Optimal Design of an Integrated Discontinuous Water-Using Network

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Ind. Eng. Chem. Res. 2009, 48, 10924–10940

Optimal Design of an Integrated Discontinuous Water-Using Network Coordinating with a Central Continuous Regeneration Unit Yongzhong Liu,*,† Guanhua Li,† Le Wang,† Jinzhi Zhang,† and Kayghobad Shams‡ Department of Chemical Engineering, Xi’an Jiaotong UniVersity, Xi’an, Shaanxi, 710049, China, and Department of Chemical Engineering, Isfahan UniVersity of Technology, Isfahan, 84156, Iran

A systematic approach is proposed to design an optimal batch water-using network with centralized regeneration in order to deal with coordination of discontinuous water-using operations with one central continuous regeneration unit during multiple repeated batch cycles. The mathematical formulation in weak forms that readily induces process dynamic expressions is established under the framework of continuous time representation, in which a rigorous dynamic model of a tank is embedded. The freshwater consumption and the regeneration flow rates are minimized by a two-stage optimization approach. The integration of the batch water-using system highlights fixed regeneration flow rates and transitional behaviors from the start-up phase to steady state in multiple batch cycles. Optimal network structures and evolution characteristics of residual water and contaminant concentrations in buffer tanks can be achieved through solutions to the nonlinear programming problems (NLP). Four scenarios, including truly batch and semicontinuous water-using systems in both multiproduct and multipurpose batch plants, are presented to demonstrate the applicability of the proposed approach. 1. Introduction As a reactant and a mass separating agent (MSA), water is indispensable for daily life and a wide spectrum of processes in chemical, biochemical, petrochemical, pharmaceutical, petroleum refining, food processing, and power generation industries. Escalating concerns arise related to environmental problems worldwide associated with water resources, wastewater discharge, and water reuse. As a result, systematic methods and techniques for water conservation have attracted much attention in both process industries and municipal districts in recent decades, and considerable efforts and progress on water-saving methodologies increasingly stimulate interest in both theoretical and applied research. Most of the systematic methodologies available for wastewater minimization have currently focused on the synthesis of a continuous water-using system rather than the discontinuous counterpart, which is often encountered in the production of specialty chemicals of high commercial value and urban water systems as well, for example. In contrast to the integration of continuous water-using systems, the synthesis of batch waterusing systems, however, involves more complications owing to both time dependency and contaminant concentration restrictions. Little attention has been paid to synthesis of discontinuous water-using systems until recent years. Since integration of continuous water-using systems in which graphical methodolo gies1-4 and mathematical programming methods5-10 prevail, wastewater minimization techniques on batch water-using systems are also dominated by graphic-based approaches11-17 in which the targets are identified prior to detailed network design and mathematical programming methods18-23 in which more complex systems can be handled. Among the existing mathematical programming methods, the formulations, in general, fall into two types in term of time representations that is the kernel in the course of optimization, * To whom correspondence should be addressed. Tel.: +86-2982664752. Fax: +86-29-83237910. E-mail: [email protected]. † Xi’an Jiaotong University. ‡ Isfahan University of Technology.

discrete time representations, and continuous time representations. The former is conceptually simpler but entails a notably larger number of variables, whereas the latter would complicate the formulation but reduce the variables effectively. For this reason, mathematical formulations based on discrete time representations may experience higher computation intensity, and this type of formulation is therefore suited to smaller-scale problems.24,25 Reuse, regeneration, and recycle are the primary means to reduce freshwater consumption and wastewater discharge in a water-using system. In contrast to continuous water-using systems, it is imperative to meet the requirements of time sequences of operations, contaminant concentration bounds, and the amount of water fed to operations simultaneously in discontinuous water-using systems. As a result, it is not unusual in optimization of batch water-using networks to install buffer tanks for exploring more possibilities of wastewater reuse in both theoretical research24-27 and industrial applications.28,29 Foo et al.13,14 indicated that a buffer storage system in repeated batch processes is the most favorable option since it minimizes the external mass separating agent and the process mass separating agent simultaneously. Majozi26 established a continuous-time mathematical model for optimization of multipurpose batch plants with only one central buffer tank, in which equipment scheduling is underscored to enhance water reuse/ recycle opportunities. Almato et al.29 presented the evolutions of levels, concentrations, and temperatures in buffer tanks associated to different production tasks for a given production plan through a mathematical model and its industrial applications. Shoaib et al.27 proposed a three-stage hierarchical approach to address the problem of synthesizing a cost-effective batch water-using network. Thus, the minimum number of tanks in the network is attained by solving a mixed integral nonlinear programming (MINLP) problem. Chen and Lee16 developed a versatile methodology for the design of batch water-using networks with one or two central buffer tanks, which is adaptive to different types of operating modes, such as fixed loads and fixed flow rates, truly batch and semicontinuous operations. Rabie and EI-Halwagi30 proposed a systematic procedure to

10.1021/ie9000053 CCC: $40.75  2009 American Chemical Society Published on Web 10/27/2009

