Superstructure-Based Synthesis Framework for a Batch Water

Article pubs.acs.org/IECR

Superstructure-Based Synthesis Framework for a Batch Water Network with Multiple Regeneration Modules Hongguang Dong,* Shuming Wang, Xiong Zou,* Zhizhong Han, and Lei Sun School of Chemical Engineering, Dalian University of Technology, No. 2 Linggong Road, 116024, Dalian, People’s Republic of China S Supporting Information *

ABSTRACT: A systematic approach for optimal batch water network synthesis is presented in this work, where multiple wastewater regeneration modules with various candidate technologies are taken into account. The integration of different regeneration technologies could further increase water reuse opportunities in the batch plants, thereby significantly reducing freshwater consumption and wastewater generation. Specifically, the regeneration modules are divided into batch and semicontinuous operating modes, which have never been considered completely in previous studies. A modified state-space superstructure incorporating a waterusing subsystem, a regeneration subsystem, and buffer tanks is adopted to capture all realistic potential configurations. Correspondingly, a mixed-integer nonlinear programming model embedded with the scheduling of regeneration subsystem and buffer tanks is formulated to minimize the total annualized cost. In addition, a rule based solution procedure is developed to reduce the computation complexity and two illustrative examples are presented to demonstrate the effectiveness of the proposed methodology.

1. INTRODUCTION Wastewater minimization has received significant attention from both academic and industrial researchers, because of the stringent legislation on effluent and the scarcity of freshwater. Over the past decades, much focus has been paid on water networks synthesis of continuous manufacturing industries, since these plants generally consume large amounts of water.1−4 Nonetheless, minimization of wastewater in batch processes is also important. Batch processes are often encountered in the production of high-valued products in small quantities, such as fine chemicals, pharmaceuticals, and agrochemicals. These industries have a tendency to generate wastewater with highconcentration toxic substances. Also, the frequent sharing of equipment by different tasks in batch plants requires more water to clean the equipment. Thus, because of the increasing application of batch production in process industries and the above-mentioned inherent nature of batch operation, it is urgent to minimize the wastewater consumption within batch processes by exploiting the opportunities for water reuse and removing hazardous material with effective regeneration facilities. Generally, synthesis of a water network is more challenging for batch processes, because of the additional time dimension. The methods for the synthesis of batch water network appearing in the literatures can be broadly classified into two categories: insight-based approaches and mathematical optimization-based approaches. The former typically involves water pinch analysis initiated by the pioneering works of Wang and Smith.5 Then, Foo et al.6 introduced a two-stage procedure to © XXXX American Chemical Society

synthesize maximum water recovery network for a batch process system based on time-dependent water cascade analysis. Apart from semicontinuous water-using processes, a graphical technique is also applied to minimize wastewater in truly batch process.7 Liu et al.8 employed concentration interval analysis to optimize batch water-using system that involves both non-mass-transfer-based and mass-transfer-based operations. Chen and Lee9 utilized graphical analysis technology to design the batch water network that includes different types of water-using operations. Furthermore, Kim10 proposed a systematic graphical design method to simultaneously consider concentration and time-interval information in batch water networks. For batch water systems with multiple contaminants, Li et al.11 developed heuristic strategies to manually design water-using networks. The graphical-based method offers insights and acceptable solutions at low computation burden; however, the significant problem simplifications are always required. Mathematical optimization techniques have been developed to handle the water network synthesis problems in their full complexity, where economic goals, multiple contaminants, and various topological structures could be taken into consideration. Early models mainly considered a single contaminant in a batch water system. For instance, Almato et al.12 developed a Received: July 22, 2016 Revised: October 1, 2016 Accepted: October 10, 2016

A

DOI: 10.1021/acs.iecr.6b02794 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

Figure 1. Schematic representations of regeneration operating modes: (a) batch, (b) single outflow semicontinuous, and (c) two outflows semicontinuous.

reused, for all types of regeneration modes. Specifically, for two outflows semicontinuous regeneration mode, there exists one other outlet stream, namely a concentrate stream, with higher contaminant concentration than the inlet stream, as shown in Figure 1c. Because of the natural extension of methodologies for continuous regeneration, single outflow semicontinuous regeneration unit is preferred in many papers, and these contributions considered only one central regeneration unit in the superstructures.35−38 Cheng and Chang39 initiated an effective formulation to integrate water-using subsystem and wastewater treatment subsystem in batch plants, where batch treatment operations with identical and nonidentical charging and discharging were incorporated. Then, Tokos and coworkers40−42 developed a systematic approach to handle the problem of a single contaminant water network synthesis in a brewery plant based on the work of Kim and Smith,43 and both batch and semicontinuous regenerators are investigated. In these papers, direct connections exist between water-using units and semicontinuous regenerator without buffer tanks, which assumed that wastewater purified in the semicontinuous regenerator was available for reuse immediately, irrespective of the amount of wastewater. In addition, it is noteworthy that the above-mentioned research has mainly focused on single outflow semicontinuous regeneration or batch regeneration: not much research has considered two outflows semicontinuous regeneration and the effect of different regeneration modes on water-using networks of batch plants in specific criteria. Recently, various regeneration technologies have been considered in the framework of a continuous water network synthesis. For example, Khor et al.44,45 developed a source− regenerator−sink superstructure to optimize a total water network with membrane separation-based regenerators; both short-cut and detailed models of these regenerators are respectively considered. Also, detailed models of reverse osmosis and electrodialysis have been extensively studied.46−48 Yang et al.49 established an optimization framework for the continuous water system, combining various regeneration technologies with shortcut models, and a realistic water network synthesis of drilling and fracturing processes in shale gas production was subsequently performed.50 Nevertheless, less attention has been paid to regeneration technologies in batch plants, where the selection of different technologies has a tendency to be more complex. Based on the operation modes, the regeneration technologies can be classified as batch regeneration technologies, such as sequential batch reaction (SBR), membrane bioreactor (MBR), and in situ regeneration; single outflow semicontinuous regeneration technologies, such as adsorption; and two outflows semicontinuous regeneration technologies, such as reverse osmosis.51 However, few works have considered the integration of

nonlinear optimization model on the basis of water indirect reuse superstructure with multiple storage tanks, and they accounted for different cost objective functions. Majozi13 formulated a mixed-integer linear programming (MILP) model that exploited all direct reuse and recycle opportunities to reduce wastewater generation. Extensive research for waterusing processes with multiple contaminants then was performed. Chen et al.14 proposed a superstructure to entail all possible recycle and reuse opportunities and discussed the impact of multiple storage tanks. Also, Dogaru and Lavric15 introduced an approach that accounted for dynamic characteristics of storage tanks to optimize the dynamic water-network topology. Besides, hybrid approaches to combine water pinch analysis and mathematical programming were also developed.16,17 For a thorough description of the graphical and mathematical approaches to batch water network synthesis, the reader is referred to the review paper by Gouws et al.18 After the synthesis of batch water network through water direct reuse/recycle and indirect reuse by buffer tanks has been deeply delved, regeneration process is introduced by many researchers to further reduce fresh water consumption and wastewater generation. As has been demonstrated by Faria and Bagajewicz,19 the wastewater regeneration subsystem and endof-pipe treatment should be part of the overall water system. It should be mentioned that numerous papers20−25 inspired by ElHalwagi and his co-workers’ seminal ideas,26,27 have been developed to address the problem of property-based water network synthesis, where interception units are involved to adjust the properties of water sources (e.g., composition, chemical oxygen demand, toxicity, etc.). In essence, interception can be seen as a more general concept of regeneration when mass integration and property integration are considered simultaneously. Most of the research related to regeneration units has been focused on the continuous water-using processes.28−30 Ahmetovic and Grossmann31 proposed a general superstructure and model for the global optimization of integrated water network, where one or multiple freshwater resources, various water-using processes and wastewater regeneration operations are combined into a single network. Later, differentiated regeneration has been utilized to minimize the freshwater consumption.32 In addition, multiobjective optimization has also been used to synthesize or retrofit the water network with multiple regenerators.33,34 Regeneration in the batch water system is different from its counterpart in continuous processes, because of the inherent features of batch operation. There are two types of regeneration technologies, according to the operation modes: batch and semicontinuous regeneration. Considering the number of the output streams, the latter type can be further divided as single outflow and two outflows semicontinuous regeneration. These three types of regeneration technologies can be seen in Figure 1. There is a purified outlet stream, which could be further B


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Figure 2. State-space superstructure of the batch water network.

