Tangent Plane Distance Function Research Articles

A unified presentation of phase stability analysis including all major specifications

The major specifications for phase equilibrium are: pressure and temperature (PT), volume, temperature and moles (VTN), pressure, enthalpy and moles (PHN), pressure, entropy and moles (PSN), internal energy, volume and moles (UVN) and entropy, volume and moles (SVN). For some of these specifications, stability testing is not well documented in the literature. It appears that derivations of stability criteria for PHN, PSN and SVN stability are presented here for the first time, as well as the possibility to solve the stability problem in their hyperspaces. Two general stability criteria are derived, which include all state functions and specifications. The general tangent plane distance (TPD) function degenerates to the TPD function corresponding to any desired specifications. The formalism is not model dependent. It is shown that stability testing for any specifications naturally reduces to either PT stability (when pressure is a specification, as in PHN and PSN stability) or VTN stability (when volume is a specification, as in UVN and SVN stability), by solving the appropriate nonlinear equation in temperature. In other words, there is no need (moreover, it is shown that there it is not even recommended) to minimize a specific TPD function in its hyperspace (defined by natural variables of the state function). This paper presents the first unified treatment of phase stability testing including all major specifications; beyond its theoretical importance, an important practical consequence is a unified calculation framework for phase stability testing, allowing a versatile numerical treatment and using existing highly robust and efficient procedures for PT, volume-based PT or VTN stability, whatever the specifications.

Phase equilibria of “syngas” components, aiming for separation by flash distillation

Syngas is a mixture containing methane, hydrogen, carbon dioxide, and carbon monoxide, and it is generated in various processes, such as gasification or methane production via steam reforming. It is desired to separate the syngas to its components in order to create a reliable clean source of energy from methane and hydrogen and to reduce emissions by capturing carbon dioxide. As for today, there is no available data on phase equilibrium of syngas in the literature, which is essential for developing separation processes by distillation. The composition of the syngas depends on its source; therefore, the method for calculating the phase equilibrium shall support any initial composition of the syngas. In the current research, we perform phase stability analysis, using tangent plane distance function and minimizing Helmholtz free energy of the system using two mixing rules: van der Waals and Wong and Sandler. The main strength of the suggested method is its ability to predict any number of phases, avoiding the need for prior knowledge. The current manuscript presents an analysis of binary mixtures of the syngas components, CH4 + CO2, CH4 + CO, CH4 + H2, CO2 + CO, CO2 + H2, and CO + H2, in order to fit the model's parameters to experimental data from the literature, at various temperatures. The aim of this research is to determine which mixing rule shall be further used and to determine the temperature dependent parameters. The parameters fitting is done by minimizing the sum of square errors using “steepest decent” and Fibonacci search method.

A reconsideration on the resolution of phase stability analysis using stochastic global optimization methods: Proposal of a reliable set of benchmark problems

Phase stability analysis is an important thermodynamic calculation in the context of process systems engineering of multiphase units (e.g., separation systems, reactors). It implies the minimization of the Tangent plane distance function (TPDF), which has been recognized as a challenging global optimization problem. Despite the importance of this thermodynamic calculation, the characterization and classification of resolution complexity of phase stability problems that are commonly used to test and compare global optimization methods have not been addressed in literature. This paper reports the first step to propose a set of benchmark TPDF problems to assess and identify the advantages and limitations of stochastic optimization methods used in phase stability calculations. A detailed review of TPDF problem reported in literature was performed and the most representative ones were studied and classified in terms of their resolution complexity via the analysis of several performance metrics associated to both reliability and efficiency to find the global TPDF minimum. Differential Evolution, Simulated Annealing, Harmony Search, Tabu Search, Particle Swarm Optimization and Genetic Algorithm were selected as the metaheuristics to perform the global minimization of selected TPDF problems. 35 phase stability problems were classified in low, medium and high difficulty TPDF problems. Results showed that, although a wide variety of phase stability problems has been reported and used for phase stability calculations, a significant number of them corresponded to global optimization problems of easy resolution. Therefore, some authors have reported biased conclusions on the performance of stochastic optimization methods. A set of 23 TPDF was proposed as benchmark problems to obtain a reliable analysis of the numerical performance of metaheuristics employed in phase stability calculations. This benchmark set contains problems with different resolution difficulties and characteristics that are considered appropriate to study and assess new methods to resolve the global TPDF optimization, as well as to improve the existing methods. This paper highlights the relevance of a continuous development and improvement of stochastic global optimizers for solving, robustly and efficiently, the phase stability analysis in multicomponent systems. TPDF minimization can be still considered a challenging problem to be faced in the context of applied thermodynamics.

