# Computational Fluid Dynamics Modelling of Two-Phase Bubble Columns: A Comprehensive Review

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## Abstract

**:**

## 1. Introduction

**Table 1.**Main applications of bubble column reactors [3].

Process | Reactants | Main Products |
---|---|---|

Oxidation | Ethilene, butane, toluene, xylene, ethylbenzene, cyclohexene, n-paraffins, glucose | Vinyl acetate, phenol, acetone, methyl ethyl ketone, benzoic acid, phthalic acid, acetophenone, acetic acid, acetic anhydride, cyclohexanol and cyclohexanone, adipic acid, sec-alcohols, glutonic acid |

Chlorination | Aliphatic hydrocarbons, aromatic hydrocarbons | Choloroparaffins, chlorinated aromatics |

Alkylation | Ethanol, propylene, benzene, toluene | Ethyle benzene, cumene, iso-butyl benzene |

Hydroformylation | Olefins | Aldehydes, alcohols |

Carbonylations | Methanol, ethanol | Acetic acid, acitic anhydride, propionic acid |

Hydrogenation | Benzene, adipic acid dinitrile, nitroaromatics, glucose, ammonium nitrate, unsaturated fatty acids | Cyclohexane, hexamethylene diamine, amines, sorbitol, hydroxyl amines |

Gas to Liquid Fuels (Fischer-=Tropsch) | Syngas | Liquid fuels |

Coal Liquification | Coal | Liquid fuels |

Desulferization | Petroleum fractions | Desulferize fractions |

Aerobic Bio-Chemical Processes | Molasses | Ethanol |

## 2. Flow Regimes

## 3. Numerical Modelling: The Eulerian–Eulerian Approach

#### 3.1. Governing Equations

#### 3.2. Interfacial Forces

**Figure 5.**Schematic representation of the interphase forces acting on a bubble. Reprinted with permission from [13].

#### 3.2.1. Drag Force

#### 3.2.2. Swarm Factor Definitions

#### 3.2.3. Lift Force

#### 3.2.4. Turbulent Dispersion Force

#### 3.2.5. Wall Lubrication Force

#### 3.2.6. Virtual Mass Force

#### 3.3. Population Balance Modelling

#### 3.3.1. Bubble Breakup Phenomena Modelling

- Turbulent fluctuations and collisions, in which breakage is mainly caused by turbulent pressure fluctuations along the surface or by particle-eddy collisions. The dominant external force is the dynamic pressure difference around the bubble, meaning that the breakage process can be studied as the balance between the dynamic pressure and surface stresses.
- Viscous shear stresses, which cause a velocity gradient around the interface that can deform or break the bubble. In addition, wake effects may be responsible for the formation of shear stresses. Breakage can be modeled as the balance between external viscous stresses and surface tension forces as expressed by means of the capillary number ${C}_{a}$.
- Shearing-off processes, which occur when small bubbles are sheared off from a large bubble through erosive breakage.
- Interfacial instabilities, which arise in the absence of a continuous phase net flow. If a significant density difference is present, as in the case of a light liquid accelerated into a heavy fluid, Rayleigh–Taylor instabilities are found; on the other hand, if the density ratio is close to unity Kelvin–Helmholtz instabilities exist.

#### 3.3.2. Bubble Coalescence Phenomena Modelling

- Turbulence-induced collisions occur as a result of the random motion of bubbles caused by turbulent fluctuations.
- Viscous shear-induced collisions are generated by global liquid velocity gradients, meaning that bubbles in a location of high liquid velocity may collide with those in a region of low liquid velocity.
- Eddy-capture, in which collision events are produced by the shear rate of the flow in the turbulent eddy.
- Bouyancy-induced collisions, in which collisions may occur because bubbles of different size have different rising velocities.
- Wake entrainment collisions, which may result when bubbles are accelerated by the wake region behind a large spherical-cap bubble.

- Film drainage model: first proposed by Shinnar and Church (1960) [57], this is a widely used model of coalescence efficiency. When two bubbles collide, a thin liquid film is trapped between their surfaces and is progressively drained. If the contact time is sufficiently high, the liquid film reaches a minimum thickness, then ruptures, causing coalescence.
- Energy model: first proposed by Howarth (1964) [58], this model is based on the concept of collision energy, in which a higher collision energy indicating a higher probability of coalescence.
- Critical approach velocity model: in this model, collisions result in coalescence phenomena if the approach velocity of the bubbles exceeds a certain critical value; otherwise, they bump into or bounce off of each other, and do not coalesce.

#### 3.3.3. Solution Methods

- Class (or Sectional) Method (CM): the internal coordinate domain is divided into a finite number of intervals (or bins), transforming the population balance equation into a set of balance equations in the physical domain. Any coalescence and/or breakup event is accompanied by the migration of particles from one class to the adjoining classes. The advantages of this method are its robust numerics and that it computes the Particle Size Distribution directly [62].
- Monte Carlo Method: this method solves the population balance equation based on the statistical ensemble approach, accurately tracking particulate changes in a multidimensional system. Nevertheless, the method accuracy strongly depends on the number of simulation particles, and requires an extensive computational time to track large numbers of particles. This makes the Monte Carlo method poorly compatible with the conceptual framework of Computational Fluid Dynamics [47].
- Standard Method of Moments (SMM): the population balance equation is turned into a set of transport equations for the moment of the particle size distribution. The primary advantage is numerical economy, as it is sufficient to solve a limited number of moment equations. Mathematically, the transformation from the space of particulate size distribution to the space of moments is extremely rigorous, and fractional moments, representing the Sauter mean diameter of the bubbles, present a serious closure problem [47]. This closure constraint can be overcome by resorting to a Quadrature Method of Moments (QMOM) approach.
- Quadrature Method of Moments (QMOM): first suggested by McGraw (1997) [63] for modelling aerosol and coagulation problems, QMOM was later applied by Marchisio et al. (2003) [64] for solving the population balance equation, becoming an attractive alternative. Compared to the SMM method, this approach solves only the transport equations of the low-order moments; however, it is able to overcome the closure problem of the SMM method [64]. With the QMOM, the integral terms in the momentum transport equations are approximated by employing an N-node Gaussian quadrature formula. This quadrature approximation requires knowledge of N weights (${\omega}_{i}$) and N nodes of abscissas ${L}_{i}$, and determines a sequence of polynomials orthogonal to the unknown number density function. The functional form (for a univariate problem with $\varphi $ as internal coordinate) reads as follows:$${\int}_{0}^{\infty}f\left(\varphi \right)n\left(\varphi \right)\mathrm{d}\varphi \approx \sum _{i=1}^{N}f\left({L}_{i}\right){w}_{i}$$When dealing with a multivariate population balance equation, for which the product–difference algorithm can not be applied, other extensions of QMOM are available, such as the Conditional Quadrature Method of Moments (CQMOM) or the Direct Quadrature Methods of Moments (DQMOM). In DQMOM, the transport equations are directly solved for the weights and nodes of the quadrature approximation, whereas CQMOM represents the multivariate extension of QMOM. Moreover, in CQMOM closure is achieved by means of multivariate quadrature approximation, and the transport equations for the moments of the distributions are solved [65].

#### 3.4. Turbulence Modelling

#### Bubble-Induced Turbulence

## 4. Literature Survey

Ref. | Year | Code | D * [m] | Sparger | AR [-] | Fluids | ${\mathit{U}}_{\mathit{G}}$ [m/s] | Flow Regime |
---|---|---|---|---|---|---|---|---|

[74] | 2001 | Ansys CFX-4.3 | 0.288 | Ring ** ${d}_{0}$ = 0.7 mm | 8.68 | Air/water | 0.5 | Homogeneous |

[78] | 2005 | Ansys FLUENT | 0.19 | Perforate plate ${d}_{0}$ = 3.3 mm | 5.05 | Air/water | 12 | Heterogeneous |

[46] | 2008 | Ansys CFX-10.0 | 0.60 | Perforated plate *** | - | Air/water | 1.2 → 9.6 | Homogeneous→ heterogeneous |

[79] | 2010 | Ansys CFX-10.0 | 0.15 | Perforated plate *** ${d}_{0}$ = 1.96 mm | 6 | Air/water | 0.2 | Homogeneous |

[45] | 2013 | Ansys CFX-13.0 | 0.288 | Perforate plate ${d}_{0}$ = 0.7 mm | 8.68 | Air/water | 0.15 → 1 | Homogeneous |

