Water Integration of Eco-Industrial Parks Using a Global Optimization

Ind. Eng. Chem. Res. 2010, 49, 9945–9960

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Water Integration of Eco-Industrial Parks Using a Global Optimization Approach Eusiel Rubio-Castro,† Jose´ Marıá Ponce-Ortega,*,† Fabricio Na´poles-Rivera,‡ Mahmoud M. El-Halwagi,§ Medardo Serna-Gonza´lez,† and Arturo Jimeńez-Gutie´rrez‡ Chemical Engineering Department, UniVersidad Michoacana de San Nicola´s de Hidalgo, Morelia, Mich., Me´xico, 58060, Chemical Engineering Department, Instituto Tecnolo´gico de Celaya, Celaya, Gto., Me´xico, 38010, and Chemical Engineering Department, Texas A&M UniVersity, College Station, Texas 77843

This Article presents a mathematical programming model for the mass integration of eco-industrial parks. The model considers the reuse of wastewater among different industries and the constraints given by the process sinks and the environmental regulations for waste streams discharged to the environment. The model allows the optimal selection of treatment units to satisfy the process and environmental regulations. The objective function consists of the minimization of the total annual cost, including the treatment unit costs, the piping costs, and the cost of fresh water. A new discretization approach is proposed for the model reformulation to handle the bilinear terms of the model as part of a global optimization strategy. Results show that significant savings can be achieved for the design of an integrated eco-industrial park with respect to the integration of each individual industry. 1. Introduction Environmental pollution is a major concern nowadays because of the adverse impacts it has raised in several aspects such as human health, weather, regional identity, food, security, biodiversity, and economics. Research efforts using process integration strategies have addressed this concern. For instance, the rational use of water in the process industry has been the subject of several papers, in which the minimization of fresh water and wastewater in single process industries has been considered. The reported methodologies dealing with this line of research can be classified as algorithmic, graphical, algebraic, and mathematical-programming approaches. Among the algorithmic methodologies, Wang and Smith1 and Almutlaq and El-Halwagi2 proposed methods for minimizing the use of fresh water. The optimal design of effluent treatment systems and water networks has been addressed by Wang and Smith3 and Foo et al.4 Ng et al.5 considered aspects over regeneration opportunities and waste treatment. Of the methods based on graphical approaches, the major contributions have considered the minimization of fresh water,6 recycling water networks,7 optimization of single-contaminant regeneration/reuse systems,8 identification of individual wastewater streams,9 and regeneration and recycle through single treatment units.10 On targeting methodologies, some works to minimize the use of fresh water,11-14 and minimize the regeneration flow water15 have been proposed. Target methods for batch water networks, and for problems with single and multiple impure fresh water feeds have been addressed by Rabie and El-Halwagi16 and Foo,17 respectively. A paper review for the methodologies techniques for the water synthesis based on pinch analysis was recently reported by Foo.18 Among the approaches based on mathematical programming techniques, Takama et al.19 proposed a method for solving the planning problem of optimal water allocation. El-Halwagi et al.20 presented a mathematical model to determine the optimal water usage and interception network. Alva-Argaéz et al.21 applied an approach to mass * To whom correspondence should be addressed. E-mail: [email protected]. † Universidad Michoacana de San Nicola´s de Hidalgo. ‡ Instituto Tecnolo´gico de Celaya. § Texas A&M University.

exchanger networks and wastewater minimization problems. Hul et al.22 established a comparison between the solution of mass integration problems using particle swarm and genetic algorithms. Karuppiah and Grossmann23 optimized the synthesis of integrated water systems considering alternatives for wastewater treatment, reuse, and recycle, and Putra and Amminudin24 proposed a strategy to generate multiple optimum solutions for the total water system design problem. Recently, Ponce-Ortega et al.25,26 have reported methodologies for the recycle and reuse networks based on stream properties. The above-mentioned formulations have considered the implementation of mass integration strategies in single plants and have not taken into account the integration of wastewater from different plants. It is worth noting that the integration of wastewater from different industries located in the same place can reduce the wastewater discharged to the environment and simultaneously reduce the fresh water usage. In particular, the concept of eco-industrial parks (EIPs) has been recently proposed by Cote and Hall.27 According to Lowe,28 an ecoindustrial park is a community of manufacturing and service businesses seeking enhanced environmental and economic performance through collaboration in managing environmental and resource issues. As one example, Figure 1 shows how three companies can be integrated into the same industrial zone with processes that share some types of raw materials, inputs, services, products, and/or wastes. The goal of this global integration is the benefit of the participating companies considering both economic and environmental aspects. Therefore, an eco-industrial park derives from attempts to apply ecological principles in industrial activities taking into account their interaction with the communities, so that benefits on pollution prevention and sustainable development can be achieved through the cooperation among the organizations. Another aspect to consider is the mass and energy integration among different processes. Olesen and Polley29 addressed the interplant water integration problem with the use of the load table based on pinch analysis; this is one of the first works reported for the interplant mass integration. Sprigg et al.30 presented a source-sink integration approach to designing EIPs and characterized the design challenges into two classes: technical/economic and organizational/commercial/political chal-

10.1021/ie100762u  2010 American Chemical Society Published on Web 09/15/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010

Figure 1. Representation of an eco-industrial park.

lenges. Yoo et al.31 developed a methodology from pinch technology for water and wastewater minimization. Wang et al.32 proposed the application of energy analysis of eco-industrial parks with power plants and applied energy analysis to the systematic evaluation of a combined heat and power plant eco-

industrial park, considering both material recirculation and energy cascade utilization. Lovelady and El-Halwagi33 introduced a mass-integration framework and mathematical formulation for the design of EIP water networks. Lovelady et al.34 developed a property-integration optimization approach for

Figure 2. Superstructure for two interceptors and two plants with two process sources and two sinks.

Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010 35

designing EIPs that are constrained by properties. Liao et al. dealt with the multiperiod problem in multiple plant water networks; this systematic approach can be applied for fixed contaminant operations and fixed flow rate operations, although it was limited to single contaminant problems. Foo36 addressed the targeting plant-wide integration using the numerical tool of water cascade analysis. Ng et al.37,38 proposed linear models to determine the minimum resource consumption for a singleimpurity resource conservation network, a methodology that was then extended to determine targets for resource conservation networks with interceptors. Chew et al.39 proposed the synthesis of direct and indirect interplant water networks integration using mixer-integer linear programming and mixer-integer nonlinear programming, respectively; in both representations, the costs of fresh water, wastewater, treatment, and cross-plant pipeline were considered. Some limitations of the work by Chew et al.39 are that it did not consider the cross-plant pipelines, the selection for the type of treatment unit was not optimized simultaneously, and only one treatment unit was allowed to be selected; in addition, the model did not consider restrictions for the pollutant concentrations discharged to the environment. These drawbacks limit the applicability of their methodology. Chew and Foo40 presented a targeting approach for the interplant water networks, and Chew et al.41 proposed an approach for the analysis of interplant water integration based on game theory. Recently, Lim and Park42 reported a nonlinear programming model based on a novel superstructure to remodel a conventional industrial park as a green eco-industrial park; their objective function was the minimization of the total consumption of industrial water. The wastewater was characterized by its flow rate, and the model did not consider the environmental constraints for the waste stream discharged to the environment, the cross-plant pipeline costs, and the regenerations cost. Furthermore, the NLP model by Lim and Park42 is non convex, and the solution by standard optimization methods cannot guarantee a global optimal solution. This Article presents a global optimization algorithm for the optimal mass integration design of eco-industrial parks. The optimal integration of interplant water networks is made using a new superstructure that contains all the potential network configurations to yield the optimal one given a set of process sources and sinks along with their flow rates and pollutant concentration specifications and/or limits. Also given are a set of units for the treatment of the process sources with specified removal capabilities and fresh water supply options with different pollutant concentrations. Maximum limits for pollutant concentrations on the wastewater to be discharged to the environment are included. The objective function consists of the minimization of the total annual cost, which includes the costs of fresh water, wastewater treatment, and cross-plant pipelines. The majors contributions of this work are: a new linear model for the water integration in eco-industrial parks is developed, which is based on a new superstructure that allows the detection of a global or near-global optimal solution; the selection of treatment units needed to meet process sinks and environmental constraints is optimized simultaneously; several types of fresh waters are considered; and streams with several pollutants, along with the environmental regulations for streams discharged to the environment, are included. 2. Model Formulation The proposed model is based on the superstructure shown in Figure 2, which shows a representation for two plants (each one with its individual mass exchange network) with two process sources, one contaminant, and several types of fresh water. In

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this representation, each source can be segregated and directed to the interceptors, to the process sinks, and/or discharged to the environment; the streams at the exit of each interceptor can be split and directed to the process sinks and/or to the environment. Also, a fictitious treatment unit is used for modeling purposes for the case when no treatment for the streams is required. In the model formulation, the subscript i is used to denote the process sources, j is used for the process sinks, r for the type of interceptors, l for the pollutants, and w for the type of fresh water. Superscripts in, out, m, max, and min are used to denote inlet, outlet, removed mass, upper, and lower limits, respectively. The model formulation is stated as follows. Global Mass Balance for Each Source. The flow rate of each process source FSi is segregated and can be directed to any process source j in the same plant or process sinks located in different plants (fssi,j). It can also be sent to any interceptor r (fsii,r) or discharged to the environment (fsei). J

FSi )

R

∑ fss

∑ fsi

+

i,j

i,r

j)1

+ fsei i ∈ I

(1)

r)1

Mass Balance for Each Process Sink. The inlet flow rate to any process sink j (FUj) is equal to the flow rate from any process source i (fssi,j) from any plant, plus the flow rate from any interceptor r (fisr,j) and the flow rate of fresh water inlet to the process sink j (fwsw,j): I

FUj )

∑ fss

R

i,j

W

∑ fis

+

r,j

i)1

+

r)1

∑ fws

w,j

j∈J

(2)

w)1

Component Balance for Each Sink. Given a maximum value for the pollutant concentration at the inlet of any process sink in any plant (cuj,l), the following constraint must be satisfied: I

cuj,lFUj g

∑

R

∑ ci

csi,lfssi,j +

out r,l fisr,j

i)1

+

r)1

W

∑ cw

w,lfwsw,j

j ∈ J;l ∈ L (3)

w)1

where csi,l is the concentration of pollutant l in process source i, ciout r,l is the concentration of pollutant l at the outlet of treatment unit r, and cww,l is the concentration of pollutant l in fresh source w. Balances for the Interceptors. Some of the mass flow rate of each source i can be segregated and sent to each interceptor: I

FIr )

∑ fsi

i,r

r∈R

(4)

i)1

in ) to any To determine the pollutant concentration inlet (cir,l treatment unit, the following component balance is used:

I

in cir,l FIr )

∑ cs

i,lfsii,r

r ∈ R;l ∈ L

(5)

i)1

The outlet flow rate from each interceptor is sent to the process sinks and/or discharged to the environment: J

FIr )

∑ fis

r,j

j)1

+ fier r ∈ R

(6)

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out To determine the outlet pollutant concentration (cir,l ) in any interceptor, we need to select a specific treatment unit. Each treatment unit is characterized by a conversion factor, RRr,l, and the outlet concentration of each treatment unit is calculated as follows:

out in ) cir,l (1 - RRr,l) r ∈ R;l ∈ L cir,l

(8)

Mass Balance for the Stream Discharged to the Environment. The mass flow rate discharged to the environment (FE) is equal to the flow rate from any process source i and the flow rate from any interceptor r discharged to environment: I

FE )

∑

R

fsei +

i)1

∑ fie

(9)

r

r)1

whereas the pollutant concentration discharged to the environment (cel) is determined through the following component balance: I

celFE )

∑

R

csi,lfsei +

i)1

∑ ci

out r,l fier

l∈L

(10)

r)1

The environmental regulations impose upper limits (cemax l ) on the allowable concentration discharged to the environment: cel e

cemax l

(11)

max 4 fier - Mfie x e 0, r ∈ R r r

(18)

min 4 x g 0, r ∈ R fier - Mfie r r

(19)

min In equations 12-19, Mmax fssi,j and Mfssi,j are upper and lower limits for the flow rate in segments connecting sources to process sinks, max min and Mfsi are upper and lower limits for the flow rate in Mfsi i,r i,r max min and Mfis segments that connect sources to interceptors, Mfis r,j r,j apply for the flow rate in the segments interceptors-process max min and Mfie are used for the flow rate in the sinks, and Mfie r r segments interceptors-waste discharged to the environment. Although one can have any positive flow rate in pipe segments, Lim and Park43,44 have suggested a lower limit of 3 ton/h for the flow rate in the pipes from practical considerations, even though this assumption slightly increases the freshwater consumption and wastewater generation by replacing reused wastewater in the eliminated interconnections with freshwater. min min min min , Mfsi , Mfis , and Mfie are taken here as 3 ton/ Therefore, Mfss i,j i,r r,j r h. Objective Function. The objective function consists of the minimization of the total annual cost, TAC, which includes the fresh water cost, WC, the regeneration cost, RC, and the crossplant pipeline capital cost, PC:

TAC ) WC + RC + PC

max 1 fssi,j - Mfss x e 0, i ∈ I;j ∈ J i,j i,j

(12)

min 1 fssi,j - Mfss x g 0, i ∈ I;j ∈ J i,j i,j

(13)

To activate the binary variable for the pipe segment between 2 ), the following the process source i and the interceptor r (xi,r relationships are used: max 2 fsii,r - Mfsi x e 0, i ∈ I;r ∈ R i,r i,r

fsii,r -

min 2 Mfsi x i,r i,r

g 0, i ∈ I;r ∈ R

(15)

