Incorporation of the Seasonal Variations in the Optimal Treatment of

Article pubs.acs.org/IECR

Incorporation of the Seasonal Variations in the Optimal Treatment of Industrial Effluents Discharged to Watersheds Oscar Burgara-Montero,† José María Ponce-Ortega,*,† Medardo Serna-González,† and Mahmoud M. El-Halwagi‡,§ †

Chemical Engineering Department, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58060, México Chemical Engineering Department, Texas A&M University, College Station, Texas 77843-3122, United States § Adjunct Faculty at the Chemical and Materials Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia ‡

ABSTRACT: This paper presents a multiperiod optimization approach to the cost-effective reduction of the negative impact caused by discharging industrial effluents into watersheds. The model considers all discharges and extractions, as well as the chemical reactions carried out in the watershed, while accounting for the seasonal variability of the system through the year. The proposed model is based on a material flow analysis applied to a distributed treatment system for industrial effluents in a macroscopic water system. The problem is formulated as a multiobjective optimization model with the objectives of simultaneously minimizing the pollutant concentrations in the final catchment areas and the total annual cost of the wastetreatment units. The model selects the place to locate the treatment units and the type of treatment technology, as well as the industrial effluents treated in each period of the year, to satisfy the environmental regulations. Two case studies are presented to show the applicability of the proposed optimization approach: one is the Bahr El-Baqar drain in Egypt and the other one is the Balsas watershed in Mexico.

1. INTRODUCTION Watersheds have played a critical role in the development of civilization because they provide several natural resources needed for the human life. Traditionally, human activities (agricultural, industrial, and living) generate significant amounts of wastewater that are discharged to the rivers and water bodies, where there are several physical−chemical−biological transformations carried out by the flora and fauna, which act on the discharged pollutants. Such natural processes are very important to achieve the sustainability of the watersheds, since they can degrade the pollutants contained in the wastewater streams. However, this symbiosis has been drastically altered recently because of the considerable increase in the global population, which, in turn, has significantly increased the industrial activity and, consequently, the amounts of highly polluted wastewater discharged to rivers and water bodies. Currently several rivers and watersheds are polluted because of numerous industrial discharges. For example, in Mexico, industries across the country discharge about 8 × 109 m3/year of effluents, with more than 9.5 million tons of biochemical oxygen demand (BOD), of which only 18% is removed by treatment systems. To solve this problem, governments have imposed stricter environmental regulations on industrial effluents and required the use of effective treatment technologies (e.g., Best et al.1 or Kennish2). However, the treatment of industrial effluents only satisfies the environmental regulations in a specific site but does not take into consideration their interactions with other discharges to the watersheds. Additionally, these regulations typically overlook the chemical and biochemical phenomena that happen inside the watersheds as well as the changes that occur during the year due to the natural and human © 2013 American Chemical Society

phenomena involved. Therefore, such an approach neglects the interactions between industrial and other discharges and uses that may cause severe pollution problems making the local regulations insufficient for meeting the broader (e.g., regional) environmental objectives. To track the pollutants through the watershed is a complicated task due to the numerous inputs, outputs, and phenomena that occur in the watersheds (see Chapra3 and Ji4); nevertheless, the local concentration for the pollutants is not required to determine the overall hazardous effect of the pollutants on the watersheds. In this regard, simplified formulations have been reported by several authors to address this problem. Cooper5 modeled a river as a plug flow reactor, and Drolc and Zagorc-Koncan6 presented a model to evaluate the dissolved oxygen concentration in the watersheds. Recently, the material flow analysis (MFA) technique has been proposed to track the pollutants through watersheds considering the main activities that happen around them and accounting only for the average phenomena in given sections of the watersheds neglecting the local fluctuations. On the basis of these ideas, Baccini and Brunner7 proposed an MFA model for analyzing ecosystems with human activities accounting for the mass and energy interactions in the system. Lamper and Brunner8 implemented a MFA model to track the average composition of the nutrients in the Danube river, and Drolc and ZagorcKoncan9 applied this model to the Krka river in Slovenia. ElBaz et al.10,11 presented a MFA model to track the sulfur and Received: Revised: Accepted: Published: 5145

November 7, 2012 January 7, 2013 March 13, 2013 March 13, 2013 dx.doi.org/10.1021/ie303065h | Ind. Eng. Chem. Res. 2013, 52, 5145−5160

Industrial & Engineering Chemistry Research

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Figure 1. Schematic definition for the addressed problem.

influences significantly the activities associated such as cultivation, harvesting, and growth (SEMARNAT19). Furthermore, several multiproduct industries discharge wastewater with different amounts and with different quality through the year because the distribution of the products changes through the year. Obviously, all these variations in natural phenomena and human activities through the year have to be taken into account to determine with confidence the performance of watersheds. Therefore, this paper presents a mathematical programming model for the optimal treatment of the industrial effluents discharged to the watersheds considering the main discharges and uses, as well as the natural phenomena, in addition to the seasonality effects on watersheds due to changes in the discharges and uses of water in the watershed throughout the year. We use a multiperiod optimization approach combined with a MFA model to describe the seasonal behavior of watersheds, which is adequate for the cases where the influence of weather changes on the watershed is considerable. In addition, this approach allows consideration of the variation of the industrial discharges in different periods of the year. The objective function consists in minimizing the total annual cost for the treatment of the industrial effluents considering the capital and operational costs to satisfy the quality constraints for the water along the watershed and to avoid the accumulation of the pollutants in the final catchment areas. This paper is organized as follows: section 2 presents the definition for the addressed problem. Section 3 presents the

nitrogen compounds in the Bar-El-Baqar drain in Egypt, and Lovelady et al.12 reformulated this MFA model to determine the maximum allowable discharges that ensure the sustainability of the system. In the same way, Lira-Barragán et al.13,14 implemented a MFA model to determine the optimal location of new industrial facilities considering the sustainability of the surrounding watershed. Then, Lira-Barragán et al.15 extended their model in terms of properties instead of composition constraints, which is more appropriate for systems constituted by several pollutants. Recently, Burgara-Montero et al.16 reported a methodology to implement a distributed treatment system for the industrial effluents discharged to a watershed and to minimize the concentration in the receiving bodies (seas, lakes, oceans) based on a MFA model. This model takes into account the interactions between the industrial discharges and other discharges and uses such as agricultural, sanitary, and natural phenomena like precipitation, filtration, and evaporation, allowing this method to track the pollutants through the watershed to avoid their accumulation. However, this model does not consider the seasonality of the system assuming that all discharges are constant through the year. This assumption yields inaccurate results for the cases where the discharges and uses change significantly through the year. It is noteworthy that the seasonal dependence for some discharges is considerable significant. Taking Mexico as an example, in January the average precipitation is 14.2 mm whereas in August it is 194.1 mm (CONAGUA17,18). In addition, the agricultural discharges change drastically through the year because the weather 5146

dx.doi.org/10.1021/ie303065h | Ind. Eng. Chem. Res. 2013, 52, 5145−5160


Article

Figure 2. Model representation to segregate the watershed in different sections through a MFA approach.

different constraints along the watershed in all of the different periods.

proposed model formulation, whereas section 4 shows the applicability of the proposed model to two case studies. Finally, the conclusions of the paper are presented in section 5.

