(132e) Robust Dual Global Optimization Algorithms for the Determination of Fluid Phase Equilibria with the SAFT Equation of State

The prediction of fluid phase behaviour is an important aspect of process simulation since assumptions of equilibrium are commonplace in unit models. The difficulty in locating these equilibrium states is often a cause of failure in simulations. In this work we examine the P,T flash: phase equilibrium at constant pressure and temperature. The solution of this problem essentially involves the determination of the global minimum of the system's Gibbs free energy, a multi-dimensional, highly non-linear and non-convex function.

We present a class of algorithm that guarantees the location of this global solution, and is capable of reliably predicting the number of phases, vapour (V) or liquid (L), present at equilibrium, along with their properties. The method is general in terms of its application to multi-component mixtures, to the calculation of any kind of fluid phase behaviour (e.g., VLE, LLE, VLLE, etc.) and to any EOS, though our focus here is on the SAFT family. The convergence is independent of initial guesses, or indeed any a priori knowledge of the behaviour of the system in question. Importantly, the number of stable phases is an output of the algorithm: it does not need to be guessed at any point.

Many chemical engineering applications now employ advanced equations of state (EOS) such as SAFT (a), due to their ability to predict accurately, and continuously, fluid phase behaviour over wide condition ranges. The proposed approach has been developed with particular consideration for numerical efficiency when working with these complex EOS. It combines aspects of the work of Mitsos and Barton (2007) (b), Nagarajan et al. (1990) (c) and Adjiman et al. (1998) (d) as a new robust methodology. The P,T flash is solved as a dual optimization problem, translated away from the Gibbs G(P,T,x) to the Helmholtz A(V,T,x) free energy (Pereira et al.) (e) . Like most EOS derived from molecular statistical mechanics, the SAFT expressions are formulated in terms of the Helmholtz free energy, and therefore implementing calculations directly in this (volume-temperature-composition) space is a far more efficient approach. In addition, the Helmholtz free energy surface is smooth and therefore numerically better behaved than the Gibbs free energy.

The infinite dual optimization is solved through the iterative solution of upper and lower bounding problems. The upper bounding problem contains only linear functions and may be solved without difficulty. The lower bounding problem, however, requires the global optimization of a non-convex function. The αBB (d) algorithm is applied to perform this optimization with the SAFT EOS. Analysis and reformulation of SAFT allows the phase equilibrium problem to be solved to global optimality with significantly improved computational efficiency. Analysis of this type may easily be applied to any molecular EOS. Different formulations and algorithms are proposed, and their performance is compared for a set of test problems.

Results are presented for multi-component systems, including associating (hydrogen bonding) mixtures and polymer solutions, modelled with the SAFT-HS EOS (f); phase equilibrium calculations for mixtures of asymmetric molecules are notoriously difficult, and convergence problems are often encountered, even with very good initial guesses. The proposed method is found to be reliable in all cases.

References:

(a) G. Jackson, W.G. Chapman and K.E. Gubbins. Mol.Phys., 65,1,1998; W.G. Chapman, G. Jackson and K.E. Gubbins. Mol.Phys., 65,1057,1998; W.G. Chapman., K.E. Gubbins, G. Jackson. and M. Radsosz. Fluid Phase Equilibria, 52, 31, 1989; W.G. Chapman, K.E. Gubbins, G. Jackson and M. Radosz. Ind. Eng. Chem. Res. 29,1709-1721, 1990.

(b) A. Mitsos and P.I. Barton. A dual extremum principle in thermodynamics. AIChE Journal, 53(8),2007.

(c) N.R. Nagarajan, A.S. Cullick and A. Griewank. New strategy for phase-equilibrium and critical point calculations by thermodynamic energy analysis. Part1. Stability analysis and flash. Fluid phase equilibria, 62(3):191,1991.

(d) C.S. Adjiman, S. Dallwig, C.A. Floudas and A. Neumaier. A global optimization method, αBB, for twice-differentiable constrained NLPs ? I. Theoretical advances. Computers and Chemical Engineering, 22(9), 1998.

(e) F.E Pereira, A. Galindo, G. Jackson, C.S. Adjiman. In preparation.

(f) A. Galindo, P. J. Whitehead, G. Jackson, A. N. Burgess, Predicting the high-pressure phase equilibria of water + n-alkanes using a simplified SAFT theory with transferable intermolecular interaction parameters, Journal of Physical Chemistry 100 (1996) 6781.