= Fourier's law
{c}
{title2=$\mathbf q=-k\nabla T$}
{wiki}
<Fourier's law> predicts the local conductive <heat flux> from the local <temperature gradient>. In a <continuum approximation> with <isotropic> <thermal conductivity> $k>0$, its <vector> form and one-dimensional form are
$$
\mathbf q=-k\nabla T,\qquad
q=-k\frac{dT}{dx},\qquad
\dot Q=Aq\quad\text{for uniform normal flux over area }A.
$$
The <heat flux> $q$ has units $\mathrm{W\,m^{-2}}$, the <temperature gradient> has units $\mathrm{K\,m^{-1}}$ and the <thermal conductivity> has units $\mathrm{W\,m^{-1}\,K^{-1}}$. The rate of <heat transfer> $\dot Q$ has units $\mathrm W$. These units distinguish a local material response from the total <power> crossing a surface.
The minus sign makes <heat> flow toward lower <temperature>: for $k>0$, $\mathbf q\cdot\nabla T=-k|\nabla T|^2\leq0$. In the worked bar in <thermal conduction>, $dT/dx=-200\,\mathrm{K\,m^{-1}}$ therefore gives $q=+400\,\mathrm{W\,m^{-2}}$. Reversing the end <temperatures> reverses the <heat flux>. A zero <temperature gradient> gives zero conductive <heat flux> in this model.
<Fourier's law> is a constitutive relation expressing measured material behavior; it is not a consequence of <conservation of energy> alone. It applies locally and does not require a <steady state>. The straight-line bar solution additionally assumes constant <thermal conductivity>, constant cross-sectional <area>, no sources or side losses, and fixed end <temperatures>. <Anisotropic> materials require the directional response described by the <thermal conductivity tensor>. The <scalar> $k$ used in the examples expresses the stated <isotropy> assumption. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node116.html[E. M. Greitzer, Z. S. Spakovszky and I. A. Waitz, MIT Thermodynamics and Propulsion, section 16.2: Fourier's law and thermal conductivity]{ref}
Historically, <Joseph Fourier>'s \i[Théorie analytique de la chaleur] was published in 1822. In Chapter II, article 112, he writes the <heat> crossing a spherical surface as the product of <thermal conductivity>, elapsed time, surface <area> and the negative radial <temperature gradient>; article 113 balances the incoming and outgoing <heat> to obtain the <heat equation> for a <sphere>. The enduring idea is this connection between a local flow rule and a global prediction of changing <temperature>. https://fr.wikisource.org/wiki/Théorie_analytique_de_la_chaleur[Joseph Fourier, Théorie analytique de la chaleur, Firmin Didot, 1822]{ref} https://fr.wikisource.org/wiki/Théorie_analytique_de_la_chaleur/Chapitre_2[Fourier, 1822, Chapter II, articles 112–113]{ref}
Back to article page
