# Investigation of mixing and gas-liquid mass transfer in a thin-gap bubble column photobioreactor with Newtonian or non-Newtonian liquid phase

Abdallah Haydar, Caroline Gentric, Walid Blel

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Abdallah Haydar, Caroline Gentric, Walid Blel. Investigation of mixing and gas-liquid mass transfer in a thin-gap bubble column photobioreactor with Newtonian or non-Newtonian liquid phase. Chemical Engineering Research and Design, 2025, 223 (1), pp.788-796. 10.1016/j.cherd.2025.10.045. hal-05391498

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# Investigation of mixing and gas-liquid mass transfer in a thin-gap bubble column photobioreactor with Newtonian or non-Newtonian liquid phase

Abdallah Haydar, Walid Blel, Caroline Gentric*

Nantes Université, Oniris, CNRS, GEPEA, UMR 6144, Saint-Nazaire F-44600, France

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

**Keywords:**

Thin-gap bubble column

Newtonian and non-Newtonian liquid phases

Mixing time

Specific interfacial area

Gas-liquid mass transfer coefficient

## ABSTRACT

Thin-gap bubble column photobioreactors (PBRs), with a gap of only a few millimeters, offer the potential for increased microalgal biomass productivity. In this study, mixing and gas-liquid mass transfer were experimentally characterized in a confined bubble column (2 mm thickness), using Newtonian (aqueous glycerol solution) and non-Newtonian (xanthan gum) fluids to mimic *Chlorella vulgaris* suspensions at high concentrations (≈ 42 g·L⁻¹). The aim was to investigate the effects of confinement and liquid rheology on mixing time (t_m) and global gas-liquid mass transfer coefficient (k_L a). Results show poor mixing in viscous and non-Newtonian liquids at low superficial gas velocity (UGsup). t_m increases with the degree of confinement, particularly at low U_Gsup, due to the reduction of agitation capacity with confinement since bubble dynamics is mostly 2D. It is also shown that increasing viscosity has limited impact on k_L a at low U_Gsup, due to compensatory effects between ‘k_L’ and ‘a’ and that gas-liquid transfer efficiency decreases with confinement, potentially due to a lower ‘k_L’, caused by reduced agitation under confinement condition.

## 1. Introduction

In recent years, the cultivation of microalgae has attracted growing interest due to its many promising applications in various fields. Among other designs, bubble columns can be used as efficient photobioreactors (PBRs) - i.e. closed microalgae cultivation systems - because they ensure good mixing and gas-liquid mass transfer. Improving their performance by increasing culture concentration for a given illuminated surface represents a promising path to achieve optimized and ecologically efficient production in such reactors (Pruvost et al., 2017). To obtain high cell concentration and high volumetric productivity (HVP), it is necessary to reduce the PBR thickness - i.e. the dimension in the direction of light gradient - to ensure light availability throughout the reactor volume. Several HVP technologies have been developed, such as the Algodfilm© technology, which operates on the principle of a falling film with a thickness of about 2 mm (Pruvost et al., 2017). This technology has achieved volumetric productivity up to 14 times the one obtained in an open raceway system. These high productivity levels are related to the high specific illuminated area (a ≈ 500 m⁻¹), which is inversely proportional to the culture thickness (Pruvost et al., 2017).

Another HVP technology, based on the bubble column principle, has been studied by Thobie et al. (2017) and Almani et al. (2023). Their technology consists in a confined bubble column with a thickness of 4 mm and a confinement ratio ‘d_b/e’ slightly lower than 1 (i.e., bubble diameter at injection is smaller than the column thickness). Hydrodynamics and gas-liquid mass transfer in various model fluids (Newtonian or non-Newtonian) have been characterized in these studies. It was shown that gas-liquid transfer was not improved within this column compared to conventional ones. This was explained by the absence of thin liquid films - which could intensify transfer (Roudet et al., 2017) - between the bubbles and the walls, but also by the weak dynamics of the bubbles in the gap given the confinement effect.

Intensification can be taken a step further by reducing the column gap up to d_b/e greater than 1, leading to specific two-phase hydrodynamics: 2D movement of bubbles, presence of a liquid film between the bubble and the column walls, increased bubble/wall interactions, bubble deformation and wall-induced constraints affecting the bubble wake (Filella et al., 2015; Ruiz-Rus et al., 2022).

It must be pointed out that at high microalgae concentrations, several specific issues may appear in such confined geometry (Soulies, 2014): (i) drastic modification of the liquid phase properties: viscosity

**Abbreviations:** CMC, carboxymethyl cellulose; HPV, high volumetric productivity; NN, non-Newtonian; PBR, photobioreactor; PIV, particle image velocimetry; PLIF, plane laser-induced fluorescence; PMMA, polymethyl methacrylate; XG, xanthan gum; 2D, Two spatial dimensions.

E-mail address: caroline.gentric@univ-nantes.fr (C. Gentric).

<footnote>* Corresponding author.</footnote>
Nomenclature

<table><tr><td>A<sub>i</sub></td><td>projected area of the contour of bubble 'i' [m<sup>2</sup>]</td></tr><tr><td>a</td><td>specific interfacial area [m<sup>-1</sup>]</td></tr><tr><td>D<sub>ax</sub></td><td>axial dispersion coefficient [m<sup>2</sup>.s<sup>-1</sup>]</td></tr><tr><td>d<sub>b</sub></td><td>bubble diameter at injection [m]</td></tr><tr><td>d<sub>i</sub></td><td>equivalent diameter of bubble 'i' [m]</td></tr><tr><td>D<sub>L</sub></td><td>diffusion coefficient of O<sub>2</sub> in the liquid phase [m<sup>2</sup>.s<sup>-1</sup>]</td></tr><tr><td>e</td><td>column gap [m]</td></tr><tr><td>g</td><td>gravity acceleration [m.s<sup>-2</sup>]</td></tr><tr><td>k<sub>L</sub>a</td><td>volumetric gas liquid mass transfer coefficient [s<sup>-1</sup>]</td></tr><tr><td>L<sub>c</sub></td><td>column width [m]</td></tr><tr><td>H<sub>L</sub></td><td>column height [m]</td></tr><tr><td>P<sub>v</sub></td><td>biomass volumetric productivity [g.L<sup>-1</sup>.j<sup>-1</sup>]</td></tr><tr><td>S<sub>light</sub></td><td>illuminated area of the reactor [m<sup>2</sup>]</td></tr><tr><td>t<sub>exp</sub></td><td>exposure or contact time [s]</td></tr><tr><td>t<sub>m</sub></td><td>mixing time [min]</td></tr><tr><td>U<sub>Gsup</sub></td><td>superficial gas velocity [m.s<sup>-1</sup>]</td></tr><tr><td>V<sub>G</sub></td><td>volume of the gaseous phase [L]</td></tr><tr><td>V<sub>L</sub></td><td>volume of the liquid phase [L]</td></tr></table>

