Water Allocation Network Design Concerning Process Disturbance

ARTICLE pubs.acs.org/IECR

Water Allocation Network Design Concerning Process Disturbance Xiao Feng,*,† Renjie Shen,‡ Xuesong Zheng,‡ and Chunxi Lu† †

State Key Laboratory of Heavy Oil Processing, Faculty of Chemical Science and Engineering, China University of Petroleum, Beijing 102249, China ‡ Department of Chemical Engineering, Xi’an Jiaotong University, Xi’an 710049, China ABSTRACT: Water system integration can effectively reduce freshwater consumption and minimize wastewater discharge of a water-using system, but integration makes the resultant water network structurally more integrated among process units. Therefore, to design a flexible water allocation network that is easy to control becomes highly desirable. In this Article, a methodology is presented for water allocation network design with process disturbance taken into account. For a new design, the synthesized network structure can guarantee that the water system consumes minimum freshwater under both normal and disturbance conditions and with minimum number of control streams under disturbance. At the same time, the information needed to adjust flow rate under disturbance will be obtained. For an existing water system, an adjustment scheme under disturbance with minimum freshwater consumption for the existing structure will be obtained. The water network designed by the new methodology has the feature of minimum freshwater consumption and high flexibility under disturbance. Some case studies are used to demonstrate the method.

1. INTRODUCTION Although water is one of many abundant natural resources on the earth, the demand for it has increased dramatically today due to rapid economic expansion in many regions worldwide. The excessive use and the pollution of water resources are big problems for human beings. For the chemical process industry, much research is focused on improving the water network to reduce freshwater consumption and wastewater discharge by using water system integration, one of the most efficient technologies for saving freshwater and reducing wastewater discharge.1-6 During the design of a water network, to minimize freshwater consumption, the resultant network will be more integrated among process units. Many water units, or even the whole network, will be affected when there are fluctuations in mass load or water quality at some process units. Adjustment is needed when there are such fluctuations, and if the adjustment is made by using freshwater directly, the water saving result will be reduced. However, in actual water systems, fluctuations of mass load and water quality are unavoidable. Therefore, it is highly desirable to design a water network with minimum freshwater consumption under both nominal and disturbance conditions. There are two major approaches for synthesizing a network operating under uncertainty: one is based on flexibility, and the other is based on stochastic programming.7 Halemane and Grossmann8 introduce the flexibility index for chemical processes, the stress of which is on ensuring feasibility of design by adjusting the control variables in the system when the uncertain parameters change. Because the flexibility index is difficult to solve, much work has been done to solve the max and min problem,9-11 but it remains a difficult task. On the other hand, for a stochastic programming approach,12 the emphasis is on achieving optimality accounting for the fact that the recourse variables can be adjusted for each parameter realization, and there exist numerous methods for the solution of several classes r 2010 American Chemical Society

of stochastic programs.7 Tan and Cruz described a procedure for the synthesis of robust water reuse networks with single-component from imprecise data using symmetric fuzzy linear programming. Yet their work can not solve the multi-concentration problem.13 Tan and Foo demonstrated the use of Monte Carlo simulation in assessing the vulnerability of water networks to noisy mass loads.14 Tolerance amount of a water network was proposed to quantify the resilience of a water network by Zhang and Feng.15 Recently, Karuppiah and Grossmann7 developed multiscenario programming models to solve the problem of water system integration under uncertainty, under the assumption that the uncertain parameters take on a finite set of known values. This novel method can solve the problem of water system integration under uncertainty to a certain extent. However, the multiscenario models associated with the integrated water networks operating under uncertainty grow in size with the number of scenarios and are computationally expensive to solve.7 In this Article, a new method for optimal water network design is presented. The goal is to make the water system have the optimum performance both under the nominal conditions and under the worst disturbed scenario. This is based on the following two thoughts. First, because at most time the system operates at the nominal status, the system should be guaranteed to be optimal under the nominal conditions. Second, if a system has optimum performance both under the nominal conditions and under the worst scenario, it will have optimum performance

Special Issue: Water Network Synthesis Received: April 8, 2010 Accepted: September 22, 2010 Revised: September 19, 2010 Published: October 05, 2010 3675

dx.doi.org/10.1021/ie100847s | Ind. Eng. Chem. Res. 2011, 50, 3675–3685

Industrial & Engineering Chemistry Research

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Figure 1. Mass transfer process in water-using units. Figure 3. Inlet and outlet stream details of each water-using unit under normal conditions.

Figure 2. Superstructure of water allocation network.

at any scenario in between. Obviously, in this way, the problem becomes much easier to solve for only dealing with two scenarios.

