CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 52, 2016 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-42-6; ISSN 2283-9216 
 

Cascade Models for Targeting and Synthesis of  

Total Water Networks 

Athanassios Nikolakopoulos*, Dimitrios Faskiotis, Antonis Kokossis 

National Technical University of Athens, Heroon Polytechneiou 9, P.C. 15780, Athens, Greece 

nikolako@mail.ntua.gr 

Success in the race for efficiency means for today’s process industries their survival. The most critical 

resources are energy and water, with the latter being a crisis factor when in shortage. Effective water savings 

are achieved by reuse, regeneration and recycling. The selection of regeneration technology cannot be 

addressed separately from targeting reuse, recycle and fresh water flowrates. This work simulates total water 

networks as cascade systems, represented by transhipment models and uses a the two step iterative 

procedure to coordinate the targeting of minimum fresh water requirements, the calculation of recycle and 

treatment flows and the selection of water treatment processes. The procedure offers the advantage of 

producing optimal solutions ahead of design by using linear mixed integer models. Two examples from 

literature are used to illustrate the new approach for single contaminant systems, one with fixed mass loads of 

contaminant and one with fixed flowrates in water using operations.  

1. Introduction 

The stringent legislative framework on discharge limits and the increased public concern on water scarcity 

force the chemical industry to adopt water management practices with ever increasing costs. The minimization 

of fresh water consumption and waste treatment costs must be synchronized since the amount of effluents is 

directly related to fresh water consumption. Additional benefits are brought by the possibility of recycling 

treated water, while treatment costs depend on the effluent flowrates and the selection of treatment 

processes. Optimization strategies for the design and synthesis of optimum water networks that enable 

maximum reuse between operations but also select treatment technologies to regenerate and recycle water 

have been known to be challenging due to the presence of nonlinearities in the formulation and no guaranty 

for the optimality of the solutions (Halim et al., 2015). Prior to detailed design it is useful to quantify the target 

for minimum water consumption together with assessing the maximum potential of a wastewater treatment 

configuration and calculate minimum wastewater treatment flowrates (Kuo and Smith, 1997). Existing 

targeting methods for total water regeneration networks developed to target the minimum freshwater, 

regenerated water, and wastewater flowrates (Parand et al., 2014) are not able to screen treatment 

technologies. Also graphical targeting methods (Wang and Smith 1994) are unable to screen treatment 

technologies and target flowrates for multiple treatment units and fresh water, while shortcut models for the 

selection of treatment processes are absent from literature (Khor et al., 2014).  

The method proposed in this paper is an iterative procedure, which links two transhipment models with 

cascade characteristics and introduces successively increasing recycle flows until convergence to the optimal 

value based on the total cost criterion. The first model - model (Ia) for fixed mass load or model (Ib) for fixed 

fllowrates - which targets the minimum required flowrate of fresh water, is extended in identifying the flowrates 

and concentrations of the wastewater streams at the effluent water mains. The second model (II) extends the 

transshipment model of (Nikolakopoulos et al, 2014), which targets treatment flowrates, in screening 

alternative treatment technologies. The link between the two models is the wastewater streams resulting from 

model (I) for which the flowrates and concentrations are allocated at the pinch points. 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1652158 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Nikolakopoulos A., Faskiotis D., Kokossis A. C., 2016, Cascade models for targeting and synthesis of total water 
networks, Chemical Engineering Transactions, 52, 943-948  DOI:10.3303/CET1652158   

943



2. Overview of the two step iterative procedure 

For the overall system, including the water using operations and the candidate treatment technologies, the 

objectives are to select the optimal set of treatment technologies, calculate recycle flows and target the 

minimum treatment flows and clear water consumption. These objectives are competitive since by recycling 

treated wastewater the consumption of clear water is reduced against an increase of the treatment flows that 

may in turn also affect the selection of treatment technologies and costs.  

Based on the availability of treated and recycled water a first targeting model (Ia) or (Ib), calculates fresh water 

targets and the flows of the effluent streams. A second model (II) selects treatment processes and calculates 

minimum treatment flows. An iterative procedure calculates the recycle flow for which the cost of the overall 

system is minimized.  

