3.1. Effect of deformed bubble shape variations on the pressure and surface energy required for bubble breakage

To illustrate the influence of pressure energy control breakup, theoretical predictions of the surface energy and the pressure energy requirements for the breakage of ellipsoidal and spherical-capped bubble are shown, respectively, in Figure 6. It can be clearly seen from Figure 6 that the energy requirement for ellipsoid bubble shifts from pressure energy to surface energy with an increase in the breakup volume fraction. This may be attributed to a higher dynamic pressure being required inside a smaller bubble for resisting the surrounding eddy pressure in order to sustain its own existence. However, the spherical bubble requires most of the surface energy for its breakage. This may mainly be due to the contribution of the large front surface of spherical-capped bubbles.

The surface energy requirement for bubble breakage in Figure 6 has taken into account the bubble shape variations. To further illustrate the significance of considering the variation of bubble shapes, a theoretical comparison of the increase in surface energy for the breakage of the original spherical bubbles versus various shapes of bubbles has been shown in Figure 7.

Figure 6. Two competitive control mechanisms of the breakage of ellipsoidal bubbles (deq,i = 16 mm) and spherical-capped bubbles (deq,i = 32 mm).

Figure 7. Normalized increase in the surface energy for the breakage of original spherical bubbles and various shapes of bubbles.

(r/R < 0.6), while underestimation is shown near the column wall for both the original breakup model and the improved breakup model. Since the standard k˜ε turbulence model is still applied in this study, the underestimation of gas holdup may be due to the slight poor prediction of the turbulence dissipation rate. The issue of underestimation on the gas holdup distribution has also been addressed by Chen et al. [27], in which case the breakup rate was artificially increased by a factor of 10 to obtain a "better" agreement with the

Figure 8. Comparison of the original breakup model and the improved breakup model for the prediction of the gas

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Figure 9 shows the radial distribution of the time-averaged turbulence dissipation rate for Case 1. The turbulence dissipation rate distribution predicted by the standard k˜ε model is smaller than the result obtained by the RNG k˜ε model. This is because the RNG k˜ε model has a specific contribution from the local strain rate as the correction to the turbulence dissipation rate. The tendency of the standard k˜ε model to underestimate the turbulence dissipation rate can also be seen in the studies carried out by Laborde-Boutet et al. [28], Chen [29] and Jakobsen et al. [30]. As a result, the standard k˜ε model is insufficient to properly estimate the turbulence dissipation rate in the regions with rapidly strained flows, which most likely corresponds to the near wall region in the bubble columns. It can be seen from Eq. (19) that the breakup rate <sup>Ω</sup>B˜ε<sup>1</sup>=<sup>3</sup> exp <sup>ε</sup><sup>2</sup>=<sup>3</sup> , which is at least equivalent to the dissipation rate <sup>ε</sup> of the order of 1/3. Therefore, the equilibrium state of bubble coalescence and breakup phenomena cannot be reasonably addressed with an inaccurate estimation of the turbulence dissipation rate and inevitably affect the predictions of gas holdup. Also, as the predicted coalescence rate is about one order of magnitude higher than the predicted breakup rate, the bubble coalescence and

experimental data.

holdup profile in the radial direction for Case 1.

The generation of spherical bubbles due to eddy collision with large ellipsoidal or sphericalcapped bubble is not covered, as the breakage volume fraction will be far smaller than 0.05. The generation of small spherical bubbles occurs more frequently due to the interaction of the shed eddies with the bubble skirt. This phenomenon was concisely described and explained by numerical modelling work carried out by Fu and Ishii [26]. It is shown in Figure 7 that the maximum increase in surface energy for ellipsoidal bubbles and spherical-capped bubbles is different. As binary breakage is assumed, a large ellipsoidal bubble breaks into two smaller ellipsoidal bubbles in most cases. The maximum increase in surface energy is demonstrated when equal-size breakage occurs, which suggests that the parent ellipsoidal bubble has been through a large deformation process itself. However, the spherical-capped bubble can break into different combinations of daughter bubble types, including one ellipsoidal and one spherical-capped bubble, two ellipsoidal bubbles, or two spherical-capped bubbles. The maximum increase in surface energy for the breakage of a spherical-capped parent bubble is found with the largest volume fraction of ellipsoidal daughter bubble. This result coincides with the existing experimental observations: the ellipsoidal bubble has a more stable structure that is able to resist bombarding eddies from both the front and the rear, whereas the sphericalcapped bubble can only resist eddies hitting from the front but is easily and rapidly ruptured by eddies hitting from the rear.

