Defining flux boundary condition: unsuccessfully results / Someone could help please?

[amnchacon]

Hello everyone, I'm trying to implement this example in Firedrake:

I had already treated this problem with both boundary conditions as Dirichlet boundary conditions, with constant temperature values. Now, my problem is with the flux boundary condition. It suppose that flux boundary conditions are embedden within the variational formulation, but after testing numerical results, these are being inconsistent. My code is running without errors, but the results are not sucessfully. The way I'm trying the variational formulation here in Firedrike is right or wrong? can you help me please?

In my variational formulation via Firedrake I'm writing:

a = -K * grad(T)[0] * v * dx (The part associated to this natural boundary condition)
...
...
L = (...) + h * (To-Ta) * v * dx

The code is running but with strange results, what I should do for traducing correctly to firedrike this flux boundary condition?

Thanks for helping me...

[amnchacon]

Another thing: such boundary condition -k(dT/dr) = h(To - Ta) contains To, which is the temperature at r = ro, i.e., a temperature on the outside surface, which is a variable.