# Mass transfer investigation in liquid-liquid two-phase flow in a corrugated millichannel

Antoinette Maarawi Chidraoui, Zoé Anxionnaz-Minvielle, Pierre Coste, Nathalie Di Miceli Raimondi, Michel Cabassud

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Antoinette Maarawi Chidraoui, Zoé Anxionnaz-Minvielle, Pierre Coste, Nathalie Di Miceli Raimondi, Michel Cabassud. Mass transfer investigation in liquid-liquid two-phase flow in a corrugated millichannel. Chemical Engineering Journal Advances, 2025, 22, pp.100739. 10.1016/j.ceja.2025.100739. cea-05043039

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# Mass transfer investigation in liquid-liquid two-phase flow in a corrugated millichannel

Antoinette Maarawi Chidraoui<sup>a</sup>, Zoé Anxionnaz-Minvielle<sup>a,* ID</sup>, Pierre Coste<sup>a</sup>, Nathalie Di Miceli Raimondi<sup>b</sup>, Michel Cabassud<sup>b ID</sup>

## ARTICLE INFO

Keywords:

Liquid-liquid mass transfer

Two-phase flow

Selective extraction

Process intensification

Corrugated channel

## ABSTRACT

The present analysis investigates mass transfer phenomena occurring in liquid-liquid two-phase flow within 2D-meandering millichannels of varying dimensions (2, 3, and 4 mm). Enhanced comprehension is facilitated by previous flow visualizations that led to establishing a flow patterns map for these millichannels. The overall mass transfer coefficient ($k_{t,a}$) is experimentally evaluated by measuring the solute concentration transferred between phases, selectively extracted from the organic phase at the reactor outlet. Additionally, to anticipate mass transfer coefficient variations in zigzag channels of various sizes at varying flowrates, aligning with industrial requirements, experimental $k_{t,a}$ results are correlated with observed flow regimes. Consequently, operational conditions and fluid system properties enable flow regime identification from the previously established flow map, facilitating corresponding mass transfer correlation identification. This innovative approach integrates flow study and mass transfer analysis within millireactors.

## 1. Introduction

Researchers have shown keen interest in milli-reactors as part of process intensification techniques, enhancing mass and heat transfer capabilities by maximizing specific surface area through miniaturization and reducing diffusion distances [1]. In addition, they allow to respond to industrial demands, by achieving high throughput and reducing energy loss and operating costs, compared to numbering-up of micro-reactors [2]. Various types of milli-reactors have been developed, some featuring corrugated geometries that induce secondary flows (Dean Vortices), thereby enhancing mixing and heat transfer efficiencies [2-7]. These intensified compact devices are hence used to overcome mass transfer limitations in conventional reactors resulting in low yields and selectivity. In fact, mass transfer between fluids inside the reactor plays a key role in multiphase reactions and, depending on the chemical kinetics, can become the limiting step controlling the chemical reaction. Therefore, the characterization of this transfer is necessary in order to understand and predict the behaviour of the multiphase reaction system when considering the implementation of reactions in these intensified technologies. This subject has been widely addressed for gas-liquid and liquid-liquid flows in microchannels [8-10]. However, very little work is

conducted in millimetric channels and in particular with liquid-liquid systems [7,11-14] that are widely used in different industrial applications such as polymerization [15], catalysis, enzymatic reactions, extraction [16], emulsification [14], chemical and pharmaceutical synthesis [17,18].

The liquid-liquid mass transfer is generally evaluated using the overall mass transfer coefficient $k_{t,a}$ (s⁻¹) calculated according to different models that require the knowledge of the concentration profile within the phases [19]. The experimental work of the study aims at measuring the concentration of the solute in one of the phases at the reactor outlet in order to be able to quantify the transferred quantity and evaluate the overall mass transfer coefficient. A phase separation step is thus necessary at the reactor exit. Due to the reduced residence time in the micro- and millichannels, phase separation must be nearly instantaneous to avoid mass transfer during sampling. Additional residence time at the sampling stage may be non-negligible compared to the residence time in the reactor, which may distort concentration measurements. Therefore, accurate characterization of mass transfer requires ideal phase separation at the reactor outlet. Different methods of phase separation are used and can be found in the literature. Most of them are based either on the gravity effect (decantation) [20-22] or on the wettability of the phases (separation membranes [23,24] and flow

<footnote><sup>a</sup> Univ. Grenoble Alpes, CEA, LITEN, DTCH, Laboratoire Composants et Sytèmes Thermiques, 38000 Grenoble, France</footnote>

<footnote><sup>b</sup> Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France</footnote>

<footnote>* Corresponding author.
E-mail address: zoe.minvielle@cea.fr (Z. Anxionnaz-Minvielle).</footnote>
Nomenclature

<table><tr><td>S<sub>p</sub></td><td>cross-sectional area (m<sup>2</sup>)</td></tr><tr><td>S</td><td>exchange area (m<sup>2</sup>)</td></tr><tr><td>d<sub>h</sub></td><td>hydraulic diameter (m)</td></tr><tr><td>Λ</td><td>Darcy friction factor</td></tr><tr><td>k<sub>L</sub></td><td>mass transfer coefficient (m·s<sup>-1</sup>)</td></tr><tr><td>a</td><td>interfacial area (m<sup>2</sup>·m<sup>-3</sup>)</td></tr><tr><td>E</td><td>extraction coefficient (%)</td></tr><tr><td>L</td><td>channel developed length (m)</td></tr><tr><td>L<sub>s</sub></td><td>straight section length (m)</td></tr><tr><td>L<sub>c</sub></td><td>characteristic length (m)</td></tr><tr><td>Q</td><td>mass flowrate (kg·h<sup>-1</sup>)</td></tr><tr><td>Re</td><td>Reynolds number</td></tr><tr><td>Nu</td><td>local Nusselt number</td></tr><tr><td>Pr</td><td>Prandtl number</td></tr><tr><td>We</td><td>Weber number</td></tr><tr><td>q</td><td>mass flowrate ratio</td></tr><tr><td>R<sub>c</sub></td><td>radius of curvature (m)</td></tr><tr><td>R<sub>ci</sub></td><td>internal radius of curvature (m)</td></tr><tr><td>Dei</td><td>internal Dean number</td></tr><tr><td>c</td><td>concentration (mol·m<sup>-3</sup>)</td></tr><tr><td>V<sub>R</sub></td><td>reactor volume (m<sup>3</sup>)</td></tr><tr><td>A<sub>max</sub></td><td>maximum absorbance</td></tr><tr><td>F</td><td>mass transfer flux (mol·s<sup>-1</sup>)</td></tr><tr><td>J</td><td>mass transfer flux per unit area (mol·s<sup>-1</sup>·m<sup>-2</sup>)</td></tr><tr><td>D</td><td>diffusion coefficient (m<sup>2</sup>·s<sup>-1</sup>)</td></tr></table>

<table><tr><td>u</td><td>velocity (m·s<sup>-1</sup>)</td></tr><tr><td>Sh*</td><td>modified Sherwood number</td></tr><tr><td>m</td><td>partition coefficient</td></tr><tr><td>r</td><td>linear correlation coefficient</td></tr><tr><td>b<sub>1</sub>, b<sub>2</sub></td><td>constants</td></tr><tr><td>z</td><td>coordinate over the channel length (m)</td></tr></table>

