Metallurgical and Materials Transactions B

, Volume 49, Issue 5, pp 2821–2834 | Cite as

A Multi-scale Mathematical Model of Growth and Coalescence of Bubbles Beneath the Anode in an Aluminum Reduction Cell

  • Meijia Sun
  • Baokuan LiEmail author
  • Linmin Li


Modeling of bubble shapes is challenging because of the wide range of length scales in aluminum reduction cells. A 3D multi-scale mathematical model was developed to understand the nucleation, growth and coalescence of bubbles beneath the anode and to investigate the transition of bubbles from the micro- to macro level. The motion of micro-bubbles is examined using the discrete bubble model (DBM) within a Lagrangian reference. An algorithm for the transition from discrete micro-bubbles to large bubbles, which are fully resolved by the volume of fluid (VOF) approach, is achieved using a user-defined function. The two-way coupling between discrete bubbles and continuous fluids is achieved by inter-phase momentum exchange. The model involves three kinds of bubble coalescence: micro-bubble coalescence is taken into account by the DBM; large bubbles swallowing up micro-bubbles are solved by the discrete-continuum transition model; the coalescence between large bubbles is handled by the VOF method. Numerical results show that the coverage and thickness of bubbles agree well with the experimental data in the literature. Bubble thickness stops increasing when the bubble elongates with the anode bottom, which is 4.0 to 4.5 mm. Meanwhile, two asymmetrical thick heads can be observed under the anode. Bubble release frequency increases with increasing current density.


\( m_{\text{b}} \)

Bubble mass

\( m^{\prime}_{1} \)

New bubble mass after coalescence

\( m_{1} \)

Larger bubble mass

\( m_{2} \)

Smaller bubble mass

\( \vec{u}_{\text{b}} \)

Bubble velocity

\( \vec{u}^{\prime}_{\text{b}} \)

New bubble velocity after coalescence

\( \vec{u}_{{_{{{\text{b}}1}} }} \)

Larger bubble velocity

\( \overrightarrow {u}_{{_{b2} }} \)

Smaller bubble velocity

\( \vec{u}^{\prime}_{{{\text{b}}1}} \)

Larger bubble velocity after rebound

\( \vec{u}^{\prime}_{\text{b2}} \)

Smaller bubble velocity after rebound

\( u_{\text{g}} \)

Resolved bubble velocity

\( \vec{u} \)

Bath velocity

\( \vec{u}_{\text{rel}} \)

Relative velocity

\( \overrightarrow {F}_{\text{b}} \)

The resultant force of buoyancy and gravity

\( \overrightarrow {F}_{\text{d}} \)

Drag force

\( \overrightarrow {F}_{\text{vir}} \)

Virtual mass force

\( \overrightarrow {F}_{p} \)

Pressure gradient force

\( \overrightarrow {F}_{\text{me}} \)

Momentum exchange

\( \overrightarrow {F}_{\text{s}} \)

Surface tension

\( V_{\text{b}} \)

Bubble volume

\( V_{\text{cell}} \)

Cell volume

\( d_{\text{b}} \)

Bubble diameter

\( \overline{d} \)

Arithmetic mean diameter of two bubbles

\( r \)

Bubble radius

\( r_{1} \)

Larger bubble radius

\( r_{2} \)

Smaller bubble radius

\( C_{\text{D}} \)

Drag coefficient



\( F \)

Faraday constant


Molar gas constant

\( p_{1} \)

Bubble pressure in the bath

\( H \)

Anode immersion depth

\( \Delta t \)

Time step

\( P \)

Collision probability


Total number of bubbles

\( \overline{N} \)

Mean collision number

\( N_{1} \)

Number of smaller bubbles

\( N_{\text{c}} \)

Number of collisions

x, y

Random numbers (0, 1)

\( b \)

Actual collision parameter

\( b_{\text{crit}} \)

Critical offset

\( {\text{We}} \)

Collisional Weber number

\( S_{1} ,S_{2} \)

Mass source

Greek Symbols

\( \rho \)

Mixture density

\( \rho_{\text{b}} \)

Bubble density

\( \rho_{\text{g}} \)

Resolved bubble density

\( \rho_{\text{l}} \)

Bath density

\( \mu \)

Mixture viscosity

\( \mu_{\text{l}} \)

Bath viscosity

\( \mu_{\text{g}} \)

Resolved bubble viscosity

\( \sigma_{ij} \)

Surface tension coefficient



\( \alpha \)

Volume fraction

\( \alpha_{\text{g}} \)

Gas volume fraction



The authors wish to thank State Key Laboratory of Complex Nonferrous Metal Resources Clean Utilization, No. CNMRCUKF1607.


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Copyright information

© The Minerals, Metals & Materials Society and ASM International 2018

Authors and Affiliations

  1. 1.School of MetallurgyNortheastern UniversityShenyangChina
  2. 2.College of Energy and ElectricityHohai UniversityNanjingChina
  3. 3.State Key Laboratory of Complex Nonferrous Metal Resources Clean UtilizationKunmingChina

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