Optimization of Multiple Freshwater Resources in a Flexible-Schedule

Article pubs.acs.org/IECR

Optimization of Multiple Freshwater Resources in a FlexibleSchedule Batch Water Network Nitin Dutt Chaturvedi and Santanu Bandyopadhyay* Department of Energy Science and Engineering, Indian Institute of TechnologyBombay, Powai, Mumbai 400076, India ABSTRACT: Utilization of multiple freshwater resources can significantly reduce the operating cost of a water allocation network. This is demonstrated in the literature for continuous processes. In this paper, a mathematical formulation is proposed to minimize the operating cost of water allocation network in a batch process by utilizing multiple freshwater resources. The proposed methodology incorporates the variable scheduling of a batch process, for a given production, and it is applicable to single as well as multicontaminant water allocation problems. The applicability of the methodology is demonstrated with two illustrative examples. In these two examples, in comparison to single freshwater use, reductions of 17% and 32% in operating costs are observed. requirement for semicontinuous batch processes with fixed-load operations. Chaturvedi and Bandyopadhyay14 developed procedures based on rigorous mathematical arguments for targeting minimum freshwater requirement for single as well as cyclic batch processes. Lee et al.15 developed a mathematical model for synthesis of interplant WANs involving continuous and batch processes. As these methodologies are restricted to fixed-schedule operations, additional freshwater reduction through changing the schedule of operations cannot be explored. For variable-schedule approaches, freshwater minimization is carried out by determining an optimal schedule of operations. Majozi16 proposed a formulation to introduce wastewater minimization constraints within an established scheduling framework. Cheng and Chang17 proposed a formulation to incorporate batch production, water reuse, and wastewater treatment into a single comprehensive model. However, all possible network configurations were not included in the superstructure, resulting suboptimal solution. Gouws and Majozi18 incorporated multiple contaminants along with multiple storage vessels in a mixed-integer nonlinear programming (MINLP) formulation. Oliver et al.19 used a hybrid method that combines the use of insight-based and mathematical optimization-based techniques to synthesize a batch water network. Gouws et al.20 proposed a freshwater minimization in multipurpose batch chemical processes with and without central reusable water storage. Zhou et al.21 presented a nonconvex MINLP formulation to address similar problem addressed by Cheng and Chang.17 Majozi and Gouws22 presented a mass-transfer-based MINLP formulation for freshwater minimization in multipurpose batch process. Li et al.23 presented a mathematical technique to address singleand multiple-contaminant batch WANs, which takes account of production, water-reuse subsystems, and wastewater treatment

1. INTRODUCTION Water is one of the major resources in process industries. Efficient use of freshwater minimizes operating costs associated with water use and its discharge, as well as reduces its environmental impact. Techniques of process integration have been successfully applied for optimization of water usage in batch and continuous process industries.1,2 Wang and Smith,3 in a seminal paper, presented a graphical method to calculate minimum water requirement for continuous processes. Since then, significant research efforts have been directed toward water minimization in continuous processes.4 Design and synthesis of water allocation networks (WANs) for a batch process are generally more complex, because of the presence of an additional time dimension. Water minimization techniques for a batch process can be classified in two categories: fixed-schedule and variable-schedule. For water minimization problems with a fixed schedule, time is treated as a parameter. In variable-schedule water minimization problem, time is treated as a variable and an optimal schedule is determined.2 For fixed-schedule batch processes, mathematical as well as conceptual methodologies were proposed to reduce freshwater consumption. Wang and Smith5 presented a graphical method to calculate minimum freshwater requirement for semicontinuous processes. Almato et al.6 proposed nonlinear programming (NLP) formulations to optimize WANs for a batch process. Kim and Smith7 proposed a methodology to minimize freshwater requirement for batch processes including network design constraints. Foo et al.8 developed an algebraic methodology for targeting minimum freshwater requirement in cyclic batch processes. Majozi et al.9 proposed a graphical approach to minimize freshwater for truly batch processes. Chen et al.10 investigated the impact of central storage facilities on freshwater reduction in WANs. Shoaib et al.11 developed a three-stage methodology, where all water reuse/recycle between water sources and sinks are obtained through two consecutive batches of operation via water storage. Chen and Lee12 proposed a graphical method to handle batch process consisting of both truly batch and semicontinuous operations. Kim13 proposed an algebraic technique to minimize freshwater © 2014 American Chemical Society

