(356i) Robustness and Efficiency of Phase Stability Testing at Vtn and Uvn Specifications

Phase equilibrium calculations at pressure and temperature (PT) specifications, consisting in the minimization of the Gibbs free energy (for phase splitting) and of the Tangent plane distance (TPD) function (for phase stability testing) with respect to mole numbers, are the most commonly used and there are well documented in the literature. However, in many practical applications, phase equilibrium calculations at isochoric-isotherm (VTN) and iso-energetic-isochoric (UVN) conditions are required. These kinds of calculations have received an increasing interest in the last decade. In UVN stability testing, the TPD (to the entropy surface in the UVN space) function must be maximized. Several formulations of the TPD function are analyzed: i) in mole fractions, molar volume and molar internal energy; ii) in mole numbers, volume and internal energy and iii) in component molar densities and internal energy density. It is shown that for each formulation, the TPD functions for UVN stability have opposite signs and the same stationary points as their VTN stability and PT stability counterparts. Furthermore, UVN stability can be analyzed totally disconnected from the UVN space, by solving simpler VTN or PT stability problems. Solving a VTN stability problem instead of a UVN one avoids the resolution at each iteration of the nonlinear equation relating the temperature to internal energy; in VTN stability, this equation must be solved only once, at the specifications. Moreover, the initialization scheme is greatly simplified, since the stationary points of the TPD function are sought in the VTN space, instead of UVN space.

A mathematical analysis of convergence properties of the successive substitution (SSI) method for VTN stability reveals its potential instability, unlike in PT stability, in which SSI is very robust. Even though a descent direction is guaranteed, SSI takes often a too large step length, leading to severe convergence problems. This problem can be theoretically overcome by using a damping factor; however, since damping tends to bring all the eigenvalues of a key matrix within the unit circle, a severe damping will bring all the eigenvalues close to unity, leading to an unacceptable slow convergence rate. A severe damping may be also required in many cases to avoid constraint violations. It is shown that the SSI method should never be used alone, undamped SSI combined with Newton method must be avoided and damped SSI used with extreme caution. According to these observations, a robust combined SSI/Newton method is proposed, using damped SSI only in early iteration stages and sculptured switching criteria to avoid all possible shortcomings of using the SSI method. Modified Newton iterations are used to minimize the TPD function with respect to component molar densities with preconditioning giving an optimal scaling. A modified Cholesky factorization guarantees a descent direction by applying a diagonal correction of the Hessian matrix to restore its positive definiteness. A two-stage line search procedure is used; in a first stage, the step length is reduced to keep iteration variables in the feasible domain and in a second stage an inexact line search based on Wolfe conditions and quadratic/cubic backtracking ensures a sufficient decrease of the TPD function, with a small number of additional function evaluations.

In this work, several domains were identified where SSI is either systematically not detecting a phase split, converging to the trivial solution from all initial guesses (at low molar densities, between the lower dew point and spinodal curves) or systematically diverges (at high molar densities/pressures and also around and outside the stability test limit locus). The SSI method may often exhibit chaotic or cyclic behavior, being unable to reduce the gradient norm and the TPD function below a certain value, far from the solution (in some cases instability is not detected, even though the global minimum is negative). In other cases, even if it progress towards a negative TPD (indicating an unstable phase in early iteration stages), SSI jumps to another valley of the objective function and eventually lands on a local trivial minimum (indicating a stable phase), which is a strong attractor (at least one eigenvalue of the key matrix takes very large absolute values).

The convergence behavior of our methods for VTN stability is analyzed for a variety of mixtures and the numbers of iterations required to achieve convergence are compared with those reported in the literature. The proposed methods are highly robust and systematically faster than previous methods, for some test points up to one or even two orders of magnitude. Moreover, our calculation procedures are able to find the global minimum of the TPD function in all test cases, unlike previously proposed SSI/Newton methods, which either are strongly attracted by local minima or they diverge.