(1) Water-using processes operate in truly batch mode, where water is charged and discharged at specific time points. (2) Water reuse from previous production cycles to the present production cycle is ignored. (3) CRs with limited flow rates are available immediately, while BRs with limited regeneration capacities are only available after the regeneration ends. The duration of the batch regeneration is given as a parameter. (4) The pipeline cost is assumed to be negligible. (5) The removal efficiency of the regeneration process is assumed to be independent of operating conditions such as flow rate and concentration. Both fixed contaminant removal ratio and fixed outlet contaminant concentration regeneration models could be adopted in the proposed synthesis framework. Furthermore, the published works for wastewater minimization in batch plants are mostly based on predefined production schedules. Meanwhile, in some recent works,39,52−58 wastewater minimization and an optimal production schedule are carried out simultaneously; the results indicated that true minimum freshwater consumption can be realized by allowing the production schedule to change. However, the computation complexity of these models increases exponentially with the problem scale. As the interaction between water-using and regeneration network has already been considered in this work, the solution of the corresponding model with variable production schedule will overwhelm the capabilities of any available solvers. Hence, this article is based on a predefined production schedule, but time is still treated as an optimization variable for the buffer tank and regeneration module. The tasks of optimizing batch production schedules, water-using network and regeneration networks will be performed simultaneously in future work. Note that the current framework has yet to consider uncertainties or variations in demand and process parameters, which could be achieved by robust optimization or stochastic programming techniques.

various regeneration technologies in a single batch water system, to the best of our knowledge. Hence, this article aims to present a general superstructure of integrated water network, which consists of a water-using network, a regeneration network, and buffer tanks, in batch plants. The selection of appropriate regeneration technologies among candidates and the effects of different regeneration modes on water network will be explored. Within the proposed mixed integer nonlinear programming (MINLP) model, the tradeoffs among freshwater consumption, regeneration cost, and end-of-pipe treatment cost are also considered. The outline of this paper is as follows. Description of the problem is defined in Section 2, and the modified state-space superstructure is illustrated in Section 3. All mathematical formulations are given in Section 4, and the rule-based optimization strategy can be found in the following section Section 5. Finally, the proposed methodology is demonstrated by two illustrative examples in Section 6, and general conclusions are provided in the last section Section 7.

2. PROBLEM STATEMENT The problem addressed in this article is stated as follows. Given are the following: (i) a set of batch water-using operations with known mass loads of each contaminant, (ii) a predefined production schedule, (iii) a set of fresh water sources and end-of-pipe treatments, (iv) a set of batch regeneration modules (BRs), (v) a set of semicontinuous regeneration modules (CRs), (vi) four sets of storage tanks: those installed before the BRs (ub ∈ Ub), after the BRs (vb ∈ Vb), before the CRs (uc ∈ Uc), and after the CRs (vc ∈ Vc) (vii) the necessary technical and economic data, and (viii) the time horizon of interest. The objective is to obtain a cost-optimal total water network embedded with a regeneration subsystem for batch processes. The solution also includes the schedules of active regeneration modules and storage tanks. Note that the capital and operational cost of the water network will be taken into consideration in the objective function. The following assumptions are made for the synthesis problem.

3. SUPERSTRUCTURE The state-space superstructure was proposed by Bagajewicz and Manousiouthakis59 and has been proven to be highly effective in handling synthesis of complex process network. The original structure has been modified in the present work to synthesize a C


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position sufficient regeneration modules and storage tanks to provide enough reuse opportunities. In this study, the initial numbers of them are chosen heuristically as the total number of candidate technologies and the actual number rests on the optimization process.

batch water-using network and regeneration network simultaneously. The overall superstructure is viewed as a system of two interconnected blocks, as shown in Figure 2. One is called the distribution network (DN), where all corresponding mixers, splitters, and connections between them are embedded. The freshwater streams and effluents to the end-of-pipe treatment are considered as the external inputs and outputs of the DN block, respectively. The other is the so-called process operator (PO), which can be further divided into three sub-blocks, i.e., water-using operator (WO), regeneration operator (RO), and storage operator (SO). WO represents a water-using operation in a truly batch pattern, wherein water is supplied from mixers attached on the upper side of the DN at the start and wastewater is discharged to splitters attached on the left side of DN at the end of its operation. RO includes multiple BRs and CRs to satisfy the purification need of the water network, and SO consists of storage tanks installed before and after the regeneration modules. Wastewater flows into RO and SO by the mixers attached on the right side of the DN, while purified and stored water is returned to the splitters attached on the lower side of the DN. In this superstructure, the splitting and mixing can only occur in the DN, where one-to-one correspondences between mixers/splitters and water users/ regeneration modules/buffer tanks are ensured. To capture an entire family of alternative structures, the fully connected state-space superstructure could be easily constructed by splitting every input stream to the DN into several branches and connecting each of them to a mixing node leading to PO blocks or to the end-of-pipe treatment. Specifically, all direct, indirect, and regeneration reuse and/or recycle opportunities can be included and modeled via the interactions between the DN and the PO. However, in essence, there are a large amount of infeasible and redundant configurations in the completed superstructure, which always lead to a combinational explosion. Based on the problem statement and the insights on the water network synthesis, some structural simplifications are adopted and listed as follows: (1) the freshwater can only feed to the mixers for water-using operations; (2) since the flow rate of wastewater should be adjusted to satisfy the operating range of CRs, as well as dampen sharp flow variation, wastewater must flow into storage tanks Uc before it is purified in the CRs and the regeneration water should be stored in the storage tanks Vc; (3) there is no connection between storage tanks Ub (Uc) and CRs (BRs); (4) for a CR with two-outflows regeneration technology, the concentrate stream flows directly to the end-of-pipe treatment facility; (5) wastewater produced by water-using operations could be stored in buffer tanks Uc and Ub, but could not be allowed to flow directly into buffer tanks Vc and Vb; and (6) the outlet streams of BRs could not be allowed to flow into buffer tanks Ub. There are various alternative regeneration technologies within each operating mode and, at most, one technology can be selected in each active regeneration module. The operation capacity of regeneration module is determined by the selected technology. Thus, the effect of regeneration modes and the selection of various regeneration technologies are also taken into consideration. Finally, we must note that it is imperative to

4. MATHEMATICAL MODEL Based on the proposed superstructure, a mathematical model is formulated to synthesize the total water network for batch plants. Because of the presence of bilinear terms, the resulting formulation gives rise to a nonconvex MINLP, or more specifically, a mixed-integer quadratically constrained quadratic program (MIQCQP). The overall integrated mathematical model is composed of five parts and will be introduced separately. For the sake of clarity, all parameters in the model start with capital letter, while the variables are presented in italic font. The constraints in the mathematical model mainly consist of mass balances (water and contaminant) for water-using operations, regeneration modules, and buffer tanks. Variable bounds and constraints associated with the selection of regeneration technologies are also formulated by relating the binary variables. In addition, the schedule constraints for regeneration modules and buffer tanks are presented. Note that the synthesis model is established on the basis of discrete-time representation, in which time horizon is uniformly divided into N − 1 equal time intervals by N time points t ∈ T. By providing a reference grid of time for both batch and semicontinuous operations, this renders the possibility of formulating the scheduling constraints in a relatively straightforward and simple manner. Furthermore, the time discretization is made fine enough such that all operations can always be considered to start and end at the boundary of time intervals. 4.1. Distribution Network. As shown in the superstructure, all interconnection streams in the DN are characterized by the splitters and mixers. For the sake of simplicity, the superscripts “in” and “out” respectively denote the inlet and outlet physical quantities of PO. Furthermore, set SP is introduced to represent all splitters in DN, while set MX is used to denote all mixers attached to DN. Obviously, the mass balances should be satisfied at every splitter and mixer. The concentrations in the outlet streams are assumed to be identical to that in the inlet stream of the splitter, which are all denoted by the same variables. Thus, there are only water balance constraints for the splitters. 4.1.1. Mass Balances for the Mixers Attached to the Upper Side of the Distribution Network. Constraints (1) and (2) are water and contaminant balances around the inlet mixers attached to the upper side of the DN for water-using operations, respectively, which show that the inlet water of a water-using operation i ∈ I may come from freshwater sources f ∈ F, storage tanks sr ∈ SR, water-using operations i′ ∈ I, and batch regeneration modules br ∈ BR. Constraint (3) is the inlet water balance for the end-of-pipe treatment facility e ∈ E, which shows that the total amount of wastewater flowing into the endof-pipe treatment facility in the time horizon contains wastewater from water-using operations and concentrate streams from two outflows semicontinuous regeneration modules. D