2. PROBLEM STATEMENT The general problem of water allocation network design concerning process disturbance may be stated as follows. In a water system, a set of water units are given. Each water-using unit with a given concentration limits contaminant loads, and upper bounds of the contaminant load increase during a disturbance. External freshwater will be used in the system. If the concentration of wastewater from a water unit is less than a predetermined maximum inlet limit of another unit, then it can be reused by the other unit. There is no limitation on the wastewater that flows to the environment of the whole water network. The objective of this work is to develop an algorithmic technique for the water networks with minimum freshwater consumption and wastewater discharge both under the nominal conditions and under the worst disturbed scenario. 3. SUPERSTRUCTURE OF WATER ALLOCATION NETWORK UNDER NOMINAL AND DISTURBED CONDITIONS 3.1. Superstructure of Water Allocation Network under Nominal Condition. Under the nominal operating condition,

inlet streams must match the demand of water-using units both on the quantity level and on the quality level. The inlet and outlet concentrations must not be greater than the maximum values. With the steady mass transfer process between the water stream and the process stream, a constant amount of contaminant is removed from the process stream and is carried away by the outlet water stream. The mass transfer process is shown in Figure 1. In a water allocation network, each water-using unit can serve as a potential water source for other units. That is, it can supply other units with its discharged water. On the other hand, each water-using unit can act as a potential water sink for other units. That is, it can receive discharged water from other units. Therefore, the superstructure of the system is shown in Figure 2. The details for inlet and outlet streams of each water-using unit are shown in Figure 3. By reusing the water between water-using units, freshwater consumption in the water-using system can be decreased significantly.

In Figure 2, (1) “F” is the symbol for the freshwater source. Arrows spreading from it denote freshwater streams allocated for water-using units. (2) “M” is the symbol for the mixing point for water streams before they enter water-using units. Arrows pointing to the mixing point denote allocated freshwater streams and the water streams coming from other water-using units. (3) “S” is the symbol for the splitting point for discharged water, including the water streams sent to other water-using units and the wastewater streams. (4) “W” is the symbol for the wastewater main. Arrows pointing to it denote the wastewater streams discharged from water-using units. Arrows in group 2 can be connected with arrows in group 1 or in group 3. Arrows in group 4 can be connected with arrows in group 3. It should be noticed that water cannot be recycled in any water-using unit. That is, the discharged water of a certain waterusing unit cannot be used again in the same water-using unit. 3.2. Superstructure of Water Allocation Network under Disturbance. In a water system, the original disturbance will come from some process streams and will lead to a variation of the contaminant mass load of water in the process. Next, the outlet contaminant concentration of water from these units will vary and in turn affect the water inlet concentration in the downstream processes so as to propagate the disturbance.16 Disturbance can cause the transferred mass load to increase or decrease. When the mass load decreases, the water-using unit and the whole water system can operate normally. When the mass load increases, the corresponding water-using unit and its downstream units will deviate from their normal operation scenarios, and adjusting the flow rate of freshwater and water streams between waterusing units should be performed, otherwise the system cannot operate normally. The disturbed water-using system may re-establish its stable operation after such adjustment. Therefore, in this Article, we only consider the situation of mass load increase. The network after adjustment is called the water-using network under disturbance in this Article. After the disturbance is removed from the system, the system will operate under the nominal conditions again, so the flow rates of all the streams should be restored to the nominal conditions. The adjustable streams include freshwater streams, water streams between water-using units, and wastewater streams. The details for inlet and outlet streams of each water-using unit under disturbance are shown in Figure 4.

4. DESIGN PROCEDURE FOR WATER ALLOCATION NETWORK CONCERNING DISTURBANCE Because the water-using system usually operates under the nominal condition, it is important to guarantee that the system consumes minimum freshwater under nominal conditions, which can reduce operating cost. So first, the minimum freshwater consumption under the nominal condition is calculated by solving the mathematical model P1, and it is a necessary reference in the following procedure. Second, a mathematical model, P2, which simultaneously describes the water-using system under the 3676

dx.doi.org/10.1021/ie100847s |Ind. Eng. Chem. Res. 2011, 50, 3675–3685


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Figure 4. Inlet and outlet stream details of each water-using unit under disturbance.

nominal conditions as well as at the worst disturbed scenario is established. Moreover, the limit of freshwater consumption is added to the constraints to specify the maximum acceptable freshwater consumption in the water-using network under nominal conditions. By solving model P2, the minimum increase of freshwater consumption for the worst disturbed scenario is obtained. To minimize the number of streams to be adjusted, another mathematical model P3 is introduced to simplify the network controllability. By solving this model, the final optimal design can be generated, including the optimal waterusing network and the optimal adjustment scheme under disturbance. 4.1. Minimum Freshwater Consumption under Nominal Condition. According to the water and contaminant balance in each water-using unit and each mixing point, the mathematical model P1 can be established for the superstructure of the water allocation network under nominal conditions. The objective is to minimize the total freshwater consumption. The constraints include the water and contaminant balance and contaminant concentration limits. P1: X FjW objective : min ð1Þ j∈P

S.t. Mass balance of inlet and outlet streams in unit j: X X Fi, j ¼ FjD þ Fj, k þ FjL j∈P FjW þ i∈P i6¼ j

ð2Þ

k∈P k6¼ j

Mass balance of contaminant s for the inlet mixing point before unit j: X X W ðFi, j 3 Cout Fi, j Þ 3 Cin ð3Þ i, s Þ ¼ ðFj þ j, s j∈P, s∈C i∈P i6¼ j

i∈P i6¼ j

Mass balance of contaminant s for mass transfer in unit j: X ðFjW þ Fi, j Þ 3 Cin j, s þ Mj, s i∈P i6¼ j

