Robustness and efficiency of phase stability testing at VTN and UVN conditions

Abstract

Beyond the well-documented conventional phase equilibrium calculations at pressure and temperature (PT) specifications, other specifications, such as volume, temperature and moles (VTN) and internal energy, volume and moles (UVN) have received an increasingly interest in the last years. Whatever the selected set of independent variables, phase stability at UVN conditions can be asserted by solving simpler VTN or PT stability problems (a non-linear equation is solved only once at UVN specifications for temperature). In VTN stability, modified Newton iterations were previously used to achieve convergence, since successive substitution iterations (SSI) may exhibit problematic convergence. A mathematical analysis of convergence properties of the successive substitution (SSI) method for VTN stability reveals its potential instability, unlike for PT stability, in which SSI is very robust. This problem can be theoretically overcome by using a damping factor, but severe damping may lead to unacceptable slow convergence rates. It is shown that the SSI method should never be used alone, combined undamped SSI-standard Newton method is neither robust nor efficient as claimed previously and must be avoided; moreover, damped SSI can be used with caution only in early iteration stages. The convergence behavior of several methods for VTN stability is analyzed for a variety of mixtures and the numbers of iterations are compared with those reported in the literature. Several domains in the molar density-temperature plane were identified, where SSI is either systematically not detecting a phase split (converging to the trivial solution from all initial guesses) or systematically diverges. Our calculation procedures are highly robust and systematically faster than previous methods (for some test points up to one or even two orders of magnitude) and 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. This paper gives for the first time a detailed mathematical and numerical analysis of the convergence behavior of SSI and combined SSI-Newton methods in VTN stability testing.