Received: Revised: Accepted: Published: 5996

October 28, 2013 February 15, 2014 March 5, 2014 March 5, 2014 dx.doi.org/10.1021/ie403638v | Ind. Eng. Chem. Res. 2014, 53, 5996−6005

Industrial & Engineering Chemistry Research

Article

subsystems. Gouws et al.2 presented a review of all the techniques for water minimization in batch processes. Recently, Nonyane and Majozi24 proposed a mathematical model for wastewater minimization with longer horizon time. Tokos et al.25 developed an approach to design a large-scale water system, to integrate water-using operations and wastewater treatment units in different production sections within the same network via a case study. Tokos et al.26 proposed a biobjective optimization method for evaluation of the environmental and economic impacts to retrofit WANs. Tokos et al.26 used benchmarking of environmental indicators to calculate the environmental impact. Multiple freshwater resources can be utilized to minimize the operating cost of the process. Significant research efforts have been observed to optimize the use of multiple freshwater resources in continuous processes. Wang and Smith27 presented a graphical methodology, based on the concept of limiting composite curve, to target multiple freshwater resources and multiple-contaminant WANs. Almutlaq et al.28 proposed another graphical approach, based on a material recycle pinch diagram,29,30 to target multiple freshwater resources in WAN. It may be noted that economic factors are not included directly in these two methodologies. Jezowski et al.31 proposed a linear programming (LP) model, which accounts for multiple freshwater resources of different prices and concentrations, along with availability limits. Shenoy and Bandyopadhyay32 introduced the concept of prioritized cost to target multiple freshwater resources to minimize the operating cost of the overall process. Liu et al.33 proposed a simultaneous procedure for design and targeting WANs with multiple resources. Deng and Feng34 extended the concept of prioritized costs for property-based water networks with multiple freshwater resources. On the other hand, limited studies are conducted to optimize the use of multiple freshwater resources in batch processes. Similar to continuous processes, multiple freshwater resources may be utilized in batch water networks to minimize the overall operating cost of the process. Through an MINLP formulation, Li and Chang35 included the possibility of multiple freshwater resources. However, the applicability of the proposed methodology was not demonstrated. Recently, Chaturvedi and Bandyopadhyay36 proposed an algebraic targeting methodology for optimization of multiple freshwater resources in a batch process in order to minimize overall operating cost. However, the methodology is restricted to single contaminant and fixed-schedule batch process. In this paper, a mixed-integer linear programming (MILP)based mathematical formulation is proposed to minimize the operating cost of a batch process, incorporating multiple freshwater resources. The overall operating cost is minimized exploring flexibilities in schedule while keeping the overall production at its maximum. The proposed methodology is applicable to single- as well as multiple-contaminant WAN. Applicability of the proposed methodology is demonstrated with two illustrative examples. In these two examples, in comparison to single freshwater use, reductions of 17% and 32% in operating costs are observed.

(i) production scheduling data, including equipment and storages capacities, durations of tasks, time horizon of interest, and product recipes, (ii) water requirement data including, flow rates for sources and demands, maximum limit of concentrations for demands for each contaminant, and concentrations for sources for each contaminant, (iii) freshwater resources specifiaction data including cost, concentrations level of each contaminant, and maximum limit of availability of each resource. The objective is to determine the minimum operating cost for water supply incorporating flexibilities in schedule, while satisfying the maximum production.

