CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 81, 2020 

A publication of 

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

Guest Editors: tbc Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš 
Copyright © 2020, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-79-2; ISSN 2283-9216 

Efficient Modeling of Electrodialysis Process for Waste Water 

Treatment through Systematic Parameter Estimation 

Alexia Voutetakia, Dimitrios Gkouletsosa, Athanasios I. Papadopoulosa,*, Panos 

Seferlisb, Konstantinos Plakasa, Dimitrios Bollasc, Symeon Parcharidisc 

aChemical Process and Energy Resources Institute (CPERI), Centre for Research and Technology Hellas (CERTH), 57001 

 Thermi, Thessaloniki, Greece 
bDepartment of Mechanical Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece 
cSystems Sunlight S.A., Xanthi, Greece 

 spapadopoulos@cperi.certh.gr 

Electrodialysis is a promising electrochemical process for separation and recovery of useful ions from waste 

waters. The large number of parameters involved in electrodialysis models makes their estimation challenging. 

This work proposes a systematic sensitivity analysis approach to identify the impact model parameter variations 

exert on multiple electrodialysis performance indicators, within a wide operating range. This enables the robust 

mapping of electrodialysis performance toward the direction of maximum variability in the multi-dimensional 

parametric space and the identification of operating regimes, where the input parameters exhibit the highest 

sensitivity. Such regimes are used to determine upper and lower limits in the parameter estimation problem. 

Parameters and ranges exerting insignificant changes on the output electrodialysis indicators are omitted from 

evaluation, which reduces the associated complexity. The approach is validated against published experimental 

results obtained for a lead ions removal electrodialysis recirculation batch process. Only four out of nine 

parameters need to be estimated, while excellent match is observed between experimental and predicted data. 

1. Introduction

Electrodialysis (ED) is an electrochemical process in which ions migrate through ion-selective membranes as a 

result of their attraction due to two electrically charged electrodes. Although initially established for water 

desalination (Vargas et al., 2011), ED has been gaining attention for treatment of dilute waste water streams 

(Scialdone et al., 2014) due to lower capital and operational costs than methods such as ettringite precipitation 

(Usinowicz et al., 2006) and to efficient ion recovery for subsequent reuse (Pilat, 2001). The development of ED 

models is very important to scale-up and optimise the process. However, ED modelling includes significant 

challenges due to the complexity of the associated electrochemical and mass transfer phenomena such as ion 

electro-migration, ion diffusion, and water transport. The accurate prediction of ED output performance 

indicators such as current density and concentration profiles may require the determination of over 30 model 

parameters pertaining to geometrical, structural, electro-chemical and other features (Rohman et al., 2010). The 

values of such parameters are often not reported in experimental studies either because they are difficult to 

measure or the scope of the studies is to assess the separation capabilities of ED through measurement of 

solely output performance indicators (Campione et al., 2018). The latter could be used for the estimation of the 

missing model parameter values, which enabling the development of ED models for various ionic species. The 

large number of model parameters and the diverse operating regimes used in ED for the removal of different 

species make parameter estimation through non-linear optimization very challenging (Ortiz et al., 2005). 

Conventionally, the entire set of parameters is considered despite that the ED performance may not be affected 

uniformly by all of them. Parameter value ranges are defined arbitrarily through trial and error (Campione et al, 

2018) with detrimental effects on estimation quality and computational effort (Ortiz et al., 2005).  

To address these challenges, a systematic approach is proposed and implemented for the first time in ED 

systems to determine parameter value ranges which may be used as upper and lower limits in optimisation-

 

  DOI: 10.3303/CET2081141 

 

 

 

 
 
 

 
 

 
 
 

 
 
 

 
 

 

 

 

 
 
 

 
 
 

 
 
 

 
 

 
 
 

 
 
 

 
 
 

Paper Received: 05/05/2020; Revised: 22/05/2020; Accepted: 31/05/2020 
Please cite this article as: 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  DOI:10.3303/CET2081141 

841



based, parameter estimation methods. The proposed approach enables the identification of the impact 

simultaneous and multiple parameter variations exert on the ED output within a wide range of condition changes, 

using a non-linear sensitivity analysis method (Seferlis and Hrymak, 1996). Parameters that exert almost 

negligible changes on the output ED performance indicators, even under large variability of conditions, are 

omitted from consideration in the parameter estimation problem (Papadopoulos et al., 2013). This allows 

estimation algorithms to be focused on the estimation of the values for fewer parameters that significantly affect 

the quality of model fitting, which obtaining reliable results with reduced computational effort. The proposed 

approach is validated against experimental results for an ED process used in the removal of lead ions from 

aqueous solutions in the form of lead nitrate (Gherasim et al., 2014). Lead is a hazardous polluting metal, which 

is found in industrial wastewaters (dyes, batteries, vehicles, manufacture and so forth) (Ghorbani et al., 2020) 

and drinking water (Harvey et al., 2016), at elevated concentrations.  

