Optimal Synthesis of Property-Based Water Networks Considering

Article pubs.acs.org/IECR

Optimal Synthesis of Property-Based Water Networks Considering Growing Demand Projections César Sotelo-Pichardo,† Hisham S. Bamufleh,‡ José M. Ponce-Ortega,*,† and Mahmoud M. El-Halwagi‡,§ †

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

S Supporting Information *

ABSTRACT: This paper presents a mathematical programming model for the optimal synthesis and retrofitting of water networks based on the properties of the streams that impact the processing in the plant and the environment. One important feature of the proposed approach is that it accounts for changes in the operation through a time horizon with growing demands. The optimization formulation considers changes in the demands and accounts for time-based variations in the flow rates required for the process sinks and constraints for properties in the process sinks and in the environment. Furthermore, the proposed model allows the installation of different units and the retrofitting of the water network over the considered time horizon. The objective function minimizes the total cost associated with the entire life of the project while accounting for the time value of money and the specific demands for the process and the environment that change through the life of the project. Two case studies are solved to show the applicability of the proposed approach.

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INTRODUCTION The refining and process industries are characterized by the enormous usage of fresh water and discharge of wastewater. Because of the continued retrofitting projects to increase the capacities of refineries and process industries, usage and discharge of water are growing. With the scarcity of fresh water resources, there is a critical need for effective water conservation strategies. Water recycle approaches are among the most effective water conservation strategies. The substitution of fresh water with recycled process- and wastewater streams may be carried out through direct recycle or following treatment of the recycled streams to meet the requirements of process units (sinks) that use fresh water. These process sinks require that the recycled water streams meet specific constraints on flow rate, concentration of impurities, and properties. Various tools have been proposed for the synthesis of waterrecycle networks. Foo1 has reviewed many of these tools. An interesting problem associated with water networks is the retrofitting of existing networks that were not properly designed or when the process and/or environmental constraints have changed. In this context, Fraser and Hallale2 presented a retrofitting approach based on the pinch point technology. Alfadala et al.3 presented a methodology based on a set of heuristic rules with the goal of minimizing the total annual cost. Cheng and Hung4 developed an mixed-integer nonlinear programming (MINLP) formulation for retrofitting single units of mass exchange considering the use of external agents to facilitate the separation. Tan and Cruz5 developed a linear programming model to retrofit a water network with one component. Tan and Manan6 presented a systematic methodology to retrofit water networks and introduced regeneration units using a sequential approximation, and Sotelo-Pichardo et al.7 proposed a mathematical programming model for the © XXXX American Chemical Society

retrofitting of water networks based on the constraints given in terms of limits for the composition of the manipulated streams. This approach was later improved to incorporate property constraints by Sotelo-Pichardo et al.8 Bishnu et al.9 developed a multiperiod synthesis approach for water networks involving multiple industrial processes with seasonal variations. Faria and Bagajewicz10 presented a mathematical programming approach for retrofitting water networks to maximize net present value. A common limitation in these foregoing research efforts is that retrofitting was carried out for a specific current requirement. In many cases, retrofitting is needed over a time horizon to meet future demand increases that are associated with capacityincrease projects. A simplistic approach to satisfy the future requirements and increasing demands for water is to overdesign the water network to meet the maximum future demand through a single retrofit. A better approach is to develop a timebased strategy for retrofitting that considers the future projections for increase in water demand over multiple periods and the possibility of stage-wise retrofitting over multiple periods while accounting for the existing infrastructure. This is the focus of this paper which is aimed at developing a systematic method for synthesizing and simultaneously retrofitting a multiyear water network that accounts for future projections. The proposed optimization formulation takes into account from the design stage future projections and expansions and the possibility of multiannual retrofitting that avoids an overdesigned network. The objective function is the minimization of the total cost in the entire horizon time Received: August 5, 2014 Revised: October 24, 2014 Accepted: November 4, 2014

A

dx.doi.org/10.1021/ie503127p | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Figure 1. Schematic representation of the addressed problem.

including capital costs for units, pipes, reconfiguration, and expansions and the operating costs. It should be noted that the constraints for the process and environment include stream properties and pollutant concentrations.

treatment units. At the center of the superstructure there is a set of treatment units that may be installed over any period of the time horizon. These treatment units may also undergo retrofitting to adjust their capacities and efficiencies as required over the different time periods after their initial installation. All the units involved in the superstructure are connected through pipes which may be installed and retrofitted over the different time periods. This representation enables the simultaneous synthesis and retrofitting of the water network while considering future production demands. The next section presents the optimization formulation for the proposed superstructure to solve the addressed problem.

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OUTLINE OF THE PROPOSED MODEL The addressed problem can be defined as follows (see Figure 1): Given a set of process units with specific water demands, which can change with respect to mass and properties in the different periods through the entire time horizon based on the production projection, the growth in water demands requires retrofitting of the water network. Instead of overdesigning a current water network that meets the maximum projected demand, there is a need to consider a multistage retrofitting strategy to cope with the increase in water demands over a time horizon. There are several water sources with given flow rates, composition, and properties which may vary over multiple periods through the entire time horizon based on the production projection. To satisfy the process sink water demands throughout the entire time horizon, a water network must be synthesized and retrofitted starting from the first period (involving pipes and treatment units for water reusing, recycling, and regeneration) and over the subsequent periods (involving new pipes, new treatment units, readjusting the existing units and changing the operation of the water network). The retrofitting strategy should be aimed at minimizing the total cost over the entire time horizon involving the capital, retrofitting, and operating costs associated with the water network. This model must properly consider the time value of money and determine the necessary actions of installing new units or readjusting the existing ones. To address the above-mentioned problem, the superstructure shown in Figure 2 is proposed. The process sources (recyclable process streams which may vary over the time periods) are split and assigned directly to process sinks or to treatment units. The split streams may be mixed prior to the process sinks or the