3. MATHEMATICAL FORMULATION To introduce the available flexibilities in schedule, it is necessary to incorporate the scheduling framework. Hence, the constraints of this formulation can be categorized under two heads: scheduling constraints and WAN-related constraints. Constraints related to allocation, capacity, duration, and sequence are included in scheduling constraints. While the constraints related to flow and concentration requirements are included in WAN-related constraints. To determine the minimum operating cost for the WAN of a batch process, the following mathematical model is proposed. The mathematical model is composed of the following sets, variables, parameters, and constraints: Sets J = {j|j = unit} N = {n|n = event point} S = {s|s = any state} Sin = {sin|sin = input state} ⊂ S Sout = {sout|sout = output state} ⊂ S Sd = {sd|sd = resource demand state} ⊂ S Sso = {sso|sso = source state} ⊂ S R = {r|r = resource} CO = {co|co = contaminant} Continuous Variables m(s,j,n) = amount of state “s” exits or enter to/from unit “j” at a time point “n” M = maximum production Tp(s,j,n) = time at which state “s” appears in unit “j” at event point “n” Tds(sd,j,n) = time at which demand related to state “sd” starts in unit “j” at event point “n” Tde(sd,j,n) = time at which demand related to state “sd” ends in unit “j” at event point “n” Tss(sso,j,n) = time at which the source related to state “sso” starts in unit “j” at event point “n” Tse(sso,j,n) = time at which the source related to state “s” starts in unit “j” at event point “n” z(sd,sso,j,j′,n,n′) = fraction of time when the source related to (sso,j,n) is available to supply the demand related to (sd,j′,n′) to the total duration of source (j′⊆J,n′⊆N) fav(sd,sso,j,j′,n,n′) = flow available from the source related to (sso,j,n) is available to supply the demand related to (sd,j′,n′) (j′⊆J,n′⊆N) fsup(sd,sso,j,j′,n,n′) = flow supplied from the source related to (sso,j,n) is available to the demand related to (sd,j′,n′) (j′⊆J,n′⊆N) Qu(s,j) = maximum allowed amount of state “s” in unit “j” q(s,j,n) = amount of state “s” in unit “j” at time point “n”

2. PROBLEM STATEMENT The problem addressed in this work can be stated as follows. Given: 5997

dx.doi.org/10.1021/ie403638v | Ind. Eng. Chem. Res. 2014, 53, 5996−6005


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Rr(sd,j,n) = freshwater requirement of resource “r” for a demand related to (sd,j,n) Binary Variables y(s,j,n) = usage of state “s” in unit “j” at time point “n” yw(sd,sso,j,j′,n,n′) = availability of the source related to (sso,j,n) related demand related to (sd,j′,n′) (j′⊆J,n′⊆N) Parameters α = constant coefficient of processing time β = variable coefficient of processing time Cr = cost of resource “r” cr(co) = concentration level of contaminant co in resource “r” cd(co,sd,j) = maximum limit of contaminant “co” for state “sd” and in unit “j” cso(co,sso,j) = outlet concentration of contaminant “co” from state “sso” and unit “j” Fd(sd,j) = flow requirement by demand related to state “sd” and in unit “j” Fs(sso,j) = flow generation by source related to state “sso” and unit “j” H = time horizon of interest RM r = specified maximum availability of a freshwater resource “r” (t) τD(sd,j) = duration of demand related to state “sd” in unit “j” τS(sso,j) = duration of the source related to state “sso” and unit “j” 3.1. Scheduling Constraints. The term “scheduling constraints” includes constraints related to allocation, capacity, duration and sequence, and time horizon. Allocation Constraints. Equation 1 allocates the tasks to units. It may be noted that that only one task can take place in any given unit at any given point in time.

∑ y(sin , j , n) ≤ 1

Tp(sout , j , n) = Tp(sin , j , n − 1) + α(sin , j)y(sin , j , n − 1) + β(sin , j) ∑ m(sin , j , n − 1) Sin

∀ j ∈ J , n ∈ N , sin ∈ Sin , sout ∈ Sout

Sequence Constraints. A state “s” can only be used in a particular unit, at any event point after all previous state being processed. If sout ′ is a previous state of sin, then the sequence constraint can be expressed as follows: Tp(sin , j , n) ≥ Tp(s′out , j , n) ∀ j ∈ J , n ∈ N , sin ∈ Sin , sout ∈ Sout , s′out ∈ Sout

Tp(s , j , n) ≤ H

q(s , j , n) ≤ Q U (s , j)

∀ j ∈ J , n ∈ N , sd ∈ Sd (10)

Tse(sso , j , n) = Tp(sso , j , n)

∀ j ∈ J , n ∈ N , sso ∈ Sso (11)

Tds(sd , j , n) = Tde(sd , j , n) − τd(sd , j)y(sd , j , n)

∑

∀ j ∈ J , n ∈ N , sd ∈ Sd

∀ j ∈ J , n ∈ N , sso ∈ Sso

m(sout , j , n)

m(sin , j , n) −

∑

(13)