¼ ðFjD þ

X

Fj, k þ FjL Þ 3 Cout j, s j∈P, s∈C

ð4Þ

k∈P k6¼ j

With the known limiting water data of the water-using system, the mathematical model can be solved, and the minimum freshwater consumption under nominal conditions will be obtained. The minimum freshwater consumption is denoted as FminW in the following. 4.2. Minimum Increase of Freshwater Consumption under Disturbance. The water system can run steadily if there is no disturbance. When a disturbance occurs in the water-using system, for example, the mass load of contaminant to be removed increases in certain units, the outlet contaminant concentration will increase if the flow rate is fixed, so that the corresponding units and their downstream units will deviate from their nominal operating conditions. Therefore, the system needs to adjust. To bring those disturbed water-using units to stable operation under disturbance, the most convenient approach is increasing the freshwater consumption of the disturbed water-using units, so that the outlet contaminant concentration can attain its acceptable level. However, the freshwater consumption will increase if one uses this approach. Another way is redesigning the water allocation network with optimization according to the limiting water data under disturbance and sequentially adjusting the existing network. Although in this method the freshwater consumption of the new design can be guaranteed to be the minimum, the network structure may be considerably different from that of the existing network. Difficulty in switching between the two networks will have a great negative effect on the feasibility of the new design, because the condition under disturbance is temporary after all. What we want is the network structure under disturbance to be the same as that under nominal conditions, and the freshwater consumptions are achieving the minimum value under both conditions. The water-using system under disturbance is shifted to another stable condition only by adjusting the streamflow rate of the existing network. Through optimization with the limiting water data under disturbance, the freshwater consumption also can be minimized under this condition. To carry the point, a mathematical model, P2, which can describe the water-using system both under nominal conditions and under disturbance, is established. With mass balance specification and the limits for inlet and outlet contaminant concentrations, basic constraints of the model can be formulated. Yet they are not sufficient for the structure specification. To confine the structure of the network under disturbance to be the same as that of the network under nominal condition, several additional constraints are introduced. F þ dF is used to describe the streamflow rate under disturbance. F is the streamflow rate under nominal condition, and dF is the flow rate adjustment under disturbance. A constant λ is introduced, which should conform to the regulation: 0 < λ < 1. With constraints 7-9, the structure of the network under disturbance is guaranteed to be the same as that under nominal conditions and can control percent change for the adjusted flow rate. - λ 3 FjW edFjW eλ 3 FjW ð7Þ

Constraints for inlet and outlet concentrations in unit j: in, max Cin j∈P, s∈C j , s e Cj , s

Cout j, s

e

out, max Cj, s

j∈P, s∈C

All of the above variables are non-negative.

ð5Þ ð6Þ

- λ 3 Fi, j edFi, j eλ 3 Fi, j

ð8Þ

- λ 3 FjD edFjD eλ 3 FjD

ð9Þ

In the optimal solution, if a certain stream exists in the network under nominal conditions, then F > 0. With the constraint 3677



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of -λ 3 F e dF e λ 3 F, it can be deduced that F þ dF g (1 - λ)F > 0. This means that the corresponding stream exists in the network under disturbance. On the other hand, if a certain stream does not exist in the water-using network under nominal conditions, then F = 0. With the constraint of -λ 3 F e dF e λ 3 F, it can be deduced that dF = 0. This means that the corresponding stream does not exist in the network under disturbance either. So it is demonstrated that in the optimal solution the structure of the network under disturbance is guaranteed to be the same as the one under nominal conditions. With the known limiting water data of the water-using system and mass load disturbance, the mathematical model can be established. P2: X dFjW objective : min ð10Þ j∈P

S.t. Constraints 2-6: Additional constraints: Mass balance of inlet and outlet streams in unit j under disturbance: dFjW þ

X

dFi, j ¼ dFjD þ

i∈P i6¼ j

X

dFj, k þ dFjL j∈P

ð11Þ

k∈P k6¼ j

Mass balance of contaminant s for the inlet mixing point before unit j under disturbance: X

0

W W ððFi, j þ dFi, j Þ 3 Ci,out s Þ ¼ ððFj þ dFj Þ

i∈P i6¼ j

þ

X 0 ðFi, j þ dFi, j ÞÞ 3 Cj,ins j∈P, s∈C

ð12Þ

i∈P i6¼ j

Mass balance of contaminant s for mass transfer in unit j under disturbance: ððFjW

