Ceramic membrane contactors for liquid-liquid
extraction: experimental and modeling aspects

Alain Aprahamian, Jean Philippe Bonnet, Véronique Wernert, Isabelle Beurroies, Renaud Denoyel, Philippe Moulin, Gilles Borda, Denis Ode, Laurent Grospelly, Jérôme Anquetil

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To cite this version:

Alain Aprahamian, Jean Philippe Bonnet, Véronique Wernert, Isabelle Beurroies, Renaud Denoyel, et al. Ceramic membrane contactors for liquid-liquid extraction: experimental and modeling aspects. Journal of Membrane Science and Research, 2025, 11, pp.2021955. 10.22079/jmsr.2024.2021955.1655. cea-04865840

HAL Id: cea-04865840

https://cea.hal.science/cea-04865840v1

Submitted on 13 Feb 2026

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Ceramic membrane contactors for liquid-liquid extraction: experimental and modeling aspects

A. Aprahamian<sup>b</sup>, J.-P. Bonnet<sup>d</sup>, V. Wernert<sup>c</sup>, I. Beurroies<sup>c</sup>, R. Denoyel<sup>c*</sup>, P. Moulin<sup>d</sup>, G. Borda<sup>b</sup>, D. Ode<sup>b</sup>, L. Grospelly<sup>c</sup> J. Anquetil<sup>a</sup>.

<sup>a</sup> TAMI Industries, Z.A. Les Laurons – CS 65 – 2611 Nyons Cedex, France

<sup>b</sup> CEA, DEN, F-30207 Bagnols-sur-Cèze, France

<sup>c</sup> Aix Marseille Univ, CNRS, MADIREL UMR 7246, Campus St Jérôme 13397 Marseille cedex 20

<sup>d</sup> Aix Marseille Univ, Centrale Marseille, M2P2 UMR 7340, Equipe Procédés Membranaires (EPM), Europôle de l’Arbois, BP80, Pavillon Laennec, Hall C, 13545 Aix en Provence Cedex, France

<sup>e</sup> SIVA, Z.A. Les Laurons – CS 65 – 2611 Nyons Cedex, France

Abstract

Pertraction is a fluid-fluid extraction process where a solid porous membrane separates two liquids phases but allows their contact inside its porous structure. This process is used here to transfer nitric acid from an aqueous solution to an organic one by using hollow fiber membranes made of ceramic that ensures a high chemical stability. In order to model the process and to predict the functioning conditions, a number of characterization methods are used to determine the material structure parameters and liquid properties parameters that will be then introduced in a mass transfer model. Contact angle, tortuosity, porosity, diffusion coefficients are thus determined by methods specially adapted to this membrane configuration such as capillary rise, liquid-liquid displacement, impedancemetry, diffusion experiments, mercury porosimetry. These membranes are composed of three layers, support, coating and active layer with pore sizes in the range 4 µm, 0.5 µm and 30-200 nm, respectively. After characterization, these membranes are tested for the transfer of nitric acid from an aqueous solution to an organic phase made of a tributylphosphate / dodecane mixture. An in-series model using material structure parameters and diffusion coefficients is developed. It shows a reasonable agreement between measured and calculated transferred flux of nitric acid. It also suggests that, in the studied cases, the liquid-liquid interface is generally situated inside the support layer.

Keywords: pertraction, wettability, tortuosity, diffusion, multilayer

*corresponding author
## 1 Introduction

One of the methods of liquid-liquid extraction, the so called « pertraction » for PERmeation-exTRACTION, is based on the use of a solid porous membrane that separates the two liquid phases but allows their contact inside its porous structure. The role of the membrane, which has here no selectivity, does not act any more as a filter but as a stabilization medium for the contacted phases. In such a process the aqueous and organic phases are flowing on each side of the membrane whereas the transport from a phase to the other occurs by diffusion of the molecules within the pores. The method is applicable to any immiscible couple of liquid as well as to liquid and gas provided the interface can be stabilized in the pores, which means, in the gas-liquid case, a liquid that is non-wetting for the pore walls. In some cases, the membrane can be filled by a liquid immiscible with the two liquids that are on each side of it but are miscible between them [1]. One of the interests of this process is the possibility to scale-up the apparatus without taking into account either the coalescence mechanisms of emulsions or the kinetics of sedimentation that depends on phase densities. The development of this process has been the subject of numerous articles since the 1970s for laboratory scale developments [2-3] and for a smaller number on an industrial scale [4], [5]. Most of the works were realized with organic membranes [6], and more especially in hollow fibers modules that offer numerous possibilities in term of outside dimensions, pore sizes, porosities, high specific area and hydrophobic characters. Many compounds can thus be extracted: organics [1, 7], drugs [8], metals [9], rare-earth elements [10]. For example, in a recent review, pertraction is presented as the most promising method to recover and purify medium-chain fatty acids [11]. However, for applications in domains where very corrosive liquids are used, ceramic membranes may be useful. This is the case, for example, in the spent fuel cycle chemistry [12], but also in rare earth recycling chemistry [13]. The interest of ceramic membranes is also that they do not swell due to organic liquids or molecules [14]. In the present work, the extraction of nitric acid by a tributylphosphate / dodecane mixture (named TPH/TBP in the following) is carried out in hollow fibers specially shaped by TAMI Industries from standard supports and layers. The interest of hollow fibers, that allows to maximize contact surface area, has been recently underlined [15] The performances of such contactor are studied as a function of membrane characteristics, operating conditions and fluids properties. In order to optimize the extraction and to be able to scale-up the procedure, it is necessary to be able to propose a mathematical model of the process that considers the various phenomena involved in the transfer as well as the material structure parameters and the fluid properties [16]. Indeed, the performance and the time stability of the process depends on (i) membrane structure, thickness, porosity, pore size distribution, topology, (ii) fluid characteristics, surface tension, viscosity and (iii) specificity of the chemical system solid/liquid1/liquid2 which defines the wettability properties. This last point is particularly important because the location of the interface inside the pores of the membrane, as illustrated in Fig. 1, depends on pore size distribution, on pressure gradient between the two immiscible phases and on contact angle, which is the result of the affinities of the two liquids for the surface. Theoretically, the liquid/liquid interface will be stabilized in a pore of size $r_p$ if the Laplace-Washburn equation is verified between the contact angle $\theta_{12}$, pressure gradient $\Delta P$ and liquid/liquid interface tension $\gamma_{12}$ [2,3]:

$$ \Delta P = \frac{2 \gamma_{12} \cos \theta_{12}}{r_p} \qquad (1) $$

When the surface is perfectly wetted by one of the phases, i.e. the absolute value $\cos\theta_{12}$ is 1, any pressure gradient lower than $\Delta P = \frac{2\gamma_{12}}{r_p}$ applied to the less wetting liquid will be enough to maintain the stability of the interface location at the entrance of the pore. At the opposite, if the contact angle is close to 90° and consequently $\cos\theta_{12}$ close to 0, the pressure gradient may be very small according to Eq. (1) making the stabilization of the system very delicate. To overcome this difficulty, it is possible to play on either the membrane surface chemistry in order to modify the contact angle, or on the pore size distribution. In the latter case, by building asymmetric multilayer membranes with various pore sizes, it is possible to both maintain the stiffness of the material and have small enough pores such that the Eq. (1) can be verified. This is this method, i.e. to play on the pore size distribution, that has been chosen in the present case because the surface chemistry cannot be easily modified. Another crucial point is the hysteresis of wetting phenomena: the pressures of displacement of a fluid by the other in a pore of given size is not the same during intrusion and extrusion by the less wetting fluid. Consequently, the stability of the position of
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the interface during the process of pertraction may depend on imbibition order by the different phases. The modeling is then based on a resistance in series approach without adjustable parameters. All material and fluid parameters are determined in independent experiments. In this paper, the process model is described in the case of a single tubular membrane used in a countercurrent configuration. Then the materials studied in this work and the methods used for their characterization are presented. In the last part the results of pertraction experiments are compared to the modeling by measuring and calculating transferred flux of nitric acid. The originality of the approach is to show that the contribution to the transport efficiency of each layer of a multilayer membrane can be evaluated and that the position of the liquid-liquid interface can be localized

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## 2. Theory

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The model describes the transfer of a solute from an aqueous phase to an organic phase through a hollow ceramic fiber. The hollow fiber is held in a carter. The aqueous phase circulates in the center of the fiber whereas the organic phase circulates counter currently into the carter. The contact between the two phases takes place only inside the membrane at the interface between the two liquids. The model is piston type with dispersion of the solute in the hollow fiber as well as in the carter and diffusion of the solute within the membrane. The diffusion of the solute is treated by a model of resistances in series (Fig. 1) each layer of the membrane having its own resistance value [2-3]. In the example of Fig. 1, the liquid/liquid interface is localized inside the active layer, the membrane being constituted of three layers: active layer, on the aqueous solution side inside the hollow fiber, precoating and support.

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<caption>Fig. 1. Modeling concentration gradients through the membrane in steady state. The center of the hollow fiber is on the left-hand side and contains the aqueous phase. $R_{j=1,4}$ are radii. $R_i$ corresponds to the position of the liquid/liquid interface: the membrane is filled by aqueous phase on the left-hand side and oil phase on the right-hand side. The lines correspond to the concentration $x$ as a function of $r$, distance from membrane center. The size of each domain is arbitrary.</caption>

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The knowledge of the geometric and topologic data of the fibers in each layer ($j = 1$ to 3, 1 for the active layer, 2 for the precoating and 3 for the support) and of the properties of each of the contacted phases makes it possible to determine the concentrations of solute at any point of the fiber, the carter and the membrane. The overall material transfer coefficient $K_T$ (m.s⁻¹) makes it possible to compare the tests between the different membranes. It takes into account the following parameters: (i) the transfer coefficients of matter in the diffusional films ($k_A$ and $k_B$) accessible by correlations of the literature; (ii) the geometry of the membrane (number of layers, thickness, tortuosity and porosity of each layer); (iii) Constrictivity, a coefficient that translates the brake to the transfer for a given solute / solvent pair and linked to the porosity of the layer, this parameter will be taken equal to 1; (iv) the partition coefficient ($K_D$) of the solute between aqueous phase and solvent; (v) the solvation coefficient ($k_e$) at the aqueous / solvent phase interphase and (vi) the bulk solute diffusion coefficients in each phase. This leads to several equations for the amount transfer through the system in steady state. $X(z,t,r)$ is the concentration of solute at time $t$, height $z$ and position $r$ along the radius of the membrane, $x_A$ indicates that the concentration is in aqueous phase and $x_B$ indicates that it is in oil phase. $D_A$ and $D_B$ are the diffusion coefficients of the probe in aqueous and oil phase, respectively. $\tau_i$ is the tortuosity of layer $i$. $L$ is the length of the membrane. $R_i$ radii are indicated in Fig. 1.
At the level of diffusional film aqueous phase side the flow rate of material (mol.s⁻¹) is given by:

$$q = 2\pi L\epsilon_1 R_1 k_A (x_A - x_A(z, t, R_1)) \qquad (2)$$

In the active layer, in the aqueous phase:

$$q = 2\pi L \frac{\epsilon_1 D_A}{\tau_1 \ln \frac{R_1}{R_2}} (x_A(z, t, R_1) - x_A(z, t, R_2)) \qquad (3)$$

In the active layer, in the oil phase:

$$q = 2\pi L \frac{\epsilon_1 D_B}{\tau_1 \ln \frac{R_2}{R_3}} (x_B(z, t, R_3) - x_B(z, t, R_1)) \qquad (4)$$

In the precoating:

$$q = 2\pi L \frac{\epsilon_2 D_B}{\tau_2 \ln \frac{R_3}{R_2}} (x_B(z, t, R_2) - x_B(z, t, R_3)) \qquad (5)$$

In the support:

$$q = 2\pi L \frac{\epsilon_3 D_B}{\tau_3 \ln \frac{R_4}{R_3}} (x_A(z, t, R_3) - x_A(z, t, R_1)) \qquad (6)$$

