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Ind. Eng. Chem. Res. 2007, 46, 4954-4963
Design Methodology for Flexible Multiple Plant Water Networks Z. W. Liao, J. T. Wu, B. B. Jiang, J. D. Wang, and Y. R. Yang* Department of Chemical and Biochemical Engineering, State Key Laboratory of Chemical Engineering, Polymer Reaction Engineering DiVision, Zhejiang UniVersity, Hangzhou 310027, People’s Republic of China
This paper deals with the design of flexible multiple plant water networks. Some practical features of multiple plant integration such as operating flexibility and expensive operating cost have made the number of cross plant interconnections (CPIs) an important parameter of water minimization problem. By combining pinch insight with mathematical programming, target freshwater usage and CPI conditions are obtained first without considering the detailed network design. Subsequently, an MILP model is established for the design of flexible water networks of individual plants. The proposed systematic approach can be applied for both fixed contaminant (FC) operations and fixed flow rate (FF) operations, limited to a single contaminant. An example from literature and a practical example of a chlor-alkali complex are used to illustrate the applicability of the approach. 1. Introduction Because the rising cost of freshwater and effluent treatment as well as the more stringent environmental regulations, considerable design techniques have been developed to minimize the freshwater usage and wastewater discharge. Early work focused on fixed contaminant (FC) operations; for instance, Wang and Smith1 introduced a graphical method for targeting maximum water reuse, which has been improved by several authors.2-5 Mathematical programming techniques have also been used for water integration problems. Huang et al.6 developed an NLP model for the design of water networks. Alva-Argaez et al.7 introduced transshipment models that enable easy screening and scoping ahead of the network development. Savelski and Bagajewicz8 reduced the NLP problem to an LP problem for single contaminant systems. Feng and Seider9 introduced internal water mains to simplify the structure of the water network. Xu et al.10 developed a three step MINLP programming method to target water networks with regeneration reuse. A comprehensive review can be found in the work by Bagajewicz.11 Subsequent work shifted to fixed flow rate (FF) operations. Dhole et al.12 introduced the water source and demand composite curves, but the resulting target depends on mixing options. Hallale13 suggested a water surplus diagram, and Manan et al.14 developed a water cascade diagram. The two methods are able to achieve the real target by ensuring pure water surplus at each purity level. Later, El-Halwagi et al.15 and Prakash and Shenoy16 developed a new graphical technique, the load versus flow composite, to achieve the target. Recently, Agrawal and Shenoy17 extended the limiting composite curve concept of the FC operations to solve the problem of the FF operations. The above-mentioned techniques have focused on water integration for individual plants. As reported recently,18 water saving opportunities can be further achieved by cross plant water integration. This suggests that the strategy of cross plant water integration should be considered. The cross plant integration has some practical features. For example, cross plant water transfer may cause high pumping and piping costs. An even more practical consideration is that the production rates of some plants may vary from * To whom all correspondence should be addressed. E-mail:
[email protected]. Fax: +86 571 87951227.
period to period as a result of market or seasonal changes. This flexibility can influence the water saving benefits. If one of the integrated plants changes its production rate, the other integrated plants may have to adjust their water networks to keep water use efficient. In this work, we define the flexibility as the ability to operate feasibly under a set of specified operating conditions. Therefore, these considerations yield the problem of multiperiod design which can be stated mathematically as
min Ctol ) C(pI, pII, qI, qII) +
Ce(pI, pII, qI, qII, zI,e, zII,e) ∑ e∈V
(P1)
s.t.
ge(pI, pII, qI, qII, zI,e, zII,e) ) 0
e ) 1, ..., NE
(1)
fe(pI, pII, qI, qII, zI,e, zII,e) e 0
e ) 1, ..., NE
(2)
V(pI, pII, qI, qII, zI,1, zI,2, ..., zI,NE, zII,1, zII,2, ..., zII,NE) e 0 (3) where NE is the number of periods and V is the set of periods. The superscripts I and II denote within plants and across plants, respectively. pI and pII are vectors of design variables representing the number of interconnections, and qI and qII are vectors of binary variables concerning the existence of an interconnection, while zI,e and zII,e are vectors of state variables representing the flow rate of process streams for period e. The objective Ctol consists of a fixed charge, C, for the capital cost of the network structure, and Ce, for the operating cost of the individual periods. ge and fe are vectors of equations and inequalities, respectively, for period e, while V is a vector of inequalities involving variables of all time periods. The multiperiod problem of water networks has not been studied yet. However, there is a rich literature in studying the multiperiod problem of heat exchanger networks.19-24 In this paper a new approach that combines the insights from water pinch with mathematical programming is developed to deal with the complicated multiperiod problem in multiple plant water networks. The approach consists of two steps. In the first step, targets including freshwater usage and number of cross plant interconnections (CPIs) are established. In the second step,
10.1021/ie061299i CCC: $37.00 © 2007 American Chemical Society Published on Web 05/31/2007
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 4955
Figure 2. Flow pattern in concentration level k of a single plant. Figure 1. Transshipment model for the single plant water minimization problem.