þ dFjW Þ þ

P i∈P i6¼ j

ððFjD þ dFjD Þ þ

0

ðFi, j þ dFi, j ÞÞ 3 Cj,ins P j∈P j6¼ k

ð13Þ

Constraints for inlet and outlet concentrations in unit j under disturbance: 0 in, max j∈P, s∈C Cj,ins e Cj, s

out, max

Cj,out s e Cj , s

j∈P, s∈C

ð14Þ ð15Þ

ð7Þ

- λFi, j e dFi, j e λFi, j j∈P

ð8Þ

- λFjD e dFjD e λFjD j∈P

ð9Þ

W FjW ¼ Fmin

ð16Þ

j∈P

In mathematical model P2: Constraints 2-6 describe the limitations in the water-using networks under nominal condition, including mass balance formulation as well as the inlet and outlet concentration limits in each unit. Constraints 11-15 describe basic limitations in the waterusing networks under disturbance, including mass balance formulation as well as the inlet and outlet concentration limits in each unit. Constraints 7-9 limit the network structure under disturbance to be the same as that under nominal condition. Constraint 16 limits the freshwater consumption for the waterusing network under nominal condition to the minimum value. The minimum increase of freshwater consumption under disturbance can be obtained by solving the model, which is denoted as dFminW in the following. The optimal solution provides the information about the water-using network under nominal conditions and streamflow rate alterations under disturbance. 4.3. Optimal Water-Using Network with Minimum Number of Control Streams. The number of adjusting streams correlates with the degree of control and cost. Having fewer such streams is better. In the optimal solution of mathematical model P2, the number of control streams may be not minimal. Therefore, another model P3 is introduced to minimize the number of control streams, so that the switch between networks under different conditions will be more convenient and economical. To properly describe the control status of water streams, several binary variables are introduced, including dyW j , dyi,j, and dyD j . The mathematical model needs to include the following constraints:

ðFj, k þ dFj, k Þ þ ðFjL þ dFjL ÞÞ 3

0

- λFjW e dFjW e λFjW j∈P

X

þ Mj, s þ dMj, s ¼

j∈P, s∈C Cj,out s

0

Accessorial constraints:

jdFjW j - U 3 dyW j e0

ð17Þ

jdFi, j j - U 3 dyi, j e0

ð18Þ

jdFjD j - U 3 dyDj e0

ð19Þ

It can be guaranteed that in the optimal solution, when the variable is equal to 1, the corresponding stream should be adjusted; when the variable is equal to 0, the corresponding stream should not be adjusted. Here, U is a constant, which is large enough, larger than any flow rate in the network. In the optimal solution, if a certain stream should be adjusted, then |dF| > 0. With the constraint |dF| - U 3 dy e 0, it can be deduced that dy = 1. If no adjustment is exerted on a certain stream, then |dF| = 0. Whether dy = 1 or dy = 0, the constraint |dF| -P U 3 dy e 0 can be satisfied. However, with the minimization of dy, it can be guaranteed in the optimal solution that dy = 0. The minimum number of control streams can be obtained by using the known limiting water data of the water-using system 3678



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and the mass load disturbance to solve the following mathematical model P3. P3: XX X X dyW dyi, j þ dyDj objective : min j þ j∈P i∈P j∈P j∈P ð20Þ

Table 1. Water Data of Example 1 process

j6¼ i

S.t. Constraints 2-9: Constraints 11-16: Additional constraints: jdFjW j -

Cin i,max (ppm)

U

W 3 dyj e0

j∈P

ð17Þ ð18Þ

jdFjD j - U 3 dyDj e0 j∈P

ð19Þ

W dFjW e dFmin ð1 þ ξÞ

ð21Þ

j∈P

In mathematical model P3: Constraints 17-19 describe the status (adjusting or not adjusting) of water streams in the water-using network. Constraint 21 specifies the acceptable increase of freshwater consumption for water-using networks under disturbance. ξ is a constant. If ξ = 0, then the acceptable increase of freshwater consumption is defined as the minimum value. A trade-off between freshwater adjustment and the number of control streams exists by choosing different ξ. The final optimal design can be constructed according to this solution. The optimal solution provides the information for the water-using network under nominal conditions and streamflow rate alterations under disturbance. In this design, both the waterusing network under nominal conditions and the one under disturbance can achieve the minimum freshwater consumption. The switch between networks under different conditions is significantly simplified by limiting the structure and minimizing the number of control streams in the water-using network under disturbance.

5. SOLVING THE MATHEMATICAL MODELS The proposed model is highly nonlinear, which arises from the bilinear terms in many of the constraints. Integer variables are introduced in the third model P3. The mathematical models can be solved by nonlinear mathematical programming. In this Article, software LINGO is used to solve the mathematical models. The initial points to nonlinear optimization heavily affect the solution. All the results yielded by the software, with the initial points chosen by the software, are indicated as local optima. 5.1. Adding Other Constraints. Some other constraints can be added to the models, so that the target network may achieve other features. For example, if the total flow rate in a certain unit should not exceed Fmax j , then the following constraints can be added. X FjW þ Fi, j eFjmax ð22Þ i∈P i6¼ j

Cout i,max (ppm)

δm (g/h)

1

2000

0

100

200

2

5000

50

100

250

3

30 000

50

800

4

4000

400

800

ðFjW þ dFjW Þ þ

jdFi, j j - U 3 dyi, j e0 i∈P; j∈P, j6¼ i X

mi (g/h)

X

ðFi, j þ dFi, j Þ e Fjmax

ð23Þ

i∈P i6¼ j

Or if connections between certain water-using units are prohibited, then the following constraint can be added: ð24Þ Fi , j ¼ 0 With the consideration of practical engineering situations, adding some proper constraints may give the target water-using network better engineering or economic features. 5.2. The Optimal Adjustment Scheme under Disturbance. For grassroots design, based on the nominal conditions and the worst disturbed scenario, mathematical models P1, P2, and P3 can be used to obtain the optimal network structure with minimum freshwater consumption both under nominal conditions and under the worst disturbed scenario. Next, for the water network operates under a certain disturbance, or for an existing network, because the connections among process units are known, by solving the mathematical models P1, P2, and P3, the adjustment scheme under disturbance with minimum freshwater consumption can be obtained.