3 is used to model the existence of the The binary variable xr,j pipe segment between any interceptor r and any process sink j:

max 3 fisr,j - Mfis x e 0, r ∈ R;j ∈ J r,j r,j

WC ) HY

(16)

W

∑ ∑ fw

(21)

w,jCUw

j)1 w)1

Here, HY represents the plant operating hours per year, CUw is the fresh water unit cost, and fwsw,j is the flow rate of the fresh water w in the process sink j. The cost of treatment units includes fixed and operating costs for the interceptors: R

RC ) KF

∑

R

CUrFIrR + HY

r)1

L

∑ ∑ CUM ci r

m r,lFIr

(22)

r)1 l)1

where KF is an annualization factor, CUr is the investment cost coefficient, R is a cost function exponent, and CUMr is the unit cost for mass removed in each interceptor. The piping cost includes fixed and operational costs for all pipe segments required in the mass integration network:

{

PC ) KF I

p

(14)

(20)

The fresh water cost is calculated using the following relationship: J

Configurations of Pipes. A set of binary variables is used to determine the existence of pipe segments in the network configuration; when a pipe segment exists, then the associated binary variable must be equal to one, and its cost is included in the objective function. A minimum value for the flow rate is included for each possible piping segment to exist. When the flow rate from the process source i to any process sink j is higher than the minimum flow rate required for a pipe segment, then the piping is required, 1 ) must be one. This and the associated binary variable (xi,j situation is modeled through the following relationships:

(17)

Finally, the pipe segment (xr4) required to discharge the flow rate to the environment from any treatment unit r is modeled as follows:

(7)

Notice that each treatment unit has a specific unit cost that is considered in the objective function. m ) for each interceptor is the The removed concentration (cir,l difference between the inlet and outlet pollutant concentration: m in out cir,l ) cir,l - cir,l r ∈ R;l ∈ L

min 3 fisr,j - Mfis x g 0 r ∈ R;j ∈ J r,j r,j

i)1 R

p

J

1 Di,j fssi,j

I

∑ ∑ 3600FV + x D CU j)1 J

3 Dr,j fisr,j

1 i,j

1 i,j

p

∑ ∑ 3600FV + x D CU r)1

j)1

3 r,j

3 r,j

p

+p

i)1 R

+p

R

2 Di,r fsii,r

∑ ∑ 3600FV + x D CU + r)1 Dr4fier

2 i,r

2 i,r

∑ 3600FV + x D CU r)1

4 r

4 r

p

p

}

(23) 1 Di,j

2 where is the distance between source i and sink j, Di,r 3 between source i and interceptor r, Dr,j between interceptor r and process sink j, and Dr4 between interceptor r and the waste discharged to the environment; F is the density, V is the velocity,


p is a parameter for cross-plant pipeline capital cost, and CUp is the pipe unit cost.

false, for each of which B is equal to zero. This disjunction is modeled using the convex hull reformulation. First the optimization variables are disjointed as follows:

3. Bilinear Terms Reformulation

Q

The model is reformulated as a mixed integer linear programming problem (MILP) by discretizing the pollutant concentrations of the streams in a finite number of intervals (t) via a convex hull reformulation.45 First, an explanation for the discretization of any bilinear term Fx is given. To discretize the variable x, the following equation is used (see Pham et al.46): max(x) - min(x) ψq ) min(x) + (q - 1) t

(24)

where t represents the number of intervals used to split x, and ψ is a constant discretized term used to model the continuous variable x. The bilinear terms in the component balances are substituted by a new variable: B ) Fψq

[

Yq q ) 1...t + 1 B ) Fψq

B)

∑B

dis q

(26)

q)1 Q

F)

∑F

dis q

(27)

q)1

Next, the equation is formulated in terms of the disjointed variables: dis Bdis q ) F q ψq

(28)

Upper limits are established for the disjointed variables as follows: max Bdis q e MB yq

(29)

max Fdis q e MF yq

(30)

(25)

where F is the stream flow rate and ψq is the value of the stream concentration. The optimization problem consists then in finding the optimal discrete value ψq. The optimal selection is modeled using the following disjunction: ∨

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Finally, the following equation is used to select the active region through the selection of one binary variable:

]

This disjunction implies that only one q (discrete choice) must be selected for the bilinear terms. Thus, only one Boolean variable Yq is true, its bilinear term B must be equal to Fψq (notice that ψq is constant), and all other Boolean variables are

Q

∑y

q

)1

(31)

q-1

Discretized Model. Equations 3, 6, 10, and 18 are rewritten in terms of the new variables to obtain an MILP model. First, the bilinear terms are substituted into these equations for a new set of variables as follows:

Figure 3. Linearization of the exponential term FIR for the capital cost of treatment units.

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Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010 I

cuj,lFUj g

∑

R

∑

csi,lfssi,j +

i)1

Q

1 Br,j,l +

1 Br,j,l )

r)1

∑B

1,dis r,j,l,q

r ∈ R;j ∈ J;l ∈ L

(32)

r ∈ R;j ∈ J;l ∈ L

(33)

q)1

W

∑ cw

w,lfwsw,j

j ∈ J;l ∈ L (3′)

w)1

Q

fisr,j )

∑ fis

dis r,j,l,q

q)1

I

∑ cs

2 ) Br,l

i,lfsii,r

r ∈ R;l ∈ L

(6′)

1,dis dis out Br,j,l,q ) fisr,j,l,q cir,l,q r ∈ R;j ∈ J;l ∈ L;q ∈ t + 1

i)1

(34)

I

R

∑ cs

B3l )

i,lfsei

i)1

+

∑B

RC ) KF

∑

l∈L

(10′)

1,dis 1 Br,j,l,q e MBmax r ∈ R;j ∈ J;l ∈ L;q ∈ t + 1 1 y r,j,l r,l,q

(35)

dis max 1 fisr,j,l,q e Mfis y r ∈ R;j ∈ J;l ∈ L;q ∈ t + 1 r,j r,l,q

(36)

r)1

R

R

CUrBr6

4 r,l

+ HY

r)1

L

∑∑

5 CUMrBr,l

(22′)

Q

∑y

r)1 l)1

where the bilinear terms apply as follows: for the component balance at the exit of interceptor r that goes to process sink j; B2r,l for the component balance in the sources going to interceptor r; B3l for the component balance in the waste discharged to the 4 for the component balance at the exit of environment; Br,l 5 for the interceptor r that is discharged to the environment; Br,l 6 mass removed in interceptor r; and Br for the fixed cost of treatment units. 1 , the following disjunction is For the bilinear product Br,j,l included: q ) 1...t + 1

[

1 Yr,l,q 1 out Br,j,l ) cir,l,q fisr,j

]

1,dis 1 dis is the disaggregated bilinear term for Br,j,l , fisr,j,l,q is Here, Br,j,l,q 1 the disaggregated flow rate for fisr,j, yr,l,q is a binary variable dis max used to select the value of ciout r,l,q and fisr,j,l,q, and MBr,j,l1 is an upper 1 limit for the value of the bilinear termBr,j,l. 2 , the disjunction and its Similarly, for the bilinear product Br,l convex hull reformulation are stated as follows:

