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An MINLP Model for Sustainable Water Management in Macroscopic Systems: Integrating Optimal Resources Management to the Synthesis of Distributed Treatment Systems Jaime Garibay-Rodriguez, Vicente Rico-Ramírez, and José María Ponce-Ortega ACS Sustainable Chem. Eng., Just Accepted Manuscript • DOI: 10.1021/ acssuschemeng.6b02128 • Publication Date (Web): 23 Jan 2017 Downloaded from http://pubs.acs.org on January 24, 2017

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ACS Sustainable Chemistry & Engineering

An MINLP Model for Sustainable Water Management in Macroscopic Systems: Integrating Optimal Resources Management to the Synthesis of Distributed Treatment Systems Jaime Garibay-Rodriguez Instituto Tecnologico de Celaya, Departamento de Ingenieria Quimica, Av. Tecnologico y Garcia Cubas S/N, Celaya, Guanajuato, Mexico 38010 [email protected]

Vicente Rico-Ramirez* Instituto Tecnologico de Celaya, Departamento de Ingenieria Quimica, Av. Tecnologico y Garcia Cubas S/N, Celaya, Guanajuato, Mexico 38010 [email protected]

Jose M. Ponce-Ortega Universidad Michoacana de San Nicolas de Hidalgo, Departamento de Ingenieria Quimica, General Francisco J. Mugica S/N, Morelia, Michoacan, Mexico, 58060 [email protected] *Corresponding author: Vicente Rico-Ramirez Tel. +52 461-6117575 x 5579 Fax. +52 461-6117744

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Abstract Recognizing the growing pressure on water resources, literature reports several efforts in the area of mathematical programming to deal with the management of industrial and macroscopic water systems. This paper presents a mathematical programming model which integrates two strategies for sustainable water management. On the one hand, the model allows finding an optimal schedule for the distribution and storage of natural and alternative water sources to satisfy the demands of different users in a macroscopic system, while maintaining sustainable levels of water in the natural water resources. On the other hand, optimal decisions also involve the number, capacity, type and location of treatment units in a macroscopic system. Our approach results in an MINLP multi-period model which has been solved through the GAMS® modeling environment. A case study with different scenarios shows the scope of the proposed approach and the significance of the results.

Keywords: Sustainable water management, mathematical programming, distributed water treatment systems, MINLP formulation, macroscopic systems.


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INTRODUCTION One of the many concerns for the sustainable development of the current and the future generations, regarding many challenges such as population growth and climate change, is the need for a safe and adequate supply of fresh water. A 40% deficit over the world’s water supply is projected under the current management and consumption by the year 20301. Furthermore, the increasing demand of water to produce energy, and the lack of control of waste generation and disposal over the water bodies, only exacerbate the problem of reduced water supply capability2. It is currently believed that there is enough fresh water available in the world to meet all of the demands from the different sectors of society3. However, in many regions, due to climate and/or concentration of population in bigger cities, the natural supply of water rarely meets the demand. To face these challenges, there have been many efforts in the area of mathematical programming to assess the feasibility of implementing different technologies to obtain water from nonconventional sources4-11. These non-conventional sources include the use of desalinated water, rainwater harvesting and wastewater reuse. The above approaches have in common that, no matter the water source, the problem to be solved is one about management of the resources, where the goal is to satisfy the demands of a given system at a minimum cost. One of these approaches8 considers the use of water collected from rain in addition to the conventional water sources, such as dams and deep wells, in order to satisfy the demand in a macroscopic system. Another challenge about water management is the one concerning wastewater. A substantial effort has been made in the area of Process Systems Engineering to reduce the environmental impact over the water bodies12-22. Burgara-Montero et al.17 proposed a distributed treatment



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system throughout a watershed to ensure the sustainability, concerning the pollution levels of the natural bodies of water. This approach uses the MFA (Material Flow Analysis) technique to assess the system. In the analysis of these distributed treatment systems, one thing that has not been considered is the use of treated water as reclaimed water for different purposes in the watershed. Therefore, this paper proposes the integration of a rain water collecting system to the optimal resources management as well as a distributed system to treat the effluents generated in a watershed. The objective of this integration is to assess the global impact of the usage and disposal of water in the whole extent of a watershed. Through this assessment, the main goal is to find the optimal distribution of the resources. That goal includes minimizing the total investment in the water storage technologies as well as the treatment technologies which can satisfy the demand of water, reducing the water consumption from the natural sources as well as the pollution levels in the natural water bodies. Scope Literature reports mathematical programming formulations for both i) the implementation of distributed treatment systems and ii) the efficient management of water resources (including water harvesting strategies). Actually, those strategies could be applied sequentially to a macroscopic water system in order to achieve a sustainable solution. This work, however, integrates both of those sustainable water management strategies into a single mathematical formulation. This approach has not been reported in previous works. The formulation not only involves a larger number of constraints but also gets more complex due to the nonlinearities resulting from the integration; for instance, linear water balance constraints used in previous formulations become nonlinear (bilinear) since now both the flows as well as the compositions of a large number of streams are unknown. 4 ACS Paragon Plus Environment

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The basic ideas provided by Napoles-Rivera et al.8 were used to formulate some aspects of the water management and the rainwater harvesting systems. They introduced a novel mathematical programming approach to assess a rainwater harvesting system at a macroscopic level. Their assessment includes the optimal rainwater harvesting infrastructure needed to fulfill the water demand in a macroscopic system (e.g. city or municipality) at the minimum cost. On the other hand, Burgara-Montero et al.17 proposed a general framework to implement a distributed treatment system to mitigate the environmental impact of the wastewater over the water bodies in a watershed. Their ideas were used to formulate the wasterwater treatment system in this paper. So, even though both ideas have been evaluated separately by using mathematical programming formulations, our approach is focused on the integration of these two strategies. Through the integration of these approaches, we seek to provide a more efficient strategy to mitigate the harmful consequences of unnapropiate water management and wastewater disposal. This results in an new extended MINLP model. A direct advantage of an integrated formulation is that we could show the interactions among the various systems and processes involved and the individual significance or impact of each of them.