Density-based phase envelope construction including capillary pressure

A recently proposed density-based phase envelope construction method (Nichita, Fluid Phase Equilib. 478, 100–113, 2018) is adapted to account for capillary effects. The set of saturation point equations is selected such as both the zero tangent plane distance (TPD) function (in terms of component molar densities and temperature) and the Young-Laplace equation are honored. The set of variables and potential specifications includes mixture molar density, temperature and the modified equilibrium constants (defined as the ratios of reference to incipient phase component molar density). The number of equations and the variables are the same as in the bulk fluid case. The density-based method including capillary pressure is not dependent on the thermodynamic model; any pressure-explicit equation of state and volume-explicit interfacial tension model can be used. The equation of state (EoS) must not be solved for volume and the elements of the Jacobian matrix have simpler expressions than those in conventional (pressure-based) methods. A code for phase envelope construction of bulk fluids can be easily modified by adding the capillary terms to the residual functions and Jacobian matrix. The additional partial derivatives of capillary terms have very simple expressions due to the explicit in volume form of the interfacial tension model. Unlike in conventional formulations, negative pressures in the reference phase can be handled. The proposed method is tested for several hydrocarbon mixtures, ranging from natural gases to heavy oils. As compared to a bulk fluid, under capillary pressure influence the bubble point pressures are suppressed and the dew point locus is expanded, with a shift of cricondentherm points towards higher temperatures. For the mixtures investigated, the computational results are practically identical to those reported in the literature. The computational procedure is robust, there are no problems neither in crossing the critical region (where interfacial tensions are very low) nor at important negative pressures and for very large capillary pressures (of the order of hundreds bar in some test examples).

Volume-based phase stability analysis including capillary pressure

The effect of capillary pressure on phase equilibrium is very important in tight formations, such as shale oil and gas reservoirs. In confined fluids, the capillary pressure induced by highly curved interfaces causes bubble points suppression and an inflated dew point locus, with a shift of cricondentherm points towards higher temperatures, as compared to the bulk fluid. In the conventional phase stability testing including capillary pressure, the problem is solved using the classical tangent plane distance (TPD) function (related to the Gibbs free energy surface) and mole numbers as primary variables. In this work, a volume-based approach (in which the equation of state needs not to be solved for volume) is used to solve this problem, as a bound and linear inequality constrained minimization of a TPD function with respect to the component molar densities. A modified Cholesky factorization (to ensure a descent direction) and a two-stage line search procedure (to ensure that iterates remain in the feasible domain and the objective function is decreased) are used in Newton iterations; a proper change of variables strengthens the robustness. The Weinaug-Katz equation (widely used in chemical and petroleum industry), which is a function of molar densities only, is used for interfacial tensions; the additional partial derivatives in the gradient vector and Hessian matrix have very simple expressions, unlike in conventional formulations. The proposed method is tested for two hydrocarbon mixtures (an oil and a gas condensate) in a wide range of pressures, temperatures and curvature radii, with a special attention paid to the convergence behavior near the singularities, that is, the stability test limit locus (STLL) and the spinodal. The method proves to be highly robust and exhibits fast convergence, showing a similar convergence behavior and almost the same computational effort as in the case of a bulk fluid. The results are only slightly different from conventional methods (except at high curvatures near the cricondentherm and at low pressures); differences are due to the fact that the dependence of capillary pressure on composition is taken into account in deriving the stationarity conditions, unlike in the conventional approach. The proposed method is not dependent on the thermodynamic model and can be used with any capillary pressure representation in which the interfacial tension model is explicit in molar densities.

