D'Ambrosio, Lee, and Wächter (2009, 2012) introduced an algorithmic approach for handling separable non-convexities in the context of global optimization. That algorithmic framework calculates lower bounds (on the optimal min objective value) by solving a sequence of convex MINLPs. We propose a method for addressing the same setting, but employing disjunctive cuts (generated via LP), and solving instead a sequence of convex NLPs. We present computational results which demonstrate the viability of our approach.