Design of Integrated Water System

University of Manchester for the degree of Master of Philosophy
in the Faculty of Engineering and Physical Sciences
2011
2
1.2 Organisation of Dissertation ......................................................... 22
Chapter 2. Literature Survey ............................................................................. 24
2.1 Targeting for Water and Effluent Minimisation .................................. 26
2.2 Network Design of Water and Effluent Systems ................................. 31
Chapter 3. Automated Design of Water Systems with the Consideration of
Pressure Drop and Pumping ............................................................................. 35
3.1 Introduction ........................................................................................ 35
3.3 Mathematical Formulation ................................................................. 40
3.4 Solution Strategy ................................................................................. 49
3.5 Case Study ........................................................................................... 56
3.5.1 Example 1 ................................................................................. 58
3.5.2 Example 2 ................................................................................. 63
4.1 Introduction ........................................................................................ 72
4.3 Mathematical Formulation ................................................................. 81
4.4 Solution Strategy ................................................................................. 92
4.5 Case Study ........................................................................................... 96
4.5.1 Example 1 ................................................................................. 97
4.5.2 Example 2 ............................................................................... 101
Figure No. Title Page No.
Fig. 2.1 Construction of the limiting composite curve 27 Fig. 2.2 Water demand and source composites 28
Fig. 2.3 (a) Water surplus diagram (b) Water surplus diagram with increased freshwater flowrate
29
Fig. 2.4 Superstructure for the integration topology in an industrial park
34
Fig. 3.1 Superstructure representation 37 Fig. 3.2 Node-superstructure representation 38
Fig. 3.3 Schematic representation of water-using operation 40 Fig. 3.4 Solution Strategy in the forth stage 55 Fig. 3.5 Network configuration for Example 1 in Scenario 1 61 Fig. 3.6 Network configuration for Example 1 in Scenario 2 61 Fig. 3.7 Network configuration for Example 1 in Scenario 3 62 Fig. 3.8 Network configuration for Example 1 in Scenario 4 62 Fig. 3.9 Network configuration for Example 2 in scenario 1 67
Fig. 3.10 Network configuration for Example 2 in scenario 2 67 Fig. 3.11 Simplified network configuration for Example 2 in
scenario 2
68
scenario 3 69
scenario 4 70
73
Fig. 4.2 Representation of the effect of buffer tank 76 Fig. 4.3 Representation of the effect of the water
comes/leaves through supplemented pipeline 76
Fig. 4.4 Superstructure representation 77
5
Fig. 4.5 Illustration of the percentage of occurrence 78 Fig. 4.6 Illustration of the maximum possible required
volume 79
Fig. 4.7 Schematic representation of water-using operation 82 Fig. 4.8 Schematic representation of water-using operation 82 Fig. 4.9 Network configuration for Example 1 in all
scenarios 100
Fig. 4.10 Network configuration for the base case of Example 1
101
Fig. 4.11 Network configuration for Example 2 in Scenario 1 105 Fig. 4.12 Network configuration for Example 2 in Scenario 2 105 Fig. 4.13 Network configuration for Example 2 in Scenario 3
106
Fig. 4.14 Network configuration for the base case of Example 2
106
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Table No. Title Page No.
Table 2.1 Process data for water-using operations 27 Table 2.2 Process data for the construction of water demand
and source composites 28
Example 1
Table 3.3 Pressure drops for water-using operations in
Example 1
example 2
Table 3.8 Pressure drops for water-using operations in
Example 2
Table 4.2 Percentage of occurrence for Example 1 98
Table 4.3 Maximum allowable inlet/outlet concentration of
each water-using operation in Example 1
98
Table 4.5 Computational data for Example 1 99
7
Table 4.7 Operating data for Example 2 102
Table 4.8 Percentage of occurrence for Example 2 103
Table 4.9 Maximum allowable inlet/outlet concentration of
each water-using operation in Example 2
103
Table 4.11 Computational data for Example 2 104
Table 4.12 Results for Example 2 104
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Abstract
The thesis is composed of two parts. In the first part, novel automated design
method for the water networks has been developed to consider impacts of
pressure drop and pumping arrangement in the network design. A
superstructure-based optimisation framework has been developed to
systematically consider all the key design issues simultaneously, including
flowrate constraints, pressure-drop constraints and other operational constraints,
as well as to fully accommodate rigorous economic trade-off between fresh
water cost, piping cost, and pumping cost. The proposed optimisation study
enables the identification of the optimal distribution of water within the
network, together with most appropriate location and capacity of pump
required for water systems, which provides a cost-effective and realistic
configuration of water networks. The network complexity has been readily
controlled by imposing design constraints in the optimisation. In the second
part, an automated design method for the water networks under uncertain
process conditions has been developed. To deal with the uncertainty problem,
the installation of buffer tank and the installation of supplementing pipelines
are systematically considered in the design of water networks. The
cost-effective and feasible distribution and reuse of water within the network is
identified from the superstructure-based optimisation model, with which all the
operational constraints and fluctuating operating characteristics for water-using
operations are fully and simultaneously reflected. The optimisation framework
provides rigorous economic evaluation of the network through trade-off
between operating and capital costs under uncertainty. The conceptual insight
related to the design of water networks is incorporated in developing a reliable
solution strategy which can effectively deal with the difficulty in solving the
mixed-integer nonlinear programming problem in both parts. Case studies are
given to validate the proposed models and demonstrate the importance of
considering pressure drop constraints in practice and the uncertain process
conditions in the design and optimisation of water networks.
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Declaration
I declare that no portion of the work referred to in the thesis has been submitted
in support of an application for another degree or qualification of this or any
other university or other institute of learning.
Szu-Wen Hung
May 2011
Copyright Statement
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this thesis) owns any copyright in it (the “Copyright”) and he has given
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Such Intellectual Property Rights and Reproductions cannot and must not
be made available for use without the prior written permission of the
owner(s) of the relevant Intellectual Property Rights and/or
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iv. Further information on the conditions under which disclosure, publication
and exploitation of this thesis, the Copyright and any Intellectual Property
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from the Head of School of Chemical Engineering & Analytical Science.
11
Acknowledgements
This thesis cannot be completed without the contribution and encouragement
from many people. These few lines of acknowledgment only slightly express
my appreciation.
First, I sincerely thank Dr. Jin-Kuk Kim for his supervision and encouragement.
His guidance and support play an essential role in the birth of this thesis.
During the discussion with him, I learn the methods to cope with problems, the
attitude toward research, and the technique of expression. Also, I acknowledge
a valuable help from Prof. Robin Smith, Prof. Krzysztof Urbaniec, and Dr.
Edward Roberts. Their suggestion makes this thesis more complete.
Moreover, I want to express my gratitude to Dr. Nan Zhang and Mr. Simon
Perry. Their classes give me some inspiration for developing the mathematical
models in this thesis and the solution strategies of solving the proposed models.
I also thank to the University of Manchester for providing abundant learning
resource including books in the library, various data bases, and many kinds of
workshops. Besides, I especially appreciate the staffs and students in the Centre
for Process Integration. During the research period of time, they provide
encouraging and welcoming environment for doing research. Many thanks to
Yongwen and Ming for their help in my struggle with GAMS. Besides, I would
like to express my appreciation to Muneeb and Mona for their great help during
the first month I came here. Thanks to Luyi, Jing, Lluvia, Maria, Yuhang, Kok
Siew and so many others for their company.
I deeply appreciate Centre for Process Integration from where I received
financial support, and thanks for giving me the chance to join the research
group in the Centre for Process Integration. Moreover, many thanks to my
parents for their love and suggestion in my life. Thanks them for giving me
freedom to explore my life, and support me all the time.
