Using a linear inverse heat conduction model to estimate the boundary heat flux with a material undergoing phase transformation

Abstract

Modeling heat conduction during steel-cooling processes is challenging because solid–solid phase transformations can take place, introducing highly non-linear phenomena to the heat equation. Using a linear model for the inverse problem can be helpful for estimating the dissipated heat flux, especially because complex phase transformations models that use empirical parameters are not always available. However, the performance and applicability of such a simplified inverse model for a strongly non-linear process is still unclear. This study presents several numerical simulations to evaluate the accuracy and limitations of solving a linear inverse heat conduction problem (i.e. constant thermophysical properties and no internal heat source) to estimate the boundary heat flux during cooling of a material undergoing phase transformations. Preliminary simulations with stable materials but with temperature-dependent thermophysical properties showed that the linear model performs well to estimate the boundary heat flux, except when the material has a highly temperature-dependent specific heat, like pure iron near its Curie temperature. Then, we performed several simulations for 42CrMo4 steel, which undergoes phase transformations and, hence, has phase- and temperature-dependent thermophysical properties and latent heat of phase transformations as an internal heat source. The latent heat has a very small effect on the heat flux estimation for fast cooling conditions; however, it can lead to an interpretation bias in medium and slow cooling conditions due to temporary underestimates of the heat flux, reaching errors up to 100%. Even when the estimated heat fluxes seem accurate (average errors smaller than 10%), further estimates of the temperature evolution or phase transformations kinetics are inaccurate (range of uncertainties as large as 200 °C and 20%, respectively) because of the phase-dependent thermophysical properties. Hence, using a linear inverse heat conduction problem for a material undergoing phase transformations is acceptable only for fast cooling conditions ( h > 1500 Wm −2 K −1 ) and exclusively to estimate the boundary heat flux without further simulations. • Numerical simulations of a coupled heat conduction-phase transformation model. • Simulated temperature data were used in a linear inverse model to estimate the heat flux. • The inverse model performs well for AISI 304 and Ni201, which do not undergo phase transformations. • For the 42CrMo4 steel, which undergoes phase transformation, it performs well only for h > 1500 Wm − 2 K − 1 . • For lower h, the latent heat plays an important role, so the simplified model is no longer acceptable.