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∑ mf , i , t + ∑

miin, t =

f ∈F

msr, i , t +

∑ mi′ ,i ,t + ∑ i ′∈ I

sr ∈ SR

+

∑

out msr, i , t csr, c ,t +

sr ∈ SR

out mbr, i , t cbr, c ,t

i ′∈ I

∀ i ∈ I, c ∈ C, t ∈ T (2)

br ∈ BR

N−1

me =

∑ ∑ mi ,e ,t + ∑ ∑ ΔTfcr,e ,t t∈T i∈I

∀e∈E

in mbr, t =

cr ∈ CR t = 1

∑ mi ,br, t + ∑

(3)

i∈I

4.1.2. Mass Balances for the Splitters Attached to the Left Side of the Distribution Network. The water balance around the outlet of the water-using operation is given in constraint (4), which indicates that the outlet splitters attached to the left side of the DN could be connected to the mixers assigned for storage tanks ub and uc, water-using operations i′, batch regeneration modules br, and the end-of-pipe treatment facility e. miout ,t =

∑ ub ∈ Ub

∑

+

∑

mi , ub , t +

uc ∈ Uc

∑ mi ,ub ,t

+

i∈I

mvb , uc , t +

∑

mvc , uc , t +

vc ∈ Vc

∑

∑

∑

ΔTfcr, v , t − 1

muinb , t cuinb , c , t =

vc ∈ Vc , t ∈ T, t > 1

c

cr ∈ CR

in in f cr, c = t cr, c , t

∑ mi ,u ,t ciout ,c ,t

(7)

(8)

muout = b,t (9)

∑ mi ,u ,t ciout ,c ,t + ∑ c

i∈I

+

∑ vc ∈ Vc

+

∑

mvinb , t cvinb , c , t =

muout = c ,t

out mbr, uc , t cbr, c ,t

br ∈ BR

vb ∈ Vb, t ∈ T

∀ cr ∈ CR, c ∈ C , t ∈ T

c

∑ mu , i , t + ∑ b

mub ,br, t

ub ∈ Ub, t ∈ T

br ∈ BR

∑ mu , i , t + ∑ c

cr ∈ CR

ΔTfu ,cr, t − 1 |t > 1 c

uc ∈ Uc, t ∈ T

mvout = b,t

(10) out mbr, vb , t cbr, c ,t

fu ,cr, t cuout c,c ,t

i∈I

br ∈ BR

∑

∑

(15)

(17)

mvb , uc , t cvout b,c ,t

uc ∈ Uc, t ∈ T

∀ cr ∈ CR, t ∈ T

c

i∈I

vb ∈ Vb

mvc , uc , t cvout c,c ,t

fu ,cr, t

(16)

i∈I

muinc , t cuinc , c , t =

(14)

where fincr,t represents the inlet flow rate of semicontinuous regeneration module cr ∈ CR at time point t ∈ T, which is equal to the sum of flow rates of streams coming from storage tanks uc ∈ Uc. 4.1.4. Mass Balances for the Splitters Attached to the Lower Side of the Distribution Network. Constraints (17)− (20) and constraint (21) represent mass balances around the splitters attached to the lower side of the DN for storage tanks and batch regeneration module br, respectively. Water stored in tank sr is transferred to the corresponding mixers in the DN, and the purified water in batch regeneration module br may flow to water-using operations i, storage tanks uc, or vb, and other batch regeneration module br′.

ub ∈ Ub, c ∈ C , t ∈ T

b

mvc ,br, t cvout c ,c ,t

out mbr ′ ,br, t cbr ′,c ,t

uc ∈ Uc

mbr, uc , t

vb ∈ Vb, t ∈ T

∑ uc ∈ Uc

br ∈ BR

br ∈ BR

mvinc , t =

∑

+

vc ∈ Vc

∑

in f cr, = t

(6)

mbr, vb , t

ub ∈ Ub

mvb ,br, t cvout b,c ,t

Similarly, constraint (15) is a water balance around the mixer attached on the right side of the DN for semicontinuous regeneration module cr, and the contaminant balance is ensured by eq 16.

(5)

uc ∈ Uc, t ∈ T

mvinb , t =

∑

mub ,br, t cuout b,c ,t

∀ br ∈ BR, c ∈ C , t ∈ T (4)

vb ∈ Vb

(13)

∑ mi ,br,t ciout ,c ,t + ∑

vb ∈ Vb

∀ i ∈ I, t ∈ T

ub ∈ Ub, t ∈ T

∑ mi ,uc ,t + ∑

mvc ,br, t

br ′∈ BR,br ′≠ br

i∈I

muinc , t =

∑ vc ∈ Vc

∀ br ∈ BR , t ∈ T

i∈I

4.1.3. Mass Balances for the Mixers Attached to the Right Side of the Distribution Network. Constraints (5)−(8) represent water balances around the inlet mixers attached to the right side of the DN for storage tanks, and the corresponding contaminant balances are shown as constraints (9)−(12). Note that the inlet mass balance for storage tank vc is based on the flow rate and is apparently different from the others, since water is transferred to other tanks in a truly batch pattern. muinb , t =

mbr ′ ,br, t

in in mbr, t c br, c , t =

+

e∈E

br ∈ BR

mvb ,br, t +

vb ∈ Vb


i ′∈ I

∑ mi ,e ,t

mi ,br, t +

∑

mub ,br, t +

ub ∈ Ub

∑

+

∑ mi ,i′ ,t

mi , uc , t +

(12)

On the right side of the DN block, mixers also exist for the batch regeneration modules, the water and contaminant balances for which are respectively given in constraints (13) and (14). The inlet water of batch regeneration module br ∈ BR may flow from storage tanks ub ∈ Ub, vb ∈ Vb, and vc ∈ Vc, water-using operation i ∈ I, and another batch regeneration module br′ ∈ BR.