a water network that can be operated in different periods is obtained. The article is organized as follows: a new transshipment model is introduced first to establish minimum freshwater cost for a single plant in which the FC operations and the FF operations coexist. Then, this transshipment model is extended to the multiple plant case. An MINLP problem is set up to obtain the minimum total cost under a number of l CPIs. Next, a sequential procedure is introduced to determine the optimum number of CPIs. Finally, an MILP problem is established to obtain the flexible water network for individual plants. To illustrate these concepts, examples from literature and a chloralkali complex are solved. This paper is restricted to problems with single contaminants. Work to extend the approach to multiple contaminants and to permit the use of regenerated water is underway. 2. Transshipment Model for a Single Plant The FC operations and the FF operations may coexist in a single plant. For the FF operations, the inlet and outlet flow rate are fixed. The outlet streams always leave at specified concentrations, while the inlet streams have maximum allowable concentrations given. For the FC operations, the contaminant loads are fixed and the maximum allowable inlet and outlet concentrations are specified. The flow rate of the inlet and outlet of the operation is the same and is determined by
L F) cout - cin
water flow as a commodity. The water flow is shipped from process sources and freshwater sources to process demands and wastewater mains through concentration levels. Figure 1 illustrates that water flows from process sources and freshwater sources to the corresponding concentration level and then to process demands and wastewater mains in the same concentration level with the remainder going to adjacent lower and higher concentration levels. The lowest concentration level is denoted by k ) 1, while the highest level is denoted by k ) K. Figure 2 illustrates the water flow pattern for level k. The summary is as follows: 1. Streams flow into a particular concentration level from process sources and freshwater sources whose composition corresponds to the concentration level. 2. Streams flow out of a particular concentration level to process sinks and wastewater mains with a concentration requirement at the same level. 3. Streams flow out of a particular concentration level to the adjacent lower and higher concentration levels. These are the residual flows that cannot be utilized in the present level and consequently have to flow to the adjacent concentration levels. 4. Streams flow into a particular level from the adjacent lower and higher concentration levels. The following sets are defined to identify the set of freshwater source and process source/sink that are located at concentration level k:
Gk ) {〈h|freshwater source h in level k〉} Sk ) {〈i|source i which is present in level k〉}
(4)
where L is the given mass load of the contaminant. For targeting purposes, the inlet and outlet concentrations of the FC operations can be assumed at their maximum values.1 If all the inlet streams are regarded as sinks and all the outlet streams are regarded as sources, the problem of targeting minimum freshwater cost for a single plant can be described as follows: The set of freshwater sources is designated G ) {h ) 1, 2, ..., NG}, each freshwater source has a specified composition, ch. The set of process sources is designated S ) {i ) 1, 2, ..., NS}, and each source has a given flow rate, FiS, and a given composition, ci. The set of process sinks is designated D ) {j ) 1, 2, ..., ND}, and each sink requires a specified feed flow rate FjD and a specified composition cj. The objective is to minimize the cost of freshwater sources that can be purchased to supply the use of process sources in process sinks. As shown in Figure 1, the whole system can be divided into several concentration levels which correspond to the concentration of freshwater sources and process sources/ demands. Following an analogy to the HEN problem,25 a transshipment model is developed in the form of considering
Dk ) {〈j|sink j which is present in level k〉} To simplify the formulations, the residual flows between level k and level k + 1 can be regarded as one simple flow rk whose flow rate is the residual flow from level k to level k + 1 minus the residual flow from the opposite direction. As contaminated water streams are composed of pure water and contaminant, the pure water concentration of level k can be expressed as (106 - ck), where the concentration is measured in parts per million. The surplus of pure water between level k - 1 and level k is defined as
Qk ) rk-1[(106 - ck-1) - (106 - ck)] ) rk-1(ck - ck-1)
(5)
As detected from the pinch insight,13 the amount of pure water should meet the demands of each concentration level. Therefore, the cumulative surplus of pure water at each concentration level k should be nonnegative: k
