MODELING GRAVITY WAVE IN 3D WITH OPENFOAM IN AN ALUMINUM REDUCTION CELL

WITH REGULAR AND IRREGULAR CATHODE SURFACES

Marc Dupuis ^{1} and Michaël Pagé, Pr. Eng. ^{2}

marc.dupuis@genisim.com

michael.page@simu-k.com

Keywords: Modeling, irregular cathode, cell stability, gravity wave, OpenFoam

Abstract

In any cases, as presented in last year review [2], it is the opinion

of the authors that a transient cell stability analysis is required to

In recent years extensive modeling work has been done to assess

make any prediction of the impact of irregular cathode surface

if the usage of an irregular cathode surface does or does not

technology on the cell stability as bath/metal interface wave

increases the MHD cell stability.

dynamic is a time dependant phenomenon.

So far, 2 types of studies have been carried out: full 3D steady-

As demonstrated in [2], MHD-Valdis is the perfect tool to model

state analysis and

2D shallow layer dynamic analysis. Cell

and hence study MHD driven cell stability in the case of regular

stability being a transient phenomenon, steady-state results are not

flat cathode surface as the model only represents and solves the

providing any direct answer to the question.

key physics required, nothing more, which make MHD-Valdis an

extremely efficient model.

2D shallow layer dynamic analysis can directly answer the

question, but unfortunately, irregular cathode surface introduces a

One of the key simplification in MHD-Valdis solver is the use of

third dimension to the flow, so this type of 2D analysis is not the

the 2D shallow layer CFD model to solve the bath and metal flow.

best suited to analyze this type of 3D flow problem.

Yet, as Figure 12 b) of [1] clearly demonstrates, for cells using

irregular cathode technology, this simplification is no longer valid

The current work presents a new way to analyze the problem and

as the flow is now fully 3D in nature.

answer the question. Lateral gravity waves have been simulated in

a 3D cell slice model using VOF formulation in OpenFoam.

MHD-Valdis also does not consider the impact of gas bubble

Results obtained for cells using regular and irregular cathode

release on the bath flow. This was demonstrated to be quite a

surfaces are compared.

valid simplification as it is clear that the dynamic of the gas

bubble release under the anodes, which strongly affects the global

Introduction

bath resistance, is decoupled from the cell stability problem.

Otherwise no cell stability model developed up to now would be

Irregular cathode surface technology is still the subject of research

valid.

in China where it is still quite popular. The most recent Chinese

paper known to the authors on the subject was published in

In [2] and in previous studies presented before that [3,4,5], MHD-

Metallurgical and Materials Transaction B in 2014 [1].

Valdis could not clearly show the impact of irregular cathode

technology on cell stability, maybe because of the

2D flow

That paper presents a very detailed 3D model based on ANSYS

structure simplification.

and CFX solvers. Four steady-state solutions are presented with

and without irregular cathode and with and without considering

Yet it is not by mistake that results of a full 3D transient MHD

the effect of the gas release under the anodes.

driven cell stability analysis was not presented in

[1]. Even

nowadays, CPU resources are still too sparse and too expensive

In addition to these four

3D steady-state solutions, two

2D

for such an analysis to be carried out without a huge R&D budget.

transient solutions were presented with and without irregular

cathode that model only the gas release.

The need for a third modeling approach

When comparing the two

2D transient solutions, no major

If reducing the metal flow to a shallow layer representation is not

difference in the global deformation or evolution of the bath/metal

a justified simplification in the case of irregular cathode surface

interface can be identified (see Figure 16 of [1]).

and if solving a full 3D transient cell stability problem is still not

practical, clearly there is a need to find a new modeling approach.

When comparing the two 3D solution of the conventional cathode

with and without the effect of the gas release, a major difference

There are two parts to the MHD driven cell stability issue: the

in the local deformation of the bath/metal interface in the small

energy source part coming from the presence of the variable

channel between anodes can be identified (comparing Figure 14 a)

Lorentz force in the metal pad as explained by Urata and

with Figure 28 a) of [1]).

