Definition of Thermal Diffusivity and Thermal Conductivity
Definition
$$ {\frac{ \partial }{ \partial t }} \mathbf{u} = \alpha \nabla^2 \mathbf{u} $$ In the above general heat equation, the heat diffusion coefficient $\alpha$ is called the thermal diffusivity. The product of the thermal diffusivity $\alpha$, the density $\rho$, and the specific heat $C_{p}$ is called the thermal conductivity. $$ k := \rho C_{p} \alpha \left[ {\frac{ \mathrm{W} }{ \mathrm{m} \cdot \mathrm{K} }} \right] $$
Explanation
According to dimensional analysis, the dimension of the thermal diffusivity $\alpha$ is $\mathsf{L}^2 \mathsf{T}^{-1}$, and its SI unit is $\mathrm{m}^2/\mathrm{s}$. In the heat equation, the left-hand side is differentiated once with respect to time, while the right-hand side is differentiated twice with respect to space.
$$ \rho C_{p} {\frac{ \partial }{ \partial t }} \mathbf{u} = k \nabla^2 \mathbf{u} $$ Writing the heat equation in terms of thermal conductivity yields the form above. Physically this is more intuitive, but for mathematical handling the thermal diffusivity is more convenient.