A general method for predicting the bank thickness of a smelting furnace with phase change

A new methodology is proposed for predicting the bank thickness covering and protecting the refractory brick walls of the smelting furnaces. The inverse method predicts the bank thickness changing with both time and coordinates, by using a two-dimensional physical model of the furnace that is more general and challenging than the previous studies. Moreover, the thermal conductivities of the phase change material are identified simultaneously because they are poorly known, which is more challenging and practical. The inverse problem is solved by the Levenberg-Marquardt Method (LMM). The direct problem of the smelting furnace and the sensitivity matrix coefficients of the LMM are calculated by ABAQUS, using a complex user-defined element that is set up based on the complex-variable-differentiation method (CVDM) and the user element subroutine (UEL). Finally, numerical examples are given to examine the performances of the approach for predicting the bank thickness covering the refractory brick walls of the smelting furnaces.