∑1 Qk g 0
∀k ) 2, 3, ..., K
(6)
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By performing a total mass balance and pure water surplus constraint at each concentration level, the transshipment model for minimum freshwater cost of a single plant case is given by
min
ah F wh ∑ h∈G
(P2)
s.t. eqs 5 and 6
FiS + ∑ F wh + rk-1 ) ∑ F Dj + Wk + rk ∑ i∈S h∈G j∈D k
k
k
∀k ) 1, 2, ..., K (7)
rK ) 0 ∀k ) 1, 2, ..., K, h ∈G
F wh , Wk g 0
(8) (9)
where F wh is the flow rate of the hth freshwater source, Wk is the flow rate of wastewater out of level k, and ah is the cost coefficient of the hth freshwater source. The above formulation is an LP problem, which can be easily solved to yield minimum freshwater cost. Moreover, the location of a pinch point is indicated when the cumulative pure water surplus approaches zero, namely, k
∑1 Qk ) 0
(10)
3. Targeting Procedure for Integration Across Plants The objective of the previous model is to determine the minimum freshwater cost in a single plant. However, for the case of multiple plants, minimizing the freshwater target is not enough. As the piping and pumping cost are high for cross plant integration, the number of CPIs becomes an important parameter. It is desired to obtain a minimum freshwater consumption design with minimum number of CPIs. Moreover, the multiperiod effect should be considered in the objective function. The flow rate of the CPIs should vary with the periods to keep water use efficiency among different periods. At these points it is assumed that the cross plant water streams are allowed to mix in the CPIs and to vary flow rate with the period changing. In addition, the concentration of each interconnection is assumed fixed among periods for ease of concentration control. Therefore, the CPI in this paper stands for a main water pipe with specified concentration level. It receives streams of different concentrations from the sources of one plant and delivers them long distances to the sinks of another plant. Note that if a CPI is selected in one of the periods, it will have to be presented during all the other periods. Let us consider the general case of NM plants that has NE periods. Given are that each period has the operating time of te and a number of l CPIs are presented among the plants. To extend the single plant model to the multiple plant case, the concentration levels of sources and sinks from all plants should be presented in the transshipment model of each plant, and the locations of CPIs need to be identified. Without loss of generality, assume the oth CPI is established between plant m and plant n. Indeed, the introduction of a CPI adds a new concentration level to the transshipment model of the two plants, and this concentration level can be settled anywhere between concentration level c1 and cK. Let Om,n,k,o denotes the oth interconnection that transfers water from plant m to plant n with its concentration located between level ck and ck+1. As the
Figure 3. Possible locations of l CPIs between plant m and plant n.
concentration of CPI is assumed fixed among periods, it is denoted by ckm,n,k,o. Also let Tm,n,k,o,e be the flow rate of Om,n,k,o in period e. Figure 3 shows the possible locations of the Om,n,k,o between plant m and n. The concentration levels ckm,n,k,o for the oth CPI between level ck and ck+1 can be different. However, because there only exists one oth CPI, only one value of such concentration levels is valid. Therefore, all these possible concentrations between level ck and level ck+1 can be assumed to be an identical value ckk,o:
ckm,n,k,o ) ckk,o
m, n ∈ M, m * n, k )1, 2, ..., K - 1 (11)
For ease of formulation, the constant concentration levels ck are also denoted by ckk,o:Assume that the number of l potential
ck ) ckk-1,l+1 ) ckk,0, k ) 1, 2, ..., K
(12)
CPIs between concentration level ck and level ck+1 are arranged in concentration increasing order:
k ) 1, 2, ..., K - 1, o ) 0, 1, ..., l (13)
ckk,o e ckk,o+1
Consider the following general binary variables:
Ym,n,k,o )
{
1 water main Om,n,k,o exists 0 otherwise
}
(14)
These binary variables are related to the variables Tm,n,k,o,e, through the inequalities
Tm,n,k,o,e - UYm,n,k,o e 0 ∑ e∈V
m, n ∈ M, m * n, o )
1, 2, ..., l, k ) 1, 2, ..., K - 1 (15)
where U is a sufficiently large number. Note that when Ym,n,k,o takes a zero value, there are no cross plant flow rates existing, but when it is set to one, any amount of flow rate that does not exceed U can exist. To guarantee that only one oth CPI exists and the total number of l CPIs exist, the following set of equations is introduced: K-1
∑ ∑ ∑ Ym,n,k,o ) 1
o ) 1, 2, ..., l
(16)
m∈M n∈M,n*m k)1
Let rm,k,o,e be the water flow from concentration level ckk,o-1 to level ckk,o in plant m during period e. An MINLP problem based
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 4957
