# Coupling reactive transport and speciation models to study nanofiltration of metallic salts from an acid leachate

T. Porqueddu, B. Laubie, A. Szymczyk, Marie-Odile Simonnot

► To cite this version:

T. Porqueddu, B. Laubie, A. Szymczyk, Marie-Odile Simonnot. Coupling reactive transport and speciation models to study nanofiltration of metallic salts from an acid leachate. Separation and Purification Technology, 2025, 378, pp.134757. 10.1016/j.seppur.2025.134757. hal-05334521

HAL Id: hal-05334521

https://hal.univ-lorraine.fr/hal-05334521v1

Submitted on 28 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Coupling reactive transport and speciation models to study nanofiltration of metallic salts from an acid leachate☆

T. Porqueddu<sup>a</sup>, B. Laubie<sup>a</sup>, A. Szymczyk<sup>b</sup>, M.O. Simonnot<sup>a, *, e</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

Editor: B. Van der Bruggen

### Keywords:

- Nanofiltration
- Hydrometallurgy
- Reactive transport modelling
- Speciation
- Hyper accumulator plant
- Phytomining
- Nickel

## ABSTRACT

This work proposes the application of membrane processes to filter the metals contained in a leachate of hyperaccumulating plant ashes. To better understand the phenomena involved in metal separation, a transport model based on the Nernst-Planck equation was used (SEDE model). The leachate was acidic and concentrated in various salts, including nickel, magnesium, and iron sulfate. It therefore exhibited a complex speciation. The objective was to study speciation influence on the modelling of the rejection rate. The leachate was diluted 2, 4, 8, 10 and 20 times to assess the quality of the nanofiltration and the influence of sulfuric acid and of salt concentration on the rejection rate of three metals of interest: nickel, iron, and magnesium. An optimum rejection rate was obtained for the leachate diluted 8-fold, at an ionic strength of 0.37 mol kg⁻¹, with a rejection rate of 85 % for the three metals. Pore swelling was observed for pure leachate nanofiltration, considerably reducing the rejection of all divalent ions. Iron was most affected by changes in speciation. By considering the speciation upstream of nanofiltration, the prediction of the retention rate of some divalent cations can be improved by up to 50 %, compared to experimental data. This work also constitutes a first step towards the evaluation of the quantity of recyclable protons after an acid leaching stage in hydrometallurgy.

## 1. Introduction

Certain raw materials are crucial to the European economy. They are essential to produce many emerging technologies, such as electric vehicles, solar panels, and wind turbines. As the need for raw materials is set to grow, the European Union (EU) has taken steps to ensure its independence: by March 2023, the EU has decided to produce 15 % of its critical materials needs through recycling. For these metals, the processes commonly chosen for this application are hydrometallurgical processes. Indeed, they offer high selectivity and low energy consumption, while producing a final product of sufficient purity to be reused in an industrial process.

However, hydrometallurgical processes present a major drawback: they consume a lot of reagents, especially acids, particularly during the leaching step. After metal recovery, the pH of the leachate must be raised with an alkaline reagent, which further increases the number of chemical species used, and leads to substantial waste production. In all these processes, the need to reduce the number and quantity of reagents used has driven the development of recycling processes, using

technologies such as liquid-liquid extraction [1,2].

Membrane processes, such as nanofiltration (NF), offer the possibility of reducing the quantity of reagent needed by concentrating the metals of interest in the retentate. In industry, these types of processes are mainly used for two reasons:

- they are often applied to treat polluted water or effluents, or to separate metals and contaminants from waste acid streams [3,4];

- the low rejection of poorly dissociated or unionised solutes has also led to a membrane recovery process ensuring high metal recovery. This application is often encountered when recovering lithium from a multi-component stream, such as a brine or battery leachate [5].

However, in hydrometallurgy, the use of membrane technologies is mostly limited to solutions with a pH not lower than 1, often to preserve membrane integrity [3,6,7], only one example, to our knowledge, concerns leachate with a pH of 0.5 [8]. This represents a significant limitation in this field, as some solutions can reach a very high proton concentration [9,10].

<footnote><sup>a</sup> Université de Lorraine, CNRS, LRGP, F-54000 Nancy, France</footnote>

<footnote><sup>b</sup> Univ Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes) – UMR 6226, F-35000 Rennes France</footnote>

<footnote>* This article is part of a special issue entitled: 'EuroMembrane 2024' published in Separation and Purification Technology.</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: marie-odile.simonnot@univ-lorraine.fr (M.O. Simonnot).</footnote>
To withstand these treatment conditions, ceramic membrane could be used. Indeed, they possess great chemical and thermic stability [11]. However, polymer membranes are preferred by the industry mainly because of their lower capital cost and for their versatility [12]. Today, polymeric NF membranes resistant to sulfuric acid leachates are commercially available [13,14], although their lifetime under these conditions is assumed to be short compared to other NF membranes [15], which can last up to 5 years, depending on the application [16].

The recovery of critical materials includes the exploitation of secondary resources, such as metal-rich soils or mine tailings. Certain plants, known as hyperaccumulators, have the capacity to capture and accumulate metals or elements of interest in their above-ground parts, through phytoextraction. This has led to the development of the agro-mining chain, which includes the cultivation of hyperaccumulator plants and the recovery of metals [17]. The first stage in metal recovery often involves burning the plant to obtain ash, which in the case of nickel can contain 15–20 % by mass of the metal. The ash is then leached to transfer the metal into solutions using the conventional hydrometallurgical approach [18].

Therefore, this work proposes to consider the filtration of a hyper- accumulator plant ash leachate with an NF membrane, to recycle the acid used in the leaching process. Taking as an example a leachate of high acidity and high salinity, the objective of this article is to explore the NF of hydrometallurgical leachates. To better understand this separation, a transport model may be helpful. Few studies propose to simulate the filtration of such an acidic solution. When they do exist, the approach chosen is based on irreversible thermodynamics [19]. As this type of model considers the membrane as a black box, it does not enable us to identify the phenomena involved in the case of hydrometallurgical leachate.

To complete the experimental approach, it was decided to use a mechanistic model, the Steric, Electric and Dielectric exclusion model (SEDE), to simulate this NF [20]. This model is based on the extended Nernst-Planck equation, accounting for the effect of finite pore size on the diffusion and convection of species through narrow pores, and on an equilibrium partitioning relationship, to take into consideration the distribution of species at the pore inlet and outlet [21].