[80] | 2013 | Ansys FLUENT-14.0 | 0.44 | Perforate plate ** ${d}_{0}$ = 0.77 mm | 4 | Air/water | 10 | Heterogeneous |

[81] | 2013 | Ansys CFX-14.5 | 0.19 | Perforated plate *** ${d}_{0}$ = 1 mm | 2.63 | Air/water | 8 → 12 | Heterogeneous |

[71] | 2014 | Ansys CFX-13.0 | 0.24 × 0.72 | Needle ** | 5.08 | Air/water | 0.3 → 1.3 | Homogeneous |

[82] | 2014 | Ansys CFX-14.0 | 0.15 × 0.15 | Full opening ** | 2.74 | Air/water | 5 → 12 | |

[83] | 2014 | OpenFOAM | 0.2 | Perforate plate ** ${d}_{0}$ = 1.2 mm | 5 | Air/water | 10 | Heterogeneous |

[88] | 2014 | Ansys CFX-14.5 | 0.24 × 0.72 | Needle ** | 5.08 | Air/water | 13 | |

[89] | 2016 | Ansys FLUENT-14.5 | 0.4 | Perforate plate ** | 4 | Air/water | 3 → 25 | Heterogeneous |

[84] | 2017 | Ansys FLUENT-17.2 | 0.138 | Perforate plate ${d}_{0}$ = 4 mm | 6.52 | Air/water | 19 → 38 | Heterogeneous |

[90] | 2017 | Ansys CFX (1) Ansys FLUENT (2) | 0.39 | Tree ** ${d}_{0}$ = 0.5 mm | 5.13 | Air/water | 14 → 28 | Heterogeneous |

[91] | 2018 | Ansys CFX-15.0 | 0.156 | Perforate plate ** ${d}_{0}$ = 1 mm | 1.9 → 5.11 | Air/water | 2.1 | Homogeneous |

[30] | 2018 | Ansys FLUENT-15.0 | 0.40 | Perforate plate ** ${d}_{0}$ = 2 mm | 4 | Air/water | 0.03 → 35 | Homogeneous → heterogeneous |

[85] | 2020 | Ansys FLUENT-17.2 | 0.15 | Perforate plate ** ${d}_{0}$ = 1.5 mm | - | Air/water | 23 | Heterogeneous |

Ref. | Dispersed Phase Model | Coalescence Model | Breakup Model | Momentum Exchange | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Frequency | Efficiency | Drag | Swarm Factor | Lift | Turbulent Dispersion | Wall lubrication | Virtual Mass | |||

[74] | Mono-dispersed | No | No | No | ${C}_{D}$ = 0.44 | No | No | No | No | No |

[78] | MUSIG 16 classes | Prince and Blanch | Chesters (A) Luo (B) Prince and Blanch (C) | Luo and Svendsen (1) Martinez-Bazan (2) | Schiller and Naumann | No | No | No | No | No |

[46] | Mono-dispersed | No | No | No | Schiller and Naumann (A) Grace (B) Ishii and Zuber sphere (C) Ishii and Zuber ellipse (D) Grevskott (E) White (F) | No | Tomiyama | Lopez | No | No |

[79] | Mono-dispersed | No | No | No | Ishii and Zuber | No | Tomiyama | Lopez (only for RANS) | Antal | - |

[45] | Mono-dispersed (A) iMUSIG 2 classes (B) | No | No | No | Ishii and Zuber | No | Tomiyama | Burns | Hosokawa | No |

[71] | Mono-dispersed (1) iMUSIG 2 classes (2) | No | No | No | Ishii and Zuber | No ($\alpha $) Riboux ($\beta $) | Tomiyama | Burns | Hosokawa | No |

[80] | Mono-dispersed (A) MUSIG (B) iMUSIG (C) | Luo | Luo | Luo and Svendsen | Shiller and Naumann (1) | No | Tomiyama ($\alpha $) No ($\beta $) | No | No | No |

[81] | Mono-dispersed ${d}_{b}$ = 4 mm (A) ${d}_{b}$ = 6 mm (B) | No | No | No | Grace | Simonnet | Tomiyama (1) No (2) | Burns | No | No |

[82] | Mono-dispersed | No | No | No | Schiller and Naumann (A) Grace (B) Ishii and Zuber (C) ${C}_{D}$ = 0.44 (D) | No | ${C}_{L}$ = 0.5 | No | No | No |

[83] | MUSIG 10 classes (A) MUSIG 20 classes (B) | Prince and Blanch | Luo | Luo and Svendsen | Tomiyama | Ishii and Zuber | Tomiyama (1) Behzadi (2) | Burns | No | No |

[88] | iMUSIG 2 classes | No | No | No | Ishii and Zuber | No | Tomiyama | Burn | Hosokawa | ${C}_{VM}$ = 0.5 |

[89] | Mono-dispersed | No | No | No | Shiller and Naumann (A) Tomiyama (B) | Simonnet (A) Simonnet * (B) No (C) | No | No | No | No |

[84] | MUSIG 14 classes | Luo ${C}_{0}$ = 0.1 (A) ${C}_{0}$ = 0.2 (B) ${C}_{0}$ = 0.3 (C) ${C}_{0}$ = 0.5 (D) ${C}_{0}$ = 0.9 (E) ${C}_{0}$ = 1.1 (F) | Coulaloglou and Tavlarides | Luo and Svendsen (1) Lehr (2) | Ishii and Zuber | No | Tomiyama | Simonin and Viollet | Antal | No |

[90] | Mono-dispersed | No | No | No | Grace | No | No | Burns | No | No |

[91] | Mono-dispersed | No | No | No | Ishii and Zuber | No | Tomiyama | Lopez | Antal | ${C}_{L}$ = 0.5 |

[30] | Mono-dispersed | No | No | No | Tomiyama | No (A) Simonnet (B) McClure, 2014 (C) McClure, 2017b (D) Gemello (E) | No | No | No | No |

[85] | MUSIG | Luo | Luo | Luo and Svendsen | Ishii and Zuber | No | Tomiyama | Burns | Frank | ${C}_{VM}=0.25$ |

Ref. | Turbulence Modelling | BIT | Numerical Aspects | Geometry | Mesh Size | |||
---|---|---|---|---|---|---|---|---|

Continuous Phase | Dispersed Phase | P-V Coupling | Spatial Discretization | Time Discretization | ||||

[74] | $k-\epsilon $ | No | Pfleger and Becker (1) No (2) | SIMPLEC | - | - | 3D cylindrical | 6150 (A) 12,300 (B) 62,400 (C) |

[78] | $k-\epsilon $ | No | - | - | - | - | 2D axisymmetric | 36,000 |

[46] | $k-\epsilon $ | No | Sato | SIMPLE | - | - | 3D cylindrical | 90,000 |

[79] | $k-\omega $ (A) $k-\epsilon $ (B) $k-\epsilon $ RNG (C) RSM (D) LES (E) | No | Sato | - | Second-order implicit | First-order implicit | 3D cylindrical | 52,330 |

[45] | $k-\omega $ SST | No | - | - | - | - | 3D cylindrical | 30,000 |

[71] | $k-\omega $ | No | Rzehak (A) No (B) Sato (C) Morel (D) Troshco (E) Politano (F) Politano varied (E) | - | - | Second-order implicit | 3D rectangular | 200,000 |

[80] | $k-\epsilon $ RNG | No | - | - | Volume fraction: QUICK Others: second-order upwind | - | 3D cylindrical | 67,392 |

[81] | $k-\epsilon $ | No | Pfleger and Becker ($\alpha $) NO ($\beta $) | - | High resolution schemes | Second-order implicit | 3D cylindrical | 58,800 |

[82] | $k-\epsilon $ RNG | No | Sato | - | - | - | 3D rectangular | 46,080 |

[83] | $k-\epsilon $ ($\alpha $) RSM ($\beta $) | $k-\epsilon $ ($\alpha $) RSM ($\beta $) | Sato | PISO | - | - | 2D axisymmetric | - |

[88] | $k-\omega $ SST | No | Rzehak (A) Sato (B) No (C) | - | High resolution schemes | second-order implicit | 3D rectangular | 200,000 |

[92] | $k-\epsilon $ RNG | $k-\epsilon $ RNG | - | SIMPLE | - | - | 3D cylindrical | 342,230 |

[89] | $k-\epsilon $ RNG | No | No | - | High resolution schemes | - | 3D cylindrical | |

[84] | Mixture $k-\epsilon $ RNG | No | Coupled | Continuity: QUICK Momentum: second-order upwind Turbulence: second-order upwind PBM: second-order upwind | Second-order implicit | 2D axisymmetric | 10,422 | |