Improved three-phase equilibrium calculation algorithm for water/hydrocarbon mixtures

Three-phase vapor-liquid-aqueous (VLA) equilibria can frequently appear in hydrocarbon reservoirs. The presence of the aqueous phase, which can commonly exist as a pure water phase, can dramatically increase the probability of encountering convergence issues in the three-phase equilibrium calculation algorithms. The primary reason is that there is a lack of good initial estimates of equilibrium ratios in both stability tests and flash calculations. In this research, we improve the three-phase VLA equilibrium calculation algorithm for water/hydrocarbon mixtures by developing a new scheme for initializing equilibrium ratios as well as a new procedure for three-phase equilibrium calculations. By applying this new initialization scheme, one can securely identify the aqueous-like stationary point on the tangent plane distance function. In our algorithm, convergence problems, which can frequently appear in the flash calculations of three-phase VLA equilibrium calculations, are avoided. We demonstrate the robustness and efficiency of the newly developed algorithm by showing a number of example calculations.

Density-based phase envelope construction

A new density-based phase envelope construction procedure is proposed. The equations for two-phase saturation molar density and temperature computations are established by equating to zero the tangent plane distance (TPD) function in terms of component molar densities and temperature to find a non-trivial set of component molar densities at various specifications: mixture molar density, temperature and the modified equilibrium constants. The latter are defined as the ratios of feed to incipient phase component molar density. The nonlinear system of equations is solved by the full Newton method. The sequence of calculations is started at some conditions where convergence is easily obtained and an extrapolation procedure provides high quality initial guesses for subsequent calculations. The phase envelope is traced in the molar density-temperature plane, where a unique saturation temperature exists at specified mixture molar density; the representation in the pressure-temperature plane is obtained by calculating explicitly the pressure from the equation of state at given temperature and component molar densities on the saturation curve. The equation of state (EoS) must not be solved for volume and the elements of the Jacobian matrix have simpler expressions than their conventional (pressure-based) counterparts. The proposed method is successfully tested for a variety of mixtures, ranging from binary and ternary mixtures to reservoir fluids, exhibiting usual (closed) and unusual phase envelopes, such as open-shaped envelopes characterized by a bubble point branch extending to infinity, branches with negative pressures constructed in a single run (unlike in the conventional case), several branches, double retrograde behavior and swallowtail pattern corresponding to liquid demixing at low temperatures. The proposed method is not dependent of the thermodynamic model and any pressure-explicit equation of state can be used, provided the required partial derivatives of fugacity and pressure with respect to mole numbers, temperature and volume are available.

A Unified Algorithm for Phase-Stability/Split Calculation for Multiphase Isobaric-Isothermal Flash

Summary The conventional method for multiphase flash is the sequential usage of phase-stability and phase-split calculations. Multiphase flash requires the conventional method to obtain multiple false solutions in phase-split calculations and correct them in phase-stability analysis. Improvement of the robustness and efficiency of multiphase flash is important for compositional flow simulation with complex phase behavior. This paper presents a new algorithm that solves for stationary points of the tangent-plane-distance (TPD) function defined at an equilibrium-phase composition for isobaric-isothermal (PT) flash. A solution from the new algorithm consists of two groups of stationary points: tangent and nontangent stationary points of the TPD function. Hence, equilibrium phases, at which the Gibbs free energy is tangent to the TPD function, are found as a subset of the solution. Unlike the conventional method, the new algorithm does not require finding false solutions for robust multiphase flash. The advantage of the new algorithm in terms of robustness is more pronounced for more-complex phase behavior, for which multiple local minima of the Gibbs free energy are present. Case studies show that the new algorithm converges to a lower Gibbs free energy compared with the conventional method for the complex fluids tested. It is straightforward to implement the algorithm because of the simple formulation, which also allows for an arbitrary number of iterative compositions. It can be robustly initialized even when no K value correlation is available for the fluid of interest. Although the main focus of this paper is on robust solution of multiphase flash, the new algorithm can be used to initialize a second-order convergent method in the vicinity of a solution.

