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

Suppose in a material of variable thermal conductivity  

  the termperature varies from point to point so that  

Suppose also that a source inside the material generates heat at a constant rate  
  Joules per unit volume.
Heat flows at a rate  
  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  
\[\mathbf{\nabla} \cdot \mathbf{h} = \phi \]

\[ \mathbf{h} = -k \mathbf{\nabla} T\]

\[ \mathbf{\nabla} \cdot (k \mathbf{\nabla} T) = - \phi\]

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