[78] | $k-\epsilon $ | No | - | Ansys CFX: coupled Ansys FLUENT: PC-SIMPLE with NITA | Ansys CFX: Turbulence: first-order Others: second-order bounded Ansys FLUENT: Momentum: QUICK Volume fraction: QUICK Scalar: second-order upwind Turbulence: first-order upwind | Ansys CFX: second-order implicit Ansys FLUENT: first-order implicit | 3D cylindrical | 36,000 |

[91] | $k-\epsilon $ | No | Sato | SIMPLE | Second-order upwind | - | 3D cylindrical | 605,802 |

[30] | $k-\epsilon $ RNG | No | - | PC-SIMPLE | Momentum: QUICK Volume fraction: QUICK Scalar: second-order upwind Turbulence: first-order upwind | First-order implicit | 3D cylindrical | 40,000 |

Ref. | Bench- Mark | Errors [%] | ||
---|---|---|---|---|

Gas Holdup | Local Void Fraction | Local Liquid Velocity | ||

[74] | [74] | (1B) 39.73 (2A) 28.77 (2B) 17.12 (2C) 7.53 | (1B) 4.21 (2A) 12.21 (2B) 11.14 (2C) 13.36 | (1B) 30.79 (2A) 38.62 (2B) 51.75 (2C) 64.63 |

[78] | [93] | - | (1A) 28.01 (1B) 30.47 (1C) 22.74 (2B) 26.04 | (1A) 61.61 (1B) 59.99 (1C) 53.01 (2B) 51.44 |

[46] | [94] | (A) 21.12 (B) 24.57 (C) 17.21 (D) 15.65 (E) 22.84 (F) 6.41 | Evaluated at ${U}_{G}$ = 1.2 cm/s: (D) 4.40 (E) 4.19 (G) 5.96 Evaluated at ${U}_{G}$ = 9.6 cm/s: (D) 12.53 (E) 10.67 (G) 5.18 | Centerline values: (A) 15.61 (B) 33 (C) 25.52 (D) 10.85 (E) 16.78 (F) 15.56 (G) 1.16 |

[79] | [95] | - | Near sparger region: (A) 36.24 (B) 34 (C) 31.44 (D) 29.55 (E) 18 Fully developed region: (A) 17.84 (B) 17.72 (C) 14.17 (D) 12.69 (E) 12.67 | Near sparger region: (A) 50.65 (B) 46.05 (C) 57.06 (D) 53.59 (E) 32.08 Fully developed region: (A) 17.04 (B) 19.64 (C) 15.77 (D) 18.17 (E) 18.48 |

[45] | [74] | Without NDF: (B) 26.65 With NDF: (A) 10.38 (B) 9.46 | Without NDF Evaluated at ${U}_{G}$ = 0.15 cm/s: (B) 18.79 Evaluated at ${U}_{G}$ = 0.5 cm/s: (B) 15.56 Evaluated at ${U}_{G}$ = 1 cm/s: (B) 19.01 With NDF Evaluated at ${U}_{G}$ = 0.15 cm/s: (A) 22.02 (B) 20.89 Evaluated at ${U}_{G}$ = 0.5 cm/s: (A) 21.68 (B) 14.03 Evaluated at ${U}_{G}$ = 1 cm/s: (A) 21.7 (B) 17.60 | Without NDF Evaluated at ${U}_{G}$ = 0.15 cm/s: (B) 32.67 Evaluated at ${U}_{G}$ = 0.5 cm/s: (B) 32.41 Evaluated at ${U}_{G}$ = 1 cm/s: (B) 30.48 With NDF Evaluated at ${U}_{G}$ = 0.15 cm/s: (A) 33.02 (B) 13.74 Evaluated at ${U}_{G}$ = 0.5 cm/s: (A) 20.80 (B) 36.39 Evaluated at ${U}_{G}$ = 1 cm/s: (A) 15.30 (B) 27.31 |

[71] | [96] | - | Evaluated at ${U}_{G}$ = 0.3 cm/s: (1A$\alpha $) 4.19 (1A$\beta $) 12.22 (1B$\alpha $) 4.19 (1C$\alpha $) 4.19 (1D$\alpha $) 4.19 (1E$\alpha $) 9.43 (1F$\alpha $) 6.98 Evaluated at ${U}_{G}$ = 1.3 cm/s: (1A$\alpha $) 14.42 (2A$\alpha $) 6.48 (2A$\beta $) 20.34 (2B$\alpha $) 8.95 (2C$\alpha $) 6.73 (2D$\alpha $) 7.49 (2E$\alpha $) 8.31 (2F$\alpha $) 7.85 (2G$\alpha $) 9.36 | Evaluated at ${U}_{G}$ = 0.3 cm/s: (1A$\alpha $) 44.27 (1A$\beta $) 44.27 (1B$\alpha $) 44.45 (1C$\alpha $) 44.45 (1D$\alpha $) 48.18 (1E$\alpha $) 43.27 (1F$\alpha $) 40.32 Evaluated at ${U}_{G}$ = 1.3 cm/s: (1A$\alpha $) 47.48 (2A$\alpha $) 30.20 (2A$\beta $) 39.28 (2B$\alpha $) 41.57 (2C$\alpha $) 57.50 (2D$\alpha $) 41.67 (2E$\alpha $) 47.58 (2F$\alpha $) 45.75 (2G$\alpha $) 80.64 |

[80] | [56] | - | (1A$\beta $) 21 (2B$\alpha $) 9.96 (2B$\beta $) 21.44 (2C$\alpha $) 9.68 (2C$\beta $) 21.53 | (1A$\beta $) 21.25 (2B$\alpha $) 29.89 (2B$\beta $) 32.31 (2C$\alpha $) 22.26 (2C$\beta $) 37.05 |

[81] | [97] | - | Evaluated at ${U}_{G}$ = 8 cm/s: (1A$\alpha $) 15.37 (1B$\alpha $) 23.31 (2A$\alpha $) 20.46 (2B$\alpha $) 22.29 Evaluated at ${U}_{G}$ = 12 cm/s: (1A$\alpha $) 20.80 (1A$\beta $) 75.80 | - |

[82] | [98] | (A) 14.76 (B) 9.45 (C) 6.18 (D) 22.75 | - | - |

[83] | [99] | - | (1A$\alpha $) 8.66 (2A$\alpha $) 10.96 (1A$\beta $) 9.60 (2A$\beta $) 27.70 (1B$\alpha $) 5.17 (1B$\beta $) 11.67 | |

[88] | [96] | - | (A) 9.44 (B) 8.88 (C) 8.65 | (A) 31.31 (B) 63.34 (C) 53.57 |

[89] | [89] | (1C) 42.01 (2C) 39.55 (2A) 35.87 (2B) 7.99 | Evaluated at ${U}_{G}$ = 16 cm/s: (2B) 6.67 | - |

[84] | [100] | (1A) 3.32 (1B) 12.63 (1C) 15.02 (1F) 32.53 (2F) 3.87 | Evaluated at ${U}_{G}$ = 1.9 cm/s: (1A) 10.85 (1B) 13.39 (1C) 13.3 (1D) 15.93 (1E) 20.57 (1F) 23.15 (2C) 3.65 (2D) 3.69 (2E) 4.11 (2F) 6.82 Evaluated at ${U}_{G}$ = 3.8 cm/s: (1A) 16.6 (1B) 29.54 (1C) 34.91 (1F) 50 (2F) 9.9 | - |

[90] | [101] | (1A) 6.69 (2B) 7.78 | Evaluated at ${U}_{G}$ = 16 cm/s: (1B) 12.48 (2B) 8.94 | Evaluated at ${U}_{G}$ = 16 cm/s: (1B) 35.25 (2B) 33.78 |

[91] | [91] | 13.67 | - | - |

[30] | [89] | (A) 84.77 (B) 44.40 (C) 54.91 (D) 29.82 (E) 2.83 | Evaluated at ${U}_{G}$ = 3 cm/s: (E) 5.70 Evaluated at ${U}_{G}$ = 16 cm/s: (E) 4.07 | Evaluated at ${U}_{G}$ = 3 cm/s: (E) 38.39 Evaluated at ${U}_{G}$ = 16 cm/s: (E) 25.18 |

[20] | [102] | - | 8.07 No lift: 42.99 No turbulent dispersion: 23.86 No wall lubrication: 12.64 No virtual mass: 8.99 No BIT: 9.32 | - |

