A random walk based methodology to calculate the sensitivity coefficients in inverse radiant boundary design problems

Abstract

Radiative heating furnaces are widely used for heat treatment applications in manufacturing industries. Strict requirements on heat treatment characteristics of the workpieces require precise control of furnace heater temperatures. Numerical techniques for the solution of inverse heat transfer problems can be used for the optimization of the radiant heater temperatures. In gradient-based optimization techniques, the optimization is carried out using the sensitivity coefficients between the heater and workpiece temperatures. In this study, we present a methodology for calculating the sensitivity coefficients for the solution of inverse boundary design problems involving transient heating of solid workpieces in a three-dimensional radiant furnace. In our methodology, the sensitivity coefficients on the selected design points within the workpieces are calculated by analytical differentiation of design point temperature formulations derived utilizing the floating random walk method and radiative heat flux formulation with exchange factors. The methodology for calculating sensitivity coefficients has been verified through applications to both a one-dimensional scenario and the three-dimensional radiant furnace model. The developed methodology was integrated into a gradient-based optimization algorithm in two test cases, to determine the furnace heater temperatures required to achieve the desired temperature histories at design points within the workpieces. First test case focuses reconstruction of an assumed heater temperature and examines the impact of design point configuration and input data noise on convergence behavior and calculated estimation. The findings reveal that increasing the number of design points in the workpieces increases solution accuracy, with transverse positioning of the design points in the workpieces resulting in heater temperatures closest to the assumed values. Second test case focused on estimating required heater temperatures for spatially uniform heating pattern on the workpieces. With 72 transversely positioned design points, we achieved uniform heating at these points with a maximum relative temperature deviation of 0.26%.