PROBLEM STATEMENT A general description of the addressed system is shown in Figure 1. To account for the seasonal variability of the water demands and the quantity of rain, the problem is divided in periods of time that could correspond to one or several years divided in months. The system involves a watershed as a natural drainage system for the different users (e.g., domestic, industrial and agricultural). These users withdraw water from various natural sources (e.g., dams, springs and



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deep wells) as well as from installed storage tanks and/or artificial ponds with an amount of water harvested from rain in a current or a previous period of time. In the context of this paper, a watershed is a hydrological macroscopic network of users and water bodies. The users and the water bodies interact with each other through the withdrawal of the available water resources and the disposal of wastewater streams generated by the different activities of the users.

Our

formulation is intended to be applicable to regions where there are water over-exploitation problems. In such regions, the natural water resources may be sufficient now, but there will be water scarcity in the future (unsustainable water management) if the community keeps relying only on natural resources. The above description corresponds to one component of the whole system in which there are two goals: i) reduce as much as possible the fresh water consumption from natural sources and ii) minimize the total costs. For the reduction of the total fresh water consumed from natural sources, the best solution, without considering the economy of the system, would be to install every possible storage tank or artificial pond available, and also, to use as much reclaimed water as possible. The goal, however, is to provide decision makers with a wide range of different solutions, so that they can choose an appropriate solution with the desired compromise between the environmental and the economic objectives. The second component of the global system is introduced as a consequence of the first one. Once the water is used by the domestic and the industrial users, it has to be treated to avoid the accumulation of hazardous pollutants in the watershed. The amount and type of pollutants present in the wastewater depend on the source of usage and has a complex composition. Domestic wastewater usually contains non-toxic organic compounds (e.g., proteins, amines and


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carbohydrates), whereas industrial wastewater, besides containing non-toxic organic compounds, can also contain toxic metals (e.g. arsenic, chromium and mercury). To account for all the inlets and outlets, as well as the chemical and biochemical phenomena occurring in the watershed, an MFA technique is used. Along with the MFA, a division of the watershed in sections, called “reaches”, is used to track the flows and compositions of the main river. In practice, the division is made in the main river of the watershed so the reaches would correspond to sections of a river. The sections of the main river also contain a number of tributaries and a number of domestic, industrial and agricultural areas, which are near to them, and are bounded by the limits of the river section, thus defining one reach. The complete resulting model is a multi-period MINLP model. The primary goal of the model is the minimization of the total costs while achieving sustainable limits on the water consumption from fresh sources and on the concentration of the pollutants in the watershed. In addition, given the associated costs in the system (rainwater harvesting devices, storage tanks, artificial ponds, piping and pumping, treatment facilities), the solution of the problem will also provide the number, size and location of the storage tanks and artificial ponds for the rainwater harvesting system and the number, type, location and size of the treatment units that minimizes the costs of installation and operation. MATHEMATICAL MODEL Model assumptions The formulation of the various constraints of the model is based on the following considerations: •

The system is at steady state and can be modeled by using the MFA technique.



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•

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The system can be represented by a deterministic model based on the most probable scenario from statistical reports and no sources of uncertainty are considered.

•

The treatment units can be described by using efficiency parameters.

•

The concentration of the selected contaminant in each section of the river is considered as uniform.

•

Transportation cost parameters are estimated through the optimum design of pipes and pumps in the water networks. These parameters correspond to averaged costs (depending upon the place where the water is sent from and its destination) per unit of flow.

Figure 2 shows a schematic representation of one reach. The sets of the model are represented with uppercase letters. The elements in the set (indexes) are represented with the corresponding lowercase letters of each set. R is used to represent the reaches of the watershed (r represents the elements of the set in the model). For every reach r there are a number of tributaries (J) that discharge to the main river, a number of domestic/industrial sinks (both represented with the set U) and a number of agricultural sinks (A) (the sinks can have one or more individual industrial, domestic or agricultural facilities respectively). The number and type of pollutant is represented with the set L. In addition, for the distributed treatment system, X stands for the set of interceptors, which represents individual interceptor x of the treatment unit (this concept will be explained further). For the rain harvesting system, S represents the set of the storage tanks, and P the set of the artificial ponds. The set T represents the period of time. Water consumption from natural sources Every sink in the watershed can satisfy its demands through consumption from natural sources, harvested rainwater from storage tanks and artificial ponds and, only for agricultural sinks, reused water from the treatment units. Equation (1) shows the main balance for the total water 8 ACS Paragon Plus Environment

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consumption from natural sources. This equation will be used to set a limit on the water consumption from natural sources. Note that the balance represents that all the water from natural sources is not withdrawn from the same source; many different sources, such as well or dams, can satisfy the water demand for one user (e.g. a city). The terms

and

represent the total amount of water used in the domestic/industrial sinks and agricultural sinks, respectively, not the number of natural sources from which these sinks withdraw water. (1)

The term

represents the summation of all the natural water consumed in every sink over a

period of time t. Rain balance in the storage system Rain is seasonal and location dependent. In this context, the total rainwater harvested directly depends on the harvesting area, the runoff coefficient of the roofs of houses and buildings and the amount of precipitation. The balance for the rainwater harvested in every reach and every storage device is given in the following equations: (2) (3) where time t.

is the total rainwater available in the reach r, for the storage tank s over the period of is used to represent the quantity of rain in the reach for the time period,

represents the area available for every storage tank s and

is the runoff coefficient. Equation

(3) is analogous to Equation (2) for the artificial ponds. 9 ACS Paragon Plus Environment


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Once the total amount of rainwater available in every reach is calculated, the balance for the storage tanks and artificial ponds are given by the following equations: (4) (5) where

and

are variables that represent the water sent to the storage tanks and artificial

ponds, respectively. The last terms of Equations (4) and (5) are used to model the quantity of water that cannot be stored over a period of time t. Balance in storage tanks There are a number of available storage tanks s that can be installed in every reach r. The model formulation allows finding the specific period of time in which these tanks must be installed and/or used. The mass balance in the storage tanks is given by the following equation: (6)

where

is the amount of water stored in the storage tank for the current period of time, is water accumulated from the previous period and

, and

are variables that

represent the amount of water sent to the agricultural and domestic/industrial sinks, respectively.