## 5. Conclusions

- Concerning the interfacial forces, the momentum transfer between the phases is dominated by the drag force. For a proper description of the drag coefficient, the models of Tomiyama et al. (1998) [19], Grace et al. (1976) [16], and Ishii and Zuber (1978) [17] can be implemented; however, they should be corrected with a swarm factor. When presenting numerical studies, a sensitivity analysis among the different models should be performed.The lift, wall lubrication, and turbulence dispersion forces should be added to the model to obtain more accurate solutions. In particular, a correlation that predicts the change in the sign of the lift coefficient should be considered.
- Concerning the turbulence modelling, RANS models, in particular the $k-\epsilon $ RNG and the $k-\omega $ SST models, provide satisfying results in terms of average quantities. The LES turbulence model provides better results in the near-sparger region, where the flow is more anisotropic. However, no remarkable differences compared to the RANS methods have been highlighted in the fully developed region.
- The modelling approach of the dispersed phase (i.e., mono-dispersed, MUSIG, iMUSIG, PBM) should always be related to the simulated flow regime. A mono-dispersed approximation applies at very low superficial gas velocities. Conversely, multiple-size models that include coalescence and breakup should be considered.
- Regarding the numerical settings, high-order resolution discretization schemes should be used in order to prevent or reduce numerical discretization errors as much as possible.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

BIT | Bubble-Induced Turbulence | |

CFD | Computational Fluid Dynamics | |

CM | Class Method | |

DNS | Direct Numerical Simulation | |

LDA | Laser Doppler Velocimetry | |

LES | Large Eddy Simulation | |

NDF | Number Density Function | |

PBM | Population Balance Model | |

PBE | Population Balance Equation | |

PIV | Particle Image Velocimetry | |

PSV | Particle Shadowgraph Velocimetry | |

QMOM | Quadrature Method of Moment | |

RANS | Reynolds-Averaged NavierStokes | |

RSM | Reynolds Stress Models | |

$\overrightarrow{u}$ | Local velocity | [m/s] |

${C}_{D}$ | Drag coefficient | [-] |

${C}_{L}$ | Lift coefficient | [-] |

${C}_{TD}$ | Turbulent dispersion coefficient | [-] |

${C}_{WL}$ | Wall lubrication coefficient | [-] |

${C}_{VM}$ | Virtual mass coefficient | [-] |

${d}_{b}$ | Bubble diameter | [m] |

${d}_{eq}$ | Equivalent bubble diameter | [m] |

${D}_{H}^{*}$ | Non-dimensional diameter | [-] |

${D}_{cr}^{*}$ | Critical non-dimensional diameter | [-] |

${D}_{H}$ | Hydraulic diameter | [m] |

E | Bubble aspect ratio | [-] |

$Eo$ | Eötvös number | [m] |

${F}_{D}$ | Drag force | [kg/${\mathrm{m}}^{2}{\mathrm{s}}^{2}$] |

${F}_{L}$ | Lift force | [kg/${\mathrm{m}}^{2}{\mathrm{s}}^{2}$] |

${F}_{TD}$ | Turbulent dispersion force | [kg/${\mathrm{m}}^{2}{\mathrm{s}}^{2}$] |

${F}_{WL}$ | Wall lubrication force | [kg/${\mathrm{m}}^{2}{\mathrm{s}}^{2}$] |

${F}_{VM}$ | Virtual mass force | [kg/${\mathrm{m}}^{2}{\mathrm{s}}^{2}$] |

g | Acceleration due to gravity | [m/${\mathrm{s}}^{2}$] |

$g\left({V}_{b}\right)$ | Breakup frequency | [$1/\mathrm{s}$] |

h | Swarm factor | [-] |

$h({d}_{b,i},{d}_{b,j})$ | Collision frequency | [$1/\mathrm{s}$] |

${J}_{T}$ | Drift flux | [m/s] |

k | Turbulent kinetic energy | [${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$] |

$\overrightarrow{{M}_{I}}$ | Momentum exchanges | [kg/${\mathrm{m}}^{2}{\mathrm{s}}^{2}$] |

$Mo$ | Morton number | [-] |

p | Pressure | [Pa] |

$R{e}_{b}$ | Bubble Reynolds number | |

S | Total source/sink term in the population balance equation | [${\mathrm{m}}^{2}$/s] |

${S}_{b}$ | Total source/sink term due to breakup | [${\mathrm{m}}^{3}$/s] |

${S}_{c}$ | Total source/sink term due to coalescence | [${\mathrm{m}}^{3}$/s] |

${S}_{m}$ | Total source/sink term due to mass transfer | [${\mathrm{m}}^{3}$/s] |

${S}_{p}$ | Total source/sink term due to pressure change | [${\mathrm{m}}^{3}$/s] |

${S}_{ph}$ | Total source/sink term due to phase change | [${\mathrm{m}}^{3}$/s] |

${S}_{r}$ | Total source/sink term due to reaction | [${\mathrm{m}}^{3}$/s] |

${U}_{G}$ | Superficial gas velocity | [m/s] |

${U}_{L}$ | Superficial liquid velocity | [m/s] |

${V}_{b}$ | Bubble volume in population balance equation | [${\mathrm{m}}^{3}$] |

${\alpha}_{G}$ | Local gas volume fraction | [-] |

$\beta $ | Daughter distribution function | [-] |

${\epsilon}_{G}$ | Global gas holdup | [-] |

$\epsilon $ | Turbulent dissipation rate | [${\mathrm{m}}^{2}/{\mathrm{s}}^{3}$] |

$\lambda ({d}_{b,i},{d}_{b,j})$ | Coalescence efficiency | [-] |

$\mu $ | Dynamic viscosity | [$\mathrm{kg}/\mathrm{m}/\mathrm{s}$] |

${\mu}_{t}$ | Turbulent viscosity | [${\mathrm{m}}^{2}/\mathrm{s}$] |

$\omega $ | Specific dissipation rate | [$1/\mathrm{s}$] |

$\rho $ | Density | [$\mathrm{kg}/{\mathrm{m}}^{3}$] |

$\sigma $ | Surface tension | [kg/${\mathrm{m}}^{2}{\mathrm{s}}^{2}$ |

$\overline{\tau}$ | Viscous stress tensor | [$\mathrm{kg}/\mathrm{m}/{\mathrm{s}}^{2}$] |

