# Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Population Balance Modeling

#### 2.1.1. Class Methods

#### 2.1.2. Quadrature-Based Moments Method

#### 2.1.3. Quadrature Method of Moments

#### 2.1.4. Direct Quadrature Method of Moments

#### 2.1.5. Extended Quadrature Method of Moments

#### 2.1.6. Closure Models for Coalescence and Breakage

## 3. Numerical Solution

**PI**SO-SI

**MPLE**) approach provided by OpenFOAM, which is a combination of the PISO (Pressure Implicit with Split Operator) and SIMPLE (Semi Implicit Method for Pressure Linked Equations) procedures.

## 4. Results and Discussion

#### 4.1. Test Case 1: Pseudo-2D Bubble Column

#### 4.2. Test Case 2: 3D Bubble Column

#### 4.3. Test Case 3: Water Electrolysis Reactor

- Reference electrode: Ag/AgCl saturated with KCl
- Counter electrode: platinum grid
- Working electrode (rotating electrode): made by embedding a platinum rod in an insulating Poly Vinylidene Fluoride (PVFD) cylinder.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Symbols | ||

$a(\zeta )$ | breakage kernel | s${}^{-1}$ |

B | birth | m${}^{-3}$ s${}^{-1}$ |

$b(\nu )$ | break-up frequency function | s${}^{-1}$ |

${C}_{D}$ | drag coefficient | − |

${C}_{l}$ | lift coefficient | − |

${C}_{{\nu}_{m}}$ | virtual mass coefficient | − |

${c}_{f}$ | increase coefficient of surface area | |

D | death | m${}^{-3}$ s${}^{-1}$ |

d | bubble diameter | m |

${d}_{32}$ | Sauter mean diameter | m |

$Eo$ | Eotvos number | − |

f | friction coefficient for flow around bubbles | − |

${f}_{i}$ | volume fraction of bubble class i | − |

$\mathbf{F}$ | volumetric force | N m${}^{-3}$ |

$\mathbf{g}$ | acceleration vector due to gravity | m s${}^{-2}$ |

$\stackrel{=}{I}$ | unit tensor | |

k | turbulent kinetic energy | j kg${}^{-1}$ |

L | bubble size | m |

$m(\nu )$ | mean number of daughter produced by breakage | − |

n | number density of bubbles | m${}^{-3}$ |

N | angular velocity | rad s${}^{-1}$ |

p | pressure | Pa |

${P}_{c}$ | coalescence efficiency or collision probability | − |

$p(\nu ,{\nu}^{\prime})$ | pressure | N m${}^{-2}$ |

$\mathbf{r}$ | position vector | m |

$\mathbf{R}$ | interphase force | N m${}^{-3}$ |

$Re$ | Reynolds number | − |

${\overline{R}}_{\varphi}^{eff}$ | Reynolds (turbulent) and viscous stress | m s${}^{-2}$ |

t | Time | s |

$\mathbf{U}$ | average velocity of phase | m s${}^{-1}$ |

${u}_{ij}$ | bubble approaching turbulent velocity | m s${}^{-1}$ |

$We$ | Weber number | − |

Greek Symbols | ||

$\alpha $ | Volume fraction | − |

$\beta $ | constant | $2.05$ or s${}^{-1}$ |

$\beta (\zeta ,{\zeta}^{\prime})$ | coalescence rate | s${}^{-1}$ |

$\u03f5$ | Turbulent kinetic energy dissipation rate | m${}^{2}$ s${}^{-3}$ |

${\theta}_{i,j}$ | collision frequency | m${}^{-3}$ s${}^{-1}$ |

$\mu $ | dynamic viscosity of the continuous phase | N/m${}^{3}$ |

$\lambda $ | eddy size | m |

$\rho $ | density | kg/m${}^{3}$ |

$\sigma $ | surface tension or variance | Nm${}^{-1}$ or m |

$\zeta $ | internal variable | − |

${\Omega}_{B}$ | breakage frequency | s${}^{-1}$ |

$\nu $ | bubble size or kinematic viscosity | m${}^{3}$ or m${}^{2}/s$ |

${\xi}_{ij}$ | size ratio | ${d}_{i}/{d}_{j}$ |

$\mathrm{\Gamma}(a,x)$ | incomplete Gamma function | − |

$\eta $ | diameter ratio | ${d}_{i}/{d}_{j}$ |

$\stackrel{=}{\tau}$ | stress tensor | kg m${}^{-1}$ s${}^{-2}$ |

Subscripts | ||

$ag$ | aggregation | |

$br$ | breakage | |

$eff$ | effective | |

G | gas phase | |

L | liquid phase | |

m | mixture | |

i | phase number | |

$lam$ | laminar | |

t | turbulent |

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**Figure 2.**Profile of axial liquid velocity through a line at y = 37 cm from the bottom of the bubble column. (

