The overall objective of this work is to examine the influence of interphase mass transfer on the composition trajectories in heterogeneous azeotropic distillation. Experiments were carried out in a bubble cap distillation column operated at total reflux with the systems: water-ethanol-cyclohexane and water-acetone-toluene. The experiments were carried out in regions of the composition space such that liquid-liquid phase splitting occurred on some of the trays. In order to model the composition trajectories, a rigorous non-equilibrium (NEQ) stage model is developed. The NEQ model incorporates the Maxwell-Stefan diffusion equations to describe the various intraphase transfers. The developed NEQ model is in good agreement with the experimental results for both experimental systems. In sharp contrast, an equilibrium (EQ) stage model fails even at the qualitative level to model the experiments. For example, for the water-ethanol-cyclohexane system the EQ model anticipates distillation boundary crossing when this does not take place in practice. For the water-acetone-toluene system the EQ model does not anticipate distillation boundary crossing when this phenomena is found in the experiments. It is concluded that for reliable design of azeotropic distillation columns we must take interphase mass transfer effects into account in a rigorous manner.