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MODEL The proposed model formulation is based on time-based mass balances, property constraints, logical constraints, and design relationships for the involved units in the superstructure shown in Figure 2. Prior to developing the model formulation, the following indices are described: i represents the process source, j the process sinks, u the treatment units, t the time periods, and p any stream property. The model is described as follows. Balances for the Process Sources. The flow rate for each process source Wi,t can be split and assigned to the different sinks w1i,j,t or to the treatment units w2i,u,t at any time period t. Wi , t =

∑

wi1, j , t +

j ∈ NSINKS

∑

wi2, u , t ,

u ∈ NTREAT

∀ i ∈ NSOURCES, t ∈ NTIME

(1)

Distribution of Fresh Sources. The fresh source Fr,t can be segregated as f r,j,t to be distributed to any process sources as follows: Fr , t =

∑ j ∈ NSINKS

fr , j , t ,

∀ r ∈ NFRESH, t ∈ NTIME

j ≠ waste

(2) B



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Figure 2. Proposed superstructure of the addressed problem.

as has been previously reported by Shelley and El-Halwagi,11 El-Halwagi et al.,12 Ng et al.,13 and Ponce-Ortega et al.14 Table 1 in the Supporting Information shows some examples of these property operators previously reported.15 Performance of Treatment Units. Each treatment unit has an efficiency factor EFp,u, which is independent of time and modifies the stream properties to satisfy the process sinks. EFp,u depends on type and size of treatment unit; the optimization process consists of determining the selection of the treatment units and treated flow rates that satisfy all the sink constraints. The value of the property operator at outlet of the treatment units is calculated as follows:

It should be noted that the fresh sources cannot be sent to the environment fr , j , t = 0,

∀ r ∈ NFRESH, j = waste, t = NTIME (3)

Balances for Each Treatment Unit. The treated flow rate in each treatment unit Hu,t is the sum of the flow rates from process sources w2i,u,t and other treatment units h2u′,u,t over any time period t

∑

Hu , t =

wi2, u , t +

i ∈ NSOURCE

∑

hu2′ , u , t ,

u ′∈ NTREAT u ′≠ u

∀ u ∈ NTREAT, t ∈ NTIME

ψuout = Efu , t , pψuin, t , p , ,t ,p

(4)

∀ u ∈ NTREAT, p ∈ NPROP, t ∈ NTIME

There are also required property balances to determine the value of the stream’s properties Hu , tψuin, t , p

=

∑ i ∈ NSOURCE

wi2, u , t ψi , t , p

+

∑ u ′∈ NTREAT u ′≠ u

Balances at the Exit of the Treatment Units. The flow rate exiting from the treatment units Hu,t is distributed to the sinks (h1u,j,t) or recycled to another treatment unit (h2u,u′,t) for improving other properties in this stream

hu2′ , u , t ψuout , ′,t ,p

∀ u ∈ NTREAT, p ∈ NPROP, t ∈ NTIME

(6)

Hu , t =

(5)

∑ j ∈ NSINKS

Notice that the property balances are stated in terms of the property operators ψ, which are different for each property, and these can be determined experimentally or through correlations

hu1, j , t +

∑ u ′∈ NTREAT u ′≠ u

∀ u ∈ NTREAT, t ∈ NTIME C

hu2, u ′ , t ,

(7)



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Notice that the exiting flow rate from a given treatment unit cannot be sent to the same treatment unit, which is modeled as follows: hu2, u ′ , t = 0,

max Gjmin , t ≤ Gj , t ≤ Gj , t ,

(11) max where Gmin j,t and Gj,t are the minimum and maximum flow rate limits for the sink j, respectively. Notice that these maximum and minimum flow rates change through time according to the projection of the demand of the plant. There are also required specific limits for the properties in any sink (ψp,j) for specific properties p, these property constraints for each process sink can be stated in terms of the property operators because these are the optimization variables in the model formulation

∀ u , u′ ∈ NTREAT, u = u′ , t ∈ NTIME (8)

Balances in the Mixers before Any Sink. The flow rate sent to a sink (Gj,t) is the sum of the flow rates from fresh sources (f r,j,t), process sources (w1i,j,t), and treatment units (h1u,j,t) Gj , t =

∑

∑

fr , j , t +

r ∈ NFRESH

∑

wi1, j , t +

i ∈ NSOURCE

hu1, j , t ,

u ∈ NTREAT

∀ j ∈ NSINKS, t ∈ NTIME

(9)

ψ jmin ≤ ψj , t , p ≤ ψ jmax , ,t ,p ,t ,p

Furthermore, there are required a set of property balances before any process sink to calculate the resulting property p inlet to any sink j (ψp,j,t); these balances require taking into account the property operator for the fresh sources (ψr,t,p), process sources (ψi,t,p), and treatment units (ψout u,t,p) as follows:

∑

Gj , t ψj , t , p =

fr , j , t ψr , t , p +

r ∈ NFRESH

+

∑

∑

∀ j ∈ NSINKS, p ∈ NPROP, t ∈ NTIME

i ∈ NSOURCE

hu1, j , t ψuout ,t ,p

u ∈ NTREAT

(10)

