A contribution to the simultaneous chemical and phase equilibrium calculation

1979 ◽  
Vol 44 (8) ◽  
pp. 2293-2301 ◽  
Author(s):  
Oldřich Coufal ◽  
Alois Fidler
Keyword(s):  
Free Energy ◽  
Phase Equilibrium ◽  
Gibbs Free Energy ◽  
Equilibrium Composition ◽  
Equilibrium Calculation ◽  
Reacting Systems

Some problems are treated in detail which are encountered during calculating the equilibrium composition in heterogeneous reacting systems with respect to the minimum of the total Gibbs free energy.

scholarly journals APPROACH TO A RELIABLE SOLUTION STRATEGY FOR PERFORMING PHASE EQUILIBRIUM CALCULATIONS USING MINLP OPTIMIZATION

2014 ◽  
Vol 44 (1) ◽  
pp. 63-70
Author(s):  
I. J. JEREZ ◽  
F. MUÑOZ ◽  
J. M. GOMEZ
Keyword(s):  
Free Energy ◽  
Phase Equilibrium ◽  
Gibbs Free Energy ◽  
Energy Minimization ◽  
Continuous Variables ◽  
Solution Strategy ◽  
Free Energy Minimization ◽  
Integer Variables ◽  
Gibbs Free Energy Minimization ◽  
Equilibrium Calculations

The objective of this contribution is to propose a reliable strategy to solver the problem of phase equilibrium calculations for non-ideal systems, using the Gibbs free energy minimization. This type of problem, using the Gibbs free energy minimization, is usually formulated as a Mixed Integer NonLinear Programming (MINLP) Optimization. This optimization problem allows the compositions to be associated with continuous variables, and the presence of phases in the equilibrium to be associated with the integer variables. The solution strategy proposes a bi-level approach. The first level combines a stochastic (Simulated Annealing – SA) and a local deterministic algorithm (Sequential Quadratic Programming – SQP), and solves a Non Linear Programming Problem (NLP). The continuous variables are considered at this level. The second level considers the integer variables. The advantage of this bilevel strategy lies in its easy implementation and in its proven efficiency to locate global optima with acceptable computational load. This article includes the study of the Water-Ethanol-Cyclohexane and Water-Ethanol-Glycerin systems. A comparative analysis was conducted using experimental data reported in published works and theoretical calculations by means of the Gamma-Phi classic method.

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.

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

Modeling Effect of Geochemical Reactions on Real-Reservoir-Fluid Mixture During Carbon Dioxide Enhanced Oil Recovery

SPE Journal ◽  
2017 ◽  
Vol 22 (05) ◽  
pp. 1519-1529 ◽  
Author(s):  
Ashwin Venkatraman ◽  
Birol Dindoruk ◽  
Hani Elshahawi ◽  
Larry W. Lake ◽  
Russell T. Johns
Keyword(s):  
Free Energy ◽  
Phase Behavior ◽  
Gibbs Free Energy ◽  
Oil Recovery ◽  
Equilibrium Composition ◽  
Co2 Injection ◽  
Free Energy Function ◽  
Fluid Mixture ◽  
Reservoir Fluid ◽  
Geochemical Reactions

Summary Carbon dioxide (CO2) injection in oil reservoirs has the dual benefit of enhancing oil recovery from declining reservoirs and sequestering a greenhouse gas to combat climate change. CO2 injected in carbonate reservoirs, such as those found in the Middle East, can react with ions present in the brine and the solid calcite in the carbonate rocks. These geochemical reactions affect the overall mole numbers and, in some extreme cases, even the number of phases at equilibrium, affecting oil-recovery predictions obtained from compositional simulations. Hence, it is important to model the effect of geochemical reactions on a real-reservoir-fluid mixture during CO2 injection. In this study, the Gibbs free-energy function is used to integrate phase-behavior computations and geochemical reactions to find equilibrium composition. The Gibbs free-energy minimization method by use of elemental-balance constraint is used to obtain equilibrium composition arising out of phase and chemical equilibrium. The solid phase is assumed to be calcite, the hydrocarbon phases are characterized by use of the Peng-Robinson (PR) equation of state (EOS) (Robinson et al. 1985), and the aqueous-phase components are described by use of the Pitzer activity-coefficient model (Pitzer 1973). The binary-interaction parameters for the EOS and the activity-coefficient model are obtained by use of experimental data. The effect of the changes in phase behavior of a real-reservoir fluid with 22 components is presented in this paper. We observe that the changes in phase behavior of the resulting reservoir-fluid mixture in the presence of geochemical reactions depend on two factors: the volume ratio (and hence molar ratio) of the aqueous phase to the hydrocarbon phase and the salinity of the brine. These changes represent a maximum effect of geochemical reactions because all reactions are assumed to be at equilibrium. This approach can be adapted to any reservoir brine and hydrocarbon as long as the initial formation-water composition and their Gibbs free energy at standard states are known. The resultant model can be integrated in any reservoir simulator because any algorithm can be used for minimizing the Gibbs free-energy function of the entire system.

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