2. Models and methods

2.1 Modeling of an ED system 

A schematic diagram of an ED process for batch recirculation is presented in Figure 1. Briefly, a salt solution is 

pumped through an ED cell from a feed tank, while an electric voltage potential is applied by the power supply 

across the stack resulting to two distinct groups of compartments-cells. The first solution is called concentrate 

(indicated as C) and includes the brine solution. The second solution is called diluate (indicated as D) and is 

related to the feed solution. The cations move towards the cathode, and the anions towards the anode. The 

cations pass through the negatively charged CEM (Cation Exchange Membranes), and the anions pass through 

the positively charged AEM (Anion Exchange Membranes). Consequently, a rise of ion concentration is 

observed in the concentrate compartment and a decrease in the dilute compartment. 

Figure 1: ED batch recirculation process 

The mathematical model that describes the ED process is given by the following equations (Ortiz et al., 2005): 

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

, 
𝑑𝐶𝑑𝑖𝑙

𝑡𝑎𝑛𝑘

𝑑𝑡
=

𝑄(𝐶𝑑𝑖𝑙 − 𝐶𝑑𝑖𝑙
𝑡𝑎𝑛𝑘)

𝑉𝑑𝑖𝑙
𝑡𝑎𝑛𝑘

(3) 

𝐶𝑐𝑜𝑛𝑐
𝐴𝐸𝑀 = 𝐶𝑐𝑜𝑛𝑐 +

𝜙𝑗

𝑧𝐹𝑘𝑚
(𝑡𝐴𝐸𝑀 − 𝑡−),     𝐶𝑑𝑖𝑙

𝐴𝐸𝑀 = 𝐶𝑑𝑖𝑙 −
𝜙𝑗

𝑧𝐹𝑘𝑚
(𝑡𝐴𝐸𝑀 − 𝑡−) (4) 

𝐶𝑐𝑜𝑛𝑐
𝐶𝐸𝑀 = 𝐶𝑐𝑜𝑛𝑐 +

𝜙𝑗

𝑧𝐹𝑘𝑚
(𝑡𝐶𝐸𝑀 − 𝑡+),    𝐶𝑑𝑖𝑙

𝐶𝐸𝑀 = 𝐶𝑑𝑖𝑙 −
𝜙𝑗

𝑧𝐹𝑘𝑚
(𝑡𝐶𝐸𝑀 − 𝑡+) (5) 

It is composed of the mass balances in the concentrate, Eq(1), and diluate, Eq(2), compartments of the ED 

stack, and the mass balances in the two tanks shown in Eq(3). 𝑁 is the number of cell pairs, 𝑉𝑐𝑜𝑚𝑝 the 

compartment volume, 𝑄 the overall flow rate of the solutions, 𝐶𝑐𝑜𝑛𝑐, 𝐶𝑑𝑖𝑙, 𝐶𝑐𝑜𝑛𝑐
𝑡𝑎𝑛𝑘, 𝐶𝑑𝑖𝑙

𝑡𝑎𝑛𝑘 the concentration of the

842



concentrate and diluate solutions at the outlet and inlet of the ED stack, 𝜙 the current efficiency, 𝑗 the current 

density, 𝐴 the membrane area, 𝑧 the ion’s charge, 𝐹 Faraday’s constant, 𝐷𝐴𝐸𝑀 and 𝐷𝐶𝐸𝑀 the diffusion coefficients 

through AEM and CEM, 𝑙𝐴𝐸𝑀 and 𝑙𝐶𝐸𝑀 the membrane thickness, and 𝐶𝑐𝑜𝑛𝑐
𝐴𝐸𝑀, 𝐶𝑑𝑖𝑙

𝐴𝐸𝑀, 𝐶𝑐𝑜𝑛𝑐
𝐶𝐸𝑀, 𝐶𝑑𝑖𝑙

𝐶𝐸𝑀 the concentrations

on the surfaces of the AEM and CEM in concentrate and dilute compartments. The latter are calculated with 

