samt textformatering. Jupyter Notebooks kan också konverteras till PDF-
format för rapporter. Exempel på några standard-kretsar som har implementerats är:
▪ Naturgas distribution
▪ Lagring och distribution av smörjolja
▪ Lagring och distribution av olika bränsleoljor
▪ Kylvatten
▪ Distribution av instrumentluft
▪ Distribution av dricksvatten
▪ Startluft
▪ Förbränningsluft
▪ Avgassystem
▪ Distribution av kemikalier
89
Fördelen med den nuvarande rapportstrukturen är att rapporterna genereras på basen
av indata som kan uppdateras på ett enda ställe i rapportstrukturen. Nya rapporter kan
därefter genereras enkelt med uppdaterade resultat. Detta ger en möjlighet att leverera
rapporter på basen av indata som finns tillgängligt under olika skeden av projekt. I
startskeden när konceptuell design utförs kan rapporter skapas på basen av sunda
antaganden och erfarenhet i liknande projekt tillsammans med det indata som finns
tillgängligt. I senare design skeden var mera indata finns tillgängligt kan rapporter med
större noggrannhet genereras.
För utveckling i framtiden kan en ekonomisk-optimering av nätverkets rörstorlekar
inkluderas i Pcalcs biblioteket vara av avsevärd användbarhet. En dylik optimering
borde först beakta de totala anskaffnings-, transport- och installationskostnaderna vid
konstruktionen av ett rörsegment i nätverket genom en fast kostnadsfunktion. Den
andra delen av kostnadsfunktionen skulle bestå av driftskostnader som följer av
pumparnas tryckökningar. Behövliga randvillkor kunde inkludera t.ex. en global gräns
på nätverkets flödeshastigheter och gränser på tillåtet tryck i nätverkets inlopps- och
utloppsnoder.
90
10 References
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Hill, 1965.
[2] “EPANET,” EPA, 31 08 2020 . [Online]. Available: https://www.epa.gov/water-
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[3] R. Manning, “"On the flow of Water in Open Channels and Pipes.",” Transactions
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[4] G. S. Williams and A. Hazen, Hydraulic tables: showing the loss of head due to the
friction of water flowing in pipes, aqueducts, sewers, etc. and the discharge over weirs,
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[5] J. W. S. Q. V. L. Ian H. Bell, “Pure and Pseudo-pure Fluid Thermophysical Property
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[6] A. International, ASTM D341-20e1, West Conshohocken: ASTM International, 2020.
[7] SAE, Engine Oil Viscosity Classification SAE J300, Warrendale, 2015.
[8] A. International, ASTM D1250-19e1, West Conshohocken: ASTM International, 2019.
[9] ISO 8216-1: Petroleum products — Fuels (class F) — Specifications of marine fuels,
Vernier, Geneva: ISO, 2017.
[10] ASTM D396-21, Standard Specification for Fuel Oils, West Conshohocken, PA: ASTM
International, 2021.
[11] R. H. &. G. D. W. PERRY, Perry's chemical engineers' handbook, New York: McGraw-Hill,
2008.
91
[12] Y. H. S.-Y. J. C.-J. C. Mohammad Kilani, “Numerical simulation of flow in a screw-type
blood pump,” Journal of Visualization, vol. 8, no. 1, pp. 33-40, 2005.
[13] CRANE, CRANE Technical Paper 410 US: Flow of fluids through valves, pipes and
fittings., STAMFORD, CT (USA): Crane Valves, 2018.
[14] C. Matthews, A Quick Guide to Pressure Relief Valves (PRVs), WILEY, 2005.
[15] S. V.-S. C. S. Jorge Parrondo, “Development of a predictive maintenance system for a
centrifugal pump,” Journal of Quality in Maintenance Engineering, vol. 4, no. 3, pp. 198-
211, 1998.
[16] V. K. Balakrishnan, Graph Theory, McGraw-Hill, 1997.
[17] K. JHA, “AUTOMATIC MINIMAL LOOP EXTRACTION AND INITIALISATION FOR WATER
PIPE NETWORK ANALYSIS,” International Journal of Simulation, vol. Vol. 8 , no. No 2 ,
2007.
[18] L. A. R. Ezio Todini, “Unified Framework for Deriving Simultaneous Equation,” JOURNAL
OF HYDRAULIC ENGINEERING, pp. 511-526, 2013.
[19] L. K. Eaton, "Hardy Cross and "The Moment Distribution Method", Retrieved April 10,
2011..
[20] H. N. A. U. L. Theory, “Don J. Wood, Carl O. A. Charles,” Journal of the Hydraulics
Division, vol. Vol. 98, no. Issue 7, pp. Pg. 1157-1170, 1972.
[21] P. P. Dejan Brkić, “An Efficient Iterative Method for Looped Pipe Network Hydraulics
Free of Flow-Corrections,” Fluids, vol. 4, no. 2, p. 73;, 2019.
[22] P. S. Todini E, “A gradient method for the solution of looped pipe networks,” System
Analysis and Simulation, vol. 1, pp. pp. 1-20, 1988.
[23] P. R. Bhave, Analysis of Flow in Water Distribution Networks, Lancaster, PA.: Technomic
Publishing. , 1991.
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[24] “Evaluation of the suitability of global gradient algorithm and inverse matrix method
for steady-state analysis of water distribution systems.,” Omer Halisdemir University
Journal of Engineering Sciences, vol. 7, no. 1, pp. 23-33, 2018.
[25] H. E. Grecco, “Pint: makes units easy,” 2021. [Online]. Available:
https://github.com/hgrecco/pint.
[26] D. Goodger, “reStructuredText,” 24 May 2016. [Online]. Available:
https://docutils.sourceforge.io/rst.html#try-it-online. [Accessed 29 Jul 2021].
[27] C. Bell, “fluids: Fluid dynamics component of Chemical Engineering Design Library
(ChEDL),” 2016. [Online]. Available: https://github.com/CalebBell/fluids.
[28] “Meteostat Developers,” 17 06 2021. [Online]. Available:
https://dev.meteostat.net/python/. [Accessed 21 Jun 2021].
[29] A. T. Ippen, “The influence of viscosity on centrifugal pump performance, Trans. ASME,
Vol. 68 (1946), pp. 823-848,” Bethlehem, PA, 1946.
93
Appendix A
This appendix displays the documentation home page for the main Pcalcs code library,
the circuit solver and other documentation are not shown for the sake of brevity.
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95
96
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