The SEDE model is also able to simulate the dielectric effects, divided into two phenomena:

- the Born effect, which considers the lowering of the dielectric constant of a solution confined in nanodimensional pores [22];

- the image force, resulting from the difference between the dielectric constant of the membrane and that of the solution inside the pore [23].

However, references proposing to model the speciation of the solution to be filtered prior to NF modelling are rare, although this approach is considered crucial for process design [24]. This type of reactive transport simulation is common for geochemical models and is seldom performed in membrane modelling. Indeed, the only example of a reactive transport model suitable for simulating ion transfer in NF is the SEDFRT model (Solution Electro Diffusion Featuring Reactive Transport) [25], based on irreversible thermodynamics. To our knowledge, no mechanistic reactive transport model is available to simulate the rejection rate of ions across an NF membrane. By coupling the SEDE transport model with the PHREEQC speciation software, this work aims to determine whether this approach is better for modelling the separation of an acidic and concentrated leachate, such as the hydrometallurgical one.

## 2. Theoretical background

### 2.1. Transport model with the Steric, Electric and Dielectric Exclusion (SEDE) model

The SEDE model, developed by Szymczyk and Fievet has been described several times [17,18], so its main features are briefly summarised here. This model is based on the extended Nernst-Planck equation (Eq. (1)).

$$j_i = -K_{i,d}D_{i,\infty} \frac{dc_i}{dx} - \frac{z_i F K_{i,d} c_i D_{i,\infty}}{RT} \frac{d\varphi}{dx} + K_{i,c} c_i \nu \quad (1)$$

where $j_i$ is the molar flow of the ion i (mol s⁻¹m⁻²); $K_{i,d}$ and $K_{i,c}$ the hydrodynamic coefficients respectively accounting for the effect of the pore walls on convective transport and for hindered diffusion inside the pores (−); $D_{i,∞}$ the diffusion coefficient of ion i in the bulk solution at infinite dilution (m² s⁻¹); v the fluid velocity inside the pores (m s⁻¹); φ the electrostatic potential of ion i (V); $z_i$ valence of ion i (−); T the temperature (K); F the Faraday constant (C mol⁻¹) and x the normalised coordinated across the membrane (m). The equations corresponding to the different terms used in the extended Nernst-Planck equation are available in supplementary material S1.

Simulation is used to study the impact of five different phenomena on separation:

- steric hindrance: this effect represents the influence of the size of a species on its rejection rate, without taking into account a potential polarisation concentration;

- Donnan effect: this phenomenon is related to the presence of fixed charges on the membrane surface;

- image force: this effect is linked to the difference in dielectric constant between the pore solution and the membrane. For example, if the dielectric constant of the membrane is lower than that of the solution, it induces an increase in the rejection of the cations and anions into the leachate. Born's dielectric effect, linked to the difference between the dielectric constant of the bulk and pore solutions is not considered here;

- solution speciation: speciation refers to the distribution of chemical forms of an element in a solution;

- pore swelling: this represents the change in the average pore size of a membrane in contact with a high-salinity solution. It occurs when the membrane load increases due to the high concentration of salt in solution. In addition, the number of counterions in the pore rises, leading to an increase in the average pore size [26].

In the SEDE model, the partition coefficient of an ion i takes into account the first four effects. This model can predict the rejection of elements from a solution over a range of permeate fluxes. It requires four parameters to simulate ion transport through the NF membrane: the average pore size, the thickness/porosity ratio, the effective volume charge density of the membrane X, the dielectric constant inside the pore εp and the dielectric constant of the membrane εm.

The active layer of the membrane is assimilated to a group of identical cylindrical ion-selective pores of radius $r_p$ and length $\Delta x$ (with $\Delta x \gg r_p$). The first two parameters are determined by the characterisation of the membrane in neutral and in acidic environments. The dielectric constant $\epsilon_p$ is considered the same as the bulk dielectric constant $\epsilon_b$ and constant for all the solutions studied. The value of these parameters is set at 84 [27]. The effective volume charge density is the fitting parameter of the simulation, fixed at 10mol m⁻³.

The information regarding the solution composition (number of elements considered, concentration and valence of the ion, atomic radius and diffusion coefficient, temperature) constitutes the input parameter of the model (supplementary material S2). The output parameters consist of all the rejection rates of the elements considered in the treated
solution. The equations for simulating ion transport are presented in supplementary material S1. The numerical equations are solved using the Runge-Kutta 4 method at each point of the membrane.

## 2.2. Speciation model: PHREEQC

PHREEQC version 3 is a software package for simulating geochemical equilibria in aqueous media, developed by the United States Geological Survey [28]. PHREEQC is designed to calculate the equilibrium state by considering acid-base reactions, complexation, redox reactions, solubilisation/precipitation, adsorption, and gas-liquid exchange. The calculation of equilibrium is based on the following equations.

Considering the reaction between a species A and B, giving a species D (Eq. (2)), the mass action law is given by Eq. (3), in the case of infinite dilution, with $a_i = \frac{C_i}{C_i \gamma_i}$, $a_i$ being the activity of the species i, $C_i$ the concentration of a species i, $C_i^0$ a reference concentration, and $\gamma_i$ the dimensionless activity coefficient of species i. $K^0$ is determined for zero ionic strength and a temperature of 298 K.

$$aA + bB = dD \qquad (2)$$

$$K^0 = \frac{(a_D)^d}{(a_A)^a (a_B)^b} \qquad (3)$$

For a given ionic strength, the reaction can be written as in Eq. (4), and to obtain $K'(T)$ at a different temperature, Eq. (5) is needed.

$$K'(T) = \frac{[D]^d}{[A]^a [B]^b} = K^0(T) \frac{(\gamma_a)^a (\gamma_b)^b}{(\gamma_D)^d} \qquad (4)$$

$$\Delta_r G(T) = \Delta_r H - T \Delta_r S = -RT \ln K'(T) \qquad (5)$$

For the sake of simplicity, the temperature chosen for all the leachates is 25 °C. To determine the activity coefficients, the following models are available in PHREEQC:

- the Davis model, valid for an ionic strength below 0.1 M;

- the extended Debye-Hückel model, valid for an ionic strength of less than 0.7 M;

- the Specific Ion Theory (SIT) model, valid for an ionic strength below 4 M;

- the Pitzer model, valid for all ionic strengths.

