DOI: 10.3303/CET2188170 
 

 
 

 

 

 
 
 

 
 
 

 
 

 
 
 

 
 
 

 

 
 

 
 

 
 
 

 
 
 

 
 
 

 
 
 

 
 
 

 

Paper Received: 9 June 2021; Revised: 16 August 2021; Accepted: 7 October 2021 
Please cite this article as: Latinis A., Papadopoulos A.I., Seferlis P., 2021, Optimal Operational Profiles in an Electrodialysis Unit for Ion 
Recovery, Chemical Engineering Transactions, 88, 1021-1026  DOI:10.3303/CET2188170 

CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 88, 2021 

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Yee Van Fan, Jiří J. Klemeš

Copyright © 2021, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-86-0; ISSN 2283-9216 

Optimal Operational Profiles in an Electrodialysis Unit 
for Ion Recovery 

Athanasios Latinisa, Athanasios I. Papadopoulosa, Panos Seferlisb,* 
a Department of Mechanical Engineering, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece 
b Chemical Process and Energy Resources Institute, Centre for Research and Technology Hellas, 57001, Thermi-

Thessaloniki, Greece 
seferlis@auth.gr 

Electrodialysis is an efficient process for the cleaning of industrial water effluent streams from toxic ionic 
substances with the simultaneous recovery of valuable ions for reuse. Energy consumption and waste recovery 
are the two key goals of the process. A dynamic optimisation program aims to determine the optimal operating 
conditions in terms of applied voltage and effluent flow rate in a batch operating scheme. A model is developed 
that accounts for the ion transfer through the ion selective membranes and the dynamics of the system. A 
combination of three objective functions targeting the minimisation of the overall batch process time, the 
minimisation of the electrical energy consumption required for the ion transfer, and the maximisation of the 
overall degree of separation is investigated. The manipulated variables applied voltage and feed stream flow 
rate are considered as piecewise constant during each time interval, spanning the duration of the entire batch. 
The dynamic optimisation problem is solved through standard non-linear programming techniques which 
calculate the optimal batch duration and condition profiles for the system. A multi-objective analysis is presented 
for various combinations of weight values for the joint objective function through the development of the Pareto 
optimal front. The current approach has been implemented in the removal and recovery of sulfuric anions from 
an aqueous solution and resulted in the achievement of a high degree of separation in a shorter period at a 
much lower energy consumption. 

1. Introduction

Electrodialysis is an electrochemical process in which ions are transferred through ion selective membranes 
under the influence of an externally applied electric field imposed by two electrically charged electrodes. 
Electrodialysis units enable the separation of anions and cations from waste discharge streams and enable the 
simultaneous recovery of valuable ions. Extensive research has been carried out on the fundamental principles 
of the electrolysis system as well as the latest technological developments that have taken place in this process 
(Al-Amshawee et al., 2018). The complexity of an electrodialysis unit is quite high due to the combined 
electrochemical and mass transfer phenomena within a complicated geometry of alternating flow channels and 
membrane modules. The efficient operation of an electrodialysis unit depends on a number of process 
parameters as stated by Voutetaki et al. (2020). 
Electrodialysis units aim to meet stringent quality specifications for the discharged stream while enabling a high 
degree of recovery for valuable ions from the effluent stream. In order to develop optimal operating policies for 
the electrodialysis process a model-based optimisation approach is necessary. To this end, Rohman et al. 
(2010) developed a model of high accuracy for hydrochloric acid recovery. Voutetaki et al. (2020) provided a 
validated process model with optimally estimated parameters using experimental process data for the recovery 
of sulfuric acid. Usually, electrodialysis operates in a batch mode and the calculation of optimal operating policies 
involves the determination of the intensity of the electric field applied to the recirculated effluent stream. Shah 
et al. (2018) investigated the optimal design of electrodialysis units for the desalination of brackish groundwater 
and concluded that capital cost dominates over energetic cost and therefore high current density units were 
preferred. Parulekar (1998) minimised energy consumption using a weighted-average resistance for the 
electrodialysis stack by obtaining optimal profiles for the electric current and the fluid flow. A more systematic 

