I am sorry for the typo, It should have been T. I’ve also now distinguished between the time step (i and i+1) and the normal vector.
However, you have to be quite careful when transforming the system to cylindrical coordinates, as the new term adds in a singularity.
Please see this old thread on how to derive the variational formulation for axi-symmetric problems, where both the derivatives and integration measure is transformed into cylindrical coordinates.