An MINLP Model for the Optimal Location of a New Industrial Plant

Ind. Eng. Chem. Res. 2011, 50, 953–964

953

An MINLP Model for the Optimal Location of a New Industrial Plant with Simultaneous Consideration of Economic and Environmental Criteria Luis Fernando Lira-Barragań,† Jose´ Marıá Ponce-Ortega,*,† Medardo Serna-Gonza´lez,† and Mahmoud M. El-Halwagi‡ Chemical Engineering Department, UniVersidad Michoacana de San Nicola´s de Hidalgo, Morelia, Michoacań, Me´xico, 58060, Chemical Engineering Department, Texas A&M UniVersity, College Station, Texas 77843, United States

This paper presents a new mixed-integer nonlinear programming model for the optimal allocation of a new industrial plant which impacts the water quality throughout a surrounding watershed. In addition to the economic aspects, the optimization approach also accounts for the environmental impact, such that the wastewater streams discharged to the environment have characteristics that ensure the sustainability of the surrounding watershed. The model was formulated to predict the behavior for the watershed impacted for the new polluted discharges through the material flow analysis technique. Therefore, all discharges and extractions are considered as well as the chemical reactions that take place in the watershed. The watershed model is combined with a disjunctive model for the optimal location of a new industrial plant and the selection of the type of treatment. The objective function is aimed at minimizing the total annual cost, which includes the wastewater treatment costs and the location-based cost of the new plant (including the transportation of raw materials, products, and services, as well as the land cost). The constraints on water quality are imposed at various locations throughout the watershed. A sensitivity analysis is carried out to generate the noninferior curve that shows the trade-off between cost and environmental impact. Furthermore, the model can be used to identify incentives needed for nonoptimal locations to become attractive enough to install the new plant. Two example problems were used to show the applicability of the proposed methodology and the effectiveness of solving the optimization formulation. 1. Introduction Nowadays, for the location of a new industrial plant, usually only economic aspects are considered, without taking into account the environmental damage that industrial effluents produce over the final wastewater disposal.1 Once the new plants are installed, the problems associated with the pollution that the new plants produce are identified, and then it is required to install new equipment to treat the industrial wastes to repair the damage and to satisfy the environmental regulations, which represents an additional cost that may yield uneconomical processes. Several methodologies have been reported to address the problem of recycle/reuse for resource conservation inside industrial facilities (see, for example, Dunn and El-Halwagi,2 El-Halwagi,3,4 and El-Halwagi and Spriggs5). These methodologies allow for yielding an efficient use of resources and reducing the waste streams discharged to the environment. These resource conservation methodologies can be classified as pinch techniques and techniques formulated as mathematical programming models.6 The pinch technologies use heuristic rules based on targeting followed by design conducting to sequential approaches. Foo7 presented a review for the pinch techniques for water network synthesis (see, for example, Wang and Smith,8 Dhole et al.,9 El-Halwagi and Spriggs,5 Polley and Polley,10 Hallale,11 Manan et al.,12 El-Halwagi et al.,13 and Feng et al.14). Other sets of methodologies based on algebraic techniques for resource conservation have been reported by Sorin and Bedard,15 Agrawal and Shenoy,16 and Gomes et al.17 Finally, mathematical * To whom correspondence should be addressed. E-mail: jmponce@ umich.mx. † Universidad Michoacana de San Nicola´s de Hidalgo. ‡ Texas A&M University.

programming formulations have been reported to automate the synthesis of recycle and reuse networks inside the plant (see, for example, Savelski and Bagajewicz,18-20 Alva-Argaez et al.,21,22 Benko et al.,23 Teles et al.,24 Gabriel and El-Halwagi,25 Kuo and Smith,26 Doyle and Smith,27 Galan and Grossmann,28 Hernandez-Suarez et al.,29 Gunaratnam et al.,30 Karuppiah and Grossmann,31 Putra and Amminudin,32 Ponce-Ortega et al.,33,34 and Na´poles-Rivera et al.35). However, all previous formulations consider exclusively the processes that happen inside the industrial facilities, ignoring the processes that occur outside the plant. It is worth notice that the wastewaters discharged by the industrial plants are transported by the drainage systems to their final disposal, and these final disposals commonly are lakes or seas that are drastically impacted by the polluted discharges from the industrial plants. On the other hand, watershed systems are the means by which the effluents are transported to their final disposal. In the watersheds, mixing and splitting processes naturally exist, which naturally change the properties of the streams, in addition to the natural degradation of the components due to chemical, biochemical, and physical transformations (Brunner and Rechberg36). Watersheds involve a number of tributaries that feed into reaches, streams, or rivers that finally lead to catchment areas like lakes, seas, or oceans; in addition, watersheds impact and are impacted drastically by their surroundings. Baccini and Brunner37 developed a material flow analysis (MFA) model for analyzing ecosystems with human activities where mass, energy, and information are being exchanged with the surroundings. Lampert and Brunner38 proposed an MFA model to track nutrient discharges into the Danube river basin. El-Baz et al.1 presented an MFA model to track the nitrogenous compounds in the Bahr El-Baqar drainage system, whereas El-Baz et al.39 presented an MFA model to include a mass integration for the

10.1021/ie101897z  2011 American Chemical Society Published on Web 12/20/2010

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Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

