Calculating binary and ternary multiphase equilibria: the tangent plane intersection method

The background and capabilities of a recently developed phase equilibrium calculation method (Tangent Plane Intersection method) are discussed. The method is based on an alternative interpretation of the tangent plane concept, such that the section of the tangent plane constrained within the Gibbs energy surface (τ) provides an indication of how close the search is to the minimum Gibbs energy solution. The T.P.I. method has been used to calculate binary liquid–liquid, vapour–liquid–liquid and vapour–liquid equilibria, and also a range of ternary vapour–liquid–liquid equilibrium behaviour. Three solution techniques (discrete, hybrid1, hybrid2), two of which are based on the Nelder–Mead simplex method, are proposed for the calculation of the ternary multiphase equilibria. These alternatives utilise the available variables by grouping them together in different ways, and are compared in terms of their reliability and efficiency. Unusually, the methods are initialised using compositions near the pure components. This means that multiphase regions distributed throughout the available composition space can be determined using a single, non-arbitrary set of initial phase estimates. Both hybrid methods were found to be sensitive to the position of the overall mixture composition and shape of the multiphase region in question, but, for the systems examined, the hybrid1 solution method was found to be both the most reliable and widely applicable of the three studied.