An empirical relationship between the conduction rate in a material and the temperature gradient in the direction of energy flow, first formulated by Fourier in 1822 [see Fourier (1955)] who concluded that "the heat flux resulting from thermal conduction is proportional to the magnitude of the temperature gradient and opposite to it in sign". For a unidirectional conduction process this observation may be expressed as:
where the vector is the heat flux (W/m2) in the positive x-direction, dT/dx is the (negative) temperature gradient (K/m) in the direction of heat flow (i.e., conduction occurs in the direction of decreasing temperature and the minus sign confirms this thermodynamic axiom) and the proportionality constant λ is the Thermal Conductivity of the material (W/mK). Fourier's Law thus provides the definition of thermal conductivity and forms the basis of many methods of determining its value. Fourier's Law, as the basic rate equation of the conduction process, when combined with the principle of conservation of energy, also forms the basis for the analysis of most Conduction problems.
Fourier. J. (1955) The Analytical Theory of Heat, Dover Publications, New York.