Bahr El-Baqar drainage system in Egypt. Lovelady et al.40 introduced an approach to integrate the discharges of an industrial process with macroscopic watershed systems based on an MFA model that they called the reverse problem formulation. In the reverse problem formulations,40 the maximum concentrations for the pollutants discharged to the environment are fixed to avoid hazardous accumulation in the final disposal to ensure sustainable processes through the MFA technique; however, this is a sequential approach that does not simultaneously optimize the model. In addition, it is worth noting that the wastewater treatment costs for the new plant are strongly dependent on the new location. Therefore, this paper presents a new model based on a mathematical programming formulation for the optimal location of a new industrial plant considering minimization of the total cost and treatment of the wastewater streams to satisfy environmental regulations as well as natural degradation of the contaminants to avoid pollution problems in the final disposal due to the accumulation of contaminants, and therefore to ensure sustainable systems. In addition, the model considers different and independent constraints for the water quality in the watershed regions (or sections) that are impacted by the new discharge; this is because water could be taken from some of these regions for human, agricultural, or other uses, or because there is a city or a place that must be protected to avoid hazardous situations. Therefore, the requirements for the water quality in these different locations of the watershed depend on the specific use. Finally, the proposed model can be used as a tool to conduct a sensitivity analysis for the case when the requirements for the water quality change. Also, the set of optimal solutions for different scenarios could be identified through a Pareto curve. Thus, the model could be used as a tool for the decision makers when a new industrial plant needs to be located. The paper is organized as follows: section 2 presents the definition of the problem addressed in this paper; section 3 presents the model formulation; section 4 presents the results and discussions of the application of the proposed model, and finally section 5 presents the conclusions of the paper. 2. Problem Statement The problem addressed in this paper can be stated as follows. Given are a set of alternative locations to install a new industrial plant P ) {p|p ) 1, 2, ..., Np}. Each allowed plant location p has associated an installation cost, CLANDp, which includes the cost associated with the transportation of the raw materials and final products, as well as the land cost. In addition, multiple pollutants can be considered (C ) {c|c ) 1, 2, ..., Nc}). The problem then consists of determining the optimal location ) for the new plant that satisfies environmental constraints (yEnvReg c for the industrial effluents and, at the same time, that does not exceed the accumulation limit of pollutants in the final disposal ), as well as satisfying independent constraints for (ysustainability c specific regions n through the watershed that depend on the type desired ). of water use (yc,n(r) The discharge at the exit of the new plant is fixed by the process conditions (total flow rate discharged and its composition), but the quality for the discharge over the river near the new plant must be determined because this must be treated to satisfy environmental regulations and to avoid accumulation of the pollutants at the final disposal. This treatment depends on the location of the new plant. To satisfy this objective, the behavior of the river is simulated considering all inlet and outlet streams through the MFA technique.

The objective function consists of the minimization of the total annual cost (TAC) that is constituted by two components: the installation costs for the new plant and the treatment cost for the wastewater stream leaving the plant. The installation cost includes the cost for land and production, as well as the transportation costs for the raw materials and the products. The second component accounts for the treatment cost of wastewater streams using the available treatment technologies; this cost is divided into two parts: one that is independent of the treated flow rate and one that depends on the treated flow rate in each treatment technology. The part of the cost for the treatment units that is independent of the treated flow rate is a very important factor in avoiding infeasible networks with a very large number of treatment units. Figure 1 shows a general watershed system, where the main stream of the river is fed by several streams or tributaries (FTr,t). The main stream of the river exchanges water during its trajectory. According to its use, the following types of water exchanges can be identified: water for agricultural use, wastewater discharged to the river with and without treatment, industrial and residential effluents, etc. In addition, the model takes into account phenomena like natural precipitation, filtration, and vaporization. These processes significantly modify the composition of the materials transported in the river; therefore, to adequately track the mean composition for the hazardous compounds, it is required that the river be sectioned in parts where the overall composition can be considered constant (these sections are represented in Figure 1 with ovals). These sections are called reaches, which identify sections where no bigger effluents are discharged and/or extracted. The flow rate and concentrations in each reach are different from those in the other reaches due to the inlet and extraction streams. The tributaries are channels or branches of the river, which may contain discharges with or without treatment, industrial discharges, etc., and their flow rates (FTr,t) and concentrations (CTc,r,t) affect the reaches where they are discharged. On the other hand, a very interesting term in the model is the reactive term. This term takes into account the possible chemical and biochemical reactions that can be carried out by the flora and fauna inside the rivers, which can decompose or produce hazardous materials. One of the objectives is to keep final disposal of the watershed clean, taking into account the current and new discharges. The wastewater discharges for the new plant must yield a concentration lower than the one that ensures a sustainable process for the final disposal. In addition, the model must satisfy constraints for the water quality through the entire watershed, which are independent of each other but depend on the specific location of water use (see Figure 1). It is common to extract water from a river to feed a city or agricultural areas; therefore, the water quality through the river is very important. Another important aspect of the model formulation is that it is able to consider the trade-offs between the pollutant concentration discharged in the final disposal and the total cost associated, simultaneously optimizing the location and the treatment system. Therefore, when there are stricter environmental regulations, the associated cost to treat the waste streams and the location cost are bigger. On the other hand, for relaxed environmental regulations, the associated total cost is lower. As a result, the final location for the new plant depends on the decision makers (investors), government, and society, and the methodology presented in this paper allows simultaneous optimization of the total cost and the quality of the water discharged. Finally, local governments usually offer incentives


955

Figure 1. Alternatives to locate the new industrial plant.

to bring industries to their areas because of the social benefits brought through the generation of jobs yielded by the installation of a new plant; therefore, the proposed model could also be a useful tool to determine the total expenses that a local government needs to pay to be an attractive location for the installation of a new industry with respect to other locations. 3. Model Formulation The sets used in the model formulation are defined first for a better understanding. c is a subindex used to denote the component, so Nc corresponds to the total number of components, and C is a set that contains all c’s. r is a subindex used to denote a section of the river, whereas Nr corresponds to the total number of sections of the river used for a specific problem. R is a set that contains all r’s. t is a subindex for the effluents discharged to the river; Nt is the total number of effluents and T, their set. p is a subindex to denote the locations for the new plant; Np denotes the total number of possible locations, and P is their set. j is a subindex for the interceptors allowed to treat the composition of the effluent of the new plant; Nj denotes the total number of interceptors (including a fictitious interceptor used for modeling the bypassing stream), and J is their set. The equations for the model are described as follows. Balance for Each Reach. The exit flow rate from each reach r (Qr) is equal to the inlet flow rate to this reach (Qr-1) plus the average precipitation (Pr), direct industrial discharges (Dr), residential discharges (Hr), the sum of all effluents inlet to the

Nt(r) reach (∑t)1 FTr,t), including the possible discharge of the new plant in this reach (QPNEWr(p)), minus the extractions due to filtration and evaporation (Lr) and the use of water in that section of the river (Ur).