∨

, r ∈ R;l ∈ L;j ∈ J

q ) 1...t + 1

The convex hull reformulation for this disjunction is

R

z

flow rate range

Az

Cz

error

0.6

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 1 2 3 4 5 6 7

0-5 5-15 15-25 25-75 75-175 175-350 350-525 525-875 875-1000 0-10 10-35 35-55 55-105 105-275 275-380 380-550 550-690 690-835 835-1000 0-20 20-95 95-240 240-420 420-550 550-825 825-1000 0-35 35-80 80-220 220-360 360-620 620-880 880-1000

0.5253 0.2451 0.1821 0.1278 0.0879 0.0651 0.0528 0.0438 0.0389 0.5012 0.2801 0.2240 0.1889 0.1463 0.1233 0.1110 0.1018 0.0956 0.0905 0.5415 0.3605 0.2887 0.2513 0.2324 0.2168 0.2047 0.6930 0.6014 0.5468 0.5109 0.4848 0.4644 0.4539

0.0000 1.4440 2.3613 3.9356 7.0035 11.0610 15.2480 20.1310 24.2580 0.1931 2.3957 4.2434 6.2860 11.2660 17.1870 21.9340 26.9500 31.1760 35.4950 0.5144 4.4896 11.4550 20.3160 28.0340 36.5850 46.5850 0.6797 3.6511 8.5389 16.1920 25.8880 38.3840 47.3520

0.0000 0.0093 0.0012 0.1067 0.2326 0.4639 0.1128 0.9602 0.0396 0.2237 0.0624 0.0064 0.0491 2.0358 0.0399 0.2464 0.0449 0.0421 0.0676 0.7076 1.2183 2.2472 1.2503 0.1058 1.7616 0.0998 0.9932 0.0668 2.1169 0.3939 3.6433 1.1617 0.0218

0.8

0.9

[

2 Yr,l,q 2 in Br,l ) cir,l,q FIr

]

, r ∈ R;l ∈ L

Q

Table 1. Constants Used To Calculate the Investment Cost for Interceptors for Different Values of r

0.7

(37)

q)1

B1r,j,l

∨

) 1 r ∈ R;l ∈ L

1 r,l,q

2 ) Br,l

∑B

2,dis r,l,q ,

r ∈ R;l ∈ L

(38)

r ∈ R;l ∈ L

(39)

q)1

Q

FIr )

∑ FI

dis r,l,q,

q)1

2,dis dis in Br,l,q ) FIr,l,q cir,l,q , r ∈ R;l ∈ L;q ) 1...t + 1

(40)

2,dis 2 Br,l,q e MBmax , r ∈ R;l ∈ L;q ) 1...t + 1 2 y r,l r,l,q

(41)

dis max 2 FIr,l,q e MFI yr,l,q, r ∈ R;l ∈ L;q ) 1...t + 1 r

(42)

Q

∑y

2 r,l,q

)1

(43)

q)1

2,dis 2 dis is the disaggregated bilinear term for Br,l , FIr,l,q is Here, Br,l,q max2 max the disaggregated flow rate for FIr, and MBr,l and MFIr are upper 2 and flow rate FIr, respectively. limits for the bilinear term Br,l 2 Also, yr,l,q is a binary variable to select the optimal values for in dis and FIr,l,q . cir,l,q


B3l :

For the bilinear product ∨

q ) 1...t + 1

[

For the bilinear product Y3l,q

B3l ) cel,qFE

]

∨

q ) 1...t + 1

, l∈L

[

9951

5 : Br,l

5 Yr,l,q 5 m Br,l ) cir,l,q FIr

]

, r ∈ R;l ∈ L

Q

∑

)

B3l

B3,dis l,q ,

l∈L

∑B

5 Br,l )

Q

5,dis r,l,q ,

q)1

∑ FE

dis l,q

l∈L

(45)

dis B3,dis q,l ) FEl,q cel,q l ∈ L;q ∈ 1 + 1

(46)

max 3 B3,dis l,q e MBl3 yl,q l ∈ L;q ∈ 1 + 1

(47)

3 Mmax FE yl,q

e

l ∈ L;q ∈ 1 + 1

3 l,q

)1 l∈L

[

4 Yr,l,q 4 out Br,l ) cir,l,q fier

(49)

]

∑

)

4,dis Br,l,q ,

5,dis 5 Br,l,q e MBmax r ∈ R;l ∈ L;q ∈ t + 1 5 y r,l r,l,q

(59)

dis,2 max 5 FIr,l,q e MFI yr,l,q r ∈ R;l ∈ L;q ∈ t + 1 r

(60)

∑y

r ∈ R;l ∈ L

) 1 r ∈ R;l ∈ L

5,dis 5 dis,2 is the disaggregated bilinear term for Br,l , FIr,l,q is Here, Br,l,q max5 the disaggregated 2 flow rate FIr, MBr,l is the upper limit for the bilinear term B5r,l, and y5r,l,q is a binary variable used to select m dis,2 and FIr,l,q . the optimal values of cir,l,q Finally, the linearization of the exponential term FIR that appears in eq 22 and that is replaced by Br6 in eq 22′ is made by splitting the curve for a fixed value of R into linear segments. For each segment of the curve, a linear regression is made to obtain a linear representation of the original curve. The piecewise linearization procedure is shown in Figure 3. Notice that only one segment of the curve can be selected; therefore, a disjunction and a convex hull reformulation are used to model this situation:

∨

z∈Z

(50)

[

min M FI r,z Br6 )

]

6 Yr,z max e FIr e M FI ,r ∈ R r,z Ar,zFIr + Cr,z Z

q)1

∑B

Br6 )

6,dis r,z ,

r∈R

(62)

r∈R

(63)

z)1

Q

∑ fie

fier )

(61)

q)1

, r ∈ R;l ∈ L

Q

4 Br,l

(58)

(48)

3,dis where Bl,q is the disaggregated bilinear term for the flow rate is the disaggregated flow rate for FE, Mmax B3l , FEdis l,q FE is an upper 3 limit for the mass flow rate to the environment discharge, MBmax l 3 is an upper limit for the bilinear term B3l , and yl,q is a binary dis and cel,q. variable to select the values of FEl,q 4 For the bilinear product Br,l:

∨

(57)

5,dis dis,2 m Br,l,q ) FIr,l,q cir,l,q r ∈ R;l ∈ L;q ∈ t + 1

5 r,l,q

q)1

q ) 1...t + 1

r ∈ R;l ∈ L

Q

Q

∑y

dis,2 r,l,q

q)1

q)1

FEdis l,q

(56)

Q

∑ FI

FIr )

Q

FE )

r ∈ R;l ∈ L

q)1

(44)

dis r,l,q

r ∈ R;l ∈ L

(51)