G | Gas phase | |

L | Liquid phase | |

k | k-th phase |

## References

- Kantarci, N.; Borak, F.; Ulgen, K.O. Bubble column reactors. Process Biochem.
**2005**, 40, 2263–2283. [Google Scholar] [CrossRef] - Buffo, A.; Marchisio, D.L. Modeling and simulation of turbulent polydisperse gas-liquid systems via the generalized population balance equation. Rev. Chem. Eng.
**2014**, 30, 73–126. [Google Scholar] [CrossRef] - Khan, K.I. Fluid Dynamic Modeling of Bubble Column Reactor. Ph.D Thesis, Politecnico di Torino, Torino, Italy, 2014. [Google Scholar]
- Wu, B.; Firouzi, M.; Mitchell, T.; Rufford, T.E.; Leonardi, C.; Towler, B. A critical review of flow maps for gas-liquid flows in vertical pipes and annuli. Chem. Eng. J.
**2017**, 326, 350–377. [Google Scholar] [CrossRef][Green Version] - Kitscha, J.; Kocamustafaogullari, G. Breakup criteria for fluid particles. Int. J. Multiph. Flow
**1989**, 15, 573–588. [Google Scholar] [CrossRef] - Brooks, C.S.; Paranjape, S.S.; Ozar, B.; Hibiki, T.; Ishii, M. Two-group drift-flux model for closure of the modified two-fluid model. Int. J. Heat Fluid Flow
**2012**, 37, 196–208. [Google Scholar] [CrossRef] - Besagni, G. Bubble column fluid dynamics: A novel perspective for flow regimes and comprehensive experimental investigations. Int. J. Multiph. Flow
**2021**, 135, 103510. [Google Scholar] [CrossRef] - Krepper, E.; Lucas, D.; Prasser, H.M. On the modelling of bubbly flow in vertical pipes. Nucl. Eng. Des.
**2005**, 235, 597–611. [Google Scholar] [CrossRef] - Lapin, A.; Lübbert, A. Numerical simulation of the dynamics of two-phase gas–liquid flows in bubble columns. Chem. Eng. Sci.
**1994**, 49, 3661–3674. [Google Scholar] [CrossRef] - Buwa, V.V.; Deo, D.S.; Ranade, V.V. Eulerian–Lagrangian simulations of unsteady gas–liquid flows in bubble columns. Int. J. Multiph. Flow
**2006**, 32, 864–885. [Google Scholar] [CrossRef] - Besbes, S.; El Hajem, M.; Aissia, H.B.; Champagne, J.; Jay, J. PIV measurements and Eulerian–Lagrangian simulations of the unsteady gas–liquid flow in a needle sparger rectangular bubble column. Chem. Eng. Sci.
**2015**, 126, 560–572. [Google Scholar] [CrossRef] - Khan, I.; Wang, M.; Zhang, Y.; Tian, W.; Su, G.; Qiu, S. Two-phase bubbly flow simulation using CFD method: A review of models for interfacial forces. Prog. Nucl. Energy
**2020**, 125, 103360. [Google Scholar] [CrossRef] - Suh, J.W.; Kim, J.W.; Choi, Y.S.; Kim, J.H.; Joo, W.G.; Lee, K.Y. Development of numerical Eulerian-Eulerian models for simulating multiphase pumps. J. Pet. Sci. Eng.
**2018**, 162, 588–601. [Google Scholar] [CrossRef] - Schiller, L. A drag coefficient correlation. Zeit. Ver. Deutsch. Ing.
**1933**, 77, 318–320. [Google Scholar] - Frank. Viscous Fluid Flow; McGraw-Hill: New York, NY, USA, 1974. [Google Scholar]
- Grace, J.R. Shapes and velocities of single drops and bubbles moving freely through immiscible liquids. Trans. Inst. Chem. Eng.
**1976**, 54, 167. [Google Scholar] - Ishii, M.; Zuber, N. Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AIChE J.
**1979**, 25, 843–855. [Google Scholar] [CrossRef] - Grevskott, S.; Sannaes, B.; Duduković, M.; Hjarbo, K.; Svendsen, H. Liquid circulation, bubble size distributions, and solids movement in two-and three-phase bubble columns. Chem. Eng. Sci.
**1996**, 51, 1703–1713. [Google Scholar] [CrossRef] - Tomiyama, A. Struggle with computational bubble dynamics. Multiph. Sci. Technol.
**1998**, 10, 369–405. [Google Scholar] - Zhang, D.; Deen, N.; Kuipers, J. Numerical simulation of the dynamic flow behavior in a bubble column: A study of closures for turbulence and interface forces. Chem. Eng. Sci.
**2006**, 61, 7593–7608. [Google Scholar] [CrossRef] - Dijkhuizen, W.; Roghair, I.; Annaland, M.V.S.; Kuipers, J. DNS of gas bubbles behaviour using an improved 3D front tracking model—Drag force on isolated bubbles and comparison with experiments. Chem. Eng. Sci.
**2010**, 65, 1415–1426. [Google Scholar] [CrossRef] - Bridge, A.; Lapidus, L.; Elgin, J. The mechanics of vertical gas-liquid fluidized system I: Countercurrent flow. AIChE J.
**1964**, 10, 819–826. [Google Scholar] [CrossRef] - Wallis, G.B. One-Dimensional Two-Phase Flow; McGraw-Hill: New York, NY, USA, 1969. [Google Scholar]
- Rusche, H.; Issa, R. The effect of voidage on the drag force on particles, droplets and bubbles in dispersed two-phase flow. In Proceedings of the Japanese European Two-Phase Flow Meeting, Tshkuba, Japan, 25–29 September 2000. [Google Scholar]
- Roghair, I.; Lau, Y.; Deen, N.; Slagter, H.; Baltussen, M.; Annaland, M.V.S.; Kuipers, J. On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers. Chem. Eng. Sci.
**2011**, 66, 3204–3211. [Google Scholar] [CrossRef] - Buffo, A.; Vanni, M.; Renze, P.; Marchisio, D.L. Empirical drag closure for polydisperse gas–liquid systems in bubbly flow regime: Bubble swarm and micro-scale turbulence. Chem. Eng. Res. Des.
**2016**, 113, 284–303. [Google Scholar] [CrossRef] - Simonnet, M.; Gentric, C.; Olmos, E.; Midoux, N. CFD simulation of the flow field in a bubble column reactor: Importance of the drag force formulation to describe regime transitions. Chem. Eng. Process. Process Intensif.
**2008**, 47, 1726–1737. [Google Scholar] [CrossRef] - McClure, D.D.; Norris, H.; Kavanagh, J.M.; Fletcher, D.F.; Barton, G.W. Validation of a computationally efficient computational fluid dynamics (CFD) model for industrial bubble column bioreactors. Ind. Eng. Chem. Res.
**2014**, 53, 14526–14543. [Google Scholar] [CrossRef] - McClure, D.D.; Kavanagh, J.M.; Fletcher, D.F.; Barton, G.W. Experimental investigation into the drag volume fraction correction term for gas-liquid bubbly flows. Chem. Eng. Sci.
**2017**, 170, 91–97. [Google Scholar] [CrossRef] - Gemello, L.; Cappello, V.; Augier, F.; Marchisio, D.; Plais, C. CFD-based scale-up of hydrodynamics and mixing in bubble columns. Chem. Eng. Res. Des.
**2018**, 136, 846–858. [Google Scholar] [CrossRef] - Yang, G.; Zhang, H.; Luo, J.; Wang, T. Drag force of bubble swarms and numerical simulations of a bubble column with a CFD-PBM coupled model. Chem. Eng. Sci.
**2018**, 192, 714–724. [Google Scholar] [CrossRef] - Tomiyama, A.; Tamai, H.; Zun, I.; Hosokawa, S. Transverse migration of single bubbles in simple shear flows. Chem. Eng. Sci.
**2002**, 57, 1849–1858. [Google Scholar] [CrossRef] - Wellek, R.; Agrawal, A.; Skelland, A. Shape of liquid drops moving in liquid media. AIChE J.
**1966**, 12, 854–862. [Google Scholar] [CrossRef] - Ziegenhein, T.; Lucas, D. The critical bubble diameter of the lift force in technical and environmental, buoyancy-driven bubbly flows. Int. J. Multiph. Flow
**2019**, 116, 26–38. [Google Scholar] [CrossRef] - Besagni, G.; Inzoli, F.; De Guido, G.; Pellegrini, L.A. Experimental investigation on the influence of ethanol on bubble column hydrodynamics. Chem. Eng. Res. Des.
**2016**, 112, 1–15. [Google Scholar] [CrossRef] - Ziegenhein, T.; Tomiyama, A.; Lucas, D. A new measuring concept to determine the lift force for distorted bubbles in low Morton number system: Results for air/water. Int. J. Multiph. Flow
**2018**, 108, 11–24. [Google Scholar] [CrossRef] - Behzadi, A.; Issa, R.; Rusche, H. Modelling of dispersed bubble and droplet flow at high phase fractions. Chem. Eng. Sci.
**2004**, 59, 759–770. [Google Scholar] [CrossRef] - Simonin, O.; Viollet, P. Modelling of turbulent two-phase jets loaded with discrete particles. In Phenomena in Multiphase Flows; Hemisphere Publishing Corporation: London, UK, 1990; Volume 1990, pp. 259–269. [Google Scholar]
- Burns, A.D.; Frank, T.; Hamill, I.; Shi, J.M. The Favre averaged drag model for turbulent dispersion in Eulerian multi-phase flows. In Proceedings of the 5th International Conference on Multiphase Flow, ICMF, Yokohama, Japan, 30 May–4 June 2004; Volume 4, pp. 1–17. [Google Scholar]
- Lopez de Bertodano, M.; Moraga, F.; Drew, D.; Lahey, R., Jr. The modeling of lift and dispersion forces in two-fluid model simulations of a bubbly jet. J. Fluids Eng.
**2004**, 126, 573–577. [Google Scholar] [CrossRef] - Antal, S.; Lahey, R., Jr.; Flaherty, J. Analysis of phase distribution in fully developed laminar bubbly two-phase flow. Int. J. Multiph. Flow
**1991**, 17, 635–652. [Google Scholar] [CrossRef] - Tomiyama, A. Effects of Eotvos number and dimensionless liquid volumetric flux on lateral motion of a bubble in a laminar duct flow. In Proceedings of the 2nd International Conference on Multiphase Flow, Kyoto, Japan, 3–7 April 1995; Volume 3. [Google Scholar]
- Hosokawa, S.; Tomiyama, A.; Misaki, S.; Hamada, T. Lateral migration of single bubbles due to the presence of wall. In Proceedings of the Fluids Engineering Division Summer Meeting, Montreal, QC, Canada, 14–18 July 2002; Volume 36150, pp. 855–860. [Google Scholar]
- Frank, T. Advances in computational fluid dynamics (CFD) of 3-dimensional gas-liquid multiphase flows. In Proceedings of the NAFEMS Seminar: Simulation of Complex Flows (CFD)–Applications and Trends, Wiesbaden, Germany, 25–26 April 2005; pp. 1–18. [Google Scholar]
- Ziegenhein, T.; Rzehak, R.; Krepper, E.; Lucas, D. Numerical simulation of polydispersed flow in bubble columns with the inhomogeneous multi-size-group model. Chem. Ing. Tech.
**2013**, 85, 1080–1091. [Google Scholar] [CrossRef] - Tabib, M.V.; Roy, S.A.; Joshi, J.B. CFD simulation of bubble column—An analysis of interphase forces and turbulence models. Chem. Eng. J.
**2008**, 139, 589–614. [Google Scholar] [CrossRef] - Yeoh, G.; Tu, J. Basic theory and conceptual framework of multiphase flows. In Handbook of Multiphase Flow Science and Technology; Springer: Berlin, Germany, 2017; pp. 1–47. [Google Scholar]
- Liao, Y.; Lucas, D. A literature review on mechanisms and models for the coalescence process of fluid particles. Chem. Eng. Sci.