∑ mi′ ,i ,t ciout ′,c ,t

f ∈F

c

∀ vc ∈ Vc , t ∈ T, t > 1

(1)

∑ mf ,i ,t Cf ,c + ∑

miin, t ciin, c , t =

out ΔTfcr, v , t − 1 ccr, c ,t−1

cr ∈ CR

br ∈ BR

∀ i ∈ I, t ∈ T

∑

mvinc , t cvinc , c , t =

mbr, i , t

∑ mv , i , t + ∑ b

i∈I

vb ∈ Vb, t ∈ T

(11) E

(18)

br ∈ BR

mvb ,br, t +

∑

mvb , uc , t

uc ∈ Uc

(19) DOI: 10.1021/acs.iecr.6b02794 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research mvout = c ,t

∑ mv , i , t + ∑

mvc ,br, t +

c

i∈I

∑ uc ∈ Uc

br ∈ BR

vc ∈ Vc , t ∈ T out mbr, t =

(20)

∑ mbr,i ,t + ∑ i∈I

mbr, uc , t +

uc ∈ Uc

∑

+

starting time nor finishing time for the operation. On the other hand, the amount of inlet and outlet water of operation i at time point t could also be bounded in a reasonable range if the operation starts or ends at that time point. Constraints (30) and (31) respectively give the maximum inlet and outlet concentrations of water-using operation i.

mvc , uc , t

mbr,br ′ , t

∑

mbr, vb , t

vb ∈ Vb

Mimin Ziin, t ≤ miin, t ≤ Mimax Ziin, t

∀ br ∈ BR, t ∈ T

out max out Mimin Ziout Zi , t , t ≤ mi , t ≤ Mi

(21)


At the splitter attached to the lower side of the DN for semicontinuous regeneration module cr, purified water is transferred to a set of storage tanks vc, stated as constraint (22). The concentrate stream of module cr directly flows to the end-of-pipe treatment facility e through the splitter attached to the lower side of the DN, shown as constraint (23). Since freshwater can only flow to water-using operations, total amount of freshwater f consumed in the horizon is calculated by eq 24. out f cr, = t

∑

fcr, v , t

con f cr, = t

∑ fcr,e ,t

(23)

∀f∈F

∑

(24)

i∈I t∈T

∀ sp ∈ SP, mx ∈ MX, t ∈ T (25)

=

miin, t

+

Ziin, tMiΔ

1 N

∀ i ∈ I, t , t ′ ∈ T, t ′ = t + Δti

=

Ziin, t Mli , c

+

(31)

(32)

t∈T

∀ br ∈ BR (33)

t∈T

∑

wbr,tbΔt tb

∀ br ∈ BR (34)

tb ∈ TB

miin, t ciin, c , t

∀ i ∈ I, t , t ′ ∈ T, c ∈ C , t ′ = t + Δti

(30)

∀ br ∈ BR

∑ wbr,t ≤ wbr ≤ ∑ wbr,t

dubr =

(26) out miout , t ′ ci , c , t ′

(29)

Since the characteristics of the batch regeneration module are dependent on the selected technology, wbr,tb will be adopted in the following constraints to ensure that the technical and economic parameters of selected technology tb could be active in the module br. 4.3.1.2. Scheduling Constraints for BRs. Constraint (33) ensures that the module br exists if br is occupied in no less than one time interval. In addition, the duration of batch regeneration in module br with the selected technology tb is defined in constraint (34). Constraint (35) further expresses the requirement that, if br starts regeneration at time point t, then no other regeneration can start in br until the former one is finished after the duration dubr. However, constraint (36) indicates that batch regeneration should start and end within the given time horizon.

4.2. Water-Using Operator. The overall water balance for a water-using operation in WO is given by constraint (26), and the mass balance for the contaminant is given by constraint (27). Since the production schedule of a water-using operation is predefined, two sets of binary parameters, Zini,t and Zout i,t , are introduced to indicate, respectively, the starting and finishing times of water-using operations. Zini,t = 1 means that operation i starts at time point t, while Zout i,t = 1 signifies that the operation i finishes at time point t. Note that the outlet flow is always prior to the inlet flow when a water-using operation is finishing and starting immediately at the same time point miout ,t′

wbr,tb = wbr

tb ∈ TB

4.1.5. Variable Bounds in the Distribution Network. The upper limit for the amount of water streams from corresponding splitter sp to the mixer mx in the DN is formulated as constraint (25). msp,mx, t ≤ Mspmax,mx

∀ i ∈ I, c ∈ C, t ∈ T

(28)

4.3. Regeneration Operator. 4.3.1. Batch Regeneration Modules. 4.3.1.1. Constraints for Technology Selection in BRs. Within the uniform discrete time formulation, binary variable wbr,t is introduced to determine whether or not batch regeneration starts in module br at time point t, while wbr indicates whether module br is selected in the optimal network configuration. Binary variable wbr,tb is introduced to specify if batch regeneration technology tb is chosen in module br. Exactly one batch regeneration technology can be selected from set TB for each active batch regeneration module br, which is described as constraint (32).

∀ cr ∈ CR, t ∈ T

∑ ∑ mf , i , t

∀ i ∈ I, t ∈ T

∀ i ∈ I, c ∈ C, t ∈ T

out,max ciout , c , t ≤ Ci , c

(22)

e∈E

mf =

ciin, c , t ≤ Ciin,max ,c

∀ cr ∈ CR, t ∈ T

c

vc ∈ Vc

∀ i ∈ I, t ∈ T

wbr, t ′ ≤ 1 − wbr, t (27)

∀ br ∈ BR, t , t ′ ∈ T, t < t ′ < t + dubr ≤ N

MΔi

where denotes the amount of the water changing in waterusing operation i and M1i,c represents the mass load of contaminant c in water-using operation i, considering the concentration effect of water gain and loss. Besides, parameter Δti denotes the distance between time points of the start and finish of water-using operation i. The lower and upper bounds for the amount of inlet and outlet water of water-using operation are respectively described as constraints (28) and (29). Those two equations ensure that no water flows in and out when the time point is neither a

wbr, t = 0

(35)

∀ br ∈ BR, t ∈ T, N − dubr < t ≤ N (36)

where N is the total number of the time points in the given time horizon H, and parameter Δttb denotes the duration of batch regeneration technology tb. By introducing a large enough positive parameter 4 , constraints (33)−(36) can be typically reformulated as constraints (37) and (38). Obviously, the reformulations could also be performed to other similar constraints in this F


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Industrial & Engineering Chemistry Research paper. For the sake of brevity, only one 4‐parameter is used in the formulations; however, multiple 4‐parameters are selected by the strategy proposed by Trespalacios and Grossmann60 to accelerate the solving procedure. It is noted that other types of reformulation methods, such as the convex hull method, could also be used.

in cbr, c ,t ≤

wbr, t ′ ≤ 1 − wbr, t + 4(1 − wbr,tb) (37)

N

∑

wbr, t ′ ≤ 4(1 − w br,tb)

t ′= N −Δt tb+ 1

∀ br ∈ BR , tb ∈ TB , t ∈ T

1 N

(38)

where 4 is a sufficiently large positive number. For a technology tb not selected in the module br, the corresponding constraints (37) and (38) become redundant. 4.3.1.3. Mass Balance Constraints for BRs. The overall water balance for batch regeneration module br is ensured by constraint (39). The outlet concentration of the regeneration module is dependent on the model of the regeneration processes. Constraint (40) denotes contaminant balance in the module br when a fixed removal ratio for a contaminant is employed to characterize the purification efficiency. However, if a fixed outlet concentration assumption is adopted, constraint (40) will be replaced by constraint (41).

(39)

∀ br ∈ BR , tb ∈ TB , c ∈ C , t , t ′ ∈ T, t ′ = t + Δt tb (40) out Cfx,out tb, c − 4(2 − w br, t − w br,tb) ≤ c br, c , t ′

≤ Cfx,out tb, c + 4(2 − w br, t − w br,tb) (41)

con in out f cr, = f cr, − f cr, t t t

(46)

(47)

∀ cr ∈ CR, t ∈ T

(48)

out in ccr, c , t − 4(2 − wcr, t − wcr,tc) ≤ ccr, c , t (1 − Rc tc, c ) out ≤ ccr, c , t + 4(2 − wcr, t − wcr,tc)

∀ cr ∈ CR , tc ∈ TC , t ∈ T

wbr,tbCapmax − 4(1 − wbr, t ) tb

tb ∈ TB

out ccr, c ,t =

(42)

∀ br ∈ BR, t ∈ T

∀ cr ∈ CR

∀ cr ∈ CR , tc ∈ TC , t ∈ T

tb ∈ TB

in −4wbr, t ≤ mbr, t ≤ 4w br, t

wcr,tc = wcr

out ≤ f cr, + 4(2 − wcr, t − wcr,tc) t

in wbr,tbCapmin − 4(1 − wbr, t ) ≤ mbr, t tb

∀ br ∈ BR, t ∈ T

(45)

out in f cr, − 4(2 − wcr, t − wcr,tc) ≤ f cr, Rrtc t t

4.3.1.4. Capacity and Concentration Limit Constraints for BRs. Constraints (42) and (43) set lower and upper bounds to the inlet water of batch regeneration module br with a selected technology tb. Also, constraint (44) gives the maximum permissible inlet concentration of contaminant c for batch regeneration module br.