Abreviations

<table><tr><td>RhB</td><td>rhodamine B</td></tr><tr><td>KH<sub>2</sub>PO<sub>4</sub></td><td>potassium phosphate</td></tr><tr><td>K<sub>2</sub>HPO<sub>4</sub></td><td>potassium hydrogen phosphate</td></tr><tr><td>2D</td><td>two-dimensional</td></tr><tr><td>UV–Vis</td><td>Ultraviolet-visible</td></tr></table>

Greek symbols

<table><tr><td>Θ</td><td>angle between two straight sections (°)</td></tr><tr><td>μ</td><td>dynamic viscosity (Pa·s)</td></tr><tr><td>ρ</td><td>fluid density (kg·m<sup>-3</sup>)</td></tr><tr><td>τ</td><td>residence time (s)</td></tr></table>

Subscripts

<table><tr><td>eq</td><td>relative to equilibrium</td></tr><tr><td>aq</td><td>relative to aqueous phase</td></tr><tr><td>org</td><td>relative to organic phase</td></tr><tr><td>z</td><td>relative to coordinate over the channel length</td></tr><tr><td>cal</td><td>relative to a calculated value</td></tr><tr><td>exp</td><td>relative to an experimental value</td></tr></table>

splitters [25–27]). The latter allows more precise concentration measurements. In fact, Liu et al. [27] performed liquid-liquid mass transfer experiments for different fluid systems using a flow splitter and a separating funnel to compare the separation efficiency of both methods. In terms of mass transfer, the results showed higher $k_{L}$a when using the settling funnel compared to the flow splitter (relative difference in $k_{L}$a between 8–80 %). This is due to the higher residence time (4–20 min) in the funnel (vs 0.1 s in the flow splitter) which induces additional mass transfer in the separation zone and hence an overestimation of $k_{L}$a.

From the experimental results, a $k_{L}$a correlation can be established and is specific to the studied channel geometry and fluid system. This will help to predict the mass transfer performance of the reactor at various operating conditions, especially during the scaling-up procedure. In the literature, several correlations of the liquid-liquid mass transfer coefficient in microchannels have been developed [21,28–30] but their rangeability remains limited compared to that in gas-liquid flow. Unlike the correlations of the Darcy coefficient ($Λ=b_{1}•Re_{b}^{2}$) and Nusselt number ($Nu=b_{1}•Re_{b}^{2}•Pr^{0.33}$), mass transfer correlations often have different forms and include different dimensionless numbers and parameters. Di Miceli Raimondi et al. [31,32] proposed a model for the mass transfer coefficient $k_{L}$ (m·s<sup>-1</sup>) of the dispersed phase during liquid-liquid flow in microchannels based on 2D numerical simulations. This model requires knowledge of the flow regime in order to estimate the droplet/plug size and interfacial area. In the case of $k_{L}$a (s<sup>-1</sup>) correlations with dimensionless numbers, phase surface velocities and/or channel geometric parameters are required. The velocity of the mixture, of one or of both phases, appears in most correlations, either explicitly or in the Reynolds number. This parameter drives the residence time of the phases in the reactor and plays a role in the flow structure, which affects the mass transfer. In addition, the flow ratio q, as well as the hydraulic diameter $d_{h}$ of the channel, emerge from the correlations, as they also have an effect on the flow regime in the channel, impacting the mass transfer. The latter is greatly influenced by flow regimes, which in turn are affected by operating conditions, channel geometry, and fluid properties [33,34]. Different liquid-liquid flow regimes are identified in

micro/milli-reactors and summarized in the review of Al-Azzawi et al. [35]. Among the observed flow typologies, plug/slug and dispersed/drop flows are the most favourable for mass transfer due to the large contact area between phases and internal recirculation in plugs/slugs/drops [8]. Most of the mass transfer studies in micro- and milli-channels figuring in the literature have targeted plug/slug type flow for ease of post-processing of experimental data [10,36].

In this work, a mass transfer study in the square cross-section zigzag millichannels with U-bends of the "DeanHex" milli-structured heat exchanger/reactor [3] is conducted and provides quantitative results of liquid-liquid mass transfer in millichannel, including scale-up in channel size. $k_{L}$a correlations are then proposed for each flow regime that was identified in our previous work [37], in which various flow regimes were observed across total flowrates ranging from 1.5 to 25 kg h<sup>-1</sup>, including annular/plug, annular, annular/dispersed, and dispersed patterns. A dimensionless flow pattern map was established based on the dispersed phase Weber number as a function of the continuous phase internal Dean number, facilitating the separation of flow regimes in the milli-channels of increasing dimensions. This hydrodynamic study helps understanding the obtained mass transfer results of the present work. The originality of this work resides then in the innovative combination of the flow study and mass transfer analysis within millireactors.

## 2. Materials

### 2.1. Experimental prototypes

This study introduces three prototypes, labelled S2, S3, and S4, which are designed to investigate liquid-liquid mass transfer within zigzag millichannels. These channels have square cross-sections with hydraulic diameters of 2, 3, and 4 mm, as illustrated in Fig. 1. The corrugated flow channel with U-bends shares the same geometry as the one used in Théron et al.’s heat transfer investigation [38] and in our previous hydrodynamic study [37].

The locations and dimensions of the five connections in the
<figure><img src="image_2.png" /><figcaption>Fig. 1. Detailed drawings of the prototypes a) S2, b) S3 and c) S4 [37], where the main channel dimensions (hydraulic diameter, external and internal radius of curvature) are highlighted in red.</figcaption></figure>

prototypes are shown in Fig. 2. As in our previous study [37], connections 1 and 2 are dedicated to the continuous and dispersed flow inlets, respectively. Table 1 provides an overview of the key dimensions of the channels and the location of connections. The ratio of hydraulic diameter to radius of curvature ($d_h/R_c$) is maintained consistently across all prototypes, resulting in an iso-Reynolds flow in the channels that corresponds to an iso-Dean flow.

Each prototype consists in four plates assembled together by means of screws inserted around the zigzag channel preventing leakage, as seen in Fig. 3. The detailed description of each prototype can be found in our previous work [37]. Only the first three rows of the channels are used in the experiments of the present work in order to avoid reaching equilibrium at the outlet. The volume of the tested channels in the prototypes are thus equal to the volume between connections 1–4 in Table 1.

## 2.2. Fluid system

Demineralized water ($\rho = 997 \text{ kg} \cdot \text{m}^{-3}$, $\mu = 1 \text{ mPa} \cdot \text{s}$) and octan-1-ol ($\rho = 826 \text{ kg} \cdot \text{m}^{-3}$, $\mu = 7.29 \text{ mPa} \cdot \text{s}$) are used in the experiments. The interfacial tension between these two liquids is equal to $8.5 \text{ mN} \cdot \text{m}^{-1}$.