6. CASE STUDY 6.1. Single Contaminant Water-Using Systems Involving 4 Units. In this section, two cases are studied. The first case is cited

from Wang and Smith.1 The limiting water data and optimal design presented in the reference are shown in Table 1 and Figure 5, respectively. By solving the mathematical model P1, the minimum freshwater consumption under nominal conditions can be obtained as 90 t/h, which accords with the design in Figure 5. Next, consider the situation under disturbance. With the mass load increase shown in Table 1, mathematical model P2 can be established and solved. The minimum increase of freshwater consumption is obtained as 4.5 t/h. By solving the mathematical model P3, the optimal water-using networks can be generated. These are shown in Figure 6 under nominal conditions and in Figure 7 under disturbance. The streams that need to be adjusted during disturbance are indicated with broken lines. The adjustment under disturbance for the design in Figure 7 involves one unit and two water streams. When the water-using network in Figure 5 is under disturbance in Table 1, the freshwater consumption of unit 1 and unit 2 should be increased to re-establish the stable operating condition of the system. Moreover, the excessive outlet flow rate of unit 1 and unit 2 should be discharged as wastewater, as shown in Figure 8. The alteration of this design involves 2 units and 4 water streams, and one stream is added (2 t/h from unit 1 to wastewater discharge). If the network structure remains unchanged, that is, no new stream is added, more freshwater will be consumed, 3679



Figure 5. Solution from Wang and Smith (1994) for example 1.

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Figure 9. Solution for disturbance condition with unchanged structure in Figure 5, which needs more fresh water.

Table 2. Water Data of Example 2

Figure 6. Optimal network under nominal conditions for example 1.

process

mi (g/h)

Cin i,max (ppm)

Cout i,max (ppm)

δm (g/h)

1 2

7000 22 400

0 100

200 500

470 1010

3

62 550

200

650

5140

4

2000

0

200

210

Figure 7. Optimal network under disturbance for example 1. Figure 10. Optimal network under nominal conditions from Tan and Cruz13 for example 2.

Figure 8. Modified solution under disturbance for network structure in Figure 5.

as shown in Figure 9. Obviously, neither of the two water systems in Figure 8 or Figure 9 is as easy to control as that of Figure 7. The second case comes from Tan and Crus.13 Design data and optimal design presented in the reference are shown in Table 2, Figure 10, and Figure 11. By solving the mathematical model P1, the minimum total freshwater demand for the system under nominal conditions can be obtained as 146.2 t/h, the same as the result in Tan and Cruz,13 and the optimal design is shown in Figure 12, which is different from Figure 10. The system running under disturbance with the mass load increase is shown in Table 2, and the optimal water-using network can be generated with the help of mathematical model P2 and mathematical model P3, which is shown in Figure 13. It gives a total freshwater as 156.43 t/h, larger than that in Figure 11. However, it should be pointed out that the minimum freshwater in Figure 11, 156.3 t/h, is not enough for the system. In the network generated by Tan and Cruz,13 the outlet concentration of the first unit is 200.27 ppm, which is a little greater than the maximum value, 200 ppm. The same as the unit four in Figure 11, if the total water is 156.3 t/h, its outlet concentration is 200.9 ppm, greater than the maximum value, 200 ppm. In fact, the minimum freshwater of the system under disturbance is 156.43 t/h, the same as the value in Figure 13.

Figure 11. Optimal network under disturbance from Tan and Cruz13 for example 2.

Figure 12. Optimal network under normal conditions for example 2.

The water network structures, which are generated by this Article, are the same under the nominal condition and disturbance by comparing Figures 12 and 13. 6.2. Single Contaminant Water-Using System Involving 10 Units. This case is cited from Bagajewicz and Savelski.4 The limiting water data and optimal design presented in the reference are shown in Tables 3 and 4, respectively. By solving mathematical model P1, the minimum freshwater consumption under nominal conditions can be obtained as 166.267 t/h, which accords with the design in Table 4.4 Next, consider the situation under disturbance. With the mass load increase shown in Table 3, mathematical model P2 can be established and solved. The minimum increase in freshwater 3680



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Table 5. Optimal Solution under Normal Condition for Example 3 process

Figure 13. Optimal network under disturbance for example 2.