Balance in artificial ponds


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The mass balance for artificial ponds is analogous to the balance for storage tanks. It is assumed that artificial ponds can only be installed in zones where there is enough area available, typically in locations outside the urban regions that do not interfere with the current uses of the territory for another purposes. Piping and pumping costs for sending water to the urban areas from artificial ponds are greater than those required from storage tanks. The mass balance for artificial ponds is given by: (7)

where

represents the amount of water available in the artificial pond. The terms

and

are variables that represent the amount of water sent to the agricultural

domestic/industrial sinks, respectively. Design of the rainwater harvesting and storage system A set of disjunctions is given for the decision of installing storage tanks and artificial ponds in terms of new variables (

and

). Disjunctions (8) and (9) model the decision of

installing or not a storage tank or artificial pond, respectively.

(8)

(9)



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where

and

In Equation (8), if

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are the lower and upper bounds for the capacity of the artificial ponds. is true, then a storage tank is installed for the reach r, in the location s

over the period of time t. In this case, the capacity of the storage tank must lie between the upper limit (

) and the lower limit (

). Also, the associated cost of installation, which includes

the cost of gutters, pipes, pumps and the appropriate treatment, is activated ( contrary, if the storage tank is not required, the variable

is false, and the term

). On the is equal

to zero, indicating that for that reach and in that location, there is not a storage tank installed or used. Although the cost functions in Equations (8) and (9) should be nonlinear, an acceptable approximation of these expressions can be accomplished by assuming that the cost functions are linear. Thus, the linear cost functions are formulated as follows: (10) (11) The variables

and

define the maximum capacity allowed for the storage devices.

The following inequalities must hold: (12) (13) (14)


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(15) The first two inequalities indicate that the amount of water stored over any period of time must be lower than the maximum capacity of the storage device. The last two inequalities state that the storage devices cannot receive more water than the maximum capacity of the storage devices over any period of time. Finally, the previous disjunctions can be reformulated using the “Big-M” reformulation23: (16) (17) (18) (19) (20) (21) The decision of installing or not installing a storage device depends on two factors: the first one is the compliance of the demands of water in the watershed; the second one is related to the associated cost of each storage device.

Balance in domestic and industrial sinks



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The demand of water in each domestic/industrial sink can be met through water from natural sources, as well as harvested rainwater. The mass balance for domestic/industrial sinks is given by: (22)

where

is a known parameter of water demand, which is divided in two terms in Equation

(23). The term

represents the consumed water and

stands for the wastewater.

For modeling purposes, it is assumed that 50% of the total demand is consumed in the sinks and the rest is disposed. This assumption mainly affects the consumption from natural resources. The assumption is justified by both water purification for drinking and losses due to leaks and other uses (which cannot be reclaimed, so that water has to be replenished from natural resources). (23) Balance in agricultural sinks The mass balance for agricultural sinks is given by Equation (24). The difference between the agricultural and the domestic/industrial balances is that, for agricultural sinks, a new variable is introduced to account for the usage of reclaimed water. The term

represents the amount

of reclaimed water sent to an agricultural sink associated to a domestic/industrial treatment unit. It is assumed that all the water sent to the agricultural sinks is consumed in the processes. (24)

Balance in reaches


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The inlets (treated or non-treated wastewater from domestic and industrial sinks, tributary contribution and precipitation) and outlets (direct usage and evaporation or filtration loss) are accounted for in the mass balance for every reach, defined in the following equation: (25)

where

is the total flow of the reach. The flow of the previous section is represented by

and the contribution of precipitation and tributaries are represented with

and

respectively. The wastewater discharges to the reach are represented by

, . The term

is used to model the excess of reclaimed water that cannot be used by the agricultural users over a period of time. Finally, reach and

represents the amount of water used directly from the

is used to account for the evaporation and/or filtration losses.

For the purpose of ensuring the quality of the water in the watershed, a component balance in the reaches is used. In this balance (Equation (26)) every term of Equation (25) is multiplied by a term of concentration of pollutants and

is used to model the natural degradation of

pollutants by chemical and biochemical phenomena occurring in the water bodies.

(26)



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The natural degradation of pollutants in the water bodies is usually modeled as a first order kinetic reaction. Thus,

, where kr,l is the Arrhenius constant for

the natural degradation of the pollutants and Vr,t is the volume of the main river in the reach at a period of time t. Please note that, the term

can be used to model complex phenomena

occurring in the water bodies with different kinetic relationships. Design of the distributed treatment system There are a number of different effluents that discharge pollutants throughout the watershed. To satisfy the environmental constraints of the system, it is necessary to treat these effluents with the appropriate technology. The optimization formulation proposed in this work must determine the location where the treatment is needed, as well as the magnitude of the flow to be treated and the type of technology necessary. This can be achieved through a distributed treatment system, in which there is a possible treatment unit in every effluent generated by the domestic and industrial sinks. Each treatment unit consists of a series of individual facilities called interceptors. Each interceptor can use a different type of technology to treat wastewater and, as consequence, the cost and efficiency of removal of the different type of pollutants varies in every one of them. The treated wastewater can be reused as reclaimed water for the agricultural sinks. However, the concentration of pollutants for the reclaimed water must be lower than a given parameter that ensures its safety. This adds a decision to be made in the design of the distributed treatment system, in which the treatment units and number and type of interceptors not only are used to maintain adequate levels of pollution in the watershed, but they are also used to reduce the water consumption from fresh sources through the usage of reclaimed water. Figure 3 shows the design of the distributed treatment system. 16 ACS Paragon Plus Environment

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(27)

In the previous disjunction, Equation (27),

is used to represent the segregated flow into

interceptor x. The segregated flow is the fraction of the total amount of the wastewater stream that is treated in one interceptor. The efficiency of removal of pollutants of each interceptor is model with the parameter

. The term

effluents before the treatment and

stands for the concentration of the domestic is a variable that represents the concentration of

pollutants after the treatment. There are three Boolean variables in the disjunction. In the first level, the variable

is used to

model the existence or non-existence of the treatment unit for the associated domestic sink. If is true, the design equations for the interceptor network are activated. If

is false, the

outlet concentration is equal to the inlet concentration, there are no segregated flows, the variable for reclaimed water (

) is equal to zero and

wastewater generated in the domestic sink

is equal to the total amount of .