increases with the biomass concentration and the rheological behavior becomes non-Newtonian at biomass concentration greater than 30 g. L⁻¹, (ii) increased bubble coalescence, (iii) increased biofilm development. These phenomena will impact bubble behavior and subsequently mixing and gas-liquid mass transfer, they will also impact light transfer. Indeed, bubble coalescence can reduce the range of homogeneous regime, which is the preferred regime for microalgae cultivation (Almani et al., 2023). Biofilms strongly affect both the quality and the availability of light penetrating the reactor, thus decreasing biomass productivity and significantly increasing cultivation costs (Zeriiouh et al., 2017). Therefore, it is essential to characterize hydrodynamics and gas-liquid mass transfer in this geometry, especially in presence of non-Newtonian liquid phase for the successful application of this bioreactor in the cultivation of microalgae at high cell concentrations.

In general, hydrodynamics in bubble columns can primarily be described by identifying the flow regimes and their transitions (Gourich et al., 2006). Three main flow regimes are observed in conventional unconfined columns: homogeneous regime, transition regime, and heterogeneous regime (Olmos et al., 2003b). Recently, in confined bubble column (2 mm gap), Haydar et al. (2023) characterized flow regimes and their transitions using both Newtonian and non-Newtonian fluids. Their findings revealed different flow regimes and specific bubble interactions compared to classical bubble columns in such confined conditions.

In the literature, there is a limited number of global hydrodynamic studies, in particular dealing with mixing, conducted on confined bubble columns. Among these, Thobie et al. (2017) and Almani et al. (2023) characterized the global hydrodynamics in a 4 mm-gap bubble column (‘d<sub>b</sub>/e’ ≤ 1), using Newtonian fluids (tap water and glycerol solutions at 10 and 50 wt%) and non-Newtonian fluids (aqueous solutions of xanthan gum (XG) at 1 g.L⁻¹ and carboxymethylcellulose (CMC) at 2 g.L⁻¹). These studies highlighted the effect of confinement, and reported similar trends as in classical bubble columns, where mixing time decreases with superficial gas velocity, and increases with viscosity. They also pointed out that mixing in viscous or non-Newtonian liquids was quite poor, particularly in homogeneous and transition regimes compared to water.

Thobie et al. (2017) and Almani et al. (2023) also compared the global gas-liquid mass transfer coefficient, k<sub>L</sub>a, obtained in the 4 mm-gap bubble column to those found in the literature with unconfined bubble columns. They found that k<sub>L</sub>a values are lower under confined conditions than in classical columns, both for Newtonian and non-Newtonian fluids. According to these authors, this result is

explained by the lower terminal velocities observed in a confined column in comparison to a classical one, which increases exposure time and subsequently decreases the mass transfer coefficient ‘k<sub>L</sub>’ and to the absence of thin liquid film between the bubbles and the walls which could intensify gas liquid transfer around a bubble (Almani et al., 2023). They also found that the rheological properties of the liquid phase significantly affect the k<sub>L</sub>a value. Indeed, k<sub>L</sub>a values are much higher for water compared to non-Newtonian and Newtonian viscous fluids.

In the present study, the objective is to further intensify the PBR by reducing the column gap compared to the one used by Thobie et al. (2017) and Almani et al. (2023) and to evaluate potential improvements in gas-liquid mass transfer over these earlier studies particularly due to the presence of liquid films between the bubble and the walls (Roudet et al., 2017). For this, a 2 mm-gap bubble column PBR having a ‘d<sub>b</sub>/e’ ratio greater than 1 is used. Previous investigations of the same 2 mm thin-gap bubble column focused on the respective influences of confinement and rheological properties of the liquid phase on gas holdup, flow regimes, and their transitions (Haydar et al., 2023). The present study aims to provide a comprehensive characterization of mixing and gas-liquid mass transfer in this thin-gap bubble column for a wide range of superficial gas velocities and various flow regimes. The case of Newtonian low viscosity liquid phase (tap water) is compared to viscous Newtonian (aqueous glycerol solution) and non-Newtonian (aqueous XG solutions) fluids. The concentration of glycerol and XG solutions is adjusted to replicate the viscosity of microalgae suspensions at a concentration of 42 g.L⁻¹.

## 2. Material and methods

### 2.1. Experimental setup

This study is conducted in a thin-gap bubble column (Fig. 1) (Haydar et al., 2023). It consists of two polymethyl methacrylate (PMMA) plates, each 5 cm thick. One of the plates is carved to a depth of 2 mm over a surface area of 30 cm wide × 120 cm high. It is filled with the liquid phase up to a height of 97 cm, corresponding to an operational liquid volume of approximately 582 mL. All experiments are conducted with a stagnant liquid phase (no liquid inlet nor outlet). Air is used as the gas phase and is injected through 15 capillaries with an inner diameter of 1.016 mm, evenly distributed at the bottom of the column. Upstream of these capillaries, a homogenization chamber is installed, ensuring the same gas pressure and thus an identical flow rate at each capillary.

The range of superficial gas velocities (U<sub>Gsup</sub>) studied is from 0.002 to 0.040 m.s⁻¹, covering all flow regimes previously identified in this column by Haydar et al. (2023), regardless of the liquid nature. U<sub>Gsup</sub> is defined as the ratio of the injected volumetric gas flow rate to the cross-sectional area of the column. The gas flow rate is precisely adjusted using a mass flow meter / controller (EL-FLOW® - BRONK-HORST®) with a capacity of 1.5 mL.min⁻¹ placed before the homogenization chamber, and connected to a LabVIEW interface for accurate real-time control of the air flow rate.

The front wall of the column is equipped with nine ports (P1-P9) (Fig. 1), which allow the installation of various sensors. For this study, a conductivity probe, used to measure mixing time, and a dissolved oxygen probe, used to determine the gas-liquid mass transfer coefficient, are both installed at port P5, located at the mid-width of the bubble column at a height of 700 mm (Fig. 1).

### 2.2. Physicochemical characterization of model fluids

Model fluids are used to mimic the rheological behavior of the microalgae Chlorella vulgaris at high cell concentrations. To this end, glycerol and xanthan gum (XG) solutions are employed. First, a concentrated glycerol solution (Glycerol, 98 +%, extra pure, SLR, Fisher Chemical) at 50 wt% having a viscosity of 5.7 × 10⁻³ Pa.s (Thobie et al., 2017) is used to study the influence of a viscous Newtonian liquid phase.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Experimental setup of the thin-gap bubble column (2 mm thickness) (adapted from Haydar et al., 2023).</figcaption></figure>

This viscosity is higher than what is typically found in the Newtonian range for microalgae cultures with concentrations up to 30 g.L⁻¹ (Souliès et al., 2013). However, this solution will be studied to observe the effect of increased viscosity on hydrodynamics and gas-liquid mass transfer in a confined bubble column, while remaining in the Newtonian regime.