Davidson among others [6,7] and the energy dissipation part

coming from viscous damping.

This local effect is far less intense in Figure 28 b) when compared

with Figure

28 a) for the irregular cathode which seems to

Clearly the aim of irregular cathode surface technology is to affect

contradict what is observed in Figure 16.

this second energy dissipation part by increasing the viscous

damping in the metal pad.

Yet for a bath/metal interface wave to move around, not only the

metal must be displaced but also the bath. Furthermore, since the

ACD layer thickness is much less than the metal pad thickness,

the required bath flow velocity needs to be much greater than the

required metal flow velocity.

So clearly, the viscous damping in the bath is very important and

must be considered in the analysis which will be the case in a 3D

analysis of the damping rate of a gravity bath/metal interface

wave in an aluminium reduction cell.

This approach reduces the difficulty of a study of the viscous

damping effect of irregular cathode surface which is perfectly

valid as this is the key effect that needs to be investigated.

Furthermore, if we choose to study a lateral gravitational wave,

the geometry of the problem can be reduced to a cell side slice as

presented in Figure 1.

Figure

2: Experimental results for a free interface wave

Figure 1: Geometry of the cell side slice model

motion between two liquids in a closed container (Figure 16 in

[10])

OpenFoam and the free interface wave

Base case model setup

Even reduced both in terms of physic and geometry, we are still

left with a quite difficult question to solve namely a transient 3D

The geometry of the base case model, with regular flat cathode

multiphases (three in this case: metal, bath and air) flow. Very

surface was already presented in Figure

1. The model depth

few codes are able to cope with this problem, the open source

extends from a front frictionless symmetry plane located at half

code OpenFoam being one of them.

the anode width to the back frictionless symmetry plane located at

half width of the small channel between two anodes. The length of

OpenFoam has quickly become a very popular code in many

the model is typical of a cell cavity width minus a 10 cm uniform

fields such as marine applications due to its free surface modeling

ledge thickness in both ends: 3.94 m. The height of the model is

capabilities

[8]. Its free surface capabilities are comparable to

enough cavity depth to leave room for

20 cm of metal pad

other VOF solvers like CFX. A direct comparison between

thickness, 20 cm of bath thickness and 7.5 cm of air on top.

experimental, OpenFoam and CFX results for a free surface study

are presented in [9].

The model mesh is presented in Figure 3. The mesh is fine enough

to resolve fairly well the boundary layer problem close to the solid

The free interface wave between a gas and a liquid or between

surfaces

(cathode, ledge and anodes). It is constituted of

two non-miscible liquids in a closed rectangular container has

hexagonal finite volumes of approximately uniform size. The

been extensively studied experimentally, as can be seen per

mesh also perfectly aligned with the initial bath-metal position in

example in Figure 2, a reproduction of Figure 16 in [10]. As

order to have a perfectly smooth initial position of that bath-metal

explained in [11], it is very difficult to measure experimentally in

interface.

a reproducible manner the damping rate of such a gravity driven,

viscous damping wave problem.

Recently, OpenFoam has been quite successfully used to model

this type of free surface wave topic, per example [12] is a Ph.D.

thesis on the subject.

The problem of modeling the damping of a gravity bath-metal

interface wave in an aluminium reduction cell is a very similar

question with the extra difficulty that there are immersed anodes

in the top liquids and that it will be very difficult to get physical

measurements for model validation.

Figure 3: Mesh of the cell side slice model

The model contains

1,180,980 hex finite volumes with an

orthogonal quality of

0.77. It uses a k-ω SST

(shear stress

transport) turbulence model because of its demonstrated capability

to well predict drag [13].

The bath and metal properties utilized where obtained using Peter

Entner’s AlWeb application

[14]. A quite standard bath

composition has been selected, see Figure 4.

Time: 0 second

Time: 15 seconds

Figure 4: Bath and metal properties from AlWeb

Time: 30 seconds

The transient evolution is starting from a resting position having a

sloped bath metal interface of -2 cm on the left side to +2 cm on

the right side as shown in Figure 5.