Figure 8 compares the time-averaged gas holdup predicted by the original breakup model and the improved breakup model. It can be found that the improved breakup model has achieved results very similar to the experimental data at the core region of the column

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Figure 8. Comparison of the original breakup model and the improved breakup model for the prediction of the gas holdup profile in the radial direction for Case 1.

(r/R < 0.6), while underestimation is shown near the column wall for both the original breakup model and the improved breakup model. Since the standard k˜ε turbulence model is still applied in this study, the underestimation of gas holdup may be due to the slight poor prediction of the turbulence dissipation rate. The issue of underestimation on the gas holdup distribution has also been addressed by Chen et al. [27], in which case the breakup rate was artificially increased by a factor of 10 to obtain a "better" agreement with the experimental data.

The generation of spherical bubbles due to eddy collision with large ellipsoidal or sphericalcapped bubble is not covered, as the breakage volume fraction will be far smaller than 0.05. The generation of small spherical bubbles occurs more frequently due to the interaction of the shed eddies with the bubble skirt. This phenomenon was concisely described and explained by numerical modelling work carried out by Fu and Ishii [26]. It is shown in Figure 7 that the maximum increase in surface energy for ellipsoidal bubbles and spherical-capped bubbles is different. As binary breakage is assumed, a large ellipsoidal bubble breaks into two smaller ellipsoidal bubbles in most cases. The maximum increase in surface energy is demonstrated when equal-size breakage occurs, which suggests that the parent ellipsoidal bubble has been through a large deformation process itself. However, the spherical-capped bubble can break into different combinations of daughter bubble types, including one ellipsoidal and one spherical-capped bubble, two ellipsoidal bubbles, or two spherical-capped bubbles. The maximum increase in surface energy for the breakage of a spherical-capped parent bubble is found with the largest volume fraction of ellipsoidal daughter bubble. This result coincides with the existing experimental observations: the ellipsoidal bubble has a more stable structure that is able to resist bombarding eddies from both the front and the rear, whereas the sphericalcapped bubble can only resist eddies hitting from the front but is easily and rapidly ruptured

Figure 7. Normalized increase in the surface energy for the breakage of original spherical bubbles and various shapes of

78 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

Figure 8 compares the time-averaged gas holdup predicted by the original breakup model and the improved breakup model. It can be found that the improved breakup model has achieved results very similar to the experimental data at the core region of the column

by eddies hitting from the rear.

bubbles.

Figure 9 shows the radial distribution of the time-averaged turbulence dissipation rate for Case 1. The turbulence dissipation rate distribution predicted by the standard k˜ε model is smaller than the result obtained by the RNG k˜ε model. This is because the RNG k˜ε model has a specific contribution from the local strain rate as the correction to the turbulence dissipation rate. The tendency of the standard k˜ε model to underestimate the turbulence dissipation rate can also be seen in the studies carried out by Laborde-Boutet et al. [28], Chen [29] and Jakobsen et al. [30]. As a result, the standard k˜ε model is insufficient to properly estimate the turbulence dissipation rate in the regions with rapidly strained flows, which most likely corresponds to the near wall region in the bubble columns. It can be seen from Eq. (19) that the breakup rate <sup>Ω</sup>B˜ε<sup>1</sup>=<sup>3</sup> exp <sup>ε</sup><sup>2</sup>=<sup>3</sup> , which is at least equivalent to the dissipation rate <sup>ε</sup> of the order of 1/3. Therefore, the equilibrium state of bubble coalescence and breakup phenomena cannot be reasonably addressed with an inaccurate estimation of the turbulence dissipation rate and inevitably affect the predictions of gas holdup. Also, as the predicted coalescence rate is about one order of magnitude higher than the predicted breakup rate, the bubble coalescence and

