Signal and systems analysis for unsteady heat conduction problems

Abstract

A general formulation of heat transfer phenomena based on signal analysis is developed in this article. Starting from the classical matrix formulation of heat transfer through a slab, normal modes are defined. The thermal normal modes are linear combinations of heat flow and temperature variations measured on the external surfaces of the slab. Although normal modes fit to a superposition principle, they do not define thermal signals since the scalar product of heat flow and temperature has no physical meaning. Starting from this classical description, a new formulation of heat transfer based on entropy concept is then stated. The ‘‘entropy production’’ and ‘‘entropy exchanged with outside’’ are expressed in terms of normal coordinates. It is shown how the time distribution of entropy production for the quasi-steady-state of flow can be characterized by a cross-correlation function between the space-average heat flow and its entropy conjugated coordinate. In the same way, the time distribution of entropy exchanged with outside results from a cross-correlation function between a normal coordinate and the corresponding entropy conjugated coordinate. Under unsteady-state conditions, the entropy exchanged with outside can be computed by convoluting the cross-correlation function representative of entropy production and the impulse response of heat transfer through the slab. This new formulation yields to consider thermal signal independently of their actual waveforms, but of their correlation function (i.e., their frequency content). In the experimental part, the theoretical results are extended to include practical problems of heat transfer in building sections exposed to natural weather conditions and internal heating.