Eq(4) and Eq(5), where 𝑡− and 𝑡+ are the solution’s transport numbers, 𝑡𝐴𝐸𝑀 and 𝑡𝐶𝐸𝑀 are the transport numbers 

in AEM and CEM, and 𝑘𝑚 the mass transfer coefficient. The first three modes of mass transport are derived 

mainly from fundamental continuity and Nernst-Planck equations. In the selected model, it is assumed that the 

water volume transferred from the diluate to the concentrate tank is negligible (Wright et al., 2018). ED highly 

relies on the applied voltage and on the corresponding current density. The relationship describing the electrical 

current flow is obtained from Kirchhoff’s 2nd law and Ohm’s law (Wright et al., 2018). 

2.2 Proposed sensitivity analysis approach 

This work employs sensitivity analysis in order a) to identify the ED process operating parameters that affect 

the process performance significantly and b) to determine the upper and lower bounds of those parameter 

values, where high sensitivity is observed. Firstly, for every time step 𝑡, out of a total 𝑁𝑡  steps, a sensitivity 

matrix 𝑃(𝑡)  is developed which incorporates the derivatives of multiple output ED operating indicators in vector 

𝐹(𝑡) (e.g., concentrations and current density) with respect to input parameters in vector 𝜀(𝑡), as follows:  

In Eq(6), 𝑁𝑝 is the total number of operating parameters and 𝑁𝑓 is the total number of ED operating indicators. 

The matrix derivatives are calculated by propagating through the ED model a vector of infinitesimal changes ∆ε 

on vector 𝜀𝑛𝑜𝑚, which includes the nominal values for the ED model parameters. The nominal values are 

obtained either from experimental data when available, or from prior knowledge of similar systems. The 

decomposition of 𝑃𝑇(𝑡)𝑃(𝑡) provides the matrix eigenvectors 𝛩𝑘(𝑡) for each parameter in vector  𝜀, where 𝑘 =

[1, 𝑁𝑒] and 𝑁𝑒 is the total number of eigenvectors. The eigenvector 𝛩
1(𝑡) that corresponds to the largest in

magnitude eigenvalue of 𝑃𝑇(𝑡)𝑃(𝑡) indicates the dominant direction of variability in the multi-parametric space 

which results from the simultaneous consideration of the combined effects of all parameters in vector 𝜀(𝑡) on all 

ED performance indicators in 𝐹(𝑡). Individual entries in the eigenvector 𝛩1(𝑡) direction indicate the contribution 

of the corresponding to the entry parameter in the dominant direction. A relatively few eigenvectors can explain 

a large percentage of the system variability and the dimensionality of the problem can be reduced. 

Subsequently, the parameter estimation problem can focus on the parameters involved in the few dominant 

eigenvector directions. For each such parameter, the range in which they exert the highest effect on the ED 

performance indicators can be identified through the following transformation:  

In Eq(7), 𝜁𝑡 represents the parameter variation magnitude coordinate. The same 𝜁𝑡 value is imposed on all 

parameters to move 𝜀(𝑡, 𝜁𝑡) in the direction of 𝛩
1(𝑡), accounting for the direction of maximum variability with

respect to 𝐹(𝑡, 𝜁𝑡). The values of 𝜀(𝑡, 𝜁𝑡) are propagated through the ED model to obtain 𝐹(𝑡, 𝜁𝑡). The values of

𝐹𝑒𝑥𝑝(𝑡) are available from experimental results. By moving 𝜁𝑡 within [𝜁𝑡
𝐿, 𝜁𝑡

𝑈], so that 𝐹(𝑡, 𝜁𝑡) approaches 𝐹
𝑒𝑥𝑝(𝑡)

to the closest possible proximity, 𝜀(𝑡, 𝜁𝑡
𝐿) and 𝜀(𝑡, 𝜁𝑡

𝑈) are obtained. These values are used as upper and lower

limits in parameter estimation. The method is implemented for every time step 𝑡. In this respect, the parameter 

estimation problem is limited systematically within a reduced range that captures the combined effect of all 

parameters on the ED performance indicators, as opposed to the arbitrary determination of such ranges 

otherwise. Having estimated the upper and lower limits, parameter estimation is implemented as follows:  