Flow Availability Constraints. A source may or may not available for a demand. A six-index binary variable is introduced as follows:

(3)

∑

(12)

Tss(sso , j , n) = Tse(sso , j , n) − τs(sso , j)y(sso , j , n)

sout ∈ Sout

sin ∈ Sin

∀ j ∈ J, n ∈ N

(9)

Tde(sd , j , n) = Tp(sd , j , n)

(2)

∀ j ∈ J, n ∈ N q(s , j , n) = q(s , j , n − 1) +

s = product

3.2. WAN-Related Constraints. Time mapping equations (i.e., time of demands and sources as mapped with respective process units are added in the second part), flow, and contaminant balance equations are included in this section. Time Mapping Constraints. Equations 10−13 map the time of demands and sources to process states.

Material Balances. Equation 3 expresses the material balance around a particular unit “j”. Similarly, eq 4 implies a general mass balance for amount of state “s” stored at a time point “n”: m(sin , j , n − 1) =

(8)

s⊂S j∈J n∈N

m(sin , j , n) ≤ Wmaxy(sin , j , n)

∀ j ∈ J, n ∈ N

∑

∀ j ∈ J, n ∈ N, s ∈ S

∑ ∑ ∑ m(s , j , n) = M

sin ∈ Sin

sin ∈ Sin

(7)

Production Constraint. Equation 9 ensures the maximum production (M) that can be produced for a given time horizon. The maximum production can be determined using published formulations.37,38 Also, considering only the scheduling constraints (eqs 1−8), the formulation can be solved with the objective of maximizing production to determine the maximum production.

Capacity Constraints. This constraint implies that the total amount of all the states at time point “n” is bounded by the capacity of the unit. The amount of material being processed should be between the limits Wmin and Wmax:

∑

∀ j ∈ J, n ∈ N, s ∈ S

Storage Constraints. Equation 8 ensures that the amount of state “s” stored at each time point cannot exceed the maximum allowed limit:

(1)

Wminy(sin , j , n) ≤

(6)

Time Horizon Constraints. Equation 7 directs that the appearance of all states should be within time horizon of interest:

∀ j ∈ J, n ∈ N

sin ∈ S

(5)

⎧1← ⎪ ⎪ yw (sd , sso , j , j′, n , n′) = ⎨ ⎪ ⎪ ⎩ 0←

m(sout , j , n)

sout ∈ Sout

(4)

Duration Constraints. The duration of a task is expressed in terms of the amount of material processed: 5998

If source related to (sso , j , n) can supply to demand related to (sd , j′, n′) partially or completely otherwise



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Figure 1. Flowsheet corresponding to a two-product batch plant (Example 1).

However, a source may be partially available for a demand. The limit of flow (z(sd,sso,j,j′,n,n′)) available from a source related to (sso,j,n) for a demand related to (sd,j′,n′) can be calculated as the fraction of time when the source is available to supply the demand to the total duration of source given in eq 14.

Fd(sd , j′, n′) = +

∑ R r(sd , j′, n′)

Fs(sso , j , n) ≥

∑ ∑ ∑

(19)

fsup (sd , sso , j , j′, n , n′)

sd ∈ Sd j ′∈ J n ′∈ N

(Tde(sd , j′, n′) − Tss(sso , j , n)) − MM(1 − yw (sd , sso , j , j′, n , n′))

∀ j ∈ J , n , ∈ N , sso ∈ Sso

(τ(s , j)) ∀ j , j′ ∈ J , n , n′ ∈ N , sd ∈ Sd , sso ∈ Sso

∀ j′ ∈ J , n′ ∈ N , sd ∈ Sd

r∈R

z(sd , sso , j , j′, n , n′) ≤

∑ ∑ ∑ fsup (sd , sso , j , j′, n , n′) sso ∈ Sso j ∈ J n ∈ N

(14)

(20)

Fd(sd , j′, n′)cd(co , sd , j′)

Equation 14 can be rearranged as follows:

≥

∑ ∑ ∑ fsup (sd , sso , j , j′, n , n′)cs(co , sso , j) sso ∈ Sso j ∈ J n ∈ N

z(sd , sso , j , j′, n , n′)(τ(s , j)) − (Tde(sd , j′, n′) − Tss(sso , j , n))