Volume-based phase stability testing at pressure and temperature specifications

Conventional phase equilibrium calculations at temperature and pressure specifications require the resolution of the equation of state (EoS). In volume-based calculations, the EoS must not be solved for volume, which is a primary variable. The tangent plane distance (TPD) function can be expressed in terms of mole numbers and volume or of component molar densities. The stationary points of the TPD function, as well as the location of the stability test limit locus (STLL) are different for the two formulations. A modified TPD function in terms of mole numbers and volume is proposed here. Using the block structure of the Hessian matrix, it is shown that Newton iterations in volume-based stability are inherently slower than those in the conventional PT stability, since the Hessian matrix is evaluated at a lower implicitness level. The minimization of several TPD functions (in volume-based and conventional stability testing) is analyzed using various sets of independent variables and scaling procedures. A modified Cholesky factorization and a two-stage line search procedure ensure a sequence of decreasing TPD functions in all cases. The proposed methods are tested for several mixtures, with emphasis on the vicinities of singularities (STLL and spinodal). With proper scaling, the modified Newton iterations are robust and converge very fast for most conditions and reasonably fast for very difficult conditions. The proposed algorithm is not model-dependent; any pressure explicit EoS can be used, provided the required partial derivatives are available.

Finding multiple stationary points of the Gibbs tangent plane distance function via the topographical global initialization

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.

Fast and robust phase stability testing at isothermal-isochoric conditions

In this paper, a robust and efficient phase stability testing procedure at isotherm-isochoric conditions is proposed. The tangent plane distance function is expressed in terms of component molar densities. Convergence behavior using various sets of independent variables is analyzed and it is shown that the choice of independent variables is very important for both robustness and computational speed. A particular attention is paid for conditions in which the Hessian matrix is notoriously ill-conditioned, that is, in the vicinity of the stability test limit locus (STLL) and of the spinodal, where the convergence is slow and problematic. It is found that poor conditioning occurs also at certain conditions in the two-phase region at low temperatures and towards high densities. A modified Cholesky factorization (to ensure a descent direction) and a line search procedure are used in Newton iterations. The equation of state (EoS) must not be solved for volume and any pressure explicit EoS can be used; only expressions of pressure, fugacity and their partial derivatives with respect to mole numbers are required. The proposed method is tested on several examples from the literature, using a refined grid in the temperature-molar density plane. The variables obtained by scaling the ideal part of the Hessian matrix lead by far to the most robust formulation, with no failure observed for the investigated mixtures, and much faster than previous methods, with a low average number of iterations required for convergence for all mixtures.

A note on effective phase stability calculations using a Gradient-Based Cuckoo Search algorithm

Stochastic global optimization methods have been successfully used to perform phase stability calculations. However, these methods may show some drawbacks in challenging phase stability problems. In this study, we made use of the gradient of the tangent plane distance function to improve the performance of Cuckoo Search (CS) algorithm, which is a promising nature-inspired stochastic global optimization method, for the calculation of phase stability analysis. The new modified algorithm, Gradient-Based Cuckoo Search (GBCS), was evaluated for solving several challenging phase stability problems. Its performance at different numerical effort levels and the effect of stopping criterion have been analyzed. GBCS was found to perform better than the original CS algorithm. In comparison with other stochastic optimization methods using an improvement objective function-based stopping criterion, GBCS proved to be the most reliable without any reduction in efficiency.