Figure 9. Radial distribution of time-averaged turbulence dissipation rate for Case 1.

breakup phenomena cannot be reasonably addressed under this scenario and will inevitably affect the predictions of gas holdup. In addition, as pointed out by Jakobsen et al. [30], despite the accuracy of calculating the local turbulence dissipation rate from the k˜ε turbulence model, this turbulence dissipation rate merely represents a fit of a turbulence length scale to singlephase pipe flow data. Therefore, the contribution of turbulence eddies that are induced by the bubbles has not being included. More importantly, the mechanism of bubble breakage caused by the interactions of bubble-induced turbulence eddies with the subsequent bubbles, which may be dominant in the core region of the bubble column, cannot be revealed through the breakage kernels that are very sensitive to the turbulence dissipation rate.

Figure 10 shows the radial distribution of time-averaged gas holdup at different cross sections in the axial direction. The results are obtained by using the improved breakup model. It can be seen clearly from Figure 10 that the predicted time-averaged gas holdup in the fully developed region (H/D > 5) has achieved self-preserving characteristics regardless of the axial positions. It appears that the inlet conditions have a weak influence on this selfpreserving nature in the bubble columns, which is a result concurring with some previous experimental findings [31, 32].

Figure 11 presents the unit volume-based interfacial area in the bulk region for each bubble class. Due to the large differences in size from the smallest to the largest bubble class, the y-axis is shown in a log10 scale. Interfacial area is a key parameter that largely affects the prediction of heat and mass transfer between gas and liquid phase in the bubble columns. Although the differences in the simulated interfacial area between the improved breakup model and the original breakup

model are not significant when the bubble size is relatively small, the influence of the bubble shapes is gradually reflected when the shape of the bubbles transforms from ellipsoid to spheri-

cal-cap, resulting in much larger interfacial areas for spherical-capped bubbles.

Figure 11. Comparison of the simulated interfacial area in the bubble column for Case 2.

Figure 10. Radial distribution of time-averaged gas holdup at different axial positions.

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Figure 10. Radial distribution of time-averaged gas holdup at different axial positions.

breakup phenomena cannot be reasonably addressed under this scenario and will inevitably affect the predictions of gas holdup. In addition, as pointed out by Jakobsen et al. [30], despite the accuracy of calculating the local turbulence dissipation rate from the k˜ε turbulence model, this turbulence dissipation rate merely represents a fit of a turbulence length scale to singlephase pipe flow data. Therefore, the contribution of turbulence eddies that are induced by the bubbles has not being included. More importantly, the mechanism of bubble breakage caused by the interactions of bubble-induced turbulence eddies with the subsequent bubbles, which may be dominant in the core region of the bubble column, cannot be revealed through the

Figure 10 shows the radial distribution of time-averaged gas holdup at different cross sections in the axial direction. The results are obtained by using the improved breakup model. It can be seen clearly from Figure 10 that the predicted time-averaged gas holdup in the fully developed region (H/D > 5) has achieved self-preserving characteristics regardless of the axial positions. It appears that the inlet conditions have a weak influence on this selfpreserving nature in the bubble columns, which is a result concurring with some previous

Figure 11 presents the unit volume-based interfacial area in the bulk region for each bubble class. Due to the large differences in size from the smallest to the largest bubble class, the y-axis is shown in a log10 scale. Interfacial area is a key parameter that largely affects the prediction of heat and mass transfer between gas and liquid phase in the bubble columns. Although the differences in the simulated interfacial area between the improved breakup model and the original breakup

breakage kernels that are very sensitive to the turbulence dissipation rate.

Figure 9. Radial distribution of time-averaged turbulence dissipation rate for Case 1.

80 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

experimental findings [31, 32].