For mass transfer characterization, a dye, Rhodamine B (RhB) soluble in both phases, is used [14]. Water and octan-1-ol are almost immiscible. However, the very low solubility of octan-1-ol in water ($0.54 \text{ g} \cdot \text{L}^{-1}$ at $25 \text{ }^{\circ}\text{C}$) is sufficient to bias the measurement of RhB concentration that estimates the solute flux transferred between the two phases. Therefore, to avoid the transfer of octan-1-ol and water,

<table><caption>**Table 1**

Geometric parameters of the studied channels [37].</caption><thead><tr><th>Prototypes</th><th>S2</th><th>S3</th><th>S4</th></tr></thead><tbody><tr><td>Hydraulic diameter d<sub>h</sub> (mm)</td><td>2</td><td>3</td><td>4</td></tr><tr><td>Radius of curvature R<sub>c</sub> (mm)</td><td>1.5</td><td>2.25</td><td>3</td></tr><tr><td>Straight length between two bends L<sub>s</sub> (mm)</td><td>7</td><td>7</td><td>7</td></tr><tr><td>Bends angle Θ (°)</td><td>90</td><td>90</td><td>90</td></tr><tr><td>Channel volume (mL) between two connections</td><td>1–2 0.18</td><td>0.46</td><td>0.91</td></tr><tr><td>1–3</td><td>0.81</td><td>2.03</td><td>4</td></tr><tr><td>1–4</td><td>1.23</td><td>3.10</td><td>6.08</td></tr><tr><td>1–5</td><td>720</td><td>809</td><td>897</td></tr></tbody></table>

octan-1-ol-saturated water and water-saturated octan-1-ol are used in all experiments performed at $25 \text{ }^{\circ}\text{C}$. For this purpose, the pure fluids (water and octan-1-ol) are mixed together at iso-volume using a magnetic stirrer overnight. Then, the mixture is left to rest in a separating funnel for at least three hours. Phase separation of this mixture yields volumes of water saturated octan-1-ol and octan-1-ol saturated water.

RhB is dissolved in octan-1-ol saturated water to have a concentration of $20 \text{ mg} \cdot \text{L}^{-1}$ (± 0.5 %). An OHAUS balance with an accuracy of ±0.1 mg is used to weigh the required mass of RhB. Phosphate buffer solution (1 M), prepared with monobasic potassium phosphate ($KH_2PO_4$) and potassium hydrogen phosphate ($K_2HPO_4$), is added to the aqueous RhB solution to have a pH equal to 7 (a few drops per 1 L of aqueous RhB solution) and to prevent mutual dissolution between the two phases [14].

<figure><img src="image_3.png" /><figcaption>Fig. 2. Location and dimensions of connections in the three prototypes a) S2, b) S3 and c) S4 [37].</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3. Front side photo of the "S4" prototype [37].</figcaption></figure>

RhB concentrations are low ($20 \text{ mg-L}^{-1}$) in order to avoid hydrodynamic flow disturbances caused by mass transfer especially related to Marangoni effects (fluid movements near the interface caused by an interfacial tension gradient induced by mass transfer). Under these conditions, the physical properties of the two-phase system are assumed to be constant during the experiments, equal to those of pure water and pure octan-1-ol.

## 2.3. Experimental setup

The schematic of the mass transfer characterization experimental setup is shown in Fig. 4, where circuits 1 and 2 are for aqueous and organic phases respectively. In each circuit, a thermal bath controls the phase temperature; a magnetic drive pump, with a frequency inverter for the flowrate adjustment, ensures the delivery of the fluid to the prototype; and a CORIOLIS flowmeter measures the mass flow of the fluid.

Fluid temperatures are monitored at the inlet of each reactor phase using thermocouples 1 and 2, and at the reactor outlet using thermocouple 3 to ensure the flow is isothermal. Temperature data is crucial for assessing fluid properties, especially density and viscosity. At the outlet

of the reactor, for the determination of the mass transfer, a selective extraction of the organic phase is realised using a syringe pump (PUMP 33 HARVARD APPARATUS). Then, the extracted sample is analysed using a UV-Visible spectrophotometer (JENWAY model 6405). The phase separation system and the analytical method will be described in more detail in the following sections. At the outlet, a tank is used to collect the mixture. The acquisition of experimental data is carried out using a NI (National Instruments) data acquisition system and the LABVIEW Signal Express software.

## 2.4. Separation system

In the water/RhB/octan-1-ol system [14], RhB is the solute that transfers from water (dispersed phase) to octan-1-ol (continuous phase). As already mentioned in Section 2.1, the reactor channels are made of PC and Teflon which are hydrophobic, so that the aqueous phase is dispersed in the continuous organic phase. The concentration of RhB in the organic phase is determined following the extraction of this phase at the reactor outlet. To avoid unexpected additional mass transfer at the time of sampling, selective extraction of one of the phases should be immediate and performed as close to the reactor outlet as possible. Also, an additional residence time after the reactor exit could be non-negligible compared to the residence time in the millichannel (few seconds) and would induce errors on the measured concentration (overestimation of mass transfer). Therefore, the sampling system shown in Fig. 5 is set up.

At the outlet of the reactor, the stainless steel connector is drilled and a Teflon capillary tube (external diameter = 1.6 mm, internal diameter = 0.5 mm) is inserted so that it touches the wall of the T-connector. The passage is sealed with a Teflon hose coupler. The length of the capillary is reduced as much as possible to minimize dead volumes. The capillary is then connected to a syringe with a luer-lock tip to avoid aspiration problems due to sealing. The syringe is operated by a syringe pump in the filling mode to allow sample collection at a constant rate (0.1 - 0.5 mL-min$^{-1}$). The extraction rate of the syringe pump is adjusted to favour the aspiration of the organic phase only.

The selective extraction of one of the phases is performed based on the affinity of the phases towards different materials. On the one hand,

<figure><img src="image_5.png" /><figcaption>Fig. 4. Schematic of the experimental setup.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5. a) Photograph and b) schematic of the sampling system at the reactor outlet.</figcaption></figure>

as the aqueous phase has a higher affinity for stainless steel, it flows in the outflow direction to the discharge tank (Fig. 5b). The organic phase, on the other hand, has a higher affinity for Teflon and is attracted to the capillary and then drawn into the syringe (Fig. 5b). The extracted organic phase samples are then analysed by UV-Vis spectrometry to determine the concentration of RhB. Note that for this sampling system, the volume of fluids between the reactor outlet and the capillary is 0.17 mL, which represents 14%, 5% and 3% of the reactor volume (between the inlet and the outlet at connection 4) of the three prototypes S2, S3 and S4 respectively. Therefore, the residence time related to sampling (<0.3 s) in the adapted system is small compared to that in the reactor (few seconds) for all tested conditions, which does not induce significant errors on calculated $k_L$ values.

## 2.5. Analytical method

The spectrophotometer is used to obtain an absorbance spectrum over a wavelength range of 450 to 600 nm. The maximum absorbance $A_{max}$ of the measured spectrum for each known concentration of RhB in octan-1-ol (ranging from 2 to 14 mg·L⁻¹) is then determined and plotted against the corresponding concentration to obtain the Beer-Lambert law. It should be noted that in the study by Zhao et al. [14] where the water/RhB/octan-1-ol system is used, the absorbance measurement in the organic phase was done at a wavelength of 554 nm which is within our measurement range.

The obtained relationship between absorbance and concentration $c_{org}$ (mg·L⁻¹) of RhB in organic phase is linear ($r^2=0.994$) in the range of 2 to 10 mg·L⁻¹ and is given by the following equation:

$$A_{max} = 0.196 \times c_{org} \quad (1)$$

Any sample with an out-of-range measurement will be diluted to fall within the linearity range. Zhao et al. [14] obtained a linear law ($r^2=0.998$) for RhB concentration in the organic phase ranging from 2 to 8 mg·L⁻¹.

## 3. Experimental methods

The thermally controlled aqueous (water saturated in octan-1-ol/ RhB) and organic (octan-1-ol saturated in water) solutions are fed into the reactor at 25 °C and flow for a few seconds before the syringe pump is turned on. Table 2 lists the set of experiments performed where the flowrates of the aqueous and organic phases, $Q_{aq}$ and $Q_{org}$, and the ratio between these flowrates $q$ are varied.

$$q = \frac{Q_{aq}}{Q_{org}} \quad (2)$$

Three different values of $q$ are tested (0.5, 1 and 2).