Nt(r)

Qr ) Qr-1 + Pr + Dr + Hr +

∑ FT

r,t

t)1

+ QPNEWr(p) -

Lr - Ur, ∀r ∈ R

(1)

where Nt(r) refers to the total number of tributaries discharged to the reach r. Component Balances for Each Reach. The exit mass from the reach for the toxic compound (QrCQc,r) is equal to the inlet mass to the reach (Qr-1CQc,r-1) plus the precipitation (PrCPc,r), industrial discharges (DrCDc,r), residential discharges (HrCHc,r), t(r)FT CT tributaries (∑Nt)1 r,t c,r,t), and the mass that can be discharged by the new plant (QPNEWr(p)CPNEWc,r(p)), minus the losses (LrCLc,r) and use (UrCUc,r). A very interesting term that appears Vr rc,r dVr. This reactive term considers is the reaction term, ∫V)0 the chemical and biochemical reactions that take place in the river due to interaction of the component with the flora and fauna inside the system, which can produce or eliminate the component. Finally, the component balance for each reach is stated as follows:

956


QrCQc,r ) Qr-1CQc,r-1 + PrCPc,r + DrCDc,r + Nt(r)

HrCHc,r +

∑ FT

r,tCTc,r,t

t)1

LrCLc,r - UrCUc,r -

+ QPNEWr(p)CPNEWc,r(p) -

∫

Vr r V)0 c,r

dVr, ∀c ∈ C, r ∈ R

(2)

Balances for Tributaries. The balance for the tributary t that discharges to the reach r is described as follows: untreated treated FTr,t ) Sr,t + Sr,t + Ir,t + Pr,t + Dr,t + QPNEWr,t(p) - Lr,t - Ur,t, ∀r ∈ R, t ∈ T

(3)

The total flow rate that discharges the tributary t and inlets to the reach r (FTr,t) is equal to the sum of the discharges with and without treatment (Streated ,Suntreated ), industrial discharges (Ir,t) r,t r,t and pluvial discharges (Pr,t), direct discharges (Dr,t), and the new discharge if the new plant is located on this tributary (QPNEWr,t(p)) minus the losses (Lr,t) and use or extraction of water (Ur,t). Component Balances for Each Tributary. The mass exits to each tributary t and directed to the reach r is calculated as follows: untreated untreated treated treated FTr,tCTc,r,t ) Sr,t CSc,r,t + Sr,t CSc,r,t + Ir,tCIc,r,t + Pr,tCPc,r,t + Dr,tCDc,r,t + QPNEWr,t(p)CPNEWc,r,t(p) - Lr,tCLc,r,t - Ur,tCUc,r,t -

∫

Vr,t r V)0 c,r,t

dVr,t, ∀c ∈ C, r ∈ R, t ∈ T

(4)

It is noteworthy that the component balances for each tributary Vr,t include the reaction term ∫V)0 rc,r,t dVr,t to consider the natural degradation of the components in the system. Agricultural Discharges and Agricultural Uses. The agricultural discharges Dr,t and agricultural uses Ur,t are related to the surrounding areas because they are proportional to the agricultural areas. These discharges and uses are calculated as follows: Dr,t ) Rr,t × Ar,t, ∀r ∈ R, t ∈ T

[

]

W1 CPNEWc,1 g 0, QPNEW1 g 0 CPNEWc,2 ) 0, QPNEW2 ) 0 ∨ l CPNEWc,p ) 0, QPNEWp ) 0 W2 CPNEWc,1 ) 0, QPNEW1 ) 0 CPNEWc,2 g 0, QPNEW2 g 0 ∨ · · · ∨ l CPNEWc,p ) 0, QPNEWp ) 0 Wp CPNEWc,1 ) 0, QPNEW1 ) 0 CPNEWc,2 ) 0, QPNEW2 ) 0 , ∀c ∈ C l CPNEWc,p g 0, QPNEWp g 0

[

[

]

In the disjunction, Wp is a Boolean variable associated with location p of the new plant; CPNEWc,p is the concentration for the discharge of the new plant to the river at location p. Because the final discharge for the new plant to the river depends on the location of the new plant, when the optimal location for the new plant is location 1, the Boolean variable W1 is set as true, and CPNEWc,1 and QPNEWc,1 must be bigger than zero. All of the others Boolean variables must be false, and the CPNEWc,p and QPNEWp associated with their locations must be set to zero. A similar situation occurs for the case when the selected location is 2 or 3 or so on for the p possible locations. It is worth mentioning that the value for the discharge concentration of the new plant is strongly influenced by the location through the material flow analysis model, satisfying the constraints given in order to maintain control of the pollutants in their final disposal as well as the maximum concentration allowable in the discharge. In this paper, since the plant design is the same for any location, it is assumed that the flow rate discharged from the new plant is a constant value specified prior to the optimization of the location. Finally, the disjunction is modeled using the convex hull reformulation (see Raman and Grossmann;41 Ponce-Ortega et al.42) as follows: The Boolean variables are transformed into a set of binary variables. When the Boolean variables are true, the associated binary variables are one; otherwise, when the Boolean variables are false, the associated binary variables are zero. To select only one location to place the new plant, the following relationship is used:

∑w

p

Ur,t ) βr,t × Ar,t, ∀r ∈ R, t ∈ T

]

)1

(5)

p∈P

where Rr,t is the flow rate per unit of area in m /acre · s, βr,t is the agricultural use of water per unit of area for the tributary t in m3/acre · s, and Ar,t is the area that covers the tributary t in acres. 3

Location of the New Plant. The values of the flow rate and the composition discharged from the new plant depend on the location of the plant. For example, if the new plant is located at position 1, the flow rate and composition discharged from the new plant in that position must be bigger than zero; otherwise, if the new plant is not located at position 1, the flow rate and composition discharged would be zero. For a set of possible locations p where the new plant can be placed, this is modeled through the following disjunction:

Then, upper limits are imposed on the variables CPNEWc,p and QPNEWp depending on whether the new plant is located in position p or not. CPNEWc,p e CPNEWUPwp, ∀c ∈ C, p ∈ P

(6)

QPNEWp e QPNEWUPwp, ∀p ∈ P

(7)

Equations 6 and 7 are used to ensure that CPNEWc,p and QPNEWp take values bigger than zero only when the new plant is located in that position, and these take values of zero when the location is not selected for placement of the new plant. Wastewater Treatment System. As shown in Figure 2, a set of interceptors is available to satisfy the discharge limits for the new plant. As the optimization process must determine


957

Figure 2. Interceptors network for wastewater treatment.

the optimal selection of the treatment system, this must be located in the place where the new plant is located. Therefore, it is important to remark that the treatment requirement depends strongly on the location of the new plant to satisfy the sustainability of the system in the final disposal. First, the concentration discharged for the new plant is calculated as the sum of the discharges for any allowable location because only one location is able to place the new plant; this situation is modeled as follows:

∑ CPNEW

c,p

) ydisc c , ∀c ∈ C

(8)

p∈P

Similarly, for the flow rate discharged for the new plant:

∑ QPNEW

p

) QPNEWDATA

(9)

p∈P

where QPNEWDATA is a constant parameter known prior to the optimization process. Figure 2 shows a representation of the interception network used to treat the wastewater for the new plant. Notice that the treatment of the wastewater stream may or may not be required according to the location of the new plant and the MFA model. On the other hand, to model the performance of the interceptors, an efficiency factor for each component (γc,j) is used to simulate the unit j. This efficiency factor depends on the design and operating parameters for each unit, as can be seen in the next equation: γc,j ) f(Design and Operating Parameters) The available treatment units can be simulated (for example, in ASPEN PLUS, using the Kremser equation, etc.), and the efficiency factors for each unit are determined prior to the optimization process. However, the final concentrations depend on the technologies selected and the inlet concentration, the coefficients for the cost functions (FCj and VCj) for each treatment unit j depend on the type of unit (with their corresponding efficiencies), and the total cost for the treatment network accounts for the type of selected unit and the treated flow rate. Therefore, the optimization consists of the selection

of the treatment units with known characteristics and the treated flow rate. In this sense, the interceptor units to be used depend on each particular process, but they can be distillation columns, adsorption columns, stripping columns, liquid-liquid extraction columns, etc. Important remarks for the treatment network are as follows: • The wastewater stream can be treated by one or several technologies available for removing the pollutant. • The flow rate at the exit of the plant is the same as the flow rate at the exit of the interceptor network, but the composition is not the same. • Each interceptor has an efficiency factor required to remove the pollutant, which is directly related to the amount of pollutant that it can remove. In addition, the cost for each interceptor is constituted by a fixed charge FCj and a variable charge VCj that depends on the flow rate treated. • The last interceptor in the interception network is a fictitious interceptor, which is included for modeling the bypass streams. This fictitious interceptor has a conversion factor of zero and an associated cost of zero. The following relationship is used to determine the concentration at the exit of each interceptor considered: in yout c,j ) yc (1 - γc,j), ∀c ∈ C, j ∈ J

Notice that the exit concentration from each interceptor for each component is a given parameter because this depends on the inlet concentration and the conversion factors; however, the optimal decision is given in terms of the flow rates inlet to each interceptor. The balances for the interceptor network are presented as follows: QPNEWDATA )

∑f

j

(10)

j∈J

∑fy

out j c,j

) QPNEWDATAydisc c , ∀c ∈ C

(11)

j∈J

where fj is the segregated wastewater flow rate to the interceptor j, γc,j is the efficiency factor required to remove the pollutant c for the interceptor j, yin c is the concentration of the pollutant c

958


out inlet to the interceptor networks and outlet to the plant, and yc,j is the concentration of the pollutants at the exit of the interceptor j. The following constraints are required for the existence of the interceptors:

fj g δzj, ∀j ∈ J

(12)

fj e f UPzj, ∀j ∈ J

(13)

where δ represents a small parameter used for modeling purposes, zj is a binary variable associated with the existence of the interceptor j, and f UP is an upper limit for fj. Notice that f UP is defined as f UP ) QPNEWDATA. Constraints for the Quality of Specific Reaches. Some reaches through the watershed require specific constraints for the water quality because they need to provide water for specific uses. The proposed model is able to consider these constraints through the following relationship: desired CQc,n(r) e yc,n(r) , ∀c ∈ C, n(r) ∈ N(R)

(14)

where N(R) represents a subset of reaches n from all of the reaches r that requires a specific water quality. Constraint for the Final Disposal. The concentration for the waste discharged to the river by the new industrial plant must be restricted by environmental regulations as follows: ydisc e yEnvReg , ∀c ∈ C c c However, environmental regulations do not prevent the accumulation of pollutants in the final disposals (i.e., lakes or seas). To get a sustainable process, the following constraint must be added to the model: e ysustainability , ∀c ∈ C CQfinal c c

(15)

Objective Function. The objective function consists of minimization of the total annual cost considering installation of the new plant as well as treatment costs. Therefore, the objective function is stated as follows: min TAC )

∑ CLAND w

p p

p∈P

+ kf

∑ FC z j

j∈J

j

+ HY

∑ VC (f ) j

cexp

j

j∈J

(16) where TAC is the total annual cost, CLANDp is the annualized installation cost for the new plant at site p, kf is an annualization factor, FCj is the fixed charge for the interceptor j, VCj is the variable charge for the interceptor j, HY is the hours per year that the new plant operates, and cexp is a exponential factor used to consider the economies of scale. Remarks. The chemical and biochemical reactions can be represented by a simple kinetic model, although several chemical and biochemical reactions may occur. The installation costs (CLANDp) include three different terms: transportation of the raw materials, transportation of the products, and the land. All of these costs depend strongly on the location. The flow rate discharged from the new plant, QPNEWDATA, is considered a constant. The same case applies for the pollutant concentration exiting the plant, yin c , while the pollutant concen-

Figure 3. Candidate sites for the location of the new industrial plant for example 1.

