Reliable Computation of High-Pressure Solid−Fluid Equilibrium

Abstract

The calculation of solid-fluid equilibrium at high pressure is important in the modeling and design of processes that use supercritical fluids to selectively extract solid solutes. We describe here a new method for reliably computing solid-fluid equilibrium at constant temperature and pressure, or for verifying the nonexistence of a solid-fluid equilibrium state at the given conditions. Difficulties that must be considered include the possibility of multiple roots to the equifugacity conditions, and multiple stationary points in the tangent plane distance analysis done for purposes of determining global phase stability. Somewhat surprisingly, these issues are often not dealt with by those who measure, model and compute high pressure solid-fluid equilibria, leading in some cases to incorrect or misinterpreted results. It is shown here how these difficulties can be addressed by using a methodology based on interval analysis, which can provide a mathematical and computational guarantee that the solid-fluid equilibrium problem is correctly solved. The technique is illustrated with several example problems in which the Peng-Robinson equation of state model is used.However, the methodology is general purpose and can be applied in connection with any model of the fluid phase.

Phase Behavior and Reliable Computation of High-Pressure Solid−Fluid Equilibrium with Cosolvents

“…The critical end-points are where two phases become identical in the presence of a third phase; e.g., vapor and liquid become critical in the presence of a solid phase. Above the sublimation pressure of the solid, only equilibrium between two phases (either solid-vapor or solid-supercritical fluid equilibrium) can exist at temperatures and pressures between the LCEP and UCEP, except for the case with a temperature minimum in the SLV curve, as shown by curve II in Figure 1 (which 7 will be discussed below). At temperatures close to the UCEP or LCEP one can observe large changes in solvent power with small changes in pressure.…”

“…However, we have shown previously that there can exist multiple solubility roots to the equifugacity equations for solid-fluid equilibrium when using cubic EOS at temperatures near the SLV lines. 7 The key to identifying the correct root(s), and, thus, the correct type of phase equilibrium (SFE, SLE or SLV), is testing the roots for thermodynamic stability. If there is no solubility root that is both stable and feasible (based on overall solute loading and material balance), then there is no solid phase present at equilibrium, and a multiphase flash calculation can be done to determine the number and composition of fluid phases present.…”

“…(6) or (7), and F 2 f is computed using the EOS with y 2 = 1. This representation of the Gibbs energy surface is discussed in much more detail by Xu et al 7 and Marcilla et al 24 The distance between the Gibbs energy surface and the tangent plane is referred to as the tangent plane distance. A plane tangent to the Gibbs energy surface at the fluid phase …”

“…If y 0 is a solution to the equifugacity conditions for solid-fluid equilibrium, then the tangent plane will pass through S 2 g (see Xu et al 7 and Marcilla et al 24 for discussion) and thus D S is zero. Otherwise, the sign of D S can be checked by a straightforward point evaluation.…”

“…Without the identification of the correct, thermodynamically stable solutions to a model, experimental data can be misinterpreted and gross design errors can be made. For instance, we have shown 7 that for binary systems in the literature, there are instances where data is reported as solid-fluid equilibrium but the equation-of-state model, with parameters regressed by the authors from their experimental data, actually predicts vapor-liquid equilibrium (VLE).…”