Aleksander NAWRAT, Janusz SKOREK

1. Silesian University of Technology in Gliwice Inverse approach for identification of the shrinkage gap thermal resistance in continuous casting of metals Aleksander NAWRAT, Janusz SKOREK Presented by Ireneusz SZCZYGIEŁ IPES 2003

2. 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

3. Geometry of the problem

4. 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.

5. Steady state equation INGOT Solid phase Liquid phase MOULD

6. Boundary conditions Condition on solid/liquid interface Energy balance at the solid/liquid interface The temperature of the solid/liquid interface, is known:

7. Additional boundary conditions Pouring and cut-off metal temperature Axial symmetry of the temperature Cooling systems: Primary Secondary

8. KNOWN QUANTITIES UNKNOWN QUANTITIES Geometry of the problem Temperature filed in the ingot and mould Thermo-physical parameters of the ingot and mould Phase change location Distribution of the thermal resistance in the gas-gap between the ingot and mould Boundary conditions Direct problem

9. KNOWN QUANTITIES UNKNOWN QUANTITIES Geometry of the problem Temperature filed in the ingot and mould Thermo-physical parameters of the ingot and mould Phase change location Distribution of the thermal resistance in the gas-gap between the ingot and mould Boundary conditions Results of temperature measurements points located in the mould Inverse problem

10. Geometry of the problem

11. 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:

12. Geometry for the test program Assumed geometry of the system for continuous casting: ingot radius Hr = 0,1 m, ingot lengthHz = 1m, mould lengthHCz = 0,22 m, mould thicknessHCr = 0,025 m, Assumed boundary conditions: cut off temperature Tk= 20C cooling systems: primary α2 = 1500 W/m²K, secondary α1 = 9000 W/m²K temperature of the cooling fluid T = 20 C Assumed calculation accuracy ε = 1 %.

13. 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.

14. Sensitivity coefficients of temperature change in respect to the change of thermal resistance (z = 0,087 m, wz=0,002 m/s, Cu)

15. Sensitivity coefficients for the ingot and mould for fixed radius coordinates (Cu, wz=0,002 m/s)

16. Algorithm of the least squaresadjustment 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(λi) is global stiffness matrix of considered problem T is vector of unknown nodal temperatures c is vector that represent boundary conditions

17. In our work the least square adjustment method with the “a priori” information about the unknown values is applied to identify unknown conductivities λi. 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 xe=[T, λ] includes identified values of thermal conductivity and both measured and unknown nodal temperatures, Ae =[A λe, ATe] is coefficient matrix, c is the vector of constants.

18. The matrix coefficients that appear in equation can be calculated from following dependences:

19. 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 = [λ0, T0]T, Ve is covariance matrix of measurementsVand “a priori” estimated quantities Gλ: Estimated quantities Gλare given by:

20. Initially assumed values of thermal conductivityin the gas-gap Under assumed values of thermal conductivity in the gas-gap the solve direct problem For calculated temperature field and phase change locationsolve the inverse problem based on calculation of the substitute thermal conductivity λi(that means reciprocal of thermal resistance) under additional information i.e. results of temperature measurements in the wall of the mould Check the required accuracy for improved values of λi STOP 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.

21. 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.

22. Thermal resistance identification

23. Results of identification

24. 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.

25. 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.