# Study of the dispersed and flooded aeration regimes in two-phase gas-liquid stirred tanks using Positron Emission Particle Tracking

W.J. Peace, R.O. Jones, F. Augier, M.J.H. Simmons, C.R.K. Windows-Yule

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W.J. Peace, R.O. Jones, F. Augier, M.J.H. Simmons, C.R.K. Windows-Yule. Study of the dispersed and flooded aeration regimes in two-phase gas-liquid stirred tanks using Positron Emission Particle Tracking. Chemical Engineering Research and Design, 2025, 219, pp.546-556. 10.1016/j.cherd.2025.06.037. hal-05215275

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# Study of the dispersed and flooded aeration regimes in two-phase gas-liquid stirred tanks using Positron Emission Particle Tracking

W.J. Peace<sup>a</sup>, R.O. Jones<sup>a</sup>, F. Augier<sup>b</sup>, M.J.H. Simmons<sup>a,*</sup>, C.R.K. Windows-Yule<sup>a</sup>

## ARTICLE INFO

**Keywords:**

Positron Emission Particle Tracking
Gas-liquid stirred tanks
Flooded
Dispersed
Mixing effectiveness

## ABSTRACT

This investigation employs Positron Emission Particle Tracking (PEPT) to provide insights into the hydrodynamics and mixing behaviour of two-phase gas-liquid stirred tank systems. Using a T = 0.2 m diameter tank agitated by a standard Rushton turbine (D = C = T/3), industrially relevant superficial gas velocities (U<sub>SG</sub> = 0–1.67 cm s<sup>-1</sup>) are investigated across the dispersed and flooded aeration regimes in a viscous Newtonian fluid. In the dispersed regime, the impeller discharge angle is quantitatively measured, revealing a non-linear dependence on gas flow rate. In the flooded regime, impeller bypassing is evident, accompanied by a stagnation zone forming at the fluid surface and a dominant re-circulation loop that deviates significantly from the classical toroidal flow structure exhibited from a Rushton turbine. Additionally, a Lagrangian-based method is applied for the first time to assess the mixing performance of aerated systems, demonstrating that an increase in gas flow rate profoundly influences the mixing dynamics, regardless of the aeration regime.

## 1. Introduction

Two-phase gas-liquid stirred tank systems are used in a wide range of industrial applications including biological cell cultivation (Nienow, 1990) and broth fermentation processes (Schell et al., 2001; Cappello et al. 2021). Understanding of the multiphase flow within these systems is paramount to optimise the homogeneity of the gas dispersion and mixing throughout the vessel; from a biological perspective, large gradients in concentration of key parameters, such as oxygen concentration and pH, can have a deleterious effect on microbiological growth rates.

Despite their ubiquity, the understanding of gas-liquid stirred tanks is quite limited since studies of the hydrodynamic behaviour in stirred vessels have primarily used optical-based methodologies. This creates a limit on the measurements which can be achieved in two-phase systems using Laser Doppler Anemometry (LDA) (Rousar and van den Akker, 1994; Morud and Hjertager, 1996) or Particle Image Velocimetry (PIV) (Deen and Hjertager, 2002; Deen et al., 2002; Khopkar et al., 2003; Hall et al., 2005) due to the opacity of the mixture which precludes optical access unless very low (<1–2 % by volume) gas loadings are used (Chung et al., 2009). Although such studies have enabled the identification of flow features such as cavity formation behind the impeller blades and determination of velocities of both gas and liquid

components in the flow, the gassing rates used are necessarily well below those used industrially, which gives scope for further measurements using methodologies which do not suffer from optical restrictions.

Positron Emission Particle Tracking (PEPT) is a non-invasive, nucleonic, three-dimensional Lagrangian measurement technique. A small tracer particle, in this investigation an ion-exchange resin particle of diameter d<sub>p</sub> = 300 µm, is radioactively labelled with the radioactive isotope Fluorine-18 and inserted into the experimental system. The radioactive particle undergoes β+ decay, emitting multiple highly penetrating gamma ray pairs, anti-parallel and colinear in nature to within ± 0.5° (Windows-Yule et al., 2020a). The positions of the particle are logged as a function of time, meaning that particle trajectories, velocities and other physical quantities and functions can be derived. The major advantages of PEPT are that it can be used to track the 3-D position of the tracer particle and that it is applicable to opaque systems. Provided the degree to which the tracer particle can be considered as a true fluid velocimeter, by faithfully following the fluid streamlines, the PEPT method has a powerful potential for discovery of new phenomena in such a complex system. Whilst PIV and LDA are pre-eminent in their application to resolve length scales in the flow due to turbulence (with particle tracers typically 0.5–10 µm in size, typically below the Kolmogorov scale), PEPT is nevertheless highly valuable for the interrogation

<footnote><sup>a</sup> School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom</footnote>

<footnote><sup>b</sup> IFP Energies Nouvelles, Rond-point de l'échangeur de Solaize, BP 3, Solaize 69360, France</footnote>

<footnote>* Corresponding author.
E-mail address: m.j.simmons@bham.ac.uk (M.J.H. Simmons).</footnote>
of average flow fields and 3-D structures in the flow via Lagrangian analysis which are not easily observed via conventional 2-D single plane PIV measurements (Adrian, 2005).