+

− MM(1 − yw (sd , sso , j , j′, n , n′)) ≤ 0 ∀ j , j′ ∈ J , n , n′ ∈ N , sd ∈ Sd , sso ∈ Sso

Freshwater Resource Availability. Maximum availability of a freshwater resource “r” can be confined using the following equation:

∑ ∑ ∑ R r(sd , j , n) ≤ R rM

z(sd , sso , j , j′, n , n′) ≤ 1 (16)

(22)

Objective Function. The objective is to minimize the operating cost given in eq 23: minimize

(17)

∑ ∑ ∑ ∑ R r(sd , j , n)Cr sd ∈ Sd j ∈ J n ∈ N r ∈ R

Equation 17 expresses the flow of a source available to the demand. Flow and Contaminant Balances. Equation 18 imposes the source flow limitation, where fsup(sd,sso,j,k,n,p) is the flow supplied from fav(sd,sso,j,k,n,p) to a demand related to (sd,j,n). Equations 19 and 20 express the flow balance for demand and limit of source supplied as per availability, respectively. Equation 21 expresses the maximum limit on contaminant level of demands.

(23)

The overall formulation can be solved as a mixed-integer linear programming (MILP) problem.

4. ILLUSTRATIVE EXAMPLES The proposed methodology is illustrated via two examples. The first example is a single contaminant example, and its results are compared with a fixed-schedule batch process. The second example illustrates the applicability of a methodology to multicontaminant WAN in a batch process. The optimization problems are solved with the XPRESS solver included in GAMS39 software (GAMS version 23.7). 4.1. Illustrative Example 1. The proposed methodology is applied to calculate minimum operating cost for the two

fsup (sd , sso , j , j′, n , n′) ≤ fav (sd , sso , j , j′, n , n′) ∀ j , j′ ∈ J , n , n′ ∈ N , sd ∈ Sd , sso ∈ Sso

∀r∈R

sd ∈ Sd j ∈ J n ∈ N

fav (sd , sso , j , j′, n , n′) = Fz s (sd , sso , j , j′ , n , n′) ∀ j , j′ ∈ J , n , n′ ∈ N , sd ∈ Sd , sso ∈ Sso

∀ j′ ∈ J , n′ ∈ N , sd ∈ Sd (21)

(15)

It may be noted that this fraction cannot be greater than 1 and this constraint is imposed in eq 16:

∀ j , j′ ∈ J , n , n′ ∈ N , sd ∈ Sd , sso ∈ Sso

∑ R r(sd , j′, n′)cr(co) r∈R

(18) 5999



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product batch plant,40 which is adopted to include water minimization aspects. Figure 1 represents the corresponding flow sheet of process. Water is required for washing of reactors at the end of each reaction. The production recipe is as follows. Feed A is heated inside in a heater (HR). A mixture of 50% feed B and 50% feed C, on a mass basis, is reacted (reaction 1). The product of this reaction is intermediate BC. A mixture of 40% hot A and 60% intermediate BC is reacted (reaction 2) to form product 1 (40%) and intermediate AB (60%). A mixture of 20% feed C and 80% intermediate AB is reacted (reaction 3). The reaction produces impure E. Impure E is purified to produce product 2 (90%) and intermediate AB (10%). A heater (HR) is used to heat feed A. Two reactors (RR1 and RR2) are available to perform three different chemical reactions (reactions 1, 2, and 3) and a separator exists to purify impure E. Table 1 shows information regarding the production process.

(products 1 and 2) is 151.3 units, comprising 70.9 units of product 1 and 80.4 units of product 2. Feed A is fed to heater, and feeds B and C are fed to reactors RR1 and RR2 at the starting point of horizon. The feed are processed as per recipe; product 1 is produced via reaction 2, and product 2 is produced after separation. Reactors are washed at the end of each reaction. Table 2 shows information regarding the washing requirement and two Table 2. Data Pertaining to the Water Requirement for Example 1 Contaminant Concentration (ppm)

Table 1. Information Regarding the Production Process for Two-Product Batch Plants processing time (h), α + β*batch size unit

max batch size (kg)

α (h)

β (× 102 h/kg)

heating, H

HR

100

0.667

0.6667

reaction 1, R1

RR1 RR2

50 80

1.334 1.334

2.6640 1.6650

reaction 2, R2

RR1 RR2

50 80

1.334 1.334

2.6640 1.6650

reaction 3, R3

RR1 RR2

50 80

0.667 0.667

1.3320 0.8325

separation, S

SR

200

1.334

0.6667

task

demand

source

flow (t)

