Tangent Plane

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Christodoulos A. Floudas - One of the best experts on this subject based on the ideXlab platform.

  • glopeq a new computational tool for the phase and chemical equilibrium problem
    Computers & Chemical Engineering, 1997
    Co-Authors: Conor M. Mcdonald, Christodoulos A. Floudas
    Abstract:

    Abstract Calculation of phase and chemical equilibrium represents a crucial phase in the modeling of many separation processes. For conditions of constant temperature and pressure, a necessary and sufficient condition for the true equilibrium solution is that (i) the total Gibbs free energy of the system be at its global minimum, or (ii) the minimum of the Tangent Plane distance function be nonnegative for all phase models used to represent the system. In this work, the goal is to obtain equilibrium solutions corresponding to a global minimum of the Gibbs free energy as efficiently as possible, for cases where the liquid phase or phases can be modeled by the NRTL, UNIQUAC, UNIFAC, Wilson, modified Wilson and ASOG equations. Vapor phases whose behavior can be described as ideal can also be handled. In achieving this goal, there are two distinct problems of relevance: (i) the minimization of the Gibbs free energy, denoted (G), and (ii) the minimization of the Tangent Plane distance function, or the Tangent Plane stability criterion, denoted (S). For all these activity coefficient models, GLOPEQ (GLobal OPtimization for the Phase and chemical EQuilibrium problem) can guarantee global solutions for problems (G) and (S), but a combined algorithm employs them in tandem, using (G) to generate candidate equilibrium solutions which can then be verified for thermodynamic stability by solving (S). Two key features of the combined algorithm are that (i) as much information as is possible is obtained from local searches, and (ii) it is preferable to verify a globally stable equilibrium solution using the Tangent Plane criterion, as this problem contains fewer variables than the minimization of the Gibbs free energy. Results for several examples are presented, and all but one of them are for the case of phase equilibrium, due to the paucity of examples for reacting systems that employ excess Gibbs free energy models.

  • global optimization for the phase stability problem
    Aiche Journal, 1995
    Co-Authors: Conor M. Mcdonald, Christodoulos A. Floudas
    Abstract:

    The Gibbs Tangent Plane criterion has become important in determining the quality of obtained solutions to the phase and chemical equilibrium problem. The ability to determine if a postulated solution is thermodynamically stable with respect to perturbations in any or all of the phases is very useful in the search for the true equilibrium solution. Previous approaches have focused on finding stationary points of the Tangent Plane distance function. Obtaining all stationary points, however, cannot be guaranteed due to the complex and nonlinear nature of the models used to predict equilibrium. Simpler formulations for the stability problem are presented for special problems where nonideal liquid phases can be adequately modeled using the NRTL and UNIQUAC activity coefficient equations. It shows how the global minimum of the Tangent Plane distance function can be obtained for these problems. A global optimization approach is advantageous because a nonnegative solution can be asserted to be the globally stable equilibrium one, unlike available local algorithms. For the NRTL equation, the GOP algorithm of Floudas and Visweswaran (1990, 1993) is used to guarantee obtaining ϵ -global convergence to the global minimum. For the UNIQUAC equation, a branch and bound algonthm based on that of Falk and Soland (1969) is used to guarantee convergence to the global solution. Computational results demonstrate the efficiency of both global optimization algorithms in solving various challenging problems.

  • Global Optimization and Analysis for the Gibbs Free Energy Function Using the UNIFAC, Wilson, and ASOG Equations
    Industrial & Engineering Chemistry Research, 1995
    Co-Authors: Conor M. Mcdonald, Christodoulos A. Floudas
    Abstract:

    The Wilson equation for the excess Gibbs energy has found wide use in successfully representing the behavior of polar and nonpolar multicomponent mixtures with only binary parameters but was incapable of predicting more than one liquid phase. The UNIFAC and ASOG group contribution methods do not have this limitation and can predict the presence of multiple liquid phases. The most important area of application of all these equations is in the prediction of phase equilibria. The calculation of phase equilibria involves two important problems: (1) the minimization of the Gibbs free energy and (2) the Tangent Plane stability criterion. Problem (2), which can be implemented as the minimization of the Tangent Plane distance function, has found wide application in aiding the search for the global minimum of the Gibbs free energy. However, a drawback of all previous approaches is that they could not provide theoretical guarantees that the true equilibrium solution will be obtained. The goal of the work is to find the equilibrium solution corresponding to the global minimum of the Gibbs free energy. A proof that the Wilson equation leads to a convex formulation for the minimization of the Gibbs energy is provided so that a local optimization technique will always converge to a global minimum. In addition, new expressions are derived for the molar Gibbs free energy function when the UNIFAC, ASOG, and modified Wilson equations are employed. These expressions are then transformed so that application of a branch and bound based global optimization algorithm originally due to Falk and Soland (1969) is possible. This allows global solutions to be obtained for both the minimization of the Gibbs free energy and the minimization of the Tangent Plane distance function. The algorithm is implemented in C as part of the package GLOPEQ, global optimization for the phase equilibrium problem (McDonald and Floudas, 1994d). Results for several examples are presented

Nelio Henderson - One of the best experts on this subject based on the ideXlab platform.

