2.1. General description

The experimental set-up is described in figure 1(a). It has been used previously by Riboux, Risso & Legendre (Reference Riboux, Risso and Legendre2010) and Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011). Bubbles are injected through stainless steel capillaries (1) in a square glass column of $15~\text{cm}\times 15~\text{cm}$ cross-section and 100 cm high. The gas line is equipped with three different rotameters (2) and one manometer (3) to deal with a large range of gas flow rates and volume fractions. A three-way valve enables the switch from nitrogen to air (4). The use of 841 capillaries of 15 cm length and $d_{c}=0.2~\text{mm}$ inner diameter ensures an homogeneous injection of bubbles of almost equal sizes.

Figure 1.

(a) Experimental installation and (b) imaging set-up.

Experiments are performed at ambient temperature and pressure ( $T=20\,^{\circ }\text{C}$ and $P=P_{atm}$ ). The liquid used for all experiments is tap water filtered to remove particles larger than $15~{\rm\mu}\text{m}$ (5). As a consequence, in the regime considered, gas–liquid interfaces can be considered to be clean (Ellingsen & Risso Reference Ellingsen and Risso2001). This point has been carefully validated by measuring the terminal velocity for single bubbles. The main physical properties of the system are summarized in table 1.

Table 1.

System properties at $T=20\,^{\circ }\text{C}$ and $P=101\,325~\text{Pa}$ .

2.2. Measurements of gas volume fraction and bubble velocity

The gas volume fraction ${\it\alpha}$ and the average vertical bubble velocity $\langle V_{z}\rangle$ are measured by means of a dual-tip optical fibre probe (RBI Instrumentation) which is introduced at the centre of the column (7). A threshold just higher than the noise level is first applied on the raw signal to define the binarized signal. An example of raw and binarized signals obtained for each fibre is presented in figure 2. Then, the volume fraction is determined from

(2.1) $$\begin{eqnarray}{\it\alpha}=\frac{\sum {\rm\Delta}t_{yi}}{t_{aqc}},\end{eqnarray}$$

where $t_{acq}$ is the acquisition duration, ${\rm\Delta}t_{yi}$ the residence time of bubble $i$ on the first fibre (see figure 2) and ${\it\Sigma}{\rm\Delta}t_{yi}$ the total time during which the gas phase is detected. The signal acquisition is performed with a sampling frequency of 10 kHz. A good statistical convergence and an overall accuracy better than 2 % is obtained for a recording time larger than 800 s.

Figure 2.

Signals from the optical probe. Symbols: raw signal from first ( $+$ ) and second ( $\times$ ) fibre. Line: binarized signals (——).

The vertical velocity $V_{zi}$ of bubble $i$ is obtained by

(2.2) $$\begin{eqnarray}V_{zi}=\frac{d_{s}}{{\rm\Delta}t_{12i}},\end{eqnarray}$$

where ${\rm\Delta}t_{12i}$ is the time elapsed between the detection of the bubble interface by the first and the second fibre (as reported in figure 2) and $d_{s}$ is the distance between the two fibre tips. The main difficulty of this technique is to match two successive rising fronts corresponding to the piercing of the same bubble. Spurious unrealistic low or large velocity measurements are detected in some cases, especially when two bubbles interact close to the probe. According to the sensitivity study of Riboux (Reference Riboux2007), values smaller than $V_{min}=0.03~\text{m}~\text{s}^{-1}$ or larger than $V_{max}=0.7~\text{m}~\text{s}^{-1}$ have been removed.

2.3. Measurement of bubble geometrical characteristics

The most reliable technique to determine the bubble shape is probably to process images obtained by means of a high-speed camera. A classic way to image the bubbles is to use a fixed focal lens with a thin depth of field, as done by Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011). However, the larger the gas volume fraction, the more numerous are blurred out-of-focus bubbles in the field of view. The use of a fixed focal lens is thus limited to moderate gas volume fractions ( ${\it\alpha}\leqslant 15\,\%$ ).

The study of collective effects in a dense bubble swarm therefore requires the development and the use of another optical technique. In the present work, we use a telecentric lens, which has the particularity to have a depth of field larger than the column width (15 cm) and a constant magnification factor all along the direction of view. The main advantage is to image bubbles with sharp contours, even in a very dense bubbly flow. The main drawback is that the increase of the field of view results in a significant reduction of the spatial resolution. In addition, it has been possible to follow individual bubbles only on a short distance. For those two reasons, the measurement of the bubble velocity is less accurate and image processing has been specifically used to measure the bubble geometrical characteristics.

The imaging set-up consists of a high-speed CMOS camera (Photron APX, figure 1 b) equipped with a telecentric lens (TC-4M-172 Opto Engineering) to visualize a window of $94~\text{mm}\times 94~\text{mm}$ located at the centre of the column at a distance of 150 mm above the injectors tips. The spacial resolution is $5.8~\text{pixel}~\text{mm}^{-1}$ . The camera is operated at 500 images per second with an exposure time varying from $1/20\,000$ to $1/500~\text{s}$ depending on the lighting intensity. Lighting is supplied by an halogen spot of 1000 W.

The recorded images are processed by using Matlab^®. The bubble edges are detected by applying a threshold to the raw images in grey levels. The interior of the bubbles is then filled and small aberrant objects detected in the picture are removed. A test of convexity is performed to identify cases for which the detected object corresponds to two superimposed bubbles. It consists of comparing the surface area $S_{obj}$ of the detected object to the area $S_{conv}$ of the smallest convex polygon that can contain the object. Only the objects with $S_{obj}/S_{conv}\geqslant 0.95$ are retained, the others being discarded. Examples of detected contours are drawn on typical raw images in figure 3 for different gas volume fractions.

Figure 3.

Typical images of the bubble swarm with detected bubble contours marked with yellow/light lines: (a)  ${\it\alpha}=12.2\,\%$ ; (b)  ${\it\alpha}=23.9\,\%$ ; (c)  ${\it\alpha}=30.6\,\%$ ; (d)  ${\it\alpha}=33.9\,\%$ .

