F.E. Pereira, A. Galindo, G. Jackson, C.S. Adjiman
Department of Chemical Engineering,
Corresponding author e-mail: firstname.lastname@example.org
The prediction of phase behaviour is an important aspect of thermodynamic modeling applications such as parameter estimation and process simulation. Moreover, the difficulty in locating these equilibrium states is often a cause of numerical failure in these tasks. This work examines the use of two duality-based algorithms to solve the P,T flash (phase equilibrium at constant pressure and temperature), and is focused on the relative merits of making use of a pressure solver or not in the evaluation of the free energy. The solution of the P,T problem requires the determination of the global minimum of the system's Gibbs free energy, a multi-dimensional, highly non-linear and non-convex function.
A duality-based formulation of phase stability at constant temperature and pressure was proposed by Mitsos and Barton . This interpretation of phase stability has attractive numerical properties since it is driven by a concave master problem, which prevents the calculation from diverging. Subsequently, we proposed a translation of this formulation to the Helmholtz free energy , and in this work, we explore the relative merits of solving the P,T flash in the two spaces of the Gibbs and Helmholtz free energies. We compare two algorithms, HELD (Helmholtz Free Energy Lagrangian Dual)  and GELD (Gibbs Free Energy Lagrangian Dual), presented here, and examine their behaviour on various case studies where the thermodynamic models are represented through the Statistical Associating Fluid Theory for Potentials of Variable Range (SAFT-VR) [4,5] and the Peng-Robinson (PR)  equations of state (EOSs). Both algorithms require only the respective free energy functions and their first derivatives with respect to composition and volume. Consequently, they are straightforward to interface with any thermodynamic method from which these properties can be obtained.
The major difference between the two algorithms is that GELD employs a pressure solver to evaluate the Gibbs free energy at constant pressure, whereas in HELD, the volume is treated as an explicit variable, and the pressure is only equal to that specified for the PT flash (P0) at key points during the algorithm. HELD has been designed with the hope of improving efficiency with complex EOS, for which constant pressure properties cannot be analytically obtained. The aim of the transformation from the Gibbs to the Helmholtz free energy is to avoid having to solve the EOS in volume a large number of times during the solution of the phase equilibrium problem.
We aim to determine whether the Helmholtz free energy-based formulation of the dual problem for P,T phase equilibrium used by HELD conveys any benefits, in terms of either efficiency or reliability, when performing calculations with the PR  and SAFT-VR [4,5] EOSs. Case studies are presented for both associating and non-associating systems of up to ten components, exhibiting VLE, LLE and VLLE. It is found that when the Gibbs free energy is available analytically, as is the case for the PR EOS, and would also be the case for instances such as liquid-phase equilibria represented through activity coefficient models, then GELD is the preferable algorithmic option. However, as the EOS becomes more expensive to evaluate, for example, when a nonlinear association system must be solved at each evaluation, the CPU requirements of HELD are less than those of GELD. Despite the differences in computational performance, the reliability of both methods is found to be high, provided a tailored and robust route to obtaining the volume roots of the EOS (where required) is available to GELD.
 A. Mitsos and P. I. Barton, A dual extremum principle in thermodynamics. AIChE Journal, 53, 2131 (2007).
 F. E. Pereira, G. Jackson, A. Galindo and C. S. Adjiman, A duality-based approach to the (P,T) phase equilibrium problem in the volume-composition space, Fluid Phase Equilibria, 299, 1 (2010).
 F. E. Pereira, G. Jackson, A. Galindo and C. S. Adjiman, The HELD algorithm for multicomponent, multiphase equilibrium calculations with generic equations of state, submitted for publication.
 A. Gil-Villegas, A. Galindo, P. J. Whitehead, S. J. Mills, G. Jackson and A. N. Burgess, Statistical associating fluid theory for chain molecules with attractive potentials of variable range, The Journal of chemical physics, 106, 4168 (1997).
 A. Galindo, L. A. Davies, A. Gil-Villegas and G. Jackson, The thermodynamics of mixtures and the corresponding mixing rules in the SAFT-VR approach for potentials of variable range, Molecular Physics, 93, 241 (1998).
 D. Y. Peng and D. B. Robinson, A new two-constant equation of state, Industrial and Engineering Chemistry Fundamentals, 15, 59 (1976).
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