## Heat Flow in a Material of Variable Thermal Conductivity With Internal Heat Source

Suppose in a material of variable thermal conductivity

$k=k(x,y,z)$
the termperature varies from point to point so that
$T=T(x,y,z)$

Suppose also that a source inside the material generates heat at a constant rate
$\phi$
Joules per unit volume.
Heat flows at a rate
$\mathbf{h}$
per unit area perpendicular to a surface drawn in the material. then
$\mathbf{\nabla} \cdot \mathbf{h} = loss \: of \: heat \: per \: unit \: volume$

For the temperature to remain steady
$\mathbf{\nabla} \cdot \mathbf{h} = \phi$

But
$\mathbf{h} = -k \mathbf{\nabla} T$

Hence
$\mathbf{\nabla} \cdot (k \mathbf{\nabla} T) = - \phi$