It is known that for general arithmetic circuits there is not much of a difference between standard model and one with division. It is true, because one can simulate circuit with divisions that computes some polynomial by a circuit without division with a polynomial blow up.

Situation changes if we move to a non-commutative world, where such a reduction is unknown and proving lower bounds for formulas with division is a big open question, while for formulas without division exponential lower bounds are known. 

Now let's consider model with division where every gate should compute a polynomial rather then rational function. Are we aware of any lower bounds for such model?