Successive Substitution Augmented for Global Minimization of the Gibbs Free Energy

2015 ◽  
Author(s):  
Sara Eghbali ◽  
Ryosuke Okuno
Keyword(s):  
Free Energy ◽  
Gibbs Free Energy ◽  
Global Minimization ◽  
Successive Substitution

Global Gibbs Free Energy Minimization in Reactive Systems via Harmony Search

2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Adrian Bonilla-Petriciolet ◽  
Ma. del Rosario Moreno-Virgen ◽  
Juan Jose Soto-Bernal
Keyword(s):  
Free Energy ◽  
Phase Equilibrium ◽  
Gibbs Free Energy ◽  
Chemical Equilibrium ◽  
Equilibrium Problems ◽  
Harmony Search ◽  
Reactive Systems ◽  
Optimization Method ◽  
Global Minimization ◽  
Equilibrium Calculations

Phase equilibrium calculations in systems subject to chemical reactions play a major role in the design of reactive separation schemes including chemical reaction engineering. Basically, these calculations involve the global minimization of Gibbs free energy constrained by the material balances and chemical equilibrium restrictions. However, Gibbs free energy function is non-convex, highly non-linear with many decision variables, and may have several local minimums including trivial and nonphysical solutions. In these conditions, conventional numerical methods are not suitable for solving reactive phase equilibrium problems. Recently, there has been a significant and increasing interest in the development of global strategies for reliably solving reactive phase equilibrium problems subject. Harmony search (HS) is a global stochastic optimization method, which has been conceptualized using the musical process of searching for a perfect state of harmony. Until now, HS has been successfully applied to solve various engineering and optimization problems. However, there are few studies concerning the application of this optimization method for chemical engineering calculations. To the best of our knowledge, the performance of HS for solving reactive phase equilibrium problems has not yet been reported. Therefore, this paper introduces the application of HS-based algorithms to the constrained global minimization of Gibbs free energy in reactive systems. Specifically, we have studied the performance of three variants of HS in reactive phase equilibrium calculations. Our results are useful to identify the capabilities and relative strengths of HS with respect to other stochastic optimization methods for the simultaneous calculation of physical and chemical equilibrium.

Equilibrium analysis of CH 4 , CO , CO 2 , H 2 O , H 2 , C mixtures in C–H–O atom space using Gibbs free energy global minimization

AIChE Journal ◽  
2020 ◽  
Vol 67 (1) ◽  
Author(s):  
Demetrios Chaconas ◽  
Patricia Pichardo ◽  
Ioannis V. Manousiouthakis ◽  
Vasilios I. Manousiouthakis
Keyword(s):  
Free Energy ◽  
Gibbs Free Energy ◽  
Equilibrium Analysis ◽  
Global Minimization

Complex Multiphase Equilibrium Calculations by Direct Minimization of Gibbs Free Energy by Use of Simulated Annealing

1998 ◽  
Vol 1 (01) ◽  
pp. 36-42 ◽  
Author(s):  
Huanquan Pan ◽  
Abbas Firoozabadi
Keyword(s):  
Free Energy ◽  
Phase Equilibrium ◽  
Equilibrium State ◽  
Gibbs Free Energy ◽  
Constant Temperature ◽  
Critical Region ◽  
Tangent Plane ◽  
Initial Guess ◽  
Stationary Points ◽  
Successive Substitution