The property operators are required to account for the nonlinear behavior of some properties in the property balances, and these can be determined experimentally or deduced mathematically. Sink Constraints. There are specific requirements for the flow rate inlet to any process sink (Gj,t) as follows: ⎡ Yu , t ⎢ ⎢ Hu , t ≥ δ ⎢ use ⎢ Hu , t ≥ Hu , t′ ∀ t ′ < t ⎢ ⎢ Ef uuse , t , p ≥ Ef u , t ′, p ∀ t ′ < t ⎢ ⎢⎡ ⎤ YuA, t1 ⎢⎢ ⎥ ⎢⎢ ⎥ use H ≥ δ ⎢⎢ u,t ⎥ ⎢⎢ ⎥ Cost Costa Costm = + u,t u,t u,t ⎢⎢ ⎥ ⎢⎢ ⎥ B1 ⎤ ⎢⎢⎡ Yu , t ⎥ ⎥ ⎡ ⎤⎥ ⎢⎢⎢ YuB2, t use ⎥ ⎥ ⎢ ⎢⎢⎢ Hu , t ≥ Hu , t ⎥ ⎥∨ ⎢ use ⎢⎢⎢ Hu , t ≥ Hu , t ⎥ ⎥ max ⎥ ⎥ ⎢ ⎢⎢⎢ Hu , t ≤ Hu ⎥ ⎢Costa = 0 ⎥ ⎥⎥∨ ⎢⎢⎢ ⎦ ⎣ u,t ⎢ ⎢ ⎢Costa = K [CF A + CV A(H − H use)]⎥ ⎥ ⎣ u,t Ft u u u,t u,t ⎦ ⎢⎢ ⎥ ⎢⎢ ⎥ C1 ⎤ ⎡ ⎢⎢ Yu,t ⎥ ⎥ ⎢ C2 ⎤ ⎡ ⎢⎢ Yu,t use ⎥ ⎢ ⎥ ⎥⎥ ⎢ ⎢⎢ Efu , t , p ≥ Ef u , t , p ⎥ ⎢ use ⎥ ⎢⎢ ⎢ ∨ ⎥ ⎢ Efu , t , p = Ef u , t , p ⎥ ⎥ ⎢⎢ ⎢ Efu , t , p = Ef uuse , t , pMeu , p ⎥ ⎢ ⎥ ⎥ ⎢⎢ ⎢ ⎣ Costmu , t = 0 ⎦ ⎥ ⎢ ⎢ ⎢Costm = K [CFM + CV MH ]⎥ ⎥ u,t Ft u u u,t ⎦ ⎦ ⎢⎣ ⎣ ⎣

(12)

min where ψp,j and ψmax are given lower and upper limits, p,j respectively, for the specific properties constrained in the process sink. Notice that different sinks require different constraints in different limits. For example, the environmental regulations impose specific limits for properties like pH, toxicity, chemical oxygen demand, etc., and some process units limit properties like density, viscosity, reactivity, pressure vapor, thermal conductivity, etc. It should be noted that the constraints for the different properties in the process sinks change through time according to the projected process conditions. Installation and Modification of Treatment Units. For modeling the installation and modification of the treatment units in the different periods of time, the following disjunction is applied:

wi1, j , t ψi , t , p

∀ j ∈ NSINKS, p ∈ NPROP, t ∈ NTIME

∀ j ∈ NSINKS, t ∈ NTIME

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∨ ⎤⎥ ⎡ YuA2 , t ⎥ ⎥ ⎢ ⎥⎥ ⎢ Huuse =0 , t ⎥⎥ ⎢ max ⎥⎥ ⎢ δ ≤ Hu , t ≤ Hu ⎥⎥ ⎢ ⎥⎥ ⎢ I I ⎣Cost u , t = KFt [CFu + CVuHu , t ]⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

In the previous disjunction, the following options were considered. First, the treatment unit u can be used (when Yu,t is true) or not (when Yu,t is false). When the treatment unit is not

⎡ ¬Yu , t ⎤ ⎥ ∀ u ∈ NTREAT ⎢ ⎢ Hu , t = 0 ⎥ , t ∈ NTIME ⎥ ⎢ ⎢⎣Cost u , t = 0 ⎥⎦

used, the inlet flow rate (Hu,t) and cost (Costu,t) are zero. On the other hand, when the treatment unit is used in a given time period, the inlet flow rate must be greater than zero (i.e., Hu,t ≥ D



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there are two options. The first option is associated with the case in which the retrofit of the unit is required (associated with the binary variable yA1 u,t ), and the other option with the case in which the unit is installed in this time period (associated with the binary variable yA2 u,t ). Then, when the treatment unit is needed (the binary variable yu,t should be 1) in a given time period, one of the above-mentioned options should be selected as follows:

δ). Then, we need to define two new variables to indicate the maximum flow rate inlet to that unit in previous times (Huse u,t ) and the maximum efficiency used in previous times (Efuse u,t,p). Then, there are two options: one (when the Boolean variable YA2 u,t is true) states that flow rate inlet to the treatment units in all previous time periods is zero (Hu,tuse = 0), which means that the treatment unit does not exist at that period and so the installation is required, and the capital cost for this unit is calculated accordingly. It should be noted that the value of money through the time is calculated taking into account the parameter KFt, which depends on the time period when the unit is installed. On the other hand, when the unit exists in previous period (i.e., when the Boolean variable YA1 u,t is true), there are four options. When the existing capacity for the treatment unit is enough for handling the required flow rate (Huse u,t ≥ Hu,t), then its cost is zero (which is associated with the Boolean variable YB2 u,t ). When the required capacity is greater than the one existing in the previous periods (Hu,t ≥ Huse u,t ), then the Boolean variable is activated and the capital cost associated with the capacity expansion is properly calculated considering the value of the money through time. It should be noticed that the overall capacity expansion for each treatment unit must be lower that a maximum limit (Hmax u ). There is another point to consider for improving the treatment units during each time period, which is related to the efficiency. Therefore, when the required efficiency for a treatment unit during a given time period corresponds to the existing one (Efu,t,p = Efuse u,t,p), the Boolean variable YC2 u,t is true and the capital cost for improving the efficiency is zero (Costmu,t = 0). On the other hand, when the required efficiency is greater than the existing one (Efu,t,p ≥ C1 Efuse u,t,p), then the Boolean variable Yu,t is true and the efficiency can be improved by a factor Meu,p and the corresponding capital cost must be calculated accounting for the value of the money through time. Notice that in the previous disjunction the index t′ is used for the time periods before t. It should be noted that this disjunction contains the main contribution of the present paper. The following relationships are obtained from reformulating the previous disjunction as a set of algebraic constraints. First, the binary variable can be one or zero yu , t ≤ 1