Ind. Eng. Chem. Res., Vol. 48, No. 24, 2009

overcome the lumped usage of water over a cycle and to eliminate recycle within the same cycle in the integration of a cost-effective batch water-using network. However, the freshwater consumption in some cases cannot be effectively lessened simply through installation of buffer tanks. That is often the case when some water-using operations that consume larger amounts of water and discharge with higher contaminant concentrations simultaneously appear in the system. Apparently, wastewater regeneration should be appended to the system for further reducing freshwater consumption. In practice, a centralized wastewater treatment plant is often favored by both industries and municipal districts because of capital cost and maintenance, and a centralized treatment unit can simultaneously perform regeneration and regeneration recycling in the system,31 notwithstanding optimum design could benefit from distributed wastewater networks.32,33 In this context, the necessity of regeneration in the system needs to be evaluated to accommodate requirements in reality. Furthermore, the capacity and operational conditions of regeneration units should be identified, provided that the entire system could operate steadily. Cheng and Chang19 initiated an all-purpose mathematical formulation based on discrete time representation that embraces modular models of schedules, water-reuse subsystems, and wastewater treatment subsystems. A highly integrated network design can be attained by adequately dealing with interactions between subsystems and properly imposing suitable constraints. Regarding the wastewater treatment units in a batch water network, Chang and Li20,21 indicated that wastewater treatment networks should be designed to be competent to deal with peak loads of inlet flow rate and contaminant concentrations, where each wastewater stream should be adjusted within a certain desirable range before entering regeneration units. In their model, evolution characteristics of residual water and contaminant concentrations in tanks are evaluated on uniformly discretized time intervals of a batch cycle to facilitate mass balances by algebraic equations instead of differential equations in nature. Regarding operations of wastewater treatment, it is worthwhile to note that wastewater treatment units in batch waterusing networks usually operate in a continuous mode, whereas water-using processes run in a discontinuous mode. Naturally, the coordination problem between continuous regeneration processes and batch water-using operations should be inevitably confronted during the integration of the system. Furthermore, a fixed network structure in multiple batch cycles is usually expected in running the system with ease, especially in a cyclic batch mode. Consequently, running these operations together needs continuous regeneration units at steady-state to be in harmony with time-dependent batch processes, and the targeting requirement of the water-using network and the minimal regeneration flow rate must be satisfied. Furthermore, the evolution characteristics of buffer tanks should be predicted under the condition of ensuring that the regeneration unit continuously operates. Unfortunately, the synthesis methodologies on the water-using network in which discontinuous waterusing operations coordinates with continuous wastewater treatment units harmoniously are rarely reported in the existing literature. Moreover, it is not unusual in practice that consecutive operating cycles are performed to achieve a certain throughput or to meet the need of a specific production order, in which the identical production schedule is carried out repeatedly in successive cycles. Nevertheless, the transitional characteristics

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from the start-up phase to steady state for these multiple repeated cycles have only been scantily addressed in the field despite their importance. The primary objective of this work is to establish a mathematical approach on the integration of a discontinuous waterusing system with one central continuous regeneration unit operating in multiple successive cycles, in which the coordination of discontinuous processes and the continuous process in the system with fixed regeneration flow rates and fixed network structures are fulfilled. Furthermore, the transitions from the start-up phase to steady state of the system for the multiple successive cycles are also highlighted. To demonstrate the adequacy of the proposed approach, the mathematical formulation in weak forms under the framework of a continuous time representations is developed in section 4, in which a rigorous dynamic model of tank is incorporated, after the problem statement described in section 2 and the superstructure defined in section 3 are presented. Thereafter, four scenarios for the synthesis of single-contaminant batch water-using systems are provided in section 5 to illustrate the effectiveness of the proposed method. 2. Problem Statement Consider a batch water-using system with a predetermined production schedule repeatedly running in multiple batch cycles. There is one central continuous regeneration unit in the system. The objectives of optimal water-using network design are the following: (i) to minimize the freshwater consumption and wastewater discharge of the system in the multiple repeated cycles; (ii) to minimize the regeneration flow rate and to operate the regeneration unit with constant flow rates in multiple operating cycles; (iii) to obtain the optimal water-using system with fixed network configurations in multiple operating cycles under the restrictions of targets on freshwater consumption and wastewater regeneration. Furthermore, the following design specifications need to be clarified in detail for coordinating operations of the optimal batch water-using network with the central continuous regeneration unit in multiple repeated cycles: (i) the start moment of the central regeneration unit during the start-up phase; (ii) the time dependent variations of flow rates and contaminant concentrations of each streams in the system; (iii) the volume profiles of buffer tanks affiliated with the central continuous regeneration unit; (iv) transitions of the water-using system from start-up phase to steady state during the multiple repeated cycles. To meet these ends, in the design of the batch water-using system with one central continuous regeneration unit, it is assumed that the following data are known: (i) the production schedule with a constant time horizon; (ii) the contaminant mass loads or the required quantity of water-using operations in the system; (iii) the allowable inlet and outlet concentrations of waterusing operations; (iv) the moments and durations of charging/discharging of water-using operations. 3. Superstructure for a Discontinuous Water-Using Network with One Central Continuous Regeneration Unit For a batch water-using system with a production schedule that is known a priori, the superstructure of the system with

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Figure 1. Superstructure of the water network in a batch process with one central continuous regeneration unit and two affiliated buffer tanks.