At the level of the diffusional film, oil phase side:

$$q = 2\pi L \epsilon_4 R_4 k_B (x_A(z, t, R_1) - x_B) \qquad (7)$$

Eq. (3-6) simply corresponds to the Fick equation applied in cylindrical geometry to the diffusion through a porous medium of porosity $\epsilon_i$ and tortuosity $\tau_i$. The transfer at the liquid/liquid interface has then to be modelled. This is done by considering that the flow of material is proportional to the gradient of concentration $x_A(z, t, R_i) - x_{A*}(z, t, R_i)$ where $x_{A*}(z, t, R_i)$ is the theoretical concentration at the interphase in equilibrium with organic phase. One then has:

$$q = 2\pi L \epsilon_1 R_i k_e (x_A(z, t, R_i) - x_{A*}(z, t, R_i)) \qquad (8)$$

$k_e$ being the transfer constant associated to this process. Considering equilibrium at the interface, it can be stated:

$$x_B(z, t, R_i) = K_d x_{A*}(z, t, R_i) \qquad (9)$$

where $K_d$ is the partition coefficient between the two phases.

Introducing the overall material transfer coefficient $K_T$ by the following equation:

$$q = 2\pi L \epsilon_1 R_1 K_T \left(x_A - \frac{x_B}{K_d}\right) \qquad (10)$$

the ensemble of Eq. (2-10) leads to:

$$\frac{1}{K_T} = \frac{1}{k_A} + \frac{R_1}{\left[ \frac{D_A}{\tau_1 \ln \frac{R_1}{R_2}} \right]} + \frac{R_1}{k_e R_i} + \frac{R_1}{K_d \left[ \frac{D_B}{\tau_1 \ln \frac{R_2}{R_3}} \right]} + \frac{R_1 \epsilon_1}{K_d \left[ \frac{\epsilon_2 D_B}{\tau_2 \ln \frac{R_3}{R_2}} \right]} + \frac{R_1 \epsilon_1}{K_d \left[ \frac{\epsilon_3 D_B}{\tau_3 \ln \frac{R_4}{R_3}} \right]} + \frac{R_1 \epsilon_1}{K_D k_B R_4 \epsilon_4} \qquad (11)$$

# 3. Materials and Methods

## 3.1. Hollow fibers membrane

Hollow fibers made of titania were prepared by using methods already described [17]. A ceramic paste composed of a titania powder, water and organic additives to bind and adjust its plasticity is prepared by mixing and then pushed through a tube-shaped die. The process makes it possible to produce hollow fibers with controlled dimensions, both in diameter and wall thickness. Once extruded and then cut, the fibers are slowly dried in rooms with controlled
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temperature and humidity. The final stage of the process is a sintering which allows, while remaining at a temperature below the melting point of titanium oxide, to weld the TiO2 particles together while maintaining control of the necessary open porosity. Images obtained by SEM (Scanning Electronic Microscopy, JEOL JMS-6340F, Japan) are given in Fig. 2 for one of these membranes. This SEM is a high-resolution scanning electron microscope with a cold field emission source and an in-lens detection. The resolution can reach 1.2 nm at 15 kV and magnifications from x 25 to x 650.000 (at 8 mm WD) are possible. They are constituted three layers: (i) a support with a thickness of around 500 µm and a mean pore size of a few µm, (ii) a second layer, called precoating in the following, with a thickness in the range 30-100 µm and a mean pore size of 1 µm, and (iii) a third layer, called active layer in the following, with variable pore sizes and thicknesses.

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<caption>Fig. 2. Example of the SEM image of a three layers titania membrane</caption>

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## 3.2. Membrane structure parameters

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### Porosity and pore size

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The porosities and pore size distributions of the tubes were obtained by mercury porosimetry [18]. Experiments were done either with the AutoPore IV (Micromeritics, USA) or with the Poremaster (Quantachrome; USA) apparatus. Intrusion and extrusion are carried out after the sample was evacuated. The cumulative intruded volume is measured as a function of intrusion pressure. This latter is transformed in pore size by applying the Laplace Washburn equation with a contact angle of 130°. An example of pore size distribution is given in Fig. 3.

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<caption>Fig. 3. Mercury porosimetry analysis of membrane M3. Differential (dots) and cumulative (squares) volumes versus pore size.</caption>

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The extrusion curves, not shown here, evidence hysteresis as usual and allows to check that the sample is not destroyed by the intrusion pressure since to each intrusion step corresponds an extrusion step. Three steps can be observed in Fig. 3 corresponding to the three layers: support, precoating and active layer. From these curves it is possible to extract the porous volumes of each layer by the difference of cumulative volume between two steps. The reported pore sizes correspond to the maxima of the different peaks of the pore size distribution. The important
parameter for the process modeling is the porosity of each layer. It is calculated from the porous volume obtained for each layer, the fabrication data and the density of material (titania) by the following equation:

$$ \varepsilon_i = \frac{V_{pi} \cdot m_t}{m_i} / \left( \frac{V_{pi} \cdot m_t}{m_i} + 1 / \rho \right) \qquad (12) $$

where $V_{pi}$ is the porous volume of the layer $i$, $m_t$ is the mass of the membrane, $m_i$ the mass of the layer $i$ and $\rho$ the density of the material.