∑

∀ cr ∈ CR

t∈T

4.3.2.3. Mass Balance Constraints for CRs. To formulate a general model for single outflow and two outflows semicontinuous regeneration modules, a fictitious concentrate stream is introduced for the single outflow module. The relationship between the flow rate of the inlet and permeate stream in the semicontinuous regeneration module cr is presented by constraint (47), and the flow rate of concentrate stream could be calculated by eq 48. The permeate flow of the module is proportional to the inlet flow, in terms of the fixed recovery ratio (Rrtc). Notice that the flow rate of the fictitious concentrate stream to the end-of-pipe treatment facility will take a value of zero. In other words, the recovery ratio of a single outflow semicontinuous regeneration technology is one. Similar to batch regeneration module, constraint (49) represents the contaminant balance in the module cr with a fixed removal ratio for a contaminant (Rctc,c). If a fixed outlet concentration (Cfx,out tc,c ) is applied, constraint (50) will be active.

out ≤ cbr, c , t ′ + 4(2 − w br, t − w br,tb)

≤

t∈T

∑

out in cbr, c , t ′ − 4(2 − w br, t − w br,tb) ≤ c br, c , t (1 − Rc tb, c )

∑

∑ wcr, t ≤ wcr ≤ ∑ wcr, t

tc ∈ TC

out ≤ mbr, t ′ + 4(2 − w br, t − w br,tb)

∀ br ∈ BR, c ∈ C, t , t ′ ∈ T, t ′ = t + Δt tb

(44)

4.3.2.2. Constraints for Technology Selection in CRs. Similarly, binary variable wcr,tc is introduced to specify if semicontinuous regeneration technology tc is chosen in module cr. Constraint (46) guarantees that exactly one semicontinuous regeneration technology is selected per active cr module.

out in mbr, t ′ − 4(2 − w br, t − w br,tb) ≤ mbr, t Rrtb

∀ br ∈ BR , tb ∈ TB , t , t ′ ∈ T, t ′ = t + Δt tb

∀ br ∈ BR, t ∈ T

4.3.2. Semicontinuous Regeneration Modules. 4.3.2.1. Scheduling Constraints for CRs. The difference between the batch and semicontinuous regeneration modes is that the water purified in the br is only available after a specified processing time, while the water purified in the cr is available for use immediately. Thus, the scheduling of cr is simpler and binary variable wcr,t is introduced to signify if semicontinuous regeneration is performed in module cr at time point t. However, wcr indicates whether module cr is selected in the optimal network configuration. Constraint (45) indicates that the module cr exists if cr is occupied in no less than one time interval.

t ′= t + 1

∀ br ∈ BR , tb ∈ TB , t ∈ T

in,max wbr,tbC tb, c

tb ∈ TB

t +Δt tb− 1

∑

∑

∑ tc ∈ TC

fx,out wcr,tcC tc, c

(49)

∀ cr ∈ CR, t ∈ T (50)

Constraints (51) and (52) present restrictions on the inlet flow rate for the semicontinuous regeneration module cr.

(43) G


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Industrial & Engineering Chemistry Research Constraint (53) ensures that wastewater flows to the regeneration module cr meet the concentration regulation.

∑

in wcr,tcFtcmin−4(1 − wcr, t ) ≤ f cr, ≤ t

tc ∈ TC

∑

⎛ min TAC = ⎜⎜ ∑ ICSsr wsr + ⎝ sr ∈ SR

wcr,tcFtcmax

+

tc ∈ TC

∀ cr ∈ CR, t ∈ T

(51)

+

⎛ ⎞ wcr,tc ICtc⎟⎟ + NTC⎜⎜ ∑ mf Cf f ⎠ tc ∈ TC ⎝ f ∈F

∑ meCee + ∑ ∑ ∑ f ∈F

in −wcr, t 4 ≤ f cr, ≤ wcr, t 4 t

∀ cr ∈ CR, t ∈ T

in ccr, c ,t ≤

∀ cr ∈ CR, c ∈ C , t ∈ T

+

in,max wcr,tcC tc, c

tc ∈ TC

4.4. Storage Operator. The net increase in the amount of the water stored in the buffer tank is given by the difference between the water flowing into and from the buffer tank and there is no water flowing into the tank at the beginning of the time horizon. The overall water balance around specific storage tanks sr in SO is given by constraints (54) and (55), and the corresponding mass balances for a contaminant are defined by constraints (56) and (57). Note that constraints (56) and (57) are based on the assumption that the contaminant concentration of the exit stream is the same as the concentration inside the storage unit. In addition, constraint (58) ensures that the water in the buffer tank sr does not exceed the maximum storage capacity and constraint (59) is imposed to guarantee the feasibility of mass balance for buffer tank uc installed ahead of semicontinuous regeneration modules. ∀ sr ∈ SR, t ∈ T, t > 1 (54) out qsr,1 = Q sr,0 − msr,1

∀ sr ∈ SR

(55)

out out in in out out qsr, t csr, c , t = qsr, t − 1csr, c , t − 1 + msr, t csr, c , t − msr, t csr, c , t

∀ sr ∈ SR, t ∈ T, t > 1 out out out qsr,1csr, c ,1 = Q sr,0Csr, c ,0 − msr,1 csr, c ,1

qsr, t ≤ wsrQ srmax

qu , t ≥ c

∑ cr ∈ CR

(56)

∀ sr ∈ SR

∀ sr ∈ SR, t ∈ T

ΔTfu ,cr, t c

∀ uc ∈ Uc, t ∈ T, t < N

(60)

5. SOLUTION PROCEDURE A dedicated optimization procedure is developed to enhance the solution quality of the aforementioned MINLP model. Specifically, the proposed solution procedure can be divided into two stages. The first stage is to eliminate regeneration technologies inferior to others, based on preprocessing rules, which could effectively reduce the solution space. The second stage is similar to that observed in the methods described by Dong et al.61 and Zhou and Li.62 In this interactive solution framework, the proposed MINLP model is solved by deterministic algorithms with initial guesses of specified variables. The selected variables for initial guesses are the binary variables wbr,tb and wcr,tc, and continuous variables fcr,vc,t, f uc,cr,t, msp,mx,t and qsr,t. The main goal at this point is to obtain good local optimal solutions by directly solving the MINLP model with the randomly generated initial values. If there is no feasible solution, a relaxed mixed-integer linear program (RMINLP) is then solved with the same initial guesses, and the corresponding feasible solution is slightly modified to generate new initials for the original problem. To determine the distribution of the local optimal solutions of the MINLP problem, the second stage is repeated until a predetermined number of feasible solutions are captured. As observed in previous works and examples in this paper, good local optimal solution might be found by stage II for a moderate size problem. To select proper regeneration candidates for the MINLP model from various available technologies, four preprocessing rules are presented and each is described in detail below, according to priority order. Rule 1: If one contaminant can only be purified by a certain regeneration technology, select this regeneration technology. Rule 2: For CRs, the selection of available technologies should be made based on the purification efficiencies and investments. Two economic indicators, namely, capital cost per unit of capacity (ICtc) and operating cost per unit of regenerated water (OCtc) are introduced and the equation definitions are shown in constraints (61) and (62), respectively.

