Application of near critical behavior of equilibrium ratios to phase equilibrium calculations

International audienceWe examine the asymptotic behavior of the equilibrium ratios (Ki) near the convergence locus in the pressure-temperature plane. When the Equation of State (EoS) is analytical, which is the case of most EoS of engineering purpose, Ki tends towards unity or, equivalently, its logarithm lnKi tends to zero, according to a power ½ of the distance to this locus. As a consequence, if lnKi is expressed as a linear combination of pure component parameters with coefficients only depending on mixture phase properties (i.e., reduction parameters), these coefficients obey a similar power law. Deviations from the ½ power law are thus fairly limited for lnKi and for the reduction parameters (at least in the negative flash window between the convergence locus and the phase boundaries), which can be exploited to speed up flash calculations and for quickly determining approximate saturation points and convergence pressures and temperatures. The chosen examples are representative synthetic and natural hydrocarbon mixtures, as well as various injection gas-hydrocarbon systems

Méthodes d'optimisation numérique pour le calcul de stabilité thermodynamique des phases

La modélisation des équilibres thermodynamiques entre phases est essentielle pour le génie des procédés et le génie pétrolier. L analyse de la stabilité des phases est un problème de la plus haute importance parmi les calculs d équilibre des phases. Le calcul de stabilité décide si un système se présente dans un état monophasique ou multiphasique ; si le système se sépare en deux ou plusieurs phases, les résultats du calcul de stabilité fournissent une initialisation de qualité pour les calculs de flash (Michelsen, 1982b), et permettent la validation des résultats des calculs de flash multiphasique. Le problème de la stabilité des phases est résolu par une minimisation sans contraintes de la fonction distance au plan tangent à la surface de l énergie libre de Gibbs ( tangent plane distance , ou TPD). Une phase est considérée comme étant thermodynamiquement stable si la fonction TPD est non- négative pour tous les points stationnaires, tandis qu une valeur négative indique une phase thermodynamiquement instable. La surface TPD dans l espace compositionnel est non- convexe et peut être hautement non linéaire, ce qui fait que les calculs de stabilité peuvent être extrêmement difficiles pour certaines conditions, notamment aux voisinages des singularités. On distingue deux types de singularités : (i) au lieu de la limite du test de stabilité (stability test limit locus, ou STLL), et ii) à la spinodale (la limite intrinsèque de la stabilité thermodynamique). Du point de vue géométrique, la surface TPD présente un point selle, correspondant à une solution non triviale (à la STLL) ou triviale (à la spinodale). Dans le voisinage de ces singularités, le nombre d itérations de toute méthode de minimisation augmente dramatiquement et la divergence peut survenir. Cet inconvénient est bien plus sévère pour la STLL que pour la spinodale. Le présent mémoire est structuré sur trois grandes lignes : (i) après la présentation du critère du plan tangent à la surface de l énergie libre de Gibbs, plusieurs solutions itératives (gradient et méthodes d accélération de la convergence, méthodes de second ordre de Newton et méthodes quasi- Newton), du problème de la stabilité des phases sont présentées et analysées, surtout du point de vue de leurs comportement près des singularités; (ii) Suivant l analyse des valeurs propres, du conditionnement de la matrice Hessienne et de l échelle du problème, ainsi que la représentation de la surface de la fonction TPD, la résolution du calcul de la stabilité des phases par la minimisation des fonctions coût modifiées est adoptée. Ces fonctions coût sont choisies de telle sorte que tout point stationnaire (y compris les points selle) de la fonction TPD soit converti en minimum global; la Hessienne à la STLL est dans ce cas positif définie, et non indéfinie, ce qui mène a une amélioration des propriétés de convergence, comme montré par plusieurs exemples pour des mélanges représentatifs, synthétiques et naturels. Finalement, (iii) les calculs de stabilité sont menés par une méthode d optimisation globale, dite de Tunneling. La méthode de Tunneling consiste à détruire (en plaçant un pôle) les minima déjà trouvés par une méthode de minimisation locale, et a tunneliser pour trouver un point situé dans une autre vallée de la surface de la fonction coût qui contient un minimum 9 à une valeur plus petite de la fonction coût; le processus continue jusqu'à ce que les critères du minimum global soient remplis. Plusieurs exemples soigneusement choisis montrent la robustesse et l efficacité de la méthode de Tunneling pour la minimisation de la fonction TPD, ainsi que pour la minimisation des fonctions coût modifiées.The thermodynamic phase equilibrium modelling is an essential issue for petroleum and process engineering. Phase stability analysis is a highly important problem among phase equilibrium calculations. The stability computation establishes whether a given mixture is in one or several phases. If a mixture splits into two or more phases, the stability calculations provide valuables initialisation sets for the flash calculations, and allow the validation of multiphase flash calculations. The phase stability problem is solved as an unconstrained minimisation of the tangent plan distance (TPD) function to the Gibbs free energy surface. A phase is thermodynamically stable if the TPD function is non-negative at all its stationary points, while a negative value indicates an unstable case. The TPD surface is non-convex and may be highly non-linear in the compositional space; for this reason, phase stability calculation may be extremely difficult for certain conditions, mainly within the vicinity of singularities. One can distinguish two types of singularities: (i) the stability test limit locus (STLL), and (ii) the intrinsic limit of stability (spinodal). Geometrically, the TPD surface exhibits a saddle point, corresponding to a non-trivial (at the STLL) or trivial solution (at the spinodal). In the immediate vicinity of these singularities, the number of iterations of all minimisation methods increases dramatically, and divergence could occur. This inconvenient is more severe for the STLL than for the spinodal. The work presented herein is structured as follow: (i) after the introduction to the concept of tangent plan distance to the Gibbs free energy surface, several iterative methods (gradient, acceleration methods, second-order Newton and quasi-Newton) are presented, and their behaviour analysed, especially near singularities. (ii) following the analysis of Hessian matrix eigenvalues and conditioning, of problem scaling, as well as of the TPD surface representation, the solution of phase stability computation using modified objective functions is adopted. The latter are chosen in such a manner that any stationary point of the TPD function becomes a global minimum of the modified function; at the STLL, the Hessian matrix is no more indefinite, but positive definite. This leads to a better scheme of convergence as will be shown in various examples for synthetic and naturally occurring mixtures. Finally, (iii) the so-called Tunneling global optimization method is used for the stability analysis. This method consists in destroying the minima already found (by placing poles), and to tunnel to another valley of the modified objective function to find a new minimum with a smaller value of the objective function. The process is resumed when criteria for the global minimum are fulfilled. Several carefully chosen examples demonstrate the robustness and the efficiency of the Tunneling method to minimize the TPD function, as well as the modified objective functions.PAU-BU Sciences (644452103) / SudocSudocFranceF