## Tortuosity

The tortuosity of the membranes was determined by conductivity measurements following already described procedures [19], but using a cell adapted to the geometry of the membranes. The porous medium is imbibed by a concentrated electrolyte and the conductivity (or resistance) of the material is measured. The tortuosity is given by:

$$ \tau = \frac{\varepsilon \sigma^{\circ}}{\sigma_{eff}} \qquad (13) $$

where $\sigma^{\circ}$ and $\sigma_{eff}$ are the electrical conductivity of the bulk electrolyte and impregnated material, respectively. The conductivity measurements were carried out in a cell whose temperature was controlled within 0.1 K thanks to a circulating thermostated bath. The electrical resistance measurements were carried out by impedance spectroscopy, from 0.01 Hz to 1 MHz with a MODULAB apparatus, using a home-built conductivity cell constituted from one cylindrical electrode inside the membrane tube and another cylindrical tubular electrode outside the membrane. The amplitude of the exciting signal was fixed to a small value of 20 mV. The real part of the impedance (the resistance) is obtained by graphical extrapolation of the Nyquist diagram to zero imaginary part. The resistance is measured in the presence and absence of the membrane. Considering the relationships between the resistance and conductivity in cylindrical geometry, it can be shown that the tortuosity in this cylindrical configuration is given by:

$$ \tau = \varepsilon \left[ 1 + \frac{R' - R}{R} \frac{\ln \frac{r_e}{r_i}}{\ln \frac{r_i}{r_o}} \right] \qquad (14) $$

where $\varepsilon$ is the membrane porosity, $R$ the resistance measured without membrane and $R'$ with membrane. $r_e$ and $r_i$ are outside and inside radius of the membrane, $r_1$ and $r_0$ are the radius of external and internal electrodes, respectively.

The measurements are done at several ionic strengths to check that the measured tortuosity is independent of it, showing that double layers effects are screened. The measured tortuosity was constant within experimental error for NaCl concentrations between 0.001M and 1M. This apparent tortuosity of the membrane can be related to the tortuosity of each layer by considering a resistance in series model in a cylindrical geometry. For example, using the notations of Fig. 1, it can be written:

$$ \frac{\tau}{\varepsilon} \ln \frac{R_4}{R_1} = \frac{\tau_1}{\varepsilon_1} \ln \frac{R_2}{R_1} + \frac{\tau_2}{\varepsilon_2} \ln \frac{R_3}{R_2} + \frac{\tau_3}{\varepsilon_3} \ln \frac{R_4}{R_3} \qquad (15) $$

Measuring the tortuosity of the membrane at different states of fabrication, support alone, support+precoating, support+precoating+active layer, it is then theoretically possible to deduce that of each layer. Unfortunately, the accuracy of the measurement as well as the reproducibility of the fabrication prevent to determine by this equation the tortuosity of the layers. In fact, due to its thickness and low porosity, the support is the principal contributor to the measured tortuosity. As a consequence, the tortuosity of the sublayer and layer was estimated from the following equation that is valid for aggregates of spheres [19-20] which is reasonable when looking at images of these layers (Fig. 2):

$$ \tau_i = 1 - p \cdot \ln \varepsilon_i \qquad (16) $$
where $\tau_i$ is the tortuosity of the layer $i$ and $\varepsilon_i$ its porosity. The values of the tortuosity of the layers calculated by this equation and that of the support, determined directly on a tube only made of the support, will be used in the model.

## Wettability

As indicated in the introduction, the wetting properties of the material will determine the position of the liquid-liquid interface in the pore structure as well as the stability of functioning. For a given pressure gradient between aqueous and organic phases, the interface will be positioned in pores that verify Eq. (1) but keeping in mind that due to hysteresis of wetting the position of this interface will depend on the order of introduction of the two phases. Consequently, we used two experimental procedures to analyze the wetting properties of the membrane: (i) a capillary rise method at the liquid-liquid interface was developed and (ii) directly on a pertraction set-up the pressure needed to displace a phase by the other through the membrane is determined.

About the capillary rise method, in a first step the porous membrane tube attached to a balance is contacted with the organic phase and the weight is measured as a function of time. After saturation the measured mass gives access to the porosity. In a second step the tube is totally immersed in the oil phase and put into contact with the aqueous phase in a vessel containing the two liquids (the aqueous phase is the denser). The capillary rise of the aqueous phase displacing the organic one is followed by the weight gain versus time.

During the first step the evolution of mass with time is given by:

$$m(t) = S \rho \varepsilon \left[ \frac{r \gamma \cos \theta}{2 \eta \tau} \right]^{1/2} t^{1/2} \quad (17)$$

where S is the section area of the porous material, r is the density of the liquid, $\gamma$ is the surface tension of the liquid, $\eta$ is the viscosity, r is the mean pore size and $\theta$ the contact angle. Classically the experiment is done for a wetting liquid assuming zero contact angle and a partially wetting liquid for which the contact angle is calculated by the following equation:

$$\cos \theta = \left( \frac{P_2}{P_1} \right)^2 \left( \frac{\rho_2}{\rho_1} \right)^2 \frac{\gamma_1 \eta_2}{\gamma_2 \eta_1} \quad (18)$$

where $P_i$ is the slope corresponding to liquid $i$ (obtained by plotting $m(t)$ versus $t^{1/2}$), 2 corresponds to the liquid for which the contact angle is calculated, 1 to the wetting liquid. In the present work during the second step where the aqueous phase replaces the organic phases, the same type of equation as Eq. 16 can be used excepted that here the liquid density is replaced by the difference of density between the two liquids. Then the contact angle in the system HNO$_3$ aqueous solution/TPH is:

$$\cos \theta = \left( \frac{P_{HNO_3}}{P_{TPH}} \right)^2 \left( \frac{\rho_2 - \rho_1}{\rho_1} \right)^2 \frac{\gamma_{TPH}}{\gamma_{TPH/HNO_3}} \frac{\eta_2}{\eta_1} \quad (19)$$

where $P_{HNO_3}/P_{TPH}$ is the slope for capillary rise at the HNO$_3$/TPH interface and $P_{TPH}$ the slope of capillary rise at the TPH/air interface. This assumes that contact angle with TPH is 0. An example of capillary rise at the various interfaces is given in Fig. 4.

## Liquid-liquid displacement

In such experiment, which uses a set-up similar to the one described in the next paragraph for pertraction, the membrane is first imbibed with the aqueous solution and then the oil phase is introduced into the system from the top of the carter in the outside part. The bottom outlet is plugged. This is a continuous experiment. The aqueous phase is first displaced from the outside part of the carter by flowing through the membrane inside the hollow fiber. This is done at low pressure. When all the water has been removed from the carter, the pressure increases since now oil phase must displace water from the support and then from the layers. The pressure at which aqueous phase is displaced can be related to the pore size and contact angle through the Laplace-Washburn equation. An example of the recording is given in Fig. 5 with the description of the different steps. Due to the highly corrosive character of HNO$_3$, only one membrane was tested by this procedure and this liquid. Because the pore size is known here by
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other techniques, the liquid-liquid method is used here to determine the contact angle contrary to what is usually done [21-23], i.e. pore size distribution is obtained assuming a null value of contact angle.