tration for the final discharge of the new plant to the river, ydisc c , is an optimization variable. 4. Results and Discussion Two cases are presented to show the applicability of the proposed methodology. Example 1. The data for this example correspond to the Bahr El-Baqar drain reported previously,40 which is one of the largest drains on the Nile Delta in Egypt and ends in Lake Manzala. The system has a total length of 106.5 km and an approximated width of 23 m at the beginning and 70 m at the end, whereas the final disposal has a total area of 1000 km2 and an average depth of 1 m. The drain receives several types of discharge including agricultural drainage, treated and untreated domestic wastewater, and industrial wastewater. Detailed information with respect to this drainage system was reported by Lovelady et al.40 A new fertilizing plant must be installed that discharges phosphorus as the major contaminant (in this case, only one key component was selected). The phosphorus discharged by the new plant must not exceed a final concentration in the lake of 1.3 ppm to maintain the system. Figure 3 shows a schematic representation of the watershed system considered in this example. In this system, there are

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011 Table 1. Installation Cost for the Different Sites to Locate the New Plant for Example 1

959

Table 2. Treatment Cost for the Available Interceptors for Example 1

site

annualized installation cost, $MM/year

interceptor

fixed cost, $

variable cost, $ h/m3

efficiency, γj

1 2 3 4

10 17 35 18

1 2 3 4

568,000 446,000 353,000 0

0.235 0.193 0.168 0

0.95 0.84 0.76 0

residential, agricultural, and industrial discharges that significantly change the composition of the phosphorus in the river. Figure 3 also shows four sites where it is possible to locate the new industrial plant; each one of these sites has different installation costs associated, as can be seen in Table 1 (land cost, costs for transportation of raw materials and products, etc.). The flow rate for the new plant (QPNEWDATA) is 2 m3/s, and the concentration for the discharge of the new plant before treatment (yin) is 12.5 ppm of phosphorus. Obviously, the installation of the new plant will alter and impact the environment through the new discharges to the watershed, increasing the composition of the contaminants in the surrounding rivers, as well as contaminating the final disposal in the lake (CQfinal). In addition, water is taken from the zone corresponding to reach 6; this water is used to feed a town, which requires that the maximum concentration of phosphorus must be lower than 2.1 ppm. In reach 11, some farmers take water to their crops; then, the water must contain a concentration lower than 3.15 ppm. Also, the maximum concentration allowed to be discharged to the lake is of 1.3 ppm to allow for the natural degradation of phosphorus in the lake. This upper value was determined from a sustainability analysis.40 It is important to mention here that the system discharges water in the final disposal with a concentration of 1.17 ppm of phosphorus prior to the installation of the new industrial plant. To model the system, the following assumptions were used in this example problem (Lovelady et al.40): 1. Insignificant precipitation. The flow rate due to precipitation was eliminated according to the weather in Egypt. 2. Insignificant losses. Water filtration and vaporization are too small compared to the convective flows. 3. The concentration for the phosphorus at the exit of the wastewater treatment plants (WWTP) was taken as 9 mg/ L. This assumption is based on a laboratory analysis for different treatment plants. 4. The concentration of the phosphorus for the wastewater without treatment is 15 mg/L. 5. The concentration of the phosphorus for the water treated is 9.75 mg/L. 6. The concentration of the phosphorus for the agricultural wastewater is 1.5 mg/L. The reaction kinetic for the degradation of the phosphorus in the reaches takes the following form:

∫

Vr r V)0 r

dVr ) k × CQr × Vr, ∀r ∈ R

where k is the kinetics constant for the reaction measured experimentally,40 CQr is the concentration of the phosphorus in the reach, and Vr is the volume of the reach. Whereas, for the tributaries, the reaction kinetics takes the following form:

∫

Vr,t r V)0 r,t

dVr,t ) k × CTr,t × Vr,t, ∀r ∈ R, t ∈ T

where k ) 9.041909 × 10-6/s, for reaches and tributaries.

Table 3. Results Comparison for a Constraint of 1.3 ppm in Example 1 site

installation cost, $/year

treatment cost, $/year

total annual cost, $/year

1 2 3 4

10 × 106 17 × 106 35 × 106 18 × 106

7.6 × 106 2.54 × 106 0 0

17.59 × 106 19.54 × 106 35 × 106 18 × 106

In addition, values of Rr,t are 0.000066 m3/acre · s for all reaches. β ) 0.000023 m3/acre · s for reaches 1-6, 8, and 12-15; β ) 0.000011 m3/acre · s for reaches 7 and 9-12. The operation time for the plant is 8600 h/year, the annualization factor is 0.1, and the exponential factor that considers the economies of scale is 1. Finally, the interceptors available with their unitary costs and efficiencies are shown in Table 2. The model was implemented in the General Algebraic Modeling System, GAMS (Brooke et al.43), and the solver DICOPT (Viswanathan and Grossmann44) was used to solve it. The optimal solution for this example corresponds to locating the new plant at site 1, including a treatment for 1.453 m3/s of the wastewater discharged from the new plant using interceptor 3, whereas 0.547 m3/s bypasses the treatment units. For the optimal solution of this example, the wastewater discharged from the new plant located at site 1 must be 5.597 ppm to ensure that the natural degradation satisfies the constraint for the discharge in the final disposal of 1.3 ppm, and to obtain concentrations for reaches 6 and 11 lower than 1.313 ppm and 2.842 ppm, respectively. The total annual cost to install the new plant at site 1 is $17.6 × 106/year. Table 3 shows a result comparison with respect to the different locations available for installing the new plant. It is worth noting that sites 1 and 2 present treatment requirements to satisfy the sustainability of the system, and sites 3 and 4 do not require treatment of the wastewater discharged for the new plant due to them being far away from the final disposal. Therefore, in these cases, it is possible for the pollutant to be disintegrated naturally through the river. However, although sites 3 and 4 do not require treatment of the new wastewater, the installation costs for these places are more expensive than for site 1, including the cost to treat the wastewater required for this last location. Additionally, if the problem is solved without taking into account the environmental aspects (i.e., considering exclusively the cost), the economic solution is given for the site with the smallest installation cost (smallest CLANDp), that is site 1. In this case, the final discharge to the lake has a concentration of 1.5 ppm, which is 15.38% bigger than the maximum concentration required to maintain the system without dangerous accumulations. In Table 3, it is possible to identify how much the other locations, different from the optimal one, have to pay as incentives to be attractive economically for the installation of the new plant. For example, site 2 has to pay 1.95 × 106 $/year, whereas site 4 has to pay 0.41 × 106 $/year to be attractive economically while satisfying environmental constraints. Therefore, the proposed model is a useful tool for governments to determine the incentives that they will have to pay to make the

