It is demonstrated that during secondary refining at the ladle furnace the carbon content of steel and the residence time of the metal in the

ladle exert a significant impact on the residual content of non-metallic inclusions (NMI) in steel. Mathematical calculations showed that

the dynamic forces have minor effect on the motion of small sized NMI, making it difficult to penetrate deep into the slag.

The generic mathematical model and computational algorithm considering hydrodynamics, heat and mass transfer processes during casting

and forming steel ingots and castings are offered. Usage domains for turbulent, convective and non-convective models are determined

depending on ingot geometry and thermal overheating of the poured melt. The expert system is developed, enabling to choose a

mathematical model depending on the physical statement of a problem.