<figure><img src="image_6.png" /></figure>

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<figure><img src="image_7.png" /><figcaption>Fig. 4. capillary rise method at the liquid/liquid interface in the case of membrane M3. Line corresponds to the experimental data. The dotted line is the best fit obtained using Eq. (16). The fit is correct until 3,000s. Beyond, the membrane is saturated by the aqueous phase that has fully displaced the oil phase.</figcaption></figure>

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263 Fig. 5. Displacement of aqueous solution by TPH in support S3 alone and membrane M3 (S3M1-0.14µm). The first step at 0.08 bars is at the same height for membrane and support (see arrows). The second step at 0.28 bars corresponds to either the sub-layer or the active layer.

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267 *Diffusion coefficients*
268 The diffusion coefficient of each pure liquid phase used is necessary to interpret the pertraction test. The value of this coefficient in the aqueous phase used (HNO₃ 1N in water) is known: 0.108 cm²h⁻¹ [24]. To acquire the data in the organic phase, a dedicated setup was designed. The measurement method was validated with the aqueous phase before applying it to the organic phase.

<figure><img src="image_8.png" /></figure>

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273 Fig. 6. Schematic representation of the experimental setup, in aqueous medium (left), and in organic medium (right)
274 - *x can be 30, 50 or 70%

275 For the measurement of the diffusion coefficient of nitric acid in an aqueous medium, the low tank is filled with 1N HNO₃. For the measurement in organic medium, this tank must be filled with TBP (at the desired percentage in the TPH), loaded with a solution of 1N HNO₃. For that, the contact of a volume of TBP (300 mL), and the double volume of acid (600 mL), must be realized. Then, in order to follow the evolution of the acid concentrations along the tube, three sampling points at 5, 10 and 15 cm allow to take 200 µL of solution.

280 The calculation of the diffusivity of nitric acid in infinite aqueous and organic media is based on Fick's second law, in which z and t represent the height and time of measurement, respectively:

$$
\frac{\partial x_A}{\partial t}(z, t) = D_A \frac{\partial^2 x_A(z, t)}{\partial z^2} \qquad (20)
$$

284 In the case of diffusion in a tube initially filled with solute at the concentration xA° and one end of which is held at the constant solute concentration xE°, the solution of the preceding equation is :

$$
x_A(z,t) = (x_A^E - x_A^0) \text{erfc} \left\{ \frac{z}{2\sqrt{D_A \sqrt{t}}} \right\} + x_A^0 \quad (21)
$$

287 Fig. 7 shows the theoretical nitric acid concentration profiles along the water-filled diffusion column. It can be seen that the experimental points (▲) follow the expected theoretical evolution

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<figure><img src="image_9.png" /><figcaption>Fig. 7. Theoretical evolution of diffusion in an infinite aqueous medium and experimental points for the diffusion of nitric acid in water (▲). Sampling is done at two heights of the set-up shown in Fig. 6.</figcaption></figure>

The average value of the diffusion coefficient of 1 N nitric acid in water is 0.120 cm²·h⁻¹, to be compared with the coefficient found in the literature (0.108 cm²·h⁻¹ [24]) which means that the experimental device developed for these measurements is valid and gives realistic orders of magnitude.

<figure><img src="image_10.png" /><figcaption>Fig. 8. Theoretical evolution of the diffusion in infinite organic media and experimental points for the diffusion of nitric acid in organic solvent.</figcaption></figure>

As for the diffusion of nitric acid in aqueous medium, the theoretical evolution of the diffusion in organic medium, for different percentages of TBP, was plotted to check if the experimental points are not too far from the expected curves (cf. Fig. 8). It appears that the results are more dispersed in TBP than in water but remain exploitable.

For each percentage of TBP tested, an average value of the mean diffusion coefficient is calculated. For TBP 30%, it is 0.310 cm²·h⁻¹, 0.219 cm²·h⁻¹ for TBP 50% and 0.171 cm²·h⁻¹ for TBP 70%, which shows the penalizing and known [25] effect of increasing solvent viscosity on the diffusion coefficient. The viscosities of the phases are 1.69, 1.99 and 2.36 mPa·s for TBP 30%, 50% and 70% in TPH, respectively.

Table 1 shows the values of the diffusion coefficients in aqueous and organic infinite media determined experimentally. It appears that contrary to what was expected, it is in the 30% TBP that the diffusion is the fastest and in water that it is the slowest. Indeed, the diffusion coefficient of 1 N nitric acid in 30% TBP is 0.310 cm²·h⁻¹ against 0.120 cm²·h⁻¹ in water, which is 2.5 times lower. However, when the concentration of TBP in the solvent increases by 20%, the diffusion coefficient decreases by 1.5 times on average.
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<caption>Table 1: Diffusion coefficients of HNO₃ in aqueous or organic infinite media.</caption>

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<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">Diffusion coefficients in infinite media determined experimentally (D)</th><th rowspan="2">D<sub>moy</sub><br>(cm<sup>2</sup>h<sup>-1</sup>)</th></tr><tr><th colspan="4">(cm<sup>2</sup>h<sup>-1</sup>)</th></tr></thead><tbody><tr><td></td><td>Water</td><td>0.122</td><td>0.110</td><td>0.112</td><td>0.136</td><td>0.120</td></tr><tr><td rowspan="3">Organic phase</td><td>TBP 30%</td><td colspan="2">0.308</td><td colspan="2">0.319</td><td>0.310</td></tr><tr><td>TBP 50%</td><td colspan="2">0.194</td><td>0.183</td><td>0.281</td><td>0.219</td></tr><tr><td>TBP 70%</td><td colspan="4">0.171</td><td>0.171</td></tr></tbody></table>

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Pertraction

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The set-up used for pertraction experiment is shown in figure 6. The core of the system is the ceramic fiber hold in a glass carter. The aqueous phase ([HNO₃] = 3N) circulates inside the fiber, whereas the organic phase ([TBP] 30% in the TPH) circulates outside; the two phases circulate in countercurrent, with flow rate ratios close to 0.2 between aqueous and organic phases.