Calculation of Joule-Thomson and isentropic expansion coefficients for two-phase mixtures

Joule–Thomson (JT) and isentropic expansion coefficients describe the temperature change induced by a pressure variation under isenthalpic and isentropic conditions, respectively. They are commonly used to model a variety of processes in which either fluid compression or expansion is involved. While a lot of work has been devoted to inferring the JT coefficient from an equation of state when the fluid is a single phase, little attention has been paid to multiphase fluids, where phase equilibrium has to be taken into account; previous work has only addressed the construction on the JT inversion curve. In the present paper, we describe and implement an approach to calculate these two coefficients for multi-component fluid systems, including when they form two different phases, liquid, and vapor, in thermodynamic equilibrium. The only ingredients are an equation of state and expressions for the ideal part of the specific heats of the fluid components. We make use of cubic equations of state, but any thermodynamic model can be used in the proposed framework. Calculations conducted with typical geofluids, some of them containing CO2, show that these coefficients are discontinuous at phase boundaries (where enthalpy and entropy variations exhibit angular points), as expected with any thermodynamic quantity built from first-order derivatives of state functions, and cannot be simply inferred from the coefficients of the liquid and vapor phases

Phase equilibrium calculations for mixtures containing water

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Phase equilibrium calculations for mixtures containing water

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Robust and efficient volume-based phase stability analysis

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Density-based phase envelope construction including capillary pressure

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A simple approximate density-based phase envelope construction method

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Phase Envelope Construction for Mixtures with Many Components

A reduction method for constructing vapor–liquid equilibrium phase envelopes with cubic equations of state is presented. The paper describes the calculation procedures for saturation (dewpoint/bubblepoint) pressures and temperatures, quality lines (for given mole fraction or volume fraction of one of the equilibrium phases), cricondentherm and cricondenbar points, the critical point, and the spinodal. The phase envelope construction is fully automatic. The saturation points are calculated throughout the critical region by stepping around the phase envelope in pressure or temperature increments. An extrapolation procedure taking advantage of the Jacobian matrix available from a previous step is used. Problem formulation in terms of reduced variables leads to simpler partial derivatives with respect to pressure and temperature. The proposed method is successfully tested for several representative hydrocarbon mixtures with various phase envelope shapes

An improved reduction method for phase stability testing in the single-phase region

A new reduction method for mixture phase stability testing is proposed, consisting in Newton iterations with a particular set of independent variables and residual functions. The dimension of the problem does not depend on the number of components but on the number of components with nonzero binary interaction parameters in the equation of state. Numerical experiments show an improved convergence behavior, mainly for the domain located outside the stability test limit locus in the pressure–temperature plane, recommending the proposed method for any applications in which the problematic domain is crossed a very large number of times during simulations