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Table 4. Total Cost for Several Sustainability Constraints in Each Site for Example 1 total annual cost, $/year for each site concentration discharged in the final disposal lower than

1

1.2 1.25 1.3 1.4 1.503

21.73 × 10 19.46 × 106 17.59 × 106 13.86 × 106 10 × 106

2 6

installation of a new industrial plant attractive while satisfying the environmental constraints for the new pollutants discharged. Table 4 shows the optimal costs associated with the installation of the new plant for different upper limits of the pollutant concentration discharged in the final disposal for all possible locations. The information reported in Table 4 allows for conducting a sensibility analysis for the case when the pollutant concentration discharged in the final disposal changes, and this could be a useful tool for the decision maker and the governments interested in analyzing several scenarios prior to installation of the new plant. For example, site 1 is the one that is less sensitive to changes for the constraint in the final disposal, whereas site 3 is always the most expensive. In addition, using the proposed model, it is also possible to develop a multiobjective optimization approach to determine the optimal trade-offs between the total annual cost and the concentration for the pollutant discharged in the final disposal, simultaneously considering the optimal location and the treatment requirements. The optimal Pareto curve can be seen in Figure 4. It is worth noticing that the solutions above the Pareto curve represent suboptimal solutions, whereas the solutions below the Pareto curve are infeasible solutions. Example 2. For this problem, the data for the Balsas watershed system located in Mexico were considered (CONAGUA45,46). This system is one of the biggest in Mexico, where several industrial zones discharge their effluents. Figure 5 shows the main streams of the river and 20 possible locations to place a new industrial plant that discharges a hazardous pollutant (in this case, the key compound is arsenic, which is very toxic). The following assumptions are made for the pollutant considered in this example: 1. The concentration of the sanitary discharges without untreated ) is 0.5 ppm. treatment (CSr,t 2. The concentration of the sanitary discharges with treatment treated ) is 0.03 ppm. (CSr,t 3. The concentration of the industrial discharges (CIr,t) is 0.06 ppm. 4. The concentration of the precipitation (CPr,t) is 0 ppm. 5. The concentration of the agricultural discharges (CDr,t) is 0.055 ppm. 6. The concentration of the industrial and sanitary discharges (CHr) is 0.07 ppm.

Figure 4. Pareto curve for example 1.

3

28.02 × 10 23.56 × 106 19.54 × 106 17 × 106 17 × 106 6

4

46.63 × 10 39.86 × 106 35 × 106 35 × 106 35 × 106

27.55 × 106 19.39 × 106 18 × 106 18 × 106 18 × 106

6

7. The concentration of the direct discharges (CDr) is 0.075 ppm. The flow rate for the new plant (QPNEWDATA) is 5 m3/s, and the concentration of the hazardous pollutant discharged is 20 ppm. Table 5 shows the data for the installation costs for the different location alternatives of the new industrial plant (remember that these installation costs include the transportation cost for the raw materials and products and the land cost). Notice in Table 5 that sites 18, 19, and 20 are the cheapest ones because these sites are the closest to the sea, and therefore the transportation costs are the smallest because that zone links with the port facilities. On the other hand, the chemical interaction between the pollutant and the environment is represented by a first-order reaction with a kinetic constant of k ) 4.52095 × 10-7/s.

Figure 5. Available locations for the new industrial plant for example 2. Table 5. Installation Costs for Example 2 site

annualized installation cost, $/year

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

12 × 106 8 × 106 9.5 × 106 8 × 106 9.5 × 106 8 × 106 12.5 × 106 10.5 × 106 7.7 × 106 7 × 106 7.8 × 106 7.5 × 106 8.2 × 106 8.5 × 106 7.6 × 106 9.5 × 106 9 × 106 5 × 106 4.8 × 106 5.5 × 106

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011 Table 6. Data for the Available Interceptors for Example 2 interceptor

fixed cost, $

variable cost, $ h/m3

efficiency factor, γj

1 2 3 4 5 6 7 8 9 10 11

195,000 186,000 179,500 175,400 170,000 162,000 158,000 144,000 137,500 126,000 0

