= Thermal diffusivity
{title2=$\kappa$}
{wiki}
<Thermal diffusivity> measures how rapidly <thermal conduction> smooths a <temperature> difference inside a material. <Thermal conductivity> measures the <heat flux> driven by a specified <temperature gradient>; <thermal diffusivity> also accounts for the <heat capacity> that must be warmed. A material can carry substantial <heat> yet change <temperature> slowly because it stores substantial <internal energy> per <kelvin>.
Write $\alpha=\kappa$ for <thermal diffusivity>, reserving $k$ for <thermal conductivity>. In a homogeneous <isotropic> material,
$$
\alpha=\frac{k}{\rho c}=\frac{k}{C_{\mathrm{vol}}},\qquad
[\alpha]=\frac{\mathrm{W\,m^{-1}\,K^{-1}}}{\mathrm{J\,m^{-3}\,K^{-1}}}
=\mathrm{m^2\,s^{-1}}.
$$
Here $\rho$ is <mass density>, $c$ is <specific heat capacity> for the relevant mechanical constraint, and $C_{\mathrm{vol}}=\rho c$ is <volumetric heat capacity>. The familiar $k/(\rho c_p)$ uses <specific heat capacity at constant pressure>. For ordinary <solids> with negligible thermal expansion, the difference between the two constrained <specific heat capacities> is often neglected. A rigid fixed-<volume (thermodynamics)> model instead uses <specific heat capacity at constant volume>; a compressible moving <fluid> requires the accompanying <work> and transport terms. Thus the constraint is part of the approximation, not a choice made solely from the units. <Light and Laser Flash Analyzers by NETZSCH Analyzing and Testing>[NETZSCH, Principle of the LFA Method: conductivity equals diffusivity times specific heat capacity times density]{ref}
Assume a stationary material in the <continuum approximation>, local thermal equilibration on times shorter than those resolved, no internal heating, no <phase transition>, and negligible <convection> and mechanical <work>. Combining <conservation of energy> with <Fourier's law> gives the one-dimensional balance
$$
\rho c\frac{\partial T}{\partial t}=-\frac{\partial q}{\partial x}
=\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right).
$$
If $k,\rho,c$ are constant over the region and <temperature> excursion, this becomes the <heat equation> $\partial_tT=\alpha\partial_x^2T$. If $k$ varies, retain the differentiated product; replacing it with $k\partial_x^2T$ loses a term. <Anisotropic> materials need a <thermal conductivity tensor>. The scalar constant-$\alpha$ examples below deliberately exclude these complications. <Rear-surface integral method for calculating thermal diffusivity from laser flash experiments by Elliot J. Carr>[Carr, section 1: homogeneous isotropic constant-property assumptions; section 2, equations (1)–(3): insulated one-dimensional heat equation]{ref}
For an original comparison, stipulate two model materials with equal <thermal conductivity> $k=2\,\mathrm{W\,m^{-1}\,K^{-1}}$ and equal <mass density> $\rho=1000\,\mathrm{kg\,m^{-3}}$, but <specific heat capacities> $c_A=1000$ and $c_B=4000\,\mathrm{J\,kg^{-1}\,K^{-1}}$. Then
$$
\alpha_A=2\times10^{-6}\,\mathrm{m^2\,s^{-1}},\qquad
\alpha_B=5\times10^{-7}\,\mathrm{m^2\,s^{-1}}.
$$
At the same <temperature gradient>, both have the same instantaneous <heat flux> by <Fourier's law>. At the same initial <temperature> profile and <boundary conditions>, material B takes four times as long to smooth a given spatial pattern. With fixed end <temperatures>, identical geometry and no sources, both eventually have the same steady profile and <heat flux>. The larger <heat capacity> changes the transient, not this <steady state>. These are stipulated parameters, not measurements of named substances.
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