yuA1 + yuA2 = yu , t ,t ,t


A2 Hu , t = HuA1 , t + Hu , t

(18)

∀ u ∈ NTREAT, t ∈ NTIME (19)

Huse u,t

It should be noted that the continuous variable is not disaggregated because this always takes the value of zero for the term A2. Then, the relationships in options A1 and A2 are stated in terms of their corresponding disaggregated continuous variables and multiplying the constant terms by the corresponding binary variables as follows: A1 Huuse , t ≥ δyu , t


max A2 HuA2 , t ≤ Hu yu , t

A2 HuA2 , t ≥ δyu , t



(20) (21) (22)

Cost uA2, t = KFt [CFuI yuA2 + CVuI Hu2, t ] ,t ∀ u ∈ NTREAT, t ∈ NTIME

(23)

Cost uA1, t = Costau , t + Costmu , t ∀ u ∈ NTREAT, t ∈ NTIME

(24)

where the terms Costau,t and Costmu,t are associated with the increment in capacity and efficiency, respectively, and this is modeled as follows. There are two options for the capacity of treatment units. When the flow rate used in the current period (yA1 u,t ) is greater than zero, one option is that the capacity of the existing treatment unit is lower than the one needed; therefore, the unit must increase its capacity (yB1 u,t ). The other case is when the existing capacity is greater than or equal to the one needed (yB2 u,t ). This is modeled through the following relationship:

(13)

(14)

∀ u ∈ NTREAT, t ∈ NTIME, t ′ < t

yuB1, t + yuB2, t = yuA1 ,t

(15)

The maximum efficiency used in all previous periods for a given treatment unit (Ef use u,t,p) is determined by Ef uuse , t , p ≥ Ef u , t ′ , p


Cost u , t = Cost uA1, t + Cost uA2, t

The maximum manipulated flow rate in all time periods for a given treatment unit (Huse u,t ) is determined by Huuse , t ≥ Hu , t ′

(17)

Then, the continuous variables involved in options A1 and A2 are disaggregated as follows:

The binary variable is activated when the treated flow rate is greater than zero Hu , t ≥ δ·yu , t



(25)

The continuous variables involved in options B1 and B2 are disaggregated as follows:

∀ u ∈ NTREAT, t ∈ NTIME, t ′ < t

AB1 AB2 HuA1 , t = Hu , t + Hu , t

(16)

∀ u ∈ NTREAT, t ∈ NTIME (26)

It should be noted that if there is required additional capacity or additional efficiency, it is required to determine the additional costs associated with the retrofit associated with the retrofit as follows as follows. There are two retrofitting options; the first one is associated with the capacity, and the second one is associated with the efficiency. For the capacity,

use B1 HuAB1 , t ≥ Hu , t · yu , t


(27)

It should be noted that the continuous variable Costau,t is not disaggregated because this is always zero for option B2. Then, the involved relationships in terms of B1 and B2 are stated E



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⎡ ⎤ Yipip ,j,t ⎢ ⎥ ⎢ ⎥ 1 ≥ δ w i,j,t ⎢ ⎥ ⎢ ⎥ 1,use 1 wi , j , t ≥ wi , j , t ′ ∀ t ′ < t ⎢ ⎥ ⎢ ⎥ ⎤ ⎡ ⎤⎥ ⎢⎡ YiD1 YiD2 , j , t , j , t ⎥ ⎢ ⎥⎥ ⎢⎢ ⎥ ⎢ ⎥⎥ ⎢⎢ 1,use 1,use = ≥ δ w w 0 i,j,t i,j,t ⎥ ⎢ ⎥⎥ ⎢⎢ ∨ ⎢ ⎢⎢ 1 1,max ⎥ 1 1,max ⎥ ⎥ ⎢ ⎢ wi , j , t ≤ wi , j , t ⎥ ⎢ δ ≤ wi , j , t ≤ wi , j , t ⎥ ⎥ ⎥ ⎢ ⎥⎥ ⎢⎢ ⎢⎣ ⎢⎣ Costtubi , j , t = 0 ⎥⎦ ⎢⎣ Costtubi , j , t = KFt CTi , j ⎥⎦ ⎥⎦ ⎡ ⎤ ¬Yipip ,j,t ⎢ ⎥ ∨ ⎢⎢ wi1, j , t = 0 ⎥⎥ ⎢ ⎥ ⎣ Costtubi , j , t = 0 ⎦

through the disaggregated variables and using the binary variables to activate the constant terms as follows: max B1 HuAB1 , t ≤ Hu · yu , t


(28)

use B1 Costa u , t = KFt [CFuA yuB1, t + CVuA(HuAB1 , t − Hu , t yu , t )]