Szu-Wen Hung May 2011
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Nomenclature
C= {| is a contaminant present in the water }, = 1,2, … ,
Set
S= {| is a fresh water source available }, = 1,2, … ,
E= {| is a discharge point }, = 1,2, … ,
U= {| is a water-using operation in the water network}, = 1,2, … ,
T= {| is a buffer tank available }, = 1,2, … ,
Fs,u = freshwater low from water source to operation
Variables for water flow used in Chapter 3
Fu = total water low through the operation
Fu,e = water low from operation to discharge point
Fu,ua = water low from operation to operation
operates at maximum water lowrate
Variables for water flow used in Chapter 4
Fs,u max
= freshwater low from water source to operation when operation
Ft,umax =
water low from buffer tank to operation when operation
Fu,t max =
water low from operation to buffer tank when operation
Fu,e max =
water low from operation to discharge point when operation
operate at maximum water lowrate
Fs,u n
operates at nominal water lowrate
13
Ft,un =
Fu,t n =
Fu,e n =
operate at nominal water lowrate
Fs,u n
operates at minimum water lowrate
Ft,umin =
Fu,t min =
Fu,e min =
operate at minimum water lowrate
Fu,ua = water low from operation to operation
Ft,u avg = average water low from buffer tank to operation
Fu,t avg = average water low from operation to buffer tank
Cc,u in = the inlet concentration c stream entering operation
Variables for concentration
Cc,u out = the outlet concentration c stream leaving operation
Cc,t = the concentration c water in the buffer tank t
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Au,ua
= cross sectional area of a pipe connecting operation u and operation

At,u = cross sectional area of a pipe connecting and buffer tank t and
operation
Au,t
= cross sectional area of a pipe connecting operation and and buffer
tank t
Au,e
= cross sectional area of a pipe connecting operation u and discharging
point
Variables associated with cross-sectional area of pipeline
As,u = cross sectional area of a pipe connecting freshwater source s and
, =
pressure drop in pipe between fresh water source s and operation
Variables for pressure calculation
= pressure at the discharge point
, =
pressure drop in pipe between operation u and operation
, =
pressure drop in pipe between operation u and discharge point
= pressure elevation of the pump at fresh water source s
, = pressure elevation of the pump between water source s and
operation u
, = pressure elevation of the pump between operation u and
operation ua
, = pressure elevation of the pump in the front of operation u
15
, = pressure elevation of the pump at the end of operation u
, = pressure elevation of the pump between operation u and
discharge point e
Costs,u pipe = piping cost of the connection between freshwater to
operation
Costttank = cost of buffer tank
Costt,u pipe = piping cost of the connection between buffer tank to
operation
Costu,t pipe = piping cost of the connection between operation to buffer
tank
Costu,ua pipe = piping cost of the connection between operation to
operation
Costu,e pipe = piping cost of the connection between operation to
discharge point
Costs pump,cap = capital cost of pump at freshwater source
Costs,u pump,cap =
capital cost of pump between supply water source and operation
Costu,ua pump,cap =
capital cost of pump between operation and operation
Costu in,pump,cap = capital cost of pump in the front of operation
Costu out,pump,cap = capital cost of pump at the end of operation
Costu,e pump,cap
= capital cost of pump between operation and discharge point
Costs pump,op = operating cost of pump at supply water source
Costs,u pump,op =
operating cost of pump between supply water source and operation
Costu,ua pump,op =
operating cost of pump between operation and operation
16
Costu in,pump,op = operating cost of pump in the front of operation
Costu out,pump,op = operating cost of pump at the end of operation
Costu,e pump,op
= operating cost of pump between operation and discharge point
, = binary variable for stream from water source to operation
Binary variables
, = binary variable for pipeline from buffer tank t to operation
, = binary variable for pipeline from operation to buffer tank t
, = binary variable for stream from operation to operation
, =
binary variable for stream from operation to discharge point
= binary variable for buffer tank t
= binary variable for pump at water source
, =
binary variable for pump between water source and operation
, =
binary variable for pump between operation and operation
, = binary variable for pump in the front of operation
, = binary variable for pump at the end of operation
, =
binary variable for pump between operation and environment
, , =
maximum limiting inlet concentration of contaminant to operation
, ,
Parameters for Concentration bounds
= average water lowrate of operation
Parameters for water flowrate used in Chapter 4
, = nominal mass load of contaminant of operation
, = maximum mass load of contaminant of operation
Parameters for mass load used in Chapter 4
, = minimum mass load of contaminant of operation
, = time duration of the minimum water lowrate of operation
, = time duration of the nominal water lowrate of operation
, = time duration of the maximum water lowrate of operation
, ,
= time duration of the minimum mass load of contaminant of operation
Parameters for time period duration used in Chapter 4

, ,
= time duration of the nominal mass load of contaminant of operation

, ,
= time duration of the maximum mass load of contaminant of operation

, = distance between water source and operation
, = distance between buffer tank and operation
, = distance between operation and buffer tank
, = distance between operation and operation
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Parameters for the flow velocity in the pipeline
, = velocity in pipes between water source and operation
, = velocity in pipes between buffer tank and operation
, = velocity in pipes between operation and buffer tank
, = velocity in pipes between operation and operation
, = velocity in pipes between operation and discharge point
Regression parameters for piping costs
, = a freshwater low from source to operation

,
= "b" freshwater low from source to operation
,
= "" of stream from buffer tank to operation
,
= "" of stream from buffer tank to operation
,
= "" of stream from operation to buffer tank
,
= ""of stream from operation to buffer tank
,
= "" low from operation to operation
,
= "" low from operation to operation
,
= "" low from operation to discharge point
,
= "" of low from operation to discharge point
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Other parameters
Cc,u out,k
= outlet concentration of contaminant in stream leaving operation
in iteration
Ctk
= outlet concentration of contaminant in stream leaving buffer tank
in iteration
= annual operating hour
, = mass load of contaminant of water using operation
= the maximum number of inlet streams entering operation
= pressure at freshwater source
P u = pressure drop of operation
= unit cost of fresh water from water source
= eficiency of pump
= density
= viscosity
Chapter 1. Introduction
In the past few decades, since the scarcity of water resources and the rising
awareness of environmental protection, the costs of supply water and effluent
treatment soar rapidly. To enhance the preservation of water, eliminating
environmental impact, and elevating the competitiveness of process industries,
design methodologies for water-using and wastewater-treating systems, based
on Process Integration techniques, have inspired academic and industrial
communities to improve water-using efficiency and, at the same time,
minimize wastewater production. Through the reduction in the amount of
wastewater, the cost of effluent treatment also decreases.
According to available design methodologies, three options are adopted to
enhance water-using efficiency, namely, water reuse, water regeneration reuse
and water regeneration recycling (Wang and Smith, 1994). Water reuse means
that the effluent from water-using operation may enter other water-using
operation to be used again. Before the effluent is reused, it is necessary to
ensure that the concentration of the water being reused is under a specific
design value which is determined according to some operating factors, such as
minimum mass transfer driving force, maximum solubility, the need to prevent
precipitation of certain chemical, the corrosion restriction, and the minimum
requirement of the throughput. The appropriate limitation imposed on the
maximum inlet and outlet concentration helps to ensure the operating
feasibility and the quality of the products. Sometimes, the effluent blends with
other streams with better quality to improve the possibility for reuse.
Moreover, the installation of water-treating operations is another way to
enhance water reuse possibility. To avoid the interference of high-level
contaminant in the effluent stream, sometimes, it is necessary to partially
remove the contaminants in the effluent by the treatment unit (e.g. filter,
adsorption, sedimentation tank, stripper, etc) before the effluent is reused. Also,
the treated stream may need to be blended with other high-quality streams
before reusing. This situation is termed as water regeneration reuse. It is worth
21
noting that, in this case, the effluent is not reused in the same water-using unit
again after regeneration. Besides, water regeneration recycling is also the case
in which effluent is regenerated to remove excess contaminant before it is
reused. However, the difference between water regeneration reuse and
recycling is that the effluent stream can re-enter the water-using unit which
uses the same stream previously (Wang and Smith, 1994). In some scenarios,
water regeneration recycling is forbidden because of the build-up of
contaminants which are not removed by the water-treating operation.
The network design determines how the water reuse, water regeneration reuse
and water regeneration recycle are implemented. Numerous design
methodologies for water-using and wastewater-treating system have been
suggested to determine the interconnections between different water-using
units and treatment units, subject to specified objective function. However, in
early scientific attempts made in the academic community, water systems and
treatment systems were often regarded as two separated systems, to simplify
the design problem. The synergy between these two systems had been
considered in the various studies (Wang and Smith 1994; Castro, Matos et al.
1999; El-Halwagi, Gabriel et al. 2003), although the possibilities of the direct
interactions between the individual elements within these two systems were
often ruled out in these early studies.
The design of total water systems, which is defined as an integrated water
network involving both water-using systems and effluent treatment systems,
has been discussed and initiated by Gunaratnam et al. (2005). This is followed
by Ng et al. (2007) and Bandyopadhyay and Cormos (2008), in which the
optimization to a specific objective function is considered by regarding
water-using systems and treatment systems as a total system. This integrated
concept provides benefits to improve the efficiency of using resources.
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1.1 Project Aim and Objectives
There are two aims in this thesis. One is to study the effect of pressure drop and
pumping arrangement in the network design. The other one is to discuss the
integrated water network under uncertainty.
