Model order reduction for thermomechanical phenomena arising in blast furnace hearth
Analysing ironmaking process using mathematical models
The origin of steelmaking can be traced back to thousands of years ago. Modern steelmaking was developed in 19th century including the development of Bessemer process. The previous stage to steelmaking is ironmaking. It is a high temperature process performed inside blast furnace. During this process hot metal is produced from iron ore. At the bottom of blast furnace, the temperature can be as high as 1500 degree Celsius. This produces significant thermal stress inside the blast furnace hearth walls, reducing blast furnace campaign period.
Coupled thermomechanical model corresponding to this process is governed by energy conservation equation and momentum conservation equation. Blast furnace hearth is made up of different materials : various types of carbon blocks, brick, mortar, steel shell. The properties of these materials depend on the temperature. Ceramic cup of the blast furnace hearth is made up of periodic assembly of bricks and mortar. By using homogenization technique [1], this periodic assembly is replaced with equivalent orthotropic material. Collectively, these efforts introduce complexities such as non-linearity, orthotropy and heterogeneity in the mathematical model [3]. In view of the prevailing physical conditions during the process, axisymmetry hypothesis was introduced.
Next, we consider the repeated solution of the coupled thermomechanical problem under variation of “parameters”. Parameters here correspond to the geometry of the hearth and the material properties (thermal conductivity, Young’s modulus, Poisson ratio and thermal expansion coefficient). In other words, we consider parametric partial differential equations solved repeatedly under variation of parameters.
For these repeated computations, we use model order reduction approach. It involves working with smaller system of equations to accelerate the computations. This makes it suitable for real-time computations and quick transfer of computational results to industrial problems. We use Artificial Neural Network (ANN) in our model reduction approach [2]. Recently, ANN has seen considerable growth in application as function approximator in numerical analysis and model order reduction. It requires to learn from the data and is considered as non-intrusive. The non-intrusive nature helps to develop interface between software library used to perform Finite Element (FE) computations and model order reduction approach without access to the matrices corresponding to system of equations of FE method.
The approach used in this work is an example of ancient physical process,
ironmaking, analysed using modern elements of model order reduction such as Artificial Neural Network. It is also an example of mathematical models used to understand and to solve complex physical problems of industrial interest and contribution to continuous evolution of real-world technologies.
Acknowledgement
This work has been carried out in collaboration with ArcelorMittal in Asturias (Spain) and Technological Institute for Industrial Mathematics (ITMATI) in Santiago de Compostela (Spain) under ROMSOC project.
References
[1] P. Barral, M. Fanjul, L. Perez, P. Quintela, and M.Teresa. Equivalent thermomechanical model for ceramic cups. Proceedings of the 139 European Study Group with Industry, p. 85–101, 2018.
[2] N. V. Shah, M. Girfoglio, P. Quintela, G. Rozza, A. Lengomin, F. Ballarin,
and P. Barral. Finite element based model order reduction for parametrized
one-way coupled steady state linear thermomechanical problems, In preparation.
[3] N. V. Shah, M. Girfoglio, and G. Rozza. Thermomechanical modelling for
industrial applications, 2021.
