Accurate transient heat flux from simple treatment of surface temperature distribution in the semi-infinite case

TL;DR

This paper introduces a simplified analytical method to accurately estimate transient heat flux from surface temperature measurements in semi-infinite materials, effectively isolating multi-dimensional effects for practical use.

Contribution

It presents a new approach that separates one-dimensional and multi-dimensional components of heat flux, simplifying calculations and improving accuracy in transient heat conduction analysis.

Findings

Recovery of heat flux within 1% accuracy for 2D distributions

Effective approximation of multi-dimensional conduction using second derivatives

Practical discretization levels sufficient for typical experimental uncertainties

Abstract

When the variations of surface temperature are measured both spatially and temporally, analytical expressions that correctly account for multi-dimensional transient conduction can be applied. To enhance the accessibility of these accurate multi-dimensional methods, expressions for converting between surface temperature and heat flux are presented as the sum of the one-dimensional component plus the multi-dimensional component. Advantage arises herein because potential numerical challenges are isolated within the one-dimensional component and practitioners are already familiar with well-established one-dimensional methods. The second derivative of the surface heat flux distribution scaled by the thermal diffusivity and the duration of the experiment delivers an approximation of the multi-dimensional conduction term. For the analysis of experiments in which multi-dimensional effects are significant, a simplified numerical approach in which the temperature within each pixel is treated as uniform is demonstrated. The approach involves convolution of temperature differences and pixel-based impulse response functions, followed by a summation of results across the region of interest, but there are no singularities that require special treatment in the multi-dimensional component. Recovery of heat flux distributions to within 1% is demonstrated for two-dimensional heat flux distributions discretized using several tens of elements, and for a three-dimensional distribution discretized using several hundred pixels. Higher accuracy can be achieved by using finer spatial resolution, but the level of discretization used herein is likely sufficient for practical applications since typical experimental uncertainties are much larger than 1%.