**a**) The variation of primary and secondary nodes and their effects on the predicted velocity profile. (

**b**) The comparison among CM (classes), DQMOM (3 nodes), EQMOM (3 nodes) and EQMOM (2 nodes).

**Figure 4.**Number density function in water zone (liquid phase) using EQMOM (

**a**) ${N}_{\alpha}=2$ and (

**b**) ${N}_{\alpha}=3$.

**Figure 6.**Comparison between EQMOM and QMOM against experimental data: (

**a**) axial gas velocity and (

**b**) axial liquid velocity for the case of Deen [64].

**Figure 7.**Color maps of time-averaged Sauter diameter along the plane located in the middle of the column of Deen [64].

**Figure 8.**(

**a**) Schematic of hexahedral mesh [67]. (

**b**) The specifications of the reactor and (

**c**) the location of volume W1.

**Figure 9.**(

**a**) The continuous distribution function imposed at the electrode surface (Nierhaus et al. [66]); (

**b**) mean axial velocity component profile ${{u}_{z}}^{*}=\frac{{u}_{z}}{\sqrt{{\omega}_{z}\nu}}$ and comparison to the analytical solution.

**Figure 10.**Bubble size distribution in W1 for (

**a**) EQMOM (three nodes) and rpm = 100; (

**b**) EQMOM (three nodes) and rpm = 250.

**Table 1.**The most remarkable studies in the field of bubbly flow modeling using Population Balance Model (PBM). CM, Method of Classes; EQMOM, Extended Quadrature Method of Moments; DQMOM, Direct Quadrature Method of Moments.

Reference | Test Case | PBM | Hypotheses | Remarks |
---|---|---|---|---|

Bannari et al. [1] | bubble column | CM | accumulation, advection, coalescence, breakage | Constant mean bubble size does not give satisfactory results compared to those based on PBM; 25 classes give better results |

Gimbun et al. [38] | gas-liquid stirred tank | QMOM | accumulation, coalescence, breakage | Better agreement is achieved using PBM compared to a uniform bubble size |

Li et al. [37] | liquid-liquid stirred tank | EQMOM and QMOM | accumulation, advection, coalescence, breakage | Similar predictions for EQMOM and QMOM; EQMOM provides a continuous BSD |

Selma et al. [33] | gas-liquid stirred tank; bubble column | DQMOM and CM | accumulation, advection, coalescence, breakage | High number of classes is required; CM is computationally heavy; DQMOM is much more efficient (computationally) compared to CM |

Gupta and Roy [39] | bubble column | DQMOM and QMOM | accumulation, coalescence, breakage | A summary of studies done on bubble columns flow modeling using PBM; no significant difference between DQMOM and QMOM |

Askari et al. [40] | gas-liquid stirred tank | EQMOM | accumulation, coalescence, breakage | The agreement between experimental data and simulation results using EQMOM; reconstruction of bubble size distribution |

PBM | Advantages | Disadvantages |
---|---|---|

CM | Intuitive and accurate | Computationally intensive |

QMOM | Wide range of bubble sizes with a reduced computational cost | Disabled in case of null internal coordinates |

DQMOM | Wide range of bubble sizes with a reduced computational cost | Shortcomings related to non-conservative quantities (weights and abscissas) |

EQMOM | Wide range of bubble sizes with a reduced computational cost (ONLYcompared to CM) and reconstruction capability of continuous NDF | Heavy computation compared to QMOM and DQMOM |

Equation | Formulation |
---|---|

Continuity (single-phase) | $\frac{\partial \rho}{\partial t}+\nabla \xb7\left(\rho \mathbf{U}\right)=0$ |

Momentum (single-phase) | $\frac{\partial}{\partial t}\left(\rho \mathbf{U}\right)+\nabla \xb7(\rho \mathbf{U}\mathbf{U})=-\nabla p+\nabla \xb7{\overline{\overline{\tau}}}_{\mathrm{effi}}$ |