Figure 2. Water-using operation.

one central continuous regeneration unit and two affiliated buffer tanks is shown in Figure 1. Since regeneration units in industry usually operate continuously, for the purposes of coordinating discontinuous waterusing operations to continuous regeneration with respect of continuity and time sequence, buffer tanks should be installed before and after regeneration to dampen flow variations. Tank T, as shown in Figure 1, plays a role of a regulator at the entrance of the central regeneration unit, whereas tank S servers as a buffer container that supplies regenerated water to the waterusing operations in the system. After receiving discharges of other operations, tank T also discharges the mixed water to other operations, only if the requirements of the contaminant concentration bounds, amount of water, and a fixed flow rate at the regeneration unit are satisfied. In this case, the two tanks serve as a set of buffer containers connecting discontinuous processes with continuous processes. For any water-using operation in the system, as shown in Figure 2, its inlet streams could be freshwater, discharges from other water-using operations, the mixed water from tank T, and/ or the water from tank S, while its outlet streams could discharge to effluent, other water-using operations, and/or tank T. The charging duration of a water-using operation is designated as a solid line, whereas the discharging duration is designated as a dashed line, as shown in Figure 2. 4. Mathematical Formulation In a water-using system, the amount of freshwater consumption impacts tremendously the operating costs directly, whereas the flow rate for regeneration will determine both investment costs and operating costs. For a water-using system with a production schedule that is known a priori, the opportunities of water conservation could be attained by both buffer tanks and regeneration units. However, if the freshwater consumption targeting can simply be reached by a set of buffer tanks, wastewater regeneration will become redundant under this circumstance. It is, therefore, natural that optimization on wastewater regeneration should be carried out after targeting on freshwater consumption is achieved for a batch water-using system in which buffer tanks and wastewater treatment units

are simultaneously involved. As a result, for a batch waterusing system it is of significance to check the necessity of wastewater regeneration. To this end, the mathematical programming model should explicitly possess this feature. It is apparent that the superstructure shown in Figure 1 covers two representative systems, in which one is a batch water-using network with one central continuous regeneration unit and two affiliated buffer tanks and the other one is a batch water-using network with only one central buffer tank, which can also be considered a degenerate system from the former one. According to the superstructure described in Figure 1 and the assumptions made in the preceding section, mass balances on water, and the contaminant for each water-using operation in the system, the central regeneration unit and two affiliated buffer tanks should be carried out. Consequently, three steps to synthesize the water-using system are spontaneously involved because of the discontinuous nature of the system. First of all, the time horizon of a batch cycle needs to be partitioned into a number of time intervals in which mass balances can be conducted appropriately. Then, the freshwater consumption of the system should be minimized. At last, on the basis of the obtained target of the freshwater consumption, the flow rates of the central regeneration unit needs to be indentified afterward. As mentioned previously, if the flow rate for regeneration in the system is zero in this step, it indicates that an optimal waterusing network can be achieved simply by installing one central buffer tank, viz., the two affiliated buffer tanks can be merged into one, and the central regeneration unit is, of course, no longer necessarily set up for the optimal water-using network. 4.1. Time Partitioning of the Horizon of a Batch Cycle. In a batch cycle, flow rates and concentrations of charging/ discharging in both water-using operations and buffer tanks are dependent on mass balances between their intake streams and exit streams. In order to calculate the inlet and outlet concentrations and the flow rates for tanks and water-using operations at any given moment within the time horizon of a batch cycle, herein, it is necessary to partition the time horizon into a number of intervals at which the charging/discharging streams available operate continuously. Afterward mass balances can be conducted either on these time intervals or on the duration of any waterusing operation, and even on the entire time horizon of the batch cycle. Consider a batch water-using system with a predetermined production schedule within a time horizon [0, tH]. A start moment sequence of charging, Tin,s, can be obtained by arranging the start moments of charging for all of the waterusing operations, and a finish moment sequence of charging, Tin,e, can be obtained by arranging the end moments of charging for all of the water-using operations. Likewise, a start moment sequence of discharging, Tout,s, and a finish moment sequence of discharging, Tout,e, can also be constructed. Thus, the sequences related to the time partitioning are defined as Tin,s ) {tin,s j |j ∈ J}

(1)

Tin,e ) {tin,e j |j ∈ J}

(2)

|j ∈ J} Tout,s ) {tout,s j

(3)

|j ∈ J} Tout,e ) {tout,e j

(4)

Hence, the sequence of charging, Tin, and the sequence of discharging, Tout, can be obtained subsequently


Tin ) Tin,s ∪ Tin,e

(5)

Tout ) Tout,s ∪ Tout,e

(6)

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where the elements of Tin and Tout are arranged by an ascending order, respectively. Consequently, a new sequence Tk that is assigned to partition the time horizon [0, tH] can be constructed as Tk ) Tin ∪ Tout

(7)

where the elements of Tk are arranged by an ascending order. The relationship among the sequence Tk, the sequence of charging Tin and the sequence of discharging Tout in the time horizon is shown in Figure 3. For any moment t within the time horizon [0, tH], there exists t ∈ [tk, tk+1] ⊂ ∪ [tk, tk+1] ⊂ [0, tH]

Figure 3. Relationship among the sequence Tk, the sequence of charging Tin, and the sequence of discharging Tout in the time horizon.