0.2 0.2

250 250

600 600

80 80

RR1 RR2

0.2 0.2

500 500

800 800

100 100

reaction 3, R3

RR1 RR2

0.2 0.2

400 400

850 850

120 120

separation, S

SR

0

task

unit

washing time (h)

heating, H

HR

0

reaction 1, R1

RR1 RR2

reaction 2, R2

freshwater resources are available for water supply; their specifications are listed in Table 3. Washing time is included in Table 3. Freshwater Resource Specifications for Example 1 FW1 FW2

The processing time of each task varies with the batch size. One of the possible schedules38 for an 8-h time of horizon with the maximum production is shown in Figure 2. Total production

quality (contaminant concentration, ppm)

cost ($/t)

0 350

9 4

Figure 2. Given production schedule for Example 1.38 6000



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Figure 3. Possible water allocation network for given schedule for Example 1.

the total operation time of the process unit. It may be noted that the minimum operating cost for the given schedule38 can be calculated to be $2504.12, using the methodology proposed by Chaturvedi and Bandyopadhyay.36 A possible water network for this case is shown in Figure 3 (the freshwater requirement (in kg) is shown in bold italic font). Adding the total production as a constraint, along with other constraints (eqs 1−8, 10−13, and 15−22), the problem is solved to minimize operating cost (eq 23). The size of this MILP problem is 5800 constraints, 4300 continuous variables, and 940 binary variables. The model is solved using the GAMS/XPRESS39 solver on the computer (Intel Core 2 Quad, 3 GHz and 2 GB RAM). The solution time is within fractions of seconds. The minimum operating cost is determined to be $2077.46. A reduction of 17% in operating cost is observed from the given schedule requirement (multiple freshwater resources) and approximately the same reduction is observed from single freshwater resource usage in a flexible scheduled frame. Summary of results is shown in Table 4, along with the

freshwater requirement and the operating cost with various integration possibilities. It can be observed from Table 4 that operating cost for the case of multiple freshwater resources and flexible schedule is the minimum. A possible schedule and water network for the example is shown in Figure 4 (freshwater requirement (in kg) is shown in bold italic). Table 5 shows the change in freshwater requirement for the fixed38 and flexible schedule cases. The change in production schedule can also be observed; compared to fixed-schedule processes, for flexibleschedule processes, reaction 2 replaces reaction 3 in reactor 1. 4.2. Illustrative Example 2. The proposed methodology can be applied for calculation of minimum operating cost where multiple contaminants are present in WANs. Multiple contaminants are incorporated in two-product batch plant illustrated in Example 140 to illustrate applicability of proposed formulation to multiple component batch process. Data pertaining to water requirements and available freshwater resources specifications are given in Tables 6 and 7, respectively. The minimum operating cost is determined to be $2086.4, using the proposed formulation. The MILP model has 7000 constraints, 4700 continuous variables, and 1200 binary variables. The model was solved using the GAMS/XPRESS39 solver on the computer (Intel Core 2 Quad, 3 GHz and 2 GB RAM). The solution time is negligible (fractions of seconds). A summary of results is shown in Table 8. A reduction of 31.7% in operating cost may be observed from Table 8 (in comparison of single resource utilization). A possible water network for the example is shown in Figure 5 (the freshwater requirement (in kg) is shown in boldface font).

Table 4. Results for Example 1: Water Minimization resource

FW1 (t)

FW2 (t)

total operating cost ($)

without integration

single resource

800

fixed schedule

single resource multiple resources

340 45.71

523.16

3060 2504.12

single resource multiple resources

280 45.71

416.5

2520 2077.46

flexible schedule

7200

6001



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Figure 4. A possible schedule and water network for Example 1.