The geometrical properties of the bubbles are determined by assuming that the bubbles are oblate spheroids with a minor semi-axis $a$ and a major semi-axis $b$ , which are measured from the two-dimensional measured contours. The bubble aspect ratio is defined as ${\it\chi}=b/a$ . The bubble volume is estimated from $V_{b}=4{\rm\pi}b^{2}a/3$ and its equivalent diameter from

(2.3) $$\begin{eqnarray}d=(8b^{2}a)^{1/3}.\end{eqnarray}$$

The bubble area $S_{b}$ is estimated by (Beyer Reference Beyer1987)

(2.4) $$\begin{eqnarray}S_{b}={\rm\pi}\frac{d^{2}}{4}\left(2{\it\chi}^{2/3}+\frac{{\it\chi}^{-4/3}}{\sqrt{1-{\it\chi}^{-2}}}\ln \left(\frac{1+\sqrt{1-{\it\chi}^{-2}}}{1-\sqrt{1-{\it\chi}^{-2}}}\right)\right).\end{eqnarray}$$

In addition, an indirect determination of the bubble equivalent diameter can be obtained from the dual-tip optical probe by assuming that all of the bubbles have the same size. As recalled by Colombet (Reference Colombet2012), for a monodispersed population of bubbles that impact the probe with null angle of attack, $d$ can be expressed as a function of the average chord length $\langle y\rangle$ ,

(2.5) $$\begin{eqnarray}d={\textstyle \frac{3}{2}}\langle y\rangle {\it\chi}^{2/3},\end{eqnarray}$$

where $\langle y\rangle$ is obtained from optical probe measurements as

(2.6) $$\begin{eqnarray}\langle y\rangle =\frac{\mathop{\sum }_{1}^{n}(V_{zi}{\rm\Delta}t_{yi})}{n},\end{eqnarray}$$

and ${\it\chi}$ from image processing. (Note that the size distribution of the bubbles will be discussed in § 3.1 from the results of image processing.)

2.4. Measurement of interfacial area

For a bubble column of total volume $V_{tot}$ , the volumetric interfacial area, $a_{I}=\sum S_{b}/V_{tot}$ , is related to the gas volume fraction, ${\it\alpha}=\sum V_{b}/V_{tot}$ , by the relation

(2.7) $$\begin{eqnarray}a_{I}={\it\alpha}\frac{\sum S_{b}}{\sum V_{b}}.\end{eqnarray}$$

As indicated above, for each bubble detected, $a$ and $b$ are obtained from the images used to determine the bubble volume $V_{b}$ and surface $S_{b}$ . The volume fraction ${\it\alpha}$ is given by the optical probe. Then, the interfacial area $a_{I}$ is determined by using (2.7).

2.5. Measurement of mass transfer

The concentration $C(z,t)$ of oxygen dissolved in water at time $t$ and elevation $z$ is measured by means of fast response probes: Clark-type microsensors (Unisense Ox50). The technique is based on the measurement of the intensity of the electric current between an anode and an oxygen-reducing cathode, which is proportional to the oxygen concentration. Calibration of oxygen probes is performed for each experiment. Since the probe response is linear on the whole range of concentration considered, a calibration is performed in situ by using the signal measured at the beginning (anoxic water) and the end (saturated water) of each experiment. The relative uncertainty on oxygen concentration measurements is $\pm 2\,\%$ . In the present configuration, as shown in figure 1(6), two oxygen probes have been placed at two different elevations $z$ , which are reported in table 2.

Table 2.

Liquid height $H$ at ${\it\alpha}=0$ and elevation of the oxygen probes above capillaries tips.

As shown in Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011), due to the moderate height of the bubble column ( ${\leqslant}70~\text{cm}$ ), the oxygen saturation concentration in the water is almost unaffected by the variation of hydrostatic pressure (6.4 %) or by the depletion of the oxygen concentration within the bubbles during the mass transfer (6 %). Moreover, the dilution of oxygen in the bubbles induced by liquid-to-gas transfer of nitrogen at the beginning of the experiments can also be neglected (1.3 %). Consequently, the oxygen mass saturation concentration $C^{\ast }$ can be considered as constant along the $z$ axis and equal to its value at the upper surface where the pressure is equal to that of the atmosphere ( $P=P_{atm}$ ), so that

(2.8) $$\begin{eqnarray}C^{\ast }=x_{\text{O}_{2}}^{G0}{\it\rho}_{\text{H}_{2}\text{O}}\frac{M_{\text{O}_{2}}}{M_{\text{H}_{2}\text{O}}}\frac{(P-P^{sat})}{\mathit{He}}\approx 9.08~\text{mg}~\text{l}^{-1},\end{eqnarray}$$

with $x_{\text{O}_{2}}^{G0}$ the molar fraction of oxygen in the gas phase (dry air), ${\it\rho}_{\text{H}_{2}\text{O}}={\it\rho}_{L}$ the density of water ( $\text{kg}~\text{m}^{-3}$ ), $M$ the molar masses ( $\text{kg}~\text{mol}^{-1}$ ), $P^{sat}$ the vapour pressure of water in the bubbles (Pa) and $\mathit{He}$ the Henry constant for oxygen in water (Pa). Equation (2.8) results from Henry’s law for oxygen in water and Raoult’s law for water in air with activity and fugacity coefficients equal to unity for both equilibria, assuming that the liquid is essentially composed of water.

The classical ‘gassing-out’ method is used to determine the time scale of the transfer of oxygen from the bubbles to the water. This method consists in first bubbling nitrogen gas in the column in order to remove the oxygen that is initially naturally present in water. Next, without changing the inlet gas flow rate in order to not disturb the dynamics of the bubble swarm, air is suddenly injected instead of nitrogen. The concentration of dissolved oxygen $C$ then increases until it reaches the saturation concentration $C^{\ast }$ .

The moderate size of the column and the bubble-induced turbulence both contribute to an efficient liquid mixing so that the liquid phase can be assumed to be perfectly mixed for each horizontal slice of the bubble column. Moreover, owing to the large gas volume fractions and interfacial areas considered in this work, the vertical mass flux of dissolved oxygen generated by the axial mixing can be neglected compared with the oxygen flux coming from the bubbles. In such conditions, the variation of the concentration of dissolved oxygen along the bubble column is given by

(2.9) $$\begin{eqnarray}\frac{\partial C(z,t)}{\partial t}=\frac{k_{L}\,a_{I}}{(1-{\it\alpha})}(C^{\ast }-C(z,t)),\end{eqnarray}$$

where $k_{L}$ is the liquid-side mass transfer coefficient and $a_{I}$ the interfacial area. In the present configuration, the only reason for which $C$ depends on $z$ comes from the delay corresponding for the time taken by the bubble to reach a given elevation $z$ . In the following the time origin is shifted by $z/\langle V_{z}\rangle$ so that the concentration no more depends on $z$ and the signals provided by the two oxygen probes are synchronized.