1021



approach was presented by Rohman and Aziz (2019) where a dynamic optimisation approach was employed 
for the minimisation of the energy requirements. Orthogonal collocation achieved the lowest energy for the 
highest hydrochloric acid recovery within manageable computational requirements. Chehayeb and Lienhard 
(2019) investigated three different modes of operation for optimal brackish water desalination and concluded 
that constant current operation is energetically superior than constant voltage or constant entropy generation. 
According to Shah et al. (2018) the optimal operating policies depend on the composition and the concentration 
of the effluent stream as well as the amount of feed flowrate. In addition, variations in the efficiency of the 
membranes and disturbances occurring in the quality of the effluent stream are key factors that need to be 
considered in the operation of an electrodialysis unit. 
The objectives of the present work are the development of a dynamic optimisation program using the method 
of orthogonal collocation on finite elements (OCFE) to determine the optimal operating conditions of a sulphate 
ion solution cleaning unit. The program seeks to calculate the optimal profile of the applied current voltage and 
effluent fluid flow in a batch electrodialysis configuration in order to meet desired specifications for the 
concentration of ions in the output stream of the process. The problem has multiple competing objectives and 
therefore their balance requires a careful consideration for the specific input stream properties. The present 
work aims to bring a systematic approach to the online optimisation of a batch electrodialysis plant. 

2. Modeling of an electrodialysis system

A schematic of a batch electrodialysis process unit that is based on recirculation of the contaminated effluent is 
shown in Figure 1 (Voutetaki et al., 2020). The solution that is processed is passed through an electrodialysis 
cell with the aid of a pump from a feedstock tank (dilute). A specific electric voltage potential is applied at the 
electrodes in the two sides of the cell stack. Two types of compartments-cells can be distinguished. The 
compartments that carry the concentrate (indicated as C) solution in which the concentration of the ions 
increases for possible ion recovery and the compartments that carry the dilute (indicated as D) in which the 
concentration of the ions decreases and is the effluent product of the electrodialysis process. The cations move 
towards the negatively charged cathode, and the anions towards the positively charged anode. The cations can 
selectively permeate through the negatively charged Cation Exchange Membranes (CEM), whereas the anions 
can permeate through the positively charged Anion Exchange Membranes (AEM) separating C and D 
successive compartments.  

Figure 1: Electrodialysis batch process with recirculation of the concentrate and dilute solutions 

The mathematical model that describes the electrodialysis process is taken by Rohman et al. (2011): 

𝑑𝐶𝑐𝑜𝑛𝑐

𝑑𝑡
=

𝑄(𝐶𝑐𝑜𝑛𝑐
𝑡𝑎𝑛𝑘−𝐶𝑐𝑜𝑛𝑐)+

𝑁𝜙𝑗𝐴

𝑧𝐹
−

𝑁𝐴𝐷𝐴𝐸𝑀(𝐶𝑐𝑜𝑛𝑐
𝐴𝐸𝑀−𝐶𝑑𝑖𝑙

𝐴𝐸𝑀)

𝑙𝐴𝐸𝑀
 −

𝑁𝐴𝐷𝐶𝐸𝑀(𝐶𝑐𝑜𝑛𝑐
𝐶𝐸𝑀 −𝐶𝑑𝑖𝑙

𝐶𝐸𝑀)

𝑙𝐶𝐸𝑀

𝑁𝑉𝑐𝑜𝑚𝑝
(1) 

𝑑𝐶𝑑𝑖𝑙

𝑑𝑡
=

𝑄(𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘−𝐶𝑑𝑖𝑙)−

𝑁𝜙𝑗𝐴

𝑧𝐹
+

𝑁𝐴𝐷𝐴𝐸𝑀(𝐶𝑐𝑜𝑛𝑐
𝐴𝐸𝑀−𝐶𝑑𝑖𝑙

𝐴𝐸𝑀)

𝑙𝐴𝐸𝑀
 +

𝑁𝐴𝐷𝐶𝐸𝑀(𝐶𝑐𝑜𝑛𝑐
𝐶𝐸𝑀 −𝐶𝑑𝑖𝑙

𝐶𝐸𝑀)

𝑙𝐶𝐸𝑀

𝑁𝑉𝑐𝑜𝑚𝑝
(2) 