Objectives are summarised as:
Simultaneous consideration of fresh water cost, piping cost, and pumping
cost in the design of water network
Investigating the effect of pressure drop and pumping arrangement in the
water network design
Modelling for integrated water network under uncertainty
Note that the discussion in this study is limited to water networks without
effluent treatment for simplification reason. In other words, no water
regeneration occurs in the cases studies discussed in this thesis. In the future, it
would be valuable to consider total water network optimization including both
water-using operations and water-treating operations using the methods
developed in this study.
1.2 Organisation of Dissertation
The thesis is organized as follows.
Chapter 1 The problem of water network synthesis is introduced. The
introduction shows the importance of enhancing water-using
efficiency, and provides some background information of
previous study. The aim and organisation of this thesis are
pointed out in this chapter.
Chapter 2 The design methodology of water network for both pinch
approach and mathematical approach are briefly introduced in
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optimization techniques have been made.
Chapter 3 An automated method for the water network design to consider
the optimisation of pump arrangement at minimum total annual
cost is established. A superstructure-based MINLP is built up.
The model consists of mass balance equations, equations to
calculate pressure drop, cost functions, operational constraints,
and logical constraints. These equations are explained in detail
in this chapter. Besides, an effective solution strategy is
suggested. Examples are implemented under four scenarios for
comparison reason.
Chapter 4 A design methodology for integrated water network under
uncertainty is discussed in this chapter. A superstructure-based
MINLP model which is composed of mass balance equations,
operational constraints, cost functions, logical constraints is
established to do the design. The corresponding solution
strategy is developed. Examples are provided to validate the
proposed model. Each example is discussed under three
scenarios.
Chapter 5 The thesis is concluded in this chapter, and the suggestions of
possible developments of future works are discussed.
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Chapter 2. Literature Survey
This chapter briefly introduces the design methodology in previous works. In
general, the design methodology mentioned above can be classified into two
broad categories: insight-based pinch analysis and mathematical-based
optimization technique. Insight-based pinch analysis is derived from pinch
analysis for heat exchanger network, proposed by Wang and Smith (1994). On
the other hand, mathematical-based optimization techniques are developed
which is able to deal with more complex problem.
It is challenging to address the complicated interactions among the cost of
different elements by using pinch approach, and therefore, insight-based pinch
analysis has been mainly applied to a simpler design problem. Conceptual
insights obtained from the insight-based pinch approach can enhance our
physical understanding of the design problem, and it gives great help in
suggesting the possible solution strategy or initial guesses for
mathematical-based optimization model which may lead to the difficulty to
find the optimal solution without knowing any physical significance
(Gunaratnam et al., 2005). The brief review of some important design
methodologies is demonstrated in the following paragraphs.
Before directly discussing the design methodologies, it is worth noting that
there are two ways to model the water-using unit. The water-using unit can be
regarded as fixed load or fixed flowrate problem. These two ways of modelling
are applied to both insight-based pinch analysis and mathematical-based
optimization method. In the fixed load problem, the mass load needed to be
removed (washed away) by the water go through each water-using operation is
given, and the maximum allowable inlet and outlet concentration are imposed.
The relationship among flow rate, mass load, and inlet/outlet concentration is
given in Eq. 2.1. Because of the variation in the inlet and outlet concentration,
the flowrate which goes through the process may vary.
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(2.1)
On the other hand, fixed flowrate problem which consists of water source and
water sink is considered on the basis of flowrate. Inlet and outlet water stream
are regarded as separated entities. Water sources, which generate water and
provide the water to be reused/recycled, are characterized with flowrate and
impurity concentration. Water sinks receives water from the water source(s) at
a specific concentration. Since water sources and water sinks are considered
separately in the fixed flowrate problem, for the same water-using operation,
both the inlet and outlet water stream may not exist at the same time, and the
flowrate of the inlet and outlet stream may not always be the same. In other
words, the operation only has inlet flow or outlet flow (i.e. evaporator) is
allowed. This feature broadens the application range of water network design.
In some cases, these two kinds of problems are convertible. The maximum
allowable inlet concentration in the fixed load problem corresponds to the
maximum allowable inlet concentration of water sinks, and the maximum
outlet concentration corresponds to source concentration. The detail
explanation has been reviewed by Foo (2009).
Moreover, in previous literature, there are also two ways for modelling the
treatment units in the context of Process Integration method. The performance
of treatment units can be defined in terms of removal ratio or fixed outlet
concentration. The performance of treatment unit defined by the removal ratio
means that the amount of impurity load removed by that unit is with a certain
ratio to the total impurity load in the inlet stream. For example, American
Petroleum Institute(API), a common device separator in petrochemical plants,
typically removes 60 to 99% of dispersed hydrocarbon liquids, 10 to 50% of
suspended solids, 5 to 40% of BOD5 and 5 to 30% of COD (Smith, 2005). On
the other hand, the performance of treatment unit can also be defined by outlet
concentration. In this situation, the treatment unit removes the impurity load to
a fixed concentration level regardless of the inlet concentration. However,
26
sometimes, limits are imposed on the maximum inlet concentration to ensure
the operating feasibility.
For insight-based pinch analysis, the design procedure is, typically, divided
into two steps: flowrate targeting and network design. Flowrate targeting aims
to obtain the minimum amount of freshwater required and minimum amount of
wastewater generated without looking into the detailed interconnections
between individual water processes. Various methods have been proposed to
target the minimum water consumption in the fixed load problem, and the
limiting water composite curve constructed according to the maximum inlet
and outlet concentration is the most widely used one. This method plots the
limiting water composite curve on a diagram with mass load as the horizontal
axis and contaminant concentration as the vertical axis. Then, the water supply
line stems from the origin and rotates counter-clockwise until it touches the
limiting water composite curve. The point in which the supply line touches the
composite curve is defined as a pinch point. The inverse of the slope of the
water supply line indicates the minimum water consumption rate, and the end
of the water supply line corresponds to the concentration of the wastewater.
An example taken from the paper written by Wang and Smith (1994) is shown
in Table 2.1 which gives the operating data, including mass load of
contaminant, the maximum inlet and outlet concentration, and water flow rate
for each process operation. The construction of the limiting composite curve
based on these data is presented in Fig. 2.1 (a) and Fig. 2.1 (b). The matching
of water supply line is shown in Fig. 2.1 (c). In this example, the minimum
water flow rate required is 90 t/h, and the pinch point is identified at 100ppm.
The research work extends to address the problem with water loss and water
gain (Wang and Smith, 1995). Water loss and water gain occur in the
above-pinch or below-pinch region leads to different results. The detail
description can be found in Wang and Smith (1995) and Foo (2009).
27
Process Mass
1 2 0 100 20
2 5 50 100 100
3 30 50 800 40
4 4 400 800 10
Fig. 2.1. Construction of the limiting composite curve (Wang and Smith
1994)
28
Table 2.2. Process data for the construction of water demand and source
composites
(t/h) (ppm) (t/h) (ppm)
Fig. 2.2. Water demand and source composites (Hallale, 2002)
29
Fig. 2.3. (a) Water surplus diagram (b) Water surplus diagram with
increased freshwater flowrate (Hallale, 2002)
On the other hand, when it comes to the fixed flowrate problem, a water
surplus diagram is widely used, which is illustrated with an example from
Hallale (2002). Firstly, a demand and source composite curves are drawn
according to the process data provided in Table 2.2. The new form of the water
demand and source composites is presented in Fig. 2.2. Note that the vertical
axis represents water purity, and horizontal axis represents flowrate. The
definition of water purity is given in Eq. 2.2. In Fig. 2.2, the solid line identifies
water sources, and the dashed line represents water demand. The initial value
of the amount of fresh water is assumed. In this example, it is taken as 50 t/h.
30
Then, the next step is to check the feasibility under the assumed flowrate. Two
criteria must be considered in the water design problem. The first one is that the
total flowrate needs to be larger or equal to the demand flowrate. The second
one is that the purity of water from water source needs to satisfy the water
demand. The water purity needs to be considered after all the mixing
possibilities are taken into account.
Purity = 1000000−C 1000000
where C is contaminant concentration (ppm).
In Fig. 2.2, the rectangular area in which the source composite is above the
demand composite has surplus freshwater, and the rectangular area in which
the source composite is below the demand composite represents water deficit.
In the water surplus diagram (Fig. 2.3 (a) and (b)), the area of these rectangles
from Fig. 2.2 is plotted against the water purity with different fresh water
supply. Note that the minimum fresh water target is obtained by adjusting the
freshwater supply.