The analysis of the measured concentrations requires to account for the response time ${\it\tau}_{p}$ of the probes (Letzel et al. Reference Letzel, Schouten, Krishna and van den Bleek1999; Martin, Montes & Galan Reference Martin, Montes and Galan2007). For this purpose, the oxygen probe is assumed to behave as a first-order system

(2.10) $$\begin{eqnarray}\frac{\partial C_{p}}{\partial t}=(1/{\it\tau}_{p})(C-C_{p}),\end{eqnarray}$$

where $C$ is the real concentration and $C_{p}$ is the value provided by the probe. The response time of each probe has been measured and found close to ${\it\tau}_{p}=0.8~\text{s}$ . Solving (2.10) and (2.9) for a sudden increase of the mass concentration from 0 to $C^{\ast }$ at $t=0$ , it yields

(2.11) $$\begin{eqnarray}\frac{C_{p}}{C^{\ast }}=1-\frac{1}{({\it\tau}-{\it\tau}_{p})}({\it\tau}\text{e}^{-t/{\it\tau}}-{\it\tau}_{p}\text{e}^{-t/{\it\tau}_{p}}),\end{eqnarray}$$

where the time scale ${\it\tau}$ is related to the mass transfer coefficient $k_{L}$ by

(2.12) $$\begin{eqnarray}{\it\tau}=\frac{(1-{\it\alpha})}{k_{L}a_{I}}.\end{eqnarray}$$

2.6. Homogeneity of the bubble swarm

Our purpose is to study a stable bubble column in which there is no gradient of volume fraction and no large-scale liquid motions induced by buoyancy. The use of an array of capillary tubes guarantees that the bubbles are uniformly injected at the bottom of the column. However, increasing the gas volume fraction may lead to the development of an instability and to the transition to a churn flow. The onset of the instability depends on both the liquid height $H$ in the column and the gas volume fraction. For $H=70~\text{cm}$ , the flow is stable up to approximately ${\it\alpha}=10\,\%$ . For larger values of ${\it\alpha}$ , the liquid height has been reduced in order to keep a stable flow. The chosen values of $H$ are reported in table 2. With this choice, the free surface at the top of the column remains still and the gas volume fraction turns out to be uniform all over the column. Figure 4 compares the superficial gas velocity $J_{G}={\it\alpha}\times \langle V_{z}\rangle$ obtained from $\langle V_{z}\rangle$ and ${\it\alpha}$ measured by the optical probe and the superficial gas velocity $J_{G}=Q_{G}/S$ obtained from a gas flow rate $Q_{G}$ measured from the flowmeters. The good agreement obtained between these two estimations for all gas volume fractions investigated ( $0.45\leqslant {\it\alpha}\leqslant 33.9\,\%$ ) confirms the homogeneity of the gas distribution over the column.

Figure 4.

Superficial gas velocity from gas volume fraction and bubble average rising velocity measured by the dual-tip optical probe $J_{G}={\it\alpha}\times \langle V_{z}\rangle$ versus superficial gas velocity from measured gas flow rate $J_{G}=Q_{G}/S$ .

Another departure to the flow homogeneity may come from the fact that the bubbles need a certain distance to reach their terminal velocity and that mass transfer needs a certain time to attain a steady state. Considering a clean spherical bubble starting from rest, the relaxation time scale of the bubble velocity can be estimated by ${\it\tau}_{V}\approx d^{2}/(72{\it\nu}_{L})\approx 0.06~\text{s}$ , which corresponds to a distance $3{\it\tau}_{V}V_{z}\approx 5.4~\text{cm}$ . Concerning the mass transfer, Figueroa & Legendre (Reference Figueroa and Legendre2010) found a transient time ${\it\tau}_{C}\approx 10(d^{3}{\it\chi}/8)^{1/3}/V_{z}$ for $\mathit{Re}=300$ , $\mathit{Sc}=10$ and ${\it\chi}=1.2$ . In our case, this leads to ${\it\tau}_{C}\approx 0.04~\text{s}$ and ${\it\tau}_{C}V_{z}\approx 1.3~\text{cm}$ . It is therefore reasonable to consider that the flow and the mass transfer are fully developed at the location of the first oxygen probes, which is at least 5.8 cm above the capillaries.

3.1. Equivalent diameter and interfacial area

Figure 5 shows the evolution of the average bubble equivalent diameter $\langle d\rangle$ measured from image processing (2.3) as a function of ${\it\alpha}$ (○, ●). The standard deviation of the equivalent diameter measured by image processing is found to range between 11 and 21 % of the average value. The bubbles are therefore almost monodisperse and (2.5) can also be used to estimate the bubble diameter from optical probe signals. The values determined by this method are also plotted in figure 5 ( $\times$ ). Despite the strong assumptions made, including that the probe is considered to be ideal (Kiambi et al. Reference Kiambi, Duquenne, Bascoul and Delmas2001; Vejrazka et al. Reference Vejrazka, Sechet, Vecer, Orvalho, Ruzicka and Cartellier2010) and that all of the bubbles impact the probe with a null angle of attack, the difference between the two experimental techniques is less than 14 %.

The bubble diameter is observed to increase with ${\it\alpha}$ because of the process of bubble formation and detachment from the capillaries. At a very low gas volume fraction, the bubble formation can be considered as quasi-static and the bubble size is controlled by the equilibrium between buoyancy and capillary forces at the tip of the capillaries. The diameter is then given by the Tate law, $d_{T}=[6{\it\sigma}d_{c}/({\rm\Delta}{\it\rho}g)]^{1/3}=2.07~\text{mm}$ , as confirmed by the measurement of the detachment of a single bubble by Riboux et al. (Reference Riboux, Risso and Legendre2010). When increasing the inlet gas velocity $u_{c}$ , the balance of the forces acting on a bubble involves drag and added-mass forces (Gaddis & Vogelpohl Reference Gaddis and Vogelpohl1986; Duhar & Colin Reference Duhar and Colin2006). For the entire range of gas volume fraction considered here, the Weber number based on the capillary inner diameter, $\mathit{We}_{c}={\it\rho}_{L}u_{c}^{2}d_{c}/{\it\sigma}$ , stays much lower than two so that the jet regime is never reached and the bubble generation corresponds either to the static regime of formation or to the dynamic one (Mersmann Reference Mersmann1977). Knowing the gas flow rate through each capillary, the bubble diameter can be estimated by using the model of Gaddis & Vogelpohl (Reference Gaddis and Vogelpohl1986). The predictions of this model, which are reported in figure 5, show the same trend as the experimental results but with an underestimation of approximately 20 %. This discrepancy can be due to a collective effect of the bubbles on the formation process and to bubble coalescence that may take place just above the capillary tip as observed by Manasseh, Riboux & Risso (Reference Manasseh, Riboux and Risso2008).