Z

∑ FI

FIr )

q)1

dis,3 r,z ,

z)1

4,dis dis out Br,l,q ) fier,l,q cir,l,q r ∈ R;l ∈ L;q ∈ t + 1

(52)

4,dis 4 Br,l,q e MBmax r ∈ R;l ∈ L;q ∈ t + 1 4 y r,l r,l,q

(53)

dis fier,l,q

e

max 4 Mfie y r r,l,q

r ∈ R;l ∈ L;q ∈ t + 1

4 r,l,q

) 1 r ∈ R;l ∈ L

(64)

6,dis 6 Br,z e MBmax 6 yr,z, r ∈ R;z ∈ Z r

(65)

dis,3 max 6 FIr,z e MFI dis,3yr,z, r ∈ R;z ∈ Z r,z

(66)

dis,3 min 6 FIr,z g MFI dis,3yr,z, r ∈ R;z ∈ Z r,z

(67)

(54)

Q

∑y

6,dis dis,3 6 Br,z ) FIr,z Ar,z + Cr,zyr,z , r ∈ R;z ∈ Z

(55)

q)1

In the above equations, B4,dis r,q,l is the disaggregated bilinear term 4 dis 4 is an , fier,l,q is the disaggregated flow rate for fier, MBmax for Br,l r,l max 4 upper limit for the bilinear term Br,l , Mfie is an upper limit for r the flow rate fier, and y4r,q,l is a binary variable to select the values dis and ceq,l. of fier,q

Z

∑y

6 r,z

) 1, r ∈ R

(68)

z)1

In the above equations, the subscript z is used to denote the 6,dis number of linear segments of the original curve, Br,z is the 6 dis,3 disaggregated bilinear term for Br , FIr,z is the disaggregated 6 is an upper limit for the bilinear term B6, flow rate FIr, MBmax r r maxdis,3 min dis,3 dis,3 , and MFIr,z and MFIr,z are upper and lower limits for FIr,z

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Table 2. Cost of Wastewater Stripping for Different RR for the Examples Examples 1 and 2 RR interceptor

pollutant 1

1 2 3 4 5 6 7 8 9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

CU (US$)

CUM (US$-kg removed)

500 7500 9200 10 500 11 300 12 000 12 400 12 600 15 000

0.540 0.695 0.850 1.005 1.160 1.460 1.760 2.060 4.300

1 4 yr,l,q ) yr,l,q r ∈ R;l ∈ L;q ∈ t + 1

Example 3 1 2 3

pollutant 1 0.95 0 0.8

pollutant 2 0 0.95 0.9

16 800 12 600 24 000

the disaggregated variable in the convex hull formulation for each bilinear product is different. We can also see that the 1 4 pollutant concentrations in the bilinear terms Br,j,l and Br,l are the same; in this case, an additional restriction should be included to ensure that the active region in both disjunctions is the same.

1 0.0067 0.033

6 yr,z is a binary variable used to select the optimal value of the slope, Ar,z, and the intersection, Cr,z. Table 1 shows values for Az and Cz for different values of R. The increment used was min dis,3 maxdis,3 and MFI were 0.005 over the range 0-1000. Both MFI r,z r,z fixed as given in the third column of Table 1. Notice that in this formulation the concentrations of the pollutants in the different streams are no longer variables but parameters calculated with eq 24. Also notice that the variable 2 5 , Br,l , and Br6; however, FIr appears in the bilinear products Br,l

(69)

Remarks. (1) The original model is a nonconvex MINLP given by eqs 1-23. Therefore, the solution of this MINLP problem does not guarantee a global optimal solution. (2) The discretized MILP model is convex, given by eqs 1, 2, 3′, 4, 5, 6′, 7-9, 10′, 11-21, 22′, 23-69. The solution to such MILP reformulation, therefore, can lead to a global optimal solution. (3) The proposed model includes the case when there are treatment units exclusive to single plants. This situation is modeled with restrictions for the flow rates for other plants (i.e., upper limits equal to zero) from the sources to the corresponding interceptors (fsii,r) and for the flow rates from the interceptors to the corresponding process sinks (fisr,j). (4) The objective function can be set as the minimization of fresh water consumption, regeneration cost, or cross-plant pipeline cost, if such policies are of interest. (5) Some limitations for the proposed model are the following: (a) pressure drops in pipe segments are not included; (b) the recirculation on the regeneration zone is not considered; (c) only parallel configurations for the interconnection of interceptors to treat different pollutants are assumed as a convenient

Table 3. Data for the Examples sinks plant

sink

flow rate (ton/h)

sources

pollutant concentration (ppm)

source

flow rate (ton/h)

pollutant concentration (ppm)

Example 1 1 2 3

1 2 3 4 5 6 7 8 9

80 120 100 130 90 150 110 160 80

70 80 100 90 110 120 115 85 125

20.00 66.67 100.00 41.67 10.00 20.00 66.67 15.63 42.86 6.67 20.00 80.00 50.00 40.00 300.00

0 50 50 80 400 0 50 80 100 400 0 25 25 50 100

1 2 3 4 5 6 7 8 9

120 80 100 130 60 110 95 115 140

80 110 90 125 95 140 120 115 100

Example 2 1

2

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

20.00 66.67 100.00 41.67 10.00 20.00 66.67 15.63 72.86 6.67 20.00 80.00 50.00 40.00 300.00

100 80 100 800 800 100 80 400 800 1000 100 50 125 800 150

Example 3 1 2

1 2

1 2 3 4 5 6

100 130 25 300 200 125

1 2 3 4 5 6

135 156 195 150 210 142

pollutant 1 3 1 3 2 8 2

pollutant 2 20 80 0 15 70 30 Example 4 5 12 9 100 50 130

1 2 3 4 5 6

100 130 25 300 200 125

1 2 3 4 5 6

100 150 130 80 75 95

pollutant 1 35 30 3 30 35 5

pollutant 2 600 500 400 0 400 550 10 15 18 800 1,200 950


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Table 4. Results for Examples interplant integration MILP

each plant integrated independently

MINLP

MILP

without integration

Example 1 total annualized cost (US$/year) fresh water cost (US$/year) regeneration cost (US$/year) cross-plant pipeline capital cost (US$/year) total fresh water (ton/h) total wastewater (ton/h) CPU time (s)

106,637.635 95,088.889 0 11,548.746 108.056 38.056 1.196

106,637.635 95,088.889 0 11,548.746 108.056 38.056 0.66

total annualized cost (US$/year) fresh water cost (US$/year) regeneration cost (US$/year) cross-plant pipeline capital cost (US$/year) total fresh water (ton/h) total wastewater (ton/h) CPU time (s)

