Simulation of gas bubble motion in aluminium electrolytic cells

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Simulation of gas bubble motion in aluminium
electrolytic cells

• Alexei Lozinski and Michel Romerio
• Chaire danalyse et simulation numériques
• Institut danalyse et calcul Scientifique
• Ecole Polytechnique Fédérale de Lausanne 
• supported by AlCan

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Aluminium production cell
Basic chemical reaction under the anode Al2O3
C Al CO2
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The model of massless non-deformable bubbles

• - domain occupied by both fluid and gas
• P(t) - subdomain of W occupied by gas

q
G(t)
P(t)
W
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Governing equations
Navier-Stokes equations for velocity and pressure
in the fluid (in W/P(t) )
Boundary conditions
Bubble propagation
Force balance
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A generalization massless deformable bubbles
Approximation of P(t) by a geometrcal forms
depending on m parameters
Parameterization of ?P(t)
Virtual displacements
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Boundary conditions on ?P(t)
The  exact  force balance equation for a
general form of bubbles
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Example deformable ellipses with constant area
a(t)
q
G(t)
P(t)
W
generalized coordinates
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Conditions on the interface
Force balance
Velocity on ?P(t)
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Weak formulation (fictitious domain)
Glowinski, Pan, Hesla, Joseph, Périeux, J. Comp.
Phys. 169, 363-426 (2001)

• extend velocity and pressure on the whole W
• introduce the Lagrange multiplier l on P(t)

Boundary conditions
Note that
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Reformulating the force balance equation
Noting that
the force balance can be rewritten as
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Gauge reformulation of NSE
E Liu, Comm. Math. Sci. 1, 317-323 (2003)
We make the change of variables
so that a and f satisfy
The pressure can be recovered via
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Summarizing the equations(non-deformable case)
Boundary conditions
Force balance
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Numerical scheme
On the n-th time step
we search for
1. Updating the coordinates of the bubble
2. Updating the gauge variable
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3. Coupled problem for an, ln, Vn, wn
4. Updating the velocity
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Coalescence
Instantaneous transition contradicts with our
assumption that the form of the new bubble is
taken among the elements of the predefined family
of surfaces

• Our approach
• The coalescence process is started when two
  bubbles are sufficiently close
• The small bubble is swallowed up by the large
  one in a small number of time steps

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Numerical tests
Simulation parameters
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Surface tension influence
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Initial number of bubbles 40 Grid on bubbles
30 nodes Grid for velocity 128 X 256 Number
of time steps 270 CPU time 291 sec
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Repulsion from the wall
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First results in 3D
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Perspectives

• Coupling with the MHD code (Alucell)
• Iterative process
• Running bubble simulation code with velocity u(i)
  given by Alucell
• Averaging
• Rerunning Alucell with new effective density,
  electric conductivity and bubble contribution to
  the moment conservation equation
• until convergence