Calculation of phase equilibria for multi-component mixtures using highly accurate Helmholtz energy equations of state

To test the thermodynamic stability and to determine the equilibrium phase compositions in case the original phase is found unstable is one of the greatest challenges associated with calculating thermodynamic properties of multi-component mixtures. The minimization of the tangent plane distance function is a widely used method to check for stability, while different approaches can be chosen to minimize the Gibbs energy in order to find the phase equilibrium. While these two problems have been applied to several different thermodynamic models, very little work has been published on such algorithms using multi-parameter Helmholtz energy equations of state. In this work, combined stability and flash calculation algorithms at given pressure and temperature (p,T), pressure and enthalpy (p,h), and pressure and entropy (p,s) are presented. The algorithms by Michelsen et al. (1982, 1982, 1987) are used as basis and are adapted to multi-parameter Helmholtz energy models. In addition, a robust and sophisticated density solver is proposed which is necessary for the calculation of properties from the Helmholtz energy model at given state variables other than temperature and density. All partial derivatives necessary to solve the isothermal, isenthalpic and isentropic flash problems using numerical methods based on the Jacobian matrix are derived analytically and given in the supplementary material to this article. Results for some multi-component systems using the GERG-2008 model (Kunz and Wagner, 2012) are shown and discussed.

Phase Stability Analysis using a Polarization Technique and the Randomness of a Stochastic Method in an Unconstrained Optimization Framework

This article deals with the phase stability analysis of multicomponent systems via the determination of all the stationary points of the so-called tangent plane distance function, which is known as an important but difficult problem of the thermodynamics of phase equilibrium, where high nonlinearities are inherent aspects of the different thermodynamic models commonly used in the description of this distance function. To analyze phase stability described from different models, we combine three basic procedures here that were selected in order to attack points that we deem of relevant difficulty. To this end, we encapsulate in a single iterative algorithm the following steps: (i) polarization of a merit function associated with first-order stationary conditions of the phase stability problem, in order to avoid (or at least minimize) the repetition of stationary points previously calculated; (ii) a stochastic optimization method, which (at each iteration of the encapsulation algorithm) minimizes m times the same merit function polarized with the predetermined stationary points, using m different seeds for a reliable random number generator (where m is a given integer), in order to calculate new stationary points; (iii) a change of variables designed to allow the numerical method to work in the unconstrained optimization framework, providing the location of stationary points near the boundary of the original feasible set. Using the NRTL and UNIQUAC models for liquid–liquid equilibria at low pressures and the Soave–Redlich–Kwong and Peng–Robinson cubic equations of state for vapor–liquid equilibria at high pressures, we analyze 9 multicomponent systems studied in the literature, equipped with different feed compositions, totaling 32 tested mixtures, whose component numbers ranged between 3 and 12. We show 14 new stationary points obtained here that were not detected by methods previously used by other authors.

Application of a global optimization algorithm to phase stability and liquid–liquid equilibrium calculations

A global optimization algorithm was developed for finding all singular points (minima, maxima and saddles) of an objective function by exploring the natural connectedness that exists between their singular points. The idea of following ridges and valleys using information gathered along the way was significantly enhanced by applying the arc length continuation method. The algorithm was applied to the Gibbs tangent plane stability test for multiphase liquid mixtures. The global optimization algorithm gives an efficient and robust scheme for locating all stationary points of the tangent plane distance function predicted by any thermodynamic model. Since it provides very good initial estimates for the liquid–liquid equilibrium calculations, it became an integral part of a combined phase equilibrium and stability algorithm. The combined algorithm is self-starting and significantly improves reliability and robustness of multiphase equilibrium calculations. It was successfully tested on a variety of problems and is applicable to any component's mixtures and any number of liquid phases. Solutions have been found for the entire phase diagram.

Phase stability analysis using a reduction method

Phase stability testing is an important sub-problem in phase equilibrium calculations, which require the most important computational effort in reservoir compositional simulations and some process simulators. In this work, a new reduction method for phase stability analysis is proposed. Based on the multi-linear expression of the logarithm of fugacity coefficients as functions of some component properties (elements of the reduction matrix) and of the coefficients depending only on the reduction parameters, a new set of independent variables (which are unbounded, unlike reduction parameters) and the related set of error equations are proposed. Second order Newton iterations are directly used, without the need of switching from successive substitution iterations. It is shown that the proposed method has the same convergence path as the conventional method (in the compositional space) using the logarithm of formal mole numbers as independent variables. Besides, the proposed primary variables are Lagrange multipliers corresponding to a constrained minimization of the modified tangent plane distance function with respect to a specific set of variables and constraints. The robustness and the efficiency of the proposed method are tested for several synthetic and reservoir mixtures, and it compares favorably to conventional and previous reduction methods. Several interesting features are discussed, and a link between conventional and reduction methods is presented.