3. MODEL FORMULATION In the proposed model, time plays a very important role due to the seasonal variability of inputs, outputs, and transformational phenomena in the watershed. To consider these effects, differential equations are needed, and to implement them into an optimization approach a discretization method is required. The discretization methods yield large algebraic problems that make it difficult to implement them into an optimization approach. Instead of that, in this paper a multiperiod optimization approach is implemented to account for the variations of the system variables with respect to time. Several multiperiod optimization formulations have been proposed for designing and planning batch processes (see, for example, Sahinidis et al.,20 Paules and Floudas,21 Varvarezos et al.,22 Papalexandri and Pistikopoulos,23 Dedopoulos and Shah,24 Sahinidis and Liu,25 and Iyer and Grossmann26,27), which can be used as references to develop the model of the problem under study in this paper. The multiperiod model formulation for the optimal distributed treatment of industrial effluents discharged to

2. PROBLEM STATEMENT The problem addressed in this paper can be described as follows. Given a watershed (see Figure 1) where several discharges (i.e., domestic wastewater and industrial and agricultural effluents) and uses, as well as natural phenomena, impact the concentration of pollutants along the watershed in different ways that depend on seasonal changes and affect the system sustainability. The problem consists of determining the configuration and operation for a distributed system to treat the industrial effluents discharged to ensure the sustainability of the macroscopic system at the minimum total annual cost, taking into consideration different constraints for the water quality along the watershed and in the final reservoir as well as the seasonal changes for the different water discharges and uses. For tracking the pollutants through the watershed, a MFA model is implemented into a multiperiod formulation. The optimization goal is to determine the industrial effluents that must be treated, as well as the treatment units used, to yield a desired concentration in the final reservoir satisfying the 5147



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watersheds proposed in this paper is based on the general representation shown in Figure 2, where the time horizon is dividen in t periods (t = {1, ..., T}). Before presenting the model formulation, the following sets are defined: I represents the number of reaches, J represents the number of tributaries, L represents the number of pollutants, X represents the number of treatment units, W represents the number of industrial discharges, and T represents the time horizon (in this case, we used 12 months). To model the representation shown in Figure 2, a MFA model in a multiperiod framework is presented; this considers the changes in all inputs and outputs of the main river and tributaries for each period. The model takes into account the natural phenomena of precipitation, filtration, and evaporation; these phenomena alter significantly the composition of the materials transported by the watershed. To provide an adequate monitoring for the pollutants, the watershed is divided in sections (called reaches) in which the average concentration is considered constant (see Figure 2). The number of sections depends on the specific system and is determined by identifying zones where the average flow rate and concentrations remain constant (for example, sections where big tributaries cut the main river). According to this modeling approach, each river section can be considered an interconnected continuous stirred tank reactor (CSTR) in unsteady state. To model the watershed using the representation shown in Figure 2, the following equations are used. 3.1. Overall Mass Balances for Each Reach at Any Period. The volumetric flow output from each reach (Qi,t) is equal to the accumulation that exists at the previous period (Si,t−1) plus the flow rate inlet from the previous reach (Qi−1,t) minus the accumulation at actual period (Si,t) plus the summation of all industrial effluents discharged to this reach at this period (∑|W| w=1INDRi,t,w), direct discharges (Di,t), the flow rate for the temporal precipitations (Pi,t) and all tributaries discharged to the reach ∑Nj=1Ef l,iETi,j,t minus the loss due to natural phenomena of filtration and evaporation (Li,t) and the use of water in each reach at actual period (Ui,t).

Q i , t × CQ i , l , t = Si , t − 1 × CQ i , l , t − 1 − Si , t × CQ i , l , t + Q i − 1, t × CQ i − 1, l , t + Pi , t × CPi , l , t + Di , t × CDi , l , t |W |

+

∑ INDR i ,t ,w × CINDRati ,l ,t ,w w=1 NEfl , i

+ −

∑ ETi ,j , t × CETi , j , l , t − Li , t × CLi , l , t − Ui , t × CUi , l ,t i=1 Vi

∫V =0 ri ,t dVi ,t ,

i ∈ I , ∀ l ∈ L, ∀ t ∈ T

3.3. Overall Mass Balances for Tributaries at Any Period. The exit flow rate from each tributary (ETi,j,t) that discharges in the main river varies in the different periods due to the variations of the different discharges and uses with respect to time. The exit flow rate from the tributaries is equal to the sum of the treated wastewater discharges (WStreated i,j,t ), untreated wastewater discharges (WSuntreated ), industrial wastei,j,t water (INDTi,j,t,w), precipitations (Pi,j,t), and agricultural discharges (Di,j,t), minus the losses due to evaporation and filtration (Li,j,t) and the water used in that tributary at each period (Ui,j,t). |W |

ETi , j , t =

WSiuntreated ,j,t

+

WSitreated ,j ,t

+ Di , j , t − Li , j , t − Ui , j , t ,

∀ i ∈ I, j ∈ J, t ∈ T

,=1

(3)

ETi , j , t × CETi , j , l , t = WSiuntreated × CWSiuntreated ,j,t ,j ,l ,t |W |

+ WSitreated × CWSitreated ,j ,t ,j,l ,t +

∑ INDR i ,t ,w + Pi ,t + Di ,t

∑ INDTi ,j ,t ,w w=1

NEfl , j

∑ ETi ,j ,t − Li ,t − Ui ,t ,

∑ INDTi ,j ,t ,w + Pi ,j ,t

3.4. Component Balances in the Tributaries at Each Time Period. The pollutants leaving each tributary at any period (ETi,j,t × CETi,j,l,t) are equal to the sum of the inlet untreated pollutants to the tributary from untreated (WSi,j,t × treated treated CWSuntreated ) and treated (WS × CWS ) discharges, i,j,l,t i,j,t i,j,l,t industrial discharges (∑w |W| = 1 INDTi,j,t,w × CINDTati,j,l,t,w), precipitation (Pi,j,t × CPi,j,l,t), and direct discharges (Di,j,t × CDi,j,l,t), minus losses (Li,j,t × CLi,j,l,t), uses (Ui,j,t × CUi,j,l,t), and i,j,t the natural consumption (∫ VV=0 ri,j,l,t dVi,j,t) in the tributary in the considered period:

w=1

+

+

w=1

|W |

Q i , t = Si , t − 1 + Q i − 1, t − Si , t +

(2)

