2011_Water Allocation Network Design Concerning Process Disturbance

Published: October 05, 2010

r 2010 American Chemical Society 3675 dx.doi.org/10.1021/ie100847s | Ind. Eng. Chem. Res. 2011, 50, 3675–3685

ARTICLE

pubs.acs.org/IECR

Water Allocation Network Design Concerning Process DisturbanceXiao 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 awater-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 ispresented for water allocation network design with process disturbance taken into account. For a new design, the synthesizednetwork structure can guarantee that the water system consumes minimum freshwater under both normal and disturbanceconditions and with minimum number of control streams under disturbance. At the same time, the information needed to adjustflow rate under disturbance will be obtained. For an existing water system, an adjustment scheme under disturbance with minimumfreshwater consumption for the existing structure will be obtained. The water network designed by the new methodology has thefeature of minimum freshwater consumption and high flexibility under disturbance. Some case studies are used to demonstrate themethod.

1. INTRODUCTION

Although water is one of many abundant natural resources onthe earth, the demand for it has increased dramatically today dueto rapid economic expansion in many regions worldwide. Theexcessive use and the pollution of water resources are bigproblems for human beings. For the chemical process industry,much research is focused on improving the water network toreduce freshwater consumption and wastewater discharge byusing water system integration, one of the most efficient techno-logies for saving freshwater and reducing wastewater discharge.1-6

During the design of a water network, to minimize freshwaterconsumption, the resultant network will be more integratedamong process units. Many water units, or even the whole net-work, will be affected when there are fluctuations in mass load orwater quality at some process units. Adjustment is needed whenthere are such fluctuations, and if the adjustment is made by usingfreshwater directly, the water saving result will be reduced.However, in actual water systems, fluctuations of mass loadand water quality are unavoidable. Therefore, it is highly desir-able to design a water network with minimum freshwater con-sumption under both nominal and disturbance conditions.

There are two major approaches for synthesizing a networkoperating under uncertainty: one is based on flexibility, and theother is based on stochastic programming.7 Halemane andGrossmann8 introduce the flexibility index for chemical pro-cesses, the stress of which is on ensuring feasibility of design byadjusting the control variables in the system when the uncertainparameters change. Because the flexibility index is difficult tosolve, much work has been done to solve the max and minproblem,9-11 but it remains a difficult task. On the other hand,for a stochastic programming approach,12 the emphasis is onachieving optimality accounting for the fact that the recoursevariables can be adjusted for each parameter realization, andthere exist numerous methods for the solution of several classes

of stochastic programs.7 Tan and Cruz described a procedure forthe synthesis of robust water reuse networks with single-componentfrom imprecise data using symmetric fuzzy linear program-ming. Yet their work can not solve the multi-concentrationproblem.13 Tan and Foo demonstrated the use of Monte Carlosimulation in assessing the vulnerability of water networks tonoisy mass loads.14 Tolerance amount of a water network wasproposed to quantify the resilience of a water network by Zhangand Feng.15

Recently, Karuppiah and Grossmann7 developed multiscenarioprogramming models to solve the problem of water systemintegration under uncertainty, under the assumption that theuncertain parameters take on a finite set of known values. Thisnovel method can solve the problem of water system integrationunder uncertainty to a certain extent. However, the multiscenariomodels associated with the integrated water networks operatingunder uncertainty grow in size with the number of scenarios andare computationally expensive to solve.7

In this Article, a newmethod for optimal water network designis presented. The goal is to make the water system have theoptimum performance both under the nominal conditions andunder the worst disturbed scenario. This is based on thefollowing two thoughts. First, because at most time the systemoperates at the nominal status, the system should be guaranteedto be optimal under the nominal conditions. Second, if a systemhas optimum performance both under the nominal conditionsand under the worst scenario, it will have optimum performance

Special Issue: Water Network Synthesis

Received: April 8, 2010Accepted: September 22, 2010Revised: September 19, 2010

3676 dx.doi.org/10.1021/ie100847s |Ind. Eng. Chem. Res. 2011, 50, 3675–3685

Industrial & Engineering Chemistry Research ARTICLE

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

2. PROBLEM STATEMENT

The general problem of water allocation network designconcerning process disturbance may be stated as follows. In awater system, a set of water units are given. Each water-using unitwith a given concentration limits contaminant loads, and upperbounds of the contaminant load increase during a disturbance.External freshwater will be used in the system. If the concentra-tion of wastewater from a water unit is less than a predeterminedmaximum inlet limit of another unit, then it can be reused by theother unit. There is no limitation on the wastewater that flows tothe environment of the whole water network. The objective ofthis work is to develop an algorithmic technique for the waternetworks with minimum freshwater consumption and waste-water discharge both under the nominal conditions and underthe worst disturbed scenario.