Different methods, including material recovery pinch diagram (El-Halwagi et
al., 2003; Prakash and Shenoy, 2005) and water cascade analysis (Foo et al.,
2006) had been further developed to overcome the iterative characteristic of the
water surplus diagram. Design methodology to target the minimum freshwater
required, was also developed when multiple water sources are used (Wang and
Smith 1995). Besides, targeting techniques for water regeneration are also
worth discussing. In the early works, the inlet concentration for regeneration is
fixed to the pinch concentration for fixed load problem (Wang and Smith, 1994;
Wang and Smith, 1995; Kuo and Smith, 1998). The water regeneration reuse
and regeneration recycling can be distinguished by different graphical methods.
Note that, in the water regeneration reuse system, fresh and regenerated water
lines share the same slope because there is no recycle in this case. In contrast,
in the water regeneration recycle system, the slope of fresh water segment is
larger than that of regenerated water segment. It means that less fresh water is
31
required. Nevertheless, the inlet concentration for the regeneration located at
pinch concentration does not ensure the optimal design. Kuo and Smith (1998)
studied to rectify the inlet concentration for regeneration in order to obtain
better results.
2.2 Network Design of Water and Effluent Systems
For the design of water network, according to Foo (2009), some of available
design methods depend on target value, but some can be implemented without
knowing the flowrate target in advance. It should be noted that even with the
same target value, many alternative designs exist.
For insight-based pinch method, one of widely-used design methods is based
on “water main concept” proposed by Kuo and Smith (1998). The minimum
water flowrate identified from the targeting is achieved through the network
design for water-using operations. This water main method determines the
matching between available water sources and sinks with appropriate mixing
and splitting of water streams, based on mass balances and limiting water
conditions. The design procedure follows five steps stated below:
Step 1: Calculate the minimum water flowrate for each design region
Step 2: Set-up the design grid
Step 3: Connect the water-using operations with the water mains
Step 4: Merge the water-using operations
Step 5: Remove the water mains and complete the water network
Apart from insight-based pinch analysis, mathematical-based optimization
techniques can be effective for dealing with multi-contaminant systems,
performing economic trade-off among the costs of different elements,
eliminating impractical connections and imposing specific operating
constraints. Superstructure is typically used in the automated method (Kim and
Smith 2003; Gunaratnam et al. 2005), which incorporates all possibilities for
32
the water network. Based on the proposed superstructure, mathematical models
are formulated with constraints. When bilinear terms are used in the
formulation, high non-linearity associated with bilinear terms makes global
optimal results difficult to acquire. Besides, in some situations, binary variables
are necessary to be employed in operating constraints. For solving complex
optimisation models, graphical insights have been of great value to the
development of solution strategies.
Automated design methods had been developed for fixed load problems (Feng
and Seider, 2001; Gunaratnam et al., 2005; Zheng et al., 2006) as well as fixed
flow rate problems (Poplewski et al., 2010; Huang et al., 1999). In fixed load
problems, the example which addresses the regeneration reuse problem in the
total water system considering the optimal distribution of water-using and
treatment system simultaneously is based on the example presented by
Gunaratnam et al. (2005). In this example, the water network consists of
water-using and water-treating operations. Based on these two operations, a
superstructure is proposed. It is worth mentioning that mixers are set up before
each operation, and splitters are set up in the end. Mixer is used to blend all the
inlet streams from different sources, and then all the inlet streams are
transported into the device together. On the other hand, splitter, which sets up
in the end, facilitates the distribution of outlet streams. All the possible options
in the design system are identified in this structure.
The key formulation required in the optimisation framework include:
Overall flow rate balance around entire water system
Solute balance for each operation
Operational constraints:
o Constraints on water flow rate
o Constraint on the quality and quantity of inlet water stream
o Environmental regulation
Logical constraints
33
o Elimination of regeneration recycling
o Elimination of direct recycling
On the other hand, a design problem as fixed flowrate problem has been dealt
with. Fig. 2.4 is the superstructure for the water network integration, which had
been applied in an industrial park study (Chew et al., 2008). It is interesting to
note that water networks in different plants are inter-connected via a centralized
utility hub which functions as a buffer to avoid uncontrollable fluctuation in the
flow rate and concentration of the process streams. All the cross-plant streams
from water sources to sinks need to be mixed in the utility hub before
redistribution. This kind of “hub” (sometimes termed as water main) is
essential to ensure the practicability through simplifying network structure and
reduce unavoidable fluctuations especially in some complicated water network
systems (Feng and Seider, 2001; Wang et al., 2003; Zheng et al., 2006; Ma et
al., 2007).
34
Fig. 2.4. Superstructure for the integration topology in an industrial park
(Chew et al., 2008)
the Consideration of Pressure Drop and Pumping
3.1 Introduction
There has been a growing perception of environmental protection and
sustainable development in society, which raises public’s concern on the use of
nature resources by industries. Water is a key natural resource, which has been
heavily extracted and used in process industries as a raw material, mass
separating agent or heat transfer medium. On the other hand, cost related to
fresh water supply and effluent disposal has been significantly increased, which
requires urgent attention for achieving efficient use of water in process
industries. Due to this economic and socio-environmental pressure, academic
and industrial communities have been actively engaged in the search of new
opportunities and technologies for fundamentally improving the efficiency of
industrial water use.
Various design issues had been addressed in the past to enhance the
applicability of water network design methodologies as stated in Chapter 2,
however, most studies carried out in industrial and academic communities have
not included information about how pressure drop should be considered in the
water network and how the pumping of water within the network should be
arranged. A conventional design practice is, first, to consider water reuse,
regeneration and recycling in the water network and to determine the water
network configuration. With the confirmed water network, the pressure drop is
calculated and, correspondingly, the design of water pumping is considered.
There has been an attempt to calculate the overall pressure drop of cooling
water in the design of cooler networks, together with designing cooling water
networks (Kim and Smith, 2003). Although the overall pressure drop of the
cooling water network was calculated during the network optimisation and was
36
used as a design guideline for the optimisation, the impacts of the network
configuration on pressure drop and the arrangement of pumping were not
systematically considered. Also, Kim and Smith’s work did not provide any
information on how pressure drop influences the pumping of water and what is
the best pumping arrangement for the network.
A large number of structural options are available in the design of water
networks when an integrated water network is desired to reduce freshwater
consumption through water reuse. The resulting water network is complicated
and it is not straightforward to identify the most appropriate location of a pump
and its optimal size. It would be ideal to simultaneously consider pressure drop
and pumping arrangement during water network design. For most published
works, the consideration of piping cost in the optimisation of water networks
has been reported (Gunaratnam et al. 2005; Chen et al. 2010), but design issues
associated with pressure drop and pumping has not been thoroughly addressed.
This chapter focuses on the development of an automated design method which
is able to simultaneously perform the calculation of pressure drop and design
the pumping of water in the context of a water network. The developed
optimisation framework systematically carries out a rigorous economic
trade-off between fresh water cost, pipeline cost and pumping cost, which
provides the optimal distribution of water, and the optimal location and
capacity of pumps for the network. The current study has been limited to study
the design problem of water reuse in the network without considering
water-treating operations
3.2 Water Network Design
The water systems discussed in this chapter consist of water-using operations
in which water is contaminated through the mass transfer of contaminants from
process streams. Several freshwater sources may be available for the water
system, while there is a sink to take the contaminated wastewater discharged
37
from water-using operations. A mixer and a splitter may be required before
and after each water-using operation. The mixer is used to blend all the inlet
streams from different fresh water sources and/or from different water-using
operations, while the splitter facilitates the distribution of exiting streams to
other water-using operations or a sink for discharged water. The cost
considered in the water-using network in this study includes operating cost
related to freshwater and electricity cost for operating pumps, and capital cost
for pipeline and pump.
The superstructure approach adopted in many previous studies has been an
effective method to synthesise and design water networks (Kim and Smith
2003; Gunaratnam et al. 2005; Chen et al. 2010). Therefore, a superstructure is
developed in this work that considers design interactions between pipeline
arrangement and pump installation. Fig. 3.1 illustrates a superstructure with
one available water source and two water-using operations. All the piping
options for connecting among freshwater sources, water-using operations, and
wastewater discharges, as well as all the potential options for installing pumps
are incorporated in the superstructure.