× CINDTat i , j , l , t , w + Pi , j , t × CPi , j , l , t + Di , j , t × CDi , j , l , t

∀ i ∈ I, ∀ t ∈ T (1)

− Li , j , t × CLi , j , l , t − Ui , j , t × CUi , j , l , t −

3.2. Overall Component Balances for Reaches at Any Period. The amount of pollutants (l) that exits each reach (Qi,t × CQi,l,t) is equal to the sum of the mass accumulated in each reach at the previous period (Si,t−1 × CQi,l,t−1) minus the pollutants accumulated at the present period (Si,t × CQi,l,t) plus the pollutants inlet from the previous reach (Qi−1,t × CQi−1,l,t), the pollutants carried out by the precipitation (Pi,t × CPi,l,t), the direct discharges (Di,t × CDi,l,t), the mass of pollutants carried out by the industrial discharges ((∑|W| w=1INDRi,t,w) × CINDRati,l,t,w), and the mass of the pollutants carried out by the tributaries (∑i N= Ef1l,iETi,j,t × CETi,j,l,t), subtracting the losses by filtration and evaporation (Li,t × CLi,l,t), uses (Ui,t × CUi,l,t), and the mass that reacts at actual period in the corresponding reach Vi (∫ V=0 ri,t dVi,t).

Vi , j , t

∫V =0 ri ,j ,l ,t dVi ,j ,t ,

∀ t ∈ T, ∀ i ∈ I, ∀ j ∈ J, ∀ l ∈ L

(4) Vi,j,t V=0 ri,j,l,t

In previous equation, the reactive term (∫ dVi,j,t) accounts for the natural degradation for the pollutants in each tributary, which considers the kinetic (ri,j,l,t) as well as the volume in the tributary at each period. 3.5. Agricultural Discharges and Uses at Each Period of the Operating Time. The agricultural discharges, Di,j,t, and uses, Ui,j,t, are proportional to the agricultural areas surrounding the watershed. These discharges and uses are calculated as follows:

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Di , j , t = λi , j , t × Ai , j ,

∀ i ∈ I, j ∈ J, t ∈ T

(5)

Ui , j , t = βi , j , t × Ai , j ,

∀ i ∈ I, j ∈ J, t ∈ T

(6)



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where λi,j,t is the flow rate per unit of area in m3/(acre s), which varies according to the agricultural cycle. βi,j,t is a parameter for the flow of water from tributary j used for irrigation, which depends on the season of the year and is given in units of m3/ (acre s). Finally, Ai,j is the area that covers the tributary j in acres. Notice that in the case studies considered in this paper, previous linear relationships between water discharged and used and the corresponding agricultural area are based on the experience and statistical reports (see references El-Baz et al.,10,11 Lovelady et al.,12 Lira-Barragán et al.,13−15 BurgaraMontero et al.,16 CONAGUA,17,18 and SEMARNAT19). However, the proposed model is able to handle different types of functions to relate the agricultural wastewater discharged and used, including the possibility to model such discharges and uses in terms of data (used in the model as parameters) taken from reported statistical information, or to include more complicated functions that correlate the agricultural wastewater discharges and uses as functions of the agricultural area as well as the type of crop, location, irrigation technique, etc. (considering linear or nonlinear relationships). Notice that linear correlations are more adequate for the solution of the optimization problem under consideration because they require a lower number of parameters; however, nonlinear relationships may provide more accurate results. Then, the type of correlation implemented depends on the accuracy required as well as the available information, and it will determine the complexity of the corresponding multiperiod mixed integer nonlinear programming (MINLP) problem. 3.6. Design of the Multiperiod Distributed Treatment System. In the past two decades, several methodologies for the optimal design of water-using networks have been reported (see, for examples, Wang and Smith,28 Doyle and Smith,29 Benko et al.,30 Savelski and Bagajewicz,31 and Teles et al.32), whereas other methodologies have been reported for the optimal treatment of wastewater streams inside the industrial facilities (see, for example, Kuo and Smith,33 Galan and Grossmann,34 Alva-Argaez et al.,35,36 Benko et al.,34 Savelski and Bagajewicz,37,38 Hernandez-Suarez et al.,39 Gunaratnam et al.,40 Gabriel and El-Halwagi,41 Karuppiah and Grossmann,42 Putra and Amminudin,43 Ng et al.,44 Ponce-Ortega et al.,45,46 and Nápoles-Rivera et al.47). In general, these methodologies found optimal solutions with distributed treatment configurations, where the industrial wastewater streams can be segregated, recycled, and reused inside the plant to reduce the associated treatment cost. However, only the water integration inside the industrial facilities was considered, thus neglecting the interactions occurring in the surrounding watershed between all of the effluents discharged to the environment. Taking into account this last aspect, in this paper a distributed macroscopic treatment for the industrial effluents to ensure system sustainability is proposed. Therefore, a formulation of a multiperiod distributed treatment system is developed to determine the treatment of industrial discharges released to the reaches and the tributaries guaranteeing the sustainability of the macroscopic system in all periods of the time horizon. This model is established through a disjunctive formulation that allows selection of the optimal treatment technologies. Figure 3 shows the schematic representation for the design of the distributed treatment system of the industrial effluents in the tributaries, where each effluent is segregated and sent to the different available treatment units to eliminate the pollutants

Figure 3. Schematic representation for the treatment of industrial effluents.

with different capital and operational costs and with different capabilities to remove them. 3.7. Treatment for the Industrial Effluents Discharged to the Tributaries. This way, the problem consists in determining the segregated flow rate of each industrial effluent at each period to be treated in each unit. To design the distributed treatment system, the following disjunction is used for each industrial effluent w discharged into the tributary j at each period t. ⎡ Yi , j , t , w ⎢ ⎢ INDTi , j , t , w = ∑ fsi , j , t , w , x ⎢ x∈X ⎢ ⎢ ∑ fs i , j , x , t , w(1 − αx , l)CINDTi , j , l , t , w ⎢ ⎢ x∈X ⎢ = INDTi , j , t , wCINDTat i , j , l , t , w , ∀ l ∈ ⎢ ⎤ Zi , j , x , t , w ⎢ ⎡ ⎥ ⎡ ¬Zi , j , x , t , w ⎤ ⎢ ⎢ ⎥ ⎢ ⎥, ⎢ ⎢ fsi , j , x , t , w ≥ Ωimin ,j,w,x ∨ ⎥ ⎢ fsi , j , x , t , w = 0 ⎥ ⎢ ⎢ ⎦ ⎥ ⎣ ⎢ ⎢⎣ fsi , j , x , t , w ≤ Ωimax ,j,w,x ⎦ ⎢ ⎣ ∀x∈X ⎡ ⎤ ¬Yi , j , t , w ⎢ ⎥ ∨ ⎢CINDTat i , j , l , t , w = CINDTi , j , l , t , w ⎥ , ⎢ ⎥ ⎢⎣ ⎥⎦ fsi , j , x , t , w = 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ L⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