Time: 45 seconds

Figure 5: Initial bath-metal interface position

The transient evolution of the system from that starting point is

calculated using an explicit solver available in OpenFoam 2.3.0

[15], the multiphase Euler solver using a maximum courant

Time: 60 seconds

number of 0.05 and a maximum time step of 0.002 seconds.

Figure 6: Position of the bath-metal interface every 15 seconds

The transient evolution of the system was calculated for a total of

from 0 to 60 seconds

60 seconds which is more than 1 total period of the lateral wave

oscillation. The calculations were performed using a Dell 28 cores

Figure 8 is showing the velocity field after 60 seconds, indicating

Xeon ES-2697 V3 computer having 128 GB of RAM at its

that the wave has been already almost completely damped down.

disposal. That computer took about 30 CPU hours to solve that

Figure 9 illustrates the turbulent viscosity after 15 seconds. Since

problem using all 28 cores.

the laminar viscosity of the metal is

3.224e-7 m^{2}/s and the

maximum turbulent viscosity

4.66e-4 m^{2}/s, the maximum

Base case model solution

turbulent viscosity is 1447 time the laminar viscosity.

Figure

6 is showing the position of the bath-metal interface

Irregular cathode surface case model setup

position every 15 seconds. That gravity lateral wave happens to

die almost completely in a single period.

The geometry of the irregular cathode surface case model is

presented in Figure 10. The geometry of the cathode surface has

The maximum velocity is reached a little before the 15 sec. mark.

been changed when compared to the base case model. But the

Figure 7 is showing the velocity field of the front plane. The

mass of metal, the mass of bath and the 4 cm ACD have remained

solver assumed continuity of the velocities at the interfaces so the

the same.

solution shows that the bath flow drags the top layer of the metal

so the flow reversal is occurring in the metal pad and not at the

bath-metal interface. The maximum bath velocity is about 3 cm/s.

Figure 11: Mesh of the cell side slice model with irregular

cathode surface

Figure 7: Velocity field after 15 seconds (bath region has gray

background)

Figure 12: Initial bath-metal interface position

Figure 8: Velocity field after 60 seconds (bath region has a

gray background)

Figure 13: Velocity field after 15 seconds (bath region has a

gray background)

Figure 9: Turbulent viscosity after 15 seconds (bath mesh is

visible)

Figure 14: Turbulent viscosity after 15 seconds (bath mesh is

visible)

Figure 10: Geometry of the cell side slice model with irregular

cathode surface

Due to the presence of the flow obstacles, the flow in the metal

pad is now quite different. Notice that flow around obstacles has

The model mesh presented in Figure 11 contains 1,152,016 hex

been extensively studied and successfully modeled using

finite volumes. Figure

12 is showing the initial bath-metal

OpenFoam [16]. Notice also that the mesh density used in [16]

interface position. Figures 13 and 14 illustrate the velocity and the

makes the one used in this study looking somewhat coarse!

turbulent viscosity after 15 seconds. Figure 15 is showing the

position of the bath-metal interface position every 15 seconds.

Time: 0 second

Time: 15 seconds

Figure 16: Evolution of the interface front left corner

There is definitively less overshoot in the case of the irregular

cathode and also less secondary ripples on the interface so clearly

the obstacles are somewhat performing as intended [17].

Yet this observation is not in contradiction with what was

previously published in [2,3,4,5] in general and in Figure 7 of [5]

in particular. The damping effect of the irregular cathode surface

technology is not very important so many other changes to the cell

Time: 30 seconds

design can have more impact on the cell stability.

Future work

The geometry of the cell side slice model is coming from the cell

design presented in Figure

17 produced using Peter Entner

CellVolt application [18]. That cell geometry was inspired from

the GY420 420 kA cell design presented in [19]. Since that cell

design has 48 anodes, modeling a longitudinal gravitational wave

in a half cell model using the same mesh refinement used in that

Time: 45 seconds

study would require a model more than 24 times bigger. Even

with a linear increased of the required CPU time, solving such a

half cell slice model would require about 750 CPU hours which is

about 1 month of CPU time on the computer used in this study.