The methodology consists of the following steps:  

Initialization: the value of the recycle flow is set to R = 0, and the Current Total Cost (CTC) to infinity.  

Step 1: Calculate minimum fresh water and wastewater flows using model (Ia) for fixed contaminant loads or    

model (Ib) for fixed flowrates. C1 is the cost of fresh water requirements. 

Step 2: Select treatment technologies and calculate minimum treatment flows by using model (II). C2 is the 

sum of the fixed and operating cost for the selected treatment technologies. Set Total Cost TC = C1 + C2. 

Step 3: If TC < CTC, then CTC = TC. Reserve the recycle flow Rres = R, the selected technologies and the 

fresh water flow. Increase the recycle flow R = R + a, and go to Step 2. 

If TC   CTC, then STOP. The optimal recycle flow is Rres. 

3. Cascade models 

3.1 Cascade model (Ia) - Water targets in systems with fixed contaminant loads 

Let us denote by  | 1,2,...,I i i No   the index set of the water using operations. Associated with each 

operation are maximum inlet and outlet concentrations ,maxin
i

C and ,maxout
i

C respectively. The mass load of the 

contaminant transferred to each water-using operation is c
i

M .  | 1,2S s s   is the set of water sources. The 

fresh and recycled water sources are represented by s = 1 and s = 2 respectively. The limiting water flow rate, 

for the water-using operation i is  ,max ,max/l c out ini i i iF M C C  . The index set of concentration intervals is

 | 1,2,...,CI k k K  , and kC is the inlet concentration of interval k. Figure 3 shows how the transhipment 

concept is adapted to the case of the water using system. The water using operations dispose contaminant to 

water sources through specific concentration intervals, defined by the concentration limits of the contaminant 

receivers and providers. Contaminant is allowed to be transferred between concentration intervals. This 

cascade feature enables the reuse of water among operations and debottlenecks the search of the minimum 

fresh water flowrate. The mass transferred from water-using operation i through the kth concentration interval 

is  , 1
P l

i k i k k
W F C C


  , and the residual mass from water using operation i leaving interval k, is ri,k. The mass 

of contaminant transferred from water-using operation i to water source s through interval k is 
, ,s i k

W . Eqs(1-7) 

are the model Ia with the objective to minimize C1: 

min 1
s s

s S

Cost c L


   (1) 

, , 1 , , ,
,,

P

i k i k s i k i k

s S

r r W W i k




     (2) 

, , ,
, ,

W

s i k s k

i

W W s k   (3) 

 , 1 , ,k
W

s k s k
W L C C s k


    (4) 

0 ,
U

s s
L L s    (5) 

,
,

i k k

i

r y M k   (6) 

 ,0 , ,0, 0, 0,1 ,i i k i k kr r r i and y k       (7) 

Since the cost of treatment is calculated by the Transhipment model 3, the recycle flow comes at no cost in 

this model (c2 = 0). Therefore, although the objective of this model is to minimize the cost of water sources, it 

actually minimizes at the same time the flow of fresh water L1. 

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Eq(2) is the mass balances around interval k for every operation i. If 
,

0
i

i k
r  , then k is one of the pinch 

points. M is a sufficiently large number. Then from (Eq(6)) ky  = 1, and k is a pinch point. Let

 | 1,...,p p pKP k k K   be the index set of all pinch points, including the pseudo pinch K. The pseudo pinch is 

the point at the total mass of contaminant and the maximum concentration (i.e. 
,maxout

I
C ). The total mass of 

contaminant is tot c
i

i

MC M  . The overarching element of the two Transhipment models is the allocation of 

the pinch points at the upper limit of the concentration intervals where the sum of the residuals of the 

contaminant obtains positive values, while at any other interval the residuals are vanishing. The flow-rate in 

the identified water mains is not influenced by the particulars of the water reuse network. Thus, let the mass of 

contaminant removed between pinch points 1
p

k  and
p

k  be p
k

W ; 

, ,

1

,

p

p

p

k k

s i kk
s S i k k

p
W kW



  

     (8) 

Then the flowrate of the wastewater main of pinch point 
p

k  is 

 
1

/ ,p pp kp k
p

kk
W C C kF


    (9) 

where 
kp

C  is the concentration of the effluent stream
p

k . The effluent streams participate at the wastewater 

treatment optimization configuration to be assessed at the next step of the procedure. 