Continuity (multi-phase) | $\frac{\partial}{\partial t}\left({\rho}_{\mathrm{i}}{\alpha}_{\mathrm{i}}\right)+\nabla \xb7\left({\alpha}_{\mathrm{i}}{\rho}_{\mathrm{i}}{\mathbf{U}}_{\mathrm{i}}\right)=0.0$ |

Reynolds stress tensor | ${\overline{\overline{\tau}}}_{\mathrm{effi}}=\left({\mu}_{\mathrm{lam}}+{\mu}_{\mathrm{t}}\right)\left(\nabla \mathbf{U}+\nabla {\mathbf{U}}^{T}\right)-\frac{2}{3}\left(\rho k+\left({\mu}_{\mathrm{lam}}+{\mu}_{\mathrm{t}}\right)\nabla \xb7\mathbf{U}\right)\overline{\overline{I}}$ |

Momentum (multi-phase) | $\frac{\partial}{\partial t}\left({\rho}_{\mathrm{i}}{\alpha}_{\mathrm{i}}{\mathbf{U}}_{\mathrm{i}}\right)+\nabla \xb7\left({\alpha}_{\mathrm{i}}{\rho}_{\mathrm{i}}{\mathbf{U}}_{\mathrm{i}}{\mathbf{U}}_{\mathrm{i}}\right)=-{\alpha}_{\mathrm{i}}\nabla p+\nabla \xb7({\alpha}_{i}{\overline{\overline{\tau}}}_{\mathrm{effi},i})+{\mathbf{R}}_{\mathrm{i}}+{\mathbf{F}}_{\mathrm{i}}+{\alpha}_{\mathrm{i}}{\rho}_{\mathrm{i}}\mathbf{g}$ |

Interfacial momentum exchange | ${\mathbf{R}}_{\mathrm{G}}=-{\mathbf{R}}_{\mathrm{L}}={\mathbf{R}}_{\mathrm{G},\mathrm{drag}}+{\mathbf{R}}_{\mathrm{G},\mathrm{lift}}+{\mathbf{R}}_{\mathrm{G},\mathrm{vm}}$ |

Liquid-gas exchange coefficient | $K=\frac{3}{4}{\rho}_{\mathrm{L}}{\alpha}_{\mathrm{G}}\frac{{C}_{\mathrm{D}}}{{d}_{32}}\mid {\mathbf{U}}_{\mathrm{G}}-{\mathbf{U}}_{\mathrm{L}}\mid ({\mathbf{U}}_{\mathrm{G}}-{\mathbf{U}}_{\mathrm{L}})+{\alpha}_{\mathrm{G}}{C}_{\mathrm{l}}{\rho}_{\mathrm{L}}{\mathbf{U}}_{\mathbf{r}}\times (\nabla \times {\mathbf{U}}_{\mathbf{L}})+{\alpha}_{\mathrm{L}}{C}_{\mathrm{vm}}{\rho}_{\mathrm{L}}\left(\frac{{D}_{L}{\mathbf{U}}_{\mathbf{L}}}{Dt}-\frac{{D}_{G}{\mathbf{U}}_{\mathbf{G}}}{Dt}\right)$ |

Schiller–Naumann drag coefficient [42] | ${C}_{D}=\left\{\begin{array}{c}\frac{24}{\mathrm{Re}}\left(1+0.15{\mathrm{Re}}^{0.687}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{Re}\le 1000\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0.44\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{otherwise}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ |

Ishii–Zuber drag coefficient [43] | ${C}_{D}=\mathrm{max}\{min[\frac{2}{3}\sqrt{\mathrm{Eo}},\frac{8}{3}],\frac{24}{\mathrm{Re}}(1+0.1{\mathrm{Re}}^{0.75})\}$ |

Tomiyama lift coefficient [44] | ${C}_{l}=\left\{\begin{array}{c}min(0.288tanh(0.121\mathrm{Re}),f({\mathrm{Eo}}_{G}))\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{\mathrm{Eo}}_{G}<4\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ f({\mathrm{Eo}}_{G})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}4\le {\mathrm{Eo}}_{G}\le 10.7\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ |

Mixture k-$\u03f5$ model [45] | $\frac{\partial}{\partial t}\left({\rho}_{\mathrm{m}}{k}_{\mathrm{m}}\right)+\nabla \xb7\left({\rho}_{\mathrm{m}}{\mathbf{U}}_{\mathrm{m}}{k}_{\mathrm{m}}\right)=\nabla \xb7\left(\frac{{\mu}_{\mathrm{t},\mathrm{m}}}{{\sigma}_{\mathrm{k}}}\nabla {k}_{\mathrm{m}}\right)+{P}_{k}^{\mathrm{m}}-{\rho}_{\mathrm{m}}{\u03f5}_{\mathrm{m}}+{S}_{k}^{\mathrm{m}}$ |