∀ u ∈ NTREAT, t ∈ NTIME use B2 HuAB2 , t ≤ Hu , t · yu , t


(29) (30)

The options for manipulating the efficiency of treatment units are modeled in the same way as that use to model the ones associated with the capacity. In this case, when the treatment unit is required in a given time period (the binary variable yAu,t is 1), and then there are two options, one that considers an increment in the capacity for the treatment unit (associated with the binary variable yC1 u,t ) and the other one that does not involve improving the efficiency (associated with the binary variable yC2 u,t ). This is modeled as follows: yuC1 + yuC2 = yuA, t ,t ,t


∀ i ∈ NSOURCE, t ∈ NTIME, j ∈ NSINKS

To explain the previous disjunction, we discuss the following: First, when the pipe segment is not required in the time period pip t, then the Boolean variable Yi,j,t is false and then the 1 corresponding used flow rate (wi,j,t) and cost (Costtubi,j,t) are zero. On the other hand, when the pipe segment is required in the time period t, first the Boolean variable Ypip i,j,t is true and the flow rate (w1i,j,t) must be greater than zero. In addition, the maximum used flow rate in previous time periods is determined (w1,use i,j,t ) to have two retrofitting options. The first one involves the case in which the required flow rate in this period of time is lower than the maximum used in the previous periods (modeled through the Boolean variable YD1 i,j,t ), which means that the existing pipe segment is enough to handle the required flow rate; in this way, no cost is added in this option. The second option (associated with the Boolean variable YD2 i,j,t ) involves the case in which there is not an existing pipe for this segment, and so the capital costs for this new pipe segment must be included; the capital cost for the pipe segment (Costtubi,j,t) accounts for the value of the money through time depending on the period when the pipe is installed (modeled through the parameter KFt) and the distance between the units considered (CTi,j). This formulation, which involves the installation and retrofit of pipes in different time periods, has not been considered previously in the synthesis and design of water networks, and this is one of the contributions of this paper. To reformulate the previous disjunction as a set of algebraic relationships, the convex hull technique15,16 was used; the following relationships result. First, if the pipe segment is used, the binary variable ypip i,j,t is activated as follows:

(31)

The continuous variables used in the terms C1 and C2 are disaggregated as follows: Efu , t , p = Ef 1u , t , p + Ef u2 , t , p ∀ u ∈ NTREAT, t ∈ NTIME, p ∈ NPROP

(32)

It should be noted that the continuous variable Costmu,t is not disaggregated because this always is zero for term C2. Then, the constraints given in terms C1 and C2 are stated in terms of the disaggregated variables and multiplying the constant terms by the binary variables as follows: C1 Ef 1u , t , p ≥ Ef uuse , t , p· yu , t

∀ u ∈ NTREAT, t ∈ NTIME, p ∈ NPROP

(33)

C1 Ef 1u , t , p = Ef uuse , t , pMeu , p · yu , t


(34)

Costm u , t = KFt [CFuM yuC1 + CVuMHu1, t ] ,t ∀ u ∈ NTREAT, t ∈ NTIME

wi1, j , t ≥ δ·yipip ,j,t

(35)

∀ i ∈ NSOURCES , j ∈ NSINKS, t ∈ NTIME

C2 Ef u2 , t , p = Ef uuse , t , pyu , t


(37)

w1i,j,t

It should be noted that the continuous variable is not disaggregated because this is always zero in the second term. Then, the continuous relationship represents the case when the required flow rate (w1,use i,j,t ) is greater than the ones used in the previous time periods (w1i,j,t′), and it is stated as

(36)

It should be noted that the previous disjunctive formulation was reformulated using the convex hull technique.16,17 Piping Costs. In the retrofitting processes, most of the tasks involve the reconfiguration of the pipe network; therefore, in the proposed approach the piping costs represent a significant contribution because this pipe network can be readjusted in the different time periods through the time horizon. This way, for example, for the segment of pipes from the process sources to the sinks, the following disjunctive formulation applies:

1 wi1,use , j , t ≥ wi , j , t ′

∀ i ∈ NSOURCES, j ∈ NSINKS, t ∈ NTIME

t′ < t (38)

Then, there are two options associated with the first term, option D1, that states that the used flow rate in this time period F



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disjunction previously explained, where the operating costs depend on the manipulated flow rate (w), the unit costs that account for the distance between the units involved (pip), the yearly operating time (HY), and the factor adjusting the value of the money through time (YRt), which is stated as follows for the different pipe segments:

is associated with an existing pipe and option D2 when there is needed a new pipe. Therefore, only one of these two options should be selected as follows: yiD1 + yiD2 = yipip ,j ,t ,j,t ,j ,t ∀ i ∈ NSOURCES, j ∈ NSINKS, t ∈ NTIME

(39)

t ∈ NTIME

≥

(40)

t ∈ NTIME

∑

Costopipr , j = HY

∑

Costtubr , j , t

t ∈ NTIME

∀ r ∈ NFRESH, j ∈ NSINK

∑

(50)

∑

YRt pipu ′ , uhu2′ , u , t +

t ∈ NTIME

(41)

Costtubu ′ , u , t

t ∈ NTIME

∀ u′, u ∈ NTREAT

1,max D1 wi1D1 , j , t ≤ wi , j , t · yi , j , t

(51)

YRt pipu , jhu1, j , t +

∑

Costopipu , j = HY

t ∈ NTIME

(42)

∑

Costtubu , j , t

t ∈ NTIME

∀ u ∈ NTREAT, j ∈ NSINK

D2 wi1D2 , j , t ≥ δ · yi , j , t


Costtubi , u , t (49)