𝑑𝐶𝑐𝑜𝑛𝑐
𝑡𝑎𝑛𝑘

𝑑𝑡
=

𝑄(𝐶𝑐𝑜𝑛𝑐−𝐶𝑐𝑜𝑛𝑐
𝑡𝑎𝑛𝑘) 

𝑉𝑐𝑜𝑛𝑐
𝑡𝑎𝑛𝑘 (3) 

𝑑𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘

𝑑𝑡
=

𝑄(𝐶𝑑𝑖𝑙−𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘)

𝑉𝑑𝑖𝑙
𝑡𝑎𝑛𝑘 (4) 

The mass balance equations express the ions’ transfer between the compartments (Eq(1) and Eq(2)) and the 
concentrate and dilute tanks (Eq(3) and Eq(4)). 𝐶𝑐𝑜𝑛𝑐 , 𝐶𝑑𝑖𝑙 , 𝐶𝑐𝑜𝑛𝑐𝑡𝑎𝑛𝑘 , 𝐶𝑑𝑖𝑙

𝑡𝑎𝑛𝑘  correspond to the ion’s concentration in
the concentrate and dilute, compartments and tanks, 𝑉𝑐𝑜𝑚𝑝 , , 𝑉𝑐𝑜𝑛𝑐𝑡𝑎𝑛𝑘 , 𝑉𝑑𝑖𝑙

𝑡𝑎𝑛𝑘 correspond to the volumes of the

1022



solutions. 𝑄 is the overall flowrate of the treated solution, j and φ are the current density and efficiency, 𝑁 is the 
number of cell pairs, 𝐴 is the membrane area per cell pair, 𝑧 is the ion’s charge, and 𝐹 is Faraday’s constant. 
𝐷𝐴𝐸𝑀 and 𝐷𝐶𝐸𝑀 denote the diffusion coefficients through AEM and CEM, 𝑙𝐴𝐸𝑀 and 𝑙𝐶𝐸𝑀 denote the membrane 
thickness. 𝐶𝑐𝑜𝑛𝑐𝐴𝐸𝑀 , 𝐶𝑑𝑖𝑙

𝐴𝐸𝑀 𝐶𝑐𝑜𝑛𝑐
𝐶𝐸𝑀 , 𝐶𝑑𝑖𝑙

𝐶𝐸𝑀 denote the concentration of ions on the surface of the corresponding
membranes. The overall energy of the system is the summation of the electrical energy consumed in the stack, 
in order to achieve the transfer of ions, and the energy required for the pumps used for the recirculation of the 
solutions, which can be considered negligible due to the absence of significant pressure drop and the low flow 
rate in the ED stack. Electrical energy consumption (𝑘𝑊ℎ

𝑚3
) is calculated by the following formula: 

𝐸𝑑𝑒𝑠 = 𝑈𝑠𝑡 ∫
𝑗𝑠𝑡 ∗ 𝐴𝑠𝑡

𝑉𝑑𝑖𝑙
𝑡𝑎𝑛𝑘

ⅆ𝑡

𝑡𝑓

𝑡0

 (5) 

where 𝑈𝑠𝑡  is the applied potential difference; 𝑗𝑠𝑡  the current density; 𝐴𝑠𝑡 the membrane area. 

2.1 Dynamic optimisation of operation 

In a batch electrodialysis operation, the factors that can directly affect the process performance are the power 
source voltage, 𝑈𝑠𝑡 , and the overall flow rate, 𝑄. Ortiz et al. (2005) carried out experiments to test the effect of 
these variables and proved their dominant impact on the dynamic behaviour of the electrodialysis process. They 
are selected as control variables. Usually, the control variables are set at an optimal value but remain constant 
throughout the duration of the batch. However, the optimal values for the source voltage and the circulation rate 
may change as the concentration of the dilute changes with time. In addition, disturbances associated with the 
membrane performance and unknown feed concentration may affect the performance of the system. A dynamic 
optimisation approach enables the calculation of the optimal control profiles in order to ensure the maximum 
degree of ion recovery at the minimum process time and energy consumption. In the present work, the following 
optimisation criteria have been selected: 
Minimisation of the process batch time: 

min
𝑥,𝑢

𝑡 (6) 

