Calculation of critical points from Helmholtz-energy-explicit mixture models

Abstract

The calculation of the critical points for a mixture of fluids is of practical interest as the calculated critical points can be used to more reliably and efficiently construct phase envelopes. The number of stable critical points found can also provide insight into whether the mixture has an open or closed phase envelope. In this work we have developed a reliable method for determining all the critical points for a mixture that is modeled with Helmholtz-energy-explicit equations of state. This method extends the algorithms developed in the literature for simpler equations of state to these more complex mixture models. These Helmholtz-energyexplicit equations of state could be either multi-fluid models or transformations of simple cubic equations of state to Helmholtz-energy-explicit forms. This algorithm locks onto the first criticality contour (the spinodal) and traces it to high density, thereby locating all relevant critical points. The necessary analytic derivatives of the residual Helmholtz energy, numerically validated values of the derivatives for validation, sample code, and additional figures and information are provided in the supplemental material. * Commercial equipment, instruments, or materials are identified only in order to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the products identified are necessarily the best available for the purpose.

On the construction of binary mixture p‐x and T‐x diagrams from isochoric thermodynamics

“…The critical lines (given by the thick black line) are calculated by the algorithm of Bell and Jäger. The isotherms and isobars are selected to yield good coverage of the range of validity of the mixture model. The sample code documents what temperatures and pressures were used.…”

“…Figure c shows selected supercritical isotherms for a mixture of methane and n ‐propane. In this case, the Peng‐Robinson model with k ij = 0 was used because the algorithm of Bell and Jäger was unable to yield a continuous critical line for the multifluid model though that of Peng‐Robinson succeeded. This figure demonstrates that without any special treatment we can trace all the way up to the critical line.…”

“…In cubic equations of state used with the multifluid model, the reducing temperature and density are both constant values (not a function of molar concentration; see Bell and Jäger) As a result, the derivatives of τ and δ with respect to the molar concentration for cubic equations of state are different from those of the multifluid models. Therefore, the derivatives (for cubic EOS only) are given by …”

“…If a reliable starting point for the isoline is available, this algorithm may be successful at tracing the entire isoline. Conversely, there are many reasons why the phase equilibrium calculation may fail at any given point, even if a neighboring point along the phase envelope was successfully calculated: A density solution (either on the vapor or liquid side) may disappear, and if the algorithm needs to solve for the density as a function of temperature, pressure, and mixture composition, that may not be possible. The guesses for the densities and/or compositions of the co‐existing phases may not be accurate enough to initialize a derivative‐based solver within its radius of convergence. There may be a region where the underlying thermodynamic model is not defined or exhibits nonphysical behavior (see for instance Bell and Jäger) The initial guess may be in the critical region, but the solver is not aware of that fact, causing the phase equilibrium calculation to fail in subtle ways. …”

An improved method for calculating critical temperatures and critical pressures in natural gas mixtures with up to nC11 hydrocarbons

“…There are rigorous and empirical methods to calculate critical properties of Vapor-Liquid Equilibrium (VLE) in multicomponent mixtures, applied to natural gas samples (Huang and Guo, 1995). Rigorous methods are based on classical thermodynamic theories, and on the concept of Helmholtz free energy minimization (Bell and Jäger, 2017), which require simultaneous resolution of non-linear equation systems. Several authors have proposed different strategies and algorithms to solve these equation systems (Heidemann and Khalil, 1980;Michelsen, 1980;Peng and Robinson, 1977).…”