  Surface heat transfer in convective and radiative environments is sometimes
measured by recording the surface temperature history in a transient experiment
and interpreting this surface temperature with the aid of a suitable model for
transient conduction within the substrate. The semi-infinite one-dimensional
model is often adopted, and several well-developed techniques for application
of this model to surface temperature data are available. However, when a
spatial variation of heat flux exists across the surface, the application of
the semi-infinite one-dimensional approach may not always be a reasonable
approximation. In this paper we introduce a method for treatment of the
measured surface temperature history that is more accurate than the
semi-infinite one-dimensional approximation when substrate lateral conduction
is significant and the relative spatial distribution of the flux is known a
priori. This new method uses the so-called \textit{Neumann heat kernel}, which
evolves a temperature over an insulated domain with unit energy initially
deposited at a specified point. A useful impulse response function is formed by
integrating this Neumann heat kernel against the spatial variation of flux over
the surface of the domain. Neumann heat kernels are constructed for the solid
box, cylinder, and sphere. By applying the heat kernel result for the sphere to
the analysis of a convective experiment using hemispherical-nosed probes, we
demonstrate how the theoretical results enhance the practical analysis of
transient surface temperature measurements. The current approach is superior to
former methods relying on semi-empirical approximations because the
multi-dimensional heat conduction within the substrate is modelled with greater
fidelity using the heat kernel analysis.
