A publication of
CHEMICAL ENGINEERING TRANSACTIONS
VOL. 35, 2013
The Italian Association
of Chemical Engineering
www.aidic.it/cet
Guest Editors: Petar Varbanov, Jiří Klemeš, Panos Seferlis, Athanasios I. Papadopoulos, Spyros Voutetakis
Copyright © 2013, AIDIC Servizi S.r.l.,
ISBN 978-88-95608-26-6; ISSN 1974-9791
Water Allocation Network Synthesis Involving
Reliability Analysis
Jian Du*a, Jing Chena, Ji- Long Lia, Qing- Wei Mengb
a
Institute of Process System Engineering, School of Chemical Engineering, Dalian University of Technology, Dalian,
116012, China
b
School of Pharmaceutical Engineering, Dalian University of Technology, Dalian, 116012, China
dujian @dlut.edu.cn
Water has been a severe problem for freshwater reduction and legislation strengthening for many years.
The purpose of water allocation network (WAN) synthesis is to reduce the freshwater consumption and the
wastewater discharge, while the network reliability is worth considering for the sake of safety and the
maintenance cost. The structure of WAN has a significant impact on reliability issues. So the issue
discussed in this paper is to integrate the reliability analysis into the WAN synthesis problem. It is desirable
to obtain a WAN which satisfies the reliability as well as consuming the minimum freshwater. An example
is presented to illustrate the application of the proposed method and demonstrate that this method is
effective.
1. Introduction
Water is a key element for the normal functioning of the industry and the society, which is now gravely
concerned about. Researchers have proceeded a large of studies about water allocation network (WAN)
synthesis to reduce freshwater consumption and wastewater discharge in chemical industry and have
made a great achievement in the past two decades.
Many methods have been presented to settle the water allocation problem. Takama et al. (1980) first
described a water allocation network (WAN) in a petroleum refinery with a superstructure of all possible
reuse possibilities and proposed a mathematical programming to solve the problem. Wang and Smith
(1994) proposed the water pinch method with the creative concepts of water pinch and limiting water
profile. Then this graphical method developed fast (Wang & Smith, 1995; Kuo & Smith, 1995; Hallae,
2002; Feng et al., 2007) and widely applied to chemical industry. However, the graphical methods are
restricted for large-scale water allocation problem and multi-contaminant problem. Then researchers took
their eye on mathematical programming methods again. Doyle and Smith (1997) presented linear
programming (LP) and non-linear programming (NLP) methods to minimize freshwater consumption for
water networks with multi-contaminant. Alva-Argaez et al. (1999) optimized water using problem by
combining water-pinch analysis technics with mathematical programming tools. Gunaratnam et al.(2005)
applied non-linear mathematical programing(NLP) methods to minimizing freshwater consumption of water
using networks. For the difficulty of calculating and the global optimization of the solution of the problem,
Savelski and Bagajewicz (2000) developed necessary conditions for the optimal water allocation problem.
Liu et al. (2009) proposed a new concept of concentration potential to deal with multiple contaminants
problem. This new method has achieved some progress in recent years. (Liu et al., 2012; Liu et al., 2013)
Researches on reliability analysis in chemical industry have begun in1970s and already have some
achievements. (Henley & Gandhi, 1975; Goel et al, 2002; Sikos & Klemes, 2010) Reliability analysis of the
water allocation network is a new and crucial issue for it closely related to safety production and the
maintenance cost. The reliability of the network mainly relates to the reliability of the units and the structure
of the network. Therefor this paper integrates the reliability analysis into the WAN synthesis problem. For
the complexity of improving reliability of single unit to improve reliability of the network, simplifying the
structure of the water allocation network is simple as well as effective. It is desirable to obtain a WAN
which satisfies the reliability as well as consuming the minimum freshwater.
2. Problem statement
The problem of water allocation network synthesis involving reliability analysis in this paper can be briefly
stated as follows. Givens are a set of water-using units
fixed mass loads
ciin,k,max and
M i ,k
i P
in which some contaminants
k K
with
require to be removed and the maximum permissible inlet concentrations
out,max
the maximum outlet concentrations ci ,k
for each unit are specified; only freshwater is
supplied as external water source; reliability of each water-using unit
Ri is specified and the reliability of
the network only relates to the reliability of the units and the structure of the network. There is neither
water loss nor reactions among contaminants or phase change. This problem targets the optimal network
configuration which determines the minimum freshwater consumption and the resultant network structure
that satisfies the reliability issues.