PEPT has been applied to study flow phenomena in single phase stirred tanks, particularly for non-Newtonian and solid-liquid systems. The pioneering study of Fangary et al. (2000) identified stagnant areas and regions of poor mixing in the agitation of non-Newtonian carboxymethyl cellulose (CMC) solutions. Analysis of the tracer trajectories enabled calculation of circulation frequency, which was used to assess the overall mixing performance, later refined by Barigou (2004). In addition, PEPT naturally lends itself to the study of solid-liquid systems where the physical properties of the tracer particle (size, density) can be matched to that of the dispersed phase. PEPT measurements have been conducted for a range of such systems using a variety of different impeller types, to identify important hydrodynamic flow features with a variety of different tracer particle sizes (Fangary et al., 2002; Fishwick et al., 2003, 2005; Guida et al., 2009, 2010, Windows-Yule et al., 2020b), including parameters such as particle slip velocities which are important for understanding solid-liquid mass transfer (Fishwick et al., 2003). Methodologies to determine Eulerian velocity distributions and vector fields from the Lagrangian data have been conducted to enable comparison with numerical models of fluid flow obtained using Computational Fluid Dynamics (Hart-Villamil et al., 2024; Sykes et al., 2024).

Previous applications of PEPT for gas-liquid or three-phase gas-solid-liquid agitated systems are very limited. Fishwick et al. explored a two-phase gas-liquid system (Fishwick et al., 2003), investigating the hydrodynamics of up- and down-pumping impellers up to gas flow rates of 1.5 vessel volumes per minute (VVM). More recent applications of PEPT in investigating aerated multi-phase systems has been focused on flotation cells, and their hydrodynamics (Cole et al., 2022; Mesa et al., 2022).

The stirred tank investigations using PEPT presented in this paper have been in part inspired by work carried out previously via an alternative Lagrangian particle tracking technique, Computer-Activated Radioactive Particle Tracking (CARPT), also applied to two-phase gas-liquid stirred vessels (Khopkar et al., 2005). In their study, the authors noted changes in particle circulation time with the increase of the gas flow rate ($U_{SG} = 0.133–0.267 \text{ cm s}^{-1}$) in a standard baffled tank equipped with a Rushton turbine, similar to this study. Nevertheless, the limited amount of understanding of gas-liquid flow in stirred tanks at industrially relevant gassing rates gives considerable scope for further investigation to discover complex 3-D flow phenomena and as a means of validating current or future numerical simulations.

CFD simulations have been conducted previously to obtain the velocity fields of both the gas- and liquid-phases in aerated systems, equipped with a variety of different impellers (Gimbun et al., 2009; Kálal et al. 2014; Cappello et al., 2021; Maluta et al., 2022; Wang et al., 2025). Experimental validation of these simulations remains relatively limited, especially so for three-dimensional flow behaviour and in highly aerated systems. The application of PEPT offers a complementary approach, enabling direct measurement of velocity distributions. Additionally, the Lagrangian nature of the statistics obtained from PEPT experiments provide an additional insight which could be used to further refine CFD models.

In this study, Positron Emission Particle Tracking has been used to obtain the hydrodynamic behaviour of two-phase gas-liquid stirred tank systems containing a viscous Newtonian fluid, at superficial gas velocities which are representative of those typically used in industrial processes. Even though studies have been performed on systems with high superficial gas velocities (Gezork et al., 2000; Cappello et al., 2020), these investigations have typically focused upon the measurement of gas hold-up fractions, bubble sizes or qualitative observations, with no quantitative investigation of the system hydrodynamics. The experiments have been conducted in the transitional flow regime, which is representative of the flow regime in systems that are used in the

production of filamentous fungi (Hardy et al., 2017). The understanding of aeration in the transitional flow regime is limited, even more so with viscous media, due to the previously mentioned limitation for interrogating opaque media and highly aerated systems.

Velocity vector fields are presented, highlighting the substantially different hydrodynamic patterns observed in the flooded and dispersed aeration regimes. Furthermore, quantitative analysis of the mixing effectiveness has been performed, highlighting which aspects have a significant impact on the mixing performance of a two-phase gas-liquid system.

## 2. Materials and methods

### 2.1. Tank configuration and PEPT installation

Fig. 1a shows a schematic of the vessel layout. All experiments were conducted in a cylindrical flat-bottomed tank with diameter $T = 0.2 \text{ m}$, with $H/T = 1$, corresponding to a fluid volume of 6.283 L. While this aspect ratio does not replicate typical industrial bioreactor geometries, the $H/T = 1$ configuration was selected to remain consistent with prior lab-scale studies and to accommodate the installation of the reactor into the PEPT facility. Four rectangular baffles of width $T/10$ were equally spaced along the perimeter of the vessel, to prevent the formation of a surface vortex and therefore induced air entrainment from the surface close to the shaft. Agitation was provided by an IKA Eurostar 200 Control overhead stirrer, with a Rushton turbine of diameter $D = T/3$ and clearance height $C = T/3$, fitted with a shaft of 8 mm diameter. The turbine was of a standard configuration, with a central disc of diameter 3D/4, and six blades equally spaced around the impeller circumference (Rushton et al., 1950). Aeration was provided by a ring sparger of diameter $d = T/4$, at a clearance height of $c = T/6$. A 5 mm diameter tube is connected to the compressed air supply. The sparger had 13 holes positioned equally along the circumference of the sparger, each of 1 mm diameter. The sparger was routed behind one of the wall baffles to avoid disrupting the flow as much as possible (Fishwick et al., 2003, 2005).

Air flow was controlled with a Bronkhorst MASS-STREAM D-6371 mass flow controller, connected to a compressed air system. The air flow was regulated via a personal computer and bespoke Python code.