𝑃(𝑡) =

[
 
 
 
 
 
𝜕𝑙𝑛𝐹1(𝑡)

𝜕𝑙𝑛𝜀1(𝑡)
⋯

𝜕𝑙𝑛𝐹1(𝑡)

𝜕𝑙𝑛𝜀𝑁𝑝(𝑡)

⋮ ⋱ ⋮
𝜕𝑙𝑛𝐹𝑁𝐹(𝑡)

𝜕𝑙𝑛𝜀1(𝑡)
⋯

𝜕𝑙𝑛𝐹𝑁𝐹(𝑡)

𝜕𝑙𝑛𝜀𝑁𝑝(𝑡)]
 
 
 
 
 

(6) 

𝜀(𝑡, 𝜁𝑡) = 𝜀
𝑛𝑜𝑚 ∙ 𝜁𝑡 ∙ 𝛩

1(𝑡) + 𝜀𝑛𝑜𝑚, 𝜁𝑡 ∈ [𝜁𝑡
𝐿, 𝜁𝑡

𝑈] (7) 

min
𝜀(𝑡)

𝐽(𝑡) = ∑ (𝑊𝑙 ∙ ∑
𝐹𝑙(𝜀(𝑡), 𝑡) − 𝐹𝑙

𝑒𝑥𝑝
(𝑡)

𝐹
𝑙
𝑒𝑥𝑝

(𝑡)

𝑁𝑡

𝑡=1

)

2𝑁𝑓

𝑙=1

𝑠. 𝑡.  𝐸𝑞(1) − 𝐸𝑞(5) 

𝜀𝐿(𝑡) ≤ 𝜀(𝑡) ≤ 𝜀𝑈(t) 

(8) 

843



The objective function depicts the changes of the performance indicators of the ED system, caused by changes 

in parameter values between the upper and lower limits as set above, compared to the experimental data points. 

Vector 𝑡 represents the time with 𝑡𝐿 and 𝑡𝑈 being lower and upper limits and 𝑊𝑙 is a vector of weights in case it

is desired to differentiate the impact of the objective functions. 𝜀𝐿(𝑡) and 𝜀𝑈(t) correspond to the previously 

determined 𝜀(t, 𝜁𝑡
𝐿) and 𝜀(𝑡, 𝜁𝑡

𝑈) at a specific 𝜁, and so the lack of notation.

3. Implementation

The proposed approach is implemented to estimate parameters required for the simulation of an ED process 

used for the separation of Pb2+ and NO3- ions from waste water (Gherasim et al., 2014). These authors examined 

the impact flow rate, applied voltage, temperature, and feed concentration have on lead nitrate removal by ED 

in a batch recirculation mode. The parameters used within the current work include 𝑉𝑐𝑜𝑚𝑝, 𝑘𝑚, 𝐷𝐶𝐸𝑀, 𝐷𝐴𝐸𝑀, 𝑅𝐶𝐸𝑀, 

𝑅𝐴𝐸𝑀, 𝑡
−, 𝑡+ and 𝐸𝑒𝑙 (electrode potential difference). After calculation of the eigenvectors, it is observed that 𝑘𝑚,

𝑅𝐶𝐸𝑀 and 𝑅𝐴𝐸𝑀 are not involved in the dominant direction, their values are considered fixed at the values 

suggested by Ortiz et al. (2005). Non-dominant behaviour is also observed for 𝑡− and 𝑡+; their values are also 

considered fixed and calculated based on Gao et al. (2009). The relationship between 𝑡−, 𝑡+ and  𝛬0 (infinite

molar conductivity) is used with the infinite ion conductivities for Pb2+, NO3- (Haynes et al., 2012). Eventually, 

the parameters involved in the dominant direction of variability are 𝑉𝑐𝑜𝑚𝑝, 𝐷𝐶𝐸𝑀, 𝐷𝐴𝐸𝑀 and 𝐸𝑒𝑙.Their initial 

(nominal) values for the parameter estimation were also obtained from Ortiz et al. (2005). Except for the latter 

four parameters, the values of all others are shown in Table 1. The parameters used as performance criteria in 

vector 𝐹 are the concentrate and diluate concentrations and the current density. In parameter estimation, the 

concentrate and diluate concentrations are assumed to have a significant effect on the system performance, so 

their weights in Eq(8) are higher than that of the current density. The applied source voltage is 10 V, the flow 

rate is 0.070 m3/h, and the geometrical features of the ED stack are from Gherasim et al. (2014). 