∀ i ∈ I, ∀ j ∈ J, ∀ t ∈ T, ∀ w ∈ W

In previous disjunction, CINDTi,j,l,t,w is the concentration of the industrial effluents discharged to the tributaries without any treatment at each period, and CINDTati,j,l,t,w is the corresponding concentration after treatment. The Boolean variable Yi,j,t,w is associated with the requirement of the treatment unit in a given period. If this is true, the design equations for the treatment unit are applied. On the other hand, if it is false (¬Yi,j,t,w), the treatment unit is not required and therefore the concentration after treatment is equal to the inlet concentration. In the nested 5149



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disjunction, the Boolean variable Zi,j,x,t,w is associated with the existence of segregated flow in the treatment unit; when Zi,j,x,t,w is true, the segregated flow exists. The segregated flow must be greater than a lower limit, Ωmin i,j,w,x, and lower than an upper limit, Ωmax i,j,w,x, to be treated. To formulate the above disjunction as a set of algebraic equations, the Boolean variables are transformed into binary variables. Thus, when the Boolean variable is true, the associated binary variable takes the value of one, and when the Boolean variable is false, the associated binary variable is equal to zero. The above disjunction is modeled using the convex hull reformulation (see Raman and Grossman48 and Ponce-Ortega et al.49,50) to obtain the following equations. The continuous variables are segregated for each disjunction as follows:

yi , j , t , w + (1 − zi , j , x , t , w) ≥ 1, ∀ i ∈ I, ∀ j ∈ J, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W

Notice that the last two equations are not redundant; eq 14 is used for modeling the case when there is a treatment system and the associated variable binary yi,j,t,w is equal to one. In this case, at least one of the zi,j,x,t,w must be equal to one to satisfy the constraint given by eq 14. Furthermore, eq 15 is used for modeling the case when a treatment unit exists and the associated binary variable zi,j,x,t,w must be one; then, the treatment system must be installed at this location, so the corresponding binary variable yi,j,t,w must be equal to one, too. If we only consider eq 14, when zi,j,x,t,w is equal to one, then yi,j,t,w can be zero or one, since in this case there would not be a constraint that sets yi,j,t,w to one, which is done by eq 15 of the proposed model. Similarly, if eq 14 is eliminated from the model and only eq 15 is maintained, the following situation would occur: if yi,j,t,w is one, then zi,j,x,t,w can be zero or one as eq 15 does not oblige zi,j,x,t,w to take the value of one in this case. Therefore, such relationships are not redundant, so they both are required in the model. The fixed cost is independent of time, and the capital costs depend only on the maximum capacity required for all periods. Then, first a new binary variable is required to indicate that the treatment unit exists (z1i,j,x,w) if this is needed in at least one period:

CINDTat i , j , l , t , w = CINDTat id,1j , l , t , w + CINDTat id,2j , l , t , w , ∀ i ∈ I , ∀ j ∈ J , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

(7)

Upper limits are required for the disaggregated variables: CINDTat id,1j , l , t , w ≤ CINDTi , j , l , t , w × yi , j , t , w , ∀ i ∈ I , ∀ j ∈ J , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

(8)

Then, the equations of each disjunction are stated in terms of the disaggregated variables: yi , j , t , w × INDTi , j , t , w =

∑ fsi ,j ,x ,t ,w , x∈X

∀ i ∈ I, ∀ j ∈ J, ∀ t ∈ T, ∀ w ∈ W

(9)

∑ fsi ,j ,x ,t ,w × (1 − αx ,l) × CINDTi ,j ,l ,t ,w x∈X

= INDTi , j , t , w × CINDTat id,1j , l , t , w , ∀ i ∈ I , ∀ j ∈ J , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

zi1, j , x , w ≥ zi , j , x , t , w ,

(10)

∀ i ∈ I, ∀ j ∈ J, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W CINDTat id,2j , l , t , w

= CINDTi , j , l , t , w × (1 − yi , j , t , w ),

∀ i ∈ I , ∀ j ∈ J , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

fsi , j , x , t , w ≥ Ωimin , j , w , x × zi , j , x , t , w ,

fs1i , j , x , w ≥ fsi , j , x , t , w , ∀ i ∈ I, ∀ j ∈ J, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W

(12)

(13)

Equations 12 and 13 indicate the minimum and maximum flow rate for the segregated flows to be treated. Finally, the following logical relationships are needed to ensure that any treatment technology can be used only when the location for the treatment facility is selected to account for the installation costs: (1 − yi , j , t , w ) +

∑ zi ,j ,x ,t ,w ≥ 1, x

∀ i ∈ I, ∀ j ∈ J, ∀ t ∈ T, ∀ w ∈ W

(17)

3.8. Treatment for the Industrial Effluents Discharged to the Reaches. For designing a distributed treatment system of industrial effluents discharged into the reaches, another disjunction is needed for each industrial effluent w discharged to each section of watershed i. This disjunction is similar to that used for the treatment of the effluents discharged into the tributaries; the difference lies only in the location of the industrial effluent discharged. The superstructure used for designing the distributed treatment system for the industrial effluents discharged into the reaches is presented in Figure 3, and the corresponding disjunction is the following:

fsi , j , x , t , w ≤ Ωimax , j , w , x × zi , j , x , t , w , ∀ i ∈ I, ∀ j ∈ J, ∀ t ∈ T, ∀ w ∈ W

(16)

Then, to determine the capital cost associated with the treatment unit, only the maximum capacity used in all periods (fs1i,j,x,w) is considered:

(11)

The equations for the nested disjunction are stated as follows:

∀ i ∈ I, ∀ j ∈ J, ∀ t ∈ T, ∀ w ∈ W

(15)

