^{3}and Aluminium Research Centre (REGAL)

This paper illustrates the results obtained from two-dimensional numerical simulations of multiple gas bubbles growing under buoyancy and electromagnetic forces in a quiescent incompressible fluid. A lattice Boltzmann method for two-phase immiscible fluids with large density difference is proposed. The difficulty in the treatment of large density difference is resolved by using nine-velocity particles. The method can be applied to simulate fluid with the density ratio up to 1000. To show the efficiency of the method, we apply the method to the simulation of bubbles formation, growth, coalescence, and flows. The effects of the density ratio and the initial bubbles configuration on the flow field induced by growing bubbles and on the evolution of bubbles shape during their coalescence are investigated. The interdependencies between gas bubbles and gas rate dissolved in fluid are also simulated.

A steadily increasing computational power, new developments, and improvement of numerical methods allow numerical simulation to model more and more physical phenomena. In this sense, the in situ gas generated by alumina particles and carbon engenders the formation of gas bubbles in aluminum cells. These gas bubbles are produced by chemical reaction of alumina with the carbon, which consists of complex process of electrochemical reduction of alumina in the cells. The continuously released gas of the alumina reaction generates nuclei underneath the surface of carbon anode and detaches to dissolve in the cryolite which subsequently grows to bubbles. The bubbles arrange into a cellular structure in the cryolite, which is preserved by a rapid cyclic flow of gas.

A major aim of the simulations presented here is to gain a better fundamental physical understanding of gas bubbles because of the complexity of gas bubbles- liquid flow in chemical system [

In contrast with actual aluminum electrolysis cell models, which focus rather on specific aspects and for the most part on gas bubbles system, a comprehensive and general gas bubbles model is needed to be developed here which includes the wealth of phenomena occurring in aluminum electrolysis cell process. The model will elucidate the influence of gas bubbles as described in [

This section gives a detailed introduction to the technological aspects and the physical phenomena of the gas bubbles process by in situ gas generation. In particular, the focus relies upon the anodic chemical reaction of carbon.

The properties of the alumina, particle size and distribution, atomization atmosphere, purity, and heat treatment are crucial for the aluminum electrolysis process. The alumina and the cryolite assumed are mixed homogeneously. The subsequent nucleation method needs to assure a high densification in order to suppress gas loss by percolation at the beginning of the process. Optionally, the compacted material can be cut into near net-shape forms to study the phenomena of channelization and penetration of fluids in fracture.

The carbon anode considered as compacted material is constituted of one or several pieces of the compacted precursor. In the smelter there are flows from cell to cell in aluminum busbars while, in each cell, the current flow (I) is set to 25 KA and run downwards through the cell composed of 8 anodes, cryolite, molten metalpad, and a cathode carbon block.

In a matter of minutes after the beginning of the electrolysis process, gas bubbles nucleate and expand underneath the anodes in the cryolite and the process is interrupted by removing the gas out of the cell in order to stabilise the system. The final product consists of a closed cell with a cyclic formation of gas bubbles generated from the periodical reaction of alumina in the cryolite from feeders and surrounded by a fully dense skin called the top crust. The density in the multiphase fluid flows ranges from 0.4 to 0.8 g/cm^{3}, which corresponds to a relative density

Focusing on the actual electrolysis process, five expansion stages, dependent on the temperature, are identified in Figure

Schematic expansion of gas divided into five regimes. Phase (0) depicts the thermal expansion of the precursor, phase (I) denotes the bubbles nucleation and expansion under surface of anodes, the expansion in the semisolid and liquid range is in phase (II), phase (III) is the expansion in the liquid temperature range, and finally phase (IV) is the gas bubbles collapse.

The beginning of the gas formation, regime 0–II in Figure

The crack-like openings in the carbon anode are problematic because the CO_{2} can percolate through these channels to the ambience being lost for the multiphase system. In addition to these cracks, the small pores over the anode serve as nuclei when the alumina melts. In the melt, heterogeneous nucleation may occur at nucleation sites such as impurities or propellant particles.