(8)

tk∈Tk

4.2. Minimization of Freshwater Consumption. The mathematical model targeting the freshwater consumption is formulated in this section. The constraints imposed include mass balances of water and the contaminant for each water-using operation in the system, the central regeneration unit, and two affiliated buffer tanks as well. The mathematical optimization problem is described as follows: Objective min

∑

QW,P j

(9)

in,e QPT,j ) ∆(t0, tH, tin,s j , tj )

∫

min{tH,tjin,e}

P in,e QS,j ) ∆(t0, tH, tin,s j , tj )

∫

min{tH,tjin,e}

max{t0,tjin,s}

max{t0,tjin,s}

∫

in,e QPi,j ) ∆(tout,s , tout,e , tin,s i i j , tj )

fPT,j dt

(13)

P fS,j dt

(14)

min{tiout,e,tjin,e}

P fi,j dt

max{tiout,s,tjin,s}

QD,P ) ∆(tout,s , tout,e , t0, tH) j j j

∫

min{tjout,e,tH}

QPj,T ) ∆(tout,s , tout,e , t0, tH) j j

∫

min{tjout,e,tH}

max{tjout,s,t0}

max{tjout,s,t0}

(15)

fD,P dt j

(16)

fPj,T dt

(17)

(b) Mass balance of contaminant on the mixing node at the inlet of any operation j

j∈J

P P CS,j + QPT,jCPT2,j + QS,j

Constraints. 1. For any water-using operation j (a) Mass balance of water P + QPT,j + QS,j + QW,P j

∑

P + QS,j

QPj,T +

∑

k∈J,j*k QG,P j ∀j

QPj,k + ∈J

max{a,c}

f dt

∫

min{tH,tjin,e} max{t0,tjin,s}

CPT2,j )

tin,e j

1 - tin,s j

∫

fW,P dt j

(12)

∑

(18)

in,e ∆(tk, tk+1, tin,s j , tj )

tk∈Tk

tk+1

tk

(11)

where the function ∆(a,b,c,d) is constructed to identify the overlapping status of two time intervals. The definition of this function and its properties are given in Appendix A. The terms max{} and min{} represent two standard functions. For the batch water-using system, the amount of water for each stream shown in Figure 2 can be written in weak forms that readily induce process dynamic expressions as in,e ) ∆(t0, tH, tin,s QW,P j j , tj )

QPi,j)Cin,P ∀j ∈ J j

P , can be where the outlet concentration of tank T, CT2,j obtained through a dynamic tank model. In this paper, a well-mixed tank model, which is disscussed in Appendix B in detail, is used. Therefore, it gives the outlet concentration of tank T as

(10)

) 0. If there is no leakage in the system, then QG,P j In general, the amount of water Q can be expressed as the integral of flow rate f with respect to time, namely

∫

∑

i∈J,i*j

+ QPi,j ) QD,P j

Q ) ∆(a, b, c, d)

(QPi,jCout,P ) ) (QW,P + QPT,j + i j

i∈J,i*j

i∈J,i*j

min{b,d}

∑

CPT2 dt

in,e ∃k, [tk, tk+1] ⊆ [tin,s j , tj ]

(19)

In particular, when the water-using operation is in,s in,e instantaneously charged, namely, tin,s f tin,e j j , [tj , tj ] f tin j , tk + 1 f tk, the outlet concentration of tank T, P , is written as CT2,j CPT2,j ) lim

tjin,eftjin,s tin,e j

1 - tin,s j

∑

in,e ∆(tk, tk+1, tin,s j , tj )

tk∈Tk

∫

tk+1

tk

CPT2 dt )

CPT2(tin j ) (c) Mass balance of comtaminant

(20)

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P (QW,P + QPT,j + QS,j + j

∑

Therefore

QPi,j)Cin,P + Mj ) (QD,P + j j

i∈J,i*j

∑

out,P QPj,k + QPj,T + QG,P ∀j ∈ J j )Cj

FP1 )

(21)

∑f

P out,s out,e , tj , tk, tk+1) j,T∆(tj

∀t, ∃k, t ∈ [tk, tk+1]

j∈J

k∈J,k*j

(26)

M,j where Mj ) ∫ttM,j fM,j dt. The term fM,j that is geners ally a function of time represents the process dynamics s and of contaminant removal in process j. Also, tM,j e tM,j represent start and finish moments of contaminant removal in process j, respectively. (d) Concentration restrictions on inlet and outlet of operation j e

∑ [f

FP2 )

tR)] ∀t, ∃k, t ∈ [tk, tk+1] QP1 )

(23)

∑ ∆(t , t k

QPT(t)

)

QP-1 T (tH)

CPT1 )

1 Q1p

∑ ∆(t

∑

∫

j∈J

∑ ∆(t , t, t 0

min{tjout,e,t}

∫

in,s in,e j , tj )

min{t,tjin,e}

max{t0,tjin,s}

j∈J

∫

∆(t0, t, tR, tH)

max{t0,tR}

)

QP-1 T (tH)

+

∑

QPT,ju(t

-

tout j )

fR,P dt

-

j∈J

∑

QPT,ju(t

j∈J R

∫

tk+1

fPj,TCout,P dt ∀t, ∃k, t ∈ [tk, tk+1] j

(30)

tk

]

F1P/(F1P-F2P)

QPT(tk) + (FP1 - FP2 )(t - tk)

∀t, ∃k, t ∈ [tk, tk+1]

(24)

-

tin j )

fR,P(t - t )u(t - tR)

-

(25)

where

CPT2(t) )

QPT(tk - )CPT2(tk - ) + QP1 CPT1 QP1 + QPT(tk - ) ∀t, ∃k, t ∈ [tk + , tk+1 - ]

{

1 tg0 0 t 1

(40)

This means that the residual water in the two tanks and the contaminant concentration in tank T at the start moment of cycle P and at the finish moment of cycle P - 1 should be identical. 4.3. Minimization of the Flow Rate for Regeneration. The problem of minimization of the flow rate for regeneration can be expressed as follows: Objective min fR,P

(41)

Along with the constraints 10-40, the problem is subject to an extra constraint. That is