Experiments were conducted using the ADAC Forte Camera in the Positron Imaging Centre at the University of Birmingham (Parker et al., 1993, 2002). Fig. 1b shows the experimental system within the ADAC Forte camera. The inherent spatial resolution of the ADAC Forte camera is 6 mm (Parker et al., 2002). The applied algorithms and high radioactive count rate mean that the cameras can track the movement of a tracer particle at speeds of approximately 1–1.4 m/s to within sub-millimetre accuracy (Parker et al., 2002; Windows-Yule et al., 2020a). As PEPT is a single-particle measurement technique, it is vital to perform the experiments for a suitably long duration to ensure ergodicity and validity of the velocity statistics obtained (Windows-Yule et al., 2020a), with the experiments performed for at least 3 hours. Particle trajectories were calculated using the PEPT-ML algorithm developed at the University of Birmingham (Nicuşan and Windows-Yule, 2020). Outputs from the algorithm comprised of positions for the particle in three-dimensional space every 5 ms (equivalent to a 200 Hz sampling rate), along with the computed velocity components in the x-, y- and z- directions and the subsequent velocity magnitude.

### 2.2. Fluids and particulate tracer

An ion exchange resin particle of diameter ($d_p = 300 \text{ µm}$) was radioactively labelled with Fluorine-18 ($^{18}$F) ions. Whilst some previous PEPT experiments have used different isotopes such as Gallium-68 ($^{68}$Ga) (Cole et al., 2022), $^{18}$F was chosen due to its availability in the laboratory and its relatively short half-life of 109.8 min (Alauddin, 2012); from a safety perspective equipment was thus only considered as
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Tank schematic; (b) Installation for PEPT measurements.</figcaption></figure>

radioactive for one day and could be handled normally or reset after that time. Previous studies with PEPT have proposed the use of alginate hydrogel tracers (Mesa et al., 2022) for studying gas-liquid flows. A resin particle was chosen due to the ease of labelling with the Fluorine-18 ions. Additionally, a smaller resin particle could be produced ($d_p = 1$ mm of the alginate particle compared to $d_p = 300$ µm of a resin particle) therefore reducing the Stokes number, improving the tracking and flow-following ability of the particle in the fluid. Since a single particle is used in the flow, the PEPT technique can be considered as minimally invasive.

A 70 % v/v aqueous glycerol solution, prepared using Fisher Scientific Glycerol (98+% Extra Pure) and tap water, was used as the continuous phase. The solution was density-matched to the tracer particle ($\rho_L = 1182$ kg m⁻³). Confirmation of neutral buoyancy of the tracer particle in the solution was obtained by performing an experiment where the motion of the particle was observed in the static fluid for two hours. No motion of the particle was observed. The viscosity of the solution was $\mu = 33.32$ mPas, representative of media used in some industrial applications. Additionally, the higher viscosity allows for higher impeller rotation rates to be used for a given value of Reynolds number when compared with a lower viscosity system, enabling a larger number of transits through the system in a given timeframe, maximising the opportunity for the necessary ergodic statistics (Windows-Yule et al., 2020a).

## 2.3. Operating conditions

Flooding occurs when the impeller is unable to pump the gas which rises axially up the impeller shaft to the free surface (Warmoeskerken and Smith, 1985), caused by the impeller speed being too low, the impeller being too small or too high a gassing rate. The occurrence of flooding is obtained via a relationship between the impeller Froude number, $Fr = \frac{N^2D}{g}$, and the gas flow number, $Fl_G = \frac{Q_G}{ND^3}$ (Warmoeskerken and Smith, 1985; Nienow et al., 1985; Nienow, 1998), which applies in the transitional and turbulent hydrodynamic regimes.

$$Fl_G > 30Fr \left(\frac{D}{T}\right)^{3.5} \qquad (1)$$

The flooded and dispersed aeration regimes were investigated as shown in Fig. 2, with the experiments performed marked. Experiments below the red dashed line are classified as operating in the flooded

<figure><img src="image_4.png" /><figcaption>Fig. 2. Aeration regime map for system calculated for $D/T = 1/3$, with experiments performed marked on graph (Derived from Handbook of Industrial Mixing (Paul et al., 2004)).</figcaption></figure>

aeration regime, and those above it are in the dispersed aeration regimes.

The experimental conditions used are shown in Table 1. Experiments in the flooded aeration regime were conducted a constant impeller speed of 190 RPM, corresponding to a Reynolds number of 500. Experiments in the dispersed aeration regime were performed at a constant impeller speed of 530 RPM, corresponding to a Reynolds number of 1400.