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<figure><img src="image_11.png" /></figure>

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<caption>Fig. 9. Pertraction set-up adapted to one tubular membrane</caption>

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At the beginning of the experiment the whole system is filled by the oil phase. Then the aqueous phase is introduced from bottom to top. Pressure is controlled so that no water is drawn in the outside part, oil phase side. It means that the pressure oil side has always to be higher than water side in agreement with wettability properties. Experimentally it was observed that the chosen procedure was the more reproducible without draining of one phase in the other.

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Operating conditions for each membrane are given in Table 2: pressures in the different parts of the system and flow rates of the liquid phases.

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An example of recording of the experiment is given in Fig. 10. The transfer of acidity from aqueous phase to oil phase is followed by refractometry. At the beginning of the experiment there is a steep step on the recording curve followed by a plateau indicated that steady state is reached. After a while, the aqueous flow rate is divided by 2 leading only to a small variation of acidity in the oil phase: it shows that the limiting step to the transfer of acid is probably diffusion.

<figure><img src="image_12.png" /></figure>

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<caption>Fig. 10. Acidity in the oil phase as a function of time</caption>

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## 4. Results and discussions

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Many membranes have been tested and their characteristics, as obtained by the various characterization methods described in the preceding paragraph, are listed in Table 2. They all are 3-layer systems: support / precoating / active layer. There are several sets that can be distinguished by the support: S2, S3... As shown by mercury porosimetry, the support is composed of pores of 3 to 4 µm, and porosities range between 0.2 and 0.33, with reasonably reproducible values inside a set. On each type of support several types of layers were deposited. The precoating is assumed to be the same for all samples: indeed, the mean pore size is 0.72 ± 0.08µm. The porosity of the precoating is around 0.5. The active layers have mean pore sizes ranging from 30nm to 250nm depending on the sample, whereas the porosity is also around 0.5 as an average, but with larger observed variations that can be due to the elaboration method but also to a less accurate measurement on such thin layers.

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For all series, tortuosity measurements were done first on the support alone and then on the complete membrane but also in a few cases (not shown here) on support+precoating. The main conclusion of these measurements, as already underlined in the experimental part, is that the largest contribution to the measured tortuosity is that of support and it was not really possible, within experimental error, to distinguish between a membrane tube made of only support and a membrane tube made of support+layers. Looking at the Table 2, it can be observed that the tortuosity is reproducible from a membrane to the other provided they belong to the same set defined by the support (S2, S3, or S5).

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The parameters given in Table 1 are used in the calculation detailed in the modeling part. However, the position of the liquid-liquid interface inside the membrane, interface *i* in Fig. 1, is not known, which prevents from positioning the concentration gradients. Consequently, the following procedure was used: it is assumed that the position of the liquid-liquid interface is somewhere in the support. In that case the apparent diffusion coefficient *D*<sub>app</sub> is related to
the characteristics of the membranes and the bulk diffusion coefficients in aqueous solution $D_A$ and oil $D_B$ by the following equation established in the case of a model of resistances in series, considering the cylindrical geometry:

$$ \frac{\tau}{\varepsilon D_{app}} \ln \frac{R_4}{R_1} = \frac{\tau_1}{\varepsilon_1 D_A} \ln \frac{R_2}{R_1} + \frac{\tau_2}{\varepsilon_2 D_A} \ln \frac{R_3}{R_2} + \frac{\tau_3}{\varepsilon_3 D_A} \ln \frac{R_i}{R_3} + \frac{\tau_3}{\varepsilon_3 D_B} \ln \frac{R_4}{R_i} \quad (22) $$

where $\tau$ is the apparent tortuosity of the whole membrane. This is equivalent to Eq (11) after neglecting first and last term due to diffusional films. Using Eq. (22), $D_{app}$ can be plotted as function of the liquid-liquid interface position in the membrane as shown in Fig. 11, whereas using Eq. (11) $D_{app}$ can be calculated from experimental data. Values are given in Table 2. As described in previous paragraph, the values of $D_A$ and $D_B$ are 0.112 and 0.3 cm²h⁻¹, respectively.

<figure><img src="image_13.png" /><figcaption>Fig. 11. Apparent diffusion coefficient (cm²h⁻¹) in the membrane as a function of liquid-liquid interface position in the case of the membrane M1. Aqueous phase is on the left. From left to right, the vertical axis corresponds to the surface of the active layer, the first vertical dot lines to the interface between active layer and coating, and the second one to the interface between coating and support.</figcaption></figure>

By comparing the range of values of $D_{app}$ given by the curve of Fig. 11 with that calculated from experimental data given in Table 2, which mainly range between 0.13 and 0.24 cm²h⁻¹, it is probable that the liquid-liquid interface is positioned inside the support, since the part of the curve of Fig. 11 situated inside the support ranges between 0.1 and 0.28 cm²h⁻¹. Only the membrane M8 has a value, 0.28 cm²h⁻¹, which is exactly at the limit between support and coating. Fig. 11 corresponds to membrane M1, but all membranes lead to approximatively the same graph and conclusions. It also shows that the contribution of the diffusional film to the transport limitation is negligible, as expected from the various tested correlations. A surprisingly information also brought by Fig. 11 is that the influence of coating and active layer on the whole diffusional process is small. In fact, these layers have a higher porosity than the support, a weak tortuosity and a thin thickness: consequently, their contribution to the overall resistance is very small. Moreover, the size of HNO₃ molecule is small as compared to the pore size whatever the layer which render frictional effects negligible [26].

