*P,T*phase equilibrium with equations of state

__F.E. Pereira__*, A. Galindo, G.
Jackson, C.S. Adjiman*

Department of Chemical Engineering,

Corresponding author e-mail: c.adjiman@imperial.ac.uk

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 [1]. 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 [2], 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) [3] 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) [6] 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 (*P ^{0}*) 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 [6] 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.

References

[1] A. Mitsos and P. I. Barton, A dual extremum
principle in thermodynamics. *AIChE** Journal*, 53, 2131 (2007).

[2] 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).

[3] 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.

[4] 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).

[5] 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).

[6] 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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