∑Q

W,P j

e (1 + ξ)QW,P min

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(42)

j∈J

This indicates that the total amount of freshwater consumption is W,P W,P , where Qmin is the restricted not exceeding over (1 + ξ)Qmin minimum of freshwater consumption obtained from the freshwater consumption model aforementioned. The constant ξ is a relaxation factor, which can be chosen according to practical limitation of freshwater consumption of the water-using system. For example, if ξ is set to be zero in calculations, the amount of freshwater consumption of the network reaches its minimum; if ξ is chosen to be 0.1 in calculations, the amount of freshwater consumption of the system is allowed to be 10% larger than its optimum. Thus, it probably leads to the aftermath of different network structures. 5. Case Studies In this section, four scenarios will be employed to demonstrate applications of the proposed mathematical programming model for the synthesis of the discontinuous water-using system with one central continuous regeneration unit. All of these problems are coded and solved by the software package GAMS on a PC with two 2.0 GHz, Xeon processors. The NLP solver GAMS 2.50/distribution 22.3/KNITRO is selected. 5.1. Scenario 1: Wastewater Direct Reuse with Only One Central Buffer Tank. The water limiting data for this scenario come from Majozi’s work.26 The system comprises five water-using operations in discrete time duration, and the corresponding water-using data are listed in Table 1. In the literature, one central buffer tank was introduced to explore the opportunity of reduction of the freshwater consumption in the water-using network. The freshwater consumption and the optimal water-using network in the first operating cycle were obtained through mathematical programming. This example problem was resolved by Cheng and his co-workers35 to obtain the target of freshwater consumption and the network structures for two consecutive operating cycles. 5.1.1. Consumption of Freshwater and Network Structures. Regarding the same water-using system, the optimal problem is solved by the proposed model in this paper. In order to rationalize the capacity of the central buffer tank within a certain desirable range, in this case, the capacity restriction of tank T in the model was set to be 1500 kg in calculations. The outlet concentration of the central regeneration unit is set to be 100 µg · g-1. The design procedure for the water-using system is carried out in three steps, as mentioned in the preceding section. After partitioning the entire time horizon according to the given charging/discharging moments of water-using processes, the amount of freshwater consumption was obtained by solving eqs 9-40, followed by minimizing the flow rate for regeneration by solving eqs 41 and 42. In this scenario, the solutions to the problem were carried out regarding the first operating cycle and the repeated operating cycles, respectively. The solutions to the problem show that the whole model in the first operating cycle entails 79 variables and 113 constraints, and the optimal freshwater requirement of 1285.49 kg was reached in 10.563 CPU s. In contrast, the model problem in the repeated cycles that includes 79 variables and 112 constraints was solved in 19.094 Table 1. Water Data for Scenario 1 operation water/kg Cjin,max/µg · g-1 Cjout,max/µg · g-1 tin/h tout/h M/kg A B C D E

1000 280 300 280 300

0 250 100 250 100

100 510 100 510 100

0 0 4 2 6

3 4 5.5 6 7.5

100 72.8 0 72.8 0

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Table 2. Optimal Results for Scenario 1 freshwater/kg

effluent /kg

flow rate for regeneration /kg · h-1

water remaining in tank T/kg

inlet concentration of process/µg · g-1

first operating cycle

1285.49

582.031

0

703.460

repeated operating cycle

1000

0

703.460

A0 B0 C 100 D0 E 100 A0 B 154 C 100 D 154 E 100

cycle

1000

CPU s resulting in an optimal objective of 1000 kg. The detailed results of the optimal water-using network for this scenario are given in Table 2. In the calculations, the flow rate for regeneration in this scenario is zero after minimizing the flow rate for regeneration was complete. This indicates that the optimal water-using network can be achieved simply by installing one central buffer tank. Thus, as expected, the regeneration processing is not necessarily for the optimal water network. The water-using networks in the first operating cycle and the repeated operating cycles are shown in Figure 4a and b, respectively. It is obvious as shown in Table 1 that, in this example case, the outlet concentrations of some waterusing operations in which the finish moments are prior to other water-using operations happen to be less than the inlet concentration of the subsequent water-using operations. This is the major reason why the wastewater regeneration in this case becomes redundant. The results show that the minimum of freshwater consumption in the first cycle, 1285.49 kg, obtained is the same as in the work of

Figure 4. Optimal network structures in scenario 1.

outlet concentration of process/µg · g-1 A 100 B 510 C 100 D 510 E 100 A100 B 484 C 100 D 305 E 100

Majozi26 and Cheng and co-worker,35 but the network structure is slightly different. In the repeated operating cycles, the network structure obtained in this paper is completely different from the literatures, despite of the identical freshwater consumption. However, the network structure obtained by the proposed model can guarantee the stability of network structure and the capacity of buffer tanks after the first operating cycle as the results of the constraints embedded in the proposed model, which is necessary for the batch water network operating in the mode of multiple repeated cycles. 5.1.2. Variations of the Residual Water and the Contaminant Concentration in the Buffer Tank. The variations of water remaining in tank T are shown in Figure 5. The bound of the capacity of tank T was restricted to 1500 kg in calculations. It can be seen that the amount of residual water in the buffer tank is not more than 900 kg in this case. It is worth noting that the residual water in tank T reaches zero at 3 h. This is a sign of optimum solution since, if the buffer tank is not evacuated at a certain moment, the amount of water remaining in the buffer tank will have to be appended to the

Figure 5. Variations of the amount of residual water in tank T in scenario 1.