In the pertraction procedure used here, the membranes are initially wetted by oil phase whereas the aqueous phase is flowed in a second time in the inner channel. The contact angle that is representative of this situation is the one calculated by the capillary rise method where the membrane tube is first immersed in organic phase and then put into contact with the aqueous phase.
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The values of contact angle obtained vary between 50 and 80° (Table 2), showing that the aqueous phase wets slightly preferentially the membrane. One of the membranes, M6, is totally hydrophilic with a null value of contact angle.

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Applying the Laplace Eq. (1), it corresponds to pressures gradient between oil and aqueous phase ranging between 91 and 24 mbars for pore sizes corresponding to the support, which is effectively the range of pressure gradient applied during the pertraction experiment to keep the interface inside the membrane.

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Using the liquid-liquid displacement method, the pressures needed to displace the aqueous solution from the membrane are much higher as shown in Fig. 5 for the system HNO₃/ solution TPH. This figure shows that it is necessary to apply 0.08 bars to fill the support with oil phase and 0.28 bars to fill the coating. It is not possible to distinguish in our experiments the coating from the active layer since only two steps are generally observed. Some other experiments (not shown here) were done with decane/water system. In that case the pressure to apply to fill support with decane is 0.4 bars, in agreement with the ratio of surface tensions: 0.05 mJ.m⁻² for water/decane interface and 0.011 mJ.m⁻² for HNO3/TPH interface. For example, in the case of membrane M3 of Fig. 5, the apparent contact angle calculated from pore size and pressure by Eq. (1) gives a value of 54° for oil/aqueous solution displacement instead of 78° for aqueous solution/oil displacement (capillary rise). This means much higher pressures than that effectively used should be applied to maintain oil phase in the membrane if the membrane was initially wetted by water. This also indicates that by using the latter procedure (water wet membrane) membranes made only of the support could be used without the need of layers.

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Looking for the possible correlations between HNO₃ rate and the various parameters given in Table 2, the best result is obtained as expected for the plot of HNO₃ rate as a function of the apparent diffusion coefficient in the membrane (Fig. 12). This agrees with the weak influence of flow rate on molecular transfer shown by Fig. 10. Then diffusion appears to be the limiting step for the transfer by this process. Despite the fact that diffusion is the limiting step and that the liquid-liquid interface is apparently inside the support, no correlation was found between transfer rate and tortuosity, porosity or thickness of the support as well as with the ratio $\frac{\epsilon_3}{\tau_3 \ln \frac{R_4}{R_3}}$ which is the last term of Eq. (22) when

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$R_i=R_3$. This is due to the fact that the final result strongly depends on the position of the liquid/liquid interface inside the support which affects the two last terms of Eq. (22). Fig. 11 illustrates that the variation of $D_{app}$ is mainly due to the support. By applying this equation and knowing transfer experimental data ($D_{app}$) and characterization parameters of Table 2, the position can actually be calculated for each membrane, see Table 2, but there is no way to verify the value. This position depends on the contact angle, the operating pressure, the homogeneity of the support and the wetting history. Further investigations are needed to localize this interface but our study has already defined operable working conditions for these membranes.

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<figure><img src="image_14.png" /><figcaption>Figure 12: HNO<sub>3</sub> molar rate as a function of apparent diffusion coefficient of each membrane.</figcaption></figure>

5. Conclusions

Various methods have been applied to characterize hollow fibers in terms of porosity, wettability, pore size and
tortuosity. Some of these methods are new in the field of membranes, notably those given information on
wettability and tortuosity. From the wettability studies it is possible to predict the working conditions of the
pertraction experiments in term of pressure gradient to control and to localize the liquid-liquid interface position in
the membrane. By coupling these methods with a model of transfer based on a resistance in series geometry, it is
possible to predict the capabilities of such membranes to be used for pertraction processes for a given liquid/liquid
system. In the present case, the liquid-liquid interface appears to be located in the support and the limitation to
transfer seems to be mainly due to molecular diffusion. The standard support from TAMI Industries has probably the
main contribution to this limitation due to its low porosity and high tortuosity and because the other layers are thin,
have higher porosity than support and pore sizes that are large as compared to the diffusing species.

Acknowledgements: This work has been supported by ANR-12-RMNR-03 MATEX grant of French National Research Agency (ANR).
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References

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450

451

[1] A.C. Blaga, A. Tucaliuc and L. Kloetzer, Applications of Ionic Liquids in Carboxylic Acids Separation, Membranes, 12 (2022) 771, https://doi.org/10.3390/membranes12080771.

452

453

[2] A. Gabelman et S.-T. Hwang, Hollow fiber membrane contactors, Journal of Membrane Science, 159 (1999) 61-106, https://doi-org.lama.univ-amu.fr/10.1016/S0376-7388(99)00040-X

454

455

[3] E. Drioli, E. Curcio & G. Di Profio, State of the art and recent progresses in membrane contactors, Chemical Engineering Research and Design, 83 (2005) 223-233, https://doi.org/10.1205/cherd.04203

456

457

[4] R. Klaassen, P.H.M. Feron & A.E. Jansen, Membrane contactors in industrial applications, *Chemical Engineering Research and Design*, 83 (2005) 234-246, https://doi.org/10.1205/cherd.04196

458

459

[5] K.K. Sirkar, Membranes, phase interfaces, and separations: Novel techniques and membranes - An overview, *Ind. Eng. Chem. Res.*, 47 (2008), 5250-5266, https://doi.org/10.1021/ie8001952

460

461

462

[6] G. Van Eygen, B. Van der Bruggen, A. Buekenhoudt, P. L. Alconero, Efficient membrane-based affinity separations for chemical applications: A review, *Chemical Engineering and Processing - Process Intensification*, 169 (2021) 108613, https://doi.org/10.1016/j.cep.2021.108613

463

464

465

[7] F. Chemarin, V. Athès, W. Buquet, M. Bedu, F. Allais, A.F. Teixeira, M. Moussa, I.C. Trelea, Predictive modeling and experimental implementation of organic acids in stream recovery by reactive extraction in membrane contactors, *Chemical Engineering Journal*, 431 (2022) 134067, https://doi.org/10.1016/j.cej.2021.134067

466

467

468

[8] M. Bourassi, M. Kárászová, M. Pasichnyk, R. Zazpe, J. Herciková, Vlastimil Fíla 7, J. M. Macak and J. Gaálová, Removal of Ibuprofen from Water by Different Types Membranes, *Polymers*, 13 (2021) 4082, https://doi.org/10.3390/polym13234082.