  • finding multiple stationary points of the gibbs Tangent Plane distance function via the topographical global initialization
    Chemical Engineering Research & Design, 2017
    Co-Authors: Nelio Henderson, Janaina Imbiriba, Marroni De Sa Rego
    Abstract:

    Abstract In order to find multiple stationary points of the Gibbs Tangent Plane distance function, often required in the stability analysis used in phase equilibrium calculations, in this article we apply a recently revisited version of the topographical global initialization. This initialization technique is a simple and ingenious approach based on elementary concepts of graph theory. Here, the topographical initialization is employed to generate good starting points to solve a constrained global minimization problem, whose solutions are the roots of a nonlinear system, which describes the first-order stationary conditions associated with the Gibbs Plane Tangent criterion for phase stability analysis. To accomplish the task of local search, in the minimization step we use a well-established interior-point method. Our methodology was compared against another robust method using benchmarks from the literature. Results indicated that the present approach is a powerful strategy for finding multiple stationary points of the Gibbs Tangent Plane distance function, having demonstrated high efficiency and robustness.

  • a deduction of the multicriticality conditions of mixtures from the gibbs Tangent Plane criterion
    Fluid Phase Equilibria, 2013
    Co-Authors: Nelio Henderson, Wagner F Sacco, Raimundo A Rodrigues
    Abstract:

    Abstract Here, we follow a classification proposed by Griffiths and Widom [1] , where the order of a multicritical point in a mixture is equal to the number of phases which simultaneously become identical, considering m phases and assuming that m − 1 of these phases become identical to a given test phase. Thus, employing Rolle's theorem and basic properties of the so-called Tangent-Plane distance function, we develop a deduction of the multicriticality conditions of mixture from Gibbs Tangent Plane criterion, which relies on the principle of mathematical induction, being appropriate for any m ≥ 2.

Conor M. Mcdonald - One of the best experts on this subject based on the ideXlab platform.

  • glopeq a new computational tool for the phase and chemical equilibrium problem
    Computers & Chemical Engineering, 1997
    Co-Authors: Conor M. Mcdonald, Christodoulos A. Floudas
    Abstract:

    Abstract Calculation of phase and chemical equilibrium represents a crucial phase in the modeling of many separation processes. For conditions of constant temperature and pressure, a necessary and sufficient condition for the true equilibrium solution is that (i) the total Gibbs free energy of the system be at its global minimum, or (ii) the minimum of the Tangent Plane distance function be nonnegative for all phase models used to represent the system. In this work, the goal is to obtain equilibrium solutions corresponding to a global minimum of the Gibbs free energy as efficiently as possible, for cases where the liquid phase or phases can be modeled by the NRTL, UNIQUAC, UNIFAC, Wilson, modified Wilson and ASOG equations. Vapor phases whose behavior can be described as ideal can also be handled. In achieving this goal, there are two distinct problems of relevance: (i) the minimization of the Gibbs free energy, denoted (G), and (ii) the minimization of the Tangent Plane distance function, or the Tangent Plane stability criterion, denoted (S). For all these activity coefficient models, GLOPEQ (GLobal OPtimization for the Phase and chemical EQuilibrium problem) can guarantee global solutions for problems (G) and (S), but a combined algorithm employs them in tandem, using (G) to generate candidate equilibrium solutions which can then be verified for thermodynamic stability by solving (S). Two key features of the combined algorithm are that (i) as much information as is possible is obtained from local searches, and (ii) it is preferable to verify a globally stable equilibrium solution using the Tangent Plane criterion, as this problem contains fewer variables than the minimization of the Gibbs free energy. Results for several examples are presented, and all but one of them are for the case of phase equilibrium, due to the paucity of examples for reacting systems that employ excess Gibbs free energy models.