Two xanthan gum (XG) solutions (SIGMA Life Science) at concentrations of 0.5 and 1 g.L⁻¹ are then used to mimic the non-Newtonian shear-thinning behavior of *Chlorella vulgaris* suspensions at high cell concentrations. At these concentrations, the rheological properties of the XG solutions are close to those of *Chlorella vulgaris* suspensions at 42 g.L⁻¹ (Haydar et al., 2023). The rheogram comparing the rheological behavior of the different fluids with that of *Chlorella vulgaris* suspension is presented in Haydar et al. (2023). To compare with the case of Newtonian liquid at low viscosity, tap water is used.

A Milli-Q Advantage A10® purification system is used to provide ultrapure water (conductivity < 0.1 µS.cm⁻¹) for the preparation of XG and glycerol solutions. The XG solutions are prepared by dissolving the required amount of polymer in deionized water under agitation at room temperature for 24 h. The aqueous solution of glycerol at 50 wt% is prepared by diluting pure glycerol (liquid form) in deionized water. The rheological and physicochemical properties of the Newtonian and non-Newtonian fluids studied are summarized in Haydar et al. (2023).

## 2.3. Methods

### 2.3.1. Mixing time

Mixing time experiments are carried out using the conductimetric method, which involves adding NaCl aqueous solution as a tracer and monitoring the local conductivity of the mixture until stabilization. A pulse injection of tracer is performed. The amount and concentration of the injected NaCl solution depends on the density (d) of the liquid phase: 2 mL of NaCl at 20 g.L⁻¹ (d = 1.012) are used for tap water and XG solutions (0.5 and 1 g.L⁻¹), and 1 mL of NaCl at 200 g.L⁻¹ (d = 1.13) is used for the 50 wt% glycerol solution. The conductivity probe is installed in position P5 (Fig. 1). The conductivity meter used is a Tacussel CD 810, with a sampling frequency of 50 Hz. The data acquisition is performed using a LabVIEW program, allowing the signal evolution to be monitored in real time until a stable value is reached. In this work, the mixing time experiments are conducted using two different injection methods:

- The injection across the width of the column is performed at 4 injection points evenly distributed across the width (orifices with a diameter of 1.5 mm). This injection system is positioned 17 cm

above the free surface of the unaerated liquid. This setup allows to estimate the axial dispersion coefficient ($D_{ax}$).

- The local injection is performed from the top of the column at a single point using a tube with a diameter of 1.5 mm. This injection system is positioned 17 cm above the free surface of the unaerated liquid.

Comparison between results obtained by both injection methods enables the assessment of the respective influences of axial and radial mixing.

### 2.3.2. Determination of gas-liquid mass transfer coefficient, $k_L a$

The gas-liquid mass transfer in the reactor is characterized by $k_L a$ measurements using the standard method: ASCE 1992; CE12255: 15 1998, also known as the deoxygenation-reoxygenation method. In this method, the physical absorption or desorption of O₂ is experimentally characterized to estimate the global $k_L a$ coefficient. The principle of this method involves removing dissolved O₂ from the solution by injecting nitrogen (N₂) until the O₂ concentration falls below 1 % of its saturation value (i.e., the O₂ concentration reaches minimal values close to zero, around 0.1 mg.L⁻¹). Air is then injected, and the increase in dissolved O₂ concentration is monitored over time until the liquid is saturated with O₂ (> 90 % of O₂ saturation) (i.e., 8.7 mg.L⁻¹ at 20°C). The decrease (deoxygenation) and increase (reoxygenation) in dissolved O₂ concentration over time are measured using an oxygen sensor (OXROB10-CL4 Pyroscience) installed in the column at position P5 (Fig. 1) at the acquisition frequency of 1 Hz.

The response time of the O₂ probe in water is 6 s. This response time is considered negligible compared to the dynamics of gas-liquid mass transfer. Additionally, in our case, the switch between nitrogen and air bubbling is very rapid, so the recovery of hydrodynamics is considered nearly instantaneous.

Firstly, the hypothesis of perfect mixing in the PBR has been considered. However, for the whole range of $U_{Gsup}$ with all the tested fluids, the ratio $(1/k_L a)/t_m$ is not greater than 10, which corresponds to the limit recommended in the literature to validate the hypothesis of perfect mixing condition (Garcia-Ochoa and Gomez, 2009), therefore the perfectly mixed reactor hypothesis is not valid in this study. This result is also observed by Almani et al. (2023) in the thin-gap bubble column ($e = 4$ mm). There are other models, besides the perfect mixing hypothesis, such as axial dispersion for the liquid phase (Dhaouadi et al., 1997; Gourich et al., 2008), which are suitable for this geometry of a bubble column. In particular, since the determination of $k_L a$ is made from deoxygenation-reoxygenation experiments, where the gas is injected over the entire bottom surface of the reactor, there are no radial
concentration gradients of O₂ in the liquid phase in this experiment, and only imperfect axial mixing has to be considered. Thus, experiments of deoxygenation-reoxygenation were analyzed using the axial dispersion model for the liquid phase.

For the gas phase, it is assumed to have a uniform composition, equal to that of the inlet, due to the low solubility of O₂ and the relatively short residence time of the bubbles (low column height). The influence of the hydrostatic pressure difference between the top and bottom of the column is also neglected.

Two codes, developed in Scilab, based on the axial dispersion model, are used to estimate the axial dispersion coefficient, D<sub>ax</sub>, and the volumetric gas-liquid mass transfer coefficient, k<sub>L</sub>a respectively. The first program uses the experimental mixing time information obtained from “width injection” data (tracer concentration as a function of time) along with the gas holdup (Haydar et al., 2023) to estimate the value of D<sub>ax</sub>. To estimate the axial dispersion coefficient under each flow condition (U<sub>Gsup</sub> and fluid), this model uses the least square method to minimize the differences between the experimental normalized signal in response to the “ width injection” and the model result. The second program uses the obtained D<sub>ax</sub> value and the experimental reoxygenation data (dis-solved O₂ concentration as a function of time) to estimate the overall k<sub>L</sub>a: as for D<sub>ax</sub>, the root mean square error (RMSE) is calculated and minimized between the experimental and numerical results to approach the value of k<sub>L</sub>a.

All experiments were conducted twice for each fluid and superficial gas velocity. As a result, four k<sub>L</sub>a values were obtained for each experimental condition. These values were then averaged, and standard deviations are indicated in the corresponding graphs. In some instances, the deviations are too small to be visible.