The higher the ionic strength, the more parameters the model requires. In this work, the extended Debye-Hückel model (Eq. (6)) and the SIT model (Eq. (7)) are used in the appropriate ionic strength ranges (from 3.22 to 0.72 mol kg⁻¹ for the SIT model, and 0.32 mol kg⁻¹ for the extended Debye-Hückel model). During this model switch, no values were interpolated, as the models were simply changed to match the ionic strength of the solution.

$$\log \gamma_i = - \frac{A z_i^2 \sqrt{I}}{1 + B a_i^0 \sqrt{I}} + b_i I \qquad (6)$$

with $A = \frac{e^2 B}{18.424 \pi c_0 \epsilon_r k_B T}$ and $B = \sqrt{\frac{2e^2 N}{\epsilon_0 \epsilon_r k_B T}}$, I the ionic strength of the solution (M), $z_i$ the charge of ion i and $\gamma_i$ the activity coefficient of the species i. The values of $a_i^0$ and $b_i$ are displayed in the WATEQ database on PHREEQC.

$$\log \gamma_i = - z_i^2 \frac{A \sqrt{I}}{1 + B a_i^0 \sqrt{I}} + \sum_k \epsilon(i, k) m_k \qquad (7)$$

with $m_k$ the molality of the ion k (mol kg⁻¹). With this model, new data are required for optimal speciation, which are the interaction parameters between two ions, $\epsilon(i, k)$, available in the PHREEQC SIT database,

supplemented by the specific value for Ni [37]. This modified database is used for the speciation simulation.

After calculating the ion activity, the log K of the reactions listed in the SIT database involving the element identified by elemental analysis are used to model the equilibrium composition of the solution.

Speciation in solution can be modeled at different stages of the transport simulation, the first being before nanofiltration, in the feed solution. Indeed, speciation in solution before transport simulation is mandatory for a reactive transport model. The concentration of ions and their valence have an impact on the ionic strength, which influences the transport of ions through an NF membrane. This approach was considered by López et al., who proposed a reactive transport model based on irreversible thermodynamics [25]. It is also possible to model the speciation of the permeate and retentate solutions. This data can be valuable for an industrial process. Due to the mechanistic model used, there is a third option: modeling the speciation of the solution at each iteration of the Runge-Kutta method, which is used to determine the concentrations at a normalised x-coordinate of the membrane. This approach is only possible for mechanistic models, as models based on irreversible thermodynamics treat the membrane as a black box, making it impossible to track solution speciation across the membrane. However, while the mechanistic model seems better suited to this approach, certain problems remain. To simulate speciation at each stage of the Runge-Kutta method, two assumptions must be made:

- the solution inside the pore is assumed to be perfectly mixed;

- if reaction kinetics are not considered, the solution inside the pore is in thermodynamic equilibrium.

These two assumptions contradict the presence of a concentration gradient inside the pores. Reaction kinetics must therefore be considered for this approach. Given the complexity of the leachate studied, the present simulation only covers the first approach and the second approach, which consists of simulating speciation solely in the feed, the permeate and the retentate solution.

## 2.3. Methodology

To estimate the importance of including speciation in the separation, an initial simulation was carried out, without taking speciation into account. All the elements were supposed to be completely dissociated. The objective was to determine which phenomenon should be considered in a leachate separation simulation using only the SEDE transport model. Next, a simulation taking speciation into account was carried out using the SEDE model coupled with PHREEQC. The results were then compared. The values used to compare the theoretical and experimental rejection rates were the data obtained when the curve representing the experimental rejection rate as a function of the permeate volume flux reached a plateau. If no plateau was reached, the last three values of the rejection rates of the solutions studied were used.

# 3. Materials and methods

## 3.1. Preparation of a hyperaccumulator plant ash leachate

The solution selected for this study was a leachate of hyper- accumulator ashes. This solution shares many similarities with hydro- metallurgical liquors, such as its acidity and high salinity. It was used as a representative example of such leachates.

Odontarrhena chalcidica (Brassicaceae family) was grown in Albania in the Pogradec region, on ultramafic soil. This type of soil, found for example in the Balkans, can contain up to 7 mg Ni per kg of dry soil. The plant can then accumulate up to around 10 g of Ni per kg of dry biomass, the Ni being mainly located in the leaves and flowers [29]. After harvesting, the plant was sun-dried and burnt at an average temperature of 900 °C in an industrial boiler (KWB Multifire MF2 D/ZI). In the
laboratory, a finishing stage was carried out to perfect the combustion, at 900 °C for 80 min in a muffle furnace (Nabertherm LE 14/11 B150). A 600 g mass of ash was placed in a 10 L jacketed reactor containing 3 L of 2 M sulfuric acid. Leaching lasted 6 h at 75 °C. Gypsum precipitation occurred during this stage, so after cooling, an initial filtration was run using a Whatman filter (20 µm), followed by ultrafiltration (Microdyn Nadir PM UP005 membrane). The resulting solution, hereafter referred to as 'leachate #1', was analysed by inductively coupled plasma emission spectroscopy (ICP-AES), according to the procedure described by Zhang *et al.* [18]. Its elemental composition is given in Table 1. PHREEQC software was used to determine the theoretical pH of the solution and the theoretical pH of the same solution diluted 100 times. Then, the pH of the diluted solution was measured using a pH meter (senSION + MM374).

Student's law was applied to calculate measurement uncertainties. Each sample was analysed 4 times. The Standard Deviation (SD) was calculated using Eq. (8), with X the different values for each measurement, M the mean value and n the number of data points. The Standard Error (SE) was obtained using Eq. (9), and finally, the 95 % Confidence Interval (CI) was determined using Eq. (10), where $t_{(n-1)}$ is Student's coefficient. For this study, its value is 2.353.

$$SD = \sqrt{\frac{\sum (X - M)^2}{n - 1}} \qquad (8)$$

$$SE = \frac{SD}{\sqrt{n}} \qquad (9)$$

$$CI = M \pm t_{(n-1)}SE \qquad (10)$$

To assess the effect of ionic strength on NF, the leachate was used pure, and diluted 2, 4, 8, 10 and 20 times with ultrapure water, with a respective sulfuric acid concentration of 1.4, 0.6, 0.35, 0.2, 0.15 and 0.1 M. The solutions studied, with ionic strengths of 3.20, 1.60, 0.70, 0.37, 0.30 and 0.15 mol kg⁻¹ respectively, are referred to as leachates #1, #2, #3, #4, #5 and #6.

