A new flux integral relationship for half-space transient heat conduction with arbitrary initial condition

This paper derives a new integral relationship between heat flux and temperature in a transient, one-dimensional heat conducting half-space having an arbitrary initial condition. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. This relationship is developed by combining Green's functions method with a formalism associated with the regularization of weakly singular, integral equations. The resulting analytic expression provides the in-depth, heat flux history through the time integration of a weakly singular integral involving the heating/cooling rate, dT/dtdT/dt at a single spatial position. If temperature data are collected at the embedded location then resulting formulation is mildly ill posed and hence requires digital filtering for removing high-frequency noise from the collected signal. On the other hand, if heating/cooling rate data are collected at the embedded location then the resulting expression is well posed. From this formalism, a new heat flux sensor strategy is produced based on the temporal variable and not the spatial variable as used in Gardon gauges. In particular, Gardon heat flux gauges are based on Fourier's law involving a spatial gradient.