**2010**, 65, 2851–2864. [Google Scholar] [CrossRef] - Liao, Y.; Lucas, D. A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem. Eng. Sci.
**2009**, 64, 3389–3406. [Google Scholar] [CrossRef] - Prince, M.J.; Blanch, H.W. Bubble coalescence and break-up in air-sparged bubble columns. AIChE J.
**1990**, 36, 1485–1499. [Google Scholar] [CrossRef] - Luo, H.; Svendsen, H.F. Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J.
**1996**, 42, 1225–1233. [Google Scholar] [CrossRef] - Martínez-Bazán, C.; Montañés, J.L.; Lasheras, J.C. On the breakup of an air bubble injected into a fully developed 881 turbulent flow. Part 1. Breakup frequency. J. Fluid Mech.
**1999**, 401, 157–182. [Google Scholar] [CrossRef][Green Version] - Martínez-Bazán, C.; Montañés, J.L.; Lasheras, J.C. On the breakup of an air bubble injected into a fully developed 883 turbulent flow. Part 2. Size PDF of the resulting daughter bubbles. J. Fluid Mech.
**1999**, 401, 183–207. [Google Scholar] [CrossRef][Green Version] - Batchelor, G.K. The Theory of Homogeneous Turbulence; Cambridge University Press: Cambridge, UK, 1953. [Google Scholar]
- Lehr, F.; Millies, M.; Mewes, D. Bubble-size distributions and flow fields in bubble columns. AIChE J.
**2002**, 48, 2426–2443. [Google Scholar] [CrossRef] - Chen, J.; Li, F.; Degaleesan, S.; Gupta, P.; Al-Dahhan, M.H.; Dudukovic, M.P.; Toseland, B.A. Fluid dynamic parameters in bubble columns with internals. Chem. Eng. Sci.
**1999**, 54, 2187–2197. [Google Scholar] [CrossRef] - Shinnar, R.; Church, J.M. Statistical theories of turbulence in predicting particle size in agitated dispersions. Ind. Eng. Chem.
**1960**, 52, 253–256. [Google Scholar] [CrossRef] - Howarth, W. Coalescence of drops in a turbulent flow field. Chem. Eng. Sci.
**1964**, 19, 33–38. [Google Scholar] [CrossRef] - Coulaloglou, C.; Tavlarides, L.L. Description of interaction processes in agitated liquid-liquid dispersions. Chem. Eng. Sci.
**1977**, 32, 1289–1297. [Google Scholar] [CrossRef] - Chesters, A. Modelling of coalescence processes in fluid-liquid dispersions: A review of current understanding. Chem. Eng. Res. Des.
**1991**, 69, 259–270. [Google Scholar] - Luo, H. Coalescence, Breakup and Liquid Circulation in Bubble Column Reactors. Ph.D. Thesis, Department of Chemical Engineering, The University of Trondheim, Trondheim, Norvey, 1993. [Google Scholar]
- Sanyal, J.; Marchisio, D.L.; Fox, R.O.; Dhanasekharan, K. On the comparison between population balance models for CFD simulation of bubble columns. Ind. Eng. Chem. Res.
**2005**, 44, 5063–5072. [Google Scholar] [CrossRef][Green Version] - McGraw, R. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol.
**1997**, 27, 255–265. [Google Scholar] [CrossRef] - Marchisio, D.L.; Vigil, R.D.; Fox, R.O. Quadrature method of moments for aggregation–breakage processes. J. Colloid Interface Sci.
**2003**, 258, 322–334. [Google Scholar] [CrossRef] - Petitti, M.; Vanni, M.; Marchisio, D.L.; Buffo, A.; Podenzani, F. Simulation of coalescence, break-up and mass transfer in a gas–liquid stirred tank with CQMOM. Chem. Eng. J.
**2013**, 228, 1182–1194. [Google Scholar] [CrossRef] - Ma, T.; Lucas, D.; Ziegenhein, T.; Fröhlich, J.; Deen, N. Scale-Adaptive Simulation of a square cross-sectional bubble column. Chem. Eng. Sci.
**2015**, 131, 101–108. [Google Scholar] [CrossRef] - Dhotre, M.; Deen, N.; Niceno, B.; Khan, Z.; Joshi, J. Large eddy simulation for dispersed bubbly flows: A review. Int. J. Chem. Eng.
**2013**, 2013, 343276. [Google Scholar] [CrossRef] - Masood, R.; Rauh, C.; Delgado, A. CFD simulation of bubble column flows: An explicit algebraic Reynolds stress model approach. Int. J. Multiph. Flow
**2014**, 66, 11–25. [Google Scholar] [CrossRef] - Parekh, J.; Rzehak, R. Euler–Euler multiphase CFD-simulation with full Reynolds stress model and anisotropic bubble-induced turbulence. Int. J. Multiph. Flow
**2018**, 99, 231–245. [Google Scholar] [CrossRef] - Joshi, J. Computational flow modelling and design of bubble column reactors. Chem. Eng. Sci.
**2001**, 56, 5893–5933. [Google Scholar] [CrossRef] - Ziegenhein, T.; Lucas, D.; Rzehak, R.; Krepper, E. Closure relations for CFD simulation of bubble columns. In Proceedings of the 8th International Conference on Multiphase Flow—ICMF 2014, Jeju, Republic of Korea, 26–31 May 2013. [Google Scholar]
- Morel, C. Turbulence modelling and first numerical simulations in turbulent two-phase flows. In Proceedings of the 11th Symposium on Turbulent Shear Flows, Grenoble, France, 8–11 September 1997; Volume 3. [Google Scholar]
- Troshko, A.; Hassan, Y. A two-equation turbulence model of turbulent bubbly flows. Int. J. Multiph. Flow
**2001**, 27, 1965–2000. [Google Scholar] [CrossRef] - Pfleger, D.; Becker, S. Modelling and simulation of the dynamic flow behaviour in a bubble column. Chem. Eng. Sci.
**2001**, 56, 1737–1747. [Google Scholar] [CrossRef] - Politano, M.; Carrica, P.; Converti, J. A model for turbulent polydisperse two-phase flow in vertical channels. Int. J. Multiph. Flow
**2003**, 29, 1153–1182. [Google Scholar] [CrossRef] - Rzehak, R.; Krepper, E. Bubble-induced turbulence: Comparison of CFD models. Nucl. Eng. Des.
**2013**, 258, 57–65. [Google Scholar] [CrossRef] - Sato, Y.; Sekoguchi, K. Liquid velocity distribution in two-phase bubble flow. Int. J. Multiph. Flow
**1975**, 2, 79–95. [Google Scholar] [CrossRef] - Chen, P.; Sanyal, J.; Duduković, M. Numerical simulation of bubble columns flows: Effect of different breakup and coalescence closures. Chem. Eng. Sci.
**2005**, 60, 1085–1101. [Google Scholar] [CrossRef] - Ekambara, K.; Dhotre, M. CFD simulation of bubble column. Nucl. Eng. Des.
**2010**, 240, 963–969. [Google Scholar] [CrossRef] - Xu, L.; Yuan, B.; Ni, H.; Chen, C. Numerical simulation of bubble column flows in churn-turbulent regime: Comparison of bubble size models. Ind. Eng. Chem. Res.
**2013**, 52, 6794–6802. [Google Scholar] [CrossRef] - McClure, D.D.; Kavanagh, J.M.; Fletcher, D.F.; Barton, G.W. Development of a CFD model of bubble column bioreactors: Part two—Comparison of experimental data and CFD predictions. Chem. Eng. Technol.
**2014**, 37, 131–140. [Google Scholar] [CrossRef] - Masood, R.; Delgado, A. Numerical investigation of the interphase forces and turbulence closure in 3D square bubble columns. Chem. Eng. Sci.
**2014**, 108, 154–168. [Google Scholar] [CrossRef] - Liu, Y.; Hinrichsen, O. Study on CFD–PBM turbulence closures based on k–ε and Reynolds stress models for heterogeneous bubble column flows. Comput. Fluids
**2014**, 105, 91–100. [Google Scholar] [CrossRef] - Syed, A.H.; Boulet, M.; Melchiori, T.; Lavoie, J.M. CFD simulations of an air-water bubble column: Effect of Luo coalescence parameter and breakup kernels. Front. Chem.
**2017**, 5, 68. [Google Scholar] [CrossRef] - Zhang, X.B.; Yan, W.C.; Luo, Z.H. CFD-PBM Simulation of Bubble Columns: Sensitivity Analysis of the Nondrag Forces. Ind. Eng. Chem. Res.
**2020**, 59, 18674–18682. [Google Scholar] [CrossRef] - Deen, N.G.; Hjertager, B.H.; Solberg, T. Comparison of PIV and LDA measurement methods applied to the gas-liquid flow in a bubble column. In Proceedings of the 10th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 10–13 July 2000; pp. 1–12. [Google Scholar]
- Ziegenhein, T.; Lucas, D.; Besagni, G.; Inzoli, F. Experimental study of the liquid velocity and turbulence in a large-scale air-water counter-current bubble column. Exp. Therm. Fluid Sci.
**2020**, 111, 109955. [Google Scholar] [CrossRef] - Ziegenhein, T.; Rzehak, R.; Lucas, D. Transient simulation for large scale flow in bubble columns. Chem. Eng. Sci.
**2015**, 122, 1–13. [Google Scholar] [CrossRef] - Raimundo, P.M. Analysis and Modelization of Local Hydrodynamics in Bubble Columns. Ph.D. Thesis, Université Grenoble Alpes, Grenoble, France, 2015. [Google Scholar]
- Fletcher, D.F.; McClure, D.D.; Kavanagh, J.M.; Barton, G.W. CFD simulation of industrial bubble columns: Numerical challenges and model validation successes. Appl. Math. Model.
**2017**, 44, 25–42. [Google Scholar] [CrossRef] - Saleh, S.N.; Mohammed, A.A.; Al-Jubory, F.K.; Barghi, S. CFD assesment of uniform bubbly flow in a bubble column. J. Pet. Sci. Eng.
**2018**, 161, 96–107. [Google Scholar] [CrossRef] - Liang, X.F.; Pan, H.; Su, Y.H.; Luo, Z.H. CFD-PBM approach with modified drag model for the gas–liquid flow in a bubble column. Chem. Eng. Res. Des.
**2016**, 112, 88–102. [Google Scholar] [CrossRef] - Degaleesan, S. Fluid Dynamic Measurements and Modeling of Liquid Mixing in Bubble Columns; Washington University in St. Louis: St. Louis, MO, USA, 1997. [Google Scholar]
- Menzel, T.; In der Weide, T.; Staudacher, O.; Wein, O.; Onken, U. Reynolds shear stress for modeling of bubble column reactors. Ind. Eng. Chem. Res.
**1990**, 29, 988–994. [Google Scholar] [CrossRef] - Kulkarni, A.; Ekambara, K.; Joshi, J. On the development of flow pattern in a bubble column reactor: Experiments and CFD. Chem. Eng. Sci.
**2007**, 62, 1049–1072. [Google Scholar] [CrossRef] - bin Mohd Akbar, M.H.; Hayashi, K.; Hosokawa, S.; Tomiyama, A. Bubble tracking simulation of bubble-induced pseudoturbulence. Multiph. Sci. Technol.
**2012**, 24, 197–222. [Google Scholar] [CrossRef] - McClure, D.D.; Kavanagh, J.M.; Fletcher, D.F.; Barton, G.W. Development of a CFD model of bubble column bioreactors: Part one—A detailed experimental study. Chem. Eng. Technol.
**2013**, 36, 2065–2070. [Google Scholar] [CrossRef] - Deen, N. An Experimental and Computational Study of Fluid Dynamics in Gas-Liquid Chemical Reactors; Aalborg University Esbjerg: Esbjerg, Denmark, 2001. [Google Scholar]
- Rampure, M.R.; Kulkarni, A.A.; Ranade, V.V. Hydrodynamics of bubble column reactors at high gas velocity: Experiments and computational fluid dynamics (CFD) simulations. Ind. Eng. Chem. Res.
**2007**, 46, 8431–8447. [Google Scholar] [CrossRef] - Hills, J.H. Radial non-uniformity of velocity and voidage in a bubble column. Trans. Inst. Chem. Eng.
**1974**, 52, 1–9. [Google Scholar] - McClure, D.D.; Wang, C.; Kavanagh, J.M.; Fletcher, D.F.; Barton, G.W. Experimental investigation into the impact of sparger design on bubble columns at high superficial velocities. Chem. Eng. Res. Des.
**2016**, 106, 205–213. [Google Scholar] [CrossRef] - Guan, X.; Yang, N. Bubble properties measurement in bubble columns: From homogeneous to heterogeneous regime. Chem. Eng. Res. Des.
**2017**, 127, 103–112. [Google Scholar] [CrossRef]