### 2.3.3. Determination of specific interfacial area ‘a’

The shadowgraphy method is used to characterize the gas phase by the determination of the bubble size distribution and velocity. Then, the bubble size distribution is used to determine the specific interfacial area ‘a’. The aim is to understand the respective influences of the mass transfer coefficient ‘k<sub>L</sub>’ and the specific interfacial area ‘a’ on k<sub>L</sub>a values.

The shadowgraphy measurements are carried out in two distinct zones (Zone 1 and Zone 2), each with the same dimensions (80 × 113 mm²). Zone 1 is located 196 mm from the bottom of the column, while Zone 2 is situated 560 mm from the bottom. The distance between the two zones is 284 mm. Four superficial gas velocities (U<sub>Gsup</sub> = 0.002, 0.003, 0.004, and 0.007 m.s⁻¹) are tested. These velocities correspond to the homogeneous regime and the beginning of the transition regime. At U<sub>Gsup</sub> values higher than 0.007 m.s⁻¹, it becomes impossible to process the images due to the presence of bubbles which become too large for the studied zone.

For each configuration (zone, fluid and U<sub>Gsup</sub>), 500 images are captured and processed using the ‘Shadow Tracking’ method of the DANTEC DYNAMICS software. For each detected bubble i in each image, this method estimates its projected area A<sub>i</sub> in a vertical plane. The equivalent diameter d<sub>i</sub> (mm) of this bubble is defined by the following equation:

$$d_i = \sqrt{\frac{4A_i}{\pi}} \qquad (1)$$

with A<sub>i</sub> being the projected area (mm²).

The specific interfacial area ‘a’ (m⁻¹) is calculated for each image as the ratio of the sum of the bubble surfaces to the volume of the liquid phase (Eq. 2) and this for the same measurement zone:

$$a = \frac{\sum_{i=1}^{n} S_{\text{bubblei}}}{V_L} \qquad (2)$$

where the bubble is assumed to be a cylinder of diameter d<sub>i</sub> and of height e which implies that:

$$S_{\text{bubblei}} = \left(2 \times \frac{\pi d_i^2}{4}\right) + \left(e \times \pi d_i\right) \qquad (3)$$

where e (column gap) = 2 mm, n is the number of bubbles in the zone and V<sub>L</sub>, in m³, the volume of the liquid phase in the measuring zone, is calculated as follows:

$$V_L = V_{\text{zone}} - V_{\text{Gas}} \qquad (4)$$

where:

$$V_{\text{zone}} = 1.81 \times 10^{-5} \quad \text{m}^3 \text{ and } V_{\text{Gas}} = \sum_{i=1}^{n} V_{\text{bubblei}} = \sum_{i=1}^{n} \left( \frac{\pi d_i^2}{4} \times e \right) \qquad (5)$$

The average value of ‘a’ of the 500 images for both studied zones is then calculated and compared for the different fluids.

## 3. Results and discussion

### 3.1. Mixing time

This section focuses on studying the impact of superficial gas velocities, confinement and rheological properties of the liquid phase on the mixing time. First, the results obtained by local injection are presented and then these results are compared to ones obtained with the width injection system.

#### 3.1.1. Influence of the liquid phase rheology on the mixing time (local injection)

Fig. 2 shows a decrease in mixing times (t<sub>m</sub>) with increasing superficial gas velocity for the four fluids used (a semi-logarithmic scale graph is used to show differences between closely spaced points).

t<sub>m</sub> for tap water decreases rapidly during the homogeneous regime (up to U<sub>Gsup</sub> = 0.007 m.s⁻¹), then stabilizes at higher gas velocities, with values ranging between 13 and 33 s. In contrast, viscous solutions (XG and 50 wt% glycerol) exhibit much higher t<sub>m</sub> values at low U<sub>Gsup</sub> (at U<sub>Gsup</sub> = 0.002 m.s⁻¹, t<sub>m</sub> is equal to 225, 750 and 408 min, respectively for XG 0.5, 1 g.L⁻¹ and 50 wt% glycerol solutions) and decreases only slowly with U<sub>Gsup</sub> during the organized regime. This can be attributed to the influence of the high viscosity, leading to a rectilinear trajectory for bubbles without interaction (Haydar et al., 2023). The decrease in t<sub>m</sub> becomes more significant as U<sub>Gsup</sub> becomes higher than 0.007 m.s⁻¹, corresponding to the appearance of the transition regime (marked by bubble coalescence) for the XG 0.5 and 1 g.L⁻¹ solutions and the homogeneous regime for the 50 wt% glycerol one (for this fluid, bubble trajectory is no longer perfectly rectilinear but without coalescence up to U<sub>Gsup</sub> = 0.021 m.s⁻¹ (Haydar et al., 2023)).

* Influence of fluid viscosity: tap water vs. viscous solutions:

The comparison of the mixing times between tap water and the

<figure><img src="image_4.png" /><figcaption>Fig. 2. Mixing time (Local injection) vs superficial gas velocity for the four fluids used.</figcaption></figure>
viscous solutions (50 wt% glycerol and XG 0.5 and 1 g.L⁻¹) highlights the strong impact of fluid viscosity, particularly at low U<sub>Gsup</sub>, as evidenced by significantly higher t<sub>m</sub> values for the viscous fluids. For instance, at U<sub>Gsup</sub> = 0.002 m.s<sup>-1</sup>, t<sub>m</sub> varies significantly: 9 min for tap water, 225 min for XG 0.5 g.L<sup>-1</sup>, 408 min for 50 wt% glycerol, and 750 min for g.L<sup>-1</sup>. This phenomenon can be explained by the bubble dynamics in the continuous phase at low viscosity (water), which generates vorticities in bubble wakes, with quite high intensity from the lowest gas flow rates.

At higher flowrates, strong fluctuations, interactions between bubbles and several deformations are observed in water (Haydar et al., 2023), which contributes to favor small mixing times. On the other hand, for viscous solutions, the mixing capacity is much lower than for water due to the high viscosity of the medium. Under these high flowrate conditions, the shape of the bubbles in these fluids does not change during their ascension and remains ellipsoidal without wobbling. These observations lead to longer mixing times for viscous fluids in comparison to water, even at higher gas velocities.