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In the second level of the disjunction, the Boolean variable

is activated. If

the flow of each segregated effluent lies between the lower limit ; on the other hand, if Boolean variable

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is true,

and the upper limit

is false, the segregated flow is equal to zero. Finally, the

is used to model the usage of reclaimed water. If

is true, the

treated concentration must be lower than the maximum concentration of pollutant allowed for reclaimed water

, the direct discharge of wastewater to the reach is equal to zero

and the total flow of treated wastewater is equal to the available reclaimed water from domestic sinks

. If

is false, the concentration achieved in the interceptors is not

sufficient to consider the use of reclaimed water; hence the variable

is equal to zero and

is equal to the total amount of the wastewater generated in the domestic sink

.

Equation (27) is reformulated using the Convex-hull reformulation24 as follows: (28) (29) (30) (31)

(32)

(33)


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(34)

(35)

(36) (37) In Equation (35),

is a positive integer number greater than one. Additionally, the following

constraints are used to ensure that only when a treatment unit is selected, there could be segregated flows and reclaimed water streams: (38)

(39)

(40) In addition to Equation (27), the following relationship is used to model the amount of water that is sent to the agricultural sinks. The term agricultural sink and the term

represents the amount of water sent to each

represents the amount of reclaimed water that cannot be

used in any period of time. (41)

The following restriction is used to activate the fixed cost of each interceptor:



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(42) where

is a binary variable that multiplies the fixed cost parameter of each interceptor and

activates such fixed cost once the interceptor is installed (See Equation (44)). Costs The capital cost of the treatment units, the rain harvesting and the storage system are the main contributors to the Total Annual Cost (TAC). Furthermore, piping and pumping costs for the rainwater harvesting system and for the usage of reclaimed water are considered. The TAC is the summation of the treatment costs for the domestic and industrial wastewater, the installation of the storage tanks and artificial ponds and the piping and pumping costs as follows: (43) Treatment costs The fixed and variable costs as well as the operational costs of the treatment units for the domestic/industrial wastewater are given in Equation (44). (44)

and

are coefficients for the capital costs (fixed and variable) of interceptor x, and

is the operational cost of interceptor x. The exponent

is used to consider the economies

of scale. In our analysis we are assuming that a total time period of one year is divided in months. Therefore, the cost relationships are defined as annual cost relationships ($/year). If the case-study considers a smaller or a larger number of periods, adjustments to the relationships


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would have to be made in order to estimate the annual cost for every year or the averaged annual cost for all of the years considered. Rainwater harvesting and storage system costs The total annual cost of installation of storage tanks and artificial ponds in the watershed is equal to the total sum of the previously defined cost functions for every reach. (45)

Piping and pumping costs The piping and pumping costs include i) the costs for sending water from the storage tanks and artificial ponds to the sinks ( units to the agricultural sinks

, ii) the cost for sending reclaimed water from the treatment and iii) the costs for the extraction, treatment and

distribution of water from fresh sources rainwater harvesting system (

. The piping and pumping costs for the

) is given by the following equation:

(46)

and

are the piping and pumping costs per flow unit to send water from a

storage tank s to the agricultural and domestic/industrial, respectively. The terms and

are parameters that account for the piping and pumping costs associated to the

distribution of water from the artificial ponds. The following equation defines the piping and pumping costs for reclaimed water: 21 ACS Paragon Plus Environment


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(47)

is the cost per flow unit to send water from a treatment unit associated to a domestic/industrial sink u to an agricultural sink a in a reach r. The costs that account for the extraction, treatment and distribution of water from fresh sources are given in the following equation: (48)

is the unitary cost per flow unit for domestic/industrial users and

is the

unitary cost for agricultural users. The total piping and pumping cost is given by the following equation: (49) Objective function The goal of our formulation involves the minimization of the TAC while achieving sustainable limits on two environmental constraints: i) the total amount of water consumed from natural sources (TotalFWC) and ii) the concentration of pollutants in the final disposal (CQdischarge) . The objective function is then expressed in Equation (49). Observe that the constraint method25 will be used as our approach to obtain a reasonable compromise among the economic objective and the environmental constraints (ϵ1 and ϵ2).

(50)


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where the TAC has been previously defined (Equation (43)). The TotalFWC and the CQdischarge are defined by Equations (51) and (52), respectively. (51)

(52) Remarks concerning the model formulation are as follows: •

The proposed model is a non-convex MINLP problem. It allows selecting the optimal configuration of a distributed system in a watershed that deals with domestic and industrial effluents (discharged to the main river) that can be used as reclaimed water for agricultural users. In addition, the model allows finding the optimal distribution of storage tanks and artificial ponds that help reduce the consumption of water from natural sources.

•

The solution strategy allows identifying a set of solutions that simultaneously considers the two environmental constraints (

,

) and the economic objective (Min

). Notice that a region concerned with water-stress issues may seek to reduce its over-exploitation of natural resources to avoid water-scarcity in the future. Further, a system conformed by a large population will have large flows of wastewater, which can be very difficult/expensive to treat to completely remove its pollutants. The ideal value of pollutants in the treated wastewater/river should be zero; however, wastewater treatment plants usually do not have 100% efficiencies in their removal of pollutants. An important reference value on this issue is of course provided by the environmental regulations imposed by the authorities. Our approach proposes to reduce the pollutant’s concentration 23 ACS Paragon Plus Environment


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in the final section of the river to have the best solution from the wastewater treatment point of view. With such optimal value we can then analyze how other solutions differ from it (e.g. imposing a wastewater concentration limit of discharge for every point source, which is the strategy followed by regulation agencies to maintain adequate levels of pollution in the water bodies). Solution strategy As mentioned before, the constraint method25 is used to obtain a set of alternative solutions to the optimization problem. There are two main environmental constraints. The first one is the preservation of the natural bodies of water regarding their pollution levels. The second one is the conservation of the fresh water sources through the reduction of the total annual consumption. These constraints relate each other through the use of reclaimed water in the system. The solution strategy used in this paper is as follows: i) A limit to the maximum allowable concentration of pollutants discharged to the river is set; then ii) a limit on the maximum allowable amount of water consumed from fresh sources is given and, finally, iii) the total annual costs are minimized. Such strategy is repeated by changing the value of the water consumed from fresh sources for the same concentration discharged and then minimizing the costs. Once an appropriate number of solutions are obtained for the same concentration value, the whole process can be repeated for different values of the pollutant concentrations with the same values of water consumed from fresh sources. The proposed strategy will allow having a wide range of solutions that will show the optimum compromises for very different scenarios regarding the environmental and economic objectives of the problem.