$\frac{\partial}{\partial t}\left({\rho}_{\mathrm{m}}{\u03f5}_{\mathrm{m}}\right)+\nabla \xb7\left({\rho}_{\mathrm{m}}{\mathbf{U}}_{\mathrm{m}}{\u03f5}_{\mathrm{m}}\right)=\nabla \xb7\left(\frac{{\mu}_{\mathrm{t},\mathrm{m}}}{{\sigma}_{\u03f5}}\nabla {k}_{\mathrm{m}}\right)+\frac{{\u03f5}_{m}}{{k}_{m}}\left({C}_{1\u03f5}{G}_{\mathrm{k},\mathrm{m}}-{C}_{2\u03f5}{\rho}_{\mathrm{m}}{\u03f5}_{\mathrm{m}}\right)+{C}_{\u03f53}\frac{{\u03f5}_{m}}{{\u03f5}_{k}}{S}_{k}^{\mathrm{m}}$ | |

Break-up rate function [46] | ${\mathsf{\Omega}}_{B}({d}_{j}:{d}_{i})=\frac{-3{k}_{1}(1-\alpha )}{11{b}^{8/11}}{n}_{j}{\left(\frac{\u03f5}{{d}_{j}^{2}}\right)}^{1/3}\{\mathrm{\Gamma}(8/11,{t}_{m})-\mathrm{\Gamma}(8/11,b)+2{b}^{3/11}(\mathrm{\Gamma}(5/11,{t}_{m})-\mathrm{\Gamma}(5/11,b))$ |

$+{b}^{6/11}(\mathrm{\Gamma}(2/11,{t}_{m})-\mathrm{\Gamma}(2/11,b))\}$ | |

Coalescence rate [47] | $\beta ({d}_{i},{d}_{j})={\theta}_{ij}{P}_{c}$ |

Coalescence frequency [48] | $\theta (i,j)=\frac{\pi}{4}{n}_{i}{n}_{j}{({d}_{i}+{d}_{j})}^{2}{\u03f5}^{1/3}{\left({d}_{i}^{2/3}+{d}_{j}^{2/3}\right)}^{1/2}$ |

Coalescence efficiency [47] | ${P}_{C}({d}_{i},{d}_{j})=exp\left(-c\frac{{\left[0.75(1+{\xi}_{ij}^{2})(1+{\xi}_{ij}^{3})\right]}^{1/2}}{{({\rho}_{d}/{\rho}_{c}+0.5)}^{1/2}{(1+{\xi}_{ij})}^{3}}{\mathrm{We}}_{ij}^{1/2}\right)$ |

Number of Classes | 7 | 11 | 15 | 25 |
---|---|---|---|---|

Value of r | 3 | 5 | 7 | 12 |

Value of s | 2 | 1.5157 | 1.3459 | 1.1892 |

${\mathit{W}}_{\mathbf{1}}$ | ${\mathit{W}}_{\mathbf{2}}$ | ${\mathit{W}}_{\mathbf{3}}$ | ${\mathit{W}}_{\mathbf{4}}$ | ${\mathit{W}}_{\mathbf{5}}$ |
---|---|---|---|---|

$\sqrt{\frac{35+2\sqrt{70}}{63}}$ | $\sqrt{\frac{35-2\sqrt{70}}{63}}$ | 0 | $-\sqrt{\frac{35-2\sqrt{70}}{63}}$ | $-\sqrt{\frac{35+2\sqrt{70}}{63}}$ |

Boundary Conditions | Initial Condition | ||
---|---|---|---|

Inlet | Wall | Outlet | Inlet value |

${f}_{i}=\left\{\begin{array}{c}1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}2\mathrm{r}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\ne \phantom{\rule{4.pt}{0ex}}2\mathrm{r}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ | Neumann | Neumann |