3.2 Cascade model (Ib) - Water targets in systems with fixed flowrates 
In model Ib instead of mass of contaminant, it is water that falls through cumulative mass load intervals as 

illustrated in Example 2. The index set of cumulative load intervals is  | 1,2,...,Cli k k K  , and kCli is the 

exiting cumulative load of interval k.  

min Fw  (10) 

s.t.  

,1 1 1,i j
i j

Fw Fout r Fin     (11) 

 1 ,1 , , 2,..., 1k i k k j
i j

r Fout r Fin k K

        (12) 

1 , ,K i K K K j
i j

r Fout r Fin

     (13) 

Balances of water in the first and last interval are given by Eqs(11) and (13). Eq(12) gives the balance of water 

in any other interval. Fout and Fin are the outlet and inlet flows of water using processes. In each equation the 

indexes i and j belong into pre-calculated sets of processes connected to each interval as described in 

Example 2.  

The algorithm proposed by (Bandyopadhyay et al., 2006) for calculating the total waste flowrate was modified 

for calculating the pinch points and wastewater flowrates originating from the pinch points. First all distinct 

concentrations are arranged in descending order: C1 > C2 > C3 > ... > Cn. Next, for each Ci, the net flow Fi is 

calculated by taking the algebraic sum of flow rates corresponding to a concentration Ci. The cumulative flow 

rates given by 


 
1

i

i j

j

Q F . The contaminant mass loads are calculated by    1 1i i i iP Q C C , where  0oQ . 

The cumulative contaminant mass loads are Δ


 
1

i

i j

j

m P . The corresponding wastewater flow rate is 

   Δ Δ  , /ww i T i i fwf m m C C  where Cfw is the concentration of the freshwater and ΔmT is the total 

contaminant load of the process. The largest entry in the last column is the minimum wastewater flow-rate 

target and it corresponds to the pinch point concentration Cpinch1, where   ,1 argmax ww i
i

pinch f . Then the 

wastewater flowrate from the first pinch will be  Δ Δ 1 1 1/pinch T pinch pinchF m m C  leaving at concentration Cpinch1. 
The wastewater flowrate from the second pinch (assuming without of loss of generality that it will be the 

maximum concentration Cmax) will be  Δ 1 1 max 1/pinch pinch pinchF m C C . 

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3.3 Cascade model (II) – Screening of of treatment processes 

The design of a waste water network can be described with reference to a set of KS wastewater mains:

 | 1,...,p pKP k k K  , and a set of JS  treatment processes  | 1,2,..., JJ j j S  . Associated with each 

wastewater main are the original inlet concentrations
,

p

in

kpo k
C C . The outlet concentrations of all treated waste 

streams are equal to the original discharge concentration limits
,

p

out

o k
C . The mass load p

k
W of the contaminant 

transferred from main k, is already calculated. The minimum water flow rate, p
k

MWF  for the waste water 

stream k in treatment process j is given by: 

 , , ,pp p p
in out p

k o k o kk
MWF C kW C   . Let  | 1,2,...,IN n n N  be the index set of the concentration intervals. nC

is defined as the outlet concentration of interval n. The mass transferred from main 
p

k  through the nth interval 

is given by:  1, p p
P

n nn k k
M MWF C C


  . Similarly, the mass transferred to stream j through the nth interval is 

 , 1j n
W

nj n
Ft C CM


  . The solution will select a cost effective set of treatment processes and determine the 

minimum treatment flowrates
j

Ft . If 
j

fc is the fixed cost and coefj is the coefficient of the variable cost of the jth 

treatment unit, the problem of selecting treatment processes and minimizing the treatment flowrates is 

represented by equations (10-16), where 
,

p
n k

u  is the residual mass of main kp leaving interval n: 

 min j jj j
j J

Cost fc x coe Ftf


   (14) 