Table 1: List of parameters 

Parameters Units Values Source Parameters Units Values Source 

𝐴 cm2 64 Gherasim et al. (2014) 𝑉𝑐𝑜𝑛𝑐
𝑡𝑎𝑛𝑘 L 5 Gherasim et al. (2014) 

𝐵0 Å mol
-1 0.327 Kortum et al. (1965) 𝑉𝑑𝑖𝑙

𝑡𝑎𝑛𝑘 L 20 Gherasim et al. (2014) 

𝐵1 mol
-1/2 0.227 Kortum  et al. (1965) 𝑅𝐴𝐸𝑀 Ω cm

2 29 Ortiz et al. (2005) 

𝐵2 Ω
-1 m2 mol-3/2 54.16 Kortum et al. (1965) 𝑅𝐶𝐸𝑀 Ω cm

2 24 Ortiz et al. (2005) 

𝑙𝐴𝐸𝑀, 𝑙𝐶𝐸𝑀  mm 0.45 Gherasim et al. (2014)  𝛾𝑐𝑜𝑛𝑐
𝐴𝐸𝑀, 𝛾𝑐𝑜𝑛𝑐

𝐶𝐸𝑀 - 0.9 Wright et al. (2018) 

𝐿 mm 0.8 Gherasim et al. (2014) 𝛾𝑑𝑖𝑙
𝐴𝐸𝑀, 𝛾𝑑𝑖𝑙

𝐶𝐸𝑀 - 0.8 Wright et al. (2018) 

𝑁 - 10 Gherasim et al (2014) 𝐹 C mol-1 96485 - 

α Å 4 Kortum et al. (1965) 𝑇 K 297 Gherasim et al. (2014) 

𝜙 - 0.92 Ortiz et al. (2005) 𝑅 J mol-1 K-1 8.314 - 

𝑡𝐴𝐸𝑀, 𝑡𝐶𝐸𝑀 - 1 Ortiz et al. (2005) 𝛬0 S m
2 mol-1 213.10-4 Gao et al. (2009) 

𝑧 - 2 - 𝑡+ - 0.66 Gao et al. (2009) 

𝑘𝑚 m s
-1 0.77.10-3 Ortiz et al. (2005) 𝑡− - 0.33 Gao et al. (2009) 

4. Results and Discussion

4.1. Sensitivity analysis 

Table 2 shows the ranking of parameter eigenvalues with respect to different time instants. For each instant, 

the parameters’ impact on the ED system decreases from left to right in the table. Parameters 𝑉𝑐𝑜𝑚𝑝, 𝐷𝐶𝐸𝑀, 𝐷𝐴𝐸𝑀 

and 𝐸𝑒𝑙 are clearly influential on ED performance at different instants, which they are all selected. 

Table 2: Parameters ordered from left to right based on eigenvalues for different time instants 

𝑡𝐿 ≤ 𝑡 ≤ 𝑡𝑈 

t (h) Parameters ordered based on eigenvalue magnitude 

0.01 𝑉𝑐𝑜𝑚𝑝 𝐸𝑒𝑙 𝐷𝐶𝐸𝑀,  𝐷𝐴𝐸𝑀 

0.28 𝐸𝑒𝑙 𝑉𝑐𝑜𝑚𝑝 𝐷𝐶𝐸𝑀,  𝐷𝐴𝐸𝑀 

0.50 𝐸𝑒𝑙 𝑉𝑐𝑜𝑚𝑝 𝐷𝐶𝐸𝑀,  𝐷𝐴𝐸𝑀 

0.83 𝐷𝐶𝐸𝑀,  𝐷𝐴𝐸𝑀 𝐸𝑒𝑙 𝑉𝑐𝑜𝑚𝑝 

1.00 𝐷𝐶𝐸𝑀,  𝐷𝐴𝐸𝑀 𝑉𝑐𝑜𝑚𝑝 𝐸𝑒𝑙 

844



Figure 2 illustrates the results of the proposed method with respect to the determination of upper and lower 

limits for the parameters of vector 𝜀. Note that the results with respect to vector 𝐹 are reported as an average 

for all time instants, i.e. �̅�(𝜁) and �̅�𝑒𝑥𝑝, with �̅�(0)= �̅�𝑒𝑥𝑝. By initially selecting a range of [-5, 0.5] as [𝜁𝑡
𝐿, 𝜁𝑡