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Figure 6. Variations of the contaminant concentration in tank T in scenario 1.

redundant volume of the tank. As a result, this will lead to an increase of the volume of the buffer tank in design. The variations of the contaminant concentration in tank T are presented in Figure 6. It can be seen that the contaminant concentrations in tank T gradually increase in the first operating cycle and fluctuate in the repeated operating cycles. 5.2. Scenario 2: Water Reuse through Buffer Tanks and Regeneration/Regeneration Recycle. This case is intended to illustrate the proposed method for systemic design of a water system in a batch process with the wastewater treatment facility and the buffer tanks. The data used this scenario are listed in

Figure 7. Optimal network structures for the system with buffer tanks and regeneration/regeneration recycle in scenario 2.

Table 3. In order to compare the effects of the different network configurations, the situation that the batch water-using network with only one central buffer tank is also involved. 5.2.1. Consumption of Freshwater and Network Structures. During the calculations, the capacity of tank T and tank S in the water network with buffer tanks and one central regeneration unit is restricted to 70 ton in order to keep the capacity of the tanks within reasonable ranges. The outlet concentration of the central

Table 3. Water Data for Scenario 2 operation

freshwater/ton

Cjin,max/µg · g-1

Cjout,max/µg · g-1

tin/h

tout/h

M/kg

A B C D E

50 30 10 24 40

0 100 200 350 450

400 400 500 600 700

0 3 3 3.5 6

2 4 6 7.5 8.5

20 9 3 6 10

Table 4. Optimal Results for Scenario 2 (System with Buffer Tanks and Regeneration/Regeneration Recycle) water remaining in tank T/ton

water remaining in tank S/ton

freshwater/ton

effluent/ton

flow rate for regeneration/ton · h-1


68.594

16.316

11.875

9.569

42.707


50

50

4.784

9.569

42.707

cycle

inlet concentration of processes/µg · g-1 A0 B 36.810 C 200 D 315.551 E 100 A0 B 100 C 200 D 350 E 428.316

outlet concentration of processes/µg · g-1 A 400 B 400 C 500 D 600 E 700 A 400 B 400 C 500 D 600 E 700

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Table 5. Optimal Results for Scenario 2 (System with Only One Central Buffer Tank) cycle

freshwater/ton

effluent/ton

flow rate for regeneration/ton · h-1

water remaining in tank T/ton

first operating cycle and repeated operating cycle

80.5

80.5

0

0

regeneration unit is set to be 100 µg · g-1. The overall model in the first operating cycle entails 118 constraints and 84 variables and the optimal freshwater requirement of 68.594 ton is reached in 5.703 CPU s; meanwhile in the repeated operating cycles, the model entails 124 constraints, 84 variables, and 0.406 s CPU time in obtaining the results. The detailed optimal results are summarized in Table 4, and the water-using network structures for the two successive cycles are presented in Figure 7.

inlet concentration of processes/µg · g-1

outlet concentration of processes/µg · g-1

A0

A 400

B0 C 200 D 350.000 E 412.514

B 400 C 500 D 600 E 700

Figure 8. Optimal network structure for the system with only one central buffer tank in scenario 2.

In contrast, the detailed solution to the problem of the waterusing network with only one central buffer tank, tank T in this case, is given in Table 5. Specifically, the identical network structure, as shown in Figure 8, is obtained for both the first operating cycle and the repeated operating cycle in this situation. It can be seen that the amount of freshwater consumption of the system with the central regeneration unit decreases to 68.594 ton in the first operating cycle and 50 ton in the repeated operating cycle, respectively, compared with the system with only one central buffer tank in which 80.5 ton freshwater is required either in the first operating cycle or in the repeated operating cycle. On the other hand, the wastewater discharge of the system with the central regeneration unit lessens subsequently owing to regeneration/regeneration recycle. 5.2.2. Variations of the Residual Water and the Contaminant Concentration in the Buffer Tanks. For the water network with the central regeneration unit, the variations of the residual water in tank T are presented in Figure 9. Note that the residual water in the tank reaches zero at 7.5 h in the first operating cycle, and it reaches valley values when the operating moments are at 2 and 8.5 h in the repeated operating cycle. A maximum 51.235 ton is attained when the operating moment is at 2 h. Likewise, the variations of the residual water in tank S are shown in Figure 10. It shows that the residual water


Figure 10. Variations of the amount of residual water in tank S in scenario 2.


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Figure 12. Optimal network structure with only one central buffer tank in the repeated operating cycle in scenario 3.


reaches zero when the operating moments are at 2, 3, and 3.5 h in the first operating cycle. The amount of residual water is kept at a higher level with a maximum 57.059 ton when the moment is 3 h in the repeated operating cycle. The profile of contaminant concentrations in tank T is shown in Figure 11. The contaminant concentrations in tank T gradually increase in the first operating cycle, while the contaminant concentrations are held around 400 µg · g-1, not much fluctuation, most of the time during the repeated operating cycles. This is beneficial for the central regeneration unit operating in a relatively steady state. 5.3. Scenario 3: Hybrid Water-Using System That Consists of Fixed Load Operations and Fixed Quantity Operations. This scenario deals with the integration of a water-using system that consists of fixed load operations and fixed quantity operations with charging and discharging durations. The water