Table 3. Water Data of Example 3 process

m (kg/h)

Cin max (ppm)

Cout max (ppm)

2.0 2.88

25 25

80 90

3

4.0

25

200

0.4

4

3.0

50

100

0.3

5

30.0

50

800

3

6

5.0

400

800

7

2.0

200

600

8

1.0

0

100

0.1

9 10

20.0 6.5

50 150

300 300

2 0.65

0

72.73

27.5

0

2 3

F2,3 = 6.984

0 25

81.82 200

35.2 15.873

0 0

4

F1,4 = 13.75

24.65

98.62

26.811

0

5

F1,5 = 13.75

50

15.25

40 10

F8,5 = 11

0.2 0.288

F10,6 = 10

300

800

0.0

7

F3,7 = 5

200

600

0.0

8

process

Fi,j (t/h)

1

(ppm)

(ppm)

0

9

F2,9 = 28.216 F4,9 = 17.152

10

F3,10 = 17.857

50 142.49

90.91

11

300

34.633

300

0.0

5 0 80 31.267

F4,10 = 23.409 total freshwater

FjW

(t/h)

FjD

166.267

Table 6. Optimal Solution under Disturbance for Example 3

Table 4. Solution from Bagajewicz and Savelski4 of Example 3 Cjout

800

6

process

Cjin

Cjin (ppm) Cjout (ppm) FjW (t/h) FjD (t/h)

1

δm (kg/h)

1 2

Fi,j (t/h)

(t/h)

Fi,j (t/h)


1

0

80

27.5

0

2

0

90

35.2

0 0 0

3 4

F2,3 = 6.984 F1,4 = 13.75

25 25

200 100

18.159* 30.25*

5

F1,5 = 13.75

50

800

19.25*

44*

0

80

25.0

0

2

F1,2 = 13.846

25

90

30.462

0

3

F2,3 = 6.349

25

200

16.508

0

6

800

0.0

10

F1,4 = 11.154

50

100

25.427

0

F10,6 = 10

300

4

7

200

600

0.0

5

0

100

11

5

F2,4 = 23.419 F4,5 = 9.514

F3,7 = 5

49.73

300

41.368*

143.96

300

F8,5 = 11

8 50

800

20.324

40

9

F2,9 = 28.216

10

F4,9 = 18.321* F3,10 = 20.143*

F8,5 = 10.0 F9,5 = 0.162 6

F9,6 = 10

300

800

0.0

10

7

F3,7 = 1.190

200

600

0.0

5

0 50

100 300

10.0 38.546

0 67.933

150

300

0.0

43.334

0.0

0 75.838* 47.889*

F4,10 = 25.679* total freshwater

182.727

F4,7 = 1.905 F9,7 = 1.905 8 9

F2,9 = 14.540 F4,9 = 26.914

10

F3,10 = 21.667 F4,10 = 21.667

total freshwater

166.267

consumption under disturbance is obtained as 16.46 t/h. By solving the mathematical model P3, the optimal water-using networks can be generated, which are shown in Table 5 under nominal conditions and Table 6 under disturbance, respectively. The streams that need to be adjusted during disturbance are indicated with a star “*”. The adjustment under disturbance for the design in Table 6 involves 5 units and 10 water streams. The optimal water-using network under disturbance in Table 6 consumes 182.727 t/h freshwater, which is the minimum. When the water-using network in Table 3 is under disturbance in Table 2, the disturbed units can be adjusted by consuming more freshwater to re-establish the stable operating condition of the system, and the excessive outlet flow rate of corresponding

units should be discharged as wastewater. The network after such alteration is shown in Table 7. The alteration involves 8 units and 16 water streams. The network under disturbance in Table 7 consumes 190.551 t/h freshwater, which is 7.824 t/h greater than the freshwater consumption of the design in Table 6. Obviously, the design in Table 6 is better than that in Table 7 on water efficiency as well as control convenience. By using the mathematical models P2 and P3, the water-using network under disturbance, which has the same network structure as the existing network in Table 4, can be optimized. First, the limiting water data and mass load disturbance in Table 3 and the existing network information in Table 4 are given to the corresponding variables in mathematical model P2. By solving this model, the minimum increase in freshwater consumption can be obtained as 16.46 t/h, which attains the same minimum as the design in Table 6. By solving the model P3, the final optimal adjustment scheme with the minimal number of control streams can be generated, which is shown in Table 8. The alteration involves 10 units and 22 water streams. Although the scheme in Table 8 has more streams to be adjusted than that in Table 7, it consumes less freshwater. Tables 9 and 10 compare these designs. 3681



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Table 7. Modified Solution under Disturbance for Network in Table 4 process

Fi,j (t/h)

