Inverse approach for identification of the shrinkage gap thermal resistance in continuous casting of

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Inverse approach for identification of the
shrinkage gap thermal resistance in continuous
casting of metals
Silesian University of Technology in Gliwice

• Aleksander NAWRAT, Janusz SKOREK

Presented by Ireneusz SZCZYGIEL
IPES 2003
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Plan of presentation
Formulation of the problem
Geometry of the problem
Assumption for mathematical model of continuous
casting process
State equation and boundary conditions
Inverse problem
Sensitivity analysis
Algorithm of the least squares adjustment method
Results of identification of thermal resistance
of the gas shrinkage
Final conclusions
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Geometry of the problem
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The Finite Element Model was adopted for solving
analysed Stefan problem under following
assumptions

• the geometry of the calculation domain is
  cylindrical,

• the problem is steady state in the co-ordinate
  system attached to the mould,

• the temperature distribution in the ingot and
  mould is axially symmetrical,

• the phase change occurs at the constant
  temperature,

• the convection in liquid metal is neglected,

• the velocity of the ingot is constant.

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Steady state equation
INGOT
Solid phase
Liquid phase
MOULD
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Boundary conditions
Condition on solid/liquid interface
Energy balance at the solid/liquid interface
The temperature of the solid/liquid interface, is
known
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Additional boundary conditions
Pouring and cut-off metal temperature
Axial symmetry of the temperature
Cooling systems
Primary
Secondary
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Direct problem
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Inverse problem
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Geometry of the problem
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Sensitivity analysis
To appropriate selection of the measurement
points and to estimate elements valence in a
sense of their usability to identification of
unknown thermal resistance it was carried
estimated sensitivity analysis of temperature
field on the resistance modification of the
gas-gap.
Sensitivity analysis was made on the ground of
the mathematical model of the direct problem of
heat conduction. Sensitivity coefficients zi,j
are calculated in an approximate way
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Geometry for the test program
Assumed geometry of the system for continuous
casting
ingot radius Hr 0,1 m,
ingot length Hz 1m,
mould length HCz 0,22 m,
mould thickness HCr 0,025 m,
Assumed boundary conditions
cut off temperature Tk 20?C
cooling systems primary a2 1500 W/m²K,
secondary a1 9000 W/m²K
temperature of the cooling fluid T? 20 ?C
Assumed calculation accuracy e 1 .
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Thermo physical properties of metal
Thermal conductivity
?L 226 W/mK, ?S 394 W/mK.
Specific heat
cL 475 J/kgK, cS 380 J/kgK.
Density
?L 8300 kg/m³, ?S 8930 kg/m³.
Temperature of phase change
Tm 1083 ?C.
? 209340 J/kg.
Latent heat of solidification
Velocity of the ingot
wz 0,002 m/s.
Temperature of liquid metal
T0 1100 ?C.
The size of the gas-gap in the direct problem was
calculated on the base of the simplified analysis
of thermal shrinkage in the solid part of ingot.
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Sensitivity coefficients of temperature change in
respect to the change of thermal resistance (z
0,087 m, wz0,002 m/s, Cu)
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Sensitivity coefficients for the ingot and mould
for fixed radius coordinates (Cu, wz0,002 m/s)
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Algorithm of the least squares adjustment method
The partial differential equation which describe
the heat flow in the mould and the gas-gap with
all boundary conditions, is transformed using
finite element methodology to the following
linear algebraic system
Where A(?i) is global stiffness
matrix of considered problem
T is vector of unknown nodal temperatures
c is vector that represent boundary conditions
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In our work the least square adjustment method
with the a priori information about the unknown
values is applied to identify unknown
conductivities ?i.
Proposed approach requires formulation of the
problem in the following form
where
vector xe includes all identified values (that
means variables actually measured and estimated
a priori). In the considered case vector xeT,
? includes identified values of thermal
conductivity and both measured and unknown nodal
temperatures,
Ae A ? e, ATe is coefficient matrix, c is the
vector of constants.
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The matrix coefficients that appear in equation
can be calculated from following dependences
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In the least square adjustment method it is the
most likelihood estimation (in the sense of least
square approximation) of the measured and
unknowns values is given by
where ?0, T0T, Ve is covariance matrix of
measurements V and a priori estimated
quantities G?
Estimated quantities G? are given by
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Algorithm of solution of inverse problem
To identify the unknown thermal resistance and to
calculate the temperature field, (phase change
location) the iterative procedure is used.
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Test examples
In our work the measurement results were
simulated on the basis of the solution of direct
problem (exact solution) corresponding to the
inverse problem. Inaccuracy of measurements was
simulated by adding to the exact solution the
random disturbance
where

• Ti0 - simulated measurement result,
• Ti - exact value (in the sense of the solution of
  direct problem),
• ? - maximal measurement disturbance,
• - random number form the range 0,1.

To check the influences of measurement error on
the accuracy of identification, following maximal
measurement discrepancy were chosen ? 1 K and ?
5 K.
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Thermal resistance identification
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Results of identification
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Final conclusions
To estimate the best location of temperature
sensors the sensitivity coefficients method has
been applied. The sensitivity coefficients method
shows that to obtain satisfactory accuracy of
identification the sensors should be located in
the wall of the mould. Results of temperature
measurements within the wall of the mould are
much more sensitive on the thermal resistance
(which are the subject of identification) than
temperatures within the ingot.
Carried out sensitivity analysis shows that the
distance between the gas-gap and temperature
sensors has also significant influence on the
accuracy of identification. Higher accuracy is
achieved for sensors located in the wall of the
mould very close to the surface of gas-gap. It
should be also stressed, that from the technical
point of view location of temperature sensors in
the wall of the mould is much easier than the
location within the ingot.
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The least square adjustment method (LSAM) was
used to solve the considered inverse problem. The
least square adjustment technique refers in
considered case to two groups of quantities
unknown (which are not measured) and measured.
These quantities are interrelated by the
equations of mathematical model (so called
constraint equations). In contrary to the
classical algebraic problems all the quantities
are here treated as stochastic. Essential aim of
calculations is to evaluate the most likelihood
estimates of unknown and measured quantities.
Calculation and numerical tests have proved that
proposed method can be effectively used for
solving inverse boundary heat conduction Stefan
problem. Result of the present inverse analysis
can be used for optimization of the process of
casting metals.