YRt pipr , j fr , j , t +

t ∈ NTIME

Costpipu ′ , u = HY


∑ t ∈ NTIME

∀ i ∈ NSOURCE, u ∈ NTREAT

δ·yiD1 ,j,t


Costtubi , j , t (48)

YRt pipi , uwi2, u , t +

∑

Costopipi , u = HY

Terms Costtubi,j,t and w1,use i,j,t are not disaggregated because these are zero for the first and the second terms, respectively. Then the constraints are stated in terms of the disaggregated variables and multiplying the constant terms by their corresponding binary variables as follows: wi1,use ,j,t

∑ t ∈ NTIME

∀ i ∈ NSOURCE, j ∈ NSINK

1D2 wi1, j , t = wi1D1 , j , t + wi , j , t


YRt pipi , j wi1, j , t +

∑

Costopipi , j = HY

The continuous variables are disaggregated for terms D1 and D2 as follows:

(52)

Finally, the total piping cost is calculated summing the costs for the different pipe segments as follows:

(43)

Costopip =

1,max D2 wi1D2 , j , t ≤ wi , j , t · yi , j , t

∑

∑

Costopipi , j

i ∈ NSOURCE j ∈ NSINK


+

(44)

∑

∑

Costopipi , u

i ∈ NSOURCE u ∈ NTREAT

Costtubi , j , t = KFt CTi , jyiD2 ,j,t

+


(45)

+

Similar disjunctions and reformulation are applied to the different pipe segments involved in the proposed superstructure. Involved Costs. The cost for the fresh sources (Costr) is determined accounting for the fresh sources used in the entire time horizon (Fr,t) as well as the unit cost for the fresh source (Costor,t), the yearly operating time (HY), and a factor that adjusts the value of the money through time depending on the period used (YRt). Then, the cost for the fresh sources is determined as follows:

∑

Cost r = HY

YRt Costor , t Fr , t

∑

+

∑

Costopipu ′ , u

∑

∑

Costopipu , j

(53)

Objective Function. The objective function consists of minimizing of the total annual cost (TC) accounting for the costs of fresh sources (Costr), capital costs (Costou) for treatment units (new and adaptations), the operating costs for the treatment units (Costu,t), and the piping costs for the retrofit of the network (Costopip)

∀ r ∈ NFRESH

min TC =

∑

Cost r +

r ∈ NFRESH

(46)

YRt COu Hu , t

∑

u ∈ NTREAT j ∈ NSINK

+

Then the operating cost for the treatment units (Costou) depends on the treated flow rate (Hu,t), the unit operating cost (COu), the yearly operating time (HY), and the adjustment for the value of the money through time (YRt), which is stated as

∑

Costopipr , j

u ′∈ NTREAT u ∈ NTREAT

t ∈ NTIME

Costou = HY

∑

r ∈ NFRESH j ∈ NSINK

∑

∑

t ∈ NTIME u ∈ NTREAT

∑

Costou

u ∈ NTREAT

Cost u , t + Costopip

(54)

The optimization formulation is posed as a mixed-integer nonlinear programming problem which was coded using the software GAMS. The next section shows the application of the proposed optimization model to two case studies. It should be noted that the proposed model can manipulate the case when the needed capacity is lower than the one existing, because in this case there is no needed any additional change to the unit and so no additional capital cost should be included. In this case, the operating cost is associated with the

∀ u ∈ NTREAT

t ∈ NTIME

(47)

For the costs associated with the different pipe segments (Costopip), there are involved operating and capital costs. The capital costs (Costtub) are determined throughout the G



Article

Figure 3. Solution for the worst case for example 1.

Figure 4. Optimal water network for the multiperiod retrofitting solution of example 1.

manipulated flow rate in the system, which can be properly considered by the proposed model. Case Studies. Two case studies are presented to show the applicability of the proposed approach, and these are described as follows. Case Study 1. This case study for a petroleum refinery consists of three process sources, three process sinks, and a sink that corresponds to the wastewater stream that is discharged to the environment. Two pollutants whose concentration affects the sinks and environment are considered. Two possible treatment units are considered to remove these pollutants, which can be installed or readjusted during four operation

periods (each period corresponds to five years to yield a total time horizon of 20 years). Also, two fresh sources are considered, each one with different costs and contaminants. The unit costs of fresh sources are Costo1 = 0.10 with an increase of 20% each period and Costo2 = 0.09 with an increase of 25% per period. The data for the problem are shown in Tables 2−8 in the Supporting Information. Table 9 in the Supporting Information shows the unit piping and pumping costs for the pipe segments considered in example 1. First, the problem without considering the multiperiod retrofitting was considered, that is, the example for the worst case was considered, and the results are shown in Figure 3. On H



Article

Figure 5. Solution for the worst case for example 2.

importance of the proposed optimization approach for solving this type of problem. Case 2. This problem consists of three sources, four process sinks and the environment, four properties, two fresh sources, and three possible treatment units. The problem involves four time periods, each of five years. In this case study, four properties are constrained. The first property corresponds to a composition of a hazardous material, the second to the toxicity, the third to the chemical oxygen demand, and the fourth to density. The costs of fresh sources are Cost1 = 0.02 and Cost1 = 0.013 ($/kg) with an increment of 15% every period. Tables 16 and 17 in the Supporting Information show the data for the flow rates of process sources and required flow rates for process sinks. Table 18 in the Supporting Information shows the constraints for the process sinks in terms of properties, and Table 19 shows the properties for the process sources. In addition, Table 20 in the Supporting Information presents the unit efficiencies for the considered treatment units and Table 21 in the Supporting Information shows the factor used to improve their efficiencies in the different time periods. The unit costs for the treatment unis are shown in Table 22 and for the pipes in Table 23 in the Supporting Information. First, Figure 5 shows the solution of the problem considering the worst case, which avoids the use of multiperiod retrofitting by overdesigning to the maximum required capacity from the first period. Then, Figure 6 shows the optimal multiannual retrofitted configuration for example 2. Notice in Figure 6 that the network configuration is retrofitted over the different periods to increase their capacities and improve their