3. Mathematical model
3.1 Superstructure
Figure 1 describes Water utilization network superstructure model of a single process. This superstructure
model includes all possibilities of water reuse.
F
j ,i
 F ，c
，c out
j ,k
i ,l
j
out
i ,k
l
ciin,k
ciout
,k
unit i
用
水
Pro
3.2 Nonlinear programming model
cess
F =min  F
i单
)  (F   F
 ( F c元
i
Fi
Wi
Figure 1 Water utilization network superstructure model of a single operation unit
A non-linear programming mathematical model is established to determine the minimum freshwater
requirement and the initial network structure.
Objective function
(1)
i
i
Mass balance of contaminants of the inlet mixing point of unit i:
j ,i
i ,jP
i j
out
j ,k
i
i ,jP
i j
j ,i
)ciin,k
(2)
Mass balance of contaminants of the out splitting point of unit i:
( Fi   Fj ,i )ciin,k +M i ,k =(Wi   Fi ,l )ciout
,k
(3)
Fi   Fj ,i  Wi   Fi ,l
(4)
i ,jP
i j
Water balance of unit i:
iP
i j
i ,jP
i j
lP
i l
Limit of inlet concentrations of contaminant of unit i:
0  ciin,k  ciin,k,max
(5)
Limit of outlet concentrations of contaminant of unit i:
out ,max
0  ciout
, k  ci ,k
Nonnegative restriction of flow:
(6)
0  Fi ,0  Fj ,i
(7)
4. Reliability analysis
Reliability issue in water allocation network is worth considering for the sake of safety production. The
problems here are how to calculate the reliability of a water network already existed and how to improve
the reliability of the network that can’t meet the process specification. After obtaining the minimum
freshwater consumption and the initial network structure, evaluate the reliability of the obtained initial
WAN. Recognize subsystems of the network from the incidence matrix based on the initial network. The
overall reliability of WAN is evaluated in terms of the network subsystems. For the subsystem whose
reliability fails to meet the process specification, its structure simplification is imperative.
Reliability of subsystem:
Rss = Ri
Reliability of the whole system:
(8)
i
Rs = min Rss
(9)
In this paper, the reliability is improved by breaking the water-reuse stream of the related water-use unit.
Each breaking to the water-reuse stream in a subsystem is theoretically allowed if the subsystem reliability
is substandard. The mass transfer mission of the broken stream is taken place by the fresh water. Reoptimize the initial WAN with breaking the specified stream, aiming at minimizing fresh water consumption.
Thus, a different breaking will generate a different new simplified WAN. To ensure the new obtained WAN
can feature the eligible reliability and simplified the calculation, two breaking conditions should be
observed: the breaking is feasible only if it can promote the network reliability; break the one which has a
higher concentration and lower flow rate if there are options.
5. Case study
This example taken from literature Wang et al (2003) is a multi-contaminant system with seven water-using
units and its limiting data is given in table1.
Table 1: Limiting data of example
Water-using unit
Ciin,k,max (ppm)
A
0
0
20
50
20
500
150
1
2
3
4
5
6
7
B
0
0
50
110
100
300
700
C
0
0
50
200
200
600
800
Ciout,max
,k
(ppm)
A B
50 100
100 300
150 400
600 450
500 650
1100 3500
900 4500
C
50
600
800
700
400
2500
3000
Fi max
(t/h)
25
70
35
40
8
50
30
Apply the mathematical model mentioned before in this paper to solve the problem above. The minimizing
freshwater consumption of the system is 140.93 (t·h-1) and the initial water network is as Figure 2.
140.93t/
h
t/h)t/h)
25t/h
t/h)t/
unit
1
h)
4.8t/h
t/h)t/
16.9t/h
h)
t/h)t/h)
70t/h
t/h)t/
h)
unit
2
24.23t/h
t/h)t/h)
9.15t/h
t3.2t/h
/h)t/h)
unit
3
t/h)t/h
)
unit
5
12.65t/
h
10.45t/
t/h)t/
h
h)
50t/h
t/h)t/h
t/h)t/h
unit
4
3.38t/h
t/h)t/h)
30t/h
t/h)t/h
Unit
7
)
8t/h
t/h)t/
30t/h
t/h)t
/h)
h)
40t/h
t/h)t/
h)
Unit
6
)
)
Figure 2: the initial water network of the example
50t/h
t/h)t/
9.55t/h
h)
t/h)t/h)
140.93t/h
t/h)t/h)
The incidence matrix of the initial network is as follow:
1
2
3
4
5
6
7
1
1
0
1
1
1
0
0
2
0
1
0
1
0
1
0
3
1
0
1
0
0
0
1
4
1
1
0
1
0
0
0
5
1
0
0
0
1
0
0
6
0
1
0
0
0
1
0
0
0
1
0
0
0
1
7
0
1
2
3
4
5
6
In this incidence matrix, numbers 1~7 in first line
and row represent water-using units; elements 1,
0 denote connections of the units or not. Here,
diagonal element means connection of water-using
unit with itself and it is commanded to be 1. Then,
recognize subsystems of the network from the
incidence matrix based on the initial network. It is
obvious from Fig.3, that there is only a subsystem
which contains all the seven water-using units.