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ILLUSTRATIVE EXAMPLE To show the scope of the model, an illustrative example is presented in this section. The system consists of a typical river serving as both a source of water and a natural drainage system for several domestic, industrial and agricultural locations. Figure 4 shows the layout of the system and its division for modeling purposes. In addition, Figure 4 provides the annual average concentrations of the wastewater generated and its impact over the main river, without the use of a distributed treatment system. Note that the agricultural areas do not have an annual wastewater concentration because their activities only include the use of water for irrigation of crops. Although those values and the configuration do not correspond to a real problem, all parameters in the example were defined and/or estimated in order to achieve realistic orders of magnitude on the variables as well as to show the scope and significance of the model; to that end, we made a search on the primary sources of information on water management in Mexico. Those are governmental institutions, such as INEGI (National Institute of Statistics and Geography; http://www.inegi.org.mx/),

CONAGUA

(National

Water

Commission;

http://www.gob.mx/conagua) and SEMARNAT (Environmental and Natural Resources Agency; http://www.gob.mx/semarnat). For the parameters involved in water harvesting approaches (such as the dimensions and costs of storage tanks and artificial ponds) we used the values reported in references 7-9. The proposed MINLP model was coded in the GAMS modeling language and the solvers26 DICOPT/CONOPT/CPLEX were used for solving the resulting MINLP/NLP/LP problems. The illustrative example consists of 19,917 equations, 25,612 continuous variables and 2,900 binary variables. Each alternative solution can be obtained in approximately 30 seconds on a computer with an Intel core i5 processor of 2.40 GHz and 8 GB of RAM. The optimality gap of the master 25 ACS Paragon Plus Environment


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MILP problem for each iteration was zero. Despite of the model having a large number of binary variables, an acceptable computational time has been achieved through the use of appropriate initial values. Initial values for the continuous variables were estimated before optimizing the discrete decisions. The solution of the mass balance equations were used for that purpose. Then, when performing the sensitivity of the optimal solution to changes on the environmental and fresh water constraints, we used a loop for solving successive optimization problems; a solution achieved with given values of the parameters was used as the initial value of the subsequent optimization problem (with a different set of values of the environmental and fresh water consumption constraints). The main river is divided into five reaches and a pseudo-single pollutant case is considered in this example. For this case, the Biological Oxygen Demand (BOD) of the wastewater is considered. We assume that, due to the origins and composition of the wastewater, its quality can be measured through such property. This property accounts for the total amount of organic matter in the water and is the most widely used parameter to assess the quality of wastewater in urban and industrial locations. This assumption avoids the highly computational-expensive problem resulting from a large number of pollutants and makes the approach practical, given the extensive use of the BOD to characterize water. Typically, the BOD ranges from 400 to 600 mg/l in domestic wastewater, and from 800 to 1200 mg/l in industrial wastewater. Three interceptors are considered in this example. The last interceptor is a fictitious one and serves for the case where only a part of the total wastewater flow is to be treated. The parameters are given in Table 1. A maximum of nine storage tanks and four artificial ponds can be installed in the extent of each reach. Their distribution depends on the location of the sinks present in the reaches. As in many urban and industrial areas, the storage tanks considered are elevated above 26 ACS Paragon Plus Environment

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the ground. Therefore, there are pumping costs associated to the operation of the storage tanks. The artificial ponds are large locations which store rainwater collected directly on it and collected from houses and buildings. Detailed technical information about the storage infrastructure is not considered in the example. The capacity and cost parameters for storage tanks and artificial ponds are given in Table 2. The water demand in the system throughout the year per sector (agricultural, domestic and industrial) and per reach is shown in Figure 5. RESULTS AND DISCUSSION The optimization approach is based on the constraint method and it is explained in the following sections. The method requires first solving the model for three different scenarios. Scenario A For this scenario, the TAC is minimized. The solution of this scenario will provide a system functioning at conditions where no technology exists to promote the sustainability of the watershed. The results are shown in Table 3. The BOD in the final discharge is the maximum possible because there are no treatment units installed in the system. Scenario B In this scenario, the concentration in the final discharge is minimized and is used as objective function. The solution is expected to involve a high value of TAC because there is no constraint with respect to the maximum costs. Furthermore, the water consumption from natural sources is also unconstrained. Because of the conditions of the problem, this scenario results in the installation of most of the treatment units and the solution will not include any storage tanks or artificial ponds. The results are shown in Table 3.



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The BOD of the final discharge is reduced 79% from the BOD of scenario A. As expected, the TAC increases by a factor of 7.34 with respect to scenario A where there are no treatment units installed. A total of 22 interceptors are installed in this scenario; all of them are of Class 1. Scenario C This scenario minimizes the water consumption from fresh sources. To be consistent with the previous two scenarios, the TAC and the BOD in the final discharge are unconstrained; however, the use of treated water as reclaimed water may also help reduce the concentration of pollutants in the final discharge. A low value on the upper bound of BOD for a safe reuse (

) might

cancel the possibility of using reclaimed water, since the wastewater streams might not reach such concentration because of the efficiency of the interceptors. A maximum BOD of 30 mg/l is then chosen for this scenario. Table 3 provides the results obtained for this case. With respect to scenario A, the water consumed from fresh sources is reduced 12.68%. This scenario shows potential since the TAC is 6 times the TAC of scenario A (a value smaller than the factor of 7.34 obtained in scenario B). Additionally, the maximum number of possible storage tanks (45) and artificial ponds (20) in the system will be installed. This scenario also results in 30 installed interceptors in the domestic and industrial sinks; however, 15 are Class 1 interceptors and 15 are fictitious interceptors (Class 3). . Alternative optimal solutions through the constraint method In most cases, environmental and economic objectives oppose to each other. There is no unique optimal solution; a wide range of solutions are to be considered in order to make the best investment decision. To illustrate the scope of the model, two widely different scenarios are analyzed. The first one corresponds to an environmentally constrained scenario with a high 28 ACS Paragon Plus Environment