3. SUPERSTRUCTURE OF WATER ALLOCATION NET-WORK UNDER NOMINAL AND DISTURBED CONDI-TIONS

3.1. Superstructure of Water Allocation Network underNominal Condition. Under the nominal operating condition,inlet streams must match the demand of water-using units bothon the quantity level and on the quality level. The inlet and outletconcentrations must not be greater than the maximum values.With the steady mass transfer process between the water streamand the process stream, a constant amount of contaminant isremoved from the process stream and is carried away by the out-let 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 supplyother units with its discharged water. On the other hand, eachwater-using unit can act as a potential water sink for other units.That is, it can receive discharged water from other units. There-fore, the superstructure of the system is shown in Figure 2. Thedetails for inlet and outlet streams of each water-using unit areshown in Figure 3. By reusing the water between water-usingunits, freshwater consumption in the water-using system can bedecreased significantly.

In Figure 2, (1) “F” is the symbol for the freshwater source.Arrows spreading from it denote freshwater streams allocated forwater-using units. (2) “M” is the symbol for themixing point forwaterstreams before they enter water-using units. Arrows pointing to themixing point denote allocated freshwater streams and the waterstreams coming from other water-using units. (3) “S” is the symbolfor the splitting point for discharged water, including the waterstreams sent to other water-using units and the wastewaterstreams. (4) “W” is the symbol for the wastewater main. Arrowspointing to it denote the wastewater streams discharged fromwater-using units.Arrows in group 2 can be connected with arrows in group 1 or in

group 3. Arrows in group 4 can be connectedwith 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 water-using 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 willcome from some process streams and will lead to a variation ofthe contaminant mass load of water in the process. Next, theoutlet contaminant concentration of water from these units willvary and in turn affect the water inlet concentration in the down-stream processes so as to propagate the disturbance.16

Disturbance can cause the transferred mass load to increase ordecrease.When themass load decreases, the water-using unit andthe whole water system can operate normally. When the massload increases, the corresponding water-using unit and its down-stream units will deviate from their normal operation scenarios, andadjusting theflow rateof freshwater andwater streamsbetweenwater-usingunits should beperformed, otherwise the systemcannot operatenormally. The disturbedwater-using systemmay re-establish its stableoperation after such adjustment. Therefore, in this Article, we onlyconsider the situation of mass load increase. The network afteradjustment is called the water-using network under disturbance inthis Article. After the disturbance is removed from the system, thesystem will operate under the nominal conditions again, so the flowrates of all the streams should be restored to the nominal conditions.The adjustable streams include freshwater streams, water streamsbetween water-using units, and wastewater streams. The details forinlet andoutlet streamsof eachwater-using unit under disturbance areshown in Figure 4.

4. DESIGN PROCEDURE FOR WATER ALLOCATIONNETWORK CONCERNING DISTURBANCE

Because the water-using system usually operates under thenominal condition, it is important to guarantee that the systemconsumes minimum freshwater under nominal conditions, whichcan reduce operating cost. So first, the minimum freshwater con-sumption under the nominal condition is calculated by solvingthe mathematical model P1, and it is a necessary reference in thefollowing procedure. Second, a mathematical model, P2, whichsimultaneously describes the water-using system under the

Figure 1. Mass transfer process in water-using units.

Figure 2. Superstructure of water allocation network.

Figure 3. Inlet and outlet stream details of each water-using unit undernormal conditions.



nominal conditions as well as at the worst disturbed scenario isestablished. Moreover, the limit of freshwater consumption isadded to the constraints to specify the maximum acceptablefreshwater consumption in the water-using network undernominal conditions. By solving model P2, the minimum increaseof freshwater consumption for the worst disturbed scenariois obtained. To minimize the number of streams to be adjusted,another mathematical model P3 is introduced to simplifythe network controllability. By solving this model, the finaloptimal design can be generated, including the optimal water-using network and the optimal adjustment scheme underdisturbance.4.1. Minimum Freshwater Consumption under Nominal

Condition. According to the water and contaminant balance ineach water-using unit and each mixing point, the mathematicalmodel P1 can be established for the superstructure of the water allo-cation network under nominal conditions. The objective is to mini-mize the total freshwater consumption. The constraints include thewater and contaminant balance and contaminant concentration limits.P1:

objective : minX

j∈PFWj ð1Þ

S.t.Mass balance of inlet and outlet streams in unit j:

FWj þX

i∈Pi 6¼ j

Fi, j ¼ FDj þX

k∈Pk 6¼ j

Fj, k þ FLj j∈P ð2Þ

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

i∈Pi 6¼ j

ðFi, j 3Couti, s Þ ¼ ðFWj þ

X

i∈Pi 6¼ j

Fi, jÞ 3Cinj, s j∈P, s∈C ð3Þ

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

ðFWj þX

i∈Pi 6¼ j

Fi, jÞ 3Cinj, s þMj, s

¼ ðFDj þX

k∈Pk 6¼ j

Fj, k þ FLj Þ 3Coutj, s j∈P, s∈C ð4Þ

Constraints for inlet and outlet concentrations in unit j:

Cinj, s e Cin, max

j, s j∈P, s∈C ð5Þ

Coutj, s e Cout, max

j, s j∈P, s∈C ð6ÞAll of the above variables are non-negative.

With the known limiting water data of the water-using system,the mathematical model can be solved, and the minimum fresh-water consumption under nominal conditions will be obtained.The minimum freshwater consumption is denoted as Fmin

W inthe following.4.2.Minimum Increase of Freshwater Consumption under

Disturbance. The water system can run steadily if there is nodisturbance. When a disturbance occurs in the water-using sys-tem, for example, the mass load of contaminant to be removedincreases in certain units, the outlet contaminant concentrationwill increase if the flow rate is fixed, so that the correspondingunits and their downstream units will deviate from their nominaloperating conditions. Therefore, the system needs to adjust.To bring those disturbed water-using units to stable operation

under disturbance, the most convenient approach is increasingthe freshwater consumption of the disturbed water-using units,so that the outlet contaminant concentration can attain its accep-table level. However, the freshwater consumption will increase ifone uses this approach.Another way is redesigning the water allocation network with

optimization according to the limiting water data under distur-bance and sequentially adjusting the existing network. Althoughin this method the freshwater consumption of the new design canbe guaranteed to be the minimum, the network structure may beconsiderably different from that of the existing network. Diffi-culty in switching between the two networks will have a greatnegative effect on the feasibility of the new design, because thecondition 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 freshwaterconsumptions are achieving the minimum value under bothconditions. The water-using system under disturbance is shiftedto another stable condition only by adjusting the streamflow rateof the existing network. Through optimization with the limitingwater data under disturbance, the freshwater consumption alsocan be minimized under this condition. To carry the point, amathematical model, P2, which can describe the water-usingsystem both under nominal conditions and under disturbance, isestablished.With mass balance specification and the limits for inlet and

outlet contaminant concentrations, basic constraints of themodel can be formulated. Yet they are not sufficient for the struc-ture specification. To confine the structure of the network underdisturbance to be the same as that of the network under nominalcondition, 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 constantλ is introduced, which should conform to the regulation: 0 < λ < 1.With constraints 7-9, the structure of the network underdisturbance is guaranteed to be the same as that under nominalconditions and can control percent change for the adjusted flowrate.

- λ 3 FWj edFWj eλ 3 F

Wj ð7Þ

- λ 3 Fi, jedFi, jeλ 3 Fi, j ð8Þ

- λ 3 FDj edFDj eλ 3 F

Dj ð9Þ

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

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



of-λ 3 FedFeλ 3 F, it canbededuced thatFþ dFg (1- λ)F>0.This means that the corresponding stream exists in thenetwork under disturbance. On the other hand, if a certainstream does not exist in the water-using network under nominalconditions, then F = 0.With the constraint of-λ 3 Fe dFe λ 3 F,it can be deduced that dF = 0. This means that the correspondingstream does not exist in the network under disturbance either. Soit is demonstrated that in the optimal solution the structure of thenetwork under disturbance is guaranteed to be the same as theone under nominal conditions.With the known limiting water data of the water-using system

and mass load disturbance, the mathematical model can beestablished.P2:

objective : minX

j∈PdFWj ð10Þ

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

disturbance:

dFWj þX

i∈Pi 6¼ j

dFi, j ¼ dFDj þX

k∈Pk 6¼ j

dFj, k þ dFLj j∈P ð11Þ

Mass balance of contaminant s for the inlet mixing point beforeunit j under disturbance:

X

i∈Pi 6¼ j

ððFi, j þ dFi, jÞ 3C0outi, s Þ ¼ ððFWj þ dFWj Þ

þX

i∈Pi 6¼ j

ðFi, j þ dFi, jÞÞ 3C0 inj, s j∈P, s∈C ð12Þ

Mass balance of contaminant s for mass transfer in unit j underdisturbance:

ððFWj þ dFWj Þ þ P

i∈Pi 6¼ j

ðFi, j þ dFi, jÞÞ 3C0 inj, s þMj, s þ dMj, s ¼

ððFDj þ dFDj Þ þP

j∈Pj 6¼k

ðFj, k þ dFj, kÞ þ ðFLj þ dFLj ÞÞ 3

C0outj, s j∈P, s∈C ð13Þ

Constraints for inlet and outlet concentrations in unit j underdisturbance:

C0inj, s e Cin, max


C0outj, s e Cout, max


Accessorial constraints:

- λFWj e dFWj e λFWj j∈P ð7Þ

- λFi, j e dFi, j e λFi, j j∈P ð8Þ

- λFDj e dFDj e λFDj j∈P ð9ÞX

j∈PFWj ¼ FWmin ð16Þ

In mathematical model P2:Constraints 2-6 describe the limitations in the water-using

networks under nominal condition, including mass balance for-mulation as well as the inlet and outlet concentration limits ineach unit.Constraints 11-15 describe basic limitations in the water-

using networks under disturbance, including mass balance for-mulation as well as the inlet and outlet concentration limits ineach unit.Constraints 7-9 limit the network structure under distur-

bance to be the same as that under nominal condition.Constraint 16 limits the freshwater consumption for the water-

using network under nominal condition to the minimum value.The minimum increase of freshwater consumption under

disturbance can be obtained by solving themodel, which is denotedas dFmin

W in the following. The optimal solution provides theinformation about the water-using network under nominal condi-tions and streamflow rate alterations under disturbance.4.3. Optimal Water-Using Network with Minimum Num-

ber of Control Streams. The number of adjusting streamscorrelates with the degree of control and cost. Having fewersuch streams is better. In the optimal solution of mathematicalmodel P2, the number of control streams may be not minimal.Therefore, another model P3 is introduced to minimize thenumber of control streams, so that the switch between networksunder different conditions will be more convenient and econom-ical. To properly describe the control status of water streams,several binary variables are introduced, including dyj

W, dyi,j, anddyj

D. The mathematical model needs to include the followingconstraints:

jdFWj j- U 3 dyWj e0 ð17Þ

jdFi, jj- U 3 dyi, je0 ð18Þ

jdFDj j- U 3 dyDj e0 ð19Þ

It can be guaranteed that in the optimal solution, when thevariable is equal to 1, the corresponding stream should beadjusted; when the variable is equal to 0, the correspondingstream should not be adjusted. Here, U is a constant, which islarge enough, larger than any flow rate in the network. In theoptimal solution, if a certain stream should be adjusted, then |dF| > 0.With the constraint |dF| - U 3 dy e 0, it can be deduced thatdy = 1. If no adjustment is exerted on a certain stream, then |dF| =0. Whether dy = 1 or dy = 0, the constraint |dF|- U 3 dye 0 canbe satisfied. However, with the minimization of

Pdy, 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



and the mass load disturbance to solve the following mathema-tical model P3.P3:

objective : minX

j∈PdyWj þ

X

i∈P

X

j∈Pj 6¼ i

dyi, j þX

j∈PdyDj

ð20Þ

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

jdFWj j- U 3 dyWj e0 j∈P ð17Þ

jdFi, jj- U 3 dyi, je0 i∈P; j∈P, j 6¼ i ð18Þ

jdFDj j- U 3 dyDj e0 j∈P ð19Þ

X

j∈PdFWj e dFWminð1þ ξÞ ð21Þ

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 aconstant. If ξ = 0, then the acceptable increase of freshwaterconsumption is defined as the minimum value. A trade-off be-tween freshwater adjustment and the number of control streamsexists by choosing different ξ.The final optimal design can be constructed according to this

solution. The optimal solution provides the information for thewater-using network under nominal conditions and streamflowrate alterations under disturbance. In this design, both the water-using network under nominal conditions and the one underdisturbance can achieve the minimum freshwater consumption.The switch between networks under different conditions is sig-nificantly simplified by limiting the structure and minimizing thenumber of control streams in the water-using network underdisturbance.

5. SOLVING THE MATHEMATICAL MODELS

The proposed model is highly nonlinear, which arises from thebilinear terms in many of the constraints. Integer variables areintroduced in the third model P3. The mathematical models can besolved by nonlinear mathematical programming. In this Article,software LINGO is used to solve the mathematical models. Theinitial points to nonlinear optimization heavily affect the solution. Allthe results yielded by the software, with the initial points chosen bythe 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 achieveother features. For example, if the total flow rate in a certain unitshould not exceed Fj

max, then the following constraints can beadded.

