A system for calculating the refractory erosion depth of the slag coating and the thickness of the pyrometallurgical was disclose a system. The system parameters associated with a particular furnace and the slag, such as the chemical composition of the slag, temperature, heat loss, refractory properties, the water temperature of the jacket inlet, jacket emitting sphere water temperature, the mass flow rate of the jacket of the molten bath temperature, the furnace shell detecting and recording, and processes the parameters that calculates a temperature change curves and calculate and store the slag of the slag. Determining the temperature change curves and solid slag is also the thickness of the slag coating and is also used to predict the refractory erosion depth.