on the preceding binary variables to find the location of a number of l of CPIs is then proposed:
min
∑ ∑∑ m∈M e∈V h∈G
K
w ahteFm,h,e +
l
Table 1. Data for FC Operations in Example 1 operation number
∑ ∑ ∑ bm,nYm,n,k,o + m,n∈M,m*n k)1 o)1 K
l
∑ ∑ ∑ ∑ tedm,nTm,n,k,o,e m,n∈M,m*n k)1 o)1 e∈V
(P3)
s.t. eqs 11-16
∑ i∈S
S Fi,e +
m,k
∑ j∈D
Plant 2 (Reference 4) 2 0 5 50 30 50 4 400
100 100 800 800
P5
Plant 1 (Reference 5) 25 200
700
demand
D Fj,e + Wm,k,e + rm,k,1,e
e ∈ V, m ∈ M, k ) 1, 2, ..., K (17)
∑
Tn,m,k,o,e )
n∈M,n*m
∑
Tm,n,k,o,e + rm,k,o+1,e
n∈M,n*m
e ∈ V, m ∈ M, o ) 1, 2, ..., l, k ) 1, 2, ..., K - 1 (18) k
l+1
∑ ∑ [rm,k,o,e(ckk,o - ckk,o-1)] k)1 o)1 u
∑ [rm,k,l-o+2,e(ckk,l-o+2 - ckk,l-o+1)] g 0 o)1
e ∈ V, m ∈ M, u ) 1, 2, ..., l, k ) 1, 2, ..., K - 1 (19) k
l+1
∑ ∑ rm,k,o,e (ckk,o - ckk,o-1) g 0
k)1 o)1
source
process number
concentration (ppm)
D5 D6 D7 D8 P5in
100 0 0 10 200
P1in P2in P3in P4in
0 50 50 400
D1 D2 D3 D4
20 50 100 200
flow rate
process number
concentration (ppm)
flow rate
1000 100 10 100 700
20 40 10 5 50
Plant 2 (Reference 4) 20 P1out 100 P2out 40 P3ourt 10 P4ourt
100 100 800 800
20 100 40 10
Plant 3 (Reference 28) 50 S1 100 S2 80 S3 70 S4
50 100 150 250
50 100 70 60
Plant 1 (Reference 5) 80 S5 10 S6 10 S7 15 S8 50 P5out
Table 3. Period Length and Production Rate of Example 1
m ∈ M, k ) 1, 2, ..., K - 1, e ∈ V (20)
w Wm,k,e, Tm,n,k,o,e, Fm,h,e g0 m, n ∈ M, k ) 1, 2, ..., K, e ∈ V, h ∈ G (21)
rm,K,1,e ) 0
m ∈ M, e ∈ V
cout,max (ppm)
P1 P2 P3 P4
m,k
rm,k,o,e +
cin,max (ppm)
Table 2. Data for Example 1 in Source and Demand Form
w + rm,k-1,l+1,e ) ∑ Fm,h,e h∈G k
contaminant load (kg/h)
(22)
where dm,n and bm,n are the cost coefficients for operating cost and annualized capital cost of CPIs between plant m and plant n, respectively. The first constraints set (eq 17) represents the mass balance around the concentration level ck, while the next set of constraints (eq 18) represents the mass balance around the concentration level ckk,o where cross plant transfer take place. The constraint sets in eqs 19 and 20 account for ensuring the nonnegativity of pure water at each level ckk,o and ck, respectively. Finally, the nonnegativity constraints are shown in the constraint set of eq 21. It should be noted that Tm,n,k,o,e is different from Tn,m,k,o,e, for Tm,n,k,o,e represents the water transfer from plant m to plant n, while Tn,m,k,o,e represents the opposite water transfer direction. Furthermore, the CPI could occur in both ways (either from m to n or from n to m). When the minimum cost under a fixed number of CPIs is obtained, it is desired to determine the optimum number of CPIs. To find the optimum value, a step by step procedure is proposed. In the procedure, the initial value of l is set to one. At each step, the resulting freshwater cost is compared with the resulting value of the previous step. The number of l will be increased until the difference approaches zero. By comparing the minimum total cost of each step, the optimum number of CPIs is obtained. 4. Design Water Networks of Single Plants The next goal is to design the flexible water networks. Individual plants are designed separately in this section, because
production rate number of periods
working time (h/yr)
plant 1
plant 2
plant 3
period 1 period 2 period 3
2000 4000 2000
1 1 1
0 1 1
1 1 0
Table 4. Targeting Results of Example 1 CPI conditions number of CPIs
source plant
demand plant
concentration (ppm)
flow ratea (t/h)
1 2
plant 3 plant 2
plant 1 plant 1
200 100
50/39.46/0 0/44.29/44.29
Freshwater Usage freshwater flow rate (t/h) a
plant 1a
plant 2a
plant 3a
54.93/24.5/52.68
0/90/90
70/70/0
Period 1/period 2/period 3.
Figure 4. Results of the total cost targeting model for example 1.
the freshwater usage as well as the CPI conditions of each plant has been determined previously by the targeting procedure. Although the targeted freshwater usages usually change from period to period, the connecting structure of the single plant is fixed. In other words, if a interconnection between oper-
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Figure 5. Water network for example 1. Table 5. Water Network Details for Figure 5 plant 2 connection
plant 3 flow ratea (t/h)
connection
connection
F w1 F1,3 F w3 F ww 3
0/20/20 0/20/20 0/20/20 0/40/40
FS1,D2 FS2,D1 FS2,D2 FS2,D3
50/50/0 10/10/0 10/10/0 80/80/0
F5wD F5ww S Fo1,5
F2,4
0/5.71/5.71
FS3,D2
10/10/0
F ww 5
F ww 4 F w2 F2,o2
0/5.71/5.71 0/50/50 0/44.29/44.29
FS3,D4 FS3,o1
35/35/0 25/19.73/0 F3ww S 0/5.27/0 35/35/0 0/5.27/0 25/19.73/0 40/40/0 30/30/0
Fo1,D5 FS6,5 FS6,D8 Fo2,D5
FS4,D4 F4ww S FS4,o1 F1wD F2wD
a
plant 1 flow ratea (t/h)
FS6,D5
F7wD FS7,D8 F8wD FS8,D5 Fw5
flow ratea (t/h) 17.75/0/0 20/20/20 32.25/39.46/0 39.50/30.71/3 0.71 44.93/48.24/3 6.97 17.75/0/0 0/8.79/8.79 0.5/0.5/0.5 0/44.29/44.29 F6wD10/10/10 10/10/10 10/10/10 4.5/4.5/4.5 5/5/5 12.68/0/28.18
Period 1/period 2/period 3.