Fig. 3.1 Superstructure representation
superstructure has been developed for the network design. For the
simplification in the installation of pumps, if a pump is operated for pumping
38
more than one stream collectively, the pump is only used for the streams
originated from the same water source or designated to the same sink. In other
words, no extra mixing is occurred, because of the introduction of a pump. This
is a reasonable and acceptable simplification because mixing between water
streams should be selected such that freshwater consumption in the network is
minimised, according to the limiting water conditions. As freshwater cost is
dominant, compared to pumping cost in most of cases, it would be reasonable
not to merge streams for sharing the pump. Another reason to avoid extra
mixing is that the required head elevation varies from a pipeline to pipeline,
and in some scenarios, extra mixing for sharing the pump causes a great loss in
pressure head.
The superstructure introduced above includes all the possible stream
connections, but it is not enough for calculating the pressure drop of the
network. The calculation of pressure drop is determined not only by individual
pipelines and water-using operations, but also by the configuration. It is
necessary to introduce a node-superstructure (Kim and Smith 2003) which is
based on the concept of “critical path algorithm” used for the calculation of
pressure drop in the network, and Fig. 3.2 shows the node-superstructure which
includes all the mixing nodes, splitting nodes, and pumping nodes. In the
proposed node-superstructure model, mixing nodes and splitting nodes
correspond to the mixer and splitter in Fig. 3.1, respectively. Pumping nodes
are the possible locations for pump installation in Fig. 3.2.
Fig. 3.2 Node-superstructure representation
39
The automated design framework is based on the superstructure given in Fig.
3.1 which incorporates all the design options for water networks, together with
a node-superstructure in Fig. 3.2 which supports the study of pressure drop and
pumping arrangement. This network design problem has been formulated as a
MINLP problem, which is solved with an objective function to minimize total
annualised cost. The key nonlinear terms in the formulation appear in the
contaminant balance equations around the mixer, splitter, and water-using
operations, as well as in the equation representing pumping cost, which causes
computational difficulties in the optimisation. To effectively deal with
non-linearity in the optimisation model, a robust solution strategy of this
MINLP problem has been also proposed by incorporating physical insights in
the design of water network and its conceptual understanding.
For the developed model, operational constraints have been imposed to ensure
the feasibility. Besides, the model is able to manage network complexity and
flowrate restriction by manipulating binary variables. There are some
assumptions made in the network design as follows:
(i) The number of water-using operations is fixed and the mass transfer
behavior of each water-using operation is specified by maximum inlet and
outlet concentration, and the mass load to be removed.
(ii) The mass load to be removed in each water-using operation is constant.
(iii) The pressure at any point in the system is above the required net positive
suction head of every pump.
(iv) The pressure loss of each water-using operation is constant.
(v) The pressure loss related to fittings, sudden change in the direction of
pipeline, valves, or measuring devices is neglected.
(vi) All the water-using operations and other equipment are located at the same
gravity level, and
40
The mathematical formulation includes all the feasible connections based on
the proposed superstructures. The first step for developing the mathematical
model is to define sets, parameters, and variables representing the water
systems which are listed following model equations and constraints.
Fig. 3.3 Schematic representation of water-using operation
The mathematical model to design the integrated water network and determine
the allocation of pumps in the network involves a nonlinear objective function,
together with linear and nonlinear constraints. The model comprises flowrate
balance equations, contaminant balance equations, equations to calculate
pressure drop, operational constraints, and logical constraints. A schematic
representation of water-using operation is shown in Fig. 3.3, in which inlet
streams may come from freshwater sources and/or other water-using operations,
and outlet streams may be transported to other water-using operations and/or a
discharge point directly. The formulated MINLP model is as follows:
Balances around mixer, splitter, and water-using operations
Flowrate balances around the mixing and splitting points
∑ Fs,u + ∑ Fua,uua∈Us∈S = Fu ∀ ∈ (3.1)
41
where:
Fs,u is the water flow rate between source s and water using operaion u;
Fua,u is the water flow rate from water using operaion ua to u;
Fu is the total water flow through the operation;
Fu,e is the water low from operation to discharge point .
Mass balances of each contaminant around the water-using operation
∑ Fs,uCc,s + ∑ Fua,uCc,ua out + Mc,uua∈Us∈S = Fu Cc,u
out ∀ ∈ , ∈ (3.3)
where:
, is the concentration of contaminant in water source ;
Cc,u out is the outlet concentration c from operation ;
, is the mass load of contaminant of water using operation .
∑ Fs,uCc,s + ∑ Fua,uCc,ua out
ua∈Us∈S ≤ Fu Cc,u in,max ∀ ∈ , ∈ (3.4)
Operational constraints on the concentration of inlet and outlet streams
Cc,u out ≤ Cc,u
where:
, , is the maximum limiting inlet concentration of contaminant to
operation ;
operation .
Logic constraints for streams in the integrated water network
Fs,u ≤ Us,u Bs,u ∀ ∈ , ∈ (3.6)
Fs,u ≥ Ls,u Bs,u ∀ ∈ , ∈ (3.7)
Fu,ua ≤ Uu,ua Bu,ua ∀ , ∈ (3.8)
Fu,ua ≥ Lu,ua Bu,ua ∀ , ∈ (3.9)
Fu,e ≤ Uu,e Bu,e ∀ ∈ , ∈ (3.10)
Fu,e ≥ Lu,e Bu,e ∀ ∈ , ∈ (3.11)
where:
,is the binary variable for stream from water source to operation ;
42
,is the binary variable for stream from operation to operation ;
,is the binary variable for stream from operation to discharge point .
Eqs. 3.6 to 3.11 specify the upper and lower bounds of water flow rates.
Us,u , Uu,ua and Uu,e are defined as upper bounds of corresponding flow rates,
while Ls,u, Lu,ua, and Lu,e represent lower bounds. The lower bounds given
eliminate the possibility of the existence of stream with a very small flowrate
which is not economic and practical.
Maximum number of inlet streams
∑ Bua,u + ∑ Bs,us∈Su∈U ≤ NSumax ∀ ∈ (3.12)
To control the complexity of the water network, Eq. 3.12 can be introduced to
restrict the total number of streams allowed to the inlet of water-using
operations. NSumax represents the maximum allowable number of inlet streams.
The formulations given so far are based on the superstructure of Fig. 3.1. When
the design problem is associated with pressure calculation, the
node-superstructure of Fig. 3.2 is introduced, and the resulting formulations for
the calculation of pressure drop are given as follows:
P u in + P u
in,pump - P u = P u out ∀ ∈ (3.13)
Pressure calculation around water-using operation
where:
is the pressure at the mixing point of operation
is thepressure at the splitting point of operation
,is the pressure elevation of the pump in the front of operation u
P u is the pressure drop of operation
The pressure drop of the water-using operation, P u , has been assumed to be
constant and pre-known in Eq. 3.13 in this study. However, the detailed
calculation of pressure can be readily added in this framework, as long as a
43
detailed model of the water-using operations and its pressure drop is available,
and the formulation for the pressure drop calculation is reasonably simple.
P u in,pump indicates the pressure elevation of the pump installed upstream of
the water-using operations.
The pressure drop in each pipe is calculated as follows. (Kim and Smith 2003)
P s,u = 4f 0 ρvs,u
2
2
2
F s,u = A s,uv s,u (3.18)
D s,u = 4F s,u
where:
, is the pressure drop in pipe between fresh water source s and
operation
,is the pressure drop in pipe between operation u and operation
,is the pressure drop in pipe between operation u and discharge
point
,is the distance between water source and operation
,is the distance between operation and operation
,is the distance between operation and discharge point
,is the velocity in pipes between water source and operation
,is the velocity in pipes between operation and operation
,is the velocity in pipes between operation and discharge point
As,uis the cross sectional area of a pipe connecting freshwater source s and operation
44
The pressure drop in each pipe depends on the Fanning friction factor, density
of the flow, flow velocity in the pipeline, the diameter of the pipeline, and the
length of the pipeline. The general formulation to calculate the pressure drop in
the pipe is shown in Eqs. 3.14 to 3.16. Eq. 3.17 is taken to estimate the Fanning
friction factor (Kim and Smith; 2003). In this study, it has been assumed that
the flow velocity is 2ms-1 in all the pipelines. Under this assumption, the
diameter of the pipeline is a function of flowrate. Eqs. 3.18 and 3.19 show how
the diameter of the pipeline between fresh water source and water-using
operation is calculated. The diameters of other pipelines are calculated in a
similar way.
According to “critical path algorithm”, the overall pressure drop of water
networks cannot be calculated until the network configuration is known. To
address the problem without knowing the network configuration in advance, Eq.
3.20, Eq. 3.21 and Eq. 3.22 for the connections between each mixing and
splitting node are formulated with the aid of binary variables which are
introduced with a sufficiently large value, LV, to consider whether this
constraint is activated or not.
