The constant pressure-temperature (P-T) flash plays an important role in the modelling of fluid-phase behaviour, and its solution is especially challenging for equations of state in which the volume is expressed as an implicit function of the pressure. We explore the relative merits of solving the P-T flash in two ensembles: mole numbers, pressure and temperature, in which each free-energy evaluation requires the use of a numerical solver; and mole numbers, volume and temperature, in which a direct evaluation of the free-energy is possible. We examine the performance of two algorithms, HELD (Helmholtz free energy Lagrangian dual), introduced in Pereira et al, (2012), and GILD (Gibbs free energy Lagrangian dual), introduced here, for the fluid-phase equilibria of 8 mixtures comprising up to 10 components, using two equations of state. While the reliability of both algorithms is comparable, the computational cost of HELD is consistently lower; this difference becomes increasingly pronounced as the number of components is increased. (C) 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license.