  • global optimization for the phase stability problem
    Aiche Journal, 1995
    Co-Authors: Conor M. Mcdonald, Christodoulos A. Floudas
    Abstract:

    The Gibbs Tangent Plane criterion has become important in determining the quality of obtained solutions to the phase and chemical equilibrium problem. The ability to determine if a postulated solution is thermodynamically stable with respect to perturbations in any or all of the phases is very useful in the search for the true equilibrium solution. Previous approaches have focused on finding stationary points of the Tangent Plane distance function. Obtaining all stationary points, however, cannot be guaranteed due to the complex and nonlinear nature of the models used to predict equilibrium. Simpler formulations for the stability problem are presented for special problems where nonideal liquid phases can be adequately modeled using the NRTL and UNIQUAC activity coefficient equations. It shows how the global minimum of the Tangent Plane distance function can be obtained for these problems. A global optimization approach is advantageous because a nonnegative solution can be asserted to be the globally stable equilibrium one, unlike available local algorithms. For the NRTL equation, the GOP algorithm of Floudas and Visweswaran (1990, 1993) is used to guarantee obtaining ϵ -global convergence to the global minimum. For the UNIQUAC equation, a branch and bound algonthm based on that of Falk and Soland (1969) is used to guarantee convergence to the global solution. Computational results demonstrate the efficiency of both global optimization algorithms in solving various challenging problems.

  • Global Optimization and Analysis for the Gibbs Free Energy Function Using the UNIFAC, Wilson, and ASOG Equations
    Industrial & Engineering Chemistry Research, 1995
    Co-Authors: Conor M. Mcdonald, Christodoulos A. Floudas
    Abstract:

    The Wilson equation for the excess Gibbs energy has found wide use in successfully representing the behavior of polar and nonpolar multicomponent mixtures with only binary parameters but was incapable of predicting more than one liquid phase. The UNIFAC and ASOG group contribution methods do not have this limitation and can predict the presence of multiple liquid phases. The most important area of application of all these equations is in the prediction of phase equilibria. The calculation of phase equilibria involves two important problems: (1) the minimization of the Gibbs free energy and (2) the Tangent Plane stability criterion. Problem (2), which can be implemented as the minimization of the Tangent Plane distance function, has found wide application in aiding the search for the global minimum of the Gibbs free energy. However, a drawback of all previous approaches is that they could not provide theoretical guarantees that the true equilibrium solution will be obtained. The goal of the work is to find the equilibrium solution corresponding to the global minimum of the Gibbs free energy. A proof that the Wilson equation leads to a convex formulation for the minimization of the Gibbs energy is provided so that a local optimization technique will always converge to a global minimum. In addition, new expressions are derived for the molar Gibbs free energy function when the UNIFAC, ASOG, and modified Wilson equations are employed. These expressions are then transformed so that application of a branch and bound based global optimization algorithm originally due to Falk and Soland (1969) is possible. This allows global solutions to be obtained for both the minimization of the Gibbs free energy and the minimization of the Tangent Plane distance function. The algorithm is implemented in C as part of the package GLOPEQ, global optimization for the phase equilibrium problem (McDonald and Floudas, 1994d). Results for several examples are presented

Raimundo A Rodrigues - One of the best experts on this subject based on the ideXlab platform.

  • a deduction of the multicriticality conditions of mixtures from the gibbs Tangent Plane criterion
    Fluid Phase Equilibria, 2013
    Co-Authors: Nelio Henderson, Wagner F Sacco, Raimundo A Rodrigues
    Abstract:

    Abstract Here, we follow a classification proposed by Griffiths and Widom [1] , where the order of a multicritical point in a mixture is equal to the number of phases which simultaneously become identical, considering m phases and assuming that m − 1 of these phases become identical to a given test phase. Thus, employing Rolle's theorem and basic properties of the so-called Tangent-Plane distance function, we develop a deduction of the multicriticality conditions of mixture from Gibbs Tangent Plane criterion, which relies on the principle of mathematical induction, being appropriate for any m ≥ 2.

Michael F Malone - One of the best experts on this subject based on the ideXlab platform.

  • global stability analysis and calculation of liquid liquid equilibrium in multicomponent mixtures
    Industrial & Engineering Chemistry Research, 1996
    Co-Authors: Stanislaw K Wasylkiewicz, Lakshmi N Sridhar, Michael F Doherty, Michael F Malone
    Abstract:

    A global algorithm for implementing the Gibbs Tangent Plane stability test in multicomponent, multiphase liquid mixtures is described. The algorithm is self-starting and significantly improves the reliability and robustness of multiphase equilibrium calculations. The main improvement is a result of a new approach for locating the stationary points of the Tangent Plane distance function. This relies on the geometric properties of the Tangent Plane distance function but is independent of the specific fluid model. For ternary mixtures, it is shown that stationary points in the Tangent Plane distance function satisfy a global topological constraint Nmax + Nmin − Nsad = 1 where Nmax is the number of maxima, Nmin is the number of minima, and Nsad is the number of saddles. This provides a global consistency check on the numerical results that is independent of the specific model or parameters used to describe the mixture. The method has been implemented and tested on a variety of problems with three and four com...