<table><caption>Table 2 Flowrate combinations used for mass transfer characterization experiments.</caption><thead><tr><th>Prototypes</th><th>Q<sub>aq</sub> (kg·h<sup>-1</sup>)</th><th>Q<sub>org</sub> (kg·h<sup>-1</sup>)</th><th>q</th></tr></thead><tbody><tr><td rowspan="3">S2</td><td>1–2</td><td>2–4</td><td>0.5</td></tr><tr><td>1–3</td><td>1–3</td><td>1</td></tr><tr><td>1–3</td><td>0.5–1.5</td><td>2</td></tr><tr><td rowspan="3">S3</td><td>1–2</td><td>2–4</td><td>0.5</td></tr><tr><td>1–3</td><td>1–3</td><td>1</td></tr><tr><td>2–4</td><td>1–2</td><td>2</td></tr><tr><td rowspan="3">S4</td><td>1–2</td><td>2–4</td><td>0.5</td></tr><tr><td>1.6–3</td><td>1.6–3</td><td>1</td></tr><tr><td>2–3</td><td>1–1.5</td><td>2</td></tr></tbody></table>

The extraction rate of the syringe pump is adjusted according to each flowrates pair (between 0.1 and 1 mL·min⁻¹). Preliminary experiments were performed in order to know at what aspiration rate the syringe pump must be set to successfully extract only one phase at the reactor outlet. It was found that, generally, the optimal extraction rate for phase flowrates with ratio $q \le 1$ is equal to 0.5 mL·min⁻¹, except in the case where the flow regime is dispersed. In this case, the aspiration flow rate must be decreased to 0.1 mL·min⁻¹ in order to selectively extract the organic phase. However, when the flowrate of the aqueous phase becomes greater than that of the organic phase ($q > 1$), the aspiration rate of the syringe pump should be set to 0.1 mL·min⁻¹.

Several samples of 3 mL of the organic phase are extracted with the syringe for each pair of phase flowrates. A blank sample consisting in octan-1-ol saturated with water is first analysed in the UV-Vis spectrophotometer to establish a baseline before each sample analysis (using quartz cells: STARNA SCIENTIFIC spectrophotometer cells & certified reference materials, type 9/Q/10, optical path 10 mm, volume 1.4 mL). This protocol is repeated for 9 to 10 different flowrate combinations for each of the 3 reactor prototypes (S2, S3 and S4).

The concentration of RhB measured in the organic phase is then used to calculate the overall mass transfer coefficient $k_L$ according to the following model.

RhB transfers from the aqueous phase to the organic phase and thus the solute transfer flux can be written:

$$J = k_L \cdot (c_{org}^{eq} - c_{org}) \quad (3)$$

Where $J$ is the mass transfer flux of RhB (mol·s⁻¹·m⁻²) per unit area, $c_{org}$ is the solute concentration in the organic phase away from the interface (mol·m⁻³), $c_{org}^{eq}$ is the solute concentration in the organic phase at equilibrium with the aqueous phase (mol·m⁻³), and $k_L$ is the overall mass transfer coefficient in the organic phase (m·s⁻¹). The mass transfer flux $F$ (mol·s⁻¹) is defined from $J$ as follows:
$$F = J \cdot S \qquad (4)$$

Where S (m²) is the exchange surface between the two liquids. S is expressed as a function of the interfacial area "a" (m⁻¹) and the total volume of the reactor V<sub>R</sub> (m³):

$$S = a \cdot V_R \qquad (5)$$

Therefore, the mass transfer flux F (mol·s⁻¹) is equal to:

$$F = k_L a \cdot (c_{\text{org}}^{\text{eq}} - c_{\text{org}}) \cdot V_R \qquad (6)$$

Thus, the mass balance in the organic phase over an element of length dz (m) (Fig. 6) can be written as follows:

$$c_{\text{org},z+\text{dz}} \cdot Q_{\text{org},z+\text{dz}} = c_{\text{org},z} \cdot Q_{\text{org},z} + \text{dF} \qquad (7)$$

$$\text{dF} = k_L a \cdot (c_{\text{org},z}^{\text{eq}} - c_{\text{org},z}) \cdot \text{d}V_R \qquad (8)$$

$$\text{d}V_R = S_p \cdot \text{d}z \qquad (9)$$

Where S<sub>p</sub> (m²) is the square section of the reaction channel.

$$c_{\text{org},z+\text{dz}} \cdot Q_{\text{org},z+\text{dz}} = c_{\text{org},z} \cdot Q_{\text{org},z} + k_L a \cdot (c_{\text{org},z}^{\text{eq}} - c_{\text{org},z}) \cdot S_p \cdot \text{d}z \qquad (10)$$

c<sup>eq</sup><sub>org,z</sub> is the solute concentration in the organic phase at equilibrium with the solute concentration in the aqueous phase away from the interface c<sub>aq,z</sub> (mol·m⁻³) and thus can be replaced by:

$$c_{\text{org},z}^{\text{eq}} = m \cdot c_{\text{aq},z} \qquad (11)$$

Where “m” is the partition coefficient that is calculated as follows:

$$m = \frac{c_{\text{org}}^{\text{eq}}}{c_{\text{aq}}^{\text{eq}}} \qquad (12)$$

Where c<sup>eq</sup><sub>org</sub> and c<sup>eq</sup><sub>aq</sub> are the concentrations of RhB in the organic and aqueous phases at equilibrium. By stirring 20 mg/L RhB aqueous solution and 1-octanol for an hour and then letting the mixture rest for three hours before UV-Vis analysis (measurements at a constant temperature of 25 °C), the partition coefficient was found to be 4.5. This value is in the same order of magnitude as that obtained by Zhao et al. (m = 3.19) [14].

By rearranging Eq. (10) and assuming that the volume flowrates of the phases are constant along the channel, the following equation is obtained:

$$\frac{\text{d}c_{\text{org}}}{\text{d}z} = \frac{k_L a \cdot S_p}{Q_{\text{org}}} \cdot (\text{m} \cdot c_{\text{aq},z} - c_{\text{org},z}) \qquad (13)$$

In order to integrate the above equation, an expression is needed for c<sub>aq,z</sub> as a function of c<sub>org,z</sub> and thus a mass balance for RhB in the reactor is made:

$$Q_{\text{org}} \cdot c_{\text{org},z=0} + Q_{\text{aq}} \cdot c_{\text{aq},z=0} = Q_{\text{org}} \cdot c_{\text{org},z} + Q_{\text{aq}} \cdot c_{\text{aq},z} \qquad (14)$$

Therefore, since c<sub>org,z</sub> = 0=0 in the experiments, the concentration of RhB in the aqueous phase c<sub>aq,z</sub> is obtained from:

$$c_{\text{aq},z} = c_{\text{aq},z=0} - \frac{Q_{\text{org}}}{Q_{\text{aq}}} \cdot c_{\text{org},z} \qquad (15)$$

By substituting c<sub>aq,z</sub> into Eq. (13) and integrating it between z = 0 and z = L (i.e., from the inlet of the second phase to the reactor outlet), the overall volumetric mass transfer coefficient k<sub>L</sub>a can be calculated, from the experimentally measured concentration of RhB in the organic phase c<sub>org,z</sub> = L, as follows:

$$k_L a = \frac{Q_{\text{org}}}{(1 + m \cdot \frac{Q_{\text{org}}}{Q_{\text{aq}}}) \cdot S_p \cdot L} \cdot \ln \left( 1 - \frac{1 + m \cdot \frac{Q_{\text{org}}}{Q_{\text{aq}}}}{m \cdot c_{\text{aq},z=0}} \cdot c_{\text{org},z=L} \right) \qquad (16)$$

## 4. Experimental results

Our previous work [37] was focused on the hydrodynamics of water/octan-1-ol two-phase flows. Four flow regimes were identified: annular/plug, annular, annular/dispersed, and agitated (emulsion) flows and the description of each type of flow was detailed.