FWj þX

i∈Pi 6¼ j

Fi, jeFmaxj ð22Þ

ðFWj þ dFWj Þ þX

i∈Pi 6¼ j

ðFi, j þ dFi, jÞ e Fmaxj ð23Þ

Or if connections between certain water-using units areprohibited, then the following constraint can be added:

Fi, j ¼ 0 ð24ÞWith the consideration of practical engineering situations,

adding some proper constraints may give the target water-usingnetwork better engineering or economic features.5.2. The Optimal Adjustment Scheme under Disturbance.

For grassroots design, based on the nominal conditions and theworst disturbed scenario, mathematical models P1, P2, and P3can be used to obtain the optimal network structure with mini-mum freshwater consumption both under nominal conditionsand under the worst disturbed scenario. Next, for the waternetwork operates under a certain disturbance, or for an existingnetwork, because the connections among process units areknown, by solving the mathematical models P1, P2, and P3,the adjustment scheme under disturbance with minimum fresh-water consumption can be obtained.

6. CASE STUDY

6.1. Single ContaminantWater-Using Systems Involving 4Units. In this section, two cases are studied. The first case is citedfrom Wang and Smith.1 The limiting water data and optimaldesign presented in the reference are shown in Table 1 andFigure 5, respectively.By solving the mathematical model P1, the minimum fresh-

water consumption under nominal conditions can be obtained as90 t/h, which accords with the design in Figure 5. Next, considerthe situation under disturbance. With the mass load increaseshown in Table 1, mathematical model P2 can be established andsolved. The minimum increase of freshwater consumption isobtained as 4.5 t/h. By solving the mathematical model P3, theoptimal water-using networks can be generated. These are shownin Figure 6 under nominal conditions and in Figure 7 underdisturbance. The streams that need to be adjusted during dis-turbance are indicated with broken lines. The adjustment underdisturbance for the design in Figure 7 involves one unit and twowater streams.When the water-using network in Figure 5 is under distur-

bance in Table 1, the freshwater consumption of unit 1 and unit 2should be increased to re-establish the stable operating conditionof the system. Moreover, the excessive outlet flow rate of unit 1and unit 2 should be discharged as wastewater, as shown inFigure 8. The alteration of this design involves 2 units and 4 waterstreams, and one stream is added (2 t/h from unit 1 to waste-water discharge). If the network structure remains unchanged,that is, no new stream is added, more freshwater will be consumed,

Table 1. Water Data of Example 1

process mi (g/h) Ci,maxin (ppm) Ci,max

out (ppm) δm (g/h)

1 2000 0 100 200

2 5000 50 100 250

3 30 000 50 800

4 4000 400 800



as shown in Figure 9. Obviously, neither of the two water systemsin 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 canbe obtained as 146.2 t/h, the same as the result in Tan andCruz,13 and the optimal design is shown in Figure 12, which isdifferent from Figure 10. The system running under disturbancewith the mass load increase is shown in Table 2, and the optimalwater-using network can be generated with the help of mathe-matical model P2 and mathematical model P3, which is shown inFigure 13. It gives a total freshwater as 156.43 t/h, larger than thatin Figure 11. However, it should be pointed out that the mini-mum freshwater in Figure 11, 156.3 t/h, is not enough for thesystem. In the network generated by Tan and Cruz,13 the outletconcentration of the first unit is 200.27 ppm, which is a littlegreater than the maximum value, 200 ppm. The same as the unitfour in Figure 11, if the total water is 156.3 t/h, its outlet con-centration is 200.9 ppm, greater than the maximum value,200 ppm. In fact, the minimum freshwater of the system underdisturbance is 156.43 t/h, the same as the value in Figure 13.

The water network structures, which are generated by this Arti-cle, are the same under the nominal condition and disturbance bycomparing Figures 12 and 13.6.2. Single Contaminant Water-Using System Involving

10 Units. This case is cited from Bagajewicz and Savelski.4 Thelimiting water data and optimal design presented in the referenceare shown in Tables 3 and 4, respectively.By solving mathematical model P1, the minimum freshwater

consumption under nominal conditions can be obtained as166.267 t/h, which accords with the design in Table 4.4 Next,consider the situation under disturbance. With the mass loadincrease shown in Table 3, mathematical model P2 can beestablished and solved. The minimum increase in freshwater

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

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

Figure 7. Optimal network under disturbance for example 1.