The major gas bubbles expansion takes place while the cryolite is liquid, regime III-IV in Figure

The stabilization of the gas bubbles films is essential. Without stable cell walls, bubbles would immediately undergo coalescence when they contact each other. By coalescence with the atmosphere enforced by buoyancy, the bubbles and consequently massive gas portions from alumina are lost from the system. A typical gas bubbles structure cannot evolve. The gas bubbles may owe stable films to the presence of additives such as oxides. Since the alumina possesses oxide, that oxide naturally resides as a network in the precursor. The repulsive force is termed disjoining pressure in the following. Additionally, network fragments make thin films less penetrable to the melt so that the cryolite remains in the films. These effects resemble a drastically enlarged viscosity. This assumption is supported by the observed shape stability of a once melted alumina precursor.

During the gas bubbles expansion, film ruptures, growth coalescence, capillary, gravitational drainage, and electromagnetic, interbubble diffusion, gas bubbles flow and gas bubbles collapse occur. They are sketched in the following.

Film ruptures are inevitable. One observes a minimal film thickness in gas bubbles of the order of 50

Gas bubbles drainage is the flow of liquid through and out of the gas system under the influence of external and capillary forces. In the gravitational drainage, the cryolite in the plateau borders is drawn downwards by graviton so that the cryolite accumulates downwards at the cryolite layer. In the capillary drainage, the different pressures in cell films and plateau borders originating from the surface tension cause the cryolite to withdraw from the films. Naturally, these processes will reduce the film thickness in the gas bubbles system which leads to cell wall ruptures. Estimating the times scale of gas bubbles expansion, it is found to be very short compared to usual bubble in fluid. Only a drastically enlarged viscosity and in particular the static disjoining pressure in the films hinder the gravitational and capillary drainage to cause gas bubbles collapse. When the stabilization is significant, the drainage may even become negligible.

Distinct gas bubble sizes and consequently distinct bubbles may lead to interbubble diffusion of oxide carbon via the cell walls. Tiny bubbles suffer from gas loss, while large bubbles earn oxide carbon, similar to Ostwald ripening. A study of the diffusion flow of oxide carbon across cell borders demonstrates the large time scales, compared to the bubbly times, of the gas transport in cryolite. Therefore, the interbubble diffusion is negligible in the cryolite bubbly evolution. During the evolution, gas bubbles are permanently exposed to mechanical forces due to the expansion itself or spatial restriction by the cell design. Deformation of pores, rearrangement or destruction of cell films, and gas bubbles flow are the consequences. Topological rearrangements and film ruptures occur often in an avalanche-like manner. In the final stage of bubbles expansion, large bubbles are present compared to the gas bubbles system size. Gas bubbles collapse at the cryolite surface leads now to severe oxide carbon loss, in the system so that the gas system decays; that is, the collapse starts.

Reviewing gas bubbles in aluminum electrolysis cell in Figure

Vertical cut short of an aluminum electrolysis cell in which the gas expands from liquids. The points

The gas bubbles are stochastically distributed and formed simultaneously at the starting point. Dissolved alumina particles reacting with the carbon anode diffuse into the nuclei over the surface of carbon anodes. The decomposition and dissolution of alumina are modeled by a homogeneous volume source

The solute oxide carbon

The liquid domain contains islands, which are the nuclei or gas bubbles. At the gas-liquid interface

A heuristic approach condenses the film stabilization in a single function, the disjoining pressure

When the film thickness

The liquid domain

The last point is disputable and constitutes a first approximation. This leads to the following formulation.

Find the velocity

This expansion rate is explicitly dependent on time

Find the velocity

The micro-macro-coupling in the aluminum electrolysis cell is carried out by closing (

Reviewing the physical model from a numerical point of view, the most challenging aspect is the parallel implementation of the large, complex, and dynamically moving interfaces between the cryolite and the gas bubble. Among the large variety of numerical methods, the lattice Boltzmann method (LBM) [

At first, the focus remains on the fluid dynamics. Since the beginning of the last century, it is known that the kinetic Boltzmann equation can approximate the Navier-Stokes equation. By numerically solving the kinetic Boltzmann equation, the lattice Boltzmann model supplies an approximate solution for the Navier-Stokes. The LBM is consequently a kinetic scheme, based on particle dynamics. In detail, the space is divided into a 2D regular lattice. On each lattice site, single-particle distribution functions

The distinct states

The force term includes the gravity g and electromagnetic force [

The presented LBM is applied in the fluid region [

Interface cell after advection step with unknowns function (dotted line) and known function (full line).