**Table 7.**Boundary conditions, initial conditions and divergence scheme corresponding ${m}_{i}$ equations.

Boundary Conditions | Initial Condition | ||
---|---|---|---|

Inlet | Wall | Outlet | Inlet value |

Equation (32) | Neumann | Neumann |

Boundary Conditions | Initial Condition | ||
---|---|---|---|

Inlet | Wall | Outlet | Inlet value |

${L}_{i}$ and ${W}_{i}$ corresponding to ${m}_{i}$ | Neumann | Neumann |

**Table 9.**Models used in the simulation of the bubble column of Pfleger et al. [62]. TFM, Two-Fluid Model.

Settings | Model |
---|---|

Two-phase flow | Two-fluid model (TFM) |

Drag | Schiller and Naumann [42] |

Lift | Tomiyama et al. [44] |

Virtual mass | ${C}_{vm}=0.25$ [33] |

Turbulence | Standard k-$\u03f5$ model |

Population balance | CM and EQMOM |

Coalescence | Hagesather et al. [47] |

Breakage | Luo and Svendsen [46] |

Boundary Conditions | Initial Condition | ||
---|---|---|---|

Inlet | Wall | Outlet | Inlet value |

${f}_{i}=\left\{\begin{array}{c}1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}12\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\ne \phantom{\rule{4.pt}{0ex}}12\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ | Neumann | Neumann |

**Table 11.**Boundary and initial conditions for ${m}_{i}$ equations used in the QMOM and EQMOM methods (Test Cases I and II).

Boundary Conditions | Initial Condition | ||
---|---|---|---|

Inlet | Wall | Outlet | Inlet value |

${m}_{i}=\left\{\begin{array}{cc}1\hfill & i=0\hfill \\ 5\hfill & i=1\hfill \\ 25\hfill & i=2\hfill \\ 125\hfill & i=3\hfill \\ 625\hfill & i=4\hfill \\ 3125\hfill & i=5\hfill \\ \mathrm{15,625}\hfill & i=5\hfill \end{array}\right.$ | Neumann | Neumann |

**Table 12.**Boundary and initial conditions for ${W}_{i}$ and ${L}_{i}$ equations used in the DQMOM method [33].

Boundary Conditions | Initial Condition | ||
---|---|---|---|

Inlet | Wall | Outlet | ${W}_{i}=\left\{\begin{array}{c}0.33\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0.33\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0.34\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ ${L}_{i}=\left\{\begin{array}{c}0.001\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0.002\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0.003\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ |

${W}_{i}=\left\{\begin{array}{c}0.1667\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0.6667\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 0.1667\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ ${L}_{i}=\left\{\begin{array}{c}3.26\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 5.00\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \\ 6.73\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill \end{array}\right.$ | Neumann | Neumann |

Settings | Model |
---|---|

Two-phase flow | Two-fluid model |

Drag | Ishii and Zuber [43] |

Lift | ${C}_{l}=0.5$ |

Virtual mass | ${C}_{vm}=0.5$ |

Turbulence | Behzadi et al. [45] |

Population balance | EQMOM and QMOM |

Coalescence | Hagesather et al. [47] |

Breakage | Luo and Svendsen [46] |

Settings | Model |
---|---|

single-phase flow | - |

laminar | - |

population balance | EQMOM |

coalescence | no |

breakage | no |

Boundary Conditions | Initial Condition | ||
---|---|---|---|

Inlet | Wall | Outlet | Inlet value |

${m}_{i}=$ $\left\{\begin{array}{cc}1\hfill & i=0\hfill \\ 145\hfill & i=1\hfill \\ \mathrm{26,801}\hfill & i=2\hfill \\ 6.31\times {10}^{6}\hfill & i=3\hfill \\ 1.89\times {10}^{9}\hfill & i=4\hfill \\ 7.26\times {10}^{11}\hfill & i=5\hfill \\ 3.54\times {10}^{14}\hfill & i=6\hfill \end{array}\right.$ | Neumann | Neumann |

Case | DQMOM [33] | CM (25 Classes) | QMOM (n = 3) | EQMOM (n = 2) | EQMOM (n = 3) |
---|---|---|---|---|---|

Test Case 1 | 1 | 5 | - | 1.3 | 1.4 |

Test Case 2 | - | - | 1 | - | 1.5 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Askari, E.; Proulx, P.; Passalacqua, A. Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications. *ChemEngineering* **2018**, *2*, 8.
https://doi.org/10.3390/chemengineering2010008

**AMA Style**

Askari E, Proulx P, Passalacqua A. Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications. *ChemEngineering*. 2018; 2(1):8.
https://doi.org/10.3390/chemengineering2010008

**Chicago/Turabian Style**

Askari, Ehsan, Pierre Proulx, and Alberto Passalacqua. 2018. "Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications" *ChemEngineering* 2, no. 1: 8.
https://doi.org/10.3390/chemengineering2010008