the other hand, Figure 4 shows the optimal network for the case in which the multiperiod retrofitting was considered. It is worth noting that in Figure 4 the treatment unit was installed during the first operating period and that the reconfiguration is given for the pipe network through the different periods considered. It should be noted in this case that the treatment units need to be retrofitted for their performance during the operational periods to properly satisfy the water and property demands for the considered sinks. To show the details for the streams involved in the optimal water network through the different time periods, Tables 10− 12 in the Supporting Information show the flow rates for the streams involved in the water network in the different operation periods through the time horizon; furthermor, Tables 13 and 14 in the Supporting Information show the values for the stream properties for the different pipe segments over the different time periods. Finally, Table 15 in the Supporting Information shows the main economic results for example 1; both scenarios have been reported, the one for the multiperiod retrofit and the one associated with the worst case. It should be noted that in this case the solution for the worst case does not require fresh sources; this is because the treatment units installed in the first period are able to regenerate the wastewater to satisfy properly the water demands for the sinks in the entire time horizon. On the other hand, the multiannual retrofit involves fresh sources but lower treatment units as well as piping costs. This way, the total annual cost for the multiannual retrofit case is 19.75% lower than that of the worst case scenario. This shows the I



Article

Figure 6. Optimal multiperiod retrofitting for example 2.

efficiencies. To show the details of the optimal solution shown in Figure 6, Table 24 in the Supporting Information shows the flow rates from the process sources to the sinks in the different periods. Table 25 in the Supporting Information shows the flow rates from process sources to the treatment units. Table 26 in the Supporting Information shows the flow rates from the treatment units to sinks. Tables 27−29 in the Supporting Information show the capacities and efficiencies for the treatment units considered over the different time periods.

The properties entering the sinks are shown in Table 30 in the Supporting Information. Table 31 in the Supporting Information shows the main economic results for example 2 for the optimal solution in the multiannual retrofit and for the worst case optimization. It should be noted that the worst case consumes 400% more water than the multiperiod retrofit solution. The installation cost for the multiannual retrofit is 24% greater than the worst case solution, and the operating cost for the treatment units in the worst case solution is 87% greater than that in the J



Article

135/1433. The authors therefore acknowledge with thanks

multiperiod retrofit. Furthermore, the solution for the worst case has a pipe cost 88% greater than that of the multiperiod retrofit solution. Finally, the total cost for the multiperiod solution is 41% lower than that associated with the worst case solution, which shows the importance of the multiperiod retrofit approach. Finally, Table 32 in the Supporting Information shows the problem size for each case study and the computational time using a computer with an i7 processor at 2.2 GHz with 6 GB of RAM. In this case, the solver DICOPT was used for solving the MINLP problem, whereas the solvers CONOPT and CPLEX were used to solve the associated NLP and LP problems, respectively. To obtain these solutions, the following approach was implemented: First, the problem without involving treatment units and for a single period (the worst case) was solved. Then, the problem was solved (using previous solution as initial guess) for the multiperiod case. Finally, the problem is solved including the treatment units. This initialization approach helps to reduce the computation time. Furthermore, the solver BARON was tested, but no numerical solution was obtained because of the difficulty associated with finding good upper and lower bounds for the variables involved in the nonconvex terms.

DSR technical and financial support.

■

Parameters

δ = Lower limit for the flow rate in the pipes CTi,j = Unitary cost for the pipe from source i to sink j, $ CTi,u = Unitary cost for the pipe between source i with unit u, $ CTr,j = Unitary cost for the pipe between the fresh source r to the sink j, $ CTu,j = Unitary cost for the pipe between the unit u and sink j, $ CTu,u′ = Unitary cost for the pipe between the unit u and unit u′, $ COu = Operational cost for unit u, per kg treated, $/kg Costor,t = Fresh source cost in period t, $/kg CFIu = Fixed cost to install the unit u, $ CFIu = Variable cost to install the unit u, $/(kg/h) CFAu = Fixed cost to increase the capacity of the unit u, $ CVAu = Variable cost to increase the capacity of the unit u, $/(kg/h) CFM u = Fixed cost to improve the performance of the unit u, $ CVM u = Variable cost to improve the performance of the unit u, $/(kg/h) Gmin j,t = Minimum flow rate for the sink j in period t, kg/h Gmax j,t = Maximum flow rate for the sink j in period t, kg/h HY = Hours per year that the plant operates, h/year KFt = Annualization factor for the capital costs in period t, 1/ year Meu,p = Factor to improve the efficiency for unit u for property p pipi,j = Pumping cost for segment i to j, $/(kg/h) pipi,u = Pumping cost for segment i to u, $/(kg/h) pipr,j = Pumping cost for segment r to j, $/(kg/h) pipu,j = Pumping cost for segment u to j, $/(kg/h) pipu′,u = Pumping cost for segment u to ú, $/(kg/h) Wi,t = Flow rate for process source i in period t, kg/h YRt = Years in period t, years ψmin j,p,t = Minimum operator for property p in sink j in period t ψmax j,p,t = Maximum operator for property p in sink j in period t ψi,p,t = Value of operator for property p in source i in period t ψr,p,t = Value of operator for property p in fresh source r in period t

■

CONCLUSION This paper has presented an optimization approach for the design and retrofitting of water networks using a multiperiod framework. An important feature of the proposed approach is that it considers the future changes in the water demands and constraints based on future projections. This approach is superior to overdesigning the water network from the first period. The incorporation of property-based characterization of the sources and constraints for the sinks accounts for realistic water-recycle strategies. The proposed optimization approach is based on a disjunctive formulation which is reformulated as a mixed integer nonlinear programming model that can be solved in a short CPU time without significant computational problems. The proposed model was applied to two case studies. The results show that significant improvements can be obtained.