Following the identification of these liquid-liquid flow regimes in the zigzag millichannels, their impact on mass transfer is evaluated in the three millichannels. An experimental study is conducted and provides quantitative results of liquid-liquid mass transfer in millichannels, including when scaling-up the channel size.

In all experiments, the solute (RhB) is initially in the aqueous phase with a concentration equal to c<sub>org,z</sub> = 0=20 mg·L⁻¹ at the reactor inlet. The mass transfer coefficient "k<sub>L</sub>a" (s⁻¹) is calculated from the concentration of RhB measured at the reactor outlet in the organic phase using Eq. (16). Then, the extraction efficiency, E, which describes the ratio between the mass of solute transferred to the maximum transferable mass (at equilibrium), is calculated (17) to ensure that the measurement is made before equilibrium is reached (E < 100 %).

$$E = 100 \cdot \frac{c_{\text{org},z=L} - c_{\text{org},z=0}}{c_{\text{org}}^{\text{eq}} - c_{\text{org},z=0}} \qquad (17)$$

Table 3 groups the results obtained in the three zigzag channels for all tested combinations of two-phase flowrates. L (m) is the developed length of the channel (between the inlet of the aqueous phase and the outlet of the reactor) where the mass transfer between the two phases takes place. The k<sub>L</sub>a values are provided along with their corresponding uncertainty values, calculated as described in the Appendix. As seen in Table 3, the error on k<sub>L</sub>a values is <20 %, which is acceptable and does not affect the conclusions.

### 4.1. Effect of the total mass flowrate and mass flowrate ratio

The total flowrate of the phases, Q<sub>tot</sub>, varies between 1.5 and 6 kg·h⁻¹ and the flowrate ratio, q, of the two phases varies between 0.5 and 2 as shown in Table 3. The analysis of the effect of each of these parameters (Q<sub>tot</sub> and q) on the mass transfer coefficient k<sub>L</sub> (s⁻¹) is detailed in the following.

#### i. On the mass transfer coefficient

Fig. 7 shows the evolution of the mass transfer coefficient k<sub>L</sub>a in the

<figure><img src="image_7.png" /><figcaption>Fig. 6. Schematic of the reactor channel (represented by a block) showing the length element dz to perform the massbalance.</figcaption></figure>
<caption>**Table 3**</caption>

Experimental results in terms of concentrations, mass transfer coefficients, $k_{L,a}$ and extraction efficiencies, E for different phase flowrates in the three studied channels.

<table><thead><tr><th>Prototypes</th><th>d<sub>h</sub> (mm)</th><th>L (m)</th><th>Q<sub>aq</sub> (kg·h<sup>-1</sup>)</th><th>Q<sub>org</sub> (kg·h<sup>-1</sup>)</th><th>q = Q<sub>aq</sub>/Q<sub>org</sub></th><th>c<sub>org,z</sub> = L (mg·L<sup>-1</sup>)</th><th>k<sub>L,a</sub> (s<sup>-1</sup>)</th><th>E (%)</th></tr></thead><tbody><tr><td rowspan="10">S2</td><td rowspan="10">2</td><td rowspan="10">0.263</td><td>1</td><td>2</td><td rowspan="10">0.5</td><td>2.67</td><td>0.0235±0.0039</td><td>16</td></tr><tr><td>1.5</td><td>3</td><td>3.78</td><td>0.0563±0.0103</td><td>23</td></tr><tr><td>2</td><td>4</td><td>5.23</td><td>0.1252±0.0166</td><td>32</td></tr><tr><td>1</td><td>1</td><td>4.36</td><td>0.0195±0.0024</td><td>27</td></tr><tr><td>2</td><td>2</td><td>5.82</td><td>0.0545±0.0067</td><td>36</td></tr><tr><td>3</td><td>3</td><td>8.06</td><td>0.1302±0.0169</td><td>49</td></tr><tr><td>1</td><td>0.5</td><td>5.80</td><td>0.0158±0.0018</td><td>35</td></tr><tr><td>2</td><td>1</td><td>8.73</td><td>0.0426±0.0053</td><td>53</td></tr><tr><td>3</td><td>1.5</td><td>11.79</td><td>0.0876±0.0114</td><td>72</td></tr><tr><td rowspan="10">S3</td><td rowspan="10">3</td><td rowspan="10">0.305</td><td>1</td><td>2</td><td rowspan="10">0.5</td><td>2.63</td><td>0.0089±0.0015</td><td>16</td></tr><tr><td>1.5</td><td>3</td><td>2.34</td><td>0.0116±0.0019</td><td>14</td></tr><tr><td>2</td><td>4</td><td>2.09</td><td>0.0135±0.0022</td><td>13</td></tr><tr><td>1</td><td>1</td><td>4.45</td><td>0.0077±0.0009</td><td>27</td></tr><tr><td>3</td><td>3</td><td>2.77</td><td>0.0129±0.0021</td><td>17</td></tr><tr><td>2</td><td>1</td><td>2.11</td><td>0.0031±0.0005</td><td>13</td></tr><tr><td>3</td><td>1.5</td><td>3.96</td><td>0.0090±0.0011</td><td>24</td></tr><tr><td>4</td><td>2</td><td>5.96</td><td>0.0192±0.0023</td><td>36</td></tr><tr><td>1.25</td><td>2.5</td><td>1.40</td><td>0.0028±0.0004</td><td>9</td></tr><tr><td>1.5</td><td>3</td><td>2.25</td><td>0.0059±0.0010</td><td>14</td></tr><tr><td>2</td><td>4</td><td>3.34</td><td>0.0129±0.0023</td><td>20</td></tr><tr><td rowspan="10">S4</td><td rowspan="10">4</td><td rowspan="10">0.326</td><td>1.6</td><td>1.6</td><td rowspan="10">0.5</td><td>2.24</td><td>0.0028±0.0004</td><td>14</td></tr><tr><td>2</td><td>2</td><td>3.98</td><td>0.0070±0.0008</td><td>24</td></tr><tr><td>3</td><td>3</td><td>5.46</td><td>0.0151±0.0018</td><td>33</td></tr><tr><td>2.5</td><td>1.25</td><td>3.32</td><td>0.0032±0.0005</td><td>20</td></tr><tr><td>3</td><td>1.5</td><td>4.33</td><td>0.0052±0.0006</td><td>26</td></tr></tbody></table>

three tested channels as a function of $Q_{tot}$ for different values of q. Some data points are missing due to technical problems, particularly in Fig. 7b and c; however, trends emerge from the presented experimental data and are discussed below. Additional experimental data points are needed to confirm these trends.

The results indicate that, regardless of the channel diameter, $k_{L,a}$ increases with the total flowrate for a constant ratio q. Both flow rates of aqueous and organic phases were increased in a way to keep q constant. This was also found by Li et al. [39] when increasing $Q_{tot}$ with the two phases flow ratio of 1. Wang et al. [40] attributed this result to a higher surface renewal rate in the case of low liquid flowrate and to the combined effects of both a greater surface renewal rate and a larger mass-transfer area in the case of high flowrates. At higher flowrates, the sharp increase of the kinetic energy of both phases generates more turbulence in the flow. This means a higher Reynolds number and therefore a higher Dean number. Stretching, deformation, and enfolding of the phase interface can occur, which enhances the mass-transfer area and surface renewal rate [39,40]. The turbulence, becoming more and more important with the increase of the mass flowrates of the phases, improves the mixing of the phases. Thus, the mass transfer is intensified and $k_{L,a}$ increases.