1,628,805.385 768,805.160 779,120.556 80,879.668 739.236 739.236 16 380.12

1,647,108.704 702,511.030 857,941.164 86,656.510 675.491 675.491 7.71


1,123,398.024 244,933.333 771,520.032 106,944.658 231.667 231.667 5472.13

1,141,694.259 261,333.333 761.835.000 118,525.415 246.667 246.667 8.13


3,016,961.157 860,702.039 2,052,27.330 103,984.788 882.608 524.608 490.400

3,085,082.255 924,914.904 2,033,085.017 127,082.334 899.726 541.726 1.11

110,244.443 110,244.443 0 0 125.277 55.277 5.33

1,317,980.493 897,600.00 304,795.161 115,585.332 1020.000 950.000

1,849,175.861 751,018.385 988,403.992 109,753.483 722.133 722.133 1128

5,175,915.191 915,376.800 4,168,101.145 92,436.745 880.170 880.170

1,248,522.456 274,343.027 852,799.346 121′380.083 250.183 250.183 3420.12

2,020,996.890 919,200.000 980,392.840 121,404.049 880.000 880.000

4,348,222.990 478,784.635 3,742,356.023 127,082.334 460.370 102.370 2304.6

8,783,112.736 922,240.000 7,791,534.101 69,338.634 988.00 630.00

Example 2

Example 3

Example 4

simplification for the model solution; and (d) the treatment units are modeled with constant conversion factors and do not provide the detailed design of such units. (6) The model formulation can accommodate any type of effluents, including water, organic and inorganic solvents, or any other type of chemical stream.

(7) Usually, a set of treatment units is available in industrial practice to remove pollutants so as to meet process and environmental constraints. Each one of these treatment units has a specific capability to remove the pollutants at a given cost. The model is set to provide the selection of the best treatment unit (or a combination of such units) to satisfy the

Figure 4. Optimal configuration for example 1 using the MILP reformulation for interplant integration.

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Figure 5. Optimal configuration for example 1 using the MINLP problem for interplant integration.

process and environmental constraints at the minimum total annual cost. The reason to eliminate the interconnections between such treatment units is to avoid another set of bilinear terms (and the corresponding set of binary variables) to keep the solution effort within reasonable CPU times. 4. Results To show the application of the proposed model, four case studies are presented. Each case was solved as an MINLP problem with the solver DICOPT, and the relaxed reformulated model as an MILP problem with the CPLEX solver included in the GAMS software.47 The values used for the parameters KF, HY, CUp, V, and F are 0.231/year, 8000 h/year, 250 US$, 1 m/s, and 1000 kg/m3, respectively. The value for the exponent R for the cost of interceptors in all examples was 0.7. In addition, the park integration achieved by integrating each plant separately was carried out to evaluate some of the incentives for an interplant integration strategy. Example 1. This example consists of three plants, each with three process sources and three process sinks, as given in Table 3. For this example, fresh water without pollutant and water with 10 ppm of pollutant are available as fresh sources. The unit cost for the clean and nonclean fresh water is 0.13 US$/ ton and 0.10 US$/ton, respectively. In addition, the maximum allowed concentration for the wastewater discharged to the 1 2 3 ,Di,r ,Dr,j ,Dr4) for all environment is 100 ppm, and the length (Di,j pipe segments is assumed as 200 m. Eight interceptors were considered; the conversion factors (RRr), investment cost

coefficients (CUr), and unit costs for mass removed (CUMr) are given in Table 2.48 For the discretized MILP model, 51 intervals were used for q, which yields a problem with 17 703 continuous variables, 1725 binary variables, and 17 703 constraints. The original MINLP problem consists of 581 continuous variables, 252 binary variables, and 590 constraints. Table 4 shows the results obtained for this example using the proposed model, with the solvers DICOPT for the MINLP formulation and CPLEX for the proposed relaxed reformulation. Figure 4 shows the solution obtained using the relaxed MILP formulation for the interplant integration, while Figure 5 shows the solution obtained for the MINLP problem. It can be noticed in Table 4 that the optimal solution is the same for both MILP and MINLP problems, even when both configurations are different. This situation shows that multiple optimal solutions can arise for these types of problems. In these cases, the selection of the optimal configuration is to be made from practical aspects such as available spaces and operational feasibility. Table 4 also shows the solution for the case when individual integration for each plant is considered (i.e., without interplant integration). One can notice that all configurations use nonclean fresh water as part of the optimal structure. A comparison of results shows that the optimal configuration for the solution without interplant integration is about 3.3% more expensive than the configuration with interplant integration; also, the solution with interplant integration presents savings by 15.7% in fresh water consumption with respect to the solution without interplant integration. Another


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Figure 6. Optimal configuration for example 2 using the MILP formulation for interplant integration.

important aspect to note in the solutions with interplant integration is the existence of interplant flow rates; process source 7 from plant 3 goes to process sink 5 from plant 2, and process source 7 from plant 3 goes to process sink 4 from plant 2 (see Figures 4 and 5). This result shows the convenience to consider the possibility of interplant exchange flow rates in the modeling of eco-industrial parks. Furthermore, the solution for the interplant integration presents a reduction in the total wastewater discharged to the environment by 30.9% with respect to the solution without interplant integration, thus reducing significantly the environmental impact. Therefore, the possibility to interchange mass between different plants provides incentives from both economic and environmental aspects. As can be seen in Figures 4 and 5, it was not necessary to include any treatment unit to satisfy the maximum pollutant concentration to be discharged to the environment. This is because the conditions in the sources and sinks in terms of the pollutant concentration are not so different among each other and have relatively small values with respect to maximum allowed pollutant concentrations. This situation aids in the solution of the problem and allows the MINLP model to obtain

the same optimal configuration as the one obtained with the discretized global optimization technique. To view the incentive for integration strategies, the solution without integration is presented in the last column of Table 4. One can notice that the solution without integration presents increases of 844%, 100%, and 900%, in the fresh water cost, treatment cost, and piping cost, respectively, thus providing an overall increase in the total annual cost of 1136%. Example 2. This example, taken from Chew et al.,39 consists of three plants with five process sources and five process sinks each one, with the data given in Table 3. The cost of the fresh water is 0.13 US$/ton, and the maximum concentration permissible for the pollutant discharged to the environment is 100 ppm; 1 2 3 ,Di,r ,Dr,j ,Dr4) for all pipe segments is taken as the length (Di,j 100 m. Eight interceptors were considered with the conversion factors (RRr), investment cost coefficients (CUr), and unit costs for mass removed (CUMr) given in Table 2.48 In this case, 41 intervals for q were used for the discretized MILP model. The discretized problem consists of 15 939 continuous variables, 1707 binary variables, and 21 735 con-

