Fourier's law of heat conduction

In continuum form, Fourier's law is written q = −k∇T, where q is heat flux, k is thermal conductivity, and ∇T is the spatial temperature gradient. The minus sign indicates that conductive heat flux points down the temperature gradient, from hotter toward cooler regions. In one dimension, qₓ = −k(dT/dx). Heat flux measures energy transferred per unit area per unit time; total heat-transfer rate through a surface depends on its area and orientation.

Thermal conductivity measures how readily a material conducts heat and is commonly expressed in watts per metre-kelvin. For an isotropic material, k can be treated as a scalar; in anisotropic materials, conductivity depends on direction and is represented by a tensor. Conductivity may also vary with temperature, composition, or other material conditions, so its appropriate value or functional form must be used in a particular problem.

Fourier's law is a constitutive relation: it describes how heat flux responds to a temperature gradient. Combined with conservation of energy and a relation for a material's heat capacity, it yields the heat-diffusion equation. For a uniform isotropic solid with constant properties and no internal heat source, this equation is ∂T/∂t = α∇²T, where α = k/(ρcₚ) is thermal diffusivity, ρ is density, and cₚ is specific heat capacity at constant pressure. Boundary and initial conditions are then needed to solve a particular heat-transfer problem.

Fourier's law is widely used to model conduction in solids, fluids at rest, building materials, electronic components, and engineered thermal systems. In practical analysis, heat conduction must often be distinguished from convection, which involves fluid motion, and radiation, which transfers energy by electromagnetic waves. The classical law is a highly useful continuum model, but in some regimes—such as very rapid heat changes, very small length scales, or strongly heterogeneous structures—experiments and research motivate non-Fourier models that account for finite response times, ballistic transport, or additional material structure.

Joseph Fourier developed mathematical theory for heat conduction in the early nineteenth century and published Théorie analytique de la chaleur in 1822. His work connected physical heat-transfer problems with mathematical methods, including the use of trigonometric series to represent temperature distributions. The law named for him remains a foundational relation in classical heat-transfer theory.