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

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


process mi (g/h) Ci,maxin (ppm) Ci,max

out (ppm) δm (g/h)

1 7000 0 200 470

2 22 400 100 500 1010

3 62 550 200 650 5140

4 2000 0 200 210

Figure 10. Optimal network under nominal conditions from Tan andCruz13 for example 2.

Figure 11. Optimal network under disturbance from Tan and Cruz13

for example 2.

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



consumption under disturbance is obtained as 16.46 t/h. Bysolving the mathematical model P3, the optimal water-usingnetworks can be generated, which are shown in Table 5 undernominal conditions and Table 6 under disturbance, respectively.The streams that need to be adjusted during disturbance areindicated with a star “*”. The adjustment under disturbance forthe design in Table 6 involves 5 units and 10 water streams. Theoptimal water-using network under disturbance in Table 6consumes 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 consumingmore freshwater to re-establish the stable operating condition ofthe system, and the excessive outlet flow rate of corresponding

units should be discharged as wastewater. The network after suchalteration is shown in Table 7. The alteration involves 8 units and16 water streams. The network under disturbance in Table 7consumes 190.551 t/h freshwater, which is 7.824 t/h greater thanthe freshwater consumption of the design in Table 6. Obviously,the design in Table 6 is better than that in Table 7 on waterefficiency as well as control convenience.By using the mathematical models P2 and P3, the water-using

network under disturbance, which has the same network struc-ture as the existing network in Table 4, can be optimized. First,the limiting water data and mass load disturbance in Table 3 andthe existing network information in Table 4 are given to thecorresponding variables in mathematical model P2. By solvingthis model, the minimum increase in freshwater consumptioncan be obtained as 16.46 t/h, which attains the sameminimum asthe design in Table 6. By solving the model P3, the final optimaladjustment scheme with the minimal number of control streamscan be generated, which is shown in Table 8. The alterationinvolves 10 units and 22 water streams. Although the scheme inTable 8 has more streams to be adjusted than that in Table 7, itconsumes less freshwater.Tables 9 and 10 compare these designs.



process m (kg/h) Cmaxin (ppm) Cmax

out (ppm) δm (kg/h)

1 2.0 25 80 0.2

2 2.88 25 90 0.288

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 20.0 50 300 2

10 6.5 150 300 0.65

Table 4. Solution from Bagajewicz and Savelski4 ofExample 3

process Fi,j (t/h) Cjin (ppm) Cj

out (ppm) FjW (t/h) Fj

D (t/h)

1 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

4 F1,4 = 11.154 50 100 25.427 0

F2,4 = 23.419

5 F4,5 = 9.514 50 800 20.324 40

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

F4,7 = 1.905

F9,7 = 1.905

8 0 100 10.0 0

9 F2,9 = 14.540 50 300 38.546 67.933

F4,9 = 26.914

10 F3,10 = 21.667 150 300 0.0 43.334

F4,10 = 21.667

total freshwater 166.267

Table 5. Optimal Solution under Normal Condition forExample 3



D (t/h)

1 0 72.73 27.5 0

2 0 81.82 35.2 0

3 F2,3 = 6.984 25 200 15.873 0

4 F1,4 = 13.75 24.65 98.62 26.811 0

5 F1,5 = 13.75 50 800 15.25 40

F8,5 = 11

6 F10,6 = 10 300 800 0.0 10

7 F3,7 = 5 200 600 0.0 5

8 0 90.91 11 0

9 F2,9 = 28.216 50 300 34.633 80

F4,9 = 17.152

10 F3,10 = 17.857 142.49 300 0.0 31.267

F4,10 = 23.409


Table 6. Optimal Solution under Disturbance for Example 3



D (t/h)

1 0 80 27.5 0

2 0 90 35.2 0

3 F2,3 = 6.984 25 200 18.159* 0

4 F1,4 = 13.75 25 100 30.25* 0

5 F1,5 = 13.75 50 800 19.25* 44*

F8,5 = 11

6 F10,6 = 10 300 800 0.0 10

7 F3,7 = 5 200 600 0.0 5

8 0 100 11 0

9 F2,9 = 28.216 49.73 300 41.368* 75.838*

F4,9 = 18.321*

10 F3,10 = 20.143* 143.96 300 0.0 47.889*

F4,10 = 25.679*




It is obvious that the design generated by the models proposedin this Article has better features on water usage and controlconvenience. The design in Table 5 attains the minimum fresh-water consumption and has a less complex structure than thedesign in Table 2. The design in Table 6 consumes minimumfreshwater under disturbance and features a simpler structure andmore convenient control than the designs in Tables 7 and 8.6.3. Multiple Contaminant Water-Using System Involving