Minimisation of energy consumption (energy for the pumps is considered constant and negligible): 

min
𝑥,𝑢

𝐸𝑑𝑒𝑠 = min 
𝑥,𝑢

 𝐸𝑠𝑡 ∫
𝑗𝑠𝑡∗𝐴𝑠𝑡

𝑉𝑑𝑖𝑙
ⅆ𝑡

𝑡𝑓

𝑡0

 (7) 

Maximisation of the degree of separation defined as: 

max
𝑥,𝑢

 Degree =  min
𝑥,𝑢

− (1 −
𝐶𝑑𝑖𝑙

𝑡𝑎𝑛𝑘(𝑡𝑓)

𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘(1)

) (8) 

In this case, the final dilute tank concentration 𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘 is bounded by an upper limit to ensure a satisfactory level

of separation.  
In addition to the aforementioned criteria, an extra term is included in the overall goals that accounts for the rate 
of change for the control variables, 𝑢𝑇 𝑅𝑢. The symbol 𝑢 corresponds to either the source voltage or the 
circulation flowrate variation between two consecutive control intervals whereas 𝑅 is a weight matrix. The 
combination of the objective criteria results to the following function: 

𝑓(𝑥, 𝑢) = 𝑄𝑡 ∗ 𝑡 + 𝑄𝐸 ∗  𝐸𝑠𝑡 ∫
𝑗𝑠𝑡∗𝐴𝑠𝑡

𝑉𝑑𝑖𝑙
ⅆ𝑡

𝑡𝑓

𝑡0

+ ∫ 𝑢𝑇 𝑅𝑢 ⅆ𝑡
𝑡𝑓

𝑡0
− 𝑄𝐷𝑒𝑔 (1 −

𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘(𝑡𝑓)

𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘(1)

) (9) 

where 𝑄𝑡, 𝑄𝐸 , 𝑅 𝑎𝑛𝑑 𝑄𝐷𝑒𝑔 are weighted factors used to normalise the different, usually competing goals of the 
objective function. 
The objective function is minimised subject to the modelling equations Eq(1) to Eq(4). The method of orthogonal 
collocation on finite elements is employed to discretise the differential equations as implemented in Kiparissides 
et al. (2002). The time domain, 𝑡 ∈ [𝑡0, 𝑡𝑓 ], is divided into a number, 𝑁𝐸, of equally spaced time intervals, 𝛥𝜁𝑛, 
namely the finite elements, with 𝜁0 = 𝑡0 and 𝜁𝑛+1 = 𝑡𝑓. Time within each interval is normalised so that it varies 
between 𝜏 ∈ [0,1]. The actual time is then derived by 𝑡 = 𝜁𝑛−1 + 𝛥𝜁𝑛𝜏. The control variables, current voltage and 
fluid circulation flowrate, are parameterised as piecewise constant variables within each time interval. In each 
finite element a number of collocation points, 𝑁𝐶, is defined as the roots of Legendre orthogonal polynomials, 

1023



The state variables, 𝐶𝑐𝑜𝑛𝑐 , 𝐶𝑑𝑖𝑙 , 𝐶𝑐𝑜𝑛𝑐𝑡𝑎𝑛𝑘 , 𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘 , are approximated within each time interval using Lagrange

interpolating polynomials, 𝜑𝑖 (𝜏).

𝜑𝑖 (𝜏) = ∏
𝜏−𝜏𝑘

𝜏𝑖−𝜏𝑘

𝑁𝐶
𝑘=0
𝑘≠𝑗

 𝑖 = 1, …  , 𝑁𝐶 (10) 

The balance equations, Eq(1) to Eq(4), are assumed to be satisfied exactly only at the collocation points. 

𝑥𝑖 (𝜏) = ∑ 𝑥𝑖𝑗 (𝜏𝑗 )𝜑𝑗 (𝜏)
𝑁𝐶
𝑗=0  (11) 

Where 𝑥𝑖 (𝜏) corresponds to any of the state variables in the 𝑖-th finite element and 𝑥𝑖𝑗 (𝜏𝑗 ) is the value of the
state variable at the 𝑗-th collocation point. The adaptive placement of the element boundaries can change the 
density of the collocation points in the time and the discretisation method can increase its accuracy when steep 
state profiles are formed during the dynamic transition. Finally, state and control variable bounds are defined 
along the time domain. 