0.058 0.052 0.049 0.047 0.043 0.039 0.035 0.031 0.028 0.025 0

0.95 0.88 0.85 0.82 0.79 0.75 0.73 0.71 0.68 0.64 0

In addition, in this case, Rr,t ) 0.000148 m3/ha · s and βr,t ) 0.000296 m3/ha · s, the operation time for the new plant is 8400 h/year, the annualization factor is 0.1, and the exponential factor that considers the economies of scale is 1. The following constraints are required to maintain the water quality under optimal conditions through the watershed: (1) In reach 5, some farmers take water from the river; therefore, the composition must be lower than 2.5 ppm. (2) There is a city in the area corresponding to site 15; therefore, the pollutant concentration needs to be lower than 0.3 ppm. (3) Another city is located near site 18, and this city requires a concentration for the water river lower than 0.15 ppm. The data for the available interceptors to treat the wastewater discharged from the new plant for this example are shown in Table 6. To adequately consider the environmental aspects, it is required to include a constraint for the concentration of the pollutant in the final disposal once the new plant was installed (i.e., the concentration of the final disposal must be less than the one that the system is naturally able to degrade, avoiding the accumulation of this pollutant). The current concentration discharged in the final disposal without the discharge of the new plant is 0.037 ppm for the pollutant considered. First, the solution of the problem considering exclusively economic aspects was obtained, which states that the new plant must be located at site 19 with a total annual cost of $4.8 × 106/year; in this case, the concentration for the pollutant in the final disposal is 0.206 ppm. Then, several scenarios have been considered for the solution of this example. Case A. The concentration allowable to be discharged in the final disposal must be lower than 0.04 ppm. The optimal solution for case A is to install the new plant at site 2, which represents a total annual cost of $11.85 × 106/year with a treatment of 0.471 m3/s in interceptor 9 and 4.529 m3/s in interceptor 10 to yield a concentration for the final disposal that is naturally degraded. In this way, the treated stream discharged for the new plant has a concentration of 7.125 ppm. Case A requires an investment of 247% more than the economic solution to obtain a reduction of 80% of the concentration in the stream discharged in the final disposal. Case B. The concentration for the final disposal must be lower than 0.045 ppm for the pollutant considered. In this case, the optimal location for the new plant is also at site 2, but the wastewater stream does not require any treatment, and the total annual cost is $8 × 106/year. Here, comparison with the economic solution shows that it is possible to have a concentration for the final disposal less than 78% (0.045 ppm), investing $8 × 106/year, which is approximately 166% of the economic solution. Case C. The concentration in the final disposal must be lower than 0.05 ppm. In this case, the minimum total annual cost is $7.35 × 106/year, and the optimal location for the new plant is

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3

at site 10 using interceptor 10 to treat 0.447 m /s of the total wastewater discharged for the new plant. The pollutant concentration for the treated discharge of the new plant is 18.855 ppm. Therefore, investing 153% more than the economic solution yields a reduction of 75% in the concentration for the final disposal with respect to the economic solution. Case D. The concentration allowable to be discharged in the final disposal must be lower than 0.06 ppm. The optimal solution for this case is to install the new plant at site 10, which represents a total annual cost of $7 × 106/year without any treatment because the pollutant is naturally degraded through the watershed. Compared with the economic solution, case D requires an investment of 146% more to obtain a reduction of 70% for the concentration discharged in the final disposal. Case E. The concentration for the final disposal must be lower than 0.14 ppm. Because in this case the constraint is less severe than the other cases, the optimal location is in site 18, a site closer to the final disposal with a lower installation cost of $6.17 × 106/year. It is required to treat 2.235 m3/s using interceptor 10 to satisfy the environmental constraints and to yield a treated discharge from the new plant with a concentration of 14.279 ppm. For this case, it is shown that with an investment of 1.28% more than the economic solution, the concentration discharged in the final disposal increases 32%. Table 7 shows a result comparison for the cases previously analyzed (including the solution without considering the environmental constraint in the last row). Table 7 also shows the concentrations for the specific locations of the river that are restricted, and one can see that all of these constraints are satisfied. Notice in Table 7 that the total annual cost increases proportionally to the regulation for the final disposal of the pollutant. On the other hand, Table 8 shows different scenarios for the installation of the new plant in different locations and for different concentrations in the final disposal. All options considered must satisfy the constraints for the concentration through the watershed. For instance, if an upper limit for the concentration in the final disposal of 0.05 ppm was fixed, then the optimal solution is at site 10. However, if the local governments for sites 15 and 17 were interested in acquiring this new industry, they would require incentives to compensate the cost of the optimal solution by $2.23 × 106/year and $3.74 × 106/year, respectively. Figure 6 shows the Pareto curve for this case. It is worth noticing that there is a trade-off between the total annual cost and the concentration discharged in the final disposal. In addition, the optimal location depends on the specific concentration in the final disposal. This Pareto curve shows a set of optimal solutions, and it is very useful to analyze different scenarios. Finally, Table 9 shows the size and CPU time for both examples using a Core 2 processor with 3 MB of RAM. It is noteworthy that the CPU time is small. 5. Conclusions This paper presents a new mixed-integer nonlinear programming model for the optimal location of a new industrial plant satisfying the environmental regulations for the wastewater streams discharged to the rivers as well as the constraints imposed to ensure sustainability for the watershed, including local constraints for specific sites where the water can be used. In the proposed model, the mass integration for the wastewater stream can be considered together with the mass integration with the watershed. The model was formulated to predict the

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Table 7. Result Comparison for Several Sustainability Constraints in Example 2 concentration for specific reaches, ppm constraint

site

ydisc(ppm)

1

2

3

installation cost, $/year

treatment cost, $/year

total annual cost, $/year

0.04 ppm X 0.045 ppm X 0.05 ppm X 0.06 ppm X 0.14 ppm X 0.206 ppm X

2 2 10 10 18 19

7.125 20 18.855 20 14.279 20

0.943 2.445 0.123 0.123 0.123 0.123

0.139 0.139 0.277 0.285 0.139 0.139

0.032 0.037 0.042 0.043 0.137 0.029

8 × 106 8 × 106 7 × 106 7 × 106 5 × 106 4.8 × 106

3.85 × 106 0 3.50 × 105 0 1.7 × 106 0

11.85 × 106 8 × 106 7.35 × 106 7 × 106 6.7 × 106 4.8 × 106

Table 8. Total Cost for Several Sustainability Constraints and Several Locations in Example 2 total annual cost for the site, $/year upper limit for the concentration in the final disposal, ppm

2

0.04 0.045 0.05 0.06 0.14 0.206

11.85 × 10 8 × 106 8 × 106 8 × 106 8 × 106 8 × 106

10 6

15

13.25 × 10 9.54 × 106 7.35 × 106 7 × 106 7 × 106 7 × 106 6

Table 9. Size and Computation Time for the Problems Considered concept

example 1

example 2

number of constraints number of continuous variables number of binary variables CPU time (s)

81 82 8 0.37

1479 1476 31 1.68

behavior for the watershed impacted for the new polluted discharges. To conduct this task, the material flow analysis model was used. In the MFA model, the industrial, residential, agricultural, and other discharges and all extractions are considered, as well as the chemical reactions that are carried out in the rivers. The watershed model is combined with a disjunctive model for the optimal location of a new industrial plant through waste integration and satisfying environmental regulations. The model allows for the optimal selection of the type of treatment, and the objective function consists of minimizing the total annual cost that is constituted by the wastewater treatment costs and the costs for the location of the new plant. The solution of the case studies shows that the location of a new industrial plant has a significant impact on the treatment requirements to yield a sustainable process. The proposed model can be used as a tool for governments and society to determinate the incentives required for their locations to be attractive for the installation of a new plant, in order to increase the prospect of jobs in their location, whereas the Pareto curve shows a set of optimal solutions for the case where the concentration in the final disposal changes. Finally, the model can identify several pollutants; however, the main purpose of this paper is to consider the effect of a key component and to track it through its trajectory in the watershed.

