Dan Vladimir Nichita

Multiphase equilibria calculation by direct minimization of Gibbs free energy with a global optimization method

Abstract This paper presents a new method for multiphase equilibria calculation by direct minimization of the Gibbs free energy of multicomponent systems. The methods for multiphase equilibria calculation based on the equality of chemical potentials cannot guarantee the convergence to the correct solution since the problem is non-convex (with several local minima), and they can find only one for a given initial guess. The global optimization methods currently available are generally very expensive. A global optimization method called Tunneling, able to escape from local minima and saddle points is used here, and has shown to be able to find efficiently the global solution for all the hypothetical and real problems tested. The Tunneling method has two phases. In phase one, a local bounded optimization method is used to minimize the objective function. In phase two (tunnelization), either global optimality is ascertained, or a feasible initial estimate for a new minimization is generated. For the minimization step, a limited-memory quasi-Newton method is used. The calculation of multiphase equilibria is organized in a stepwise manner, combining phase stability analysis by minimization of the tangent plane distance function with phase splitting calculations. The problems addressed here are the vapor–liquid and liquid–liquid two-phase equilibria, three-phase vapor–liquid–liquid equilibria, and three-phase vapor–liquid–solid equilibria, for a variety of representative systems. The examples show the robustness of the proposed method even in the most difficult situations. The Tunneling method is found to be more efficient than other global optimization methods. The results showed the efficiency and reliability of the novel method for solving the multiphase equilibria and the global stability problems. Although we have used here a cubic equation of state model for Gibbs free energy, any other approach can be used, as the method is model independent.

Phase stability analysis with cubic equations of state by using a global optimization method

Abstract In this paper, we propose a new method for phase stability analysis with cubic equations of state by minimization of the tangent plane distance (TPD) function. A global optimization method called tunneling, able to escape from local minima and saddle points is used here. The tunneling method has two phases. In phase one, a local bounded optimization method is used to minimize the TPD function. In phase two (tunneling), either global optimality is ascertained, or a feasible initial estimate for a new minimization is generated. For the minimization step, a limited-memory quasi-Newton method is used. The tunneling method is used for the conventional approach, and for the reduced variables approach. In the latter case, the number of independent variables does not depend on the number of components in the mixture. The solution of the minimization of TPD function should be found in a space with a significantly reduced number of dimensions. The new method is used for testing phase stability for a variety of representative systems. The examples show the robustness of the method even in the most difficult situations. The proposed method is more efficient than other global optimization methods. Furthermore, in many cases, the reduced approach is faster than the conventional approach. The results showed the efficiency and reliability of the novel method for solving the global stability problem.

VOLUME-BASED THERMODYNAMICS GLOBAL PHASE STABILITY ANALYSIS

In this study we propose a method for phase stability analysis at pressure and temperature specifications, in the frame of a “volume-based” thermodynamics. The formulation of the tangent plane distance (TPD) criterion in terms of the Helmholtz free energy is used in this work for testing phase thermodynamic stability at p-T conditions, using component molar densities as primary variables. The phase stability problem is non-convex; the TPD function may exhibit multiple local minima and saddle points, the use of global optimization methods for its minimization being appropriate. For the unconstrained minimization of the TPD function we use the tunneling global optimization method, which has shown its ability in efficiently solving difficult non-convex, highly nonlinear problems. The method is tested for a variety of mixtures ranging from binaries to mixtures with many components, with emphasis on difficult conditions. The proposed method proved to be an efficient and reliable tool for phase stability analysis.

A reduction method for phase equilibrium calculations with cubic equations of state

In this work we propose a new reduction method for phase equilibrium calculations using a general form of cubic equations of state (CEOS). The energy term in the CEOS is a quadratic form, which is diagonalized by applying a linear transformation. The number of the reduction parameters is related to the rank of the matrix C with elements (1-Cij), where Cij denotes the binary interaction parameters (BIPs). The dimensionality of the problem depends only on the number of reduction parameters, and is independent of the number of components in the mixture.

Consistent delumping of multiphase flash results

An analytical and consistent delumping method, recently formulated and implemented for processes involving two-phase flash calculations, is extended to flash calculations involving any number of coexisting phases. The equation of state considered is a two-parameter cubic equation of state (EoS) with van der Waals mixing rules and non-zero binary interaction parameters (BIPs). The method is used for calculating three- and four-phase equilibria in a series of hydrocarbon mixtures containing different amounts of carbon dioxide, nitrogen, hydrogen sulphide and water. In all examples, the location, compositions and proportions of the coexisting phases obtained by the delumping procedure are extremely close to those obtained from calculations with the detailed fluid, even in cases where the multi-phase region is very tiny. To the best of our knowledge, this is the first time that a delumping procedure is formulated and successfully used for multiphase equilibria.

PARAMETRIC GENERATION OF SINGLE-PHASE PROPERTIES (P-T CURVES) FOR MOST CUBIC EQUATIONS OF STATE AND ANY MIXING RULES

A simple method is proposed for the calculation of any type of P-T curve for the most general case of two-parameter equations of state in the single-phase area. For each component parameter, temperature dependency can be expressed on the basis of any analytical function; for the mixture, any mixing rule can be considered for any parameter. For the construction of these curves, the numerical algorithm is completely described. This method is illustrated for the Joule-Thomson inversion and Boyle curves.

Numerical properties of equations involving high-order derivatives of pressure with respect to volume

This paper presents some unexpected features related to the solution of equations containing a high-order derivative of pressure with respect to volume equated to zero. For pure components, such equations define, in the pressure-temperature plane, nodal curves similar in shape to mixture spinodal curves. The analysis was made for a general form of two-parameter cubic equations of state and various numerical aspects for the Redlich-Kwong equation of state are exemplified.