(14) 5150


Industrial & Engineering Chemistry Research ⎡ Yri , t , w ⎢ ⎢ INDR i , t , w = ∑ fsri , t , w , x ⎢ x∈X ⎢ ⎢ ∑ fsr i , x , t , w(1 − αx , l)CINDR i , l , t , w ⎢ x∈X ⎢ ⎢ = INDR i , t , wCINDRat i , l , t , w , ∀ l ∈ ⎢ ⎤ Zri , x , t , w ⎢ ⎡ ⎥ ⎡ ¬Zri , x , t , w ⎤ ⎢ ⎢ ⎥∨⎢ ⎥, ⎢ ⎢ fsri , x , t , w ≥ Ωmin x ⎥ ⎢⎣ fsri , x , t , w = 0 ⎥⎦ ⎢ ⎢ ⎥ ⎢ ⎢⎣ fsri , x , t , w ≤ Ωmax x ⎦ ⎢ ⎣ ∀x∈X ⎡ ⎤ ¬Yri , t , w ⎢ ⎥ ∨ ⎢CINDRat i , l , t , w = CINDR i , l , t , w ⎥ , ⎢ ⎥ fsri , x , t , w = 0 ⎢⎣ ⎥⎦

Article

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ L⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

fsri , x , t , w ≥ Ωmin × zri , x , t , w, x ∀ i ∈ I, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W fsri , x , t , w ≤ Ωmax × zri , x , t , w, x ∀ i ∈ I, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W

(1 − yri , t , w) +

+

∀ i ∈ I, ∀ t ∈ T, ∀ w ∈ W

yri , t , w + (1 − zri , x , t , w) ≥ 1, ∀ i ∈ I, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W

zr1i , x , w ≥ zri , x , t , w,

(27)

(18)

fsr1i , x , w ≥ fsri , x , t , w, ∀ i ∈ I, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W

(19)

∑ fsri ,x ,t ,w, x∈X

OF = {min TAC; min CQdis}

(20)

x∈X

= INDR i , t , w × CINDRat id,1l , t , w , (21)

CINDRat id,2l , t , w ≤ CINDR i , l , t , w × (1 − yri , t , w), ∀ i ∈ I , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

(29)

These two goals contradict each other, which means that when the concentration in the final disposition decreases, the total annual cost for treating the industrial effluents increases; on the other hand, when the concentration of the effluent discharged to the final disposition is relaxed, the total annual cost decreases. The restrictions placed on the final disposal must be satisfied at all time periods; however, because the discharges and uses vary in each period, the treated flow rate can vary between the different periods, and this must be simultaneously optimized. The total annual cost is the sum of the capital and operational costs for the treatment units in the

∑ fsri ,x ,t ,w × (1 − αx ,l) × CINDR i ,l ,t ,w

∀ i ∈ I , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

(28)

3.9. Objective Function. The model is formulated as a multiobjective optimization problem in a multiperiod framework for the simultaneous minimization of the total annual cost and the concentration of the hazardous components discharged to the final disposal of the watershed as follows:

Then, the equations for each disjunction are fixed in terms of the segregated variables:

∀ i ∈ I, ∀ t ∈ T, ∀ w ∈ W

∀ i ∈ I, ∀ x ∈ X, ∀ t ∈ T, ∀ w ∈ W

The next relationship is required to determine the maximum flow rate treated at any period of time, and this flow rate is used to calculate the corresponding capital cost for the treatment unit:

CINDRat id,1l , t , w ≤ CINDR i , l , t , w × yri , t , w,

yri , t , w × INDR i , t , w =

(26)

Notice that these two previous relationships (eqs 25 and 26) are not redundant, and the same explanation given for relationships 14 and 15 applies. To consider the capital costs associated with the existence of the treatment units, if these units are required at any period (i.e., zri,x,t,w = 1), then the fixed capital costs must be considered (i.e., zr1i,x,w = 1); this is modeled as follows:

Upper limits for the disaggregated variables are needed:

∀ i ∈ I , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

(25)

In addition, when a technology is selected to treat an industrial effluent discharged to the tributary, the associated binary variable zri,x,t,w is one and then a treatment facility must be installed in this location and the corresponding binary variable yri,t,w must be one. This is modeled as follows:

CINDRat id,2l , t , w ,

∀ i ∈ I , ∀ l ∈ L, ∀ t ∈ T , ∀ w ∈ W

∑ zri ,x ,t ,w ≥ 1, x

In the preceding nested disjunction, CINDRi,l,t,w is the concentration for the industrial effluents discharged w to different sections of the watershed without treatment, and CINDRati,l,t,w is the corresponding concentration after treatment. In the first disjunction, the Boolean variable Yri,t,w is associated with the existence of the treatment unit; if it is true, the design equations apply, but if it is false (¬yri,l,t,w), the treatment does not exist; therefore, in this case, CINDRi,l,t,w is equal to CINDRati,l,t,w. In the nested disjunction, the Boolean variable Zri,x,t,w is associated with the existence of segregated flow in the treatment unit; when Zri,x,t,w is true, the segregated flows exist. Using the convex hull reformulation, the following algebraic equations are obtained. The continuous variables are segregated for each disjunction as follows: CINDRat i , l , t , w =

(24)

The following logical relationships are needed to ensure that any treatment technology can be used only when the location for the treatment is selected. First, when a site is selected to install a treatment facility, the binary variable yri,t,w is one and at least one technology zri,x,t,w must be selected; this is modeled as follows:

∀ i ∈ I, ∀ t ∈ T, ∀ w ∈ W

CINDRat id,1l , t , w

(23)

(22)

The equations for the nested disjunctions are stated as follows: 5151



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Figure 4. Drainage Bahr El-Baqar.

the set of Pareto optimal solutions. The basic strategy of this method is to convert the multiobjective problem into a series of problems with a single objective, selecting one objective function (in this case the TAC) and the other one is included in the model constraints as an upper bound (in this case, the discharge concentration CQdis in the final disposal). The procedure of calculating is repeated for several feasible upper bounds of the final concentration for the key pollutant in the catchment area. This strategy allows identification of the lowest cost for a given upper concentration in the final disposal and, at the same time, is able to determine the number of treatment units required, as well as the flow rates treated at each period.

distributed treatment system used throughout the time horizon. Therefore, the total annual cost is expressed as follows: TAC = k f × (∑ ∑

∑ ∑

[FCx × zi1, j , x , w + VCx × fs1i , j , x , w]γ

i∈I j∈J x∈X w∈W

+∑

∑ ∑

[FCx × zr1i , x , w + VCx × fsr1i , x , w]γ )

i∈I x∈X w∈W

+ HY × (∑ ∑

∑∑ ∑

CU op x × fsi , j , x , t , w

i∈I j∈J x∈X t∈T w∈W

+

∑∑∑ ∑ i∈I x∈X t∈T w∈W

CU op x × fsi , x , t , w)

(30)