(53)

in out qsr, t = qsr, t − 1 + msr, t − msr, t

⎞ in ⎟ ΔTf cr, w OC tc⎟ t cr,tc tc ∈ TC ⎠

∑ ∑ ∑ cr ∈ CR t = 1

∑

in mbr, t w br,tbOCtb

br ∈ BR t ∈ T tb ∈ TB N−1

(52)

wbr,tbICtb

∑ ∑ cr ∈ CR

+ 4(1 − wcr, t )

∑ ∑ br ∈ BR tb ∈ TB

(57)

(58)

(59)

where Qsr,0, and Csr,c,0 denote, respectively, the initial amount and concentration of water stored in the tank sr. The term wsr is a binary variable representing the installation of storage tank sr. 4.5. Objective Function. The objective function of the problem is to minimize the total annual cost (TAC) of the entire water network. It consists of the expected operating cost and the annualized investment cost. The operating cost of integrated water network includes freshwater cost, end-of-pipe treatment cost, and the operating cost of selected batch and semicontinuous regeneration modules. Besides, the investment cost is given by the sum of capital costs of storage tanks and regeneration modules capital costs.

ICtc = OCtc =

ICtc Rrtc·Ftcmax OCtc Rrtc

∀ tc ∈ TC

∀ tc ∈ TC

(61)

(62)

Given two semicontinuous regeneration technologies, the candidate with fewer ICtc , fewer OCtc , and better purification efficiency (higher removal ratios/lower outlet concentrations) should be preferentially selected. More generally, the alternative H


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considering the costs of regeneration and end-of-pipe treatment. Maximum inlet flow rates of both adsorption and ultrafiltration are equally set as 11.875 t/h. For SBR with the durations of 1 and 0.5 h, their capacities are given as 11.875 t/ batch and 5.938 t/batch, respectively. For these technologies, the maximum inlet concentrations is set to be 400 μg/g. Outlet concentrations of all the regeneration technologies are set to be 100 μg/g, and the recovery ratio (Rr) for ultrafiltration is determined to be 0.7. The results of this case for five scenarios are listed in Table 1 (detailed configurations for all comparable scenarios in two

with better performances in certain indicators would be chosen when the performances of its other indicators are not worse than those of other candidates. Rule 3: For BRs, appropriate technologies should be selected based on the durations of regeneration, purification efficiencies and investments. Two economic indicators, namely, the capital cost per unit of capacity (ICtb) and the operating cost per unit amount of regenerated water (OCtb) are introduced and the definition equations are shown in constraints (63) and (64), respectively. ICtb =

ICtbΔt tb Rrtb· Capmax tb

OCtb OCtb = Rrtb

∀ tb ∈ TB

Table 1. Optimization Results for Case I in Example 1

(63)

∀ tb ∈ TB

scenario

technology

freshwater (t)

A B C D E

SBR (ΔT = 1 h) SBR (ΔT = 0.5 h) adsorption (Rr = 1) ultrafiltration (Rr = 0.7) adsorption and SBR (ΔT = 1 h)

71.594 68.594 68.594 71.266 63.03

(64)

Given two batch regeneration technologies, the alternative with shorter operation durations, fewer ICtb, fewer OCtb, and higher purification efficiencies (higher removal ratios/lower outlet concentrations) should be preferentially selected. More generally, the candidate with better performances in certain indicators would be chosen when the performances of its other indicators are identical with those of other candidates. Rule 4: If ICtc , OCtc , and purification efficiencies of the given semicontinuous regeneration technology are as good as those of the batch counterpart, the former is preferred, because of the fact that the batch regeneration is intermittently available and may limit the opportunities for further water reuse in some time intervals. Rule 1 has priority over all following rules and is quite useful in actual projects, even though it is simple and obvious. Rules 2, 3, and 4 are selection criteria for available technologies, which could eliminate the same types of contaminants and are used in repeated one-to-one comparisons. The bad ones will be weeded out and the new sets of candidate regeneration technologies will be adopted in the proposed MINLP model.

examples are shown in the Supporting Information). The minimum freshwater requirement of 68.594 t per cycle (Scenario C) is attained when only one regeneration unit is involved in the water network, which is the same as the result given in the paper by Liu et al.35 Note that the fresh water consumption for the batch water network involving SBR and adsorption is the lowest, as a synergistic effect of multiple regeneration modes integration. The freshwater consumption of a water system involving adsorption (Rr = 1) is lower than that of a water network with ultrafiltration (Rr = 0.7) when the inlet capacities of both regeneration processes are identical. This result is consistent with Rule 2. Besides, adsorption has advantages over SBR (ΔT = 1), with regard to freshwater consumption with the same average regeneration capacities, which validates the preprocessing Rule 4. Similarly, SBR (ΔT = 0.5) is superior to SBR (ΔT = 1) in freshwater consumption while their average regeneration capacities are equal, verifying preprocessing Rule 3. Table 2 shows the computational statistics of all scenarios, which is solved using BARON with a 0.01 relative gap. Note

6. ILLUSTRATIVE EXAMPLES Two examples are introduced to illustrate the proposed method for the synthesis of the total water network in batch plants. The resulting MINLP problems are implemented in GAMS 23.4 on an Intel Core i3−4130 3.40 GHz machine with 4GB memory and solved by DICOPT and BARON. DICOPT is a local optimum solver that can use CPLEX as the MILP solver and CONOPT/MINOS as the NLP solver, while BARON is a global optimization solver. 6.1. Example 1. The first example comes from Liu et al.,35 which consists of five batch water-using processes merely involving a single contaminant. The process data used in this example are given in the Supporting Information (Table S1). For this instance, only one tank is considered for each set of tanks and the maximum capacity is specified as 70 t. Furthermore, the duration of time interval is set to be 0.5 h. Case I: Comparison of Different Regeneration Operating Modes. This case intends to verify the aforementioned preprocessing rules and explore the effects of different regeneration operating modes. SBR, adsorption, and ultrafiltration are individually selected as a representative for batch, single-outflow, and two-outflows semicontinuous modes in this case. To compare with the results in the literature,35 this case is solved to minimize the freshwater consumption without

Table 2. Computational Statistics of Example 1 Case I Scenario

No. of continuous variables

No. of binary variables

No. of constraints

CPU time

A B C D E

1295 1296 972 972 1726

16 17 17 17 33

4375 4393 3672 3672 4896

599.85 s 20.11 s 9.65 s 17.33 s >10 h

that the solution time of a water system with adsorption is 9.65 CPU s, which is fewer than that of a water system with SBR (599.85 CPU s) or ultrafiltration (17.33 CPU s). When both SBR and adsorption are involved in the water system, the global optimal solution requires expensive computational efforts (>10 h). In the next case, we will show the effectiveness of the proposed solution strategy, which is based on DICOPT solver and iterative initial guesses and random perturbation. Case II: Selection of Proper Regeneration Technologies. As an extension of case I, this case aims to apply the proposed methodology to determine proper regeneration technologies and optimize the total water network for batch processes. In I


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Industrial & Engineering Chemistry Research Table 3. Process Parameters for Regeneration Modules in Example 1 Case II BRs Rr Cfx,out (g/t) IC ($/yr) OC ($/t) capacity (t and t/h) duration (h) IC ($ h/t) OC (t/h)

CRs

in situ reaction

SBR

MBR

reveres osmosis

ultrafiltration

adsorption

1 150 5000 0.1 [5, 12] 1 417 0.1

1 70 10 000 0.2 [5, 18] 1.5 833 0.2

1 70 10 000 0.2 [5, 12] 1 833 0.2

0.7 20 23 000 0.6 [5, 12]

0.7 50 14 000 0.5 [5, 12]

1 50 15 000 0.4 [5, 12]

