Equation of heat conduction for spherical solid

Question

What will be the correct Equation of heat conduction for a homogeneous spherical solid with constant thermal diffusivity K and no heat source?

Answer 1

For a medium with constant thermal diffusivity $K$ and no heat source the heat equation is $$ \nabla^2 T(\pmb r, t)=\frac{1}{K}\frac{\partial T(\pmb r, t)}{\partial t} $$ and in spherical coordinates the equation becomes $$ {1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial T \over \partial r}\right) \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial T \over \partial \theta}\right) \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 T \over \partial \phi^2}=\frac{1}{K}\frac{\partial T}{\partial t} $$