ation i and operation j is selected in one of the periods, it will have to be presented in all the other periods whether in the state of busy (in use) or idle. Therefore, for individual periods, the network structure will often be more complex
than optimal solutions for the benefit of the overall design. Under such a condition, all possible connecting choices should be included in the mathematical model to obtain the overall optimal solution.
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 4959
The assumptions of fixed inlet and outlet concentrations of the FC operations and fixed inlet concentrations of the FF operations made in the previous section are relaxed in this section to obtain water networks with simpler structure and lower water transfer flow rate. Therefore, the two kinds of operations should be described separately in the following mathematical model. Let Pm be the set of the FC operations in plant m and Lj,e be the mass load of the jth FC operation in period e. The following MILP model is presented to obtain the minimum number of interconnections required for the mulitiperiod scheme of plant m.
min
{
∑
∑
w lh,jyh,j
∑
lo,jyo,j +
li,jyi,j +
i∈Pm∪RmS,j∈Pm∪RmD
∑
+
j∈Pm∪RmD,h∈G
liyww + i
i∈Pm∪RmS
o∈Om,inj∈Pm∪RmD
∑
li,oyi,o
i∈Pm∪Rm
S,o∈O
m,out
}
(P4)
∑
Fi,o,e ) Fo,e
o ∈ Om,out, e ∈ V
w (F h,j,e c h) + ∑ ∑ h∈G
(Fi,j,eci) +
S
∑ o∈O
i∈Rm
(Fi,j,ecmax ∑ ,out ) + i∈P i
m
∑
w F h,j,e ) Rm,h,e
j∈Pm∪Rm
Fw ∑ h∈G
+ h,j,e
D
∑ o∈O
h ∈ G, e ∈ V, m ∈ M (23)
∑
Fo,j,e +
m,in
∑
(Fi,o,eci) +
i∈RmS
(Fi,o,ecmax ∑ ,out) ) Fo,eco i∈P i
m
o ∈ Om,out, e ∈ V (32) Fi,j,e - yi,jUi,j e 0 ∑ e∈V
i ∈ Pm ∪ RmS, j ∈ Pm ∪ RmD
Fi,o,e - yi,oUi,o e 0 ∑ e∈V
i ∈ Pm ∪ RmS, o ∈ Om,out
(34)
Fo,j,e - yo,jUo,j e 0 ∑ e∈V
j ∈ Pm ∪ RmD, o ∈ Om,in
(35)
(33)
∑
∑
Fj,i,e +
Fj,o,e + Wj,e
y ∈ {0, 1}
o∈Om,out
j ∈ Pm, e ∈ V (24) max max Fi,j,e(ci,out - cj,in )+ ∑ ∑ i∈P
∑F h∈G
w h,j,e
(ch -
max Fi,j,e(ci - cj,in )+
i∈RmS max cj,in ) + o∈Om,in
∑
max Fo,j,e(co - cj,in )e0
j ∈ Pm, e ∈ V (25) max max Fi,j,e(ci,out - cj,out )+ ∑ ∑ i∈P
∑ h∈G
max Fi,j,e(ci - cj,out )+
S
i∈Rm
m
w max F h,j,e (ch - cj,out )+
∑ o∈O
max Fo,j,e(co - cj,out ) + Lj,e ) 0
m,in
j ∈ Pm, e ∈ V (26)
∑
Fi,j,e + Wi,e +
j∈Pm∪RmD
∑ o∈O
j ∈ Pm ∪ RmD
(36)
i ∈ Pm ∪ RmS
(37)
i ∈ Pm ∪ RmS, j ∈ Pm ∪
RmD, h ∈ G, o ∈ Om, e ∈ V (38)
i∈Pm∪RmS
m,in
i∈Pm∪RmD
m
Wi,e - yww ∑ i Ui e 0 e∈V w , Fo,j,e, Fi,o,e, Fi,j,e g 0 Wi,e, F h,j,e
Fi,j,e )
j ∈ RmD, e ∈ V (31)
(Fo,j,eco) e Fj,eDcj
w w F h,j,e - yh,j Uh,j e 0 ∑ e∈V
s.t.:
(30)
i∈Pm∪RmS
Fi,o,e ) Fi,eS
m,out
i ∈ RmS, e ∈ V (27)
(39)
where l is the weight that can account for the piping cost of the matches between sources and sinks. In the case that these cost coefficients are set to one, the above problem will yield a solution for minimum number of interconnections. Rm,h,e is the target flow rate of the hth freshwater in plant m during period e, and Fo,e and co are also the target flow rate and concentration of the oth CPI. The set of constraints in eq 23 corresponds to the freshwater consumption constraint. The set of constraints in eq 24 represents the mass balance around the FC operations, while the constraints in eqs 25 and 26 show the inlet and outlet contaminant balance of these operations. Constraint sets in eqs 27 and 28 calculate the mass balance of the sources and demands of the FF operations, while constraint sets in eqs 29 and 30 represent the mass balance of the inlet and outlet CPIs. The contaminant balance for the demands and outlet CPIs are shown in the constraint sets in eqs 31 and 32. The next series of sets (eqs 33-37) relates the continuous variables to the binary variables, where the constants U are sufficiently large numbers. These constraints are used to count the number of interconnections. Finally, the set of constraints in eq 38 represents the nonnegativity constraints. 5. Examples
∑
Fi,j,e +
i∈Pm∪RmS
∑
h∈G
w F h,j,e
+
∑
Fo,j,e ) Fj,e
D
o∈Om,in
