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A thermoelectrical model arising in aluminium electrolytic cells

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Title: A thermoelectrical model arising in aluminium electrolytic cells


1
A thermoelectrical model arising in aluminium
electrolytic cells
M.Carmen Muñiz
Dpt. Applied Mathematics Santiago de Compostela
University (Spain)
2
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)

A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 2/43
3
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.

A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 3/43
4
The Alcoa-Inespal plant in La Coruña (Spain)
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 4/43
5
The Alcoa-Inespal plant in La Coruña (Spain)
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 5/43
6
The Alcoa-Inespal plant in La Coruña (Spain)
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 6/43
7
Hall-Héroult process
Bayer process
bauxite
alumina
aluminium
The main chemical reaction of the Hall-Héroult
process
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 7/43
8
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 8/43
9
The ledge a mixture of solid bath and aluminium
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 9/43
10
2D view of the part of the cell we model
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 10/43
11
The electric and thermal domains
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 11/43
12
The physical equations
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 12/43
13
The physical equations
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 13/43
14
The physical equations
The electric problem and the thermal one are
COUPLED
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 14/43
15
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.

A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 15/43
16
Boundary conditions
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 16/43
17
Notation
Hereafter
A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 17/43
18
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

A thermoelectrical model arising in aluminium
electrolytic cells 1/10/2004 18/43
19
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

A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 19/43
20
Fixed domain method
We embed this problem into another one with a
fixed domain containing S
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 20/43
21
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
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 21/43
22
Weak formulation
We get
Therefore the weak formulation of the problem
is....
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 22/43
23
Fixed domain method
and
where
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 23/43
24
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.

A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 24/43
25
Finite element discretization
  • The reference finite element a pentahedral
  • with vertices

A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 25/43
26
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

A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 26/43
27
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.
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 27/43
28
The discretized problem
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 28/43
29
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
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 29/43
30
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 30/43
31
Isolines for temperature (ºC)
32
Numerical results
Isolines for temperature at steel shell (ºC)
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 32/43
33
Numerical results
Isolines for electric potential (V)
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 33/43
34
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
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 34/43
35
Gauging the model
Method modify either the thermal conductivity or
the radiation and convection coefficients
Key idea identification of the following
parameters
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 35/43
36
Gauging the model
We minimize the cost function
where
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 36/43
37
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
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 37/43
38
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.
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 38/43
39
results
40
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.

A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 40/43
41
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)
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 41/43
42
Theoretical review
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 42/43
43
Theoretical review
A thermoelectrical model arising in aluminium
electrolytic cells, Lausanne 1/10/2004 43/43
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