469

470

471

472

[9] A. M. Cimbru, A. A. K. Klaif Rikabi, O. Oprea, A. R. Grosu, S.-K. Tanczos, M. C. Simonescu, D. Pascu, V.-A Grosu, F. Dumitru and G. Nechifor, pH and pCl Operational Parameters in Some Metallic Ions Separation with Composite Chitosan/Sulfonated Polyether Ether Ketone/Polypropylene Hollow Fibers Membranes, *Membranes*, 12 (2022) 833, https://doi.org/10.3390/membranes12090833.

473

474

475

[10] M. Alemrajabi, J. Ricknell, S. Samak, R. Rodriguez Varela, J. Martinez, F. Hedman, K. Forsberg,* and A. C. Rasmuson, Separation of Rare-Earth Elements Using Supported Liquid Membrane Extraction in Pilot Scale, *Ind. Eng. Chem. Res.* 61 (2022) 18475-18491, https://doi.org/10.1021/acs.iecr.2c03268.

476

477

478

[11] F. Battista and D. Bolzonella, Beyond Anaerobic Digestion: New Perspectives for the Development of a Biorefinery Platform for the Simultaneous Production of Medium-Chain Fatty Acids by Chain Elongation and Biogas from Food Wastes, ACS Sustainable Chemistry and Engineering, 12 (2024) 15294-15306.

479

https://doi.org/10.1021/acssuschemeng.4c05724

480

481

482

483

484

[12] R.S. Herbst, P. Baron, M. Nilsson, (2011). Standard and advanced separation: PUREX processes for nuclear fuel reprocessing, In K. L. Nash, G. J. Lumetta (Eds.), Advanced separation techniques for nuclear fuel reprocessing and radioactive waste treatment. 1st edition. Chapter 6. Woodhead Publishing Series in Energy: Number 2. (pp. 141-175). Cambridge, UK. Woodhead Publishing Limited.

485

486

487

488

[13] Ho-Sung Yoon, Chul-Joo Kim, Kyeong-Woo Chung, Sung-Don Kim, Jin-Young Lee, Jyothi Rajesh Kumar, Solvent extraction, separation and recovery of dysprosium (Dy) and neodymium (Nd) from aqueous solutions: Waste recycling strategies for permanent magnet processing, *Hydrometallurgy*, Volume 165, Part 1, October 2016, Pages 27-43

489

490

491

492

[14] Hong Joo Lee, Chan-gyu Park, In-seol Yeo, Jung Hoon Park, E. Magnone, Successful removal of phenol from industrial wastewater using novel hydrophobic modified ceramic hollow fiber membrane contactors with remarkably high stability, *Journal of Industrial and Engineering Chemistry*, 114 (2022) 402-408, https://doi.org/10.1016/j.jiec.2022.07.029.

493

494

[15] M.A. Kaczorowska, The Latest Achievements of Liquid Membranes for Rare Earth Elements Recovery from Aqueous Solutions—A Mini Review, *Membranes*, 13 (2023) 839. https://doi.org/10.3390/membranes13100839

495

496

497

[16] E. Bringas, M.F. San Román, J.A. Irabien and I. Ortiz, An overview of the mathematical modelling of liquidmembrane separation processes in hollow fibre contactors, *J. Chem. Technol. Biotechnol.*, 84 (2009) 1583-1614, https://doi.org/10.1002/jctb.2231
498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

[17] A. Grangeon and P. Lescoche, Patent, "Support poreux inorganique pour membrane et procédé de Fabrication », Brevet d'invention FR2729584, janvier 1995

[18] H. Giesche, Mercury porosimetry: A general (practical) overview, Particle and Particle Systems Characterization, 23 (2006) 9-19, https://doi.org/10.1002/ppsc.200601009

[19] M. Barrande, R. Bouchet, and R. Denoyel, Tortuosity of porous particles, _Anal. Chem., 79 (2007) 9115-9121, https://doi.org/10.1021/ac071377r

[20] H.L.J. Weissberg, Effective diffusion coefficient in porous media, Appl. Phys., 34 (1963) 2636-2639.

[21] A. Bottino, G. Capannelli, A. Grosso, O. Monticelli and M. Nicchia, Porosimetric Characterization of inorganic membranes, Separation Science and Technology, 29 (1994) 985-999, https://doi.org/10.1080/01496399408005612

[22] L. Germic, K. Ebert, R.H.B. Bouma, Z. Borneman, M.H.V. Mulder, H. Strathmann, Characterization of polyacrylonitrile ultrafiltration membranes, Journal of Membrane Science, 132 (1997) 131-145, https://doi.org/10.1016/S0376-7388(97)00058-6

[23] J.I. Calvo, R.I. Peinador, P. Prádanos, A. Bottino, A. Comite, R. Firpo and A. Hernández, Porosimetric characterization of polysulfone ultrafiltration membranes by image analysis and liquid-liquid displacement technique, Desalination, 357 (2015) 84-92, http://dx.doi.org/10.1016/j.desal.2014.11.012

[24] International critical tables of numerical data, National Academic Press, Washington DC, 5 (1930) 63.

[25] W. Y. Fei, H. J. Bart - Predicting diffusivities in liquids by the group contribution method, Chemical Engineering and Processing-Process Intentsification, 40 (2001) 531-535, http://dx.doi.org/10.1016/S0255-2701(00)00151-3.

[26] V. Wernert, B. Coasne, P. Levitz, K. Long Nguyen, E.J. Garcia, R. Denoyel, Tortuosity of hierarchical porous materials: diffusion experiments and random walk simulations, Chemical Engineering Science, 264 (2022) 118136, https://doi.org/10.1016/j.ces.2022.118136