𝑈], it is

observed that the diluate concentration and the current density approach closest to the experimental values 

around 𝜁=-3. It further appears that these two performance indicators are affected more intensely by the 

simultaneous variation of 𝑉𝑐𝑜𝑚𝑝, 𝐷𝐶𝐸𝑀, 𝐷𝐴𝐸𝑀 and 𝐸𝑒𝑙 than the concentrate concentration. The latter remains very 

close to its experimental values, exhibiting very little sensitivity to changes. The upper and lower limits are 

determined within a range of [-3.5, -2.5], i.e. at ±16% from 𝜁=-3. The corresponding values for 𝑉𝑐𝑜𝑚𝑝, 𝐷𝐶𝐸𝑀, 𝐷𝐴𝐸𝑀 

and 𝐸𝑒𝑙 are shown in Table 3. Different ranges can also be considered and tested.  

Figure 2: Combined effect of parameters on performance indicators 

4.2. Parameter estimation and validation 

Using the limits of Table 3, the values for 𝑉𝑐𝑜𝑚𝑝, 𝐷𝐶𝐸𝑀, 𝐷𝐴𝐸𝑀 and 𝐸𝑒𝑙 are obtained from parameter estimation, 

also in Table 3 (last column). The values used as nominal from Ortiz et al. (2005) are selected initially due to 

the lack of data. Despite similarities in the species, that system has different features and is used for a different 

purpose (desalination). Results in Table 3 indicate that the estimated parameter values are considerably 

different. The ED process investigated here requires a smaller compartment volume, and higher electrode 

potentials and diffusion coefficients. Smaller volumes and higher diffusion coefficients account for a faster ion 

contact to the membrane surfaces, while the electrode potential values are proportional to the system kinetics. 

Table 3: Upper and lower limits and parameter estimation results 

Parameter Units Nominal value (Ortiz et al., 2005) Upper limit Lower limit Value from parameter estimation 

𝑉𝑐𝑜𝑚𝑝 m
3 33.10-6  40.8.10-6 3.34.10-6 3.56.10-6

𝐸𝑒𝑙 V 1.5  5.57 0.91 3.89 

𝐷𝐶𝐸𝑀,  𝐷𝐴𝐸𝑀 m
2s-1 3.28.10-11 19.1.10-11 1.37.10-11 18.5.10-11

(a) (b) 

Figure 3: Model validation vs. experimental results of a) concentration profiles and b) the current density profile 

The results of the concentration and current density profiles are verified against experimental data in Figure 3. 

The latter illustrates experimental and predicted results for all performance indicators, using both the nominal 

and estimated values of Table 3. It appears that the estimated values provide a very close match with the 

experimental results, as opposed to the nominal values in the cases of the concentrate and of the current 

density. This indicates that the values of Ortiz et al. (2005) are not suitable for the ED process case investigated 

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in this work and highlights the necessity of the proposed approach. Regarding the ED results, the system’s fast 

kinetics is noticeable. The lead’s remaining concentrations (1- 2 mg/L) are estimated to be below the toxicity 

limits for irrigation water (5 mg/L, WHO/FAO, 2007). Also, the decreasing current density observed in Figure 3b 

is due to the increase of the diluate’s resistance. The latter is likely due to a white precipitate formed on the AEM 

surface, possibly consisting of Pb(OH)2, based on experimental observations reported by Gherasim et al. (2014). 

5. Conclusions

The proposed approach unveiled the sensitivity of each one of 9 parameters with respect to ED performance 

indicators. It was shown that only 4 of them are necessary for parameter estimation, which the use of additional 

parameters would only complicate the estimation problem. The sensitivity analysis results are then used to 

determine the upper and lower limits in parameter estimation. The narrow range indicated by these limits 

enabled very good fit of the performance indicators with respect to experimental results from literature. Apart 

from its common use for desalination purposes, ED can provide a promising solution for wastewater treatment, 

applicable to the removal of several different types of heavy metal ions. In the present case, the concentration 

of lead ions is reduced to levels acceptable for plant irrigation. From the total of 140 g of lead entering the ED 

process, 139.72 g are removed, indicating a reduction of 99.8 % in what is released to the environment. 

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). 

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