A thermoelectrical model arising in aluminium electrolytic cells

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A thermoelectrical model arising in aluminium
electrolytic cells
M.Carmen Muñiz
Dpt. Applied Mathematics Santiago de Compostela
University (Spain)
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Research group

• This work was done by the research group of A.
  Bermúdez
• de Castro from 1990 to 1997
• P. Quintela.
• M.B. Cid.
• L. Carpintero.
• I.C. Area.
• P. Salgado.
• E. Seoane.
• It was supported by ALCOA-INESPAL S.A.
• (La Coruña - Spain)

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Outlines

• A mathematical model for the thermoelectrical
  problem.
• The weak formulation.
• Finite element discretization.
• The algorithm.
• Numerical results.
• A method to gauge the thermoelectrical model.
• Thelsi3D.
• Theoretical review.

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The Alcoa-Inespal plant in La Coruña (Spain)
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electrolytic cells 1/10/2004 4/43
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The Alcoa-Inespal plant in La Coruña (Spain)
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electrolytic cells 1/10/2004 5/43
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The Alcoa-Inespal plant in La Coruña (Spain)
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Hall-Héroult process
Bayer process
bauxite
alumina
aluminium
The main chemical reaction of the Hall-Héroult
process
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electrolytic cells 1/10/2004 8/43
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The ledge a mixture of solid bath and aluminium
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2D view of the part of the cell we model
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electrolytic cells 1/10/2004 10/43
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The electric and thermal domains
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electrolytic cells 1/10/2004 11/43
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The physical equations
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electrolytic cells 1/10/2004 12/43
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The physical equations
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electrolytic cells 1/10/2004 13/43
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The physical equations
The electric problem and the thermal one are
COUPLED
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References thermistor problem

• Cimatti, G.1989 Remark on existence and
  uniqueness for the
• thermistor problem, Quart. Appl. Math.
• Howison, S.D., Rodrigues, J.F. and Shillor,
  .1993 M. Stationary
• Solution to the thermistor problem, J. Math.
  Anal. Appl.
• Antontsev, S.N. and Chipot, M.1994 The
  thermistor problem
• Existence, smoothness, uniqueness, blowup, SIAM
  J. Math. Anal.

• Two additional dificulties appear in this
  problem
• The domain is not homogeneous, then the physical
  parameters
• depend not only on the temperature but on
  position as well.
• The free boundaryThe profile of the legde,
  called S.

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Boundary conditions
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Notation
Hereafter
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Conditions at the free boundary S

• S is a priori unknown.
• S is the interface between solid and liquid
  phases.
• The operation temperature of the bath drops up to
  the solidus
• temperature

ledge
melts
Mushy zone

• Arita, Y., Urata, N. and Ikeuchi, H. 1978
  Estimation of frozen bath
• shape in aluminium reduction cell by computer
  simulation, Light Metals
• Taylor, M.P. and Welch, B.J. 1987 Melt/freeze
  heat transfer
• Measurements in cryolite-based electrolytes,
  Metallurgical Trans. B
• Bruggemen, N.J. and Danka, D. J.1990
  Two-dimensional thermal
• Modelling of the Hall-Héroult cell, Light Metals

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Conditions at the free boundary S
We assume the following boundary conditions at
the free boundary
The greater the slope of S, the greater the heat
transfer.

• Bermúdez, A., Carpintero, L., Muñiz, M.C. and
  Quintela, P. 1993 An inverse
• problem related to the three-dimensional
  modelling of aluminium electrolytic
• cells, Computational Modelling of Free and
  Moving Boundary Problems

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Fixed domain method
We embed this problem into another one with a
fixed domain containing S
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Fixed domain method
Multiplying
by a test function and integrating we get
Extending T by the operation temperature in the
fictitious domain, and taking into account
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Weak formulation
We get
Therefore the weak formulation of the problem
is....
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Fixed domain method
and
where
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Weak formulation
For a solution of this problem, we define the
ledge and the free boundary as

• The theoretical analysis of this problem is
  difficult due to
• The coupling between thermal and electric
  equations.
• The nonhomogeneity of the domain.
• The physical nonlinearities.
• The free boundary.

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Finite element discretization

• The reference finite element a pentahedral
• with vertices

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Finite element discretization

• The space of real functions defined on the
  reference finite
• element with basis given by

• The degrees of freedom the values of the
  function at the vertices
• of the pentahedroms

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Finite element discretization
Associated to a family of pentahedral meshes
of the domain, we consider the finite element
spaces
where
and denotes the map transforming the
reference finite element into the element K.
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The discretized problem
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An iterative algorithm

• Bermúdez, A., Moreno, C. 1981 Duality methods
  for solving variational inequalities, Comput.
  Math. Appl.

Where is the Yosida regularization of the
Heaviside function given by
Therefore it is quite natural to try the
following iterative algoritm
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Isolines for temperature (ºC)
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Numerical results
Isolines for temperature at steel shell (ºC)
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Numerical results
Isolines for electric potential (V)
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Gauging the model
Goal to reduce the differences between
thermocouple and heat flux measurements with the
values obtained from the thermoelectrical model.
Thermocouple measurements
Heat flux measurements
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Gauging the model
Method modify either the thermal conductivity or
the radiation and convection coefficients
Key idea identification of the following
parameters
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Gauging the model
We minimize the cost function
where
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Gauging the model
Taking into account the control (c,d,e), the
state of the system TT(c,d,e) is obtained
solving the so-called state equation consisting
on the following boundary value problem
where
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Gauging the model
In order to minimize the cost function, we
consider the lagrangian functional
where p is the Lagrange multiplier related with
the constraint given by the state equation.
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results
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thelsi 3D

• Generates 2D and 3D meshes of the cell using
  Modulef Library.
• Computes function h related to heat flux at the
  free boundary.
• Makes the thermoelectrical simulation.
• Computes the coefficients to gauge the model.
• Plots the results using Modulef Library.

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Theoretical review
1 A. Bermúdez, M.C. Muñiz and P. Quintela,
Numerical solution of a three-dimensional
thermoelectric problem taking place in an
aluminium electrolytic cell, Computer Methods in
Applied Mechanics and Engineering 106, (1993),
129-142. We numerically solve the thermoelectric
problem using finite elements.
Isolines for electric potential (V)
Isolines for temperature (ºC)
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Theoretical review
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Theoretical review
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