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straints, whereas the MINLP problem contains 1088 continuous variables, 504 binary variables, and 1112 constraints. Figure 6 shows the configuration obtained using the MILP model for the solution with interplant integration, while Table 4 gives the main economic results for the MILP and MINLP solutions. Notice in Table 4 that the configuration obtained with the linear reformulation is 1.11% cheaper than the one obtained with the original MINLP problem; this result seems to indicate that the relaxed technique may be particularly useful for big and difficult problems, in which there may be process sources with significant differences in pollutant concentrations, including some with high values. The savings obtained for the MILP with respect to the MINLP solution arise because of its lower treatment and piping costs, even when higher fresh water costs are observed. In addition, the configurations of both solutions have different distributions and flow rates. Additionally, Table 4 shows the results for the solution for the case in which interplant integration is not allowed. The configuration obtained for the case of interplant integration is 11% cheaper than the configuration without interplant integration, as a consequence of the lower regeneration and cross-plant pipeline costs. Notice in Figure 6 that the interplant integration solution uses two interceptors, while the structure without interplant integration uses five interceptors to meet the pollutant concentrations required because of the process sinks limits and the environmental constraints. Notice in the configuration of Figure 6 that several interceptors with different removal capabilities are selected to meet the problem constraints; to increase the purity of the wastewater, it is possible to use parallel and series configurations. In addition, there are economic incentives for the simultaneous selection of treatment units with respect to the case when such units are fixed. Finally, the last column of Table 4 shows the results for the case prior to integration; the results show that the solution without integration shows increases of 19%, 434%, 14%, and 217% for the fresh sources, treatment, piping, and total annual costs. Example 3. This example consists of six process sources and six process sinks distributed in two plants. Two pollutants are contained in the process sources, and the values of flow rate and pollutants concentration in the process sources and sinks are shown in Table 3. Two types of fresh sources are available: one is clean with a unit cost of 0.15 US$/ton, and the other one is free of pollutant 1 with 10 ppm of pollutant 2, and its unit cost is 0.13 US$/ton. The limits for the pollutants 1 and 2 in the waste that is discharged to the environment are 10 and 100 ppm, respectively. In addition, the lengths of the pipe segments 1 2 3 , Di,r , and Dr,j , while the considered are given in Table 5 for Di,j value for D4r is taken as 200 m. Three types of interceptors were considered, with the values for conversion factors, investment cost coefficients, and unit costs for mass removed given in Table 2.48 The interceptors are distributed as follows: a set formed by interceptors of type 1, 2, and 3 can be shared for both plants, a set formed by interceptors type 1 and 2 can only be used in plant 1, and a set formed by interceptors type 1 and 2 can only be used in plant 2. This gives a total of 7 interceptors, with labels 1-3 for commons interceptors, 4-5 for interceptors for plant 1, and 6-7 for plant 2. A total of 21 intervals were used for the discretized MILP model, which gave rise to 1184 binary variables, 7686 continuous variables, and 9978 restrictions; the MINLP included 140 binary variables, 368 continuous variables, and 382 restrictions. The results obtained for this example are shown in Table 4.

Table 5. Matrix of Distribution For Example 3 (m) 1 Di,j

j i

1

2

3

4

5

6

1 2 3 4 5 6

50 50 50 200 200 200

50 50 50 250 250 250

50 50 50 250 250 250

250 250 250 50 50 50

250 250 250 50 50 50

250 250 250 50 50 50

3 Dr,j

j r

1

2

3

4

5

6

1 2 3 4 5 6 7

200 200 200 50 50

200 200 200 50 50

200 200 200 50 50

200 200 200

200 200 200

200 200 200

50 50

50 50

50 50

2 Di,r

r i

1

2

3

4

5

1 2 3 4 5 6

200 200 200 200 200 200

200 200 200 200 200 200

200 200 200 200 200 200

100 100 100

100 100 100

6

7

100 100 100

100 100 100

Notice that the total annual cost for the solution of the MILP problem for the interplant integration (Figure 7) is 1.6% cheaper than the solution of the MINLP problem for the interplant integration, and 10% cheaper with respect to the solution under a policy with individual plant integration. In addition, the MILP solution uses 6.1% and 7.6% less fresh water with respect to the MINLP or no interplant integration solutions. One can notice that the optimal structure includes interceptor 6, which is exclusive for plant 2, and interceptors 2 and 3, which are part of an interplant regeneration zone. This example shows the applicability of the proposed model for the case when there are treatment units that are exclusive to specific plants in the mass integration of eco-industrial parks. As a last comparison, one can notice that the total annual cost for the optimal interplant integration is 44% lower than the solution in which no integration is performed (see last column of Table 4). Example 4. This example is included to show the case when different plants discharge streams with very different concentration levels. In this case, a problem with two plants, each with three process sources and three sinks, is considered. The data for this case are shown in Table 3, and the data for regenerators costs and distances are the same as in example 1. This MINLP problem consists of 153 binary variables, 374 continuous variables, and 383 constraints, whereas the respective numbers for the MILP reformulated problem are 6021, 4722, and 6021. The optimal solution for this problem is shown in Figure 8, and a summary of results, as well as comparisons with respect to the solutions of the MILP, MINLP, and the no interplant integration problems, are shown in Table 4. It can be seen that the configuration for the interplant integration with the MILP model is 2.21% and 30.62% cheaper than the MINLP solution and the single plant integration, respectively. These savings are obtained because of lower costs in fresh water and pipes with respect to the MINLP solution, and because of a reduction in


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Figure 7. Optimal configuration for example 3 using the MILP formulation for interplant integration.

piping and regeneration costs with respect to the single plant integration solution. In addition, the optimal configuration (Figure 8) shows that interplant flow rates are not needed, because the streams of plant 2 are too contaminated to be recycled to the other plant (which has stricter constraints for process sinks), and the streams from plant 1 directly satisfy the environmental constraints. However, in this case, the savings are obtained due to the integration of the waste stream discharged to the environment; the effluents from plant 1 help to dilute the effluents from plant 2, thus reducing the treatment costs for the effluents of plant 2 to satisfy the environmental regulations, which is then translated into significant savings in the total annual cost.

We finally notice that the optimal solution has a total annual cost 66% lower than the structure obtained without an integration policy (see Table 4). 5. Conclusions A global optimization technique for the optimal design of eco-industrial parks has been presented. The formulation is based on a new superstructure representation that allows the interplant mass integration for a given set of process sources and process sinks with fixed flow rates and pollutant concentrations. The formulation considers the simultaneous selection of the intercep-

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Figure 8. Optimal configuration for example 4 for interplant integration.

tors needed to satisfy the process and environmental constraints. A discretization approach is used to handle the bilinear terms that appear for the component balances in several parts of the model. Such a discretization approach may be particularly useful for difficult problems, for example, large problems or cases with tight constraints for the pollutant concentration in the process sinks and the environment discharge. Although the proposed global optimization formulation is more computing intensive, it can find the global (or near global) optimal solution for cases in which the MINLP model is likely to get trapped into a local optimal solution.