Figure 11. Comparison of the simulated interfacial area in the bubble column for Case 2.

model are not significant when the bubble size is relatively small, the influence of the bubble shapes is gradually reflected when the shape of the bubbles transforms from ellipsoid to spherical-cap, resulting in much larger interfacial areas for spherical-capped bubbles.

### 3.2. Effect of deformed bubble shape variations on the interfacial mass transfer across bubble surfaces

The interfacial area obtained by the improved breakup model is based on the statistical model of bubble shapes. The results will be slightly different when a more realistic model, which considers the dynamic deformations that occur during bubble motions, is implanted into the simulations. Indeed, the current results have implied that assuming all bubbles to be of a spherical shape may lead to significant underestimation of the interfacial area and hence affect the predictions of the heat and mass transfer rate when chemical reactions are considered in the bubble column reactors. To further address this issue, the volumetric mass transfer coefficient, kLa, estimated based on the improved breakup model for each bubble class is presented in Figure 12.

The convective mass transfer film coefficient can be defined by Eq. (37)

$$k\_{\rm L} = \frac{\overline{D}}{d}Sh\tag{37}$$

Re in Eq. (38) is the bubble Reynolds number and Sc is the Schmidt number. The Schmidt

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Sc <sup>¼</sup> <sup>υ</sup>

According to the analogy between heat and mass transport phenomena, a similar method can be applied to calculate the Nusselt number by simply replacing the Schmidt number with the Prandtl number. By doing so, the ratio of convective heat transfer to conductive heat transfer

It is observed that the volumetric mass transfer coefficient is greatly increased due to the contribution of ellipsoidal and spherical-capped bubbles. However, the peak value obtained based on the improved breakup model may be attributed to the predicted number density of the corresponding bubble class. As illustrated in Figure 7, the improved breakup model requires a higher increase in surface energy at the boundary of ellipsoidal and sphericalcapped bubbles, which makes the smallest spherical-capped bubbles more difficult to break. The results for this particular bubble class may not be a good reflection of the physical phenomenon in reality, but the overall enhancement of the mass transfer coefficient is still very significant. The predictions on the overall mass transfer coefficient are shown in Figure 13. Figure 14 presents the local mass transfer coefficient at different cross sections along the height of the bubble column. It can be seen from Figure 14 that the mass transfer rate estimations based on Luo and Svendsen model and the improved breakup model are obviously very different. The results based on the Luo and Svendsen model may imply that the mass transfer is mainly associated with the regions where the larger Sauter mean bubble diameter has been predicted. The results based on the improved breakup model suggest that the mass transfer is

Figure 13. Volumetric mass transfer coefficient predicted using both the improved breakup model and Luo and Svendsen

<sup>D</sup> (39)

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number is the ratio of momentum diffusivity to mass diffusivity, defined by Eq. (39)

can be characterized.

model [3].

where D is mass diffusivity, d is the bubble diameter and Sh is the Sherwood number. The Sherwood number represents the ratio of the convective mass transfer to the rate of diffusive mass transfer. It can be determined by using the Frossling equation described by Eq. (38)

$$Sh = 2 + 0.552 \text{Re}^{\frac{1}{2}} \text{Sc}^{\frac{1}{3}} \tag{38}$$

Figure 12. Comparison of the volumetric mass transfer coefficient for each bubble class.

Re in Eq. (38) is the bubble Reynolds number and Sc is the Schmidt number. The Schmidt number is the ratio of momentum diffusivity to mass diffusivity, defined by Eq. (39)

3.2. Effect of deformed bubble shape variations on the interfacial mass transfer across

82 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

The convective mass transfer film coefficient can be defined by Eq. (37)

Figure 12. Comparison of the volumetric mass transfer coefficient for each bubble class.