* Comparison between viscous solutions:

The comparison between t<sub>m</sub>, in the organized regime, reveals distinct behaviors among the viscous fluids (Fig. 2). For instance, at U<sub>Gsup</sub> = 0.004 m.s<sup>-1</sup>, t<sub>m</sub> is significantly lower for the XG 0.5 g.L<sup>-1</sup> solution (85 min) compared to the 50 wt% glycerol solution (202 min), despite both exhibit similar flow patterns characterized by a straight and weakly oscillating bubble path, as observed in Haydar et al. (2023). This difference in t<sub>m</sub> may be explained by the slightly higher average vertical bubble velocity in the XG solution (0.11 m.s<sup>-1</sup>) compared to the glycerol solution (0.095 m.s<sup>-1</sup>) and this for a range of U<sub>Gsup</sub> between 0.002 and 0.004 m.s<sup>-1</sup> (Zone 1), as shown in the shadowgraphy analysis of Haydar (2023). Likewise, the XG 1 g.L<sup>-1</sup> solution exhibits the longest mixing time among all tested fluids during the organized regime (Fig. 2): this is primarily attributed to its higher viscosity across a wide shear rate range (Haydar et al., 2023).

As U<sub>Gsup</sub> increases (> 0.01 m.s<sup>-1</sup>) entering the transition regime, t<sub>m</sub> for XG 1 g.L<sup>-1</sup> becomes lower than that of the 50 wt% glycerol solution (Fig. 2). This behavior is likely due to the emergence of bubble coalescence in the XG solution, which enhances mixing efficiency. In contrast, the glycerol solution remains in the homogeneous regime up to U<sub>Gsup</sub> 0.021 m.s<sup>-1</sup>, with no observable coalescence, resulting in a less agitated flow and consequently longer mixing times.

Considering the application to photobioreactors, which are predominantly operated at low U<sub>Gsup</sub> (homogeneous regime in classical bubble column), the mixing times can be quite poor, which may potentially impact the even distribution of nutrients and dissolved gases. To illustrate, within the organized regime, the observed mixing time values are up to 407 times higher compared to those observed within the heterogeneous regime for XG solution at 1 g.L<sup>-1</sup>. When comparing the homogeneous and heterogeneous regimes, this ratio stands at 22 for the tap water. Consequently, these poor mixing times within organized regime, using XG and glycerol solutions (corresponding to microalgae at higher concentration), hold the potential for adverse consequences that could impact the culture's growth.

### 3.1.2. Influence of injection type for all liquid phase rheologies

In the previous part, it was observed that t<sub>m</sub> is very high with local injection. This may be attributed to quite low horizontal interaction between bubbles especially at low U<sub>Gsup</sub> and thus poor radial mixing. Thus, the width injection is used which can ensure a uniform supply of tracer over the entire width of the column so that only the axial mixing is quantified. The mixing time results with the width injection method are compared with those obtained by the local injection method (Fig. 3).

Fig. 3 shows that mixing times at low U<sub>Gsup</sub> are greater with the local injection method for all fluids. For example, at U<sub>Gsup</sub> = 0.002 m.s<sup>-1</sup>, the ratio between the two types of injection is 1.1, 3.9, 3.3 and 4.5 respectively for water, glycerol 50 wt%, XG 0.5 and 1 g.L<sup>-1</sup>. This decrease in

<figure><img src="image_5.png" /><figcaption>Fig. 3. Mixing time vs superficial gas velocity with both tracer injection methods: local and width injection for all tested fluids.</figcaption></figure>

mixing times observed using the width injection method emphasizes the poor “radial” mixing (i.e., in the width direction) at low U<sub>Gsup</sub>. In the transition and heterogeneous regimes, the mixing times using the width injection method remain lower than those with the local one, especially for viscous fluids, but the difference is lower than for small U<sub>Gsup</sub>, showing that radial and axial mixing both play a role in mixing times. For instance, at U<sub>Gsup</sub> = 0.025 m.s<sup>-1</sup>, the ratio between the two types of injection becomes 1.2, 1.14, 1.18 and 1.9 respectively for water, glycerol 50 wt%, XG 0.5 and 1 g.L<sup>-1</sup> (Fig. 3).

It is thus established that the poor mixing times of this viscous fluids cannot be exclusively attributed to the poor “radial” mixing since these high mixing times were not only observed during experiments with local injection but also in experiments involving tracer injection across the entire width of the reactor. For tap water, the mixing times are quite similar with both injection methods, suggesting that the issue primarily lies in poor axial mixing, while radial mixing remains effective, likely due to the oscillatory motion of the bubbles in tap water.

### 3.1.3. Influence of confinement on the mixing time (local and width injection)

Comparing the mixing times of confined bubble columns and conventional ones is challenging due to several factors. These include differences in column geometry, gas sparger type and geometry, tracer injection mode and location as well as probe location (McClure et al., 2015). However, a similar trend is observed: t<sub>m</sub> decreases with increasing U<sub>Gsup</sub> in both confined and conventional columns (McClure et al., 2015). In our study, t<sub>m</sub> significantly decreases up to the transition regime (Fig. 3). Beyond this point, it remains relatively constant at higher U<sub>Gsup</sub> values, which is consistent with observations in conventional columns, due to the increased agitation in the flow.

Mixing time analyses were previously conducted by Almani et al. (2023) in a confined bubble column with a gap of e = 4 mm ('d<sub>b</sub>/e' < 1), using tap water and non-Newtonian fluids including XG 1 g.L<sup>-1</sup>, with two tracer injection methods: local injection at the bottom of the column and width injection just above the free surface.

A comparison with the present study (e = 2 mm, 'd<sub>b</sub>/e' > 1) shows that, for XG 1 g.L<sup>-1</sup>, t<sub>m</sub> increases significantly with the degree of confinement, regardless of the injection type and the U<sub>Gsup</sub> value (Fig. 4). For instance, at U<sub>Gsup</sub> = 0.002 m.s<sup>-1</sup>, t<sub>m</sub> reaches 750 min with local injection vs. 10 min in Almani et al. (2023) (×75), and 167 min with width injection versus 11 min (×15) (Figs. 4a and 4b). At higher U<sub>Gsup</sub> values (> 0.007 m.s<sup>-1</sup>), corresponding to the transition and heterogeneous regimes, t<sub>m</sub> for XG 1 g.L<sup>-1</sup> remains consistently higher in this study than in Almani et al. (2023) for both injection methods: at U<sub>Gsup</sub> = 0.016 m.s<sup>-1</sup>, t<sub>m</sub> is 1.9 min vs. 0.7 min for local injection, and 1.7 min vs. 1.2 min for width injection (Figs. 4a and 4b).