A log–log representation (see the inset in figure 5) reveals that the evolution of $\langle d\rangle$ is well described by the succession of two power laws:

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\langle d\rangle -d_{0}}{d_{0}}\approx 15{\it\alpha}\quad \text{for }{\it\alpha}\leqslant 2.3\,\% & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\langle d\rangle -d_{0}}{d_{0}}\approx 2.3{\it\alpha}^{0.5}\quad \text{for }{\it\alpha}>2.3\,\% & \displaystyle\end{eqnarray}$$

$d_{0}=2.1~\text{mm}$

${\it\alpha}=0$

Figure 6 shows the evolution of the interfacial area as a function of the gas volume fraction. It is found to regularly increase with ${\it\alpha}$ according to the following empirical power laws

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{a_{I}}{a_{I0}}\approx 0.402{\it\alpha}^{0.85}\,\quad \text{for }{\it\alpha}\leqslant 2.3\,\% & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{a_{I}}{a_{I0}}\approx 0.336{\it\alpha}^{0.8}\,\quad \text{for }{\it\alpha}>2.3\,\% & \displaystyle\end{eqnarray}$$

$a_{I0}=S_{b0}/V_{b0}=3011~\text{m}^{-1}$

3.2. Bubble velocity

3.2.1. Average velocity

During the last decade, many works have investigated the velocity of bubbles rising in a swarm (Rusche & Issa Reference Rusche and Issa2000; Zenit, Koch & Sangani Reference Zenit, Koch and Sangani2001; Garnier et al. Reference Garnier, Lance and Marié2002; Riboux et al. Reference Riboux, Risso and Legendre2010). All of these studies report a significant decrease of the average bubble vertical velocity as the gas volume fraction increases.

Figure 7 shows the average vertical bubble velocity $\langle V_{z}\rangle$ as a function of ${\it\alpha}$ . The present results obtained with a dual optical probe (●) are compared with those of Riboux et al. (Reference Riboux, Risso and Legendre2010) (*) and Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011) (○) that were obtained with the same technique, and to those of Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011) (▫) that were determined by image processing with a fixed focal lens. The velocity obtained from image processing is slightly lower, probably because the detected bubbles are not far enough from the column wall. However, all of the results obtained with an optical probe collapse onto a master curve of equation

(3.5) $$\begin{eqnarray}\langle V_{z}\rangle =V_{z0}\left[0.28+0.72\exp (-15{\it\alpha})\right]^{0.5}\!,\end{eqnarray}$$

where $V_{z0}=0.32~\text{m}~\text{s}^{-1}$ is the rise velocity of an isolated bubble formed on a single capillary in the quasi-static bubbling regime, measured by Riboux et al. (Reference Riboux, Risso and Legendre2010). It is remarkable that a single simple correlation is able to describe the evolution of the average bubble velocity on a such large range of gas volume fraction ( $0.45\leqslant {\it\alpha}\leqslant 29.6\,\%$ ) along which $\langle V_{z}\rangle$ is reduced by almost a factor of two (from 0.32 to $0.17~\text{m s}^{-1}$ ).

It is important to stress that all empirical relations relating the properties of the gas phase to the gas volume fraction that have been introduced above ((3.1)–(3.5), (4.1), (4.3) and (4.5)) may depend on the particular system of gas injection used here and are therefore not universal. However, they will be of great interest to analyse and discuss the results of the present work in the following.

3.2.2. Velocity fluctuations

Figure 8 shows the variances of the bubble velocity fluctuations. Let us first discuss the variance of the velocity signal provided by the dual optical probe from this work (○) for ${\it\alpha}$ up to 30 % and from Riboux et al. (Reference Riboux, Risso and Legendre2010) (*) for ${\it\alpha}$ up to 12 %. As noted by Riboux et al. (Reference Riboux, Risso and Legendre2010), if the dual optical probe is an accurate technique to measure the average vertical bubble velocity, it does not provide a reliable value of the variance of the bubble vertical velocity. The reason comes from the fact that the considered bubbles are oblate spheroid which move with oscillating velocity and orientation. The fluctuations that are recorded by the dual probe are thus a complex combination of the fluctuations of the vertical velocity, orientation and shape. For that reason, the measured variance is observed to depend on the exact probe geometry. The values obtained by Riboux et al. (Reference Riboux, Risso and Legendre2010) with a distance between the two fibre tips of 0.5 mm is indeed significantly larger than that obtained in the present work with a fibre tip separation of 1 mm. However, the variance provided by the dual optical probe can be used to characterize the evolution with the gas volume fraction of the overall energy of agitation of the bubbles in the vertical direction. It was already noticed that the bubble vertical agitation keeps a constant value up to a gas volume fraction of around 10 % by Martínez-Mercado, Palacios-Morales & Zenit (Reference Martínez-Mercado, Palacios-Morales and Zenit2007) and Riboux et al. (Reference Riboux, Risso and Legendre2010), which suggested that the energy of bubble agitation remains controlled by wake instabilities. The present results seem to show that this result holds up to ${\it\alpha}=30\,\%$ .

Figure 8.

Variances of bubble velocity against gas volume fraction. Variance obtained from the dual optical probe in this work (○) and by Riboux et al. (Reference Riboux, Risso and Legendre2010) (*). Variances of the vertical velocity (▫) and the horizontal velocity (▪) measured in this work from particle tracking on images taken with a fixed focal lens.

Figure 9.

(a) Bubble Reynolds number ( $\mathit{Re}=\langle V_{z}\rangle \langle d\rangle /{\it\nu}_{L}$ ) and (b) Eötvös number ( $\mathit{Eo}={\rm\Delta}{\it\rho}g\,\langle d\rangle ^{2}/{\it\sigma}$ ) versus gas volume fraction: ○, method using a fixed focal lens; ●, method using a telecentric lens; ▴, result for a single bubble from Riboux et al. (Reference Riboux, Risso and Legendre2010); ——, Reynolds number determined form fitted data ((3.1), (3.2) and (3.5)); – – –, Eötvös number determined form fitted data ((3.1) and (3.2)).

Figure 10.

(a) Bubble average aspect ratio and (b) Weber number ( $\mathit{We}={\it\rho}_{L}\langle V_{z}\rangle ^{2}\langle d\rangle /{\it\sigma}$ ) versus gas volume fraction: ○, method using a fixed focal lens; ●, method using a telecentric lens; ▴, result for a single bubble from Riboux et al. (Reference Riboux, Risso and Legendre2010); ——, aspect ratio estimated from (3.6) (Legendre, Zenit & Velez-Cordero Reference Legendre, Zenit and Velez-Cordero2012); – – –, Weber number determined form fitted data ((3.1), (3.2) and (3.5)).