Phase stability analysis using a modified affine arithmetic

Phase stability analysis is a crucial step in the determination of multiphase equilibrium. This analysis by the tangent plane distance (TPD) minimization is a well-known technique, as well as the difficulties in providing guarantees that the global minimum has been found. On this regard, interval methods are powerful tools since they provide such guarantees. In this work, an interval Newton method plus generalized bisection, based on a modified affine arithmetic, is used to reliably find all possible stationary points of the TPD function. Additionally, an improved convergence test is suggested as well as a special treatment for mole fraction weighted averages. Several mixtures with up to 5 components, including LLE island type ternary systems, were studied. Both activity coefficient models and cubic equations of state were considered. For all the cases tested, the proposed modified affine arithmetic method was superior to other interval-based methods.

A new Newton homotopy based method for the robust determination of all the stationary points of the tangent plane distance function

A new Newton homotopy based method is presented that is able to determine robustly all the stationary points for the tangent plane distance function (TPDF). The solving method is equipped with the features of homotopy parameter and variables bounding that together result in a continuous homotopy path. All the stationary points of TPDF can be approached on one path, assuming only that the temperature, pressure, and composition of the component system are known. The performance of the method is examined with various LLE cases gathered from literature. The liquid phase behavior is described with NRTL and UNIQUAC excess Gibbs energy models. The consistency of stationary points is verified with the topological criterion. Based on this consideration, the proposed modified Newton homotopy method is found to be a robust computational tool for solving phase stability problems.

The Least Dot Products method: A new numerical paradigm for phase stability analysis of thermodynamic mixtures

This work introduces the Least Dot Products (LDP) method, a new algorithm for phase stability analysis of thermodynamic mixtures. Starting with the fact that tangent plane distance function (the objective function used in global stability tests) is, essentially, the dot product between the vector of compositions of a trial phase and the vector that describes the differences among the chemical potential of the components in the phases, our approach tries to obtain the least values of an auxiliary function that represents an appropriate dot product on a unitary sphere of the n-dimensional space, which is a good approximation for the stability test function. In agreement with the foundations of the Gibbs plane tangent criterion, the new algorithm simply tries to find points where the objective function is negative, which are not (necessarily) stationary points or global minima. Thus, our main contribution is not a new method for general nonlinear problems, or a rule-based termination criteria for a classical optimization method. Consequently, if such points do not exist, then LDP method is not capable to recognize the stability condition. To overcome this problem, we develop also a powerful safeguard algorithm, denominated Projected Simulated Annealing (PJSA) algorithm, which is obtained by projecting the Simulated Annealing algorithm onto the unitary sphere. In the present article, firstly we consider liquid–liquid equilibria at low or moderate pressures, where the excess Gibbs energy is described by NRTL or UNIQUAC models. Secondly, we address vapor–liquid equilibria at high pressures with cubic equations of state. To illustrate the performance of the new methodology, we use here 20 systems studied by other authors. Such problems possess 2–12 variables and constitute severe tests for many optimization methods. For some mixtures, we show that LDP method is capable of determining the instability condition using just two iterations.

Generalized Phase Boundary Determination Algorithm for Multicomponent Mixtures

An improved method to determine phase boundaries for multicomponent mixtures is described. This technique uses procedures based upon Gibbs minimization that lead to a more efficient algorithm that can also converge closer to the mixtures’ critical points than other algorithms (e.g., K-value method). Moreover, all the curves reported correspond to stable points according to the stability analysis suggested by Michelsen that implies minimization of the tangent plane distance function. Several examples compare the efficiency of this technique to commercial programs that often fail at certain conditions around the critical point, cricondenbar, and/or cricondentherm.