5Units and 3 Contaminants. This case study is cited fromKuoand Smith.17 The limiting water data and optimal designpresented in the reference are shown in Table 11 and Figure 14,respectively.By solving the mathematical model P1, the minimum fresh-

water consumption under nominal conditions can be obtained as111.79 t/h, which is a little larger than that of the design inFigure 14. It should, however, be the minimum freshwaterconsumption for the water network, because in the networkgenerated by Kuo and Smith,17 the outlet concentration of thethird contaminant in the third unit is 9504 ppm, which is a littlegreater than the maximum value, 9500 ppm. The freshwaterconsumption will increase a little if the outlet concentrationreturns to 9500 ppm.For the worst disturbed scenario of the system, which corre-

sponds to as each possible disturbance reaches its maximumvalue as shown in the last column in Table 10, solving themathematical model P2 shows that the minimum increase infreshwater consumption is 2.28 t/h. By solving the mathematicalmodel P3, the optimal water-using networks with the minimumnumber of adjusted streams can be generated. These are shownin Figure 15 under nominal conditions and in Figure 16 at theworst disturbed scenario, respectively. Six streams, denoted bydashed lines, will be adjusted at the worst disturbed scenario. Forthe network in Figure 14, the minimum freshwater consumptionat the worst disturbed scenario is 114.09 t/h, the same as that inFigure 16. The corresponding water adjustment scheme is shownin Figure 17. Comparing the networks in Figures 16 and 17, it canbe seen that, although they have the same freshwater consumption,

and the same number of connections between processes, thenumber of control steams (denoted by dashed lines) is different.The number of control steams in the network in Figure 17 iseight, two more than that in Figure 16, which correlates withmore 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 bedeleted to simplify the water network. So Figure 15 will turn intoFigure 18, in which the freshwater consumption is 0.06 t/h morethan the minimum value, but the network is simpler. When thewater network is at the worst disturbed scenario of Table 10, theoptimal adjustment solution will be generated by solving models P2and P3, as shown in Figure 19, which needs to consume 2.27 t/hmore freshwater. There are five streams that need to be adjusted. Alldesigns 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

Table 7. Modified Solution under Disturbance for Networkin Table 4



D (t/h)

1 0 80 27.5* 2.5*

2 F1,2 = 13.846 25 90 33.662* 3.2*

3 F2,3 = 6.349 25 200 18.508* 2*

4 F1,4 = 11.154 50 100 28.427* 3*

F2,4 = 23.419

5 F4,5 = 9.514 50 800 24.074* 43.75*

F8,5 = 10.0

F9,5 = 0.162

6 F9,6 = 10 300 800 0 10

7 F3,7 = 1.190 200 600 0 5

F4,7 = 1.905

F9,7 = 1.905

8 0 100 11* 1*

9 F2,9 = 14.540 50 300 45.213* 74.6*

F4,9 = 26.914

10 F3,10 = 21.667 150 300 2.167* 45.5*

F4,10 = 21.667


Table 8. Optimal Adjustment Solution underDisturbance forNetwork in Table 4



D (t/h)

1 0 80 27.5* 0

2 F1,2 = 13.846 25 90 33.662* 0

3 F2,3 = 6.923 * 25 200 18.192* 0

4 F1,4 = 13.654* 50 100 27.927* 0

F2,4 = 23.419

5 F4,5 = 10.757* 50 800 22.162* 44*

F8,5 = 11.0*

F9,5 = 0.081*

6 F9,6 = 10 300 800 0 10

7 F3,7 = 1.282* 200 600 0 5

F4,7 = 1.859*

F9,7 = 1.859*

8 0 100 11.0* 0

9 F2,9 = 17.166* 50 300 42.284* 76.060*

F4,9 = 28.551*

10 F3,10 = 23.833* 150 300 0 47.667*

F4,10 = 23.833*


Table 9. Comparison of Different Solutions under NominalCondition for Example 3

solution FminW (t/h) number of streams

design in Table 4 by

Bagajewicz and Savelski4166.667 27

design in Table 5 by

the authors

166.667 22

Table 10. Comparison of Different Solutions under Distur-bance for Example 3

solution FminW (t/h)

number

of streams

number of

adjusted streams

design in Table 5 182.727 22 10





freshwater consumption, but it also has the fewest adjustedstreams. The network will be simpler if the stream with verylow 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 mathe-matical model P2 and P3, the optimal adjustment scheme canbe obtained. For the water-using networks in Figures 16 and 18,the corresponding networks under disturbance are shown inFigures 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

minimum freshwater consumption both under nominal condi-tions and under disturbance, and a simpler adjustment config-uration.6.4. Some Information for Solving the Examples. All the

examples were solved via the software LINGO, and the resultswere local optimum. For LP, the algorithm was decided byLINGO. For NLP, successive LP was used. However, it shouldbe pointed out that the computational time and iterations are


process contaminant m (kg/h) Cmaxin (ppm) Cmax

out (ppm) disturbance δm (kg/h) maximum disturbance δmmax (kg/h)

