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

Geometry of the problem State equation and boundary conditions Assumption for mathematical model of continuous casting process Formulation of the problem Inverse problem Sensitivity analysis Algorithm of the least squares adjustment method Results of identification of thermal resistance of the gas shrinkage Final conclusions Plan of presentation:

Geometry of the problem

 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. The Finite Element Model was adopted for solving analysed Stefan problem under following assumptions:

Steady state equation INGOT MOULD Solid phase Liquid phase

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

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

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

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

Geometry of the problem

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 z i,j are calculated in an approximate way:

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 T k = 20  C cooling systems: primary α 2 = 1500 W/m²K, secondary α 1 = 9000 W/m²K temperature of the cooling fluid T  = 20  C Assumed calculation accuracy ε = 1 %.

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. Thermo physical properties of metal Thermal conductivity: L = 226 W/mK, S = 394 W/mK. Specific heat:c L = 475 J/kgK, c S = 380 J/kgK. Density:  L = 8300 kg/m³,  S = 8930 kg/m³. Temperature of phase change: T m = 1083  C. Latent heat of solidification:  = J/kg. Velocity of the ingot: w z = 0,002 m/s. Temperature of liquid metal: T 0 = 1100  C.

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

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

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: c is vector that represent boundary conditions Where: A(λ i ) is global stiffness matrix of considered problem T is vector of unknown nodal temperatures

In our work the least square adjustment method with the “a priori” information about the unknown values is applied to identify unknown conductivities λ i. Proposed approach requires formulation of the problem in the following form: vector x e includes all identified values (that means variables actually measured and estimated “a priori”). In the considered case vector x e =[T, λ] includes identified values of thermal conductivity and both measured and unknown nodal temperatures, A e =[A λ e, A T e ] is coefficient matrix, c is the vector of constants. where:

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

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 = [λ 0, T 0 ] T, V e is covariance matrix of measurements V and “a priori” estimated quantities G λ : Estimated quantities G λ are given by:

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

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: T i 0 - simulated measurement result, T i - 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.

Thermal resistance identification

Results of identification

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.

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.