In the case of tap water, the effect of confinement is relatively different. At low U<sub>Gsup</sub> (0.002 m.s<sup>-1</sup>), t<sub>m</sub> is still higher under high confined conditions ('d<sub>b</sub>/e' > 1) in comparison with Almani et al.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Comparison between mixing time values in the two thin-gap bubble columns (2 mm and 4 mm) in tap water and XG at 1 g.L⁻¹ with two tracer injection methods: a) Local injection and b) Width injection.</figcaption></figure>

(2023): 8.7 min vs. 6.4 min for local injection (×1.35), and 7.6 min vs. 3.2 min for width injection (×2.4) (Fig. 4). However, as U<sub>Gsup</sub> increases, t<sub>m</sub> becomes lower under higher confinement ('db/e' > 1), especially in the homogeneous regime (II), i.e., for U<sub>Gsup</sub> > 0.004 m.s<sup>-1</sup>. This is explained by the earlier appearance of bubble coalescence in the more confined column (2 mm, 'db/e' > 1), due to enhanced vertical bubble interactions, which promote mixing more effectively (Haydar et al., 2023). This behavior holds up to U<sub>Gsup</sub> = 0.007 m.s<sup>-1</sup> for local injection (Fig. 4a). Interestingly, for width injection, t<sub>m</sub> for tap water remains lower in the more confined column ('db/e' > 1) across all U<sub>Gsup</sub> values (Fig. 4b). These results suggest that the confinement effect (described above), characterized by the earlier occurrence of the coalescence phenomenon in a confined geometry ('db/e' > 1), predominantly impacts “radial” mixing.

This rather high mixing time observed in this study can be explained by the reduction of the agitation capacity with confinement, which leads to essentially 2D bubble movement. In particular, wall-induced effects can attenuate the flow disturbances generated by the bubbles and prevent the development of turbulence and agitation.

3.2. Specific interfacial area ‘a’

The specific interfacial area ‘a’ is determined for superficial gas ve-
locities ranging from 0.002 to 0.007 m.s⁻¹ for the four solutions. Fig. 5,
which shows the average value of ‘a’ for both studied zones, demon-
strates that the value of ‘a’ increases with U<sub>Gsup</sub> for all tested fluids. For
water, the rate of increase is lower than for the other fluids due to the
occurrence of coalescence which appears sooner for water, i.e. from
lower U<sub>Gsup</sub>. For the viscous Newtonian and non-Newtonian fluids, the
value of ‘a’ increases quickly with increasing U<sub>Gsup</sub>. For instance, when
U<sub>Gsup</sub> increases from 0.002 to 0.007 m.s⁻¹, the value of ‘a’ is multiplied
by 2.5 or 3, depending on the fluid.

This information about ‘a’ evolution as a function of U<sub>Gsup</sub> can
contribute to a better understanding of the respective impact of ‘k<sub>L</sub>’ and
‘a’ on the k<sub>L</sub>a value. This latter parameter will be discussed in the
following section.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Specific interfacial area ‘a’ as a function of superficial gas velocity for the tested fluids (average value between the two zones (1 and 2)).</figcaption></figure>

3.3. Gas-liquid mass transfer coefficient k<sub>L</sub>a

The k_L a values obtained in this confined column are compared with the k_L a values obtained in literature to investigate the effect of confinement and of the rheological properties of the liquid phase on this parameter.

3.3.1. Effect of the rheological properties of the liquid phase on k<sub>L</sub>a

The k_L a experiments were performed for the four fluids (Fig. 6). For a given liquid phase, k_L a increases with the increase of U_Gsup and for the same U_Gsup, this parameter decreases slightly with the increase in the liquid viscosity. At low U_Gsup (U_Gsup < 0.004 m.s⁻¹), the increase in the viscosity for Newtonian and non-Newtonian liquids has almost no influence on the k_L a. This result can be attributed to the compensatory effects between ‘k_L’ and ‘a’. In fact, referring to the specific interfacial area ‘a’ results in Fig. 5, it is possible to see that despite that water has a lower value of ‘a’ compared to viscous solutions (XG and glycerol 50 wt %), the k_L a of water remains slightly higher as shown in Fig. 6. This suggests that the transfer coefficient in the liquid phase ‘k_L’ might be greater in water than in viscous solutions. To support this interpretation,

<figure><img src="image_8.png" /><figcaption>Fig. 6. Evolution of the global gas-liquid mass transfer coefficient, k_L a, with the superficial gas velocity for the four liquid phases.</figcaption></figure>
$k_L$ values were calculated from the measured $k_L a$ values and the interfacial area $a$ obtained via shadowgraphy. The results (Fig. 7) confirm that $k_L$ is consistently higher in water than in viscous fluids. Thus, in viscous solutions, the larger $a$ is counterbalanced by a lower $k_L$, leading to comparable $k_L a$ values at low $U_{Gsup}$. The reduction in $k_L$ with increasing viscosity can be attributed to the bubbles’ rising dynamics, since agitation is more important in water in comparison to the viscous solutions. Low $k_L$ values for glycerol 50 wt% and both XG solutions can also be analyzed in light of the correlation of Higbie (1935) ($k_L$ = $2 \times \sqrt{\frac{D_L}{\pi t_{exp}}}$). The high viscosity of the liquid phase can make the contact time ($t_{exp}$) larger. Consequently, this longer exposure time may contribute to the decrease of ‘$k_L$’ in these solutions. Besides the effect of viscosity, the alcohol functions in the glycerol molecule can limit the gas exchanges between the bubbles and the liquid phase, since the diffusivity of oxygen across the interface is reduced compared to aqueous solutions. In fact, the presence of alcohol has a two-fold effect on $k_L a$: the presence of alcohol tends to increase ‘$a$’, due to a decrease in bubble diameter, and increase of the gas holdup. At the same time, molecules of alcohol tend to adsorb and accumulate around bubbles forming a rigid layer leading to a decrease of ‘$k_L$’ (Kojic et al., 2016). Therefore, despite the consistently higher ‘$a$’ value in the glycerol solution compared to the other fluids (cf. Fig. 5), the $k_L a$ value of glycerol remains lower. This is particularly visible at low $U_{Gsup}$ as illustrated in Fig. 6.

In transition and heterogeneous regimes, water maintains a higher $k_L a$ for all flowrates in comparison with viscous solutions (Fig. 6). Indeed, in these conditions of high $U_{Gsup}$, determining the ‘$a$’ value was not possible because of too large bubbles for the studied zone. Nevertheless, at these $U_{Gsup}$, the higher tendency of coalescence in viscous solutions in comparison to water (Haydar et al., 2023), results in an increase in the large bubbles volume. Consequently, this can lead to a much lower ‘$a$’ in viscous solutions compared to water. Also, the ‘$k_L$’ of viscous solutions (XG and glycerol 50 wt%) is expected to be lower than that observed in water. This phenomenon attributed to the influence of viscosity on the agitation process. These combined effects of reduced ‘$k_L$’ and ‘$a$’ in viscous solutions, can explain why the $k_L a$ is lower in viscous solutions in comparison to water, particularly at high $U_{Gsup}$ (Fig. 6).

