Computational Fluid Dynamics Modelling of Two-Phase Bubble Columns: A Comprehensive Review
Abstract
:1. Introduction
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
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 .
- 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 () and N nodes of abscissas , and determines a sequence of polynomials orthogonal to the unknown number density function. The functional form (for a univariate problem with as internal coordinate) reads as follows: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 | [m/s] | Flow Regime |
---|---|---|---|---|---|---|---|---|
[74] | 2001 | Ansys CFX-4.3 | 0.288 | Ring ** = 0.7 mm | 8.68 | Air/water | 0.5 | Homogeneous |
[78] | 2005 | Ansys FLUENT | 0.19 | Perforate plate = 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 *** = 1.96 mm | 6 | Air/water | 0.2 | Homogeneous |
[45] | 2013 | Ansys CFX-13.0 | 0.288 | Perforate plate = 0.7 mm | 8.68 | Air/water | 0.15 → 1 | Homogeneous |
[80] | 2013 | Ansys FLUENT-14.0 | 0.44 | Perforate plate ** = 0.77 mm | 4 | Air/water | 10 | Heterogeneous |
[81] | 2013 | Ansys CFX-14.5 | 0.19 | Perforated plate *** = 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 ** = 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 = 4 mm | 6.52 | Air/water | 19 → 38 | Heterogeneous |
[90] | 2017 | Ansys CFX (1) Ansys FLUENT (2) | 0.39 | Tree ** = 0.5 mm | 5.13 | Air/water | 14 → 28 | Heterogeneous |
[91] | 2018 | Ansys CFX-15.0 | 0.156 | Perforate plate ** = 1 mm | 1.9 → 5.11 | Air/water | 2.1 | Homogeneous |
[30] | 2018 | Ansys FLUENT-15.0 | 0.40 | Perforate plate ** = 2 mm | 4 | Air/water | 0.03 → 35 | Homogeneous → heterogeneous |
[85] | 2020 | Ansys FLUENT-17.2 | 0.15 | Perforate plate ** = 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 | = 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 () Riboux () | Tomiyama | Burns | Hosokawa | No |
[80] | Mono-dispersed (A) MUSIG (B) iMUSIG (C) | Luo | Luo | Luo and Svendsen | Shiller and Naumann (1) | No | Tomiyama () No () | No | No | No |
[81] | Mono-dispersed = 4 mm (A) = 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) = 0.44 (D) | No | = 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 | = 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 = 0.1 (A) = 0.2 (B) = 0.3 (C) = 0.5 (D) = 0.9 (E) = 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 | = 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 |
Ref. | Turbulence Modelling | BIT | Numerical Aspects | Geometry | Mesh Size | |||
---|---|---|---|---|---|---|---|---|
Continuous Phase | Dispersed Phase | P-V Coupling | Spatial Discretization | Time Discretization | ||||
[74] | No | Pfleger and Becker (1) No (2) | SIMPLEC | - | - | 3D cylindrical | 6150 (A) 12,300 (B) 62,400 (C) | |
[78] | No | - | - | - | - | 2D axisymmetric | 36,000 | |
[46] | No | Sato | SIMPLE | - | - | 3D cylindrical | 90,000 | |
[79] | (A) (B) RNG (C) RSM (D) LES (E) | No | Sato | - | Second-order implicit | First-order implicit | 3D cylindrical | 52,330 |
[45] | SST | No | - | - | - | - | 3D cylindrical | 30,000 |
[71] | No | Rzehak (A) No (B) Sato (C) Morel (D) Troshco (E) Politano (F) Politano varied (E) | - | - | Second-order implicit | 3D rectangular | 200,000 | |
[80] | RNG | No | - | - | Volume fraction: QUICK Others: second-order upwind | - | 3D cylindrical | 67,392 |
[81] | No | Pfleger and Becker () NO () | - | High resolution schemes | Second-order implicit | 3D cylindrical | 58,800 | |
[82] | RNG | No | Sato | - | - | - | 3D rectangular | 46,080 |
[83] | () RSM () | () RSM () | Sato | PISO | - | - | 2D axisymmetric | - |
[88] | SST | No | Rzehak (A) Sato (B) No (C) | - | High resolution schemes | second-order implicit | 3D rectangular | 200,000 |
[92] | RNG | RNG | - | SIMPLE | - | - | 3D cylindrical | 342,230 |
[89] | RNG | No | No | - | High resolution schemes | - | 3D cylindrical | |
[84] | Mixture 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] | 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] | No | Sato | SIMPLE | Second-order upwind | - | 3D cylindrical | 605,802 | |