j ∈ RmD, e ∈ V (28)
∑
j∈Pm∪RmD
Fo,j,e ) Fo,e
o ∈ Om,in, e ∈ V
(29)
An example adopted from the literature and an example from a chlor-alkali complex are supplied to demonstrate the efficiency of the proposed two step approach. The MINLP and MILP models are coded in GAMS.26 The MINLP problem is solved using the algorithm of Viswanathan and Grossmann,27 which is available in GAMS as the solver DICOPT. The MINLP problem is decomposed into a series of NLP and MILP subproblems. These subproblems can be solved using any NLP
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Table 6. Limiting Process Data for Example 2 demand process name
process number
source concentration
flow rate
process name
brine saturation demineralized water pump seals inlet flocculant preparation resin regeneration
D1 D2 D3 D4 D5
100 1 1 100 1
Brine Circuit Plant 20 steam condensate 20 resin regeneration 5 pump seals outlet 5 4
HCl acid synthesis scrubber hot water vat pump seals inlet
D6 D7 D8 D9
70 50 50 50
Chlorine-Product Plant 15 scrubber 10 pump seals outlet 7 hot water vat 5
D10 D11
1000 50
Bleaching Powder Plant 10 steam condensate 1.5 washing
reaction washing
and MILP solvers that run in the GAMS environment. In this paper, the SNOPT solver is used for the NLP problem and the CPLEX solver is used for the MILP problem. Note that the solution does not provide the global optimum because the MINLP problem is nonconvex. Example 1. In this example, three plants operating over three time periods are considered. The plant from Wang and Smith (1995)5 is designated as plant 1, and the plant from Wang and Smith (1994)4 is plant 2, while the plant from Polley and Polley28 is plant 3. Plant 2 and plant 3 are composed of the FC operations and the FF operations respectively, while plant 1 consists of a combination of the two kinds of operations. The data for the FC operations of plant 1 and plant 2 are presented in Table 1. Then, these limiting data are converted to the source/ demand form in Table 2 which also presents the limiting water data for the FF operations of plant 1 and plant 3. As shown in Table 3, plant 1 is busy in all the three periods, while plants 2 and 3 are idle in periods 1 and 3, respectively. Assume that the cost coefficients between plants are identical: all bm,n’s are specified at $3750/yr while all dm,n’s are specified at $0.013/t. The whole system has only one freshwater source with its concentration located at 0 ppm and cost specified at $0.125/t. If no CPI is adopted for the cross plant integration, eq P2 is applied to calculate the targets. The resulting freshwater usage and the total cost are 1.425 million ton/yr and k$178.1/yr, respectively. Applying eq P3 to this example, total costs under a different number of CPIs are derived and presented in Figure 4, where the number 0 represents no cross plant integration taking place. As shown in the figure, the number of 2 CPIs is the optimum solution. The target total cost is k$173.2/yr with a freshwater consumption of 1.273 million ton/yr. Compared with the no cross plant integration case, a 2.7% reduction in total cost as well as a 10.6% reduction in freshwater consumption is achieved. The detailed results are shown in Table 4. Note that the resulting CPIs vary flow rate from period to period because of the start up and shut downs of the plants. Next, the resulting targets are used for the detailed network design. Equation P4 is applied to the individual plants by assuming all l ’s are set to one. The solution to plant 1 results in a network containing 15 interconnections. Similarly, the results of plant 2 and plant 3 include 8 and 13 interconnections. Figure 5 shows the resulting design of the water network of the entire system. The specifications for the network of Figure 5 are presented in Table 5. Example 2. This example deals with a chlor-alkali complex. Central to the activities of the complex is a brine circuit which converts raw salt to caustic soda, chlorine, and hydrogen. Part of the chlorine and hydrogen generated from the circuit is used