The results show some of the benefits for interplant integration as opposed to carrying out the integration of each plant separately, such as lower costs and lower amounts of waste streams, and that eco-industrial parks could be an important alternative to provide structures with integrated processes that are both economically and environmentally efficient. Nomenclature B ) variable that substitutes the bilinear terms Bl3 ) bilinear term for the component balance in the environment discharge, ppm*ton/h

Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010 Bqdis ) disaggregated bilinear term for B 3,dis Bl,q ) disaggregated bilinear term for Bl3, ppm*ton/h Br6 ) bilinear term for the fix cost of the treatment units, US$/year 2 Br,l ) bilinear term for the component balance in the sources and directed to interceptor r, ppm*ton/h 4 ) bilinear term for the component balance outlet to the Br,l interceptor r and directed to environment discharge, ppm*ton/h 5 Br,l ) bilinear term for the mass removed in the interceptor r, kg/s 1 Br,j,l ) bilinear term for the component balance outlet to the interceptor r and directed to process sink j, ppm*ton/h 2,dis 2 ) disaggregated bilinear term for Br,l , ppm*ton/h Br,l,q 4,dis 4 Br,l,q ) disaggregated bilinear term for Br,l , ppm*ton/h 5,dis 5 Br,l,q ) disaggregated bilinear term for Br,l , kg/s 1,dis 1 Br,j,l,q ) disaggregated bilinear term for Br,j,l , ppm*ton/h 6,dis ) disaggregated bilinear term for Br6, US$/year Br,z cel ) concentration of the pollutant l in the environment, ppm cel,q ) disaggregated pollutant concentration cel, ppm in cir,l ) inlet concentration of the pollutant l in the interceptor r, ppm m cir,l ) removed concentration of the pollutant l in the interceptor r, ppm out ) outlet concentration of the pollutant l in the interceptor r, cir,l ppm in in cir,l,q ) disaggregated pollutant concentration cir,l , ppm m m cir,l,q ) disaggregated pollutant concentration cir,l, ppm out out ) disaggregated pollutant concentration cir,l , ppm cir,l,q csi,l ) concentration of the pollutant l in the source i, ppm cuj,l ) concentration of the pollutant l in the sink j, ppm CUMr ) mass removed unit cost in the interceptor r, US$/kg CUp ) unit cost of the pipe, US$ CUr ) unit cost for interceptor r, US$ CUw ) unit cost for fresh water w, US$/ton cww,l ) concentration of the pollutant l in the fresh water w, ppm D ) length of pipe segments, m F ) flow rate of stream, ton/h FE ) flow rate in the environment discharge, ton/h dis FEl,q ) disaggregated flow rate FE, ton/h Fi ) flow rate of stream i, ton/h fier ) flow rate of the interceptor r in the environment, ton/h dis fier,l,q ) disaggregated flow rate fier, ton/h FIr ) flow rate in the interceptor r, ton/h dis FIr,q,l ) disaggregated 1 flow rate FIr, ton/h dis,2 FIr,q,l ) disaggregated 2 flow rate FIr, ton/h dis,3 FIr,z ) disaggregated 3 flow rate FIr, ton/h fisr,j ) flow rate of the interceptor r in the sink j, ton/h dis fisr,j,l,q ) disaggregated flow rate fisr,j, ton/h Fqdis ) disaggregated flow rate F, ton/h FSi ) flow rate of the source i, ton/h FUj ) flow rate in the sink j, ton/h fsei ) flow rate of the source i in the environment, ton/h fsii,r ) flow rate of the source i in the interceptor r, ton/h fssi,j ) flow rate of the source i in the sink j, ton/h fwsw,j ) flow rate of fresh water w in the sink j, ton/h HY ) annual working hours, h/year I ) {i ) 1,2, ..., Nsources| i is a set of process sources} J ) {j ) 1,2, ..., Nsinks| j is a set of process sinks} KF ) annualized factor, year-1 Mmax ) upper limit for B B Mmax ) upper limit for F F 1 ) upper limit for B1 MBmax r,j,l r,j,l 2 ) upper limit for B2 MBmax r,l r,l 3 ) upper limit B3 MBmax l l max4 4 MBr,l ) upper limit for Br,l max5 5 MBr,l ) upper limit for Br,l

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6 ) upper limit for B6 MBmax r r max MFIr ) upper limit for FIr Mmax FE ) upper limit for FE max Mfss ) upper limit for fssi,j i,j min Mfss ) lower limit for fssi,j i,j max Mfsi ) upper limit for fsii,r i,r min Mfsi ) lower limit for fsii,r i,r max Mfis ) upper limit for fisr,j r,j min Mfis ) lower limit for fisr,j r,j max Mfie ) upper limit for fier r min Mfie ) lower limit for fier r maxdis,3 dis,3 MFI ) upper limit for FIr,z r,z min dis,3 dis,3 MFIr,z ) lower limit for FIr,z

p ) parameter for capital cost for cross-plant piping PC ) cross plant pipeline capital cost, US$/year Q ) {q ) 1,2, ..., Nintervals for discretization| q is a set of interval for disaggregate variables} R ) {r ) 1,2, ..., Ninterceptor| r is a set of interceptors} RC ) regeneration cost, US$/year RRr ) conversion factor of the interceptor r, dimensionless t ) number of intervals used to split x TAC ) total annual cost, US$/year V ) velocity, m/s W ) {w ) 1,2, ..., Nwater type| w is a set of fresh water type} WC ) fresh water cost, US$/year x ) continuous variable Y ) Boolean variable used to determine the value of B 1 1 Yr,l,q ) Boolean variable used to determine the value of Br,j,l 2 2 Yr,l,q ) Boolean variable used to determine the value of Br,l 3 Yl,q ) Boolean variable used to determine the value of Bl3 4 4 Yr,l,q ) Boolean variable used to determine the value of Br,l 5 5 Yr,l,q ) Boolean variable used to determine the value of Br,l 6 6 Yr,z ) Boolean variable used to determine the value of Br Greek Symbols F ) density, kg/m3 ψq ) discrete value of variable x in the interval q Binary Variables 1 xi,j ) used to determine the existence of pipe from source i to process sink j 2 xi,r ) used to determine the existence of pipe from source i to interceptor r 3 xr,j ) used to determine the existence of pipe from interceptor r to process sink j xr4 ) used to determine the existence of pipe from interceptor r to environment discharge yq ) used to select the optimal value of ψq and Fqdis 1 out dis yr,l,q ) used to select the value of cir,l,q and fisr,j,l,q 2 in dis yr,l,q ) used to select the value of cir,l,q and FIr,l,q 3 dis yl,q ) used to select the value of FEl,q and cel,q 4 dis out yr,q,l ) used to select the value of fier,q and cir,l,q 5 m dis,2 yr,l,q ) used to select the value of cir,l,q and FIr,l,q 6 dis,3 yr,z ) used to select the value of Ar,z, Cr,z, and FIr,z

Subscripts i ) source j ) sink l ) pollutant q ) intervals for disaggregation r ) interceptor w ) type of fresh water

9960


Superscripts in ) inlet m ) mass removed max ) upper limit out ) outlet dis ) disaggregated

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ReceiVed for reView March 30, 2010 ReVised manuscript receiVed August 22, 2010 Accepted August 30, 2010 IE100762U

Water Integration of Eco-Industrial Parks Using a Global Optimization

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