[30] | 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 = 1.2 cm/s: (D) 4.40 (E) 4.19 (G) 5.96 Evaluated at = 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 = 0.15 cm/s: (B) 18.79 Evaluated at = 0.5 cm/s: (B) 15.56 Evaluated at = 1 cm/s: (B) 19.01 With NDF Evaluated at = 0.15 cm/s: (A) 22.02 (B) 20.89 Evaluated at = 0.5 cm/s: (A) 21.68 (B) 14.03 Evaluated at = 1 cm/s: (A) 21.7 (B) 17.60 | Without NDF Evaluated at = 0.15 cm/s: (B) 32.67 Evaluated at = 0.5 cm/s: (B) 32.41 Evaluated at = 1 cm/s: (B) 30.48 With NDF Evaluated at = 0.15 cm/s: (A) 33.02 (B) 13.74 Evaluated at = 0.5 cm/s: (A) 20.80 (B) 36.39 Evaluated at = 1 cm/s: (A) 15.30 (B) 27.31 |
[71] | [96] | - | Evaluated at = 0.3 cm/s: (1A) 4.19 (1A) 12.22 (1B) 4.19 (1C) 4.19 (1D) 4.19 (1E) 9.43 (1F) 6.98 Evaluated at = 1.3 cm/s: (1A) 14.42 (2A) 6.48 (2A) 20.34 (2B) 8.95 (2C) 6.73 (2D) 7.49 (2E) 8.31 (2F) 7.85 (2G) 9.36 | Evaluated at = 0.3 cm/s: (1A) 44.27 (1A) 44.27 (1B) 44.45 (1C) 44.45 (1D) 48.18 (1E) 43.27 (1F) 40.32 Evaluated at = 1.3 cm/s: (1A) 47.48 (2A) 30.20 (2A) 39.28 (2B) 41.57 (2C) 57.50 (2D) 41.67 (2E) 47.58 (2F) 45.75 (2G) 80.64 |
[80] | [56] | - | (1A) 21 (2B) 9.96 (2B) 21.44 (2C) 9.68 (2C) 21.53 | (1A) 21.25 (2B) 29.89 (2B) 32.31 (2C) 22.26 (2C) 37.05 |
[81] | [97] | - | Evaluated at = 8 cm/s: (1A) 15.37 (1B) 23.31 (2A) 20.46 (2B) 22.29 Evaluated at = 12 cm/s: (1A) 20.80 (1A) 75.80 | - |
[82] | [98] | (A) 14.76 (B) 9.45 (C) 6.18 (D) 22.75 | - | - |
[83] | [99] | - | (1A) 8.66 (2A) 10.96 (1A) 9.60 (2A) 27.70 (1B) 5.17 (1B) 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 = 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 = 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 = 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 = 16 cm/s: (1B) 12.48 (2B) 8.94 | Evaluated at = 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 = 3 cm/s: (E) 5.70 Evaluated at = 16 cm/s: (E) 4.07 | Evaluated at = 3 cm/s: (E) 38.39 Evaluated at = 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 RNG and the 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 | |
Local velocity | [m/s] | |
Drag coefficient | [-] | |
Lift coefficient | [-] | |
Turbulent dispersion coefficient | [-] | |
Wall lubrication coefficient | [-] | |
Virtual mass coefficient | [-] | |
Bubble diameter | [m] | |
Equivalent bubble diameter | [m] | |
Non-dimensional diameter | [-] | |
Critical non-dimensional diameter | [-] | |
Hydraulic diameter | [m] | |
E | Bubble aspect ratio | [-] |
Eötvös number | [m] | |
Drag force | [kg/] | |
Lift force | [kg/] | |
Turbulent dispersion force | [kg/] | |
Wall lubrication force | [kg/] | |
Virtual mass force | [kg/] | |
g | Acceleration due to gravity | [m/] |
Breakup frequency | [] | |
h | Swarm factor | [-] |
Collision frequency | [] | |
Drift flux | [m/s] | |
k | Turbulent kinetic energy | [] |
Momentum exchanges | [kg/] | |
Morton number | [-] | |
p | Pressure | [Pa] |
Bubble Reynolds number | ||
S | Total source/sink term in the population balance equation | [/s] |
Total source/sink term due to breakup | [/s] | |
Total source/sink term due to coalescence | [/s] | |
Total source/sink term due to mass transfer | [/s] | |
Total source/sink term due to pressure change | [/s] | |
Total source/sink term due to phase change | [/s] | |
Total source/sink term due to reaction | [/s] | |
Superficial gas velocity | [m/s] | |
Superficial liquid velocity | [m/s] | |
Bubble volume in population balance equation | [] | |
Local gas volume fraction | [-] | |
Daughter distribution function | [-] | |
Global gas holdup | [-] | |
Turbulent dissipation rate | [] | |
Coalescence efficiency | [-] | |
Dynamic viscosity | [] | |
Turbulent viscosity | [] | |
Specific dissipation rate | [] | |
Density | [] | |
Surface tension | [kg/ | |
Viscous stress tensor | [] | |
G | Gas phase | |
L | Liquid phase | |
k | k-th phase |
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Author | Breakage Frequency (g) | Daughter Size Distribution () |
---|---|---|
Prince and Blanch (1990) [50] | Uniform | |
Luo and Svendsen (1996) [51] | with | with |
Martïnez and Bazän (1999) [52,53] | with [54], | with , , [54] |
Lehr et al. (2002) [55] |
Author | Collision Frequency (h) | Collision Efficiency () |
---|---|---|
Coulaloglou and Tavlarides (1977) [59] | ||
Prince and Blanch (1990) [50] | ||
Chesters (1991) [60] | with , | |
Luo (1993) [61] | with , |
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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
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 StyleBesagni, 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