1, , , , ,
,,p p p p

P p

n k n k n j k i k
j

u u M M n k


     (15) 

, , ,
, ,

W

k j n j n

k

M M j n   (16) 

 , 1 ,
W

j n n nj
M Ft C C n


    (17) 

0 ,
U

j j
Ft Ft j    (18) 

,
jj

Ftx M j   (19) 

 
1, 1, ,

0, 0, , 0,1 ,p p p
p

jN k k n k
u u u n k and x j


       (20) 

The constraint set (10) is the mass balance of the contaminant around each interval. 
, ,k j n

M is the mass of 

contaminant transferred from waste stream k to through interval n to treatment process j. For treatment 
processes with fixed outlet concentration, before the construction of the concentration intervals the 
concentrations of all waste streams must be shifted by the minimum of the outlet concentration of the 

treatment units  min min jout outjC C , thus the concentration leaving the treatment processes will have virtually 
zero value and can be assigned to the first concentrations interval: 

, minp p
in in

k o k

out
C C C   and 

, min
,p p

out out p

k o k

out
C C C k   . The same shift applies for the inlet and outlet concentrations of all treatment 

processes
minj

in in out

s j
C C C  and

min
,

j

out out out

s j
C C C j  . The intervals are defined by the shifted limiting 

concentrations. 

4. Examples 

4.1 Example 1 – System with fixed loads of contaminant at water using operations 
The system of the first example consists of five water using operations and three candidate treatment 

technologies with fixed outlet concentration (Smith, 2005). Table 1 contains the data for Example 1. 

The first iteration of model (Ia) results in three pinch points leading to three wastewater streams; STI with a 

flow of 20 t/h at 200 ppm, STII with a flow of 20 t/h at 400 ppm, and STIII with a flow of 40 t/h at 600 ppm. The 

minimum fresh water flow is 80 t/h. 

 Solving with model (II) for process selection, the results show that 80 t/h will have to be processed by process 

I και 50.91 t/h by process IIb. Treatment processes I and IIb are selected, at a cost of 2.08 M€. If the flowrate 

of the ST III was 53 t/h then TP IIa would have been selected instead of TP IIb, with a treatment flowrate of 

49.45 t/h. At the optimal solution the fresh water flowrate is 40 t/h, the recycle flowrate is 47.06 t/h, and the 

fresh water cost is 800 k€. 

 

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Table 1: Example with five water using processes and three waste treatment processes 

Water using operations 

Operation No 
Cont. Mass 
load (kg/h) 

Cin (ppm) Cout (ppm) Limiting flowrate (t/h) 

1 8 0 200 40 
2 5 100 200 50 
3 9 100 400 30 
4 6 300 400 60 
5 8 400 600 40 

Wastewater treatment processes 

Process No Cin (ppm) Cout (ppm) 
Fixed cost 

K€ 
Variable cost  

coefficient K€/t/h 
 

I 200 5 200 15 
IIa 1,000 50 50 10 
IIb 1,000 50 150 8 

Environmental discharge limit    :     Ce = 30 ppm 

 

4.2 Example 2 – System with fixed water flowrates at water using operations 

Table 2 contains the flowrates and concentrations of inlet and outlet streams of four water using processes 

(Polley and Polley, 2000). The environmental discharge limit is 50 ppm. Two are the candidate treatment 

processes, one with an oulet concentration of 30 ppm, fixed cost 700 k€ and variable cost 6 k€/t/h, and 

another one with 25 ppm outlet concentration fixed cost 860 k€ and variable cost 5 k€/t/h. 