**Figure 2.**Representation of flow regimes in vertical pipes. Reprinted with permission from [4].

**Figure 3.**Flow regimes and regime transitions in a large-diameter bubble column. Reprinted with permission from [7].

**Figure 4.**Flow pattern evolution in two-phase bubble columns. Reprinted with permission from [8].

**Figure 6.**Evolution of the drag coefficient with respect to the bubble diameter at $|\overrightarrow{{u}_{G}}-\overrightarrow{{u}_{L}}|=0.1$ m/s for air bubbles in water. Comparison between the correlations of Schiller and Naumann (1933) [14], Grace et al. (1976) [16], Ishii and Zuber (1978) [17], Grevskott et al. (1996) [18] and Tomiyama (1998) [19].

**Figure 7.**Comparison of published Swarm factor as a function of local void fraction for bubbles with ${d}_{b}$ of 6.5 mm. Comparison between the correlations of Bridge et al. (1964) [22], Wallis (1969) [23], Ishii and Zuber (1979) [17], Rusche and Issa (2000) [24], Simonnet et al. (2008) [27], Roghair et al. (2011) [25] and Buffo et al. (2016) [26]. Note the discrepancy between the correlation proposed by Simonnet et al. (2008) [27] and the others.

**Figure 10.**Influence of Non-Drag-Forces (NDF), (with the exception of the virtual mass force) on the local void fraction profile. Reprinted with permission from [45].

