An improved branch and bound algorithm for phase stability testing of multicomponent mixtures

Abstract

We investigate the phase stability of a multicomponent mixture at constant volume, temperature and moles (VTN stability). Our work is based on the TPD criterion derived in Mikyška and Firoozabadi (2012), Investigation of Mixture Stability at Given Volume, temperature and number of moles, Fluid Phase Equilibria and the branch and bound algorithm from Smejkal and Mikyška (2020), VTN-phase stability testing using the Branch and Bound strategy and the convex–concave splitting of Helmholtz free energy density, Fluid Phase Equilibria. In this contribution, we improve the algorithm with more effective bounding strategy. This improvement is achieved using the necessary condition of optimality. In the bounding step of the algorithm, before solving an underestimated convex optimization, we check whether the pressure (given by the Peng–Robinson equation of state) is feasible. If it is not the case, we can exclude the corresponding part of the feasible set from the search. The pressure function given by the Peng–Robinson equation of state is not convex and therefore leads to a nonconvex optimization problem which is computationally expensive. We propose to use a less precise estimate of the global maximum of the pressure. This estimate can be found by comparing the finite number of the values of the tangent plane to a concave overestimate of the Peng–Robinson equation of state. Another benefit of this additional step is to avoid the optimization of the underestimated objective function. The proposed method is tested on several specific examples.