Dynamic simulation of chemical processes is important for optimization and control of transient process behavior, e.g., batch processes, startup and shut-down. Most chemical processes contain separation of mixtures. These unit operations are commonly described using a phase equilibrium assumption, e.g., vapor-liquid equilibrium (VLE) for distillation, liquid-liquid equilibrium (LLE) for extraction, or vapor-liquid-liquid equilibrium (VLLE) for heteroazeotropic distillation. Fundamentally, phase equilibrium at given pressure and temperature is found at the global minimum of the Gibbs free energy of the system [1, 2, 3]. In dynamic simulation, the number of phase varies due to vanishing and (re)appearing phases. This poses a major challenge for simulation. The different approaches for phase equilibrium calculation can be classified according to the most popular modes for chemical process simulation as sequential-modular (SM) and equation-oriented (EO) . In the SM approach, the correct phase regime and the corresponding phase compositions are determined in an iterative procedure by repeatedly solving flash equations with a different phase number until a physical solution determines the actual state of the system. A major challenge in coupling these sequential-modular phase calculations to dynamic equation-oriented simulation is the handling of vanishing and (re)appearing phases and their composition. For dynamic simulation, we ideally aim at continuous and differentiable formulations. Thus, assigning arbitrary value for mole fractions of non-existing phases is not desirable because it leads to discontinuities at phase transition. In addition, dynamic simulation is more efficient if derivatives are used that, however, are not directly available from SM flash calculations. Using finite difference may lead to inaccuracy and round-off errors . Within the past decades, different works contributed to the development of EO formulations starting with the formulation of Gibbs free energy minimization and derivation of first-order (necessary but not sufficient) optimality conditions thereof . They then further relax the (necessary but not sufficient) equilibrium condition if a phase vanishes. This allows tracking a local solution with continuous and differentiable equations. The resulting algebraic system of equation enables dynamic simulation in all phase regimes of most VLE systems. We examine the application of this approach to LLE problems as proposed in literature [6, 7, 8] and show limitations of the formulation. The limitations arise because the equilibrium condition for each component is relaxed by a single auxiliary variable and there is no desired solution in certain areas of the single-phase regime. In addition, we propose a hybrid continuous formulation to overcome these limitations for dynamic simulation of phase changes in LLE systems and demonstrate this in an illustrative case study.
 Baker LE, Pierce AC, Luks KD. Gibbs Energy Analysis of Phase Equilibria. Society of Petroleum Engineers Journal. 1982;22(05):731-742.
 Michelsen ML. The isothermal ash problem. Part I. Stability. Fluid Phase Equilibria. 1982;9(1):1-19.
 Mitsos A, Barton PI. A dual extremum principle in thermodynamics. AIChE Journal. 2007;53(8):2131-2147.
 Biegler LT, Grossmann IE, Westerberg AW. Systematic methods of chemical process design. Upper Saddle River, New Jersey: Prentice Hall. 1997.
 Kamath RS, Biegler LT, Grossmann IE. An equation-oriented approach for handling thermodynamics based on cubic equation of state in process optimization. Computers & Chemical Engineering. 2010;34(12):2085-2096.
 Gopal V, Biegler LT. Smoothing methods for complementarity problems in process engineering. AIChE Journal. 1999;45(7):1535-1547.
 Biegler LT. Nonlinear programming: Concepts, algorithms, and applications to chemical processes. MOS-SIAM series on optimization. Philadelphia: Society for Industrial and Applied Mathematics and Mathematical Optimization Society. 2010.
 Sahlodin AM, Watson HAJ, Barton PI. Nonsmooth model for dynamic simulation of phase changes. AIChE Journal. 2016;62(9):3334â3351.