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.
In recent years, molecularly based equations of state, as typified by the SAFT (statistical associating fluid theory) approach, have become increasingly popular tools for the modeling of phase behavior. However, whether this, or even a much simpler model, is used, the reliable calculation of phase behavior from a given model can be a very challenging computational problem. A new methodology is described that is the first completely reliable technique for computing phase stability and equilibrium from the SAFT model. The method is based on interval analysis, in particular an interval Newton/generalized bisection algorithm, that provides a mathematical and computational guarantee of reliability and is demonstrated using nonassociating, self-associating, and cross-associating systems. New techniques are presented that can also be exploited when conventional point-valued solution methods are used. These include the use of a volume-based problem formulation, in which the core thermodynamic function for phase equilibrium at constant temperature and pressure is the Helmholtz energy, and an approach for dealing with the internal iteration needed when there are association effects. This approach provides for direct, as opposed to iterative, determination of the derivatives of the internal variables.
A deterministic technique for reliable phase stability analysis is described for the case in which asymmetric modeling (different models for vapor and liquid phases) is used. In comparison to the symmetric modeling case, the use of multiple thermodynamic models in the asymmetric case adds an additional layer of complexity to the phase stability problem. To deal with this additional complexity we formulate the phase stability problem in terms of a new type of tangent plane distance function, which uses a binary variable to account for the presence of different liquid and vapor phase models. To then solve the problem deterministically, we use an approach based on interval analysis, which provides a mathematical and computational guarantee that the phase stability problem is correctly solved, and that thus the global minimum in the total Gibbs energy is found in the phase equilibrium problem. The new methodology is tested using several examples, involving as many as eight components, with NRTL as the liquid phase model and a cubic equation of state as the vapor phase model. In two cases, published phase equilibrium computations were found to be incorrect (not stable).
The identification of the correct, stable solution to a phase equilibrium problem, given a particular thermodynamic model, is essential for the design of separation processes.It is also important in the selection of an appropriate model to represent experimental data.The need for a completely reliable method to test for phase stability is particularly pressing when the number of phases likely to be present is not intuitive to the user, as is frequently the case with high-pressure systems. Previously, we have a presented a completely reliable computational technique, based on interval analysis, to correctly identify phase equilibrium and test for phase stability in binary solvent-solute systems, that include the possibility of a solid phase, using any of a variety of cubic equations of state as the thermodynamic model.Here we extend the methodology to include multicomponent solvent-solute-cosolvent systems where the likelihood of additional phase formation is even greater than in the binary case. Gaseous or liquid cosolvents are frequently used in supercritical fluid extraction processes, and are integral in processes such as the gas anti-solvent process (GAS) to precipitate uniform solid particles. Using several examples from the literature, we demonstrate how the new computational technique can be used to identify experimental data that may have been misinterpreted and to identify models that do not predict what the modeler intended.
The reliable calculation of phase equilibrium is a critical issue in the simulation, optimization and design of a wide variety of industrial processes, especially those involving separation operations such as distillation and extraction. However, even when accurate models of the necessary thermodynamic properties are available, it is often very difficult to actually solve the phase equilibrium problem reliably. In this paper, we will discuss a deterministic method, based on interval analysis, that provides a mathematical and computational guarantee that the phase equilibrium problem has been correctly solved, and will highlight recent results using such an approach. Cases are considered in which published computational results are shown to be incorrect.
Perturbation Chain Statistical Association Fluid Theory (PC-SAFT) has been developed to handle the thermodynamics of polymeric systems for two decades. In this paper, we present a new approach to determine the vapor−liquid equilibrium (VLE) for an industrial-scale (high number counts of components) mixture especially close to or beyond the fluid's supercritical region. The newly developed calculation approach involves a reliable computation method based on a unique tangent point on the PC-SAFT equation of state. Such a tangent point method resolves the supercritical region density ambiguity problem for the PC-SAFT model.
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