In order to have a more complete description of the bubble agitation, we have also determined the velocity variance of the horizontal and the vertical bubble velocity fluctuations by PTV based on images taken with a fixed focal lens. As stated before, this imaging technique already used by Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011) is limited to moderate volume fractions. The corresponding results are also plotted in figure 8 for ${\it\alpha}$ up to 16 %. Both the horizontal and the vertical variances are found to be almost constant, $\langle {v_{z}^{\prime }}^{2}\rangle \approx 0.003~\text{m}^{2}~\text{s}^{-2}$ and $\langle {v_{x}^{\prime }}^{2}\rangle \approx 0.0075~\text{m}^{2}~\text{s}^{-2}$ , for ${\it\alpha}$ up to 10 %. As shown by Ellingsen & Risso (Reference Ellingsen and Risso2001), the horizontal component of the fluctuant velocity of an isolated bubble evolves as $v_{x}^{\prime }={\it\omega}A_{x}\cos ({\it\omega}t)$ . For the present bubble size, Riboux (Reference Riboux2007) measured an angular frequency of ${\it\omega}=29~\text{rad}~\text{s}^{-1}$ and a path amplitude $A_{x}$ varying from 3.5 to 4.9 mm, which gives a variance $\langle {v_{x}^{\prime }}^{2}\rangle =(A_{x}{\it\omega})^{2}/2$ from 0.005 to $0.01~\text{m}^{2}~\text{s}^{-2}$ , in agreement with the values found here at moderate volume fraction. When ${\it\alpha}$ is increased beyond 10 %, the vertical variance remains constant, while the horizontal one decreases down to match the vertical value around ${\it\alpha}=12\,\%$ . Such a decrease of the horizontal fluctuation of the dispersed phase has already been reported in a solid/liquid fluidized bed by Aguilar Corona (Reference Aguilar Corona2008) and Aguilar Corona, Zenit & Masbernat (Reference Aguilar Corona, Zenit and Masbernat2011). It may result from hindrance effects on bubble paths when increasing ${\it\alpha}$ .

3.3. Bubble Reynolds, Eötvös and Weber numbers

In order to fully characterize the present flow regime, it is useful to consider the values taken by the relevant dimensionless numbers in the range of volume fractions investigated. These values can be computed either from the raw values of the measured dimensional quantities or from the empirical fits proposed in the previous sections. In the following figures, plots systematically represent raw data whereas lines corresponds to values obtained from fitted data.

Figure 9(a) shows the Reynolds number, $\mathit{Re}=\langle V_{z}\rangle \langle d\rangle /{\it\nu}_{L}$ . It first increases from 670 to 780 as ${\it\alpha}$ increases from 0 to 2.5 % and then keeps a constant value as ${\it\alpha}$ is further increased. The constance of the Reynolds number for ${\it\alpha}\geqslant 2.5\,\%$ results from the fact that the increase in the bubble diameter (figure 5) is compensated by the decrease of the rise velocity (figure 7). A similar result was observed for a volume fraction up to 10 % by Martínez-Mercado et al. (Reference Martínez-Mercado, Palacios-Morales and Zenit2007), who also used a bank of capillaries to inject the bubbles. This is an interesting property of this type of experimental set-up, which allows the volume fraction to be varied while keeping the Reynolds number constant.

Figure 9(b) shows the Eötvös number: $\mathit{Eo}={\rm\Delta}{\it\rho}g\langle d\rangle ^{2}/{\it\sigma}$ . As expected from the evolution of $\langle d\rangle$ , it regularly increases from 0.5 to 3.2 as ${\it\alpha}$ varies from 0 to 30 %.

Figure 10(a) presents the mean bubble aspect ratio, $\langle {\it\chi}\rangle$ , which is found to slightly decrease from 1.7 to 1.4. The bubble deformation is known to be controlled by both the Weber number (Moore Reference Moore1965) and the Morton number (Legendre et al. Reference Legendre, Zenit and Velez-Cordero2012). Here, since we are considering a single system of fluids with constant physical properties, the Morton number is constant: $\mathit{Mo}=g{\it\nu}_{L}^{4}{\it\rho}_{L}^{2}{\rm\Delta}{\it\rho}/{\it\sigma}^{3}=2.5\times 10^{-11}$ . The measured Weber number, $\mathit{We}={\it\rho}_{L}\langle V_{z}\rangle ^{2}\langle d\rangle /{\it\sigma}$ , is plotted in figure 10(b). It is found to decrease from approximately 3.25 down to 1.8. Since the Reynolds number is almost constant, the Weber number turns out to be proportional to $\langle V_{z}\rangle$ . The decrease of the average aspect ratio, by approximately 30 %, is of the same order as that of $\mathit{We}$ , and both $\langle {\it\chi}\rangle$ and $\mathit{We}$ keep an almost constant value for ${\it\alpha}>15\,\%$ . These results are in good agreement, within 14 %, with the relation proposed by Legendre et al. (Reference Legendre, Zenit and Velez-Cordero2012) for a single bubble at low Morton number:

(3.6) $$\begin{eqnarray}{\it\chi}=\frac{1}{1-{\textstyle \frac{9}{64}}\,\mathit{We}}.\end{eqnarray}$$

3.4. Collective effect on bubble drag coefficient

We consider now the evolution of the bubble drag coefficient $C_{d}$ with the gas volume fraction in order to analyse the collective effect of the bubbles on their rise velocity. Here $C_{d}$ is determined from the balance between drag and buoyancy forces as

(3.7) $$\begin{eqnarray}C_{d}=\frac{4}{3}\,\frac{{\rm\Delta}{\it\rho}}{{\it\rho}_{L}}\frac{g\langle d\rangle }{\langle V_{z}\rangle ^{2}},\end{eqnarray}$$

where $g$ is the acceleration of gravity, the average equivalent diameter $\langle d\rangle$ is measured from image processing and the average rise velocity $\langle V_{z}\rangle$ from the dual-tip optical probe. The experimental results are shown in figure 11 (○, ●) as a function of the gas volume fraction. Here $C_{d}$ is observed to increase from 0.26 for ${\it\alpha}=0.45\,\%$ and $\langle d\rangle \approx 2.5~\text{mm}$ to 2.4 at ${\it\alpha}=34\,\%$ and $\langle d\rangle \approx 5~\text{mm}$ .

Figure 11.

Experimental drag coefficient (3.7) against gas volume fraction:

, method using a fixed focal lens; ●, method using a telecentric lens. Drag coefficient for a single bubble of same equivalent diameter rising at its terminal velocity (3.8): ▫, Mendelson (Reference Mendelson1967); ▿, Comolet (Reference Comolet1979); ▵, Dijkhuizen et al. (Reference Dijkhuizen, Roghair, Van Sint Annaland and Kuipers2010). Drag coefficient accounting for the collective effect of the bubbles: ——, Wallis (Reference Wallis1961) (3.9); — ⋅ —, Ishii & Chawla (Reference Ishii and Chawla1979) (3.10);  $\cdots \cdots$ , Garnier et al. (Reference Garnier, Lance and Marié2002) (3.11); – – –, Roghair et al. (Reference Roghair, Lau, Deen, Slagter, Baltussen, Van Sint Annaland and Kuipers2011) (3.12).