3. Optimisation results and discussion

The selected modelling approach is based on fundamental principles of electrodialysis and consists of a set of 
general, universal and theoretical steps regardless of the structural and functional details of the stack or the 
breaks and concentrations used. In the present study, the general mathematical model developed describes 
the separation of sulfuric acid solution. The model and the parameters used were validated using experimental 
results (Voutetaki et al., 2020). 
The initial control variables are selected as: 𝐸𝑠𝑡𝑎𝑐𝑘  = 20 [𝑉] and 𝑞 = 0.15 [

𝑚3

ℎ
]. The initial conditions of the 

concentrations are defined by Voutetaki et al. (2020): 𝐶𝑐𝑜𝑛𝑐 (0) = 𝐶𝑑𝑖𝑙 (0) = 𝐶𝑐𝑜𝑛𝑐𝑡𝑎𝑛𝑘 (0) = 𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘 (0) = 2000 ∗

(
1

98
) [

𝑚𝑜𝑙

𝑚3
]. It is considered that the volume of the solution in the concentrate and diluent tanks are equal (𝑉𝑡𝑎𝑛𝑘

𝑐𝑜𝑛𝑐 =

𝑉𝑡𝑎𝑛𝑘
𝑑𝑖𝑙 = 0.02 [𝑚3]) with identical sulfuric acid concentration at the beginning of the batch process. The

electrodialysis compartments are considered as flooded with the initial solution. The goal is to achieve a desired 
final dilution tank concentration which is introduced into the optimisation problem in the form of a constraint: 
𝐶𝑁𝐸,𝑐𝑛

𝑑𝑖𝑙,𝑡𝑎𝑛𝑘
≤  600 [

𝑚𝑔

𝐿
]. The time domain is divided into 10 finite elements with 4 interior collocation points in each 

element. Bounds have been set for the control variables: 0 ≤ 𝐸𝑠𝑡𝑎𝑐𝑘 ≤ 80  [𝑉] and 0 ≤ 𝑞 ≤ 0.4  [
𝑚3

ℎ
]. 

Several optimisation runs were performed using Eq(9) as the objective function with different sets of values for 
the weighting factors. The analysis of the converged optimal solutions was utilised in the construction of Pareto 
optimal fronts for the determination of the non-dominated optimal solutions as shown in Figure 2. In this way, 
the set of weighting factors that achieved the lowest value in the aggregate objective function, Eq(9), is obtained. 
The desired point in the Pareto front exhibits relatively low sensitivity (i.e. moderate slope) of the Pareto front. 
Case 76 is the selected optimal solution. The steady state solution in Figure 2 shows the achieved performance 
for the case operating at an optimal but constant profile for the control variables. 
The optimal current voltage and dilute flow rate profiles are shown in Figure 3. The current voltage reduced at 
its lower bound for about the first half of the batch as the dilute compartments have a high concentration of 
sulfuric ions. Gradually the voltage increased in the second half of the batch to push the smaller amount of 
sulfuric anions to the concentrate side as shown in Figure 4. Regarding the circulation flow rate, the optimal 
profile started with the lowest possible flow rate when the dilute concentrate was high in sulfuric ions to increase 
the residence time in the electrodialysis compartments. Progressively as the dilute concentration was dropping 
(Figure 4), the circulation flow rate was increased which in interaction with the strong applied electric field the 
degree of separation eventually reached the desired level. The combination of the two profiles enabled the 
completion of the desired degree of separation at the lowest possible batch time. This is also significant from 
an economic point of view as the shorter the batch time the greater the amount of processed effluent stream 
and the greatest the amount of recovered sulfuric ions. 
The final concentrations of the concentrate and dilute tanks and compartments achieved at the end of the batch 
are shown in Table 1. The case is considered acceptable as it achieves the final dilution tank concentration 
target set. Comparison of the optimal result with the steady state case reveals that the optimised process 
achieved a 75 % faster separation of the solution, in addition to a lower separation level which was acceptable 
but closer to the constraints. Finally, a slight decrease by 15 % was observed in the required energy. 