## 3.2. Nanofiltration set-up

The NF set-up pilot (SEPROSYS 19–16) was composed of:

- PVDF and PEEK hydraulic systems and PVDF valves to ensure high chemical resistance;

- a 5 L feed tank, fitted with a heat exchanger made of hastelloy, a corrosion-resistant material;

- a hastelloy gear pump, up to 250 L h⁻¹, covering a pressure range from 5 to 15 bar;

- a flat membrane housing, containing two 1.51 10⁻² m² thin film NF membranes Suez (GE) JX in polyamine (with an estimated cut off molecular weight of 200 Da) (these membranes are used for continuous filtration of solutions up to a mass concentration of 20 % (2 M) sulfuric acid) [14]. The performance of these membranes has been evaluated by Suez and shows high rejection rates (> 90 %) for all the elements contained in a leachate with a 5 % H₂SO₄ mass concentration with 2 000 ppm of Fe²⁺, Al³⁺, Cu²⁺ and 100 ppm of Cd²⁺. Despite the technical data assuring that these membranes are resistant to solutions with a pH below 0, studies on the stability of membranes under such acidic condition are quite scarce. They can

only be considered stable for a couple of months, depending on the pressure applied and the temperature of the solution [15].

Both nanofiltrations were used in parallel to ensure the reproducibility of the experiments. Steady state was reached after one hour of filtration, and a refrigerant (LAUDA ECO RE630 S) was used to maintain the temperature of the feed solution at 25 °C, measured using a probe thermometer (Testo 925).

## 3.3. Membrane characterisation

The mean pore size of the membrane, $r_p$, was characterised by the NF of three solutions of polyethylene glycol (PEG) (purity 95 %, Sigma Aldrich) of different molar masses (200, 400 and 600 Da), in ultrapure water at a concentration of 4 g L⁻¹ [30]. Their rejection rate for PEG 200, 400 and 600 were 92, 96 and 98 % respectively at high permeate flux. The results were analysed using Eqs. (11) and (12) (the terms are defined in the nomenclature available in Supplementary material S3). When the permeate volume flux $J_v$ is sufficiently high, Eq. (11) can be simplified to Eq. (12). The rejection rate is then independent of the permeate flux. Knowing the Stokes radius $r_{Stokes}$ of the PEG used and combining Eqs. (5) and (8) from S1 with Eq. (12), it is possible to write the polynomial Eq. (13), and to determine $r_p$. The average pore sizes are displayed in Table 2.

$$R = 1 - \frac{C_{i,p}}{C_{i,m}} = 1 - \frac{\phi_i K_{i,c}}{1 - [(1 - \phi_i K_{i,c}) e^{-Pe}]} \quad \text{with } P_e = \frac{J_v K_{i,c}}{A_K K_{i,D} D_C} \Delta x \qquad (11)$$

$$R = 1 - \phi_i K_{i,c} \qquad (12)$$

Both $\phi_i$ and $K_{i,c}$ are functions of $r_p$. Numerical resolutions give the average pore size for each PEG.

The thickness porosity ratio of the membrane is linked to the permeate flux $J_v$ by Eq. (13):

$$J_v = \frac{r_p^2}{8\eta(\Delta x/A_k)\Delta P} \qquad (13)$$

It has already been shown that a single value of $(\Delta x/A_k)$ cannot be determined [31,32]. These references, as well as the results presented in Table 2, indicate a decrease in the ratio between thickness and porosity as the solute size increased.

To ensure that the pore size did not increase during filtration of a more acidic leachate, boric acid (Sigma Aldrich, 95 %) at 4 g L⁻¹ was added to the leachate. Since boric acid is neutral under the pH conditions studied (in the form of H₃BO₃), monitoring its rejection rate enabled the detection of any pore swelling during filtration. The rejection rate of boric acid in the leachate #1, 2, 3 and 4 can be compared to further confirm the hypothesis of pore swelling (Fig. 1).

For leachate #1, pore swelling was involved:

<table><caption>**Table 2**
Average pore size and thickness porosity ratio of the membrane.</caption><thead><tr><th>Molar masses of PEG (Da)</th><th>r<sub>s</sub> (nm)</th><th>r<sub>p</sub> (nm)</th><th>Δx/A<sub>k</sub>(μm)</th></tr></thead><tbody><tr><td>PEG 200</td><td>0.43</td><td>0.77 ± 0.09</td><td>81.7 ± 12.20</td></tr><tr><td>PEG 400</td><td>0.53</td><td>0.64 ± 0.08</td><td>67.5 ± 11.34</td></tr><tr><td>PEG 600</td><td>0.63</td><td>0.69 ± 0.14</td><td>58.4 ± 14.35</td></tr></tbody></table>

<table><caption>**Table 1**
Elemental composition of leachate #1.</caption><thead><tr><th>Element</th><th>Al</th><th>Ca</th><th>Fe</th><th>K</th><th>Mg</th><th>Mn</th><th>Ni</th><th>S</th><th>Zn</th></tr></thead><tbody><tr><td>Concentration (mg L<sup>-1</sup>)</td><td>170 ± 6</td><td>660 ± 24</td><td>500 ± 19</td><td>500 ± 46</td><td>4 760 ± 380</td><td>20 ± 1.3</td><td>8 790 ± 216</td><td>144 000 ± 13,112</td><td>30 ± 3.6</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>Fig. 1. Comparison of experimental rejection rates of boric acid in leachates #1 (♦), #2(◊), #3(■), and #4(□) with the theoretical rejection rates with an average pore size of 0.7 nm (–) and 1.1 nm (---).</figcaption></figure>

- using the same method to determine the average pore size of the membrane with PEG, it was possible to determine that an average pore size radius of 1.1 nm was at least necessary to reach a rejection rate of 3 %.

- this fact is confirmed by comparing the boric acid rejection rate of the first four leachates: the first one has a lower rejection rate, which can be matched by simulating the rejection considering an average pore size of 1.1 nm, while the rejection rate of the other leachates fit to a simulation considering an average pore size of 0.7 nm.

Therefore, this study considers that the average membrane pore size to be 1.1 nm for the leachate #1, even though it may be higher than this value. The value of the ratio ($\Delta x/A_k$) was set at $20.1 \times 10^{-5}$ to take the new average pore size into consideration.

### 3.4. Leachate filtration

Each leachate was filtered following the same procedure:

- before any experiment, the NF membrane was cleaned with ultra-pure water at 25 °C for 1 h at 15 bars. This method has proven effective thanks to the monitoring of successive pure leachate filtration and the reproducibility of the element rejection rate and the permeate flow rate.

