A new three-dimensional heat flux–temperature integral relationship for half-space transient diffusion

This paper derives a new integral relationship between heat flux and temperature in a transient, three-dimensional heat conducting Cartesian half space ( x > 0 , y ∈ ( − ∞ , ∞ ) , z ∈ ( − ∞ , ∞ ) ). A unified mathematical treatment has been developed based on operational and transform methods; and singular integral equation regularization. Regularization is accomplished based on a series of observations involving the diffusive nature of the operator. This newly developed relationship provides the local heat flux perpendicular to the front surface at any location within the half space. This expression suggests that an embedded plane of temperature sensors parallel to the surface can be used to acquire the local, in-depth heat flux in the x-direction. The relationship does not require a priori knowledge of the surface boundary condition which has analytically been removed in the process. The ill-posed nature of diffusion is highlighted owing to the appearance of the heating/cooling rate (°C/s) in the integrand of the new relationship. Integral relationships of this type are highly useful for experimental investigations since the in-depth heat flux can be extracted from well-established temperature transducers.