Assume reliability of each unit is 0.98, which means
that the failure rate is 0.02 and reliability asked for
the system is 0.90. For this case,
Rs =Rss = Ri =0.987 =0.87<0.90
7
i
Reliability of the system fails to meet the process
specification, if not considering about improving
1
0
1
1
1
0
0
reliability of single unit, structure simplification of the
system is imperative. Here, the reliability can be
2
0
1
0
1
0
1
0
0
improved by breaking any water-reuse stream in the
system. Each breaking to the water-reuse stream in
3
1
0
1
0
0
0
1
1
the system is theoretically allowed which probably
01
adds freshwater consumption.
4
1
0
1
0
0
0
Results of minimizing freshwater consumption and
11
reliability improvement of all possible breakings show
5
0
0
0
1
0
0
in Table 2. It is apparently that the best choice is the
0
breaking of S1-3 which meets the reliability
6
0
1
0
0
0
1
0
specification 0.90 with the minimum freshwater
1
7
0
0
1
0
0
1
consumption 143.41 t/h. The final structure of the
30
water network corresponding to the breaking of the
3
Figure 3 Procedure of recognizing
system is as Figure 4.
1subsystems of the initial network
For this example, reliability of the water allocation
01
work is improved
to appropriate value (from 0.87 to 0.9) with 1.75% increasing of freshwater consumption
0
(from140.93 t/h to 143.41 t/h) . It is absolutely that improving reliability of single unit to improve reliability
1 is complex. So this method is effective and can be accepted.
of the network
0
21
01
0
0
0
0
Freshwater
0
consumption
t/h 11
System
Reliability44
1
11
1
0
0
1
1
0
0
0
0
0
50
5
1
Table 2: result of all possible breakings for improving reliability of the system
Breaking
S1-4
Breaking
S2-4
Breaking
S1-3
Breaking
S1-5
Breaking
S3-7
Breaking
S2-6
147.47
149.05
143.41
141.68
163.55
179.50
0.92
0.90
0.90
0.89
0.89
0.89
32.81t/h
t/h)t/h)
143.41t/
h
t/h)t/h)
25t/h
t/h)t/
unit
1
h)
)
4.8t/h
t/h)t/
10.8t/h
h)
t/h)t/h)
70t/h
t/h)t/
h)
3.2t/h
t/h)t/h
unit
2
21.8t/h
t/h)t/
7.4t/h
h)
t/h)t/h
50t/h
)
t/h)t/h
unit
3
unit
5
unit
4
2.81t/h
t/h)t/h)
30t/h
t/h)t/h
Unit
7
)
8t/h
t/h)t/
30t/h
t/h)t
/h)
143.41t/h
t/h)t/h)
h)
40t/h
t/h)t/
h)
Unit
6
Figure 4:) The final network of the example
50t/h
t/h)t/
12.6t/h
h)
t/h)t/h)
6. Conclusions
Reliability issue in water allocation network is worth considering for the sake of safety production. The
problems here are how to calculate the reliability of a water network already existed and how to improve
the reliability of the network that can’t meet the process specification. It is absolutely that improving
reliability of single unit to improve reliability of the network is complex and may difficult to realize. This
paper deals with this problem by simplifying the structure of the network which means breaking some reuse stream. Theoretically, any water reuse stream can be broken to improve the reliability of the network.
While, for water all0cation network, freshwater consumption is always the target. So the final water
allocation network should consume the minimum freshwater with the appropriate reliability. To calculate
easily, two breaking conditions should be observed: the breaking is feasible only if it can promote the
network reliability; break the one which has a higher concentration and lower flow rate if there are options.
The example in this paper shows this procedure in detail, reliability improved to appropriate value with
1.75% increasing of freshwater consumption. This result means this method is effective.
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