In preceding equation, TAC is the total annual cost ($/year), which includes the capital cost for the treatment units required to treat the industrial effluents discharged to the tributaries and to the reaches plus the operational costs for the treatment units accounting for all periods, kf is the factor used to annualize the capital costs (year−1), HY is the number of operating hours for each period (hours/period), FCx is the fixed cost and VCx is the unit variable cost for interceptor x, CUop x is the unit cost for the operation of the treatment unit x, and finally γ is the exponent for the capital cost that it is used to consider the effect of the economies of scale. To solve this multiobjective optimization problem, the constraint method is used in this work (for details about the constraint method, see the work by Diwekar51) to determine

4. RESULTS AND DISCUSSIONS Two case studies are presented to show the applicability of the proposed model; these were solved as MINLP problems using the solver SBB included in the software GAMS (Brooke et al.52) using a computer with an i7-3520 M processor at 2.9 GHz and 8 GB of RAM. The first case considers two key components and corresponds to the drain Bahr El-Baqar located in Egypt. The second case also considers two components in the Balsas watershed, which is located in the southwestern part of Mexico. The optimization process depends on the amount of contaminants to be removed, so several treatments units with given configurations and operating 5152



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Table 1. Data for the Interceptors of Example 1 interceptor

fixed cost, $

unit variable cost, $/m3

efficiency for pollutant 1, α1


unit operational cost, $/m3

1 2 3 4

2,000 1,500 1,300 0

0.235 0.193 0.168 0

0.8 0.9 0.55 0

0.65 0.7 0.75 0

1.67 × 10−3 1.3 × 10−3 0.89 × 10−3 0

Figure 5. Pareto curve for example 1 constraining component A.

of 27 753 binary variables, 66 534 continuous variables, and 126 149 constraints. The approach then consists in determining the optimal set of Pareto solutions that trades-off both objectives (TAC and CQdis for each pollutant considered). To determine the Pareto solutions, the constraint method is implemented,51 and different curves for the pollutants considered are presented (notice that each solution satisfies the environmental constraints through the watershed for both pollutants considered). The solution of the MINLP problem for each point of the Pareto curve requires an average of 16 min. First, the Pareto curve for the associated cost of a given concentration of pollutant A in the final discharge is determined (see Figure 5). Each point of the Pareto curve indicates the number of treatment units (ZTOT) used throughout the operating time (12 months) including the total flow rate treated (FTOT) and the corresponding total annual cost. Notice in Figure 5 that the objectives TAC and CQ dis contradict each other; restricting the discharge concentration requires more treatment units, and this is reflected with a significant increment in the total annual cost. Notice in Figure 5 that the stricter environmental solution (located on the left-hand side) requires 18 additional treatment units and a total flow rate treated 8.24 m3/s greater than the more relaxed environmental solution for pollutant A at the final disposal (located on the right-hand side). Notice also in this Figure 5 that for a reduction from 1.893 ppm to 1.845 ppm of pollutant A at the final disposal, four additional treatment units are needed together with an increment of 34.4% in the total flow rate treated, which represents an increment of $142,350/ year in the TAC, whereas for a reduction of the concentration at the final disposal of pollutant A from 2.193 ppm to 2.043, just four additional treatment units are required with an increment in the total flow rate treated of 2.2 m3/s, which corresponds to an increment of $46,467/year in the TAC.

conditions were simulated to determine the removal efficiency, as well as their corresponding unit capital and operational costs. 4.1. Case Study 1. Lake Manzala is a shallow lake of brackish water; it has an area of approximately 1000 km2 and is located in the northeastern Nile Delta. This lake is separated from the Mediterranean Sea by a sandy crest. Contaminated water from agricultural, domestic, and industrial activities is transported by the drain Bahr El-Bar (see Figure 4) that ends in the south of Lake Manzala, which has caused negative impacts to human health due to economic activities such as fishing, cattle ranching, and farming. Phosphorus and sulfur are the key pollutants considered; these contaminants provide the conditions for the exponential growth of biomass decreasing the concentration of oxygen and causing eutrophication in water bodies. The discharges vary from period to period over the operating time (in this case one year); each period (in this case one month) is characterized by different flows and different concentrations of discharges from the various activities throughout the drainage. The characteristics of the drain Bahr El-Baqar for each period were used as input parameters for the multiperiod MINLP model described in this paper. The distributed treatment system is placed along the drain to remove the contaminants and to minimize the pollutant concentrations discharged into the final disposal. The initial concentration in the final disposal for the pollutants A and B (phosphorus and sulfur) are 2.193 and 1.955 ppm, respectively (Lovelady et al.12). Table 1 shows the data for the interceptors considered in this case study. The concentration at the final disposal must be lower than the one that naturally the system is able to degrade to prevent the accumulation of contaminants in the system assuring its sustainability. Without treatment, considering the different discharges and uses through the time horizon, the contaminants cannot be removed naturally, since the actual concentration exceeds the self-remediation capacity. This way, the proposed approach is implemented to solve this problem, which consists 5153



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Figure 6. Distributed treatment system for a discharge concentration of component A, CQdis < 1.843 ppm for example 1.

Figure 7. Pareto curve for Example 1 constraining component B.

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Figure 8. Distributed treatment system for a discharge concentration of component B CQdis < 1.595 ppm for Example 1.

B discharged to the catchment area at the initial conditions is 1.955 ppm. To reduce this concentration, it is necessary to place a distributed treatment system throughout the drainage considering the variability through the year. Each point on the Pareto curve of Figure 7 represents the associated total annual cost generated by the total number of treatment units (ZTOT) installed along the watershed and the total flow rate of industrial effluents treated (FTOT) during the operating time. Notice in Figure 7 that the strictest environmental solution (located on the left-hand side) represents 16 treatment units to treat 7.65 m3/s for a concentration of component B at the final disposal of 1.595 ppm. The solution for a concentration of component B at the final disposal of 1.64 ppm requires six treatment units less; also the flow rate treated decreases 21%, yielding a reduction in the TAC of 48%. Additionally, for decreasing the concentration of component B at the final disposal from 1.955 ppm to 1.79 ppm just five treatment units are required to handle a total flow rate of 2.442 m3/s, yielding a total cost of $68,759/year. This way, the Pareto curve shown in

Therefore, this curve shown in Figure 5 can be useful to select the best solution that trades-off the objectives considered. As an example, Figure 6 shows the configuration for the distributed treatment system (point A) required for satisfying a constraint of CQ dis ≥ 1.843 ppm for the discharge concentration at the final disposal for component A; the total annual cost associated with this case is $413,850/year, and the total number of treatment units required are 18 (from which 10 are interceptors type one, 4 are interceptors type two, and 4 are interceptors type three, with the corresponding efficiencies and costs reported in Table 1). In addition, the total flow rate treated in all periods is 8.241 m3/s; it is noteworthy that in this case 95% of the treated effluents are discharged into the reaches instead of the tributaries. Notice that the effluents treated are distributed along the watershed to satisfy the constraints placed through the watershed for the different water uses. On the other hand, the Pareto curve for trade-off of the TAC and CQdis for component B and satisfying the environmental constraints for components A and B through the watershed is presented in Figure 7. The initial concentration of component 5155