Constraints of “critical path algorithm” (Kim and Smith 2003)
P s + P s pump + P s,u
pump − P u in + LV(1 − B s,u) ≥ P s,u ∀ ∈ , ∈
(3.20)
out,pump + P u,ua pump − P ua
in + LV(1 − B u,ua) ≥ P u,ua ∀ , ∈
(3.21)
P u out + P u
out,pump + P u,e pump − P e + LV(1 − B u,e) ≥ P u,e ∀ ∈ , ∈
(3.22)
where:
is the pressure at the discharge point ;
is the pressure elevation of the pump at fresh water source s;
, is the pressure elevation of the pump between water source s and
operation u;
, is the pressure elevation of the pump between operation u and
45
operation ua;
,is the pressure elevation of the pump at the end of operation u;
, is the pressure elevation of the pump between operation u and
discharge point e;
LV is the sufficiently large value.
In the model considering the allocation of pumps, the pumping term which
represents the pressure elevation of each pumping node, the available places to
put a pump between the mixing node and splitting node, are included. Take Eq.
3.20 as an example, when the connection between water source s and
water-using operation u exists (, = 1), LV is multiplied by zero. In this
situation, the constraint must be satisfied. In other words, the pressure
difference between the splitting node of fresh water source and the mixing node
of water-using operation needs to be larger than the pressure drop caused by the
friction in between. If the pressure difference between the splitting node and
mixing node is not large enough, the pump is introduced to compensate for
pressure drop. On the other hand, when the connection between water source s
and water-using operation u does not exist (, = 0), the role of LV, which
must be greater than the largest possible pressure drop in the system, comes
into play. Therefore, no pumps are necessary to be introduced to compensate
for the pressure drop.
For the connection between a fresh water source and a water-using operation,
there are two possible places to install a pump. The first possible place to
install a pump is at the fresh water source. In other words, water is pumped
before the water is distributed. Another choice is to install a pump at the
beginning of the connection between fresh water source and water-using
operation. The pressure elevated from these two pumps corresponds to
and ,
equations related to the connection between different water-using operations, as
well as the connection between water-using operations and discharge point are
given in Eqs. 3.21 and 3.22.
46
Eqs. 3.23 to 3.34 give the upper and lower bound for the pressure elevation of
the pump. Us pump, Us,u
pump, Uu,ua pump, Uu
pump are the
pump, Lu,ua pump, Lu
pump
represent the lower bounds. Binary variables which indicate the existence of
the pump are incorporated in these equations.
Ps pump ≤ Us
Ps pump ≥ Ls
Ps,u pump ≤ Us,u
Ps,u pump ≥ Ls,u
Pu,ua pump ≤ Uu,ua
Pu,ua pump ≥ Lu,ua
Pu in,pump ≤ Uu
Pu in,pump ≥ Lu
Pu out,pump ≤ Uu
Pu out,pump ≥ Lu
Pu,e pump ≤ Uu,e
Pu,e pump ≥ Lu,e
where:
is the binary variable for pump at water source ;
, is the binary variable for pump between water source and operation ;
, is the binary variable for pump between operation and operation ;
,is the binary variable for pump in the front of operation ;
,is the binary variable for pump at the end of operation ;
, is the binary variable for pump between operation and environment .
The objective of this optimisation model is set to minimize total annualised
cost. The objective function is represented in terms of the cost of freshwater,
Objective function
47
the cost of pipelines, the capital cost of pumps, and the operating cost of the
pumps.
u∈U∈ )
pump,cap ua∈Uu∈U +

pump,op ua∈Uu∈U +
(3.35)
The annualisation factor used in Eq. 3.35 was adopted to express capital costs
on an annual basis under the assumption that capital has been borrowed over a
fixed period of time.
Annualisation factor, AF = i(1+i)n
(1+i)n−1 (Smith. 2005)
Because the calculated values of annualisation factors for most of the
equipments in the chemical plant are between 0.1 and 0.2, the annualisation
factors adopted in case study in this chapter are assumed to be 0.1.
where:
n is the number of year
Costs is the cost of freshwater ;
Costs,u pipeis the piping cost of the connection between freshwater s to
operation ;
Costu,ua pipeis the piping cost of the connection between operation to
operation ;
Costu,e pipeis the piping cost of the connection between operation to
discharge point ;
Costs pump,capis the capital cost of pump at freshwater source ;
48
Costs,u pump,capis the capital cost of pump between supply water source and operation ;
Costu,ua pump,capis the capital cost of pump between operation and operation ;
Costu in,pump,capis the capital cost of pump in the front of operation ;
Costu out,pump,capis the capital cost of pump at the end of operation ;
Costu,e pump,capis the capital cost of pump between operation and discharge point ;
Costs pump,opis the operating cost of pump at supply water source ;
Costs,u pump,opis the operating cost of pump between supply water source and operation ;
Costu,ua pump,opis the operating cost of pump between operation and operation ;
Costu in,pump,opis the operating cost of pump in the front of operation ;
Costu out,pump,opis the operating cost of pump at the end of operation ;
Costu,e pump,opis the operating cost of pump between operation and discharge point ;
AF is the annualisation factor (an example of how to calculate the value of AF
can be found in the case study in Section 3.5).
Fresh water cost (Gunaratnam et al. 2005)
Costs = × ∑ ∑ ∈∈ Fs,u (3.36)
Piping cost (Gunaratnam et al. 2005)
Costs,u pipe = [as,uAs,u + bs,uBs,u]ds,u (3.37)
Costu,ua pipe = [au,uaAu,ua + bu,uaBu,ua]du,ua (3.38)
Costu,e pipe = [au,eAu,e + bu,eBu,e]du,e (3.39)
where:
as,u is the cost parameters "a" of freshwater low from source to operation
bs,uis the cost parameters "b" freshwater low from source to operation
au,uais the cost parameters "a" of low from operation to operation
bu,uais the cost parameters "b" of low from operation to operation
au,eis the cost parameters "a" of low from operation to discharge point
bu,eis the cost parameters "b" of low from operation to discharge point
49
The piping cost is related to the cross-sectional area and the length of the
pipeline. The values of the cost parameters, a and b, in Eqs. 3.37, 3.38 and 3.39
depend on the type of pipeline which can be obtained from some previous
works. Note that the relationship between cross-sectional area and the flowrate
has been specified in Eq. 3.18.
Pumping cost (Smith, 2005)
The pumping cost includes the capital cost of each pump and its operating cost.
Capital cost depends on the power required to drive the pump. The way to
calculate the cost of the pump at freshwater source is demonstrated in Eqs. 3.40,
3.41 and 3.42. The costs associated with other pumps are calculated in a similar
manner.
3600×ρ (3.42)
is thedensity.
where Qs (kW) is the power required for the pump. r, w, and k are cost
parameters which depend on the type of pump. The assumptions given in the
previous section, the electricity required is calculated using Eq. 3.42, which
relies on the water flowrate to be pumped and the pressure to be elevated
through the pump.
3.4 Solution Strategy
Highly non-linear terms included in the model present a major difficulties to
solve the optimisation problem directly by using a standard solver, and a
solution strategy to cope with the nonlinear equations is necessary. In the
proposed model, there are four sets of nonlinear equations: the mass balance
50
equations of contaminants, the equations to calculate the pressure drop in pipes,
the equations to calculate the capital costs of the pumps, and the equations to
calculate the supply of power to the pump. The proposed solution strategy
includes the following four elements:
The equations to calculate the capital cost of pumps are linearized by the
regression in the format of Eq. 3.43.
Costs pump,cap ≈ hQs + gBs
h, g: cost parameter from the regression of Eq. 3.40
The linearization of equations related to the calculation of pressure drop in
pipes (Eq. 3.14, Eq. 3.15 and Eq. 3.16). At first, Eqs. 3.17 and 3.19 are
substituted into Eqs. 3.14, 3.15 and 3.16, as given in Eqs. 3.44, 3.45 and
3.46, respectively.
Fu,ua 1.4 (3.45)
Fu,e 1.4 (3.46)
Note that Eqs. 3.44, 3.45 and 3.46 are univariate equations of the
corresponding flowrates. However, in this case, they cannot be fitted
properly by only one linear equation. Therefore, SOS2 technique
(Alva-Argaez, 1999) has been adopted to approximate these equations by
dividing them into several linear segments. For example, Eq. 3.44 can be
approximate by
q=1 λqw (3.47)
(3.49)
P s,u q , the pressure drop in the pipe between a fresh water source ∈
51
and water-using operation u ∈ , is obtained from Eq. 3.44 when the
flowrate is Fs,u q . λqw is from a set of SOS2 variable for which at most two
adjacent variables may be non-zero. Eq. 3.44 is replaced by Eqs. 3.47,
3.48 and 3.49. Eqs. 3.45 and 3.46 are also substituted in the same way, as
shown in Eqs. 3.50 and 3.53.