1

Table 8. Optimal Adjustment Solution under Disturbance for Network in Table 4

Cjin (ppm) Cjout (ppm) FjW (t/h) FjD (t/h) 27.5*


Fi,j (t/h)

process

0

80

2.5*

1

0

80

27.5*

0

2 3

F1,2 = 13.846 F2,3 = 6.349

25 25

90 200

33.662* 18.508*

3.2* 2*

2 3

F1,2 = 13.846 F2,3 = 6.923 *

25 25

90 200

33.662* 18.192*

0 0

4

F1,4 = 11.154

50

100

28.427*

3*

4

F1,4 = 13.654*

50

100

27.927*

0

50

800

22.162*

44*

F2,4 = 23.419 5

F4,5 = 9.514

F2,4 = 23.419 50

800

24.074*

43.75*

F4,5 = 10.757*

5

F8,5 = 10.0

F8,5 = 11.0*

F9,5 = 0.162

F9,5 = 0.081*

6

F9,6 = 10

300

800

0

10

6

F9,6 = 10

300

800

0

10

7

F3,7 = 1.190 F4,7 = 1.905

200

600

0

5

7

F3,7 = 1.282* F4,7 = 1.859*

200

600

0

5

0

100

11*

1*

8

50

300

45.213*

74.6*

9

150

300

2.167*

45.5*

10

F9,7 = 1.905 8 9

F2,9 = 14.540

F9,7 = 1.859* F2,9 = 17.166*

F4,9 = 26.914 10

F3,10 = 21.667

0

100

11.0*

50

300

42.284*

0

150

300

76.060*

F4,9 = 28.551*

F4,10 = 21.667

F3,10 = 23.833*

0

47.667*

F4,10 = 23.833* total freshwater

190.551

It is obvious that the design generated by the models proposed in this Article has better features on water usage and control convenience. The design in Table 5 attains the minimum freshwater consumption and has a less complex structure than the design in Table 2. The design in Table 6 consumes minimum freshwater under disturbance and features a simpler structure and more convenient control than the designs in Tables 7 and 8. 6.3. Multiple Contaminant Water-Using System Involving 5 Units and 3 Contaminants. This case study is cited from Kuo and Smith.17 The limiting water data and optimal design presented in the reference are shown in Table 11 and Figure 14, respectively. By solving the mathematical model P1, the minimum freshwater consumption under nominal conditions can be obtained as 111.79 t/h, which is a little larger than that of the design in Figure 14. It should, however, be the minimum freshwater consumption for the water network, because in the network generated by Kuo and Smith,17 the outlet concentration of the third contaminant in the third unit is 9504 ppm, which is a little greater than the maximum value, 9500 ppm. The freshwater consumption will increase a little if the outlet concentration returns to 9500 ppm. For the worst disturbed scenario of the system, which corresponds to as each possible disturbance reaches its maximum value as shown in the last column in Table 10, solving the mathematical model P2 shows that the minimum increase in freshwater consumption is 2.28 t/h. By solving the mathematical model P3, the optimal water-using networks with the minimum number of adjusted streams can be generated. These are shown in Figure 15 under nominal conditions and in Figure 16 at the worst disturbed scenario, respectively. Six streams, denoted by dashed lines, will be adjusted at the worst disturbed scenario. For the network in Figure 14, the minimum freshwater consumption at the worst disturbed scenario is 114.09 t/h, the same as that in Figure 16. The corresponding water adjustment scheme is shown in Figure 17. Comparing the networks in Figures 16 and 17, it can be seen that, although they have the same freshwater consumption,

total freshwater

182.727

Table 9. Comparison of Different Solutions under Nominal Condition for Example 3 solution design in Table 4 by

FminW (t/h)

number of streams

166.667

27

166.667

22

Bagajewicz and Savelski4 design in Table 5 by the authors

Table 10. Comparison of Different Solutions under Disturbance for Example 3 number

number of

solution

FminW (t/h)

of streams

adjusted streams

design in Table 5

182.727

22

10

design in Table 6

190.551

33

16

design in Table 7

182.727

27

22

and the same number of connections between processes, the number of control steams (denoted by dashed lines) is different. The number of control steams in the network in Figure 17 is eight, two more than that in Figure 16, which correlates with more cost for control. Because under nominal conditions, the flow rate of the stream between unit 2 and unit 3 is very low, the connection can be deleted to simplify the water network. So Figure 15 will turn into Figure 18, in which the freshwater consumption is 0.06 t/h more than the minimum value, but the network is simpler. When the water network is at the worst disturbed scenario of Table 10, the optimal adjustment solution will be generated by solving models P2 and P3, as shown in Figure 19, which needs to consume 2.27 t/h more freshwater. There are five streams that need to be adjusted. All designs are compared at the worst disturbed scenario in Table 12. From Table 12 it can be seen that at the worst disturbed scenario, the design in this Article not only has the minimum 3682



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Table 11. Water Data of Example 4 m (kg/h)

Cin max(ppm)

Cout max(ppm)

disturbance δm (kg/h)

maximum disturbance δmmax (kg/h)

process

contaminant

1

HC

750

0

15

30

50

H2S

20 000

0

400

150

200 50

2

3

4

5

SS

1750

0

35

20

HC

3400

20

120

0

0

H2S

414 800

300

12 500

0

0

SS

4590

45

180

0

0

HC

5600

120

220

10

20

H2S SS

1400 520 800

20 200

45 9500

70 1700

100 2000

HC

160

0

20

3

10

H2S

480

0

60

0

0

SS

160

0

20

5

10

HC

800

50

150

30

40

H2S

60 800

400

8000

1000

3040

480

60

120

5

24

SS

Figure 14. Solution from Kuo and Smith (1998) of example 4.