The findings mentioned above hold significant relevance for microalgae culture, particularly in an intensified PBRs. In fact, as the biomass concentration rises, the culture viscosity in the reactor is expected to increase. However, these results indicate that the increase in the viscosity does not affect $k_L a$ at low flowrates, which are typical operating conditions for PBRs.

### 3.3.2. Effect of the confinement on $k_L a$

To highlight the effect of confinement on $k_L a$, the results obtained in the thin-gap bubble column (2 mm) are compared with those of classical and confined columns with low confinement ratio ‘$d_b/e^<$ 1. Fig. 8a compares the $k_L a$ values observed for water in the confined column with the results of Deckwer et al. (1974), Gourich et al. (2008), Han and Al-Dahhan (2007) and Koide et al. (1983) for classical columns and with

<figure><img src="image_9.png" /><figcaption>Fig. 7. Evolution of the transfer coefficient in the liquid phase ‘$k_L$’ with the superficial gas velocity for all tested fluids.</figcaption></figure>

<figure><img src="image_10.png" /></figure>

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

<figure><img src="image_12.png" /><figcaption>Fig. 8. Comparison between $k_L a$ values in this thin-gap bubble column with literature for: (a): Water, (b): 50 wt% aqueous glycerol solution and (c): XG at 0.5 and 1 g.L$^{-1}$.</figcaption></figure>

the results of Almani et al. (2023) in confined column of 4 mm gap with ‘$d_b/e^<$ 1. Fig. 8b compares the $k_L a$ values of 50 wt% glycerol solution with the $k_L a$ values for the classical column used by Koide et al. (1983) and for the confined column of 4 mm gap with ‘$d_b/e^<$ 1 used by Thobie et al. (2017) for the same tested solution. Concerning XG, the $k_L a$ values of the two concentrations of XG (0.5 and 1 g.L$^{-1}$) are compared to those obtained by Almani et al. (2023) for confined bubble column using XG 1 g.L$^{-1}$ (Fig. 8c). Table 1 summarizes the different studies used in these comparisons.

These figures show that under confined conditions, the $k_L a$ values are lower than in classical bubble columns, for water (Fig. 8a) and non-Newtonian solutions (XG) (Fig. 8c), $k_L a$ decreases with the increase of the degree of confinement ‘$d_b/e^<$’. For glycerol solution (Fig. 8b), a different conclusion was observed and will be discussed later.

For water and the two XG solutions (Fig. 8a and c), at low $U_{Gsup}$ (homogeneous regime for water and organized regime for XG solutions), the specific interfacial area ‘$a$’ ($a$ = $S_{bubble}/V_L$) could be greater in this study in comparison to the classical and confined conditions with ‘$d_b/e^<$ 1. This can be explained by the fact that at a given bubble volume, the $S_{bubble}/V_L$ ratio is greater for a flattened bubble (this study) than that for a spherical or ellipsoidal bubble (classical and confined column 4 mm
<table><caption>**Table 1**
Studies used to compare the gas-liquid mass transfer coefficient results obtained in this study with literature.</caption><thead><tr><th>Reference</th><th>Fluid</th><th>Bubble column</th><th>L<sub>c</sub> [m]</th><th>H<sub>L</sub> [m]</th><th>e [mm]</th><th>Gas injection system</th></tr></thead><tbody><tr><td>This study</td><td>Tap water (Fig. 8a) and XG 0.5 and 1 g.L<sup>-1</sup> (Fig. 8c)</td><td>parallelepipedic</td><td>0.3</td><td>0.97</td><td>2</td><td>capillaries 15*1.016 mm</td></tr><tr><td>Almani et al. (2023)</td><td>Tap water (Fig. 8a) and XG 1 g.L<sup>-1</sup> (Fig. 8c)</td><td>parallelepipedic</td><td>0.3</td><td>0.95</td><td>4</td><td>capillaries 15*1.016 mm</td></tr><tr><td>Gourich et al. (2008)</td><td>Tap water (Fig. 8a)</td><td>cylindrical</td><td>0.1</td><td>3</td><td>-</td><td>perforated plate (76*0.5 mm)</td></tr><tr><td>Han et al. (2007)</td><td>Tap water (Fig. 8a)</td><td>parallelepipedic</td><td>0.16</td><td>1.8</td><td>-</td><td>capillaries 163*0.5 mm</td></tr><tr><td>Deckwer et al. (1974)</td><td>Tap water (Fig. 8a)</td><td>cylindrical</td><td>0.15</td><td>4.4</td><td>-</td><td>porous plate (d<sub>p</sub> = 0.15 mm)</td></tr><tr><td>Koide et al. (1983)</td><td>Tap water (Fig. 8a) and Glycerol 50 wt% (Fig. 8b)</td><td>cylindrical</td><td>0.14</td><td>1.54</td><td>-</td><td>multi-nozzles (12*2 mm)</td></tr><tr><td>Thobie et al. (2017)</td><td>Glycerol 50 wt% (Fig. 8b)</td><td>parallelepipedic</td><td>0.3</td><td>0.95</td><td>4</td><td>capillaries 8*0.508 mm</td></tr></tbody></table>

L<sub>c</sub>: column width; H<sub>L</sub>: column height; e: column gap

gap with '$d_b/e$' < 1). Furthermore, a greater 'a' value in this study could be also explained by the higher gas holdup observed in this thin-gap column and which increases with the degree of confinement (Haydar et al., 2023). Despite the expected higher 'a' in this study, $k_L a$ is always lower, indicating that '$k_L$' seems to be lower under confined conditions. The lower '$k_L$' in this study can be explained by the reduction of agitation around the bubble with confinement i.e., the vortices generated in the wake of the bubbles exhibit 2D motion, since bubbles can only move in a plane and not in 3D (Ruiz-Rus et al., 2022). For these reasons, $k_L a$ is lower in this study at low $U_{Gsup}$ for water and XG solutions in comparison with literature studies (Fig. 8a and c).

In transition and heterogeneous regimes, for water (Fig. 8a), the $k_L a$ value consistently remains lower in confined conditions compared to the results of other studies carried out at similar $U_{Gsup}$. This lower $k_L a$ for water at high $U_{Gsup}$ can also be explained by the lower '$k_L$' under confined conditions. It is important to note that a direct comparison of the 'a' value with literature results could be challenging due to the earlier transition between the regimes observed in this confined column at '$d_b/e$' > 1 (Haydar et al., 2023). For XG solutions (Fig. 8c), the $k_L a$ curve exhibits a linear increase with increasing $U_{Gsup}$ during the transition and heterogeneous regimes ($U_{Gsup} > 0.007 m.s^{-1}$), contrary to the study of Almani et al. (2023) where the $k_L a$ curve displayed a slower increase, particularly during the heterogeneous regime ($U_{Gsup} \approx 0.030 m.s^{-1}$).