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investment cost. The second one is a much less constrained scenario from the environmental point of view. Such scenario is evaluated at the same conditions for the consumption of water from fresh sources that the ones in the first scenario. One of the environmental goals is to achieve adequate levels of pollution in the watershed. To attain this result, some works suggest monitoring only the concentration of the final discharge18. This approach allows finding the distribution, number and type of treatment units necessary in the system to perform under a given bound of BOD in the final discharge. This optimal distribution of treatment units can be obtained within a wide range of possibilities, including the installation of just a few scattered units throughout the watershed. Even though omitting the use of treatment units in some wastewater streams will result in a less expensive distributed system, such situation, in a realistic scenario, can cause management, normative and even social conflicts. For this reason, instead of only limiting the BOD of the final discharge, we propose imposing an upper bound on the BOD concentration in every wastewater stream, whether domestic or industrial, to achieve this environmental constraint. Despite the TAC may increase when comparing this approach to the conventional strategy, a more robust and controllable treatment system is achieved. To reduce water consumption from fresh sources, a combination of harvested rainwater from storage devices and reclaimed water from treatment units can be used. This is where the integration proposed in this work becomes important. By limiting both the BOD in the treated water and the maximum allowable consumption from fresh sources, a reduction in the optimal cost of the system is achieved. If the BOD limit is low (environmental constraint is tight), the use of reclaimed (treated) water becomes less expensive than using storage devices; the water consumption constraint from fresh sources is therefore met at a lower cost. 29 ACS Paragon Plus Environment


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Scenario I A BOD of 150 mg/l is assumed as the maximum allowable discharge to the main river. In fact, the monthly average allowable BOD discharge for domestic and industrial wastewater in Mexico is 150 mg/l27. The maximum annual demand of fresh water obtained in scenario B is 285,085,267 m3/year. As shown in scenario C, the minimum achievable consumption from fresh sources is 248,939,788 m3/year. The solutions for this scenario will then be obtained within this range of water consumption rates for a fixed BOD in the treated wastewater of at most 150 mg/l. The upper bound of BOD for reclaimed water (

) is the same as before (30 mg/l). Figure 6 shows

the sensibility of the model to changes on the constraint of maximum allowable consumption of water from fresh sources. The behavior of the key variables is presented, which includes the behavior of the water treated in alternative technologies, the wastewater discharges, the concentration of pollutant in the final discharge of the river and the design of the distributed treatment system. Figure 6b shows the behavior of such variables in dimensionless form as the water consumed from fresh sources varies. In the figures, for better representation, CQ discharge is the value of the concentration of pollutants in the final discharge of the main river (See Equation (52)). A critical value of 265 hm3/year of Total FWC is observed; at higher values of maximum consumption of water from natural sources there is no need to treat a large number of wastewater streams for their reuse as reclaimed water. Therefore, most of the streams are discharged to the main river at 150 mg/l of BOD and the CQ discharge suddenly increases. As a consequence, at the same critical value of


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265 hm3/year of Total FWC, there is a dramatic fall in the use of reclaimed water (Total RW) as shown in Figure 6a. On the other hand, the value of Total FWC does not reach the maximum value of 285,085,267 m3/year. Starting from 272,888,109 m3/year of Total FWC, the TAC is lower when using reclaimed water than when using storage devices or fresh sources. This is only feasible due to the value of the BOD limit of treated water imposed to the problem. Within the range from 249 x 106 m3/year to 265 x 106 m3/year of Total FWC, the amount of reclaimed water used remains practically constant. Figure 6a also shows the total amount of water from storage tanks and artificial ponds, respectively. Storage tanks have less capacity at the benefit of being cheaper; on the contrary, artificial ponds can store more rainwater at a higher cost. Notice that the same tendency of reclaimed water use continues, since, at high values of Total FWC, there is practically no water consumed from rainwater harvesting devices. There is at least one interceptor installed in every domestic/industrial sink. This is due to the value of the BOD limit, since the BOD concentration of all of the wastewater streams are between 400 mg/l and 1200 mg/l. If the concentration of wastewater streams were below 150 mg/l, that behavior would not occur. Because of the BOD limit, only Class 1 interceptors are selected for this scenario, with only a portion of the wastewater streams being segregated into Class 3 interceptors (fictitious). From values of Total FWC of 249 to 264 x 106 m3/y, the same number of Class 1 and Class 3 interceptors is kept, until the Total FWC value is low enough (265 x 106 m3/y) so there can be a greater amount of water segregated into Class 3 interceptors. This case, however, does not imply that the optimal distribution of the treatment units losses



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significance; the optimal treatment distribution is still useful to determine the number of interceptors of each class that must be installed. The same trend with respect to the rest of the variables is followed by the averaged value of BOD of domestic treated wastewater (CTRD). The BOD of industrial treated wastewater is kept at 150 mg/l because the efficiency of interceptors is not enough to achieve a BOD of 30 mg/l for its use as reclaimed water. To compute the values in dimensionless form, the actual values were divided by the maximum value obtained for each variable and used as a reference. The values that correspond to the dimensionless value of one (maximum) in Figure 6b are as follows: 1.24 m3/s (total wastewater treated in Class 1 interceptors), 0.21 m3/s (total wastewater treated in Class 3 interceptors), 114.64 mg/l (average concentration of domestic discharges) and 7.59 mg/l (CQ discharge). To further illustrate the significance of the results, Figures 7 and 8 show the improvement on the quality of the water in the system at a value of 265 hm3/year of the water consumption from natural sources. The TAC for this value of water consumption is $62,183,177/year. Figure 7 shows how the various wastewater discharges change their concentration with the distributed treatment system (See Figure 4). Furthermore, Figure 8 shows the improvement of the water quality of the river with the distributed treatment system throughout a year. Scenario II As an alternative analysis, the BOD upper bound is now set at 600 mg/l. Although the value appears as arbitrary, it was selected because is much larger than the value of 150 mg/l used in the previous scenario and, at the same time, water treatment would still be needed in the system. The most interesting aspect of this scenario is that there will be some wastewater streams below the