data for this scenario are given in Table 6, which are adopted from Chen and Lee’s work.16 In this case, the hybrid waterusing system consists of fixed load operations and fixed quantity operations. As shown in Table 6, the minimum water requirements are specified in operations C and E. Therefore, these two operations can be considered fixed quantity operations. 5.3.1. Hybrid Water-Using System with Only One Central Buffer Tank. For the single operating cycle of the optimal system with only one buffer tank, we obtained the same results as those obtained by the graphical analysis that Chen and Lee proposed.16 Namely, the freshwater consumption in the cycle is 44.5 ton, and the network configurations are identical, in which recycling in operation C through tank T is allowable. The only difference is that, during the calculation, 7.5 ton of freshwater is stored in tank T in advance before operation A starts to discharge because there is no connection between freshwater and tank T in our superstructure. For the repeated operating cycle, first, we resolved the problem using the same initial conditions given in the literature, and the same results were obtained through the proposed model in this paper. Then we added constraints on recycling in operation C to improve performance of the system. In the calculations, there is no capacity restriction on tank T. The overall model in this cycle entails 246 constraints and 168 variables, and 30.297 CPU s were consumed. The results are presented in Table 7. In contrast to the result obtained by the graphical analysis,16 the water recycling in operation C through tank T is partly eliminated by mixing 5.27 ton of freshwater to operation C, as shown in Figure 12. Nevertheless, it is advantageous to be able of avoiding contaminant accumulation in operation C at the cost of increasing freshwater consumption by 5.27 ton, and the network structure is a little different from that of the literature. In some circumstances, it is crucial to deal with the issue of the contaminant build-up in water-

Table 6. Water Data for Scenario 3 operation

Cjin,max/mg · g-1

Cjout,max/mg · g-1

charging duration/h

discharging duration/h

mass load/ton

minimum requirement/ton

A B C D E

0 250 100 250 100

200 500

0.0-1.0 0.0-0.5 5.0-6.5 2.0-2.5 7.0-8.5

4.0-5.0 4.5-5.0 5.0-6.5 6.5-7.0 7.0-8.5

4.0 4.0 0 3.6 0.3

15

400

15

Table 7. Optimal Results for Scenario 3 (System with Only One Central Buffer Tank) cycle

freshwater/ton

water remaining in tank T/ton

inlet concentration of processes/mg · g-1


31.021

31.33

A0 B 250 C 100 D 206.256 E 100

outlet concentration of processes/mg · g-1 A 200 B 500 C 100 D 400 E 120

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using operations by prevention of recycling because it may lead to severe problem to the operations. Moreover, the amount of water in tank T and the concentrations varying with time are shown in Figures 13 and 14, respectively. From Figure 13, we can see that, however, the variation of amount of water in tank T is a little different from the literature during the second half of the cycle, which is indicated by the start moment of discharging in operation A. The amount of water remaining in tank T calculated by the proposed model is less than that of the counterpart predicted by the graphical method. The major reason is that the improvement of recycling in operation C may lead to a different network structure. From the above calculations, we can see that the proposed approach can be readily used to deal with the batch water-using system that consists of fixed load operations and fixed quantity operations, as the graphical analysis available in the literature does. 5.3.2. Hybrid Water-Using System with One Central Regeneration Unit and Two Affiliated Buffer Tanks. Likewise, we use the proposed superstructure and mathematical model in this paper to solve the problem. The major difference to the previous section is that the central regeneration unit is

Figure 15. Optimal network structures in scenario 3.

embedded in the proposed model. In the calculations with the central regeneration unit, the outlet concentration of the central regeneration unit is set to be 100 mg · g-1. In this case, there are also no capacity restrictions on tank T and tank S. For the first operating cycle, the overall model entails 236 constraints and 168 variables, and 1.969 CPU s were consumed. For the repeated operating cycle, the overall model in the cycle entails 244 constraints and 168 variables, and 32.656 CPU s were consumed to obtain the solutions. The detailed results are summarized in Table 8. The optimal network structures in the multiple repeated operating cycles are shown in Figure 15.

Table 8. Optimal Results for Scenario 3 (System with Buffer Tanks and Regeneration/Regeneration Recycle) cycle

freshwater/ton

water remaining in water remaining in inlet concentration of outlet concentration of flow rate for tank T/ton tank S/ton processes/mg · g-1 processes/mg · g-1 regeneration/ton · h-1


37

21.409

15.591


22.314

21.409

15.591

A0 B0 C 100 D0 E 100 A0 B 100 C 100 D 100 E 100

A 200 B 500 C 100 D 400 E 120 A 200 B 500 C 100 D 294.493 E 120

10.131

5.347



It can be seen that, in the first operating cycle, the recycles in operation C and operation E are eliminated by introducing the regeneration unit into the system, which will be starting at 5 h in the cycle. It is worth noting that this kind of network structure is superior to the system that includes recycle streams because the accumulated contaminant can be eliminated through the installation of regeneration facilities. Furthermore, during the repeated operating cycle, recycling in operation E can be completely eliminated through the central regeneration unit, although recycling in operation C can be partly eliminated by mixing freshwater with streams from tanks S and T. However, the freshwater consumption can be reduced to 37 ton in the first operating cycle and 22.314 ton in the repeated operating cycle, respectively. Correspondingly, the profiles of the amount of residual water in tank T and its concentrations in the two successive cycles are shown in Figures 16 and 17, respectively. It appears that the fluctuation of contaminant concentration in tank T is not remarkable in the repeated operating cycle except for the last segment of the cycle. Similarly, the variations of residual water in tank S varying with time are shown in Figure 18. It is interesting that in tank S a gradual increase of the amount of water during the first operating cycle and an intensive fluctuation of the amount of water during the repeated operating cycle are observed compared to variations of the amount of water remaining in tank T in multiple batch cycles. From the discussions in this scenario, it can be easily concluded that the installation of the central regeneration unit can not only reduce freshwater consumption of the system but also be beneficial to removing unexpected recycling in the system. Consequently, from a practical point of view, it is of