- 3 L of leachate was then placed in the feed tank, and passed through the membrane for 1 h, connecting the retentate and permeate to the tank;

- once steady state reached, a pressure of 15 bars was applied and the leachate was collected in a 50 mL flask to monitor the permeate flowrate;

- to lower the permeate flux, the pressure was reduced using a PVDF valve;

- to test the reproducibility of the results, the experiments were reproduced twice;

- between each variation of leachate and each variation of pressure, the membrane was washed to avoid contamination using ultra-pure water for 1 h.

The permeate and feed solution were then analysed by ICP-OES ICAP

6300 DUO, to determine the experimental rejection. The compositions of these solutions were used to determine the concentration of the retentate.

## 4. Results and discussion

### 4.1. Experimentally obtained divalent cation rejection rate

The experimental results show that, up to leachate #4, the decrease in ionic strength led to an increase in the Ni rejection rate, from 35 % for leachate #1 to 85 % for #4 (Fig. 2). If ionic strength continued to decrease, the rejection rate decreased, reaching 68 % for leachate #6. The same behaviour was observed for Mg and Fe: the rejection rate was highest for leachate #4. For leachate #1, the rejection rate for Fe (40 %) was slightly higher than for Ni (35 %) and Mg, but this difference faded away as ionic strength decreased. In all experiments, no difference in pH was recorded between the permeate and the retentate, indicating that $H^+$ ions were not rejected by the membrane.

The filtration results showed that the decrease in ionic strength maximised rejection rates until an optimum was reached. This optimum was 85 % rejection for Ni, Mg and Fe, achieved at an ionic strength of $0.37 \text{ mol kg}^{-1}$. As this study focuses on solution with high salt concentration and low pH, leachates #5 and #6 were not studied further.

The bars representing uncertainties have been added to Fig. 2. They correspond to confidence intervals (Student's t-distribution) and show that the differences observed are indeed significant. For all figures, the confidence intervals remain within the + or - 6 % range.

The relationship between pressure and flowrate by membrane surface is shown in Supplementary material S4. A slight decrease in hydraulic productivity was observed for the most concentrated solution, but it stabilized for solutions #4, #5, and #6. This change can be explained by the viscosity of these solutions.

### 4.2. Assessing the contribution of different exclusion phenomena

In this section, the importance of the various exclusion phenomena is assessed, by comparing the experimental curves with those obtained with the SEDE transport model.
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Experimental rejection rates of nickel, magnesium and iron for leachates #1 (♦), #2(◊), #3(■), #4(□), #5(●) and #6 (○) at ionic strengths from 0.15 to 3.20mol kg⁻¹. The error bars are confidence intervals (Student's t-distribution).</figcaption></figure>

### 4.2.1. Steric hindrance

Steric hindrance theoretically occurs independently of ionic strength since it is linked to pore size. For leachate #1, only steric hindrance was considered. To assess the impact of pore swelling on the results, Fig. 3a compares the experimental rejection rates of Ni, Mg and Fe with the simulated results without pore swelling and with an average pore size adjusted from 0.7 (mean pore radius obtained with PEG) to 1.1 nm.

If steric hindrance alone was considered, the calculated rejection rates were higher than those measured with higher ionic strength solutions. The differences were 33, 35 and 45 % respectively for Ni, Fe and Mg in leachate #1 (Fig. 3a).

The addition of a pore swelling effect greatly improved the simulation: the differences between the calculated and experimental rejection rates were reduced to 9, 8 and 19 % respectively for Ni, Fe and Mg. For leachates #2 and #3, no pore swelling effect is observed. Fig. 3b and c show the comparison between experimental and calculated rejection rates for Ni, Mg and Fe for these solutions, considering only steric hindrance.

The differences between the experimental and calculated results were 1 % for Ni, 2.5 % for Fe and 6 % for Mg. For this solution, the simulation was considered to provide satisfactory results for the divalent ions observed.

### 4.2.2. Effect related to the ionic strength

The high ionic strength of the solutions acts as a screen for the image force and the Donnan effect. Indeed, for leachates #1, 2 and 3, the variation in Donnan potential between the feed solution and the permeate was negligible (around 0.1 mV) (Fig. 4a). For leachate #4, this difference rose to 20 mV (Fig. 4b).

Because of this increase in the Donnan potential difference (Fig. 4b), the effect linked to the ionic strength of the solution should have a significant impact on the separation. Fig. 5 shows a comparison between a simulation carried out with the effects ionic strength effects and a simulation considering only the effects of steric hindrance.

The image force simulation was performed assuming a membrane dielectric constant of 3, calculated for polyamide membranes [33] and a pore solution dielectric constant of the same value as the bulk solution, i. e. 84 (dielectric constant of 2 M sulfuric acid).

Adding the Donnan and image force effects fixed the difference between the experimental and calculated results at 10, 10 and 11 % respectively for Ni, Fe and Mg. The added effect seemed to increase this difference.

Even at a fairly low ionic strength (0.37 mol kg⁻¹), the Donnan and image force effects appeared to overestimate the rejection rates for the metals analysed. However, the rejection rate with steric hindrance alone was far from the experimental value. The differences were 15, 7 and 3 % for Ni, Fe and Mg respectively. It can be concluded that the SEDE model
<figure><img src="image_6.png" /><figcaption>Fig. 3. Comparison of experimental rejection rates as a function of the permeate flux for nickel (▲), magnesium (◆) and iron (●) with (++++) and without (–) pore swelling, and with (––) speciation for leachate #1 (a), #2 (b) and #3 (c).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Comparison of the variation of Donnan potential of leachates #1 (––), #2 (···) and #3 (–) (a) and #4(–·) (b) as a function of the permeate flux.</figcaption></figure>

alone was not sufficient to calculate the rejection rates for the three metals under these conditions. It was therefore proposed that to study the influence of speciation on the simulations, paying particular attention to the formation of ion pairs, such as NiHSO₄⁻, MgHSO₄⁻, FeHSO₄⁻, NiSO₄, MgSO₄ and FeSO₄ [19]. Their Stokes radius were used in the simulation.