P u,ua = ∑ P u,ua qNL
q=1 λqua (3.50)
F u,ua = ∑ Fu,ua qNL
q=1 λqua (3.51)
q=1 λqout (3.53)
F u,e = ∑ Fu,e qNL
q=1 λqout (3.54)
(3.55)
Eq. 3.42 is substituted in Eqs 3.40 and 3.41. Then, Eqs. 3.40 and 3.41 are
substituted in the corresponding terms of the objective function. By these
substitutions, the non-linear terms to calculate the power required for
pumping no longer appear in the constraints of the model. All of them are
given in the objective function. This decreases the computational
difficulty in solving the optimisation model.
With the linearization of equations above, the only remaining set of
nonlinear constraints in the model is the mass balance equations of
contaminants. The nonlinear terms come from the multiplication of
outlet concentration and flowrate. A concept to decompose the MINLP
problem into MILP and LP subproblems has been developed by
Gunaratnam et al. (2005), which has been extended for proposing a new
solution strategy in this study. The assumption is made such that the value
of the outlet concentration is fixed, which helps to eliminate the nonlinear
terms in the mass balance equations. With this strategy, the MINLP
problem is decomposed into two subproblems, P1 (MINLP model) and P2
(LP model). In P1, the projection of the nonlinear constraints onto the
52
concentration domain is made and the maximum outlet concentrations are
assumed as initial guesses for the outlet concentrations for all the
contaminants (Cc,u out,k = Cc,u
for concentration variables is reasonable and effective, as outlet
concentrations of water-using operations in the optimal solution are likely
to be close to the maximum outlet concentration allowed, in order to
reduce freshwater consumptions of individual operations.
Fixing outlet concentration in the model leads to an infeasible solution
because not all the contaminants are required to be at the maximum outlet
concentration. An iteration mechanism has been introduced to regain the
feasibility of the model by relaxing Eq 3.3 to Eq 3.56 with the
introduction of slack variables. Slack variable in Eq 3.56 assists to
compensate the effect from the non-limiting contaminants. The physical
interpretation of sc,u represents the “excess capacity” of non-limiting
contaminant here. From subproblem P1, the water flowrate is optimised
with fixed outlet concentration. The water flowrate obtained from P1 is
now fixed in the subproblem P2, and the outlet concentration is optimised
with minimising slack variables. The values of Cc,ua out,k are updated
through iteration until the convergence criteria are met.
∑ Fs,uCc,s + ∑ Fua,uCc,ua out,k + Mc,uua∈Us∈S + sc,u = Fu Cc,u
out
where:
Cc,u out,kis the outlet concentration of contaminant in stream leaving operation
in iteration ;
The decomposition strategy has been coupled with the linearization and
the substitution together, which completes the overall solution strategy for
the optimisation. The reformulated objective function for P1 is given as:
53
u∈U∈ ×
pump + ∑ ∑ vPs,u pumpFs,u + zBs,u
pumpu∈Us∈S∈ +
∑ ∑ vPu,ua pumpFu,ua + zBu,ua
in,pump] +
pump} × AFe∈Eu∈U
pumpFs,uu∈Us∈S∈ + ∑ ∑ Pu,ua pumpFu,uaua∈Uu∈U
+ ∑ Pu in,pumpFu∈ + ∑ Pu
+ ω∑ ∑ sc,uc∈C∈ (3.57)
where ω is a weighting factor for infeasibilities. The weighting factor
here helps to solve the problem in GAMS. Different values of weighting
factors may lead to different results in GAMS under the same solver. Some
values may even lead to infeasible solutions. Since the code embedded in
GAMS cannot be accessed, suitable values of the weighting factors are
determined by trial and error.
In Eq. 3.57, the following parameters were used.
v = 0.00014; z = 3401.1;α = 3.55 × 10−8
The subproblem P1 is formulated as an MINLP problem which is subject
to constraints Eqs. 3.1 and 3.2, Eqs. 3.4 to 3.11, Eq. 3.13, Eqs. 3.20 to
3.34, Eqs. 3.36 to 3.39, and Eqs. 3.47 to 3.56 with an objective function of
Eq. 3.57. With the exception of the objective function, all the constraints
in this problem are linear.
Subproblem P2 is the LP model which includes Eq. 3.59 with projection
on to the flowrate domain. The outlet concentration of the water-using
operation becomes variable. The objective function of P2 is set as:
Obj. function = ∑ ∑ sc,uc∈C∈ (3.58)
Subproblem P2 is subject to Eq. 3.59.
∑ Fs,u k Cc,s + ∑ Fua,u
out ∀ ∈ , ∈ (3.59)
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where Fs,u k , Fua,u
k , and Fuk are the fixed flowrates obtained from P1 in iteration
k. New values for the outlet concentrations are forwarded to subproblem P1 and
to be used for Cc,ua out,k+1 in the next iteration. Note that the new outlet
concentration may violate the constraint stated in Eq. 3.5 in subproblem P1.
Cc,u out is, then, set as Cc,u
out,max when Cc,u out ≥ Cc,u
out,max.
In overall, the iterative procedure proposed in this study is summarised as
illustrated in Fig 3.4. (MINLP)k, P1, and (LP)k, P2, are the formulation of
(MINLP) and (LP) at iteration k, respectively.
Step 1: In the first iteration (k=1), the outlet concentration of the stream leaving
water-using operations are fixed with its maximum allowable values
(Cc,u out,k = Cc,u
Fs,u w∗, Fua,u
ua∗ , and Fut∗.
∗ ; Fua,u k =
out∗, is obtained from solving
(P2)k.
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Step 3. Check the value of Cc,u out. If Cc,u
out is greater than Cc,u out,max, Cc,u
out is
adjusted to be Cc,u out,max.
Step 4. The new set of concentration values replaces the concentration values in
subprobelm P1 in next iteration ( k = k + 1) , Cc,u out,k+1 = Cc,u
out∗ . The
corresponding flowrate value is obtained from solving (P1)k+1. The iterative
procedure continues until the sum of slack variables is small enough, which
indicates that all the constraints in the model are not violated.
It should be noted that decomposition method developed by Gunaratnam et al.
(2005) requires additional MINLP optimisation after successive iterations
between MILP and LP subproblems, while the developed decomposition
method is based on iterative optimisation of MINLP (P1) and LP (P2)
subproblems, which effectively deals with nonlinearity imposed from the
simultaneous optimisation of water network and pumping systems.
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3.5 Case Study
Two examples are given in this chapter to illustrate the proposed approach, and
four scenarios are discussed in each example for the comparison. These two
examples were selected from previous works reported in the literature, as they
provide typical examples of water network optimisation situations. The
methodology proposed in this study to synthesis water network is robust. It can
be adopted to all the water network design. To prove this proposed model is
validated in both single and multiple component cases, one of the examples is a
single component problem, and the other one is a multiple component problem.
Besides, scenario 1 and 2 have been discussed in Alva-Argaez’s work (1999).
Optimal pumping arrangement which is the main improvement in designing
water network in this study is taken into consideration in Scenario 3 and 4. The
improvements are clearly shown by comparing the results from scenario 2, 3,
and 4.
Scenario 1: An objective function is to minimize freshwater consumption.
The water network is designed without water reuse and recycling and
optimization of pumping systems. The results in this scenario are taken as
base case for comparison.
Scenario 2: An objective function is to minimize freshwater consumption.
The reuse and recycling of water is allowed, while pumping systems are
not optimized.
Scenario 3. Total annualised cost is minimised while water recycling is
allowed. Pumping systems optimization is carried out, and all the
available locations of pumps identified in the node-superstructure are fully
considered.
Scenario 4. Minimum total annualised cost is sought when water recycle is
allowed, but water can only be pumped together at the beginning of the
network.