In order to disentangle the effect of the bubble size to that of the gas volume fraction, it is interesting to compare the present results with those obtained for single rising bubbles of the same diameter. The drag coefficient $C_{d_{0}}$ of a deformed single bubble at terminal velocity is commonly estimated by (Tomiyama et al. Reference Tomiyama, Kataoka, Zun and Sakaguchi1998)

(3.8) $$\begin{eqnarray}C_{d_{0}}=\frac{8}{3}\,\frac{\mathit{Eo}}{c_{1}+c_{2}\,\mathit{Eo}},\end{eqnarray}$$

with $c_{1}=4$ and $c_{2}=1$ according to Mendelson (Reference Mendelson1967), $c_{1}=4.28$ and $c_{2}=1.02$ according to Comolet (Reference Comolet1979) and $c_{1}=19/3$ and $c_{2}=2/3$ for $\mathit{Re}>600$ with air/water systems according to Dijkhuizen et al. (Reference Dijkhuizen, Roghair, Van Sint Annaland and Kuipers2010).

The corresponding values are represented by empty squares and triangles in figure 11. Starting from similar values at low gas volume fractions, $C_{d}$ and $C_{d_{0}}$ quickly diverge as ${\it\alpha}$ increases. In the present experiments, the increase of $C_{d}$ turns out to mainly result from hydrodynamic bubble interactions. The collective effect of bubbles is really important and leads to a drag coefficient 2.4 times larger than that of an isolated bubble at ${\it\alpha}=34\,\%$ .

In the literature devoted to bubbly flows, numerous relations have been proposed to describe the evolution of the drag coefficient with the gas volume fraction. Considering air bubbles injected through a porous sparger in a column of 9.5 cm diameter filled with a soapy water solution, Wallis (Reference Wallis1961) proposed the following correlation, for $3<{\it\alpha}<30\,\%$

(3.9) $$\begin{eqnarray}C_{d}=C_{d_{0}}(1-{\it\alpha})^{-2},\end{eqnarray}$$

which was later used for one-dimensional gas–liquid modelling (Wallis Reference Wallis1969, p. 52). We have computed $C_{d}$ using relation (3.9) with $C_{d_{0}}$ from Mendelson (Reference Mendelson1967). The corresponding values are represented by a plain line in figure 11. They are in fairly good agreement with the present measurements.

Applying a mixture viscosity model to their experimental results, Ishii & Chawla (Reference Ishii and Chawla1979) (see also Ishii & Zuber Reference Ishii and Zuber1979) found the following correction to account for the effect of the gas volume fraction on the bubble drag coefficient:

(3.10) $$\begin{eqnarray}C_{d}=\frac{C_{d_{0}}}{(1-{\it\alpha})}\left(\frac{1+17.67\,[f({\it\alpha})]^{6/7}}{18.67\,f({\it\alpha})}\right)^{2}\quad \text{with }f({\it\alpha})=(1-{\it\alpha})^{1.5}.\end{eqnarray}$$

This relation is also reported in figure 11 (dashed-dotted line) by using the expression proposed by Mendelson (Reference Mendelson1967) for $C_{d_{0}}$ . It predicts an evolution of $C_{d}$ that is close to that of Wallis (Reference Wallis1961) so that it is difficult to conclude which is in best agreement with the present results.

Garnier et al. (Reference Garnier, Lance and Marié2002) experimentally investigated a homogeneous air/water bubbly flow in the presence of a co-current liquid flow for volume fractions up to 30 % and Reynolds numbers from 300 to 500. They results led to

(3.11) $$\begin{eqnarray}C_{d}=C_{d_{0}}\left(1-{\it\alpha}^{1/3}\right)^{-2}.\end{eqnarray}$$

Using again the expression proposed by Mendelson (Reference Mendelson1967) for $C_{d_{0}}$ , this relation (dotted line in figure 11) is found to considerably over-predict the effect of the gas volume fraction upon the drag coefficient compared with the present results.

Roghair et al. (Reference Roghair, Lau, Deen, Slagter, Baltussen, Van Sint Annaland and Kuipers2011) performed numerical simulations of a bubble swarm in a periodic cubic domain for $1\leqslant \mathit{Eo}\leqslant 5$ , $4\times 10^{-12}\leqslant \mathit{Mo}\leqslant 2\times 10^{-9}$ and ${\it\alpha}\leqslant 45\,\%$ . From their results, they proposed the following relation

(3.12) $$\begin{eqnarray}C_{d}=C_{d_{0}}\left(1+\frac{18}{\mathit{Eo}}{\it\alpha}\right),\end{eqnarray}$$

where $C_{d_{0}}$ is given by the relation proposed by Dijkhuizen et al. (Reference Dijkhuizen, Roghair, Van Sint Annaland and Kuipers2010). This relation suggests that the collective effect of the bubbles on the drag coefficient may depend on other parameters than the gas volume fraction, such as the Eötvös number. The corresponding values of $C_{d}$ are represented by a dashed line in figure 11. They are approximately 30 % higher than the present experimental data, just at the limit of the measurement uncertainty.

4.1. Experimental results

The time evolutions of the oxygen concentration are presented in figure 12 for ${\it\alpha}=1.46\,\%$ (a), ${\it\alpha}=15.1\,\%$ (b) and ${\it\alpha}=26.9\,\%$ (c). In this figure, the time origin has not been shifted by $z/\langle V_{z}\rangle$ so that the signal of the upper probe is delayed compared with the first. The least-squares method is used to fit each set of experimental data by (2.11) in order to obtain the transfer time scale ${\it\tau}$ . The corresponding fitting curves, represented by lines in figure 12, describe accurately the experimental results, confirming that the assumptions made about the probe response and the fact that the flow is well mixed are fulfilled.

Figure 12.

Typical measured time evolutions of the concentration of dissolved oxygen of (a)  ${\it\alpha}=1.46\,\%$ , (b)  ${\it\alpha}=15.1\,\%$ and (c)  ${\it\alpha}=26.9\,\%$ : ●, lower probe; ○ upper probe; – – – and ——, (2.11).

A total of 38 experimental runs have been conducted in the range of $0.7\leqslant {\it\alpha}\leqslant 29.6\,\%$ . The corresponding values of ${\it\tau}$ are reported in figure 13 together with the 29 values measured by Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011) in the range $0.45\leqslant {\it\alpha}\leqslant 16.5\,\%$ . The time necessary to reach the saturation is significantly affected by the void fraction since it decreases by more than one order of magnitude between ${\it\alpha}=1\,\%$ and ${\it\alpha}=30\,\%$ . Such a strong decrease is expected from the strong increase of the interfacial area with ${\it\alpha}$ (figure 6). As it is clearly visible in the log–log plot proposed in the inset of figure 13, the experimental values of ${\it\tau}$ nicely follow a simple power law,

(4.1) $$\begin{eqnarray}{\it\tau}\approx {\it\tau}_{0}\,{\it\alpha}^{-0.8}\quad \text{with}~{\it\tau}_{0}=2.22~\text{s}.\end{eqnarray}$$

Figure 13.