Furthermore, Fig. 7 shows that generally by increasing the flow ratio q (at constant total flow rate), the mass transfer coefficient increases, due to the fact that the interfacial area available for mass transfer is larger. However, an opposite tendency is sometimes observed, due to the change of the flow regime by increasing q. For example, for the 3 mm and 4 mm diameter channel, according to the visualizations and the flow patterns map from the experiments [37], at a total flow rate of 3 and 4.5 kg·h⁻¹ respectively, increasing q from 0.5 to 2 led to a decrease in $k_{L,a}$ but the flow regime changes from annular/plug to annular. When passing from $q = 0.5$ to $q = 2$, the superficial velocity of the dispersed phase becomes about 2 times higher than that of the continuous phase, so the kinetic energy of the aqueous (dispersed) phase stabilizes the aqueous core flow and lengthens it. The continuous phase can no longer break the aqueous core flow into plugs and the flow becomes annular. Thus the interfacial area decreases when switching from annular/plug flow to annular flow and therefore the $k_{L,a}$ is reduced. In the case of annular/plug flow, the presence of an octan-1-ol film between the aqueous plug and the channel wall ensures that the entire surface of the plug

participates in the mass transfer. In addition, internal circulation within the liquid plugs and interfacial diffusion between the liquid phases [25, 40] contribute to a higher $k_{L,a}$. The change from annular/plug flow to annular flow is associated with a reduction in interfacial area, which is probably the main cause of a lower $k_{L,a}$.

## 4.2. Effect of the channel diameter

When comparing results at the same total flow rate and same phase ratio, mass transfer coefficient increase with the decrease of the channel diameter (cf. Table 3). In fact, decreasing the channel diameter considerably enhances the specific surface area between the two phases, reduces the liquid-liquid mass transfer distance, and leads to a thinner boundary layer, ultimately improving the mass transfer rate [41]. In addition, to compare the mass transfer performance of the three channels, the mass transfer coefficient, $k_{L,a}$, is plotted as a function of the residence time of the phases in the reactor, $\tau$ (s). The latter is defined by the ratio of the total reactor volume to the total volumetric flowrate. Fig. 8 shows the evolution of $k_{L,a}$ as a function of $\tau$ at different q and for the three channels.

It is observed that the curves of the three channels follow approximately the same trend. At iso-residence time and iso-q, the total phase flowrate is the only variable parameter between the points due to the fact that the measurements are conducted at iso-channel length. The factor that seems to affect the results, turns out to be the flow velocity and thus the flow regime. In fact, Fig. 8 shows that as the residence time increases, the flow switches from one flow regime to another. At iso-velocity and therefore at iso-residence time, the points seem to have the same flow regime despite the difference in channel diameter and the values of $k_{L,a}$ are similar.

This result is interesting when considering a scale-up procedure. Nevertheless, extrapolation laws for the mass transfer coefficient are needed. The established laws in the present work in function of dimensionless numbers are presented in the following paragraph.

## 4.3. Mass transfer correlations

Generally, mass transfer correlations are expressed in the form of the dimensionless Sherwood number that represents the ratio of convective
<figure><img src="image_8.png" /><figcaption>Fig. 7. Effects of $Q_{tot}$ and $q$ on the mass transfer coefficient "k$_{L}$a" in the square cross-section channel of diameters: a) 2 mm, b) 3 mm and c) 4 mm (● for $q = 0.5$, ■ for $q = 1$ and ▲ for $q = 2$), indicating with colours the flow regime observed for each experimental point (orange: annular/plug, blue: annular and yellow: annular/dispersed).</figcaption></figure>

to diffusive mass transport:

$$Sh = \frac{k_L \cdot L_c}{D} \qquad (18)$$

Where $k_L$ (m·s⁻¹) is the mass transfer coefficient, $L_c$ (m) is the characteristic flow length, and D is the diffusion coefficient (m²·s⁻¹).

Nevertheless, the present work proposes a modified Sherwood number, including $k_{L}$a instead of $k_{L}$, since the interfacial area "a" is difficult to separate:

$$Sh^* = \frac{k_L \cdot a \cdot d_h^2}{D} \qquad (19)$$

Where $d_h$ (m) is the hydraulic diameter of the channel.

In the following, the diffusion coefficient D of the solute in the aqueous phase is equal to 3.2·10⁻¹⁰ m²·s⁻¹ (at 25 °C), based on the

<figure><img src="image_9.png" /><figcaption>Fig. 8. Comparison of the mass transfer coefficient $k_{L}$a in the three channels of increasing dimensions: 2 mm (dashed lines), 3 mm (solid lines) and 4 mm (dotted lines) for $q = 0.5$ (●), $q = 1$ (■) and $q = 2$ (▲), indicating with colours the flow regime observed for each experimental point (orange: annular/plug, blue: annular and yellow: annular/dispersed).</figcaption></figure>

correlation proposed by Wilke and Chang [42].

The modified Sherwood number (19) is correlated with the internal Dean number “Dei$_{org}$”, calculated with Eq. (20), and the Weber number “We$_{aq}$”, calculated with Eq. (21), which were used to establish the flow patterns map in our previous work [37]; the ratio of mass flowrates “q” which has an effect on the two-phase flow regime; and which represents the ratio of the inertial forces that stabilize the dispersed aqueous phase to the interfacial tension forces that destabilize it; and the ratio $R_{ci}/L_s$ where $R_{ci}$ (m) is the internal radius of curvature of the channel and $L_s$ (m) is the straight length between bends to take into account the number of bends per unit length of the channel. The radius of curvature destabilizes the flow which re-stabilizes in the straight length between two consecutive bends [3].

$$Dei_{org} = \frac{\rho_{org} \cdot u_{org,in} \cdot d_h}{\mu_{org}} \cdot \sqrt{\frac{d_h}{R_{ci}}} \qquad (20)$$

$$We_{aq} = \frac{\rho_{aq} \cdot u_{aq,in} \cdot d_h}{\gamma} \qquad (21)$$

The correlation of the modified Sherwood number proposed below (22) is established from all the experimental data obtained in the three channels. An optimization method on the "EES" software, which minimizes the root mean square of the relative errors between the experimental values and those predicted by the correlation, has been used.

$$Sh^* = \frac{k_L \cdot a \cdot d_h^2}{D} = 220.7 \cdot Dei_{org}^{0.18} \cdot q^{-0.68} \cdot We_{aq}^{0.63} \cdot \left(\frac{R_{ci}}{L_s}\right)^{0.3} \qquad (22)$$

Fig. 9 compares the experimental Sherwood numbers with those predicted by the correlation (22), distinguishing the observed flow regimes.