<table><caption>Table 1
Experimental conditions.</caption><thead><tr><th>Aeration regime</th><th>N (RPM)</th><th>Re</th><th>Fr (-)</th><th>U<sub>SG</sub> (cm s<sup>-1</sup>)</th><th>Q<sub>G</sub> (L min<sup>-1</sup>)</th><th>VVM (min<sup>-1</sup>)</th><th>Fl<sub>G</sub> (-)</th></tr></thead><tbody><tr><td rowspan="3">Flooded</td><td rowspan="3">190</td><td rowspan="3">500</td><td rowspan="3">0.07</td><td>0.53</td><td>10.05</td><td>1.60</td><td>0.18</td></tr><tr><td>1.00</td><td>18.89</td><td>3.00</td><td>0.34</td></tr><tr><td>1.67</td><td>31.42</td><td>5.00</td><td>0.56</td></tr><tr><td rowspan="3">Dispersed</td><td rowspan="3">530</td><td rowspan="3">1400</td><td rowspan="3">0.53</td><td>0.53</td><td>10.05</td><td>1.60</td><td>0.06</td></tr><tr><td>1.00</td><td>18.89</td><td>3.00</td><td>0.12</td></tr><tr><td>1.67</td><td>31.42</td><td>5.00</td><td>0.20</td></tr></tbody></table>
Gas flow rates, Qg, of 10.05, 18.89 and 31.42 L min⁻¹ respectively were used (Qg = 1.68 × 10⁻⁴, 3.14 × 10⁻⁴ and 5.24 × 10⁻⁴ m³ s⁻¹). The minimum gas flow rate was chosen due to the flow controller having a minimum permissible volumetric flow rate of L min⁻¹. The volumetric gas flow rate (VVM) and superficial gas velocities were selected to fall within ranges that are typical of industrial bioreactors. This ensures that hydrodynamic behaviour remains representative of practical applications, despite the use of a lab-scale system. Single-phase control cases were used for comparison in both the flooded and dispersed aeration regime.

Consideration of whether the particle tracer will faithfully follow the flow is necessary to ensure correct interpretation of the data generated. The Stokes number, Stk, is the ratio of the particle relaxation time to the fluid relaxation time, the former calculated assuming Stokes law applies, shown in Eq. (2):

$$Stk = \frac{\tau_p}{\tau_f} = \frac{\rho_p d_p^2 U_{TIP}}{18 \mu \cdot D} \qquad (2)$$

The corresponding Stokes numbers were calculated as 0.00177 and 0.00494 at N = 190 and 530 RPM respectively, significantly below the critical value of Stk << 0.1. As a result, the particle faithfully follows the flow under these conditions (Maxey and Riley, 1983; Fishwick et al., 2005; Toschi and Bodenschatz, 2009).

## 2.4. Gas hold-up

The average gas hold-up for each system was determined through measuring the change of the height of the fluid free surface (ΔH) and comparing it to the original height of the fluid (H). Three measurements were taken per operating condition, with this result averaged to obtain a final gas hold-up value. This method was deemed appropriate as there were no surface vortices present on any cases, due to a combination of a viscous media as well as the inclusion of baffles within the tank. Visual measurement of the free surface in the flooded cases was difficult due to the chaotic and ever-oscillating nature of the free surface. The percentage gas-hold-up, φg, was obtained from the below equation.

$$\phi_g = 100 \times \frac{\Delta H}{H + \Delta H} \qquad (3)$$

## 2.5. Power input (torque)

Power input and power numbers were determined by measuring the torque imparted onto the impeller. A SGR520 torque transducer (Sensor Technology Limited, Banbury, UK) was installed on the impeller shaft. Three measurements were undertaken at each of the experimental conditions for a duration of 180 s. Further measurements were undertaken without fluid, to determine the losses because of mechanical drag from the bearings and motor, enabling the torque due solely to the transfer of mechanical energy of the fluid to be calculated.

# 3. Results and discussion

## 3.1. Gas hold-up and power input

Values of gas hold-up and power input for each of the conducted experiments are shown in Table 2. Gas hold-up, φg, is reported as the average of triplicate experiments, with the associated uncertainty representing a fixed absolute error of ± 0.5 %, based on the resolution limit of the measurement technique. Similarly, the reported free surface expansion heights (ΔH) carry a fixed uncertainty of ± 1 mm to reflect the measurement precision. As expected, the overall gas hold-up increases with an increasing gas flow rate. Furthermore, the difference between the flooded cases (Re = 500) and dispersed cases (Re = 1400) are also consistent with expectations; flooded cases experience regular surface breakthrough and degassing, with the dispersed cases exhibiting

<table><caption>**Table 2**
Power number and gas hold up measurements.</caption><thead><tr><th>Aeration regime</th><th>U<sub>SG</sub><br>(cm s<sup>-1</sup>)</th><th>ΔH<br>(mm)</th><th>Φ<sub>g</sub> (%)</th><th>Np<br>(-)</th><th>P<sub>I</sub><br>(W)</th><th>P<sub>T</sub><br>(W)</th></tr></thead><tbody><tr><td>Single-Phase (190 RPM)</td><td>0.00</td><td>0</td><td>0.0</td><td>3.18</td><td>0.16</td><td>0.16</td></tr><tr><td>Single-Phase (530 RPM)</td><td>0.00</td><td>0</td><td>0.0</td><td>3.98</td><td>4.22</td><td>4.22</td></tr><tr><td rowspan="3">Flooded (190 RPM)</td><td>0.53</td><td>1 ± 1</td><td>0.5 ± 0.5</td><td>2.69</td><td>0.13</td><td>0.52</td></tr><tr><td>1.00</td><td>2 ± 1</td><td>1.0 ± 0.5</td><td>2.27</td><td>0.11</td><td>0.84</td></tr><tr><td>1.67</td><td>5 ± 1</td><td>2.5 ± 0.5</td><td>2.06</td><td>0.10</td><td>1.32</td></tr><tr><td rowspan="3">Dispersed<br>(530 RPM)</td><td>0.53</td><td>2 ± 1</td><td>1.0 ± 0.5</td><td>1.85</td><td>1.98</td><td>2.37</td></tr><tr><td>1.00</td><td>5 ± 1</td><td>2.5 ± 0.5</td><td>1.72</td><td>1.85</td><td>2.58</td></tr><tr><td>1.67</td><td>8 ± 1</td><td>4.0 ± 0.5</td><td>1.64</td><td>1.76</td><td>2.98</td></tr></tbody></table>

greater gas entrapment. Fig. 3 shows the values compared to previously established correlations shown below:

Flooded aeration regime (Smith and Warmoeskerken (1985)):

$$\phi_g = 1.67 Q_g^{0.75} N^{0.7} \qquad (4)$$

Dispersed aeration regime: (Smith and Warmoeskerken (1985)):

$$\phi_g = 0.62 Q_g^{0.45} N^{1.6} \qquad (5)$$

Any aeration regime: (Greaves and Barigou (1987))

$$\phi_g = 4.07 N^{0.62} Q_g^{0.64} \left(\frac{D}{T}\right)^{1.39} \qquad (6)$$

A±1 mm variation in the expanded height corresponds to approximately a±0.5 % change in calculated gas hold-up. As such, differences between the observed gas hold-up and literature correlations may appear more pronounced. Nevertheless, the overall trends observed are consistent with those reported in the literature.

The total power, P<sub>T</sub>, for an aerated vessel has contributions from the impeller, P<sub>I</sub> (from the torque measurements) and due to the motion of the gas into the vessel; Eq. (7) (Bouaifi et al., 2001) shows the relationship of these contributions, where V is the vessel volume (m³), ρ<sub>I</sub> is the fluid density (kg m⁻³) and g is the acceleration due to gravity (9.81 m s⁻²). Values of power draw and power number are given in Table 2.

$$P_T = P_I + V \rho_I U_{SG} g \qquad (7)$$

As shown in Table 2, increasing the aeration results in a significant reduction of the impeller power input (P<sub>I</sub>), with a corresponding drop in the power number. Despite this, total power input (P<sub>T</sub>) increases with an increase in the U<sub>SG</sub>, indicating a growing contribution from gas-induced motion.

## 3.2. Lagrangian data

Fig. 4 shows two representative trajectory samples of the Lagrangian particle in a single-phase (Fig. 4a; N = 190 RPM, U<sub>SG</sub> = 0 cm s⁻¹) and two-phase gas-liquid stirred tank (Fig. 4b; N = 190 RPM, U<sub>SG</sub> = 1.67 cm s⁻¹) obtained from the PEPT-ML algorithm (Nicuşan and Windows-Yule, 2020). In the aerated case. The data show the unsteady nature of the particle motion due to periodic contributions from the impeller blade passage causing large scale low frequency structures on the order of 1 Hz or lower within the flow, independent of the degree of aeration for the cases chosen. Due to the transitional flow regime, there appears to be some evidence of higher frequency disturbances which may be attributed to the initiation of turbulence. Kolmogorov length scales (λ<sub>K</sub>) were estimated for representative single-phase conditions. Based on the calculated energy dissipation rates, λ<sub>K</sub> was found to be approximately 0.93 mm at N = 190 RPM and 0.45 mm at N = 530 RPM. Given that the d<sub>p</sub> < λ<sub>K</sub>, this lies below the estimated dissipation length scale. However, given the spatial resolution of the PEPT technique (see §2.1), flow features at or below the Kolmogorov scale cannot be directly resolved.
<figure><img src="image_5.png" /><figcaption>Fig. 3. Percentage gas hold-up measurements with comparison to correlations (Smith and Warmoeskerken (1985); Greaves and Barigou (1987)). Error bars represent the standard error from the mean. (a) dispersed aeration regime and (b) flooded aeration regime.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Representative sample of PEPT trajectories in a stirred tank; (a) N = 190 RPM, U<sub>SG</sub> = 0 cm s<sup>-1</sup>; (b) N = 190 RPM, U<sub>SG</sub> = 1.67 cm s<sup>-1</sup> Z is the axial direction.</figcaption></figure>

The Lagrangian data are further processed to produce Eulerian ve-
locity distributions. Instantaneous velocities are captured using a two-
dimensional, fitted polynomial approach (Nicuşan and Windows-Yule,
2020). Data was divided into a cylindrical grid, with ensemble aver-
aging conducted per voxel, to compute the velocity components u<sub>r</sub>
(radial), u<sub>z</sub> (axial) and u<sub>θ</sub> (azimuthal) velocities respectively. This was
used to illustrate the hydrodynamic behaviour through the remainder of
this paper. Unless otherwise stated, reported velocity magnitudes
(U<sub>MAG</sub>) are calculated as U<sub>MAG</sub> = √(u<sub>r</sub><sup>2</sup> + u<sub>θ</sub><sup>2</sup> + u<sub>z</sub><sup>2</sup>).

3.3. Hydrodynamics of dispersed system Re=1400, N=530 RPM)

Fig. 5 shows the axial velocity profiles of the normalised velocity magnitude (U<sub>MAG</sub>) for (a) U<sub>SG</sub> = 0 cm s<sup>-1</sup> (single-phase) and (b) U<sub>SG</sub> = 1.67 cm s<sup>-1</sup> cases at r/R = 0.25 (inside the volume swept by the impeller), 0.33 (at the blade tips), 0.50 and 0.75 (both in the fluid bulk flow region). The peak velocities at r/R = 0.25 and 0.33 are similar in both the single-phase and aerated cases. A faster velocity decrease is observed in the aerated case. Notable differences are present at r/R = 0.50 and 0.75, with a decrease in the velocities in these areas. This can

be attributed to a combination of the reduced mechanical power input
by the impeller and changes in the bulk fluid behaviour resulting from
gas dispersion, both of which impact the system's ability to circulate and
mix effectively. Table 2 shows a reduction in power of 29.4 % compared
with the ungassed case.

