Nonlinear model predictive control (NMPC) is an advanced control strategy that uses a rigorous dynamic model of the process, usually described by differential-algebraic equations (DAE), to predict its future behavior. The economic optimization of the process can be included directly in the objectives of the NMPC controller, which make it attractive for application in highly nonlinear and energy-intensive processes that are subject to fluctuating operational conditions. Currently, the implementation of NMPC controllers with large-scale dynamic models is limited by the computational difficulty of solving the associated dynamic optimization problem at each period of time. In this work, we propose an index two DAE model based on fundamental principles that uses its equivalent reduced index one model to define consistent initial conditions (index hybrid DAE) with the aim of representing the dynamics of a distillation column. We demonstrate that this model describes the same physical behavior of its equivalent index one reduced model and that it decreases significantly the computational complexity of the online optimization associated with the NMPC controller. Finally, we use the index hybrid DAE model in the economic oriented NMPC (EO-NMPC) of an extractive distillation column considering different disturbances over the inlet conditions. Also, we compare the EO-NMPC with a classic PI (proportional and integral) controller to show that the first one has a better dynamic performance and improves the economic profit of the process.