Time scale of the mass transfer versus the gas volume fraction: ▫, lower probe; ▵, upper probe; ○, Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011) (using a single oxygen probe); ——, experimental fit (4.1). Inset: log–log representation.

In order to analyse the collective effect of the bubbles on the mass transfer, we have to consider the mass transfer coefficient by unit of area, $k_{L}$ . The experimental value of $k_{L}$ is obtained from the measured values of ${\it\tau}$ , ${\it\alpha}$ and $a_{I}$ by

(4.2) $$\begin{eqnarray}k_{L}=\frac{(1-{\it\alpha})}{{\it\tau}\,a_{I}}.\end{eqnarray}$$

Combining relations (3.4) for $a_{I}$ and (4.1) for ${\it\tau}$ , the following simple empirical relation is found for the mass transfer coefficient, for ${\it\alpha}>2.3\,\%$

(4.3) $$\begin{eqnarray}k_{L}\approx k_{L_{0}}(1-{\it\alpha})\quad \text{with}~k_{L_{0}}=4.45\times 10^{-4}~\text{m s}^{-1}.\end{eqnarray}$$

Figure 14(a) shows the evolution of the experimental values of $k_{L}$ as a function of the gas volume fraction. The decrease is considerably lower compared with that of ${\it\tau}$ , which indicates that most of the evolution of the total rate of transfer results from the trivial effect of the augmentation of the interfacial area and justifies the efforts made to obtain an accurate determination of $a_{I}$ .

Figure 14.

Mass transfer coefficient (a) and Sherwood number (b) versus gas volume fraction. Experiments: ▫, this work (average values); ○, Colombet et al. (Reference Colombet, Legendre, Cockx, Guiraud, Risso, Daniel and Galinat2011); 

, empirical fits ((4.3) and (4.5)). Predictions for an isolated bubble: ▴, single bubble ((4.12) and (4.13) with the parameters measured for an isolated bubble detached from a single capillary); — ⋅ —, (4.7) (Boussinesq Reference Boussinesq1905);  $\cdots \cdots$ , (4.8) (Winnikow Reference Winnikow1967); – – –, (4.9) (Takemura & Yabe Reference Takemura and Yabe1998); ——, (4.10) (Colombet et al. Reference Colombet, Legendre, Cockx and Guiraud2013); ——, (4.12) and (4.13) (Figueroa & Legendre Reference Figueroa and Legendre2010).

Figure 15.

Comparison of the measured Sherwood number with mass transfer models for turbulent flows. Present experiments: same legend as in figure 14; (4.14) and (4.15) with —— $c_{1}=2/\sqrt{{\rm\pi}}$ (Kawase et al. Reference Kawase, Halard and Moo-Young1987), – – – $c_{1}=0.523$ (Linek et al. Reference Linek, Kordac, Fujasova and Moucha2004), — ⋅ — $c_{1}=0.4$ (Lamont & Scott Reference Lamont and Scott1970).

To go further in the analysis of the physical mechanism underlying the mass transfer, we have to make dimensionless the rate of mass transfer by introducing the Sherwood number

(4.4) $$\begin{eqnarray}\mathit{Sh}=\frac{k_{L}\left\langle d\right\rangle }{D_{L}},\end{eqnarray}$$

where $D_{L}$ is the diffusion coefficient of dissolved oxygen in water. Figure 14(b) shows the evolution of the experimental Sherwood number as a function of ${\it\alpha}$ , which using empirical fits (3.1), (3.2), (4.3) and (4.4) can be described by the following empirical relation, for ${\it\alpha}>2.3\,\%$

(4.5) $$\begin{eqnarray}\mathit{Sh}\approx \mathit{Sh}_{0}(1-{\it\alpha})(1+2.3\,{\it\alpha}^{0.5})\quad \text{with}~\mathit{Sh}_{0}=445.\end{eqnarray}$$

The increase of $\langle d\rangle$ almost compensates for the decrease of $k_{L}$ so that $\mathit{Sh}$ turns out to increase moderately with the gas volume fraction, its values (4.5) remaining between 600 and 750 in the whole range of ${\it\alpha}$ investigated.

The present results therefore suggest that the collective effect of the bubbles has a relatively weak influence on the interfacial mass transfer considering the huge effect observed on the transfer time scale ${\it\tau}$ . However, it is difficult to conclude from the sole evolution of the Sherwood number since variations of bubble size, velocity and shape are associated with variation of gas volume fractions. In next section, we will compare the present results with those corresponding to a single bubble in the same flow regime and with the same geometrical properties.

4.2. Comparison with a single bubble rising in a liquid at rest

In most studies of mass transfer in bubble columns, the rate of transfer is estimated by using Higbie’s penetration theory (Higbie Reference Higbie1935),

(4.6) $$\begin{eqnarray}k_{L}=\frac{2}{\sqrt{{\rm\pi}}}\sqrt{\frac{D_{L}}{t_{c}}},\end{eqnarray}$$

where $t_{c}$ is taken equal to $\langle d\rangle /\langle V_{z}\rangle$ . In fact this is equivalent to the analytical solution obtained by Boussinesq (Reference Boussinesq1905) by considering that the flow around the bubble can be approximated by the potential flow and a very thin concentration layer on the bubble

(4.7) $$\begin{eqnarray}\mathit{Sh}=\frac{2}{\sqrt{{\rm\pi}}}\mathit{Pe}^{1/2},\end{eqnarray}$$

where $\mathit{Pe}=\langle d\rangle \langle V_{z}\rangle /D_{L}$ is the Péclet number. This solution is thus valid in the limit of large $\mathit{Re}$ and $\mathit{Pe}$ .

Various improvements have been proposed to account for the effect of bubble deformation or finite Reynolds number upon the mass transfer from a single bubble. Considering the flow approximation of Moore (Reference Moore1963), Winnikow (Reference Winnikow1967) derived the following expression that includes the effect of the Reynolds number:

(4.8) $$\begin{eqnarray}\mathit{Sh}=\frac{2}{\sqrt{{\rm\pi}}}\left[1-\frac{2.89}{\sqrt{\mathit{Re}}}\right]^{1/2}\mathit{Pe}^{1/2}.\end{eqnarray}$$

Measuring the mass transfer of almost spherical millimetre-sized bubbles from volume variations, Takemura & Yabe (Reference Takemura and Yabe1998) proposed the following relation,

(4.9) $$\begin{eqnarray}\mathit{Sh}=\frac{2}{\sqrt{{\rm\pi}}}\left[1-\frac{2}{3}\frac{1}{(1+0.09\mathit{Re}^{2/3})^{3/4}}\right]^{1/2}(2.5+\mathit{Pe}^{1/2})\end{eqnarray}$$

which was found to be in good agreement with both experiments and numerical simulations at moderate $\mathit{Re}$ numbers and large $\mathit{Pe}$ .