Below the impeller (z/H < 0.33), velocities are generally reduced by
20 %–25 % in the aerated case than when compared to the single-phase
equivalent, with much less influence of gassing above the impeller. At r/
R = 0.25, the location of the maximum velocity in the impeller discharge
is slightly higher, this progression increases further away from the
impeller indicating an upward turn of the discharge jet under gassing.
This suggests the presence of buoyancy effects due to the gas phase
impacting on the overall hydrodynamics.

Fig. 6 shows an azimuthally averaged vector field for the single-phase case (a) and the USG = 1.67 cm s⁻¹ case (b). The single-phase vector field obtained via PEPT in Fig. 6a follows the classical flow field expected for a Rushton turbine (Paul et al., 2004). For the aerated case (Fig. 6b), a weak (barely visible) secondary recirculation loop is present close to the fluid free surface; the likely cause is the presence of the upwards moving gas phase disengaging at the free surface. The velocity in this region is low compared to the impeller discharge region,
<figure><img src="image_7.png" /><figcaption>Fig. 5. Normalised velocity magnitude at N = 530 RPM located at r/R = 0.25, 0.33, 0.50 and 0.75. Dashed line indicates impeller plane. (a) U<sub>SG</sub> = 0 cm s<sup>-1</sup>, (b) U<sub>SG</sub> = 1.67 cm s<sup>-1</sup>.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Azimuthally averaged velocity vector field: (N = 530 RPM, Re = 1400); (a) U<sub>SG</sub> = 0 cm s<sup>-1</sup>, (b) U<sub>SG</sub> = 1.67 cm s<sup>-1</sup> (Dispersed aeration regime).</figcaption></figure>

which is reflected in the small magnitude of the vectors in the figure. In addition, the central stagnation point within the upper fluid recirculation loops has been translated slightly closer to the centre of the vessel; the converse is true for the lower loop, with both loops becoming more elliptical in shape.

The angle of the impeller discharge jet can be approximated using Eq. (8), with the maximum values of the time-averaged velocity magnitude at r/R = 0.33, 0.50 and 0.75 being used to calculate the values. Angles of 4.5°, 4.9° and 12.9° were obtained at values of U<sub>SG</sub> =

0.53, 1.00 and 1.67 cm s<sup>-1</sup> respectively.

$$ \alpha_d = \tan^{-1} \left( \frac{\Delta h}{\Delta r} \right) \qquad (8) $$

where Δh and Δr are the changes in the normalised axial and radial positions. Although the increase in the discharge angle as a function of an increasing gas flow rate has been qualitatively observed before in previous studies (Nienow et al., 1985; Nienow, 1998; Deen
et al., 2002) under different operating conditions, the measurements of discharge angle are presented here quantitatively for the first time to the authors' knowledge.

### 3.4. Hydrodynamics of the flooded system Re = 500, N = 190 RPM)

Fig. 7 shows the velocity distribution for the flooded system for (a) the single-phase case and (b) at a superficial gas velocity of $U_{SG}$ = 1.67 cm s⁻¹. The normalised velocity magnitudes shown in Figs. 5a and 7a show that the normalised velocity distributions are not self-similar at the values of Re chosen in the transitional regime. This is not unexpected since literature has highlighted that self-similarity is only prevalent in the very high-transitional regime and above (Re ≈ 15,000) (Yoon et al., 2005). Previous work using PEPT on single-phase systems has discovered that the normalised velocities are directionally proportional to a weakly scaled Reynolds number (Chiti et al., 2011).

Upon flooding, the hydrodynamics of the tank are radically different and dominated by an upward moving jet of fluid; with little to no impact of the impeller on the hydrodynamics. The fluid velocities from z/H = 0.66 in the flooded case shown in Fig. 7 are at least twice that of the single-phase case in Fig. 7a, highlighting the significant additional momentum transfer caused by the rising gas. Fig. 8 shows the profound differences in the velocity vector fields between the single-phase (a) and flooded (b) cases. In Fig. 8b, below the impeller, significant stagnant regions are noticeable in the bottom section of the tank. Above the impeller, the fluid is experiencing significant agitation due to the throughput of gas. A single circulation loop is present, elongated axially in the flow field. The stagnation point of the circulation loop is at approximately z/H = 0.6, much higher than any of the circulation loops shown in single-phase cases, or in the aerated dispersed case. Around the impeller plane at z/H = 0.33, there is a noticeable jet of the fluid bypassing the impeller. This occurs at the radial position of the blades, indicating a presence of large cavities behind the impeller blades (Warmoeskerken and Smith, 1985; Nienow, 1998).

### 3.5. Assessment of mixing quality via particle dispersion

The mixing performance of a stirred tank system has important implications for productivity; for biological processes such as cell cultivation and broth fermentation, it governs the gradients of dissolved oxygen and the homogeneity of the media which have a profound effect on microbiological growth (and death). Assessment of mixing performance is typically based on mixing or circulation times, with these measurements typically being grouped into invasive and non-invasive measurement techniques (Nere et al., 2003; Asciano, 2015). Whilst invasive

measurements using conductivity probes have been proven to be successful (Grenville, 1992), the impact on the flow field can be significant (Charalambidou et al., 2024). Non-invasive measurements typically use colorimetric methods such as dyes (Alméras et al., 2016; Delbridge et al., 2023), which are only suitable for transparent systems.

