Convergence behaviour of single stage flash calculations

Abstract

In this dissertation the convergence behaviour of single stage phase split calculations is investigated to determine which factors cause a flash routine (1) to converge to the trivial solution and (2) to converge very slowly or not at all. For four different mixtures, approximately 50,000 independent flash calculations were executed at variable temperatures, pressures and initial guesses for the phase distribution coefficients. For all four mixtures, the conventional successive substitution algorithm as described by Null (1970) was implemented. For one mixture the calculations were repeated using an accelerated version of the successive substitution algorithm proposed by Mehra et al (1983). At low pressures, convergence to a correct solution can readily be achieved by applying Raoult's law initial guesses, independent of the temperature and the algorithm used to converge the flash calculation, at least for the mixtures studied here. At pressures above the mixture critical pressure the trivial solution was encountered only rarely with Raoult's law initial guesses when the conventional successive substitution method was used. If an accelerated method was used, however, no trivial solutions occured. In order to test the speed of convergence and the computational efficiency of flash calculations, a large number of flash calculations were performed within the two-phase region of a 7 component mixture. Three different but related first order methods and one second order method (Newton-Raphson iteration) were used to solve flash calculations throughout the two-phase region, starting with a Raoult's law initial guess. Both in the low and high pressure regions points of slow convergence were observed. At low pressures, retardation in the speed of convergence was caused by the appearance of trivial roots in the Peng-Robinson equation of state. At high pressures slow convergence was apparently related to the nature of the free energy surface. In terms of computational efficiency the accelerated methods performed best. The Newton- Raphson method exhibited numerical problems in the critical and isobaric regions, which resulted in occurences of the trivial solution and convergence failures. Convergence problems also occured when poor guesses were provided for the phase distribution coefficients. In some cases an oscillatory behaviour of the iteration variables between the correct solution and the trivial solution was observed.