Figure 6. Pareto curve for example 2.

17

14.84 × 10 11.13 × 106 9.58 × 106 7.6 × 106 7.6 × 106 7.6 × 106 6

16.30 × 10 12.60 × 106 11.09 × 106 9 × 106 9 × 106 9 × 106 6

18

19

infeasible 13.73 × 106 13.14 × 106 11.96 × 106 6.70 × 106 5 × 106

infeasible infeasible 13.16 × 106 12.15 × 106 7.13 × 106 4.8 × 106

Nomenclature Ar,t ) area cover by effluent j in reach r, acre or ha C ) set for the components, {c|c ) 1, ..., Nc} cexp ) exponential factor useful to consider the economies of scale. CDc,r ) concentration of direct discharges to the reach r, ppm CDc,r,t ) concentration of agricultural discharges to the tributary t to the reach r, ppm CHc,r ) concentration of total discharge (i.e., industrial + sanitary) to the reach r, ppm CIc,r,t ) concentration of industrial discharge from the tributary t to the reach r, ppm CLc,r ) concentration of total losses (filtration and evaporation) from the reach r, ppm CLc,r,t ) concentration of total losses (filtration and evaporation) from tributary t, ppm CLANDp ) annualized installation cost for the new plant at site p, $/year CPc,r ) concentration of precipitation discharged to the reach r, ppm CPc,r,t ) concentration of precipitation discharged to the tributary t to the reach r, ppm CPNEWc,p ) concentration for the contaminant discharged for the new plant installed at site p, ppm CQc,r ) concentration of flow rate exit to the reach r, ppm CQc,r-1 ) concentration of flow rate inlet to the reach r, ppm CQfinal ) pollutant concentration discharged in the final disposal, c ppm CTc,r,t ) concentration of discharge for the tributary t to the reach r, ppm CUc,r ) concentration of water used from reach r, ppm CUc,r,t ) concentration of water used from tributary t discharge to reach r, ppm CStreated c,r,t ) concentration of residual wastewater discharged without treatment to the reach r for tributary t, ppm CSuntreated ) concentration of residual treated wastewater discharged c,r,t to the reach r for tributary t, ppm Dr,t ) agricultural discharges to the tributary t to the reach r, m3/s Dr ) direct discharges to the reach r, m3/s FCj ) fixed cost for interceptor j, $/year FTr,t ) discharge for the tributary t to the reach r, m3/s fj ) segregated flow rate from the wastewater of the new plant to the interceptor j, m3/s UP f ) upper limit for f1 HY ) operation time per year, h/year

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011 Hr ) total discharge (i.e., industrial + sanitary) to the reach r, m /s Ir,t ) industrial discharge from the tributary t to the reach r, m3/s J ) set for the interceptors k ) kinetic constant for the degradation of the pollutant in the system kf ) annualization factor Lr,t ) total losses (filtration and evaporation) from tributary t, m3/s Lr ) total losses (filtration and evaporation) from the reach r, m3/s Nc ) total number of components Nj ) total number of interceptors Np ) total number of allowable sites to locate the new plant Nr ) total number of reaches Nt ) total number of tributaries N(R) ) subset for specific reaches that requires composition constraints P ) set for the allowable sites to locate the new plant Pr,t ) precipitation discharged for the tributary t to the reach r, m3/s Pr ) precipitation discharged to the reach r, m3/s Qr ) flow rate exit to the reach r, m3/s Qr-1 ) flow rate inlet to the reach r, m3/s QPNEWp ) flow rate discharged for the new plant to the location p QPNEWDATA ) constant flow rate discharged for the new plant R ) set for the reaches rr ) reaction carries out in the reach r, g/s rr,t ) reaction carries out in the tributary t that discharges to the reach r, g/s untreated Sr,t ) residual wastewater discharged without treatment to the reach r for tributary t, m3/s treated Sr,t ) residual treated wastewater discharged to the reach r for tributary t, m3/s T ) set for the tributaries TAC ) total annual cost, $/year Ur,t ) water used from tributary t discharged to reach r, m3/s Ur ) water used from reach r, m3/s VCj ) variable cost for interceptor j, $/m3 year Vr,t ) volume for tributary t discharged to reach r, m3 Vr ) volume for reach r, m3 Wp ) Boolean variable for the location of the new plant wp ) binary variable for the location of the new plant desired yc,n(r) ) limit for the desired concentration in some reaches, ppm ydisc ) pollutant concentration for the final discharge of the new c plant to the river, ppm yEnvReg ) environmental regulation for the pollutant concentration c to be discharged by the new plant, ppm yin c ) pollutant concentration exits to the plant and inlets to the treatment system, ppm out yc,j ) pollutant concentration at the exit of interceptor j, ppm ysustainability ) sustainability constraint at the final disposal, ppm c zj ) binary variable associated with the existence of the interceptor j 3

Greek symbols Rr,t ) agricultural flow rate per area, m3/ha*s βr,t ) agricultural use of water from tributary t, m3/ha s γj ) efficiency factor to remove the pollutant for the interceptor j δ ) small number Indexes c ) component j ) interceptor n(r) ) reaches that requires a composition constraint p ) site to locate the new plant

963

r ) reach t ) tributary UP ) upper limit

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ReceiVed for reView September 14, 2010 ReVised manuscript receiVed November 26, 2010 Accepted December 2, 2010 IE101897Z

An MINLP Model for the Optimal Location of a New Industrial Plant

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