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600 mg/l that will not need any treatment to meet the environmental objective; nevertheless, they will probably still require treatment if reclaimed water is needed to reduce fresh water consumption, since the constraint for reclaimed water is the same as in the previous scenarios (30 mg/l). In Figure 9b, opposed to scenario I, changes in CQ discharge are smoother as the Total FWC increases its value. This indicates that reclaimed water is not preferred over harvested rainwater to reduce water consumption from fresh sources. Therefore, as the BOD constraint is relaxed, the installation of storage infrastructure is less expensive than water treatment. Figure 9a provides the behavior of the TAC against water consumption from fresh and alternative sources. Storage tanks play a much more significant role in this scenario than the one in scenario I. The same is not true for artificial ponds. Figure 9a shows that water stored in ponds drops rapidly and it is significantly smaller than the amount of storage tanks. Finally, Figure 9b shows the amount of wastewater treated in Class 1, Class 2 and Class 3 interceptors, respectively. The values that correspond to the dimensionless value of one (maximum) in Figure 9b are as follows: 0.76 m3/s (total wastewater treated in Class 1 interceptors), 0.13 m3/s (total wastewater treated in Class 2 interceptors), 0.65 m3/s (total wastewater treated in Class 3 interceptors), 552.06 mg/l (average concentration of domestic discharges) and 20.06 mg/l (CQ discharge). The number of treatment units is calculated to satisfy the upper bound of water consumption from natural sources. Then, as Total FWC increases, the number of Class 1 interceptors rapidly and steadily decreases. The purpose of analyzing scenario II is not to provide a better alternative for decision makers that the one shown on scenario I. Scenario II is used to show the implications of using the integrated approach for water management.



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CONCLUSIONS This paper has proposed a multi-period optimization model for the sustainable management of water in macroscopic systems. The resulting MINLP uses an MFA approach to account for all the inlets and outlets as well as the natural phenomena occurring in a watershed that considers different domestic, industrial and agricultural users. The interactions among these users in the watershed, the different domestic and industrial discharges, the surrounding areas and the disposal of wastewater to the environment are taken into account. Furthermore, the optimal management of the resources, from traditional (e.g. deep wells or dams) to alternative sources (rain harvested water), is considered through an optimal rainwater harvesting and storage system. On the other hand, the model formulation allows selecting an optimal distributed treatment system that deals with domestic and industrial wastewater which can be used as reclaimed water for agricultural users to further ensure the sustainability of the system. The integration of these two approaches is required to ensure the sustainability of the watershed from two different perspectives: the preservation of the natural sources of water and the maintenance of adequate levels of pollution in the water bodies of the watershed. A set of alternative optimal solutions for the problem is obtained by the constraint method, considering a wide range of compromises among the minimization of the total annual costs and the limits on the water consumption from traditional sources and on the pollution levels in the water bodies. Results of the illustrative example show that the integration of a distributed treatment system to the optimal management of the resources is an interesting approach to ensure safe levels of pollution in the water bodies; further, with the use of reclaimed water, the approach can also reduce the consumption from fresh sources along with rainwater harvesting technologies.


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Acknowledgments The authors would like to thank the financial support provided by the Mexican National Council for Science and Technology (CONACYT, Grants 257018 and 253660).

Supporting Information 1. Mathematical programming model developed in the GAMS® modeling environment 2. 19 Tables with the values of parameters of the case-study not included in the paper

Nomenclature Sets R = {1, 2, 3 … nR}

Reaches

J = {1, 2, 3 … nJ}

Tributaries

U = {1, 2, 3 … nU}

Domestic and Industrial sinks

A = {1, 2, 3 … nA}

Agricultural sinks

L = {1, 2, 3 … nL}

Pollutants

X = {1, 2, 3 … nX}

Interceptors

S = {1, 2, 3 … nS}

Storage Tanks

P = {1, 2, 3 … nP}

Artificial Ponds

T = {1, 2, 3 … nT}

Periods of time

Indices r

represents the elements of set R

j

represents the elements of set J

u

represents the elements of set U

a

represents the elements of set A

l

represents the elements of set L 35 ACS Paragon Plus Environment


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x

represents the elements of set X

s

represents the elements of set S

p

represents the elements of set P

t

represents the elements of set T

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Variables rainwater collected in storage tank s in reach r at time t maximum capacity of artificial pond p in reach r available amount of water of artificial pond p in reach r at time period t water sent from artificial pond p to agricultural sink a in reach r at time period t water sent from artificial pond p to domestic/industrial sink u in reach r at time period t cost of artificial pond p in reach r cost of storage tank s in reach r concentration of pollutant l in reach r at time period t untreated wastewater concentration of domestic/industrial sink u of pollutant l in reach r at time period t concentration of pollutant l of treated wastewater from domestic/industrial sink u in reach r at time period t concentration of tributary j of pollutant l in reach r at time period t disaggregated variable used for modelling disaggregated variable used for modelling reclaimed water of a treatment unit associated to domestic/industrial sink u in reach r at time period t reclaimed water from domestic/industrial sink u sent to agricultural sink a in reach r at time period t


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segregated flow to interceptor x associated to a treatment unit in domestic/industrial sink u in reach r at time period t water sent from natural sources to agricultural sink a in reach r at time period t water sent from natural sources to domestic/industrial sink u in reach r at time period t water consumed from natural sources at time period t flow of reach r at time period t rainwater collected in artificial pond p in reach r at time t available amount of water of storage tank s in reach r at time period t water sent from storage tank s to agricultural sink a in reach r at time period t water sent from storage tank s to domestic/industrial sink u in reach r at time period t maximum capacity of storage tank s in reach r rainwater excess of artificial pond p in reach r at time period t rainwater excess of storage tank s in reach r at time period t wastewater disposed to the environment from domestic/industrial sink u in reach r at time period t non-reclaimed water excess from a unit treatment associated to a domestic/industrial sink u in a reach r at a time period t Binary variables binary variable associated to the existence of reclaimed water from the treatment unit of domestic/industrial sink u binary variable associated to the existence of a treatment unit in domestic/industrial sink u binary variable associated to the existence of interceptor x in the treatment unit of domestic/industrial sink u