2738 0.857

1667 0.714

1250 0.4

consumption, compared to the water-using network with a central storage tank. Compared to the water network with three BRs, it yields a 4.8% reduction in annual cost and an 8.7% reduction in freshwater consumption, because the duration of batch regeneration limits the opportunities of further water reuse. Furthermore, a 4.5% reduction in annual cost is also obtained with identical freshwater consumption, in comparison with the water network that consists of three CRs in the RO. The main reason for an improved performance in economics is the lower investment of MBR in a BR than that of adsorption in a CR. The details of these results are shown in the Supporting Information. Case II illustrates that the proposed synthesis methodology can be applied to select appropriate regeneration technologies, as well as obtain better tradeoffs among freshwater consumption, regeneration cost, and the end-ofpipe treatment cost than the framework with a single type of regeneration. Because of the complexity of the network, the computational effort of global optimization based on BARON is nontrivial (more than 5 days). Hence, all of the scenarios in this case are solved using DICOPT (CPLEX/MINOS) instead of BARON, which means that the global optimality cannot be guaranteed. Thus, iterative initial guesses and random perturbation are adopted to enhance the quality of solutions. In this case, 300 MINLP problems based on random initial guesses have been solved for each scenario. The computational statistics are given in Table 5, and the best solution is obtained from the model of Scenario B, which consists of 3245 continuous variables, 62 binary variables, and 8579 constraints. Table 5 also indicates that the proposed solution procedure could capture a satisfactory solution within an acceptable computation time for all scenarios, and the problem size increases with the specified number of regeneration modules. 6.2. Example 2. The second example is studied to demonstrate the capabilities of the proposed methodology to address a more complicated problem with multicontaminants. The water-using process data are taken from the work by Kim and Smith43 (see Table S2 in the Supporting Information). As shown in Table 6, six regeneration technologies are incorporated in the framework. In this example, the inlet concentrations of all candidate regeneration technologies are assumed to be unlimited. Each category of storage tanks (Ub, Vb, Uc, Vc) consists of two tanks with volumes restricted to 200 t, and their investment costs are determined as $1000/yr. The costs of freshwater and wastewater treatment are determined as $1/t and $5/t, respectively. Total number of batch cycles every year is perceived as 800 and the duration of a time interval sets as 0.5 h. According to the preprocessing Rule 2 and Rule 3, ultrafiltration and SBR are first removed from the synthesis

addition, the regeneration subsystem involves multiple modules for both batch and semicontinuous regeneration in order to incorporate all configurations among different regeneration technologies, as well as satisfy the regeneration need of the water network. The operation and economic parameters for available regeneration technologies are presented in Table 3. It can be seen that this case involves multiple BRs and CRs, each with three candidate regeneration technologies. The maximum inlet concentrations of all the regeneration technologies are set to be 400 μg/g. The annual cost for each storage tank is equal, which is assumed to be $1000/yr. Moreover, the costs of freshwater and wastewater treatment are determined to be $1/t and $5/t, respectively. The total number of batch cycles every year is perceived as 1000. Before addressing the problem through mathematical programming, preprocessing rules are adopted to eliminate redundant regeneration technologies that are appreciably inferior to other technologies, in order to decrease computational efforts. First, the investments per unit of capacity and operating costs per unit of regenerated water for different regeneration technologies are calculated, and the results are also listed in Table 3. Then, with the application of preprocessing Rule 2 and Rule 3, ultrafiltration and SBR should be removed from the CRs and BRs, respectively. The proper number of regeneration modules in the superstructure and its influence on the obtained network structure are also considered in this case. The results in Table 4 show that the superstructure with two modules for Table 4. Optimization Results for Example 1, Case II regeneration modules in the superstructure

overall cost ($/yr)

freshwater consumption (t/yr)

central storage tank only (Liu et al.35) one BR + one CR (Scenario A) two BRs + two CRs (Scenario B) three BRs + three CRs (Scenario C) three BRs (Scenario D) three CRs (Scenario E)

483 000

80 500

386 400 351 789 351 789

57 100 50 000 50 000

369 523 368 551

54 750 50 000

both batch regeneration and semicontinuous regeneration can obtain the optimal total water network configuration. As shown in Figure 3, two MBR modules and an adsorption module are selected for regeneration. Also, the variations of the amount of residual water in storage tank ub1, uc1, and vc1 are depicted in Figures 4a−c, and the schedule of regeneration modules shown in Figure 4d. The total annual investment cost of $351 789/yr and freshwater consumption of 50 000 t/y for the total water network are achieved, which corresponds to a 27.17% reduction in annualized cost and a 37.89% reduction in freshwater J


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Figure 3. Optimal network configuration for case II in Example 1.

Figure 4. Optimal schedule of buffer tanks and regeneration modules for case II in Example 1.

Table 5. Computational Statistics of Example 1, Case II

Table 7. Optimization Results for Example 2

Scenario



No. of constraints

CPU time (s)

regeneration modules in the superstructure

overall cost ($/yr)

freshwater consumption (t/yr)

A B C D E

2273 3245 4361 3363 1305

32 62 97 50 47

5829 8579 12 609 11 310 4216

658 1337 1394 583 276

central storage tank only (Scenario A) one BR + one CR (Scenario B) two BRs + two CRs (Scenario C) three BRs + three CRs (Scenario D)

4 037 547

671 400

3 850 989 3 749 306 3 749 306

612 755 585 664 585 664

framework. The retained four types of technologies then are incorporated into the superstructure, based on which the problem can be efficiently solved. Table 7 provides some optimization results generated by superstructures with varying regeneration modules. A total annual cost (TAC) of $3 749 306/yr and freshwater consumption of 585 664 t/yr

for the total water system are obtained in Scenario C and Scenario D, which corresponds to a 7.14% reduction in annualized cost and a 12.77% reduction in freshwater consumption, compared to the water-using network with a central storage tank (Scenario A).

Table 6. Process Parameters for Regeneration Modules in Example 2 BRs Rr Rc (c1,c2,c3) IC ($/yr) OC ($/t) capacity (t and t/h) duration (h) IC ($ h/t) OC (t/h)

CRs

in situ reaction

SBR

MBR

reveres osmosis

ultrafiltration

adsorption

1 0.6/0.6/0.6 45 000 0.01 [50, 200] 1 225 0.01

1 0.85/0.85/0.85 80 000 0.02 [50, 300] 1.5 400 0.02

1 0.85/0.85/0.85 80 000 0.02 [50, 200] 1 400 0.02

0.7 0.95/0.95/0.95 250 000 0.06 [50, 300]

0.6 0.85/0.85/0.85 100 000 0.02 [50, 300]

1 0.85/0.85/0.85 90 000 0.03 [50, 200]

1190 0.086

556 0.033

450 0.03

K


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Figure 5. Optimal network configuration for Example 2.

Figure 6. Optimal schedule of buffer tanks and regeneration modules for Example 2.

7. CONCLUSION

The optimal batch total water network configuration is described in Figure 5, where an in situ reaction module and two adsorption modules are selected and there exist connections between modules with different technologies. Furthermore, tank uc1 and vc1 are chosen from set Uc and Vc (two tanks in each set), while tanks in sets Ub and Vb are all inactive. In addition, the storage profiles of tanks (see Figures 6a and 6b) show that the maximum amount of water stored in tanks uc1 and vc1 reach 200 t at 0.5 and 1 h, respectively. The schedule of regeneration modules given in Figure 6c indicates that the majority of wastewater is purified through adsorption, because of its higher removal ratio for contaminants and superiority in operation modes. The details of the configurations of other scenarios are given in the Supporting Information. All scenarios in this example are solved using DICOPT (CPLEX/ CONOPT), and the computational statistics are shown in Table 8.

The state-space superstructure has been modified to provide an integrated framework for synthesizing a water-using subsystem, a regeneration subsystem, and buffer tanks simultaneously. The advantage of this approach is that the effects of different regeneration modes, the selection of regeneration technologies, and the tradeoff between capital and operating cost can be easily incorporated in the proposed MINLP model formulation. Furthermore, the scheduling of multiple regeneration modules and buffer tanks were also considered in the methodology. A rule-based two-stage solution procedure has been developed to guarantee the solution efficiency. It can be clearly observed that the preprocessing rules can reduce redundant topological structures of the synthesis problem and results obtained so far in two illustrative examples have showed that better overall network designs with less TAC and fewer wastewater can be obtained with the proposed approach.

