A question from model theory

Assume consistency of (Peano) arithmetic. Add the negation of a statement expressing its consistency to this arithmetic, the result should still be consistent. Then translate the resulting logic into a first-order logic. By the compactness theorem, the resulting logic has a model.

Are there any interesting models resulting from this? Are there any models, which would illuminate the nature of the resulting logic? Or, are there any such models which are easily understood?