Some points are close to the bisector ($y = x$) and therefore are well predicted but some are either overestimated or underestimated (up to 80 % error) by the correlation. As a result, not all of the experimental points can be accurately predicted by the correlation (22). As already discussed, the flow regime is a key determinant of the mass transfer performance between phases. Therefore, in order to successfully predict the mass transfer coefficient, $k_{L}$a, in the corrugated millichannel, a mass transfer correlation for each of the observed regimes in our previous work [37] is proposed.

i. Annular flow
<figure><img src="image_10.png" /><figcaption>Fig. 9. Comparison between experimental and correlation-predicted kL,a values in the three zigzag channels (grey dashed lines represent 20 % error).</figcaption></figure>

Considering the above mentioned parameters and using the "EES" software, the correlation of the modified Sherwood number established with the points where the flow is annular is the following:

$$Sh^* = \frac{k_L a \cdot d_h^2}{D} = 1.84 \cdot Dei_{org}^{3.43} \cdot q^{1.97} \cdot We_{aq}^{-0.41} \cdot \left(\frac{R_{ci}}{L_s}\right)^{3.88} \quad (23)$$

The kL,a values calculated with the correlation (kL,a, cal) are compared with the experimental values (kL,a, exp) in Fig. 10.

A very good agreement between calculated and experimental results is observed (±10 % error), which suggests an accurate prediction of the proposed correlation.

According to the correlation (23), if Dei<sub>org</sub> increases, the Sherwood increases, all other parameters being equal. Indeed, the increase in Dei<sub>org</sub> is linked to an increase in turbulence on the organic phase side. This will destabilize the aqueous phase and deform the interface between the phases. Waves/ripples can be created, which improve the mixing (pulsation effects) and intensify the mass transfer. Thus, the Sherwood number increases. This is also the case when the We<sub>aq</sub> decreases (the inertial forces of the aqueous phase decrease, which destabilizes the aqueous core flow). As for the phase flow ratio, q, its increase means that the flowrate of the aqueous phase increases relative to that of the organic phase. The thickness of the aqueous core flow becomes higher, which increases the interfacial area available for mass transfer. This is why an increase in q enhances the Sherwood. Moreover, according to the correlation, when R<sub>ci</sub>/L<sub>s</sub> increases, the Sherwood increases. Indeed, the straight length between two bends is in this case

<figure><img src="image_11.png" /><figcaption>Fig. 10. Comparison between experimental and correlation-predicted kL,a values in zigzag channels for an annular flow regime (grey dashed lines represent 10 % error).</figcaption></figure>

reduced. At the exit of a bend, the flow is destabilized again by the passage through the next bend. This generates more turbulence, which promotes mixing and intensifies the mass transfer.

### i. Annular/plug flow

The correlation obtained for this flow regime is the following:

$$Sh^* = \frac{k_L a \cdot d_h^2}{D} = 5.54 \cdot 10^{-3} \cdot Dei_{org}^{3.10} \cdot q^{2.90} \cdot We_{aq}^{-0.58} \cdot \left(\frac{R_{ci}}{L_s}\right)^{0.14} \quad (24)$$

A comparison between the calculated kL,a with the correlation (kL,a, cal) and the experimental values (kL,a, exp) is shown in Fig. 11.

In this figure, all points are within an error of ±20 % (average error is 12 %), showing good agreement between the experimental data and the values calculated by the proposed correlation (24).

As We<sub>aq</sub> decreases, since our fluid system is unchanged, the interfacial tension dominates the inertial force of the aqueous phase which stabilizes the plugs. This intensifies the mass transfer since the interfacial area becomes larger and thus Sherwood number increases. On the other hand, if Dei<sub>org</sub> increases, the inertia of the continuous phase will be sufficient to destabilize the interface, break the aqueous core flow and generate plugs. This also improves the mass transfer. Regarding the flow ratio of the phases, its increase means the increase of the velocity of the aqueous phase which is present as plugs. This intensifies the internal recirculations in the plugs, whose effect, coupled with the increase in the interfacial area, improves the mass transfer and therefore the Sherwood is enhanced. Regarding the R<sub>ci</sub>/L<sub>s</sub> term, the same trend as in the case of annular flow is obtained but with a less important effect. This is consistent with the fact that the effect of bends (generation of Dean vortices) on mass transfer is less important in a plug flow than in an annular flow (exponent of 0.14 and 3.88 respectively). Indeed, in plug flow, the transversal mixing in the channel is governed by the plugs.

It should be noted that, despite the negative exponent of We<sub>aq</sub> in Eqs. (23) and (24), an increase in We<sub>aq</sub> leads to an increase in Q<sub>to</sub><sup>k</sup> due to the increase in Dei<sub>org</sub>—which has a significantly higher exponent than We<sub>aq</sub>—at constant q. Consequently, kL,a increases, as observed in Fig. 7.

### i. Annular/dispersed flow

Similarly, a correlation of the modified Sherwood is established with the experimental points for which the flow is annular/dispersed.

$$Sh^* = \frac{k_L a \cdot d_h^2}{D} = 13.53 \cdot Dei_{org}^{0.30} \cdot q^{-0.44} \cdot We_{aq}^{0.89} \cdot \left(\frac{R_{ci}}{L_s}\right)^{-0.50} \quad (25)$$

Fig. 12 shows the results predicted by the correlation in function of the experimental results.

The proposed correlation successfully predicts all experimental points with a maximum error of 20 % and an average error of 12 %. It

<figure><img src="image_12.png" /><figcaption>Fig. 11. Comparison between experimental kL,a values and those predicted by the correlation in zigzag channels, for an annular/plug flow (grey dotted lines represent 20 % error).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 12.** Comparison between experimental $k_{L,a}$ values and those predicted by the correlation in zigzag channels, for an annular/dispersed flow (grey dotted lines represent 20 % error).</figcaption></figure>

should be noted that three of these experimental points (with the highest $k_{L,a}$ values) belong to the agitated flow regime. Their $k_{L,a}$ values are well predicted by the proposed annular/dispersed flow correlation (25). It was not possible, with only three points, to establish a correlation specific to the agitated flow.

According to the correlation, increasing $Dei_{org}$ enhances the Sherwood, all other parameters being equal. The inertia of the continuous phase will tear drops out of the dispersed phase. The interfacial area is thus larger and the mass transfer is improved. At a higher $We_{aq}$, the velocity of the dispersed phase increases and in this type of flow, this creates more turbulence, especially at the interface between the two phases, thus increasing the number of drops generated. The interfacial area becomes more important, which increases the Sherwood. When $R_{ci}/L_s$ decreases, the curvature of the channel decreases ($R_{ci}$ decreases). Therefore, the formation of drops will be favoured and the Sherwood increases. When $q$ increases, the Sherwood decreases. This can be explained by the fact that the thickness of the boundary layer on the organic phase side (mass transfer zone near the interface) decreases when $q$ increases. The organic phase, which is the extraction phase, will be more rapidly concentrated in solute, which decreases the driving force (concentration difference) of mass transfer across the interface. This reduces the mass transfer and thus Sherwood decreases. It should be noted that at constant $Q_{tot}$, despite the negative exponent of $q$ in Eq. (25), $k_{L,a}$ increases with $q$ as concluded from results in Fig. 7. This is due to the dependence of $Dei_{org}$ on $u_{org}$ and $We_{aq}$ on $u_{aq}$ and to their corresponding exponents in the equation.