### 4.2.3. Speciation

The speciation of the input solution is shown in Table 3. Only the two major ions for a given species are reported and considered in the

transport modelling, except for Ni, Ni and Mg, where the neutral forms were considered. Several changes in speciation were recorded in the leachates studied:

- SO₄²⁻ became the second most abundant anion, with an ionic strength of 3.22 decreasing to 1.60 mol kg⁻¹. Similarly, the H(SO₄)²⁻/SO₄²⁻ molar ratio decreased as ionic strength increased. For a positively charged membrane, it is therefore understandable that the effects of ionic strength decreased the sulfate rejection rate (over the ionic
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Comparison of Ni(▲), Mg(◆) and Fe(●) experimental rejection rate as a function of the permeate flux with simulations for leachate #4, considering only the steric hindrance phenomenon without (–) and with (---) speciation, and considering Donnan and image force effects without (–) and with (—–) speciation.</figcaption></figure>

strength range studied, as long as the membrane polarity did not change);

- the H₃PO₄/H₂PO₄⁻ ratio decreased, indicating a lower rate of phosphate release for the same reasons;

- for Ni, although the major ion remained Ni²⁺, the NiHSO₄⁻/NiSO₄ ratio decreased with the ionic strength. For Mg, a similar phenomenon was recorded: the MgHSO₄⁻/MgSO₄ ratio was, however, lower than NiHSO₄⁻/NiSO₄;

<table><caption>**Table 3**

Speciation of leachates #1 to 4, in mol kg⁻¹.</caption><thead><tr><th></th><th></th><th>Leachate #1</th><th>Leachate #2</th><th>Leachate #3</th><th>Leachate #4</th></tr></thead><tbody><tr><td></td><td>Ionic strength</td><td>3.22</td><td>1.60</td><td>0.72</td><td>0.37</td></tr><tr><td rowspan="2">Sulfur speciation</td><td>H(SO₄)⁻</td><td>1.34</td><td>6.66 10⁻¹</td><td>3.23 10⁻¹</td><td>1.56 10⁻¹</td></tr><tr><td>SO₄²⁻</td><td>4.55 10⁻²</td><td>3.25 10⁻²</td><td>2.82 10⁻²</td><td>2.00 10⁻²</td></tr><tr><td rowspan="2">Phosphorus speciation</td><td>H₃(PO₄)³⁻</td><td>4.40 10⁻²</td><td>2.21 10⁻²</td><td>1.08 10⁻²</td><td>5.33 10⁻³</td></tr><tr><td>H₂(PO₄)³⁻</td><td>2.86 10⁻⁴</td><td>2.68 10⁻⁴</td><td>2.58 10⁻⁴</td><td>2.21 10⁻⁴</td></tr><tr><td rowspan="3">Nickel speciation</td><td>Ni²⁺</td><td>1.23 10⁻¹</td><td>6.53 10⁻²</td><td>3.26 10⁻²</td><td>1.67 10⁻²</td></tr><tr><td>NiHSO₄⁺</td><td>3.68 10⁻¹</td><td>1.22 10⁻²</td><td>9.83 10⁻³</td><td>4.56 10⁻³</td></tr><tr><td>NiSO₄</td><td>2⁰</td><td>7.75 10⁻³</td><td>7.21 10⁻⁴</td><td>4.25 10⁻³</td></tr><tr><td rowspan="3">Magnesium speciation</td><td>Mg²⁺</td><td>1.88 10⁻¹</td><td>9.89 10⁻²</td><td>4.76 10⁻²</td><td>2.35 10⁻²</td></tr><tr><td>MgHSO₄⁻</td><td>4.55 10⁻¹</td><td>1.51 10⁻²</td><td>1.36 10⁻²</td><td>7.18 10⁻³</td></tr><tr><td>MgSO₄</td><td>2⁰</td><td>8.54 10⁻³</td><td>1.10 10⁻²</td><td>5.83 10⁻³</td></tr><tr><td>Potassium speciation</td><td>K⁺</td><td>4.36 10⁻²</td><td>2.21 10⁻²</td><td>1.09 10⁻²</td><td>5.46 10⁻³</td></tr><tr><td rowspan="3">Iron speciation</td><td>KHSO₄</td><td>1.38 10⁻³</td><td>3.91 10⁻⁴</td><td>3.51 10⁻⁴</td><td>1.68 10⁻⁴</td></tr><tr><td>Fe</td><td>5.63 10⁻³</td><td>2.29 10⁻³</td><td>1.05 10⁻³</td><td>3.66 10⁻⁴</td></tr><tr><td>(HSO₄)⁺</td><td>3.35 10⁻¹</td><td>2.18 10⁻³</td><td>1.21 10⁻³</td><td>7.14 10⁻⁴</td></tr><tr><td></td><td>Fe²⁺</td><td>3⁰</td><td>1.6 10⁻⁴</td><td>1.80 10⁻⁴</td><td>1.30 10⁻⁴</td></tr><tr><td></td><td>FeSO₄</td><td></td><td></td><td></td><td></td></tr></tbody></table>

- for K, from an ionic strength of 1.60 to 0.72 mol kg⁻¹, the secondary ion changed from KHSO₄ to KSO₄, inducing a higher release rate for K for leachates with an ionic strength of 0.72 mol kg⁻¹ or less;

- Fe exhibited the most significant speciation changes. Indeed, for leachates with ionic strength of 3.22 and 1.60 mol kg⁻¹, the major ion was Fe(HSO₄)⁺, the second being Fe²⁺. The situation was reversed for leachates of ionic strength of 0.72 mol kg⁻¹;

- the neutral forms were predominant in leachate #3 and #4, less in leachate #2 and absent from leachate #1. Their presence contributed to the decrease in the rejection rate, thanks to their higher size and the fact that they were not influenced by the Donnan effect and image forces for leachate #4.

The impact of speciation on the rejection rate is illustrated in Fig. 3. All the chemical forms considered in Table 3 were used in the simulations. The model did not predict any precipitation of any element during the separation, in accordance with experimental observation. The rejection rate for each form was calculated individually and then grouped by elements to compare the simulation with and without speciation.

These results show that adding the speciation calculation to the SEDE code improved simulations for all solutions. However, some simulations were more affected than others:

- for leachates #1 and #2, the Mg rejection rate was marginally impacted. For Ni and Fe, the difference between experimental and calculated results was smaller with speciation than without (45 and 56 % less respectively);

- for leachate #3, the results were more contrasted: the difference increased for Ni but decreased drastically for Mg and Fe.