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As water reuse and recycling is forbidden, the upper bounds of water reuse and
recycling streams are set to be zero for Scenario 1. When the optimisation of
pumping systems is not considered in the model, for example, Scenarios 1 and
2, Eqs. 3.20 to 3.34 are not included in the optimisation accordingly. The
results from Scenario 1 are served as a base case to compare with other
scenarios. It is worth mentioning that the pumping cost in Scenarios 1 and 2 is
calculated under the assumption that all the pipelines and water-using
operations have their own pump and the pressure head elevation of the pumps
is the same as the pressure drop of the water-using operations/pipelines
attached to the pumps. The results from Scenario 3 include water network
design information, including optimal distribution of water, as well as optimal
location of pump and its capacity simultaneously. In Scenario 4, the restriction
on the location of pump has been implemented by setting the upper bound of
the pressure elevation of the non-existing pump to be zero(Us,u pump = Uu,ua
pump =
out,pump = Uu,e pump = 0).
For both examples, it is assumed that a single uncontaminated freshwater
source is available at the cost of $1.47/ton. It has been assumed that the annual
operating hour is 8600 hours, and an annualisation factor is 0.1. The flow
velocity in all the pipes has been assumed to be 2m/s, and the parameters for
piping cost are as,u = au,ua = au,e = 3603.4 ; bs,u = bu,ua = bu,e = 124.6
(CE Index of Equipment of 2010 = 666.0, Chemical Engineering’s Plant Cost
Index database) (Gunaratnam et al. 2005). The data of for cost parameters can
come from many different reports published in different years. CE Indexes are
used to update the cost parameters and put them on a common basis as 2010
costs. Carbon steel has been chosen as the material of construction. Besides,
the pump in this example has been assumed as a centrifugal pump. The
parameters for the capital cost of pumps are given by r = 1.97 × 103 and
w = 0.35 (CE Index of Equipment of 2010 = 666.0, Chemical Engineering’s
Plant Cost Index database), and 10% of purchased cost is added for installation
cost (Smith. 2005). After the linear regression, the values of h and g in Eq. 3.43
were obtained: h=431.69; g=3401.1. Moreover, the parameters for operating
58
cost of pump are given by k = 0.1085 which relates to the price of electricity.
The optimization platform employed in this work is GAMS® (General
Algebraic Modeling System) (Brooke et al., 2005). In this study, CPLEX® is
chosen as a solver for LP problem, and DICOPT® is used as a solver for
MINLP problem with CONOPT as NLP solver and CPLEX as MIP solver.
3.5.1 Example 1
The first example is a single contaminant problem which is taken from
Alva-Argaez (1999). Table 3.1 shows the operating data for the water-using
operations, which includes the maximum inlet and outlet concentration, and
mass load. Table 3.2 presents the distance matrix for Example 1. The pressure
drops of water-using operations are given in Table 3.3. Note that this simple
example can be solved without iterative solution strategies in all scenarios. The
reason is that this example is a single contaminant problem in which the outlet
concentrations of that single contaminant tend to reach the maximum allowable
concentrations in the design of optimal network structure. In this example, the
assigned values of outlet concentrations (Cc,u out,k = Cc,u
out,max) which have been
given in the first step of solution strategy meet the requirement in the optimal
network. Therefore, the slack variables become zero, and further iteration is
unnecessary. Although the iteration methodology has not been tested in this
example, in more complicated case the iteration may be required.The
computational data and results for Example 1 are shown in Table 3.4. and
Table 3.5., respectively.
Table 3.1. Operating data for water-using operation in Example 1
Table 3.2. Distance matrix (m) for example 1
Table 3.3. Pressure drops for water-using operations in Example 1
Max. inlet Max. outlet Mass
Operation conc. conc. load
S1 50 70 210 370 400
O1 0 30 80 90 270
O2 30 0 65 35 320
O3 80 65 0 80 95
O4 90 35 80 0 75
Operation Pressure drop (Pa)
Table 3.4. Computational data for Example 1
scenario 1 scenario 2 scenario 3 scenario 4 MINLP LP MINLP LP MINLP LP MINLP LP
no. of continuous variables
no. of binary variables
24 – 24 – 53 – 53 –
overall cost freshwater piping pumping scenario 1 $1,489,190 $1,422,225 $55,574 $11,391 scenario 2 $1,195,905 $1,137,780 $47,785 $10,340 scenario 3 $1,180,067 $1,137,780 $36,081 $6,206 scenario 4 $1,180,366 $1,137,780 $36,081 $6,505
Scenario 1: Freshwater is supplied in a parallel configuration, as shown in Fig.
3.5. In Fig. 3.5, the numbers without underline represent flowrate of each
stream(unit: ton/hr), and the number with underline means the pressure head
elevation of the pump (unit: kPa).
Scenario 2: The resultant network configuration is shown in Fig. 3.6.
Compared with the design in Scenario 1, it shows 19.7% reduction in total
annualised cost.
Scenario 3. From the optimization, it has been identified to install two pumps
in the networks, which is a much simpler pumping arrangement compared to
Scenario 2, as shown in Fig. 3.7.
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Scenario 4. The network configuration with the restriction on the location of
pump is shown in Fig. 3.8. In this example, although the restriction on the
location of pump is imposed, the optimal water distribution is the same as that
of Scenario three. Therefore, the network configurations in Fig. 3.7 and Fig. 3.8
have the same freshwater cost and piping cost, while the pumping cost in this
scenario is increased from $6,206 to $6,505.
In this example, it is observed that freshwater cost plays a key role in water
network design. Therefore, the design networks in scenario 2, 3, and 4 have the
same network configuration, and the result of scenario 1 (without water
recycling) has much higher total annual cost compared to others.
The underlined numbers shown below
each pump is the pressure head (kPa)
provided by the pump. The numbers
above each pipeline are the flow rate
through the pipe (t h−1)
: pump
Fig. 3.5 Network configuration for Example 1 in Scenario 1
The underlined numbers shown
the flow rate through the pipe (t h−1)
: pump
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the flow rate through the pipe (t h−1)
: pump
The underlined numbers shown below
each pump is the pressure head (kPa)
provided by the pump. The numbers
above each pipeline are the flow rate
through the pipe (t h−1)
: pump
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A multiple-contaminant water network consisting of seven water-using
operations and three types of contaminants are considered, which had been
adapted from (Chen et al., 2010). The operating data for water-using operations
is presented in Table 3.6. Table 3.7 shows the distance matrix for Example 2.
The pressure drops for water-using operations are given in Table 3.8. Also,
optimization is carried out with four scenarios. For all the scenarios, one
iteration is required to converge the solution between subproblems P1 and P2.
The computational data and results for Example 2 are shown in Table 3.9. and
Table 3.10., respectively.
Scenario 1. The network configuration with one-through use of water is
illustrated in Fig. 3.9. The expenditure of freshwater, pipelines, capital and
operating cost of pump are $3,323,256, $42,849, and $13,737, respectively.
The total annual cost is determined to be $3,379,842.
Scenario 2. As shown in Fig. 3.10, the network design is very complicated. To
control the complexity of the network, the additional constraint Eq. 3.12 is
introduced, and the maximum number of inlet stream for a single water-using
operation is set to be two. A simplified network configuration is provided in
Fig. 3.11. Note that the piping cost reduces from $69,629 to $61,403, and
pumping cost including capital cost and operating cost reduces from $17,525 to
$15,856. However, the total annual cost increase from $2,408,159 to
$2,482,935 due to the growth in freshwater consumption.
Scenario 3. Fig. 3.12 shows the optimal design network considering water
distribution, capacity of pumps, and location of pumps at the same time. To
reduce the complexity of the network, the simplified network by imposing the
constraint on the maximum number of inlet stream (NSumax = 2) is obtained as
given in Fig. 3.13. The total annual cost increases by 3.2% compared with the
result before network simplification is made, and the rise in total annualised
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cost mainly results from the increase in freshwater intake.
Scenario 4. The optimal configuration is given in Fig. 3.14. After limiting the
maximum number of inlet streams ( NSumax = 2) , a different network
configuration is obtained as shown in Fig. 3.15, which results in a increase of
3.2 % in total annualised cost.
Table 3.6. Operating data for water-using operation in example 2
Max. inlet Max. outlet Mass
Operation Contaminant conc. conc. load
(ppm) (ppm) (g/h)
B 0 100 2500
C 0 50 1250
B 0 300 21000
C 0 600 42000
B 50 400 12250
C 50 800 26250
B 110 450 13600
C 200 700 20000
B 100 650 4400
C 200 400 1600
B 300 3500 160000
C 600 2500 95000
B 0 400 18000
C 0 35 1575
Table 3.8. Pressure drops for water-using operations in Example 2
O1 O2 O3 O4 O5 O6 O7 discharge
S1 50 63 70 87 75 37 62 200
O1 0 125 175 200 225 150 75 150
O2 125 0 15 50 100 125 150 60
O3 175 15 0 50 25 125 150 50
O4 200 50 50 0 75

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