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contains several types of contaminants (minerals, organics, heavy metals, etc.); in this case study, two components are considered. Component one corresponds to heavy metals (key component 1), whereas component two corresponds to mercury (key component 2). These substances are very toxic and difficult to eliminate naturally and are easily accumulated in the tissue of living organisms in the ecosystem. Furthermore, they go from one to another organism in the food network increasing their concentration in each link, reaching the highest levels at the ends of the chain (human) and causing irreversible damage to human health. The purpose of this example is to determine the optimal configuration for the distributed treatment system of the industrial effluents to decrease the level of pollutants in the watershed at the minimum possible cost. Table 2 shows the data for the treatment units considered in this example, which include the fixed and unit variable capital costs, the efficiency to remove the pollutants, and the unit operational cost. The chemical interaction between the pollutants and the environment is represented by a first-order reaction with rate constants of k1 = 0.904 190 9 × 10−5/s and k2 = 0.883 245 2 × 10−5/s for the key pollutants 1 and 2, respectively. In addition, the factor used to annualize the capital costs, kf, is 0.1 year−1. Discharge concentrations in the final disposal area for the key pollutants at the initial conditions (t = 0) are 29,448 ppm and 35,946 ppm for component 1 and 2, respectively. It is noteworthy that these concentrations change through the different periods in the time horizon. The information provided by CONAGUA17,18 for precipitation, industrial discharges, domestic and agricultural discharges, evaporation, and filtration for each month was used as input data to the proposed MINLP model, which consists of 31 119 binary variables, 116 379 continuous variables, and 203 648 constraints. The time required for solving each point of the Pareto curve is 10 076 s. Three scenarios are presented to observe the behavior of the proposed model. The first scenario consists in minimizing the concentration of component 1 and leaving aside the concentration of component 2 in the final reservoir. The Pareto curve obtained for this scenario is shown in Figure 10, where each point presents the number of units needed to meet the restrictions with the associated TAC. Notice on the left-hand side of Figure 10 that when the discharge concentration is stricter, more treatment units are needed, increasing considerably the total annual cost, while on the right-hand side of Figure 10, the discharge concentrations are relaxed and a lower number of treatment units are needed, decreasing this way the total annual cost. Notice on the right-hand side of Figure 10 that there is a section where the concentration decreases significantly but the increment in the TAC is not significant. On the other hand, on the right-hand side of Figure 10 the TAC increases drastically for small decrements in the concentration of pollutant 1 at the final discharge. The second scenario consists in minimizing the concentration of component 1 and fixing an upper bound for the

Figure 7 can be useful for the decision maker to choose the solution that best balances the objectives considered. As an additional example, from the Pareto curve of Figure 7 is selected the point for a discharge concentration of the key component B of CQdis ≥ 1.595 ppm (point B) to represent schematically the configuration of the distributed treatment system obtained to satisfy this restriction during all the time of operation. Figure 8 shows the distributed treatment system, which consists of 16 treatment units with different removal efficiencies (one type 1, fourteen type 2, and one type 3) distributed throughout the watershed for treating a total flow of 7.65 m3/s, which represents a total annual cost of $418,420/ year. Notice that the number of treatment units for this case is lower than the one for the case in which the concentration of component A is decreased but the associated cost is similar. 4.2. Case Study 2. The Balsas watershed (CONAGUA17,18) is one of the largest and most important watersheds in Mexico with a surface of about 112.320 km2, which is a depression east−west in south-central Mexico. The Balsas watershed is divided into three regions: high, middle, and lower Balsas. For this case study, only the middle Balsas (see Figure 9) was considered because it corresponds to the place where

Figure 9. Middle part of the Balsas watershed (considered in example 2).

there are more industrial activities. To model the system, the most important inlet and outlet streams and the variation with respect to the season of the year are taken into account. Currently, the Balsas watershed is highly polluted because several industrial effluents are discharged to the system without adequate treatment, reducing its capacity for self-purification and allowing the accumulation of pollutants. This has severe effects on the flora and fauna, as well as in the life quality of the surrounding cities. Wastewater from industrial activities Table 2. Data for the Interceptors Used in Example 2 interceptor

fixed cost, $

unit variable cost, $/m3



unit operational cost, $/m3

1 2

1950 0

0.50 0

0.95 0

0.65 0

1.7 × 10−3 0

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Figure 10. Pareto curves for example 2.

Figure 11. Optimal configuration for the distributed treatment system in the Balsas watershed.

concentration of component 2 in the final reservoir (i.e., CQdis ≥ 25 ppm). Figure 10 shows the curve of optimal solutions to meet these constraints; as can be observed, the associated cost

is significantly higher than the one corresponding to the scenario 1. For example, for a concentration of component 1 of 18 ppm (corresponding to a concentration of component 2 of 5157



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discharges have a major influence on the optimal design of the system. Furthermore, the results indicate that the number of required treatment units increases significantly with respect to the desired concentration in the final discharge. It has also been observed that there are important trade-offs between the efficiency and cost of the treatment technologies. Finally, the chemical and biochemical phenomena occurring in the watersheds play an important role that can be synergized with the external treatment technologies.

29 ppm), 33 treatment units are required to meet this restriction and the total treated flow rate is 284 m3/s, which represents a total annual cost of $1.61 MM/year. For the second scenario, the concentration of component 1 is 15 ppm and that of component 2 is 25 ppm; to satisfy this restriction, 44 treatment units are employed with a total treated flow rate of 330.5 m3/s, and the total annual cost for this configuration is $1.71 MM/year. The total annual cost to satisfy the restriction of the second scenario increments 6% respect to the first scenario. The third scenario consists in minimizing the concentration of component 1 and fixing an upper bound for the concentration of component 2 in the catchment area of CQdis ≥ 30 ppm. Notice that this constraint for component 2 is more relaxed in scenario 3 compared with scenario 2. It is noteworthy that the Pareto curve corresponding to scenario 3 is similar to the one corresponding to scenario 1; this is because the corresponding constraint for component 2 is not activated because of the stricter constraints for component 1. It should be noted in Figure 10 that for stricter constraints of the pollutant concentrations in the final catchment area, additional treatment units are required, which, in turn, yields higher total annual costs; in addition, this cost increases exponentially with respect to the reduction in the concentration in the catchment area. This way, the Pareto curve shown in Figure 10 can be useful for the decision maker to choose the solution that best balances the opposite objectives. Point I taken from the curve of Figure 10 was obtained from the third scenario ,and the corresponding configuration of the treatment system is shown in Figure 11 to meet the discharge concentrations of components 1 and 2, respectively. The concentration discharged for component 1 is 20 ppm and for component 2 is 30 ppm; to satisfy the constraints, 21 treatment units are required, of which 17 units are used to treat effluents discharged into the tributaries and only 4 units are used to treat effluents discharged into the reaches, and the total treated flow rate is 137 m3/s, generating a TAC of $8.14 MM/year. Notice in Figure 11 that the treated flow rate changes in the different periods of the year because the concentrations and volumes discharged also vary in the different periods.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge the financial support from the Mexican Council of Science and Technology (CONACYT), as well as the Council of Scientific Research of the Universidad Michoacana de San Nicolas de Hidalgo.