process number
concentration
flow rate
S1 S2 S3
1 100 70
14 15 5
S4 S5 S6
70 80 90
10 5 7
S7 S8
1 120
4 1.5
Table 7. Period Length and Production Rate of Example 2 production rate number of period
working time (h/yr)
brine circuit plant
chlorine-product plant
bleaching powder plant
500 5500 2000
1 1 1
1 1 0.8
0 1 1
period 1 period 2 period 3
Table 8. Freshwater Source Condition in Example 2 source item
cost ($/t)
concentration (ppm)
demineralized water freshwater
1.25 0.13
1 50
Table 9. Cost Coefficients between Plants in Example 2 from
to
bm,n ($/yr)
dm,n ($/t)
brine circuit plant brine circuit plant chlorine-product plant
chlorine-product plant bleaching powder plant bleaching powder plant
3750 4375 3375
0.013 0.015 0.01
for production of hydrochloric acid and white carbon black in the chlorine-product plant. Chlorine is further utilized, along with calcium hydroxide, in the manufacture of calcium hypochlorite in the bleaching powder plant. The water system of the whole complex only has FF operations with a single contaminant. The operations mainly care about the concentration of total hardness. Table 6 gives the limiting water data of the whole complex. For the reason of market changing, the bleaching powder plant will shut down for one period, and the production rate of the chlorine-product plant will reduce for another period; detailed information is presented in Table 7. The complex has two freshwater sources, freshwater and demineralized water, whose concentration and cost are shown in Table 8. Table 9 presents the cost coefficients between plants.
Figure 6. Results of the total cost targeting model for example 2.
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 4961 Table 10. Targeting Results of Example 2 CPI conditions number of CPIs
source plant
demand plant
concentration (ppm)
flow ratea (t/h)
1 2 3
bleaching powder chlorine-product brine circuit
brine circuit brine circuit bleaching powder
1 88 1000
0/4/4 8.67/8.67/6.93 0/8.5/8.5
Freshwater Usage brine circuit demineralized water freshwater flow rate (t/h) a
planta
chlorine-product planta
bleaching powder planta
0/0/0 23.67/23.67/18.94
0/0/0 0/1.5/1.5
15/11/11 10.47/10.47/12.13
Period 1/period 2/period 3.
Table 11. Water Network Details for Figure 7 bleaching powder plant connection FS8,D10 FS7,o1 Fo3,D10 F11wD
a
flow ratea (t/h) 1.5/1.5/0 4/4/0 8.5/8.5/0 1.5/1.5/0
brine circuit plant
chlorine-product plant
connection
flow ratea (t/h)
connection
flow ratea (t/h)
FS1,D2 FS2,D1 FS2,D4 FS3,D1 F4wD FS2,o3 F2ww S F1wD Fo1,D3 Fo2,D1 F2dw D F3dw D F5dw D
14/14/14 0.6/0.6/0.67 0.26/0.26/0.26 5/5/5 4.74/4.74/4.74 0/8.5/8.5 14.14/5.64/5.57 5.73/5.73/7.40 0/4/4 8.67/8.67/6.93 6/6/6 5/1/1 4/4/4
FS4,D6 FS5,D6 FS5,o2 FS6,o2 F6wD F7wD F8wD F9wD
10/10/8 3.33/3.33/2.67 1.67/1.67/1.33 7/7/5.6 1.67/1.67/1.34 10/10/8 7/7/5.6 5/5/4
Period 1/period 2/period 3.
Figure 7. Water network for example 2.
Assuming that the flow rate of process sources and demands vary with the production rate proportionally, the minimum total cost would be k$162.1/yr. Compared with the k$196.9/yr total cost of no cross plant integration, a reduction of 17.7% is achieved. As shown in Figure 6, the optimum solution is obtained under the number of 3 CPIs, and the detailed results of the target are presented in Table 10. Then, the water networks
are obtained under the assumption of all l ’s set to one. The results are shown in Figure 7. The solution to the brine circuit results in a network containing 13 interconnections. Similarly, the resulting network of the chlorine-product plant and the bleaching powder plant include 8 and 4 interconnections, respectively. Table 11 shows the interconnection specifications for the network of Figure 7.