An alternative method of assessing mixing performance can be obtained by consideration of the dispersion of tracer particles through calculation of a mixing effectiveness. This provides an alternative perspective of the mixing performance both locally and overall, by using the Lagrangian trajectories obtained by PEPT. Although this work focuses on mixing effectiveness as a metric, the PEPT technique could also be applied to estimate mixing time by tracking the evolution of tracer dispersion. Initially proposed by Martin et al. (2007), the quantity of dispersion is a technique that measures the extent to which a group of particles sharing an approximately identical starting position at an arbitrary start time (t = t₀) become separated from one another at some subsequent time (t₁ = t₀ + Δt). In this investigation, a Δt of 100 ms is used.

The quality of mixing is obtained by measuring the rate at which these particles disperse from their initial point. This is performed by segmenting the original continuous single-particle trajectory into a series of shorter sub-trajectories, allowing for the statistical properties to be calculated locally from repeated transits throughout each voxel. Given that the tank is axisymmetric, a modified version of the equation proposed by Martin et al. (2007) is used, where the azimuthal component is azimuthally averaged. This approach is justified because variations in the azimuthal direction are negligible compared to those in the radial and axial directions.

Firstly, the particle dispersion is computed by integrating over each voxel in the PEPT measurements using Eq. (9) (Martin et al., 2007).

$$ \sigma^2 = \frac{1}{n_v} \sum_{i=1}^{n_v} (r_i - \bar{r})^2 + (z_i - \bar{z})^2 \quad (9) $$

Where $\sigma^2$ is the variance of the particle position (a measure of the particle spread from a loosely grouped region, with units of length²), i is the number of the voxel in which the calculation is referring to, $r_i$ and $z_i$ to the r and z positions of the i-th voxel, $\bar{r}$ and $\bar{z}$ are the mean positions of the particles in the r and z co-ordinates, respectively and $n_v$ is the number of voxels to describe the system, in this case, $n_v = 512,000$ voxels. This calculation was performed using the Up⁴ open-source software package (Werner, 2022). The choice of the number of voxels represents a trade-off between fidelity in the identification of the flow features within the system and ensuring ergodicity in the particle statistics.

<figure><img src="image_9.png" /><figcaption>Fig. 7. N = 190 RPM, Normalised velocity magnitude at r/R = 0.25, 0.33, 0.50 and 0.75. Dashed line indicates impeller plane. (a) $U_{SG}$ = 0 cm s⁻¹, (b) $U_{SG}$ = 1.67 cm s⁻¹.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8. Azimuthally averaged velocity vector field: (N = 190 RPM,Re = 500); (a) U<sub>SG</sub> = 0 cm s<sup>-1</sup>; (b) U<sub>SG</sub> = 1.67 cm s<sup>-1</sup> (Flooded aeration regime).</figcaption></figure>

The variance of the particles is then utilised to calculate the mixing effectiveness (ME, mm<sup>2</sup>) of the system (Martin et al., 2007).

$$ME = \frac{1}{n_p} \sum_{i=1}^{n_v} \sigma^2 n_{p,i} \qquad (10)$$

Where n<sub>p,i</sub> is the number of passes per particle through each 1<sup>th</sup> voxel. The weighting of the mixing effectiveness in Eq. (9) has no impact on the units assigned. Fig. 9 shows the evolution of the mixing effectiveness over time for the (a) dispersed and (b) flooded aeration regimes respectively for the whole tank.

The rate of increase of ME with time, i.e. the slope of the curves presented in Fig. 9, is an indicator of the mixing performance in the system. While steady-state is not reached within the time window shown, the trends in the early-stage mixing provide meaningful insight into the influence of gas flow rate on mixing performance. In Fig. 9a, the initial rate is greater for U<sub>SG</sub> = 0 cm s<sup>-1</sup>. This is to be expected with the increased power input into the system, as shown in Table 2. This also correlates with the lowest mixing effectiveness being observed at the U<sub>SG</sub> = 0.53 cm s<sup>-1</sup> operating condition. When increasing the superficial gas velocity, the mixing effectiveness increases. A greater final ME value

<figure><img src="image_11.png" /><figcaption>Fig. 9. Evolution of ME in the (a) dispersed aeration regime (b) flooded aeration regime.</figcaption></figure>
is achieved at the end of the measurement window (t = 1500 ms), indicating that the input of gas increased the maximum extent of the particle movement in the system. These trends are to be expected given the increased power input into the system, with Table 2 illustrating the dual contribution of the impeller-driven and gas-induced mixing, resulting in a greater mixing efficiency.

The flooded systems shown in Fig. 9b illustrate a different trend, with the increase of gas having a significant impact on the overall mixing effectiveness. Similar mixing efficiencies are observed at the $U_{SG} = 0.53 \text{ cm s}^{-1}$ and the $U_{SG} = 1.00 \text{ cm s}^{-1}$ operating conditions, but there is a noticeable difference between those, and the $U_{SG} = 1.67 \text{ cm s}^{-1}$ case. The trend here illustrates that the mixing is strongly governed by the gassing rate, with the power in the gassed cases being at least double compared to the single-phase equivalent case.

While the slope reflects the initial rate of mixing, the final value of ME at t = 1500 ms also provides insight into the extent of dispersion achieved. Although steady state is not reached, both the slope and the asymptotic behaviour of the ME curves highlight the influence of gas flow rate on mixing performance.

