Nonlinear thermal parameter estimation for embedded internal Joule heaters

Abstract

We propose a novel inverse scheme, which allows for estimation of thermal parameters of internal Joule heaters through measurements of surface temperature distributions during a Joule heating process. The inverse scheme is based on the governing nonlinear, inhomogeneous heat conduction and generation equation and solely assumes knowledge of the electric resistivity of the Joule heater. Polynomial forms are assumed for the thermal conductivity κ=κ(T) and cpρ=:λ=λ(T), while the method can be easily generalized to estimate parameters of any suitable form. Both the sensitivity and the adjoint methods are developed and compared. Owing to the ill-conditioning of the inverse scheme, the performance of relaxation methods and regularization schemes are analyzed (to improve numerical conditioning). A verification was conducted using polydimethylsiloxane (PDMS) embedded with a strip of conductive propylene-based elastomer (cPBE). Good agreement was achieved between theoretical predictions by the inverse scheme and experimental measurements regardless of the approximated effective potential difference across the cPBE. While constant parameter estimations sufficed to approximate one reference temperature, the inclusion of multiple instants of time required an increase in the polynomial order. The improved parameter estimation is shown to remain of the same order of magnitude for the temperature range encountered when compared with the constant approximation, i.e. κ=10.7 and 12.0Wm−1K−1, and λ=19.9 and 16.2Jm−1K−1, respectively.