Recently, considering numerical results from various previous works, Colombet et al. (Reference Colombet, Legendre, Cockx and Guiraud2013) proposed the following relation that is valid for a spherical bubble whatever the value of $\mathit{Re}$ and $\mathit{Pe}$ ,

(4.10) $$\begin{eqnarray}\mathit{Sh}=1+\left[1+\left(\frac{4}{3\,{\rm\pi}}\right)^{2/3}\left(2\,\mathit{Pe}_{max}\right)^{2/3}\right]^{3/4},\end{eqnarray}$$

where $\mathit{Pe}_{max}$ is the Péclet number based on the maximal velocity $U_{max}$ of the liquid at the interface instead of the bubble velocity $V_{z}$ , which is obtained from the correlation proposed by Legendre (Reference Legendre2007),

(4.11) $$\begin{eqnarray}\frac{U_{max}}{V_{z}}=\frac{1}{2}\,\frac{16+3.315\mathit{Re}^{1/2}+3\mathit{Re}}{16+3.315\mathit{Re}^{1/2}+\mathit{Re}}.\end{eqnarray}$$

When $\mathit{Pe}$ tends to zero, relation (4.10) tends to the diffusive solution in the absence of flow ( $\mathit{Sh}=2$ ) while it tends towards the Boussinesq solution when $\mathit{Re}$ and $\mathit{Pe}$ become very large.

The effect of the bubble deformation has been studied by Lochiel & Calderbank (Reference Lochiel and Calderbank1964), who considered the potential flow around a spheroidal bubble. They proposed to correct the Boussinesq expression by the introduction of a function $f$ of the aspect ratio  ${\it\chi}$ ,

(4.12) $$\begin{eqnarray}\mathit{Sh}({\it\chi})=\frac{2}{\sqrt{{\rm\pi}}}\mathit{Pe}^{1/2}f({\it\chi}).\end{eqnarray}$$

The validity of this solution has been recently discussed by Figueroa & Legendre (Reference Figueroa and Legendre2010), who proposed the following expression

(4.13) $$\begin{eqnarray}f({\it\chi})=0.524+0.88{\it\chi}-0.49{\it\chi}^{2}+0.086{\it\chi}^{3},\end{eqnarray}$$

which is based on the results of direct numerical simulations, and proved to be valid for $500<({\it\chi}/8)^{1/3}\mathit{Re}<1000$ , ${\it\nu}_{L}/D_{L}>100$ and $1\leqslant {\it\chi}\leqslant 3$ .

The values of $k_{L}$ predicted by all of these expressions derived for an isolated bubble are reported in figure 14(a) while the corresponding values of $\mathit{Sh}$ are reported in figure 14(b). Because the Reynolds number remains almost constant and the bubble shape does not evolve so much in the present experiments, the predictions of all of these correlations are close to each other. Moreover, these predictions are all in agreement with the experiments, considering the measurement uncertainty.

We can therefore conclude that the mass transfer in a homogeneous bubble swarm at high Péclet number is almost independent of the gas volume fraction. It has been proved to remain similar to that of a single bubble rising in a fluid at rest up to a volume fraction of 30 %. This conclusion is in agreement with the trends of the numerical simulations of Roghair (Reference Roghair2012).

4.3. Comparison with the interfacial transfer in highly turbulent flows

The bubbles generate strong velocity fluctuations in the liquid phase. It is thus interesting to compare the rate of transfer measured here with that induced at a plane interface by a turbulence of similar intensity. It has been shown that turbulent eddies can enhance the mass transfer by causing the renewal of the liquid close to the interface (Magnaudet & Calmet Reference Magnaudet and Calmet2006). Considering that the timescale $t_{c}$ of renewal of the liquid at the interface is proportional to $({\it\nu}_{L}/\langle {\it\epsilon}_{L}\rangle )^{1/2}$ , where ${\it\epsilon}_{L}$ is the rate of dissipation of the turbulence, (4.6) gives

(4.14) $$\begin{eqnarray}\mathit{Sh}=c_{1}\left(\frac{d\langle {\it\epsilon}_{L}\rangle ^{1/4}}{{\it\nu}_{L}^{3/4}}\right)\mathit{Sc}^{1/2},\end{eqnarray}$$

where $\mathit{Sc}={\it\nu}_{L}/D_{L}$ is the Schmidt number. Several values have been proposed for the prefactor $c_{1}$ : 0.4 (Lamont & Scott Reference Lamont and Scott1970), 0.523 (Linek et al. Reference Linek, Kordac, Fujasova and Moucha2004) or $2/\sqrt{{\rm\pi}}$ (Kawase et al. Reference Kawase, Halard and Moo-Young1987).

In an homogeneous bubbly flow, Riboux et al. (Reference Riboux, Risso and Legendre2010) showed that the rate of dissipation of the energy is given by

(4.15) $$\begin{eqnarray}\langle {\it\epsilon}_{L}\rangle \approx \frac{{\rm\Delta}{\it\rho}}{{\it\rho}_{L}}\,{\it\alpha}\langle V_{z}\rangle g.\end{eqnarray}$$

According to (4.15), $\langle {\it\epsilon}_{L}\rangle$ ranges from 0.02 to $0.5~\text{m}^{2}~\text{s}^{-3}$ for the range of gas volume fraction considered here. The Sherwood numbers given by relation (4.14) are plotted in figure 15. They are clearly not in agreement with the present measurements. Combining (4.15) and (4.14), it yields

(4.16) $$\begin{eqnarray}\mathit{Sh}=c_{1}\left(\mathit{Eo}^{3}/\mathit{Mo}\right)^{1/8}{\it\alpha}^{1/4}\mathit{Re}^{1/4}\mathit{Sc}^{1/2}.\end{eqnarray}$$

The evolution of Sherwood number with the gas volume fraction predicted by this relation ( ${\it\alpha}^{1/4}$ ) is not compatible with the experimental result. Moreover, the predicted evolution with the Reynolds number ( $\mathit{Re}^{1/4}$ ) is contradictory to the scaling expected considering the evolution for an isolated bubble ( $\mathit{Re}^{1/2}$ ).

This analysis confirms that the mass transfer in the bubble column is controlled by the mass transfer around a single bubble in fluid at rest. The fact that the liquid agitation may play a negligible role in the mass transfer at a bubble interface has already been noticed by Alves et al. (Reference Alves, Vasconcelos and Orvalho2006), who investigated the case of a single bubble in a turbulent flow with a dissipation rate of one order of magnitude smaller than in the present configuration.