Purpose - creation of a mathematical model of the process of energy input into an ore-thermal furnace and its distribution in volume taking into account both the electrical and occurring thermal phenomena and the representation of the bath as a set of elementary components. The latter ones, due to electrical resistance, consume electricity which is converted into heat, and then it is redistributed in volume through heat transferring. This will enable to obtain 3D changing picture in the temperature field of the working space and to solve various issues and practical problems related to energy efficiency improvement in the process of obtaining ferroalloys and in the short power supply of the furnace. The mathematical model creation is based on a real sequence of processes occurring in the furnace working space and the continuous interconnection between thermal and electrical transformations. This approach makes it possible to implement the description of the process in the form of an algorithm, whose calculated blocks are constructed on the basis of the following methods: the theory of electric circuits; the description of the volume space based on cylindrical coordinates; the analysis of the processes of heat removing in materials with different aggregate state due to heat transferring. A 3D picture of the temperature field in the furnace bath in the process is created, which in real time allows determining the temperature state of elemental volumes and the moments of melt appearance in them as well separating furnace zones, which require additional energy input to calculate the volume and dynamics of the finished melt formation and the time of its discharging. Due to the information obtained about the physical condition of the charge at any place of the furnace working space, it was possible to track the changing dynamics of the material surface relief and to determine the place of the charge filling in dynamics. A possibility of energy distribution in the bath of ore-thermal furnace in dynamics and calculation of the temperature field in it was determined. A mathematical model of the smelting ferroalloys process where the relationship of electrical and thermal processes in the volume of the charge can be considered was proposed. All these allow determining the indicators which affect the power efficiency melting. The use of the suggested model allows calculating the amount of energy input at any elemental volume of the ore-furnace bath over a certain period of time, and solving the practical problems of power efficiency process.

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