The mixing effectiveness can also be computed on a voxel-by-voxel basis to enable spatial distributions to be obtained. Fig. 10 shows the spatial distributions of the mixing effectiveness as azimuthally averaged scalar fields at t = 5000 ms, for the dispersed (left) and flooded (right) aeration regimes.

The areas of highest mixing effectiveness, unsurprisingly, are localised towards the areas of high velocity in each of the two aeration regimes. In the dispersed aeration regime, the recirculation loops above and below the discharge region show a relatively high level of mixing effectiveness, highlighting an additional insight of the homogeneity of the gas dispersion in the vessel. Poor mixing effectiveness is observed at the tank walls and base, as expected given the lower fluid velocities in those regions. In the flooded aeration regime, higher areas of mixing effectiveness are localised towards the vessel axis, providing an additional indicator that the gas rising from the sparger is driving all the

mixing occurring in the vessel. A substantial dead zone is at the bottom outer region of the vessel, with no statistics present. This indicates that the particle did not transit to this area of the tank, highlighting the local mixing performance in this area was extremely poor.

To enable a direct comparison of the overall mixing performance between operating conditions, we have compared scalar values of the ME at t = 1500 ms, taken from the curves presented in Fig. 9. While this time point does not represent a full steady state, it provides a consistent basis for comparison across aeration regimes and gas flow rates. The resulting “mixing index” values are summarised in Table 3. In the dispersed aeration regime, a non-monotonic relationship between gas input and mixing behaviour is observed. ME initially decreases at $U_{SG} = 0.53 \text{ cm s}^{-1}$ but then increases as the gas flow rate increases. The flooded regime shows a more gradual and consistent increase in mixing index with an increase in $U_{SG}$. These results align with observations made in Fig. 9 and support the conclusion that dispersed systems exhibit improved mixing efficiency, with gas-induced buoyancy being the major driving factor of mixing in the flooded aeration regime.

## 4. Conclusions

Positron Emission Particle Tracking has been used to obtain

<table><caption>**Table 3**
Comparison of ME as a function of aeration regime at t = 1500 ms.</caption><thead><tr><th>Aeration regime</th><th>U<sub>SG</sub> [cm s<sup>-1</sup>]</th><th>Mixing index (ME at 1500 ms) [mm<sup>2</sup>]</th></tr></thead><tbody><tr><td rowspan="4">Flooded (190 RPM)</td><td>0.00</td><td>2650</td></tr><tr><td>0.53</td><td>3870</td></tr><tr><td>1.00</td><td>3660</td></tr><tr><td>1.67</td><td>4110</td></tr><tr><td rowspan="4">Dispersed (530 RPM)</td><td>0.00</td><td>5380</td></tr><tr><td>0.53</td><td>4900</td></tr><tr><td>1.00</td><td>5740</td></tr><tr><td>1.67</td><td>5700</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Azimuthally Averaged Mixing Effectiveness Fields at t = 5000 ms: (a) Dispersed Aeration Regime (N = 530 RPM, Re = 1400, $U_{SG} = 1.67 \text{ cm s}^{-1}$) (b) Flooded Aeration Regime (N = 190 RPM, Re = 500, $U_{SG} = 1.67 \text{ cm s}^{-1}$).</figcaption></figure>
quantitative measurements of the hydrodynamics and the mixing effectiveness in two-phase gas-liquid stirred tanks in the dispersed and flooding regimes in transitional flow.

In the dispersed aeration regime, the angle of the discharge jet from the impeller, as a function of gas input have been measured. PEPT measurements show that the velocities are more homogeneous in the bulk region above the impeller, with the inclusion of gas. In the flooded regime, a high velocity plume of media rises about the central axis of the tank for all cases, with significant regions of fluid stagnation present in the lower portion of the tank with the hydrodynamics being radically different from the single-phase case.

A Lagrangian analysis has been applied to investigate mixing performance, via the mixing effectiveness calculation from the cases investigated. In the dispersed aeration regime, adding a small quantity of gas reduces the mixing effectiveness compared to a single-phase control case. However, increasing the gas flow rate further causes the mixing effectiveness to improve. In the flooded cases, adding gas significantly improves the mixing performance, however, as would be expected, mixing in the dispersed aeration regime is considerably better than in the flooded regime.

## CRediT authorship contribution statement

**William James Peace:** Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Conceptualization. **Ryan Jones:** Resources, Methodology. **Frédéric Augier:** Writing – review & editing, Supervision, Funding acquisition, Conceptualization. **Mark Simmons:** Writing – review & editing, Supervision, Project administration, Funding acquisition, Conceptualization. **Christopher Robert Kit Windows-Yule:** Writing – review & editing, Supervision, Project administration, Funding acquisition, Conceptualization.

## Declaration of Competing Interest

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

## Acknowledgements

The authors would like to thank the University of Birmingham's Positron Imaging Centre for the radiolabelling of the particles. WJP is funded by a University PhD scholarship from the School of Chemical Engineering at the University of Birmingham and IFP Energies Nouvelles. Funding for the torque experiments is acknowledged through the EPSRC PREMIRE Programme Grant EP/T000414/1.

Computational resources for the PEPT data processing were provided by the University of Birmingham's Birmingham Environment for Academic Research (BEAR) service. BEAR provides flexible computational resources for the University of Birmingham research community. Additional information can be provided here: http://www.birmingham.ac.uk/bear

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