Time: 60 seconds

Figure

15: Position of the bath-metal interface every

15

seconds from 0 to 60 seconds

Comparison of the damping rate

The comparison between Figure

6 and Figure

15 interface

Figure 17: Sketch of the GY420 cell design that inspired the

positions reveals very little difference. Figure 16 is more useful

cell side slice model geometry

for that, for it shows the transient evolution of the vertical position

of the front left corner of the interface for the two cases.

A model optimization study might reveal that a coarser mesh and

a bigger time step could be used without losing much accuracy, so

performing such a model optimization would be important. Yet, it

is probable that a bigger computer than the Dell 28 cores Xeon

ES-2697 V3 computer used in this study would be required in

order to obtained a practical turn around time to solve a transient

[7] P. Davidson, “An Introduction to Magnetohydrodynamics”,

3D full cell gravitational wave VOF OpenFoam model.

Cambridge Texts in Applied Mathematics, Cambridge

University Press 2001, 363-386.

Adding the MHD physic to an even bigger 3D full cell OpenFoam

model is also quite possible to do. OpenFoam has already been

[8] H. Jasak, “OpenFOAM: Introduction, Capabilities and HPC

successfully used to solve MHD flows [20,21].

Needs”, Cyprus Advanced HPC Workshop Winter 2012.

Conclusions

[9] S. Hansch, D. Lucas, T. Hohne, E. Krepper and G. Montoya,

“Comparative Simulations of Free Surface Flows Using

Lateral gravity wave can be successfully simulated in a 3D cell

VOF-Methods and a New Approach for Multi-Scale

side slice model using VOF formulation in OpenFoam.

Interfacial Structures”, Proceedings of the ASME

2013

Fluids Engineering Division Summer Meeting.

Solving for just 60 seconds of transient evolution using a Dell 28

cores Xeon ES-2697 V3 computer took about 30 CPU hours.

[10] S. Y. Lee1, C. E. Park and V. H. Ransom, “On The Gravity

Driven Force Terms of Single Pressure One-Dimensional

Comparing regular flat cathode case model results with the

Multi-Fluid Flow Model in Horizontal Channel and Their

irregular cathode surface case model results revealed that there is

Validation”, Proceedings of ICAPP 2013 Jeju Island, Korea,

definitively less overshoot in the case of the irregular cathode so

April 14-18, 2013.

clearly there is somewhat more damping in that second case.

[11] D. R. Howell, B. Buhrow, T. Heath, C. McKenna, W. Hwang

Yet this observation is not in contradiction with what was

and M. F. Schatza,

“Measurements of Surface-wave

previously published using MHD-Valdis 2D shallow layer model

Damping in a Container”, Physics of Fluids, Vol. 12, no 2,

as this new study confirms that the extra damping effect of the

February 2000, 322-326.

irregular cathode surface technology is not that significant. Many

other changes to the cell design can have more impact on the cell

[12] G. C. J. Morgan, “Application of the InterFoam VOF Code

stability.

to Coastal Wave/Structure Interaction”, Ph. D. thesis,

University of Bath, Department of Architecture and Civil

A bigger computer than the Dell 28 cores Xeon ES-2697 V3

Engineering, September 2012.

computer used in the present study would be required in order to

obtain a practical turn around time to solve a transient 3D half cell

[13] F. R. Menter,

“Review of the Shear-stress Transport

VOF model to study a longitudinal gravitational wave.

Turbulence Model Experience from an Industrial

Perspective”, International Journal of Computational Fluid

Adding the MHD physic to an even bigger 3D full cell OpenFoam

Dynamics, Vol. 23, No. 4, April-May 2009, 305-316.

model is also quite possible to do. OpenFoam has already been

successfully used to solve MHD flows.

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