1 HC 750 0 15 30 50

H2S 20 000 0 400 150 200

SS 1750 0 35 20 50

2 HC 3400 20 120 0 0

H2S 414 800 300 12 500 0 0

SS 4590 45 180 0 0

3 HC 5600 120 220 10 20

H2S 1400 20 45 70 100

SS 520 800 200 9500 1700 2000

4 HC 160 0 20 3 10

H2S 480 0 60 0 0

SS 160 0 20 5 10

5 HC 800 50 150 30 40

H2S 60 800 400 8000 1000 3040

SS 480 60 120 5 24

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

Figure 15. Optimal network under nominal condition for example 4.

Figure 16. Optimal network at the worst disturbed scenario forexample 4.

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

Figure 18. Optimal network under nominal conditions after reducingstream for example 4.

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



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

7. CONCLUSIONS

In this Article, a new method is proposed to design water-using networks concerning disturbance. Based on the nominalconditions and on the worst disturbed scenario, a three-stepmathematical model can be used to obtain the optimal networkstructure with minimum freshwater consumption both undernominal conditions and at the worst disturbed scenario, and theadjustment scheme with the minimum number of adjustedstreams under disturbance can also be obtained under any dis-turbance. For an existing water allocation network, the adjust-ment scheme can be obtained with minimum freshwaterincrease and minimum adjusted streams in the correspondingconfiguration. The case studies demonstrated that the proposed

method is effective for water allocation networks concerningdisturbance both on freshwater conservation and on convenientcontrol.

The idea can be extended to other networks, such as heat exchan-ger networks or hydrogen networks, to concern process disturbanceby making the system have the optimum performance both underthe nominal conditions and under the worst disturbed scenario.

’AUTHOR INFORMATION

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

’ACKNOWLEDGMENT

Financial support provided by the National Natural ScienceFoundation of China under Grant No. 20936004 is gratefully

acknowledged.

’NOMENCLATURECj,sin = inlet concentration of contaminant s in unit j, ppm

Cj,sin,max = maximum inlet concentration of contaminant s in unit j,

ppmCj,s

0in = inlet concentration of contaminant s in unit j underdisturbance, ppm

Cj,sout = outlet concentration of contaminant s in unit j, ppm

Cj,sout,max = maximum outlet concentration of contaminant s in

unit j, ppmCj,s

0out = outlet concentration of contaminant s in unit j underdisturbance, ppm

dFjW = adjusting flow rate of freshwater for unit j, t/h

dFi,j = adjusting flow rate of water stream flowing from unit i tounit j, t/h

dFjD = adjusting flow rate of wastewater for unit j, t/h

dFj,k = adjusting flow rate of water stream flowing from unit j tounit k, t/h

dMj,s = mass load increase of contaminant s in unit j, g/h

FjW = freshwater consumption of unit j, t/h

FminW = minimum freshwater consumption, t/h

Fi,j = flow rate of water stream flowing from unit i to unit j, t/h

FjL = flow rate of water loss in unit j, t/h

FjD = 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

Indexi,j,k = index of water-using units = index of contaminant


Figure 21. Optimal network under disturbance after reducing streamfor example 4.

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

solution

number of

streams

number of connections

between processes

FminW under normal

condition (t/h)

FminW at the worst

disturbed scenario (t/h)

number of

control streams

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

Table 13. Computational Time and Iteration Times forExamples

example 1 example 2 example 3 example 4

nominalcondition

network Figure 6 Figure 12 Table 5 Figure 15computational

time/s0a 0a 135 4

iterations/time 549 184 8693 1163underdisturbance

network Figure 7 Figure 13 Table 6 Figure 16computational

time/s47 55 988 72

iterations/time 53 237 54 537 402 019 50 219aMeans the computational time is very short.



SetsC = set of contaminantsP = set of water-using units

Constantλ = constant between 0 and 1

Binary Variablesdyj

W= binary variable denoting the adjustment of stream betweenfreshwater and unit j

dyi,j = binary variable denoting the adjustment of stream betweenunit i and unit j

dyjD = binary variable denoting the adjustment of stream between

unit j and wastewater

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