■

ASSOCIATED CONTENT

S Supporting Information *

All data and detailed results for the presented examples. This material is available free of charge via the Internet at http:// pubs.acs.org.

■

NOMENCLATURE

Variables

AUTHOR INFORMATION

Costu,t = Total cost for use of treatment unit u in period t, $ Costau,t = Cost of increase size of treatment unit u in period t, $ Costmu,t = Cost of improve performance in unit u in period t, $ Costtubi,j,t = Cost of pipes between sources i and sinks j in period d, $ Costr = Total cost of fresh sources r, $/year Costou = Total cost of treatment units u, $/year Costopipi,j = Total cost of pipes between sources i and sinks j, $/year Costopipi,u = Total cost of pipes between sources i and treatment units u, $/year Costopipr,j = Total cost of pipes between fresh sources r and sinks j, $/year

Corresponding Author

*E-mail: [email protected]. Tel.: +52 443 3223500, ext. 1277. Fax.: +52 443 3273584. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge the financial support from the Mexican Council for Science and Technology (CONACyT) and the Scientific Research Council of the Universidad Michoacana de San Nicolás de Hidalgo in Mexico. In addition, this project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under Grant 263/ K



Article

Costopipu,j = Total cost of pipes between treatment units u and sinks j, $/year Costopipu′,u′ = Total cost of pipes between treatment units u and other treatment units u′, $/year Costopip = Total cost of piping and pipes, $/year Efuse u,t,p = Maximum efficiency used in the treatment unit u before period t Ef1u,t,p, Ef2u,t,p = Disaggregated efficiency used in treatment unit u in period t Fr,t = Flow rate used of fresh sources r in the plant in period t, kg/h f r,j,t = Flow rate send from fresh source r to sinks j in period t, kg/h Gj,t = Flow rate send to sinks j in period t, kg/h Hu,t = Flow rate treated in treatment unit u in period t, kg/h h1u,j,t = Flow rate send from treatment units u to sinks j in period t, kg/h h2u′,u,t = Flow rate send from treatment units u to other treatment unit ú in period t, kg/h A2 HA1 u,t , Hu,t = Disaggregated flow rates in treatment unit u in period t, kg/h Huse u,t = Maximum flow rate treated in treatment unit u before period t, kg/h AB2 AB HAB1 u,t , Hu,t = Disaggregated flow rates of Hu,t , kg/h 1 wi,j,t = Flow rate sent from process source i to sinks j in period t, kg/h w2i,u,t = Flow rate send from process source i to treatment unit u in period t, kg/h w1,use i,j,t = Maximum flow rate send from source i to sink j, kg/h Yu,t = Boolean variable for existence of a treatment unit u in period t yu,t = Binary variable for existence of a treatment unit u in period t A2 YA1 u,t , Yu,t = Boolean variables for the previous use of treatment unit u in period t A2 yA1 u,t , yu,t = Binary variables for the previous use of treatment unit u in period t B2 YB1 u,t , Yu,t = Boolean variables for increase size of treatment unit u in period t B2 yB1 u,t , yu,t = Binary variables for increase size use of treatment unit u in period t C2 YC1 u,t ,Yu,t = Boolean variables for improve performance of treatment unit u in period t C2 yC1 u,t ,yu,t = Binary variables for improve performance of treatment unit u in period t Ypip i,j,t = Boolean variable for use of pipes between sources i and sinks j in period t ypip i,j,t = Binary variable for use of pipes between sources i and sinks j in period t D2 YD1 i,j,t ,Yi,j,t = Boolean variables for installation of pipes between sources i and sinks j in period d D2 yD1 i,j,t,yi,j,t = Binary variables for installation of pipes between sources i and sinks j in period d TC = Total cost of plant, $/year ψinu,p,t = Value of operator property p to inlet at the treatment unit u in period t ψout u,p,t = Value of operator property p to outlet at the treatment unit u in period t ψj,p,t = Value of operator property p in sinks j in period t

NSOURCES = Set for the process sources (i|i = 1,..., NSOURCES) NTIME = Set for the periods (t|t = 1,..., NTIME) NTREAT = Set for the treatment units (u|u = 1,..., NTREAT) Indexes

■

D = Disaggregated i = Process sources in = Inlet j = Sinks max = Maximum min = Minimum out = Outlet conditions p = Properties r = Fresh sources u = Treatment units u′ = Treatment units connect with other treatment unit

REFERENCES

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Sets

NFRESH = Set for the fresh sources (r|r = 1,..., NFRESH) NPROP = Set for the properties (p|p = 1,..., NPROP) NSINKS = Set for the sinks (j|j = 1,..., NSINKS) L



Article

(16) Raman, R.; Grossmann, I. E. Modeling and computational techniques for logic based integer programming. Comput. Chem. Eng. 1994, 18, 563. (17) Ponce-Ortega, J. M.; Jiménez-Gutierrez, A.; Grossmann, I. E. Simultaneous retrofit and heat integration of chemical processes. Ind. Eng. Chem. Res. 2008, 47, 5512.

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Optimal Synthesis of Property-Based Water Networks Considering

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