Summary The computational problems in reservoir fluid systems are mainly in the critical region and in liquid-liquid (LL), vapor-liquid-liquid (VLL), and higher-phase equilibria. The conventional methods to perform phase-equilibrium calculations with the equality of chemical potentials cannot guarantee a correct solution. In this study, we propose a simple method to calculate the equilibrium state by direct minimization of the Gibbs free energy of the system at constant temperature and pressure. We use the simulated annealing (SA) algorithm to perform the global minimization. Estimates of key parameters of the SA algorithm are also made for phase-behavior calculations. Several examples, including (1) VL equilibria in the critical region, (2) VLL equilibria for reservoir fluid systems, (3) VLL equilibria for an H2S-containing mixture, and (4) VL-multisolid equilibria for reservoir fluids, show the reliability of the method. Introduction Consider the multicomponent-multiphase flash at constant temperature and pressure sketched in Fig. 1. The equilibrium state (the right side of Fig. 1) consists of np phases; each Phase j consists of n1j,n2j,n3j,. . . nncj, moles. From the second law of thermodynamics, the equilibrium state is a state in which the Gibbs free energy of the system is a minimum. The minimum of Gibbs free energy is a sufficient and necessary condition for the equilibrium state. At constant temperature and pressure (note that all calculations will be performed at this condition), the Gibbs free energy of the system in Fig. 1 can be written asEquation 1 where Gj is the Gibbs free energy of Phase j, and G is the total Gibbs free energy of the system. When G is minimized with respect to nij (i=1, 2, . . ., nc; j=1, 2, . . ., np) subject to the following constraints:material balance of Component i,Equation 2the non-negative mole number of Component i in Phase j,Equation 3 The optimized values, ni(i=1, 2, . . ., nc; j=1, 2, . . ., np) are the mole numbers of the equilibrium state. The global minimization with the constraints is difficult to implement; as a consequence, direct minimization of the Gibbs free energy has not been widely applied. Conventional Approach for Phase-Equilibrium Calculations The equality of chemical potentials of each species in all phases is often used to perform the phase-equilibrium calculations:Equation 4 The number of equations in Eq. 4 is nc×(np-1), plus nc material-balance equations given by Eq. 2; a total of nc×np equations are provided. The mole numbers nij (i=1, 2, . . ., nc; j=1, 2, . . ., np) of the equilibrium state are determined by solving these nc×np nonlinear equations. The widely used solution methods are the successive substitution method through phase-equilibrium constants Ki (i=1, 2, . . ., nc) and direct application of the Newton method. Both approaches require an initial guess and work quite well for VL equilibria except in the near-critical region. In the critical region, the successive substitution becomes intolerably slow and the Newton method may fail when the initial guess is not close to the true solution. In LL and VLL equilibria, both methods may compute false solutions. The falseness is because Eq. 4 is only a necessary condition for an equilibrium state.1 The tangent-plane-distance (TPD) approach has been introduced to recognize the false solution.1,2 The concept of stability analysis is used to derive the TPD. Tangent-Plane-Distance Approach Suppose w is a given overall composition. The mathematical expression of the TPD function isEquation 5a where D(u) is the distance function between the Gibbs free energy surface and its tangent plane at composition w. When D(u) is minimized with respect to ui(i=1, 2, . . ., nc) subject toEquations 5b and 5c the optimized value, D*, provides the stability analysis of the mixture at composition w. If D* 0, the system is absolutely stable; if D*<0, the system is unstable. The optimized composition u* is a good approximation of the incipient phase composition. The application of TPD criterion improves the reliability of conventional-phase equilibrium by providing a guideline to judge that the mixture is absolutely stable. When unstable, a good initial composition u* strengthens the convergence of the Newton or the successive substitution methods. Unfortunately, the solution to Eq. 5 is also an optimization problem with constraints. Michelson2 has solved the problem by locating the stationary points of the TPD function. This approach needs to solve (nc-1) nonlinear equations. A good initial guess is required to avoid the trivial solution. Because not all stationary points can be found with this method, phase stability cannot always be guaranteed.3 Later, we will give an example of a CO2-crude system for which the approach of locating the stationary points misses the true solution in spite of its novelty and strengths. Several methods have been proposed to improve the calculation of the TPD function. These include homotopy-continuation,4 branch and bound3 and differential geometry, and the theory of differential equations.5

Chemical equilibrium and chemical kinetics

2018 ◽  
Author(s):  
Dennis Sherwood ◽  
Paul Dalby
Keyword(s):  
Free Energy ◽  
Equilibrium Constant ◽  
Gibbs Free Energy ◽  
Chemical Kinetics ◽  
Chemical Equilibrium ◽  
First Principles ◽  
Effect Of Temperature ◽  
Total System ◽  
Free Energies

Building on the previous chapter, this chapter examines gas phase chemical equilibrium, and the equilibrium constant. This chapter takes a rigorous, yet very clear, ‘first principles’ approach, expressing the total Gibbs free energy of a reaction mixture at any time as the sum of the instantaneous Gibbs free energies of each component, as expressed in terms of the extent-of-reaction. The equilibrium reaction mixture is then defined as the point at which the total system Gibbs free energy is a minimum, from which concepts such as the equilibrium constant emerge. The chapter also explores the temperature dependence of equilibrium, this being one example of Le Chatelier’s principle. Finally, the chapter links thermodynamics to chemical kinetics by showing how the equilibrium constant is the ratio of the forward and backward rate constants. We also introduce the Arrhenius equation, closing with a discussion of the overall effect of temperature on chemical equilibrium.

Extended Gibbs Free Energy and Laplace Pressure of Ordered Hexagonal Close-Packed Spherical Particles: A Wettability Study

Langmuir ◽  
2021 ◽  
Author(s):  
Amir Bayat ◽  
Mahdi Ebrahimi ◽  
Saeed Rahemi Ardekani ◽  
Esmaiel Saievar Iranizad ◽  
Alireza Zaker Moshfegh
Keyword(s):  
Free Energy ◽  
Gibbs Free Energy ◽  
Laplace Pressure ◽  
Spherical Particles ◽  
Hexagonal Close Packed
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