Table 5. Freshwater Requirements for Fixed-Schedule and Flexible-Schedule Cases for Example 1

Table 7. Freshwater Resources Specifications for Example 2 Quality (Contaminant Concentration, ppm)

Freshwater Requirement (t) Fixed Schedule task, i reaction 1 reaction 2 reaction 3 reaction 2

RR1 RR2 RR1 RR2 RR1 RR2 RR1 RR1 RR2

Flexible Schedule

FW1

FW2

FW1

FW2

22.85 22.85 0 0 0 0 0 0 0

57.15 57.15 40 66.67 96 96

22.85 22.85 0 0 0 0 0 0 0

57.15 57.15 40 40

66.67 43.55

FW1 FW2 FW3

C1

C2

C3

C4

cost ($/t)

0 150 350

0 100 200

0 150 300

0 100 150

9 5 3

Table 8. Summary of Results for Example 2 resource

96 40 59.55 66.67

cost($/ t)

single resource, FW1

multiple resources

9 5 3

339.48

53.34 157.47 273.01

FW1(t) FW2 (t) FW3 (t) total operating cost ($)

5. CONCLUSIONS Similar to continuous processes, multiple freshwater resources can be utilized in batch water networks to minimize the overall

3055.31

2086.4

operating cost related to freshwater utilization. A mixed-integer linear programming (MILP)-based mathematical formulation to minimize the operating cost of a water allocation network

Table 6. Data Pertaining to the Water Requirement for Illustrative Example 2 Maximum Inlet Concentration (ppm)

Outlet Concentration (ppm)

task, i

unit, j

C1

C2

C3

C4

C1

C2

C3

C4

washing time (h)

flow (kg)

reaction 1

RR1 RR2 RR1 RR2 RR1 RR2

100 100 700 700 900 900

200 200 600 600 800 800

350 350 300 300 600 600

200 200 400 400 750 750

450 450 800 800 1200 1200

500 500 700 700 1000 1000

400 400 750 750 700 700

300 300 650 650 900 900

0.25 0.3 0.25 0.3 0.25 0.3

80 80 100 100 120 120

reaction 2 reaction 3

6002



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Figure 5. A possible schedule and water network for Example 2.

cr = concentration of contaminant in resource “r” (ppm) co = contaminant Fd = flow requirement by demand (kg) Fs = flow generation by a source (kg) fav = flow available from a source to supply a demand (kg) fsup = flow supplied from source to the demand (kg) G = production demand (kg) H = time horizon of interest (h) J = set of available units j = unit where a task is performed M = maximum production in specified time horizon (kg) m = amount of state exits or enter to/from a unit (kg) MM = any large number N = set of event points n = event point QU = maximum amount of state can be stored within time of interest (kg) q = amount of a state at a time point (kg) r = resource R = set of available resources Rr = freshwater resource requirement of resource “r” (kg) RM r = specified maximum availability of a freshwater resource “r” (t) Sd = set of demand states Sin = set of input states Sout = set of output states S = set of states Sso = set of source states

(WAN) in a batch process by utilizing multiple freshwater resources is proposed in this paper. The methodology explores flexibilities in the schedule of a batch process without compromising the overall production, while minimizing the overall operating cost of a WAN in the batch process. The proposed methodology is applicable to single as well as multicontaminant water allocation problems. Applicability of the methodology is demonstrated with two illustrative examples. In these two examples, in comparison to single freshwater use, reductions of 17% and 32% in operating costs are observed. The proposed formulation may be utilized include other objective functions (such as minimum make-span for given batch size, etc.). Current research is directed toward such issues.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +91-22-25767894. Fax: +91-22-25726875. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE cd = maximum limit of contaminant inlet (ppm) cs = outlet concentration of contaminant (ppm) CO = set of contaminants Cr = cost of resource “r” ($) 6003



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s = state sd = demand state sin = input state sout = output state sso = source state T = time (h) Tde = time at which demand ends (h) Tds = time at which demand starts (h) Tse = time at which a source generation ends (h) Tss = time at which source generation starts (h) Tp = time at which a state appears (h) Qu = maximum allowed amount of a state in a unit (kg) q = amount of a state in a unit (kg) Wmax = minimum capacity that can be utilized of a unit (kg) Wmin = maximum capacity that can be utilized of a unit (kg) y = binary variable associated with usage of a state in a unit at a time point yw = binary variable associated with availability of a water source for a water demand z = fraction of time when a source is available to supply a demand to its total duration α = constant coefficient of processing time (h) β = variable coefficient of processing time (h/kg) τD = duration of demand related to a state in a unit (h) τS = duration of source related to a state in a unit (h)

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