■

NOMENCLATURE

Greek Symbols

α βi,j,t λi,j,t Ω

efficiency factor to remove the pollutant for the interceptor j. agricultural use of water from tributary i, m3/(ha s) agricultural flow rate per area, m3/(ha s) small number

Indexes

i j l x t w

reach tributary pollutant interceptor time period industrial discharge

Parameters

Ai,j CDi,j,t

5. CONCLUSIONS This paper has presented a multiobjective multiperiod MINLP optimization model for the optimal design of distributed treatment systems for industrial effluents discharged to watersheds. The model considers the seasonality effects of the different wastewater discharges and uses and accounts for the changes in the system performance over time and throughout the watershed. This seasonality aspect also affects the magnitude of water accumulation in the different sections of the watershed, which can significantly alter the self-purification capacity of watersheds. The MFA portion of the model tracks the flows and concentrations over spatial and temporal coordinates and provides a linkage among the various optimization variables. Two case studies have been solved to show the applicability of the proposed model for the sustainable design of distributed treatment systems of the effluents discharged to the watersheds. One case study deals with the Bahr El-Baqar drain that discharges to Lake Manzala located in Egypt and connected to the Mediterranean Sea. The other one involves the Balsas watershed located in Mexico that discharges to the Pacific Ocean. The results show that the seasonality effects of the

CLi,t CLi,j,t CPi,j,t CQi−1,t CSuntreated i,j,t CStreated i,j,t CUi,j,t Di,t Di,j,t ETi,j,t FC HY INDi,j,t 5158

area covered by effluent j in reach i, acre or ha concentration of agricultural discharges to tributary j to reach i, ppm concentration of total losses (filtration and evaporation) from reach i, ppm concentration of total losses (filtration and evaporation) from tributary j, ppm concentration of precipitation discharged to tributary j to reach i, ppm concentration of flow rate inlet to reach i, ppm concentration of residual wastewater discharged without treatment to reach i for tributary j, ppm concentration of residual treated wastewater discharged to reach i for tributary j, ppm concentration of water used from tributary j discharge to reach i, ppm direct discharges to the reach i, m3/s agricultural discharges to tributary j associated with reach i, m3/s discharge to tributary j associated with reach i, m3/s fixed cost for interceptor x, $ operation time per year, h/year industrial discharge to tributary j associated with reach i, m3/s dx.doi.org/10.1021/ie303065h | Ind. Eng. Chem. Res. 2013, 52, 5145−5160

Industrial & Engineering Chemistry Research k kf Li,j,t Li,t Ni Pi,t Pi,j,t Qi−1,t Suntreated i,j,t Streated i,j,t Ui,j,t Ui,t Vi,t Vi,j,t VCγ

Article

fs1i,j,x,w

kinetic constant for the degradation of the pollutant in the system factor used to annualize the capital costs, year−1 total losses (filtration and evaporation) for tributary j, m3/s total losses (filtration and evaporation) for reach i, m3/s total number of reaches precipitation discharged to reach i, m3/s precipitation discharged to tributary j associated with reach i, m3/s inlet flow rate to reach i, m3/s residual wastewater discharged without treatment to reach i from tributary j, m3/s residual treated wastewater discharged to reach i for tributary j, m3/s water used from tributary j associated with reach i, m3/s water used from reach i, m3/s volume for reach i, m3 volume for tributary j associated with reach i, m3 unit variable capital cost for interceptors, $/m3

fsri,x,t,w 1 fsri,x,w

ri,l,t ri,j,l,t Qi,t TAC yi,j,t,w zi,j,x,t,w z1i,j,x,w

Sets

I set for reaches J set for tributaries L set for pollutants W set for industrial discharges X set for treatment units T set for time periods

yri,t,w zri,x,t,w

Symbols

1 zri,x,w

FTOT total flow of industrial effluents treated in the time horizon WWTP waste water treatment plant ZTOT total number of treatment units installed in the time horizon

■

Variables

CETi,j,l,t CINDTati,j,l,t,w CINDRati,j,l,t,w CINDTatd1 i,j,l,t,w d1 CINDRati,j,l,t,w

CINDTatd2 i,j,l,t,w d2 CINDRati,j,l,t,w

dis CQi,l,t

CQi,t CUi,l,t fsi,j,x,t,w

segregated flow rate from the industrial effluents discharged into tributary j used to determine the variable capital cost. segregated flow rate from industrial effluents discharged into reach i and sent to interceptor x. segregated flow rate from the industrial effluents discharged into reach i used to determine the variable capital cost. reaction carried out in reach i. reaction carried out in tributary j that discharges to reach i. exit flow rate from reach i, m3/s total annual cost, $/year binary variable associated with the existence of the treatment plant used for the industrial effluent discharged in tributary j binary variable associated with the existence of the interceptor x used for the industrial effluent discharged in the tributary j binary variable used for activating the fixed capital cost for installing the treatment unit for the industrial effluent discharged in tributary j in all operation time binary variable associated with the existence of the treatment plant used for the industrial effluent discharged in reach i binary variable associated with the existence of the interceptor x used for the industrial effluent discharged in reach i binary variable used for activating the fixed capital cost for installing the treatment unit for the industrial effluent discharged in the reach in all operation time

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concentration discharged for tributary j to reach i, ppm concentration for the industrial effluent discharged into the tributary after treatment, ppm concentration of industrial effluent discharged into the reach after treatment, ppm concentration of industrial effluent discharged into the tributary after treatment, disaggregated in the first disjunction, ppm concentration of industrial effluent discharged into the reach after treatment, disaggregated in the first disjunction, ppm concentration of industrial effluent discharged into the tributary after treatment, disaggregated in the second disjunction, ppm concentration of industrial effluent discharged into the reach after treatment, disaggregated in the second disjunction, ppm pollutant concentration at the final disposal, ppm concentration of flow rate exit from reach i, ppm concentration of water used from reach i, ppm segregated flow rate from the industrial effluents discharged into tributary j and sent to interceptor x. 5159



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