In the case of the 50 wt% glycerol solution (Fig. 8b), the $k_L a$ values observed in this study are similar to those reported by Thobie et al. (2017) at low $U_{Gsup}$ (< 0.005 m.s<sup>-1</sup>). This similarity can be explained by the compensatory effect between the two parameters '$k_L$' and 'a': in comparison with Thobie et al. (2017), '$k_L$' can be lower in this study due to higher degree of confinement but in the same time, 'a' is expected to be higher in this study due to the use of 15 injection capillaries (versus 8 capillaries used in the study of Thobie et al. (2017)). For this reason, the two $k_L a$ curves (blue and yellow curves) are superimposed at low $U_{Gsup}$ (Fig. 8b). By increasing $U_{Gsup}$ (> 0.005 m.s<sup>-1</sup>), the coalescence starts earlier in the study of Thobie et al. (2017), due to the interactions between bubbles which are more pronounced with 8 capillaries, whereas in our study where 15 capillaries are used, the bubble coalescence phenomenon starts later at $U_{Gsup} = 0.021 m.s^{-1}$ (Haydar et al., 2023). Consequently, our study consistently maintains higher 'a' value when the $U_{Gsup}$ increases, explaining that the $k_L a$ values in this study are higher than that obtained by Thobie et al. (2017) in transition and heterogeneous regimes as shown in Fig. 8b. Furthermore, in comparison with the classical column (Koide et al., 1983), the $k_L a$ values for glycerol are similar in organized and homogeneous regimes (until $U_{Gsup} = 0.021 m.s^{-1}$), after that, the bubble coalescence starts in our case, so the 'a' is expected to reduce. Thus, the $k_L a$ curve increases slowly in the heterogeneous regime ($U_{Gsup} > 0.021 m.s^{-1}$). For Koide et al. (1983), at the same flowrate, they are still in the homogeneous regime with a more rapid increase in $k_L a$.

In conclusion, the characterization of gas-liquid mass transfer in a confined column by measuring the overall mass transfer coefficient, $k_L a$, revealed a decrease in transfer performance compared to conventional columns. This result is inconsistent with the observations made by Roudet et al. (2017), who characterized gas-liquid transfer locally

around a single bubble using the PLIF method. The separation of the $k_L a$ coefficient into its components, '$k_L$' and 'a', offers a potential explanation for this disparity. It suggests that confined conditions may result in a lower '$k_L$' coefficient due to the 2D activity of vortices in the wake of the bubbles. Consequently, although Roudet et al. (2017) have demonstrated the significance of local transfer within the liquid film between the wall and the bubble, the overall impact at the reactor scale appears less significant. Hence, the overall $k_L a$ coefficient is low, primarily because of the limited agitation induced by the bubbles in confined column and thus limited dissolved $O_2$ transport.

## 4. Conclusion

This study experimentally investigated a thin-gap bubble column (e = 2 mm), with a confinement ratio '$d_b/e$' > 1, for its potential application as an intensified photobioreactor for *Chlorella vulgaris* cultivation at high biomass concentrations. The influence of liquid rheology and confinement on key hydrodynamic parameters, mixing time (t<sub>m</sub>), specific interfacial area (a), and global gas-liquid mass transfer coefficient ($k_L a$), was characterized across a range of superficial gas velocities and flow regimes. Newtonian (tap water, 50 wt% glycerol solution) and non-Newtonian (xanthan gum solution at 0.5 and 1 g.L<sup>-1</sup>) fluids were used to mimic microalgal cultures.

Mixing time was measured using the conductimetric method, with two tracer injection techniques: local and width injection. Both methods demonstrated poor mixing in viscous and non-Newtonian liquids at low $U_{Gsup}$, particularly within the organized regime, with values up to 407 (local injection) and 120 times (width injection) higher compared to those observed in the heterogeneous regime. t<sub>m</sub> was also found to increase with '$d_b/e$', especially at low $U_{Gsup}$, due to reduced agitation capacity in confined conditions (2D bubble motion vs. 3D bubble motion).

Using the bubble diameter data obtained with the shadowgraphy method, the specific interfacial area 'a' was calculated for the homogeneous and the beginning of transition regimes. The results showed that in tap water 'a' increased only moderately with $U_{Gsup}$ due to significant bubble coalescence, while in viscous fluids, 'a' increased more rapidly with $U_{Gsup}$.

Concerning gas-liquid mass transfer, results showed that increasing viscosity for Newtonian and non-Newtonian liquids had limited influence on $k_L a$, especially at low $U_{Gsup}$, attributed to the compensatory effect between '$k_L$' and 'a'. In addition, the $k_L a$ results demonstrated that the gas-liquid transfer is less efficient in a confined column compared to a conventional one and it decreases with the degree of confinement (lower $k_L a$ than the column of 4 mm gap), likely due to lower '$k_L$' caused by reduced agitation under confinement condition.

These findings highlight the importance of carefully selecting operational parameters in confined PBRs. In the homogeneous regime (or organized regime for viscous fluids), poor mixing may hinder nutrient distribution and light availability. Similarly, limited gas-liquid transfer at low $U_{Gsup}$ can lead to $CO_2$ deficiency and $O_2$ accumulation, both harmful to culture growth. To mitigate these issues, operating in the early transition regime ($U_{Gsup} \approx 0.007 m.s^{-1}$) is recommended when biomass concentration becomes high. At this velocity, for the 0.5 g.L<sup>-1</sup> XG solution, t<sub>m</sub> decreased from 85 to 5 min (a 17-fold improvement),
while $k_1a$ increased from 0.0018 to 0.0032 s⁻¹ (a 1.8-fold gain),
compared to values in the organized regime.

Further studies should focus on near-wall phenomena to assess wall
shear stress induced by bubble passage and prevent biofilm formation
through improved bubbling strategies. Local liquid-phase hydrody-
namics should also be analyzed using particle image velocimetry (PIV)
to characterize wake behavior and evaluate the role of bubble size and
rheology. Finally, gas-liquid transfer within bubble swarms should be
examined to explain why confinement reduces transfer efficiency at the
reactor scale, contrary to what is observed around isolated bubbles.

CRediT authorship contribution statement

**Wald BLEL:** Writing – review & editing, Supervision, Methodology, Conceptualization.
**Gentric Caroline:** Writing – review & editing, Validation, Supervision, Methodology, Funding acquisition, Conceptualization.
**Abdallah HAYDAR:** Writing – original draft, Methodology, Investigation.

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.

Acknowledgments

The authors would like to gratefully acknowledge ANR (Agence
Nationale de la Recherche) for financial support of Abdallah Haydar's
Ph.D.

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