The interfacial area obtained by the improved breakup model is based on the statistical model of bubble shapes. The results will be slightly different when a more realistic model, which considers the dynamic deformations that occur during bubble motions, is implanted into the simulations. Indeed, the current results have implied that assuming all bubbles to be of a spherical shape may lead to significant underestimation of the interfacial area and hence affect the predictions of the heat and mass transfer rate when chemical reactions are considered in the bubble column reactors. To further address this issue, the volumetric mass transfer coefficient, kLa, estimated based on the improved breakup model for each bubble class is presented

kL <sup>¼</sup> <sup>D</sup>

where D is mass diffusivity, d is the bubble diameter and Sh is the Sherwood number. The Sherwood number represents the ratio of the convective mass transfer to the rate of diffusive mass transfer. It can be determined by using the Frossling equation described by Eq. (38)

Sh <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>0</sup>:552Re<sup>1</sup>

2Sc<sup>1</sup>

<sup>d</sup> Sh (37)

<sup>3</sup> (38)

bubble surfaces

in Figure 12.

$$S\mathcal{c} = \frac{\upsilon}{\overline{D}}\tag{39}$$

According to the analogy between heat and mass transport phenomena, a similar method can be applied to calculate the Nusselt number by simply replacing the Schmidt number with the Prandtl number. By doing so, the ratio of convective heat transfer to conductive heat transfer can be characterized.

It is observed that the volumetric mass transfer coefficient is greatly increased due to the contribution of ellipsoidal and spherical-capped bubbles. However, the peak value obtained based on the improved breakup model may be attributed to the predicted number density of the corresponding bubble class. As illustrated in Figure 7, the improved breakup model requires a higher increase in surface energy at the boundary of ellipsoidal and sphericalcapped bubbles, which makes the smallest spherical-capped bubbles more difficult to break. The results for this particular bubble class may not be a good reflection of the physical phenomenon in reality, but the overall enhancement of the mass transfer coefficient is still very significant. The predictions on the overall mass transfer coefficient are shown in Figure 13. Figure 14 presents the local mass transfer coefficient at different cross sections along the height of the bubble column. It can be seen from Figure 14 that the mass transfer rate estimations based on Luo and Svendsen model and the improved breakup model are obviously very different. The results based on the Luo and Svendsen model may imply that the mass transfer is mainly associated with the regions where the larger Sauter mean bubble diameter has been predicted. The results based on the improved breakup model suggest that the mass transfer is

Figure 13. Volumetric mass transfer coefficient predicted using both the improved breakup model and Luo and Svendsen model [3].

Acknowledgements

Nomenclature

support of EPSRC (Grant no. EP/G037345/1).

a long half axis length of a ellipse, m c short half axis length of a ellipse, m

D bubble column diameter, m

deq equivalent bubble diameter, m

Eo Eötvös number, dimensionless

es increase in surface energy, kgm2

<sup>ē</sup> mean turbulence kinetic energy, kgm2

fV breakage volume fraction, dimensionless

kL convective mass transfer film coefficient, m/s

H distance from the bottom surface, m

D mass diffusivity, m<sup>2</sup>

d bubble diameter, m

FD drag force, N/m<sup>3</sup> FLift lift force, N/m<sup>3</sup>

FVM virtual mass force, N/m<sup>3</sup>

g gravity acceleration, m/s<sup>2</sup>

k turbulence kinetic energy, m<sup>2</sup>

dV length of virtual axis, m

This work was supported by the National Natural Science Foundation of China (Grant no. 91534118). Weibin Shi would also like to acknowledge the Ph.D. scholarship of the International Doctoral Innovation Centre (IDIC) of University of Nottingham Ningbo China and the

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This chapter is an extension of a conference paper that was presented at the 13th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics (HEFAT2017) and was nominated for invitation into the HEFAT2017 Special Issue of Heat Transfer Engineering based on its designation as a high-quality paper of relevance to the modelling of fluids based systems.

C<sup>D</sup> effective drag coefficient for a bubble around a swarm, dimensionless

/s2

/s2

/s2

/s

Figure 14. Distribution of estimated volumetric mass transfer coefficient at different cross sections in the bubble column for Case 2. (a) Luo and Svendsen model; (b) improved breakup model. (from top to bottom: H = 0.6, 0.5, 0.4, 0.3 and 0.2 m.).

more uniformly distributed, in which case the enhanced overall mass transfer estimation comes from the statistical sum of the contributions of each bubble class.