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binary variable used to activate the fixed cost of interceptor x in the treatment unit of domestic/industrial sink u binary variable associated to the existence of storage tank s binary variable associated to the existence of artificial pond p Parameters cost factor for storage tanks demand in agricultural sink a in reach r at time period t harvesting area for artificial pond p harvesting area for storage tank s cost factor for storage tanks cost factor for artificial ponds runoff coefficient for artificial pond p runoff coefficient for storage tank s concentration of losses of pollutant l in reach r at time period t rainwater concentration of pollutant l in reach r at time period t maximum concentration of reclaimed water of pollutant l concentration of direct usage of pollutant l in reach r at time period t operational cost for interceptor x consumed water in domestic/industrial sink u in reach r at time period t cost factor for artificial ponds fixed cost parameter for interceptor x positive integer number greater than one used for modelling filtration and evaporation losses in reach r at time period t available rainwater in reach r at time period t piping and pumping cost to send water from artificial pond p to agricultural sink a 38 ACS Paragon Plus Environment

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piping and pumping cost to send water from artificial pond p to domestic/industrial sink u piping and pumping cost to send water from natural sources to agricultural sink a piping and pumping cost to send water from natural sources to domestic/industrial sink u piping and pumping cost to send water from storage tank s to agricultural sink a piping and pumping cost to send water from storage tank s to domestic/industrial sink u piping and pumping cost to send reclaimed water from the treatment unit associated to domestic/industrial sink u to agricultural sink a available rainwater in artificial pond p in reach r at time period t rainwater contribution in reach r at time period t available rainwater in storage tank s in reach r at time period t reaction carried out in reach r for pollutant l contribution of tributary j to reach r at time period t direct usage in reach r at time period t demand in domestic/industrial sink u in reach r at time period t wastewater of domestic/industrial sink u in reach r at time period t volume of reach r at time period t variable cost parameter for interceptor x Greek symbols efficiency of interceptor x with respect to pollutant l maximum capacity of storage tank s minimum capacity of storage tank s maximum capacity of artificial pond p 39 ACS Paragon Plus Environment


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minimum capacity of artificial pond p parameter for considering the economies of scale upper bound for the segregated flow of interceptor x associated to a treatment unit of domestic/industrial sink u in reach r lower bound for the segregated flow of interceptor x associated to a treatment unit of domestic/industrial sink u in reach r

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Martinez-Gomez, J.; Burgara-Montero, O.; Ponce-Ortega, J. M.; Nápoles-Rivera, F.; Serna-González, M.; El-Halwagi, M. M. On the environmental, economic and safety optimization of distributed treatment systems for industrial effluents discharged to watersheds. Journal of Loss Prevention in the Process Industries 2013, 26(5), 908-923.

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Hortua, A.; El-Halwagi, M.; Ng, D.; Foo, D. Integrated approach for simultaneous mass and property integration for resource conservation. ACS Sustainable Chemistry and Engineering 2013, 1(1), pp. 29-38.

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Burgara-Montero, O.; El-Baz, A. A.; Ponce-Ortega, J. M.; El-Halwagi, M. M. Optimal design of a distributed treatment system for increasing dissolved oxygen in watersheds through self-rotating discs. ACS Sustainable Chemistry and Engineering 2013, 1(10), 1267-1279.

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Burgara-Montero, O.; Ponce-Ortega, J. M.; Serna-González, M.; El-Halwagi, M. M. Incorporation of the seasonal variations in the optimal treatment of industrial effluents discharged to watersheds. Industrial and Engineering Chemistry Research 2013, 52(14), 5145-5160. 41 ACS Paragon Plus Environment


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List of Tables: Table 1. Parameters of interceptors of the distributed treatment system. Table 2. Parameters of storage devices in the rain harvesting system. Table 3. Results of Scenarios A, B and C.



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Table 1. Parameters of interceptors of the distributed treatment system. Interceptor

Fixed cost ($)

Variable cost ($/m3)

Efficiency

Operating cost ($/m3)

1

195,000

0.058

0.95

0.00170

2

144,000

0.031

0.71

0.00144

3

0

0

0

0

Table 2. Parameters of storage devices in the rain harvesting system. Storage tank

Artificial pond

50,000

600,000

($)

13,080

-

($)

1.8135

-

($)

-

111,968

($)

-

0.8895

Maximum capacity (m3)

Table 3. Results of Scenarios A, B and C. Concept

Scenario A

Scenario B

Scenario C

8,552,558

62,791,101

52,157,086

285,085,267

285,085,267

248,939,788

26.99

5.47

23.33

Total number of interceptors installed

0

22

30

Total number of storage tanks installed

0

0

45

Total number of artificial ponds installed

0

0

20

TAC ($)

Water consumption from fresh sources (m3) Discharge BOD (mg/l)


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List of Figures: Figure 1. Global sustainability in a watershed. Figure 2. Superstructure for one reach on the watershed Figure 3. Superstructure of the Distributed Treatment System Figure 4. Layout of the system in the illustrative example. Figure 5. Water demand of the illustrative example. Figure 6. a) Water consumption from different sources vs TAC, b) Behavior of different variables vs Water consumption from fresh sources (scenario I). Figure 7. Annual average concentration of pollutants with wastewater treatment (scenario I). Figure 8. Concentration of pollutants in the main river with and without a distributed treatment system (scenario I). Figure 9. a) Water consumption from different sources vs TAC, b) Behavior of different variables vs Water consumption from fresh sources (scenario II).



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Figure 1. Global sustainability in a watershed.


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Figure 2. Superstructure for one reach on the watershed



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Figure 3. Superstructure of the Distributed Treatment System


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Figure 4. Layout of the system in the illustrative example.



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Figure 5. Water demand of the illustrative example.


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Figure 6. a) Water consumption from different sources vs TAC, b) Behavior of different variables vs Water consumption from fresh sources (scenario I). 51 ACS Paragon Plus Environment


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Figure 7. Annual average concentration of pollutants with wastewater treatment (scenario I).


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Figure 8. Concentration of pollutants in the main river with and without a distributed treatment system (scenario I).



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Figure 9. a) Water consumption from different sources vs TAC, b) Behavior of different variables vs Water consumption from fresh sources (scenario II). 54 ACS Paragon Plus Environment

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For Table of Contents Use Only

An MINLP Model for Sustainable Water Management in Macroscopic Systems: Integrating Optimal Resources Management to the Synthesis of Distributed Treatment Systems

Jaime Garibay-Rodriguez, Vicente Rico-Ramirez and Jose M. Ponce-Ortega.

Synopsis: This paper presents an optimization approach for the sustainable water management in a macroscopic level


Mixed Integer Nonlinear Programming Model for ... - ACS Publications

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