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Table 8. Computational Statistics of Example 2 Scenario



No. of constraints

CPU time (s)

A B C D

1997 6635 8557 8261

0 84 171 252

8171 14 864 19 984 26 168

90 1427 1336 1908

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b02794. Water-using process data for examples; the detailed water network configurations and schedules for all comparable results in examples (PDF) L


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■

AUTHOR INFORMATION

Vb = set of storage tanks installation after batch regeneration modules Vc = set of storage tanks installation after semicontinuous regeneration modules

Corresponding Authors

* Tel.: 086-411-84986068. E-mail: [email protected] (H. Dong). * Tel.: 086-411-84986068. E-mail: [email protected] (X. Zou).

Positive Variables in cbr,c,t = inlet concentration of contaminant c in batch regeneration module br at time point t cout br,c,t = outlet concentration of contaminant c in batch regeneration module br at time point t in ccr,c,t = concentration of contaminant c of inlet stream in semicontinuous regeneration module cr at time point t out ccr,c,t = concentration of contaminant c of permeate stream in semicontinuous regeneration module cr at time point t cini,c,t = inlet concentration of contaminant c for water-using operation i at time point t cout i,c,t = outlet concentration of contaminant c for water-using operation i at time point t cinsr,c,t = inlet concentration of contaminant c in storage tank sr = (ub, uc, vb, vc) at time point t cout sr,c,t = outlet concentration of contaminant c in storage tank sr = (ub, uc, vb, vc) at time point t dubr = duration of batch regeneration in module br in = flow rate of inlet streams in semicontinuous fcr,t regeneration module cr at time point t fout cr,t = flow rate of permeate streams in semicontinuous regeneration module cr at time point t fcom cr,t = flow rate of concentrate streams in semicontinuous regeneration module cr at time point t fcr,e,t = flow rate of stream from semicontinuous regeneration module cr to end-of-pipe treatment facility e at time point t fcr,vc,t = flow rate of stream from semicontinuous regeneration module cr to storage tank vc at time point t f uc,cr,t = flow rate of stream from storage tank uc to semicontinuous regeneration module cr at time point t minbr,t = amount of inlet water for batch regeneration module br at time point t mout br,t = amount of outlet water for batch regeneration module br at time point t mini,t = amount of inlet water for water-using operation i at time point t mout i,t = amount of outlet water for water-using operation i at time point t minsr,t = amount of inlet water in storage tank sr = (ub, uc, vb, vc) at time point t mout sr,t = amount of outlet water in storage tank sr = (ub, uc, vb, vc) at time point t mbr,mx,t = amount of water from module br to corresponding mixer mx = (i, vb, uc, br′) at time point t me = total amount of effluent to the end-of-pipe treatment e in the time horizon mf = total amount of freshwater f consumed in the time horizon mf,i,t = amount of water from freshwater source f to waterusing operation i at time point t mi,mx,t = amount of water from water-using operation i to corresponding mixer mx = (i′, ub, uc, br, e) at time point t msp,mx,t = amount of water from splitter sp to mixer mx at time point t mub,mx,t = amount of water from tank ub to corresponding mixer mx = (i, br) at time point t muc,i,t = amount of water from tank uc to water-using operation i at time point t

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the National Natural Science Foundation of China, under Grant Nos. 21276039 and 20876020.

■

NOMENCLATURE

Abbreviations

BR = batch regeneration module CR = semicontinuous regeneration module DN = distribution network MBR = membrane bioreactor MILP = mixed-integer linear programming MINLP = mixed-integer nonlinear programming MIQCQP = mixed-integer quadratically constrained quadratic program PO = process operator RMINLP = relaxed MINLP RO = regeneration operator SBR = sequential batch reaction SO = storage operator TAC = total annual cost WO = water-using operator Indices

b = installation for batch regeneration module c = installation for semicontinuous regeneration module con = concentrate stream of semicontinuous regeneration module fx = fixed in = inlet of PO max = maximum min = minimum out = outlet of PO Sets

BR = set of batch regeneration modules C = set of contaminants CR = set of semicontinuous regeneration modules E = set of end-of-pipe treatments F = set of fresh water sources I = set of water-using operations SP = set of all splitters in the DN; SP = F ∪ Ub ∪ Uc ∪ Vb ∪ Vc ∪ TB ∪ TC ∪ I MX = set of all mixers in the DN; MX = E ∪ Ub ∪ Uc ∪ Vb ∪ Vc ∪ TB ∪ TC ∪ I SR = set of all storage tanks; SR = Ub ∪ Uc ∪ Vb ∪ Vc T = set of time points TB = set of batch regeneration technologies TC = set of semicontinuous regeneration technologies Ub = set of storage tanks installation before batch regeneration modules Uc = set of storage tanks installation before semicontinuous regeneration modules M


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MΔi = amount of water changing in water-using operation i, positive means gain and negative means loss Mli,c = mass load of contaminant c in water-using operation i N = total number of time points in the time horizon H NTC = total number of batch cycles every year OCtb = cost per regeneration capacity of module with technology tb OCtc = cost per regeneration capacity of module with technology tc OCtb = cost per unit purified water in regeneration module with technology tb OCtc = cost per unit purified water in regeneration module with technology tc Qmax sr = maximum amount of water stored in storage tank sr Qsr,0 = initial amount of water stored in the tank sr Rrtb = recovery ratio of batch regeneration module with technology tb Rctb,c = removal ratio for contaminant c of batch regeneration module with technology tb Rrtc = recovery ratio of semicontinuous regeneration module with technology tc Rctc,c = removal ratio for contaminant c of semicontinuous regeneration module with technology tc ΔT = length of the time interval Δti = distance between the time points of the start and finish of water-using operation i Δttb = distance between the time points of the start and finish of batch regeneration with technology tb Zini,t = denoting water-using operation i starts at time point t Zout i,t = denoting wastewater generated by operation i at time point t

mvb,mx,t = amount of water from tank vb to mixer mx = (i, uc, br) at time point t mvc,mx,t = amount of water from tank vc to mixer mx = (i, uc, br) at time point t qsr,t = amount of water stored in storage tank sr = (ub, uc, vb, vc) at time point t Binary Variables

wbr = binary variable denotes if module br is selected in the time horizon H wbr,t = binary variable denotes if batch regeneration starts in module br at time point t wbr,tb = binary variable denotes if technology tb is selected in the module br wcr = binary variable denotes if module cr is selected in the time horizon H wcr,t = binary variable denotes if semicontinuous regeneration is performed in module cr at time point t wcr,tc = binary variable denotes if technology tc is selected in the module cr wsr = denoting the installation of storage tank sr = (ub, uc, vb, vc ) Parameters

Cf,c = concentration of contaminant c in freshwater source f Cin,max = maximum inlet concentration of contaminant c in i,c water-using operation i Cout,max = maximum outlet concentration of contaminant c in i,c water-using operation i Csr,c,0 = initial concentration of contaminant c in the tank sr Cfx,out tb,c = fixed outlet concentration for contaminant c of batch regeneration module with technology tb Cin,max = maximum inlet concentration for contaminant c of tb,c batch regeneration module with technology tb Cfx,out = fixed outlet concentration for contaminant c of tc,c semicontinuous regeneration module with technology tc Cin,max = maximum inlet concentration for contaminant c of tc,c semicontinuous regeneration module with technology tc Capmax tb = maximum capacity of batch regeneration module with technology tb Capmin tb = minimum capacity of batch regeneration module with technology tb Cff = cost of per unit freshwater f Cee = cost of end-of-pipe treatment e Fmax tc = maximum flow rate of semicontinuous regeneration module with technology tc Fmin tc = minimum flow rate of semicontinuous regeneration module with technology tc H = fixed time horizon ICtb = annualized capital cost of regeneration module with technology tb ICtc = annualized capital cost of regeneration module with technology tc ICtb = capital cost per unit capacity of regeneration module with technology tb ICtc = capital cost per unit capacity of regeneration module with technology tc ICSsr = annualized installation cost of storage tank sr 4 = a sufficiently large positive number Mmax = maximum amount of water in water-using operation i i Mmin = minimum amount of water in water-using operation i i Mmax sp,mx = maximum amount of water from splitter sp to mixers mx

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