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Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007
6. Conclusion The incentives of the total cost reduction and the zero liquid discharge have made the multiple plant water integration turn into reality. Getting the optimum water network of the multiple plants can be a very complex problem when the costs of fresh water and network structure as well as the operating flexibility have to be considered simultaneously. By combining pinch insight with mathematical programming, the target freshwater usage and CPI conditions can be obtained without considering the detailed network design. With this resulting target, the subsequent design of water networks of individual plants can be treated as separate tasks. The synthesis of the flexible multiple plant water networks is now carried out in two stages: targeting and design. In the targeting stage, the MINLP problem is established to provide the target freshwater usage and CPI conditions. In the design stage, the MILP problem is proposed to obtain a flexible water network that meets the freshwater target in all periods for individual plants. Examples from the literature and a chlor-alkali complex illustrate that this systematic approach reduces the design complexities and suites for complex industrial applications. Notation a ) cost coefficient of freshwater ($/t) b ) cost coefficient for capital cost of each CPI ($/yr) C ) capital cost c ) concentration of contaminant (ppm) ck ) concentration of CPIs CPI ) cross plant interconnection d ) cost coefficient for operating cost of CPI ($/t) D ) set of process demands e ) operating period F ) water flow rate (t/h) FD ) water flow rate of process demands (t/h) FS ) water flow rate of process sources (t/h) FC ) fixed contaminant FF ) fixed flow rate fe ) vector of inequalities for period e G ) set of freshwater sources ge ) vector of equations for period e K ) number of concentration levels L ) mass load of FC operations (kg/h) l ) number of CPIs M ) set of individual plants ND ) number of demands NE ) number of periods NM ) number of individual plants NS ) number of process sources O ) cross plant interconnection (CPI) P ) set of FC operations p ) vector of design variables Q ) surplus of pure water (g/h) q ) vector of binary variables concerning the existence of an interconnection r ) water flow from one concentration level to an adjacent lower one (t/h) RD ) set of demands for FF operations RS ) set of sources for FF operations S ) set of process sources t ) operating time (h) T ) cross plant transfer flow rate (t/h) U ) upper bond for cross plant flow rate
V ) set of operating periods V ) vector of inequalities involving variables of all time periods W ) wastewater flow rate (t/h) Y ) binary variable denoting a CPI that transfers water from one plant to another in a particular concentration y ) binary variable denoting a interconnection between two operations in individual plant z ) vector of state variables R ) target freshwater usage (t/h) l ) piping cost coefficient of the match between sources and sinks Subscripts h ) freshwater source i, j ) process operation k ) concentration level m, n ) individual plant in ) at the inlet of a process o,u ) cross plant interconnection (CPI) out ) at the outlet of a process Superscripts max ) at the maximum dw ) demineralized water tol ) total w ) freshwater ww ) wastewater I ) within plants II ) across plants Acknowledgment The financial support provided by National Natural Science Foundation of China (20490205) and National High Technology Research and Development Program of China (G20070040) are gratefully acknowledged. Literature Cited (1) Wang, Y.; Smith, R. Wastewater Minimization. Chem. Eng. Sci. 1994, 49 (7), 981-1006. (2) Kuo, W.; Smith, R. Effluent treatment system design. Chem. Eng. Sci. 1997, 52 (23), 4273-4290. (3) Kuo, W.; Smith, R. Designing for the interactions between wateruse and effluent treatment. Chem. Eng. Res. Des. 1998, 76 (A3), 287301. (4) Wang, Y.; Smith, R. Design of distributed effluent treatment systems. Chem. Eng. Sci. 1994, 49 (18), 3127-3145. (5) Wang, Y.; Smith, R. Wastewater minimisation with flow rate constraints. Chem. Eng. Res. Des. 1995, 73, 889-904. (6) Huang, C. H.; Chang, C. T.; Ling, H. C.; Chang, C. C. A mathematical programming model for water usage and treatment network design. Ind. Eng. Chem. Res. 1999, 38 (7), 2666-2679. (7) Alva-Argaez, A.; Vallianatos, A.; Kokossis, A. A multi-contaminant transhipment model for mass exchange networks and wastewater minimisation problems. Comput. Chem. Eng. 1999, 23 (10), 1439-1453. (8) Savelski, M.; Bagajewicz, M. On the optimality conditions of water utilization systems in process plants with single contaminants. Chem. Eng. Sci. 2000, 55 (21), 5035-5048. (9) Feng, X.; Seider, W. New structure and design methodology for water networks. Ind. Eng. Chem. Res. 2001, 40 (26), 6140-6146. (10) Xu, D. M.; Hu, Y. D.; Hua, B.; Wang, X. L. Minimization of the flowrate of fresh water and corresponding regenerated water in water-using system with regeneration reuse. Chin. J. Chem. Eng. 2003, 11 (3), 257263. (11) Bagajewicz, M. A review of recent design procedures for water networks in refineries and process plants. Comput. Chem. Eng. 2000, 24 (9-10), 2093-2113. (12) Dhole, V.; Ramchandani, N.; Tainsh, R.; Wasilewski, M. Make Your Process Water Pay for Itself. Chem. Eng. 1996, (103), 100-103.
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ReceiVed for reView October 11, 2006 ReVised manuscript receiVed March 30, 2007 Accepted April 2, 2007 IE061299I