Table 2: Data for fixed flowrate operations in Example 2 

Inlet    Concentration (ppm)    Flowrate (t/h)    Outlet    Concentration (ppm)   Flowrate (t/h) 

In1 20 50 Out1 50 50 

In2 50 100 Out2 100 100 

In3 100 80 Out3 150 70 

In4 200 70 Out4 250 60 

 

Demands and sources are arranged in increasing order of contaminant concentration. Figure 1(a), contains 

the cascade of water flows and Figure 1(b) presents the contaminant load intervals. From left to right water is 

transferred from outlets to inlets, from the top fresh water enters the cascade and the residual of the last 

interval is the wastewater flow. The fresh water target calculated by model (Ib) is 70 t/h, and the total 

wastewater flowrate is 50 t/h. By applying the iterative methodology involving models (Ib) and (II), the fresh 

water target is calculated at 20 t/h. The final pinch wastewater flowrates are 10.24 t/h at 100 ppm and 60 t/h at 

250 ppm. The wastewater is treated only at the first treatment process with a flowrate of 70.24 t/h which is fully 

recycled and the treatment cost is 1.12 M€. There is no discharge since all the fresh water is used inside the 

system. 

5. Conclusions 

The paper tackles water integration in process industry for systems that include a set of water using 

operations and a set of candidate treatment technologies. The objective is to identify targets for water use and 

select treatment technologies exploiting the possibility of recycles in systems with fixed loads or fixed 

flowrates. The total cost consisting of the clear water and treatment costs. The proposed methodology 

produces ahead of design optimal targets for clear water use, recycle flows and treatment flowrates, while at 

the same time it selects the best configuration for regeneration technologies. Future research may expand the 

models for treatment processes characterized by removal ratios instead of fixed outlet concentrations.  

Acknowledgments 

Financial support from the Marie Curie European Research Program RENESENG (FP7- 607415) is gratefully 

acknowledged. 

 

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Int 1

0-1 kg/h

Int 2

1-2.5 kg/h

Int 3

2.5-6  kg/h

Int 4

6-12.5 kg/h

Int 5

12.5-14 kg/h

Int 6

14-23 kg/h

Int 7

23-28 kg/h

Int 8

28-38 kg/h

O1

O2

O3

O4

I1

I2

I3

I4

20 t/h

30 t/h

35 t/h

65 t/h

10 t/h

60 t/h

20 t/h

40 t/h

50 t/h

30 t/h70 t/h

65 t/h15 t/h

45 t/h25 t/h

FwS

WwT

Fw

r8

r1

r2

r3

r4

r5

r6

r7

                           
(a) 

 

 

 

 
(b) 

Figure 1: The water cascade (a), and the concentration intervals (b) for model Ib 

Reference  

Alva-Argáez A., Vallianatos A., Kokossis A., 1999, A multi-contaminant transhipment model for mass 

exchange networks and wastewater minimisation problems, Comp. Chem. Eng., 23(10), 1439-1453. 

Bandyopadhyay S., Ghanekar, M.D., Pillai, H.K., 2006, Process water management, Ind. Eng. Chem. Res., 

45(15), 5287-5297. 

Halim I., Adhitya A., Srinivasan R., 2015, A novel application of genetic algorithm for synthesizing optimal 

water reuse network with multiple objectives, Chem. Eng. Res. Des. 100, 39–56. 

Khor C.S., Chachuat B., Shah N., 2014, Optimization of water network synthesis: Milestones, challenges, and 

future directions, Ind. Eng. Chem. Res., 53(25), 10257-10275. 

Kuo W.-C.J., Smith R., 1997, Effluent treatment system design, Chem. Eng. Sci., 52(23), 4273–90. 

Nikolakopoulos A., Thomaidis, L., Kokossis, A., 2014, Mathematical programming shortcut screening models 

for the design of integrated waste treatment systems, Comp. Aid. Chem. Eng., 34, 381-386. 

Parand R., Yao H.M., Pareek V., Tadé M.O., 2014, Use of pinch concept to optimize the total water 

regeneration network, Ind. Eng. Chem. Res., 53(8), 3222-3235. 

Polley G.T., Polley, H.L., 2000. Design better water networks, Chem. Eng. Prog., 96, 47–52.  

Smith R., 2005, Chemical process: design and integration. John Wiley & Sons, Chichester, England 

Wang Y.P., Smith R., 1994, Wastewater minimisation, Chem. Eng. Sci., 49(7), 981–1006. 

 

 

 

 

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