## 5. Conclusions

The current study presents experimental findings on mass transfer characteristics in liquid-liquid two-phase flow through three zigzag millichannels. It emphasizes the direct influence of identified flow regimes on local transfer mechanisms within the channels. Specifically, the flow regime determines the interfacial area for phase exchange, significantly impacting reactor performance, particularly in mass transfer efficiency. The water/rhodamine B (RhB)/octan-1-ol system

was used in the experiments, where RhB transfers from the aqueous phase to the organic phase. Measurements of RhB concentration in the organic phase were used to calculate the mass transfer coefficient $k_{L,a}$ characterizing the mass transfer process in the channels. The values of $k_{L,a}$, obtained in the three corrugated millichannels, range from 0.003 to 0.130 s⁻¹ for total flowrates ranging from 1.5 to 6 kg·h⁻¹ with phase flowrate ratios of 0.5 to 2. Generally, an increase in $k_{L,a}$ was observed with increasing total flowrate and the ratio of "aqueous" to "organic" phase flow rates. However, this is not the case when the flow regime changes from one regime where the interfacial area is large to another one where it is small, thus inducing a decrease in mass transfer.

The effect of channel diameter was also addressed by comparing the mass transfer results in the three millichannels ($d_h=2, 3$ and $4$ mm). A remarkable result is obtained: at isovelocity, the flow regime is conserved, so that the mass transfer coefficient is conserved. Therefore, the flow velocity seems to be the parameter that drives the mass transfer regardless of the size of the corrugated channels. This is interesting for considering channel up-scaling. By increasing the channel diameter from 2 mm to 4 mm, the flow rate is increased six-fold for a constant pumping power for example, achieving industrial-scale flow rates with a 4 mm channel. Consequently, scaling up the reactor to 4 mm at the milliscale already meets industrial demands and reduces the need for numbering-up compared to using millireactors. This facilitates the direct translation of these reactor technologies to industrial applications. In addition, the fluid-structure interactions in the zigzag channels promote phase dispersion and interfacial mass transfer, thereby enhancing the overall mass-transfer performance of the reactor. This effect of the structure of the channel is particularly significant at a large hydraulic diameter, which facilitates the scalability of the reactor to an industrial scale [40].

Nevertheless, in order to predict the mass transfer coefficient in zigzag channels at the industrial scale, the experimental results in terms of $k_{L,a}$ have been correlated and this according to each observed flow regime. The flow regime can be determined from the proposed flow map in our previous work [37] using the operating conditions and the characteristics of the fluids, which will later enable the determination of the relevant Sherwood correlation. This study provides quantitative findings for liquid-liquid mass transfer in millichannels, as well as scale-up to larger channels.

## CRediT authorship contribution statement

**Antoinette Maarawi Chidraoui:** Writing – original draft, Visualization, Validation, Methodology, Investigation. **Zoé Anxionnaz-Minielle:** Writing – review & editing, Supervision, Project administration, Funding acquisition, Conceptualization. **Pierre Coste:** Writing – review & editing, Supervision, Project administration, Funding acquisition. **Nathalie Di Miceli Raimondi:** Writing – review & editing, Supervision, Conceptualization. **Michel Cabassud:** Writing – review & editing, Supervision, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix: Uncertainty Analysis of Experimental $k_{L,A}$ Measurements

As above-mentioned, the experimentally obtained mass transfer coefficient is calculated according to the Eq. (16). Considering $X_i$ as the experimental variables for all the parameters in the Eq. (16) above except $S_p$ and L (channel section and length) ($X_1=C_{org,z} = L$, $X_2=C_{aq}=0$, $X_3=m$, $X_4=Q_{org}$ et $X_5=Q_{aq}$), the derivation of $k_{L,a}$ with respect to each of these experimental variables is performed. The results are as follows:
$$dk_{L}a = dc_{org,z=L}A_{1} + dc_{aq,z=0}A_{2} + dmA_{3} + dQ_{org}A_{4} + dQ_{aq}A_{5} \qquad (26)$$

Where $A_i$ ($i$ = 1 to 5) represents the partial derivative of $k_{L,a}$ with respect to the variable $X_i$:

$$A_1 = \frac{Q_{org}}{S_p \cdot L \cdot \left[ m \cdot c_{aq,z=0} - \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right) \cdot c_{org,z=L} \right]} \qquad (27)$$

$$A_2 = \frac{-Q_{org} \cdot c_{org,z=L}}{S_p \cdot L \cdot c_{aq,z=0} \left[ m \cdot c_{aq,z=0} - \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right) \cdot c_{org,z=L} \right]} \qquad (28)$$

$$A_3 = \frac{-Q_{org} \cdot c_{org,z=L}}{S_p \cdot L \cdot m^2 \cdot \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right) \left[ c_{aq,z=0} - \left( \frac{1}{m} + \frac{Q_{org}}{Q_{aq}} \right) \cdot c_{org,z=L} \right]} - \frac{Q_{aq} \cdot \ln \left[ 1 - \left( \frac{\frac{1}{m} \cdot \frac{Q_{org}}{Q_{aq}}}{c_{aq,z=0}} \right) \cdot c_{org,z=L} \right]}{S_p \cdot L \cdot \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right)^2} \qquad (29)$$

$$A_4 = \frac{m \cdot Q_{org} \cdot c_{org,z=L}}{S_p \cdot L \cdot Q_{aq} \cdot \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right) \left[ m \cdot c_{aq,z=0} - \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right) \cdot c_{org,z=L} \right]} - \frac{\ln \left[ 1 - \left( \frac{1+m \cdot \frac{Q_{org}}{Q_{aq}}}{m \cdot c_{aq,z=0}} \right) \cdot c_{org,z=L} \right]}{Q_{org}^2 \cdot S_p \cdot L \cdot \left( \frac{1}{Q_{org}} + \frac{m}{Q_{aq}} \right)^2} \qquad (30)$$

$$A_5 = \frac{m^2 \cdot Q_{org}^2 \cdot c_{org,z=L} \cdot c_{aq,z=0}}{S_p \cdot L \cdot Q_{aq}^2 \cdot \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right) \left[ m \cdot c_{aq,z=0} - \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right) \cdot c_{org,z=L} \right]} - \frac{m \cdot Q_{org}^2 \cdot \ln \left[ 1 - \left( \frac{1+m \cdot \frac{Q_{org}}{Q_{aq}}}{m \cdot c_{aq,z=0}} \right) \cdot c_{org,z=L} \right]}{Q_{aq}^2 \cdot S_p \cdot L \cdot \left( 1 + m \cdot \frac{Q_{org}}{Q_{aq}} \right)^2} \qquad (31)$$

The uncertainty u on $k_{L,a}$ will then be equal to:

$$u(k_{L,a}) = \sqrt{\sum_{i=1}^{5} A_i^2 \cdot u^2(X_i)} \qquad (32)$$

Where $u(X_i)$ is the uncertainty on the variable $X_i$. In Table 4, the intervals of uncertainty values are provided as percentages for each variable $X$.

<caption>Table 4</caption>

Intervals of uncertainty values for each measured variable.

<table><thead><tr><th>d<sub>h</sub> (mm)</th><th>u(Q<sub>aq</sub>)</th><th>u(Q<sub>org</sub>)</th><th>u(c<sub>org,z = L</sub>)</th><th>u(m)</th><th>u(c<sub>aq,z = 0</sub>)</th></tr></thead><tbody><tr><td>2</td><td>±0.4 – 3.8 %</td><td>±0.4 – 1.8 %</td><td>±2.9 – 10 %</td><td>± 11 %</td><td>± 0.5 %</td></tr><tr><td>3</td><td>±0.3 – 0.8 %</td><td>±0.2 – 0.6 %</td><td>±3 – 10.1 %</td><td></td><td></td></tr><tr><td>4</td><td>±0.4 – 1.6 %</td><td>±0.2 – 0.6 %</td><td>±3 – 10.5 %</td><td></td><td></td></tr></tbody></table>

## Data availability

Data will be made available on request.

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