Seminars & Events

Mathematics and Computer Science Division
"The reliable solution of (P,T) phase equilibrium and stability in the volume-composition space"

DATE: October 15, 2010
TIME: 10:30 AM - 11:30 AM
SPEAKER: Claire S. Adjiman, Reader in Chemical Engineering, Imperial College London
LOCATION: Bldg 240 Conference Center 1404-1405, Argonne National Laboratory

Description:
An algorithm for the solution of phase equilibrium at constant pressure and temperature, also known as (P,T) flash, is presented. The solution of this problem corresponds to the global minimum of the system Gibbs free energy, a multi-dimensional, highly non-linear and non-convex function. It has been shown that this can be interpreted using duality theory [1]. In this talk, we present a formulation of (P,T) phase equilibrium as a dual optimisation problem in the volume-composition space, translated away from the Gibbs free energy to the Helmholtz free energy. Working in this space brings considerable benefits [2], especially when using complex equations of state (EOS) that are higher than cubic functions in volume and are formulated in the Helmholtz free energy, such as SAFT (statistical associating fluid theory). In addition, the Helmholtz free energy surface is smooth and therefore numerically better behaved than that for the Gibbs free energy. We develop an algorithm for the solution of the problem that is based on a combination of local and global optimisation, in which the number of subproblems solved globally is kept at a minimum. Using this approach, one is guaranteed to predict the number of phases present at equilibrium, along with their properties, without any need for initial guesses, or indeed any a priori knowledge about the behaviour of the system in question. The method is applicable to multi-component mixtures and to the calculation of any kind of fluid phase behaviour (e.g. vapour-liquid (VLE), liquid-liquid (LLE), vapour-liquid-liquid (VLLE) etc.) and to any EOS explicitly expressed in terms of the Helmholtz free energy. Several algorithmic options are investigated to improve computational efficiency. Results are presented for the VLE and VLLE for mixtures modelled with the augmented van der Waals EOS, a non-cubic EOS that incorporates the Carnahan-Starling Equation to account for the repulsive interactions. Further examples are presented for VLE and VLLE in polymer systems, modelled with the SAFT-HS EOS. Fluid phase equilibrium calculations for polymer systems are notoriously difficult, and convergence problems are often encountered, even with very good initial guesses. The new method is found to be reliable in all cases.

References
[1] A. Mitsos and P. I. Barton, AIChE Journal, (2007), 53, 2131.
[2] N.R. Nagarajan, A. S. Cullick, and A. Griewank, Fluid Phase Equilibria, (1991), 62, 191.

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