Figure 17. Network of Kuo and Smith17 at the worst disturbed scenario for example 4.

Figure 15. Optimal network under nominal condition for example 4. Figure 18. Optimal network under nominal conditions after reducing stream for example 4.

Figure 16. Optimal network at the worst disturbed scenario for example 4.

freshwater consumption, but it also has the fewest adjusted streams. The network will be simpler if the stream with very low flow rate is eliminated. When there is any disturbance in the system, such as that shown in the sixth column in Table 10, by solving the mathematical model P2 and P3, the optimal adjustment scheme can be obtained. For the water-using networks in Figures 16 and 18, the corresponding networks under disturbance are shown in Figures 20 and 21, respectively. From the above, it can be seen that the networks generated by the method proposed in this Article have the features of

Figure 19. Optimal network at the worst disturbed scenario after reducing stream for example 4.

minimum freshwater consumption both under nominal conditions and under disturbance, and a simpler adjustment configuration. 6.4. Some Information for Solving the Examples. All the examples were solved via the software LINGO, and the results were local optimum. For LP, the algorithm was decided by LINGO. For NLP, successive LP was used. However, it should be pointed out that the computational time and iterations are 3683



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Table 12. Comparison of Different Solutions at the Worst Disturbed Scenario for Example 4 number of number of connections solution

FminW under normal

FminW at the worst

number of

streams

between processes

condition (t/h)

network of Kuo and Smith17 (Figures 14 and 17)

13

5

111.79

114.09

8

optimal network (Figures 15 and 16)

11

5

111.81

114.09

6

optimal network after reducing stream (Figures 18 and 19)

10

4

111.87

114.14

5

disturbed scenario (t/h) control streams

method is effective for water allocation networks concerning disturbance both on freshwater conservation and on convenient control. The idea can be extended to other networks, such as heat exchanger networks or hydrogen networks, to concern process disturbance by making the system have the optimum performance both under the nominal conditions and under the worst disturbed scenario. Figure 20. Optimal network under disturbance for example 4.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ86 10 89731556. E-mail: [email protected].

’ ACKNOWLEDGMENT Financial support provided by the National Natural Science Foundation of China under Grant No. 20936004 is gratefully Figure 21. Optimal network under disturbance after reducing stream for example 4.

Table 13. Computational Time and Iteration Times for Examples example 1 example 2 example 3 example 4 nominal condition

network computational time/s iterations/time under network disturbance computational time/s iterations/time

a

Figure 6 0a

Figure 12 0a

Table 5 135

Figure 15 4

549 Figure 7 47

184 Figure 13 55

8693 Table 6 988

1163 Figure 16 72

53 237

54 537

402 019

50 219

Means the computational time is very short.

different in every time for the same example. Maybe the reason is the solutions are local optimum. The computational time and iterations are shown in Table 13.

7. CONCLUSIONS In this Article, a new method is proposed to design waterusing networks concerning disturbance. Based on the nominal conditions and on the worst disturbed scenario, a three-step mathematical model can be used to obtain the optimal network structure with minimum freshwater consumption both under nominal conditions and at the worst disturbed scenario, and the adjustment scheme with the minimum number of adjusted streams under disturbance can also be obtained under any disturbance. For an existing water allocation network, the adjustment scheme can be obtained with minimum freshwater increase and minimum adjusted streams in the corresponding configuration. The case studies demonstrated that the proposed

acknowledged.

’ NOMENCLATURE Cin j,s = inlet concentration of contaminant s in unit j, ppm = maximum inlet concentration of contaminant s in unit j, Cin,max j,s ppm 0 Cj,sin = inlet concentration of contaminant s in unit j under disturbance, ppm out Cj,s = outlet concentration of contaminant s in unit j, ppm = maximum outlet concentration of contaminant s in Cout,max j,s unit j, ppm 0 Cj,sout = outlet concentration of contaminant s in unit j under disturbance, ppm dFW j = adjusting flow rate of freshwater for unit j, t/h dFi,j = adjusting flow rate of water stream flowing from unit i to unit j, t/h dFD j = adjusting flow rate of wastewater for unit j, t/h dFj,k = adjusting flow rate of water stream flowing from unit j to unit k, t/h dMj,s = mass load increase of contaminant s in unit j, g/h FW j = freshwater consumption of unit j, t/h FW min = minimum freshwater consumption, t/h Fi,j = flow rate of water stream flowing from unit i to unit j, t/h FLj = flow rate of water loss in unit j, t/h FD j = wastewater flow rate from unit j, t/h Fj,k = flow rate of water stream flowing from unit j to unit k, t/h Mj,s = mass load of contaminant s to be removed in unit j, g/h Index

i,j,k = index of water-using unit s = index of contaminant 3684



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Sets

C = set of contaminants P = set of water-using units Constant

λ = constant between 0 and 1 Binary Variables

dyW j = binary variable denoting the adjustment of stream between freshwater and unit j dyi,j = binary variable denoting the adjustment of stream between unit i and unit j dyD j = binary variable denoting the adjustment of stream between unit j and wastewater

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Water Allocation Network Design Concerning Process Disturbance

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