Author | Breakage Frequency (g) | Daughter Size Distribution ($\mathit{\beta}$) |
---|---|---|

Prince and Blanch (1990) [50] | $g={\int}_{0.2{d}_{b,j}}^{{d}_{b,j}}\frac{0.07{\pi}^{4}}{{\lambda}_{\epsilon}}{\left({d}_{b,j}+{\lambda}_{\epsilon}\right)}^{2}{\left({d}_{b,j}^{2/3}+{\lambda}_{\epsilon}^{2/3}\right)}^{0.5}{\epsilon}^{1/3}$$\times exp\left(-\frac{1.18}{{\lambda}_{\epsilon}^{2/3}}\frac{\sigma}{{\rho}_{L}{d}_{b,j}{\epsilon}^{2/3}}\right)d{\lambda}_{\epsilon}$ | Uniform |

Luo and Svendsen (1996) [51] | $g=\frac{1}{2}{\int}_{0}^{1}0.923\left(1-{\alpha}_{G}\right)n{\left(\frac{\epsilon}{{d}_{b,j}^{2}}\right)}^{1/3}$$\times {\int}_{{\xi}_{min}}^{1}\frac{{(1+\xi )}^{2}}{{\xi}^{11/13}}exp\left(-\frac{12{c}_{f}\sigma}{{c}_{\beta}{\rho}_{L}{\epsilon}^{2/3}{d}_{b,j}^{5/3}{\xi}^{11/13}}\right)d\xi d{f}_{bv}$ with $\xi =\frac{{\lambda}_{\epsilon}}{{d}_{b}},{c}_{f}={f}_{bv}^{2/3}+{\left(1-{f}_{bv}\right)}^{2/3}-1,{f}_{bv}=\frac{{d}_{b,i}^{3}}{{d}_{b,i}^{3}+{d}_{b,j}^{3}},{c}_{\beta}=2$ | $\beta =\frac{2{\int}_{{\xi}_{min}}^{1}{(1+\xi )}^{2}{\xi}^{-11/3}exp\left(-{\chi}_{c}\right)d\xi}{{V}_{i}{\int}_{0}^{1}{\int}_{{\xi}_{min}}^{1}{(1+\xi )}^{2}{\xi}^{-11/3}exp\left(-{\chi}_{c}\right)d\xi d{f}_{bv}}$ with ${\chi}_{c}=\frac{12{c}_{f}\sigma}{{c}_{\beta}{\rho}_{1}{\epsilon}^{2/3}{d}_{b,i}^{5/3}{\xi}^{11/3}}$ |

Martïnez and Bazän (1999) [52,53] | $g={K}_{g}n\frac{\sqrt{{\beta}^{\prime}{\left(\epsilon {d}_{b,j}\right)}^{2/3}-12\sigma /\left({\rho}_{c}{d}_{b,j}\right)}}{{d}_{b,j}}$ with ${\beta}^{\prime}=8.2$ [54], ${K}_{g}=0.25$ | $\beta =\frac{\left[{D}^{*2/3}-{\Lambda}^{5/3}\right]\left[{\left(1-{D}^{*3}\right)}^{2/9}-{\Lambda}^{5/3}\right]}{{\int}_{{D}_{min}^{*}}^{{D}_{max}}\left[{D}^{*2/3}-{\Lambda}^{5/3}\right]\left[{\left(1-{D}^{*3}\right)}^{2/9}-{\Lambda}^{5/3}\right]d\left({D}^{*}\right)}$ with ${D}^{*}=\frac{{d}_{b,j}}{{d}_{b,i}}$, $\Lambda =\left(\frac{12\sigma}{{\beta}^{\prime}{\rho}_{L}}\right)$, ${\beta}^{\prime}=8.2$ [54] |

Lehr et al. (2002) [55] | $g=0.5\frac{{d}_{b,j}^{5/3}{\epsilon}^{19/15}{\rho}_{L}^{7/5}}{{\sigma}^{7/5}}exp\left(-\frac{\sqrt{2}{\sigma}^{9/5}}{{\rho}_{L}{d}_{b,j}^{9/5}{\epsilon}^{6/5}{L}^{3}}\right)$ | $\beta =\frac{1}{\sqrt{\pi}{f}_{bv}}\frac{exp\left\{-\frac{9}{4}{\left[ln\left(\frac{{2}^{2/5}{d}_{b,j}{\rho}_{L}^{3/5}{\epsilon}^{2/5}}{{\sigma}^{3}}\right)\right]}^{2}\right\}}{1+erf\left[\frac{3}{2}ln\left(\frac{{2}^{1/15}{d}_{b,i}{\rho}_{1}^{3/5}{\epsilon}^{2/5}}{{\sigma}^{3/5}}\right)\right]}$ |

Author | Collision Frequency (h) | Collision Efficiency ($\mathit{\lambda}$) |
---|---|---|

Coulaloglou and Tavlarides (1977) [59] | $h={c}_{h}\frac{{\epsilon}^{1/3}}{1+{\alpha}_{G}}{\left({d}_{b,i}+{d}_{b,j}\right)}^{2}{\left({d}_{b,i}^{2/3}+{d}_{b,j}^{2/3}\right)}^{0.5}$ | $\lambda =exp\left\{-{c}_{\lambda}\frac{{\mu}_{L}{\rho}_{L}\epsilon}{{\sigma}^{2}{\left(1+{\alpha}_{G}\right)}^{3}}{\left(\frac{{d}_{b,i}{d}_{b,j}}{{d}_{b,i}+{d}_{b,j}}\right)}^{4}\right\}$ |

Prince and Blanch (1990) [50] | $h=0.089\pi {\epsilon}^{1/3}{\left({d}_{b,i}+{d}_{b,j}\right)}^{2}{\left({d}_{b,i}^{2/3}+{d}_{b,j}^{2/3}\right)}^{0.5}$ | $\lambda =exp\left[-\frac{{r}_{eq}^{5/6}{\rho}_{L}^{0.5}{\epsilon}^{1/3}}{4{\sigma}^{0.5}}ln\left(\frac{{h}_{0}}{{h}_{f}}\right)\right]$ |

Chesters (1991) [60] | $h=0.618{\left(\frac{{d}_{b,i}}{2}+\frac{{d}_{b,j}}{2}\right)}^{3}\sqrt{\epsilon /\nu}$ | $\lambda =exp\left\{-{c}_{\lambda}{\left(\frac{We}{2}\right)}^{1/2}\right\}$ with $We=\frac{{\rho}_{c}{\left(\epsilon {d}_{ij}\right)}^{2/3}{d}_{ij}}{2\sigma}$, ${d}_{ij}={\left[\frac{1}{2}\left(\frac{1}{{d}_{b,i}}+\frac{1}{{d}_{b,j}}\right)\right]}^{-1}$ |

Luo (1993) [61] | $h=\frac{\pi}{4}\sqrt{2}{({d}_{b,i}+{d}_{b,j})}^{2}{\epsilon}^{1/3}{({d}_{b,i}^{2/3}+{d}_{b,j}^{2/3})}^{0.5}$ | $\lambda =$$\mathrm{exp}\left\{-{c}_{\lambda}\frac{0.75\left(1+{\left(\frac{{d}_{b,i}}{{d}_{b,j}}\right)}^{2}\right)}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}+{C}_{VM}\right)}\frac{{\left(1+{\left(\frac{{d}_{b,i}}{{d}_{b,j}}\right)}^{3}\right)}^{0.5}}{{\left(1+\frac{{d}_{b,i}}{{d}_{b,j}}\right)}^{3}}W{e}_{ij}^{0.5}\right\}$ with $W{e}_{ij}=\frac{{\rho}_{L}{d}_{b}\left({u}_{bi}^{2}+{u}_{bj}^{2}\right)}{\sigma}$, ${u}_{b}={\left(\frac{2.14\sigma}{{\rho}_{L}{d}_{b}}+0.505g{d}_{b}\right)}^{0.5}$ |

Author | $1/{\mathit{\tau}}_{\mathit{p}}$ | ${\mathit{C}}_{\mathit{\epsilon}\mathit{B}}$ |
---|---|---|

Morel (1997) [72] | $\propto {\left({\epsilon}_{L}/{d}_{b}^{2}\right)}^{1/3}$ | 2 |

Troshko et al. (2001) [73] | $\propto \left|\overrightarrow{{u}_{G}}-\overrightarrow{{u}_{L}}\right|/{d}_{b}$ | 0.45 |

Pfleger and Becker (2001) [74] | $\propto {\epsilon}_{L}/{k}_{L}$ | 1.44 |

Politano et al. (2003) [75] | $\propto {\epsilon}_{L}/{k}_{L}$ | 1.92 |

Rzehak et al. (2012) [76] | $\propto \sqrt{{k}_{L}}/{d}_{b}$ | 2 |

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**MDPI and ACS Style**

Besagni, G.; Varallo, N.; Mereu, R.
Computational Fluid Dynamics Modelling of Two-Phase Bubble Columns: A Comprehensive Review. *Fluids* **2023**, *8*, 91.
https://doi.org/10.3390/fluids8030091

**AMA Style**

Besagni G, Varallo N, Mereu R.
Computational Fluid Dynamics Modelling of Two-Phase Bubble Columns: A Comprehensive Review. *Fluids*. 2023; 8(3):91.
https://doi.org/10.3390/fluids8030091

**Chicago/Turabian Style**

Besagni, Giorgio, Nicolò Varallo, and Riccardo Mereu.
2023. "Computational Fluid Dynamics Modelling of Two-Phase Bubble Columns: A Comprehensive Review" *Fluids* 8, no. 3: 91.
https://doi.org/10.3390/fluids8030091