After reconstruction the usual collision operation takes place. Yet, the interface advection remains to be discussed. For this reason, interface cells contain additional information which indicates the amount of fluid inside each cell. This liquid volume fraction

With

Empty or full interface cells are detected when

Bubble cells surrounded by interface cells build connected and closed areas of cells which represent the bubbles. The volume and amount of gas in the bubbles need to be determined in each time step. Initially, the bubble volume is known. Then, it is sufficient to calculate the change of volume per time step. It is given by the change of the volume fractions

These numerical implementations need to be validated. For this, comparisons between numerical and analytical results give information on the validity range of the parameters and the accuracy of this scheme. This test will merely focus on newly developed implementation issues and on single relaxation time, whereas the interface advection is tested by streaming a gas bubble in uniform flow field. Distortions of the round bubble shape and the small deviations from analytically predetermined way of the bubble center appear after the advection. Overall, the advection appears to be very promising results.

The same holds for the curvature algorithm. The results of template sphere methods are better than those of standard finite difference methods (FDM) often applied to volume of fluid methods (VOF).

The Laplace law has been used for several parameters by considering bubble in liquid flow. In Figure

Numerical tests of Laplace law. The dashed lines represent the theoretical equation.

They remain small overall the experiment. Dynamically, the free surface boundary conditions are investigated by standing capillary and external forces waves such as gravity and the electromagnetic force. These waves are damped by the viscosity of the liquid. Frequency and damping rate obtained numerically are in good agreement with theoretical expectations. Good agreement is also depicted by a numerical investigation on the free surface boundary condition as shown in (

Having provided and tested the physical model and its numerical implementation, the simulations of gas bubbles expansion in aluminum cell process are finally performed. The tests carried out show that the consistency of the liquid intervenes as multiplicative factor in the expression of the viscosity in the system bubbles liquid

Function

Finally, when the gas fraction tends towards 1, the viscosity also tends asymptotically towards zero. Thus, we have put in evidence, on one hand, the nonlinear dependence of the viscosity of mixture and, on the other hand, the dependence of the viscosity on the bubbles density. One remarks that the density of bubbles is consequent when the viscosity tends to an asymptotic curve.

There are several numerical simulations of bubbles-liquid structure by varying the bubbles density. The initial gas fraction is the same for each simulation,

Figure

Evolution of gas rate over simulations of volume expansion of bubbles liquid with 84 to 1500 bubbles.

Evolution of gas rate during expansion of bubbles-liquid structure. The points correspond to the obtained values by computation of the value of parameter. The curves are the approximations estimated from (

Evolution of gas fraction, viscosity, velocity, and of the pressure at different points in the cell during the gas expansion.

The computational domain

The micro-macro-coupling used in this study is based on the fact that the simulation of the expansion of the representative volume needs a macroscopic variable, that is, the ambient pressure. The expansion happens through the fluctuations of pressure

In this study, a new scheme has been developed to simulate the expansion of gas bubbles in aluminum electrolysis cell in order to describe the evolution of the viscosity and the volume mass of a representative volume of bubbles-liquid mixture during the expansion of the gas. It is found that the dependence of these two parameters is on the bubbles density in the mixture. The influence of gas bubbles interactions on these parameters has been revealed. From these results, we have carried out a micro-macro-coupling to express the evolution of the mixture volume mass during expansion in function of pressures fluctuation in the mixture. This approach helps to take into account the electrolysis process and also the influence of the mixture structure during the simulations of expansion. These simulations provide the variations of gas fraction in the cell. The simulations also show that, close to the cooled edges, the gas expansion is slower and even stopped. Thus, a fine surface skin is created on the cryolite.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada and Alcoa. A part of the research presented in this paper was financed by the Fonds de Recherche du Quebec-Nature et Technologies (FRQ-NT) by the intermediary of the Aluminium Research Center-REGAL. The authors wish also to thank Dr. Donald Ziegler from the Alcoa Canada Primary Metals for the scientific discussions.