1024



(a) (b) 

(c) 

Figure 2: Pareto optimal fronts for the multiple optimisation criteria 

Figure 3: Control actions: Voltage (left-blue) and 

flowrate (right-red) during the batch duration 

Figure 4: Concentrations of solutions in 

electrodialysis compartments and tanks during the 

batch duration 

1025



Table 1: Final concentration values of the selected case at the end of the batch 

Final Time 
[min] 

Final Concentration of Dilute 
Tank 
[𝑚𝑔

𝐿
] 

Final Concentration of 
Concentrate Tank 

[𝑚𝑔
𝐿

]

Energy Consumption  
[
𝑘𝑊ℎ

𝑚3
]

Degree of 
Separation 

13.5 519 3,481 2.1 0.74 

4. Conclusions

The present study investigates the application of dynamic optimisation in a batch electrodialysis system with 
recirculation. A multi-objective framework was utilised to investigate the behaviour of the optimal control 
variables with respect to the various performance criteria. The dynamic optimisation enables the optimal 
adjustment of the control profiles to accommodate the variation in the feed concentration and feed amount in 
the electrodialysis unit. The optimisation achieves the completion of the batch in a shorter period of time by 
approximately 50 % while satisfying the effluent quality criteria and achieving a considerable reduction in energy 
requirements. On the meantime, the concentrated solution enables the recovery of valuable sulphuric ions. In 
addition, the proposed approach can potentially handle the optimisation of the operating conditions in the case 
that the performance of the ion exchange membranes changes with time. The incorporation of the proposed 
optimisation framework in a control scheme that also includes model prediction updates would further enhance 
the ability of the electrodialysis unit to operate under a wide range of effluent composition and concentration 
ranges.  

Acknowledgements 

This research has been co‐financed by the European Union and Greek national funds through the Operational 
Program Competitiveness, Entrepreneurship and Innovation, under the call RESEARCH–CREATE– 
INNOVATE (project code: T1EDK- 02677). 

References 

Al-Amshawee S., Yunus M.Y.B.M., Azoddein A.A.M., Hassell D.G., Dakhil, I.H., Hasan H.A., 2020, 
Electrodialysis desalination for water and wastewater: A review. Chemical Engineering J., 380, 122231. 

Chehayeb K.M., Lienhard J.H., 2019, On the electrical operation of batch electrodialysis for reduced energy 
consumption, Environmental Science: Water Research & Technology, 5(6), 1172-1182. 

Kiparissides C., Seferlis P., Mourikas G, Morris A.J., 2002, On-Line Optimizing Control of Molecular Weight 
Properties in a Batch Free-Radical Polymerization Reactor, Industrial & Engineering Chemistry Research, 
41, 6120-6131. 

Ortiz J.M., Sotoca J.A., Expósito E., Gallud F., Garcia-Garcia V., Montiel V., Aldaz A., 2005, Brackish water 
desalination by electrodialysis: batch recirculation operation modelling, Journal of Membrane Science, 
252(1-2), 65-75. 

Parulekar S.J., 1998, Optimal current and voltage trajectories for minimum energy consumption in batch 
electrodialysis, Journal of Membrane Science 148, 91–103. 

Rohman F.S., Aziz N., 2011, Optimization of batch electrodialysis for hydrochloric acid recovery using 
orthogonal collocation method, Desalination, 275(1-3), 37-49. 

Rohman F.S., Aziz N., 2019, Comparison of orthogonal collocation, control vector parameterization and multiple 
shooting for optimization of acid recovery in batch electrodialysis, AIP Conference Proceedings, 2124, 
020013. 

Rohman F.S., Othman M.R., Aziz N., 2010, Modeling of batch electrodialysis for hydrochloric acid recovery. 
Chemical Engineering J., 162(2), 466-479. 

Shah S., Wright N., Nepsky P., Amos G., Winter V., 2018, Cost-optimal design of a batch electrodialysis system 
for domestic desalination of brackish groundwater, Desalination, 443, 198-211. 

Voutetaki A., Gkouletsos D., Papadopoulos A.I., Seferlis P., Plakas K., Bollas D., Parcharidis S., 2020, Efficient 
Modeling of Electrodialysis Process for Waste Water Treatment through Systematic Parameter Estimation, 
Chemical Engineering Transactions, 81, 841-846. 

1026