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Figure 18. Variations of the amount of residual water in tank S in scenario 3.

significance in the synthesis of highly integrated water-using systems to install wastewater treatment facilities as they could

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Figure 19. Gantt chart for the production schedule in scenario 4.

serve as regulators to adjust the water quality of different streams in the system like freshwater does. 5.4. Scenario 4: Multipurpose Batch Water-Using System. As an extension of the scenario 3, this scenario aims at applying the proposed methodology to the integration of a waterusing system in a multipurpose batch plant, which consists of general purpose water-using operations to manufacture a variety of products, each product having different water-using requirements. The Gantt chart for the production schedule in this example case is shown in Figure 19, in which four general purpose water-using operations are involved. The water data given in Table 9, including the allowable inlet and outlet concentrations, charging and discharging durations, water-using operation durations, mass loads, and the fixed quantity required by the operation, are adapted from the water data in scenario 3, which is again a hybrid system that consists of fixed load operations and fixed quantity operations with charging and discharging durations. In this scenario, the proposed superstructure and mathematical model were used to solve this more complex problem. In the calculations, the outlet concentration of the central regeneration unit is also set to be 100 mg · g-1, and there are also no capacity restrictions on tanks T and S. In the calculations, the overall model in the first operating cycle entails 992 constraints and 567 variables, whereas the overall model in the repeated operating cycle entails 997 constraints and 570 variables. The optimal network structures in the first operating cycle and the repeated operating cycles are shown in Figure 20. It can be seen in Figure 20a that for the first operating cycle 46 ton of freshwater is consumed and 18.625 ton of wastewater is discharged. The regeneration flow rate of the central wastewater treatment unit will be 14 ton · h-1, and the regeneration processing will be starting at 1 h. In contrast to the successive repeated operating cycle, as shown in Figure 20b, the freshwater consumption is reduced to 32 ton, and the regeneration flow rate will be 12.375 ton · h-1, which is almost at the same level compared to the flow rate in the first operating cycle. From a practical perspective of running a wastewater treatment unit steadily, it is also a favorable operation mode. In this scenario, the profiles of the amount of water remaining in tank T and the contaminant concentrations in the two successive cycles are presented in Figures 21 and 22, respectively. In addition, the amount of residual water

Figure 20. Optimal network structure of the first operating cycle in scenario 4.

in tank S varying with time is shown in Figure 23. As we can see, the amount of residual water in tank T fluctuates significantly either in the first operating cycle or in the repeated operating cycle. 6. Conclusions Buffer tanks and regeneration facilities in water-using networks in batch plants are effective means to reduce freshwater consumption and wastewater emission. In fact, regeneration units generally operate in continuous mode, despite of the discontinuous nature of water-using operations. As a result, the coordination between the continuous regeneration process and discontinuous water-using operations in the batch water-using network deserves to be addressed when the optimal system is designed. In this paper, a mathematical programming based design approach for the batch waterusing network with one central continuous regeneration unit has been proposed under the framework of continuous time representations. The primary feature of the proposed methodology is to enhance much more opportunities of wastewater recovery by regeneration/regeneration recycle coupled with

Table 9. Water Data for Scenario 4 water-using operation A B C D

Cjin,max/mg · g-1

Cjout,max/mg · g-1

0 250 100 250

350 500 400

charging duration/h

discharging duration/h

operation duration/h

mass load/ton

1.0 0.5 1.0 0.5

1.0 0.5 1.0 0.5

2.0 2.5 1.0 3.0

2.8 4.0 0 3.6

minimum requirement/ton

15


Figure 21. Optimal network structure of the repeated operating cycle in scenario 4.

10937


the simultaneous coordination among discontinuous waterusing operations, buffer tanks, and the central continuous regeneration unit. Moreover, the proposed approach highlights fixed flow rates of regeneration and transitional behaviors from the start-up phase to steady state in multiple batch cycles. Four scenarios are presented to illustrate the applications of the proposed approach. The illustrative examples show that the proposed approach is capable of designing and optimizing truly batch and semicontinuous water-using systems in both multiproduct and multipurpose batch plants under long-term operation. It also provides insights into the fundamental details for design and operation of buffer tanks and wastewater treatment facilities in batch plants. Acknowledgment The authors gratefully acknowledge the financial support from the Natural Science Foundation of China (NSFC), Grant No. 20936004, No. 20876123, and Xi’an Municipal Project for Industrial Research, Grant No. GG06015. Sincere appreciation is extended to the anonymous reviewers of this paper for their constructive and valuable comments. The authors are also indebted to Dr. Dominic C. Y. Foo for helpful discussions. Appendix A: Definition of the Function ∆(a,b,c,d) The function ∆(a,b,c,d) is defined as ∆(a, b, c, d) ) u(d - a)u(b - c) ) Figure 22. Variations of the amount of residual water in tank T in scenario 4.

where

{

1 [a, b] ∩ [c, d] * L 0 [a, b] ∩ [c, d] ) L (A.1)

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u(t) )

{

∫

1 tg0 0 t

Optimal Design of an Integrated Discontinuous Water-Using Network

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