For leachate #4, it was necessary to model the impact of speciation both on the simulation considering the Donnan and image force effects and on the one with only steric hindrance (Fig. 5). The simulation was considerably improved by taking speciation into account in both cases. However, considering the Donnan and image force effects has led to the best simulation among those presented, with a theoretical/experimental deviation in the rejection rate of 2, 3 and 2 % respectively for Ni, Mg and Fe. This last simulation showed that, although the Donnan and image force effects tend to have a weaker effect at high ionic strength, they should not be neglected when dealing with concentrated leachates in the modelling of reactive transport. Iron speciation being the most varied, and changing greatly between dilutions, unsurprisingly showed more difference than the other elements when simulated by a reactive transport model.
## 5. Conclusion

The SEDE model was applied to investigate the filtration of a hyper accumulator ash leachate by a NF polyamide membrane. Several dilutions were performed to study this solution at different ionic strengths. The simulation without speciation produced results relatively far from the experimental results, except for the filtration of leachate #3. Considering the effect of ionic strength for leachate #4, such as the Donnan and image force effect, led to an overestimation of rejection rates. Modelling of the feed solution speciation improved the simulation of the rejection rates for all the filtered solutions, especially when the Donnan and image force effect were considered. Considering solution speciation of the solution, as global led to a reduction in rejection rate. Consideration of the solution speciation was particularly important when dealing with multi-ionic solution but should not be neglected when dealing with low pH solution, even with a low salt concentration, because the speciation of sulfuric acid and the formation of an ion pair is not trivial, especially if some species in solution are present in multiple chemical forms. For leachates with a higher ionic strength, considering the phenomenon of polarisation concentration, as well as speciation in this layer, can be a promising way of better simulating ion transport.

The rejection rates observed for an agromining leachate pave the way for a membrane process to recycle the acid used in hydrometal-lurgy. Although the ionic strength of a solution is strongly related to membrane performance, it has been observed that this relationship is not linear, and that dilution of the solution is not necessarily synon-yous with better performance. However, no reason was found to explain the decrease in rejection rates after reaching the optimum. It is assumed that the rise in pH may have approached the isoelectric point of the membrane. A study on the structure of these new membranes could help answer this question.

If the ionic strength in the solution changes between the feed and the permeate, they are also likely to vary within the membrane itself. The impact of a variation in ionic strength within the membrane should vary considerably depending on the composition of the stream studied: in the present case, the speciation of the divalent ions could change due to the variation in composition induced by the higher rejection rate of divalent ions, inducing a variation in speciation within the membrane. However, the change could be even more significant if several counterions are present. In the case of industrial leachates using a mixture of different types of acids (e.g., sulfuric acid and hydrochloric acid), the difference in permeance between the main anions present in the stream could lead to a radical change in speciation within the membrane, inducing a varia-tion in the valence and/or size of the elements, which would result in a variation in the overall rejection rate.

Future studies should therefore focus on analyzing compositional changes across the membrane. To achieve this, it will be necessary to incorporate reaction kinetics and determine the residence time of the solution within the pores. Furthermore, the properties of the fluid within the nanometric pores, such as viscosity, will need to be re-evaluated and updated accordingly.

Finally, if it is possible to obtain a satisfactory rejection rate with agromining leachate, the filtration of Acid Mine Drainage (AMD) with high salt concentration, solutions generally treated with neutralising agent, could be handled by a membrane process.

### CRediT authorship contribution statement

**T. Porqueddu:** Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Conceptualization.
**B. Laubie:** Writing – review & editing, Visualization, Validation, Super-vision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Data curation, Conceptualization.
**A. Szymczyk:** Writing – review & editing, Validation, Supervision, Software, Meth-odology, Conceptualization.
**M.O. Simonnot:** Writing – review & edit-ing, Validation, Supervision, Resources, Project administration,

Methodology, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare the following financial interests/personal re- lationships which may be considered as potential competing interests: Marie-Odile Simonnot reports a relationship with Econick that includes: board membership. Baptiste Laubie reports a relationship with Econick that includes: board membership and employment. If there are other authors, they declare that they have no known competing financial in- terests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work is included in the scientific program of the GISFI research consortium (Groupement d'Intérêt Scientifique sur les Friches Industrielles - http://www.gisfi.univ-lorraine.fr). The authors would like to thank the French Ministry of Higher Education, Research and Innovation for funding Thomas Porqueddu's PhD thesis. They also thank Prof. Laurence Muhr for lending them the NF set-up, Steve Pontvianne from SAMPL - LRGP (Université de Lorraine-CNRS - https://lrgp-nancy.cnrs.fr/) and the membrane team from LRGP for their advice and assistance.

## Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.seppur.2025.134757.

### References

[1] E.S.A. Manaa, T.A. Lasheen, Liquid-liquid extraction of sulfuric acid from aqueous sulfate waste solution using Alamine 336/Kerosene/TBP solvent, J. Dispers. Sci. Technol. 37 (2016) 137-143, https://doi.org/10.1080/01932691.2015.1034362.

[2] A. Agrawal, S. Kumari, K.K. Sahu, Liquid-liquid extraction of sulphuric acid from zinc bleed stream, Hydrometall. 92 (2008) 42-47, https://doi.org/10.1016/j.hydromet.2008.01.008.

[3] A.A. Manis, K.H. Soldenhoff, E.M. Ho, P.D. Macintosh, Nanofiltration in hydrometallurgy, in: A.I. Schäfer, A.G. Fane (Eds.), Nanofiltration, first ed., Wiley, 2021, pp. 759-803, https://doi.org/10.1002/9783527824984.ch18.

[4] S. Wu, H. Zhu, Y. Wu, S. Li, G. Zhang, Z. Miao, Resourceful treatment of battery recycling wastewater containing $H_2SO_4$ and $NiSO_4$ by diffusion dialysis and electrodialysis, Membranes 13 (2023) 57, https://doi.org/10.3390/membranes13060570.

[5] L. Li, V.G. Desmane, M.P. Paranthaman, R. Bhave, B.A. Moyer, S. Harrison, Lithium recovery from aqueous resources and batteries: a brief review: a review of the methods to produce lithium and approaches to recycling from end-of-life lithium-ion batteries, Johns. Matthey Technol. Rev. 62 (2018) 161-176, https://doi.org/10.1595/205651317X696676.

[6] E. Cséfalvay, V. Pauer, P. Mizsey, Recovery of copper from process waters by nanofiltration and reverse osmosis, Desalination 240 (2009) 132-142, https://doi.org/10.1016/j.desal.2007.11.070.

[7] S. Chesters, P. Morton, M. Fazel, Membranes and minewater – waste or revenue stream. Proceedings IMWA 2016 (2016) 1-14.

[8] J. López, O. Gibert, J.L. Cortina, Evaluation of an extreme acid-resistant sulphonamide based nanofiltration membrane for the valorisation of copper acidic effluents, Chem. Eng. J. 405 (2021) 127015, https://doi.org/10.1016/j.cej.2020.127015.

