An analytical inverse model for predicting the critical thickness of the protective bank inside a cylindrical smelting furnace

Abstract

A protective bank is essential for ensuring the integrity and the energy efficiency of high temperature electric ilmenite smelting furnaces. If the thickness of the protective bank is insufficient or unstable, the refractory walls are directly exposed to molten materials, extreme thermal loads and chemical interactions, accelerating degradation and increasing heat losses. Conventional inverse models rely on iterative processes with high computational costs, limiting real-time capabilities. To overcome this limitation, an analytical inverse model is developed, enabling real-time assessment of furnace integrity based on sensor measurements.

In this paper, a direct numerical model is first developed and validated to simulate the thermal behavior of a cylindrical smelting furnace. This 2D axisymmetric model accounts for the presence of two superimposed melted layers, iron and slag, while incorporating phase change and endothermic chemical reactions. Next, a 1D analytical inverse model is introduced to predict the critical thickness of the protective bank. Sensor placement is analyzed, demonstrating that deep insertion into the refractory wall enhances measurement accuracy.

The proposed inverse model successfully determines the critical bank thickness and is further extended to reconstruct its profile by integrating multiple sensors along the furnace height. To account for 2D heat transfer effects in the iron and slag layers, a model correction is introduced, significantly improving prediction accuracy. The results confirm the effectiveness of the proposed approach. Additionally, a measurement error analysis highlights the model's robustness, reinforcing its potential for real-time monitoring and control of high-temperature smelting furnaces.