The inert model was created to avoid having to solve an inert stream with a second Mixture Fraction (and its variance.) If you remove the second stream from the PDF, you will be able to activate the inert model to add a transport equation for the inert. In this situation, the PDF gets accessed with conditioned variables that account for the presence of the inert. It is a valid approach, since the inert is just a diluent, and saves you a transport equation.