[9] D.K. Nordstrom, C.N. Alpers, Negative pH, efflorescent mineralogy, and consequences for environmental restoration at the Iron Mountain Superfund site, California, Proc. Natl. Acad. Sci. 96 (1999) 3455-3462, https://doi.org/10.1073/pnas.96.7.3455.

[10] B. Dold, Evolution of acid mine drainage formation in sulphidic mine tailings, Minerals 4 (2014) 621-641, https://doi.org/10.3390/min4030621.

[11] S.M. Samaei, S. Gato-Trinidad, A. Altaee, The application of pressure-driven ceramic membrane technology for the treatment of industrial wastewaters – a review, Sep. Purif. Technol. 200 (2018) 198-220, https://doi.org/10.1016/j.seppur.2018.02.041.

[12] M.B. Asif, Z. Zhang, Ceramic membrane technology for water and wastewater treatment: a critical review of performance, full-scale applications, membrane fouling and prospects, Chem. Eng. J. 418 (2021) 129481, https://doi.org/10.1016/j.cej.2021.129481.

[13] A.R. Guastalli, J. Labanda, J. Llorens, Separation of phosphoric acid from an industrial rinsing water by means of nanofiltration, Desalination 243 (2009) 218-228, https://doi.org/10.1016/j.desal.2008.04.024.

[14] Lenntech, Duracid series fact sheets. https://www.lenntech.com/Data-sh-eets/GE-Duracid-L.pdf, 2023 (accessed 10 July 2023).
[15] J. Tanninen, S. Platt, A. Weis, M. Nyström, Long-term acid resistance and selectivity of NF membranes in very acidic conditions, J. Membr. Sci. 240 (2004) 11–18, https://doi.org/10.1016/j.memsci.2004.04.006.

[16] J.M. Berland, C. Juery. Les procédés membranaires pour le traitement de l'eau, FNDAE, no 14, 2002.

[17] A. Van Der Ent, A.J.M. Baker, G. Echevarria, M.-O. Simonnot, J.L. Morel, Agromining: farming for metals: extracting unconventional resources using plants, 2nd ed, Springer, 2021.

[18] X. Zhang, B. Laubie, V. Houzelot, E. Plasari, G. Echevarria, M.-O. Simonnot, Increasing purity of ammonium nickel sulfate hexahydrate and production sustainability in a nickel phytomining process, Chem. Eng. Res. Des. 106 (2016) 26–32, https://doi.org/10.1016/j.cherd.2015.12.009.

[19] J. López, M. Reig, O. Gilbert, J.L. Cortina, Increasing sustainability on the metallurgical industry by integration of membrane nanofiltration processes: acid recovery, Sep. Purif. Technol. 226 (2019) 267–277, https://doi.org/10.1016/j. seppur.2019.05.100.

[20] A. Szymczyk, P. Fievet, Investigating transport properties of nanofiltration membranes by means of a steric, electric and dielectric exclusion model, J. Membr. Sci. 252 (2005) 77–88, https://doi.org/10.1016/j.memsci.2004.12.002.

[21] Y. Lanteri, P. Fievet, A. Szymczyk, Evaluation of the steric, electric, and dielectric exclusion model on the basis of salt rejection rate and membrane potential measurements, J. Colloid Interface Sci. 331 (2009) 148–155, https://doi.org/ 10.1016/j.jcis.2008.11.014.

[22] W.R. Bowen, J.S. Welfoot, Modelling the performance of membrane nanofiltration - critical assessment and model development, Chem. Eng. Sci. 57 (2002) 1121–1137, https://doi.org/10.1016/S0009-2509(01)00413-4.

[23] A. Yaroshchuk, Dielectric exclusion of ions from membrane, Adv. Colloid Interface Sci. 85 (2000) 193–230, https://doi.org/10.1016/s0001-8686(99)00021-4.

[24] O. Nir, Y. Oren, M.W. Atsbha, A. Chandra, Y. Geller, M. Chaudhary, L. Monat, P. Singh, R. Zevenhoven, Reactive transport in membrane separation modelling: a

perspective, Chem. Eng. Res. Des. 188 (2022) 342–353, https://doi.org/10.1016/j. cherd.2022.09.054.

[25] J. López, A. Yaroshchuk, M. Reig, O. Gilbert, J. Cortina, An engineering model for solute transport in semi-aromatic polymeric nanofiltration membranes: extension of solution-electro-diffusion model to complex mixtures, J. Environ. Chem. Eng. 9 (2021) 105262, https://doi.org/10.1016/j.jece.2021.105262.

[26] A.I. Cavaco Morão, A. Szymczyk, P. Fievet, A.M. Brites Alves, Modelling the separation by nanofiltration of a multi-ionic solution relevant to an industrial process, J. Membr. Sci. 322 (2008) 320–330, https://doi.org/10.1016/j. memsci.2008.06.003.

[27] Y.I. Dirir, Y. Hanafi, A. Ghoufi, A. Szymczyk, Theoretical investigation of the ionic selectivity of polyelectrolyte multilayer membranes in nanofiltration, Langmuir 31 (2015) 451–457, https://doi.org/10.1021/la5044188.

[28] USGS, PHREEQC version 3. https://www.usgs.gov/software/phreeqc-version-3, 2024 (accessed 4 March 2024).

[29] A. Bani, G. Echevarria, S. Sulçe, J.L. Morel, Improving the agronomy of *alyssum murale* for extensive phytomining: a five-year field study, Int. J. Phytoremediation 17 (2015) 117–127, https://doi.org/10.1080/15226514.2013.862204.

[30] A. Escoda, S. Bouranene, P. Fievet, S. Deon, A. Szymczyk, Dehydration and pore swelling effects on the transfer of PEG through NF membranes, Membr. Water Treat. 4 (2013) 127–142, https://doi.org/10.12989/MWT.2013.4.2.127.

[31] S. Platt, M. Nyström, A. Bottino, G. Capannelli, Stability of NF membranes under extreme acidic conditions, J. Membr. Sci. 239 (2004) 91–103.

[32] C. Labbez, P. Fievet, A. Szymczyk, A. Vidonne, A. Foissy, J. Pagetti, Analysis of the salt retention of a titania membrane using the “DSPM” model: effect of pH, salt concentration and nature, J. Membr. Sci. 208 (2002) 315–329, https://doi.org/ 10.1016/S0376-7388(02)00310-1.

[33] W.R. Bowen, H. Mukhtar, Characterisation and prediction of separation performance of nanofiltration membranes, J. Membr. Sci. 112 (1996) 263–274, https://doi.org/10.1016/0376-7388(95)00302-9.