Calculation Of Critical Points Research Articles

Prediction of the critical points of natural gas mixtures by rigorous and semi-empirical methods

In the first part of this work, the modified Patel-Teja equation-of-state was applied to the rigorous critical point calculation method developed by Heidemann and Khalil. Two binary interaction coefficients ( k a ij and k b ij ) were used to fit the binary critical point data and the composition dependency of k ij -values is evident. Considering the composition characteristics of natural gas mixtures, an optimal k ij -matrix is recommended. The test on 46 natural gases (including a rich gas condensate), and other multicomponent hydrocarbon mixtures, shows a satisfactory prediction of critical properties. In the second part, the simpler semi-empirical methods based on excess critical property functions were systematically studied and a new interaction parameter ( A ij ) matrix designed for natural gas mixtures was developed. Test results on 93 binary and multicomponent mixtures are reported. Critical point data of mixtures are important parameters for phase behavior measurements of reservoir fluids.

Liquid-liquid equilibria for solutions of polydisperse polymers. Continuous thermodynamics for the lattice-fluid model

Abstract Hu, Y., Ying, X., Wu, D.T. and Prausnitz, J.M., 1994. Liquid-liquid equilibria for solutions of polydisperse polymers. Continuous thermodynamics for the lattice-fluid model. Fluid Phase Equilibria, 98: 113-128. A continuous-thermodynamics framework is presented for phase equilibrium calculations for solutions of polydisperse polymers using a lattice-fluid model. A two-step process is designed to form a real solution containing a solvent and a polydisperse polymer solute at fixed temperature and pressure. In the first step, close-packed pure components are mixed to form a close-packed polymer solution. In the second step, the close-packed mixture, considered to be a pseudo-pure substance, is mixed with holes to form a real polymer solution whose volume depends on temperature and pressure. The simplified Freed model developed previously is adopted for both steps. Besides pure-component parameters, the theory uses a binary size parameter cr and a binary energy parameter ϵ12; these binary parameters may be temperature-dependent. The functional approach is adopted to derive expressions for chemical potentials, spinodals and critical points. Computation procedures are established for cloud-point-curve, shadow-curve, spinodal and critical-point calculations for polymer solutions using either a standard distribution or an arbitrary distribution for the polymer molar masses or chain lengths. Calculations are shown for the effect of polydispersity on upper critical solution temperature (UCST), lower critical solution temperature (LCST) and hour-glass-shaped phase behavior for the system acetone-polydisperse polystyrene.

Critical point calculations for oil reservoir fluid systems using the SPHCT equation of state

Garcia-Sánchez, F., Ruiz-Cortina, J.L., Lira-Galeana, C. and Ponce-Ramirez, L., 1992. Critical point calculations for oil reservoir fluid systems using the SPHCT equation of state. Fluid Phase Equilibria, 81: 39-84. The ability for predicting the critical points of reservoir fluids using the Simplified Perturbed Hard-Chain Theory (SPHCT) equation of state proposed by Kim et al. is analyzed. The computational procedure developed by Heidemann and Khalil was used for locating the critical points in any multicomponent mixture described by this equation of state. Experimental binary vapor-liquid equilibrium data of interest in the petroleum industry have been used to evaluate the interaction coefficients in this equation for 164 binary mixtures. The performance of the SPHCT EOS to predict critical points is demonstrated on four oil reservoir fluid systems containing up to forty-eight components. Convergence to all the critical points existing in these large systems was obtained in a few iterations without any difficulty.

New strategy for phase equilibrium and critical point calculations by thermodynamic energy analysis. Part II. Critical point calculations

The thermodynamic energy analysis of Michelsen is reformulated in terms of component molar densities as the primary variables. In this formulation, we can provide analytical expressions for any higher order derivatives for the energy function. This feature enables us to enhance the speed and the reliability of the critical point search method. It is designed to find the critical composition and the critical pressure for fluid mixtures at a fixed temperature (pressure-composition diagram), as well as critical points in the composition space at the specified pressure and temperature (pseudo-ternary diagrams). This has proved to be very useful in finding the miscibility composition for complex and near-critical petroleum reservoir fluids subjected to gas injection. A number of examples illustrate the successful use of this method.

New strategy for phase equilibrium and critical point calculations by thermodynamic energy analysis. Part I. Stability analysis and flash

We have reformulated Michelsen's method of phase stability analysis, which employs the thermodynamic energy minimization in terms of the tangent plane criterion. In our method, we use the component molar densities as the primary variables in place of the mole numbers or mole fractions. Furthermore, we use a modified stability test procedure called the ‘semi-local’ stability test and a higher order initialization procedure in the two-phase region. We believe that these modifications improve the numerical stability and the speed of convergence of the iterative technique, particularly in the near-critical and the liquid-liquid regions. We demonstrate the successful use of this method with well-defined example fluid systems and near-critical and retrograde condensate petroleum reservoir fluids.

Spinoidal and Critical Points in Polydisperse Random Copolymer Blends

Abstract Spinoidal and critical points of the liquid-liquid equilibrium in random copolymer blends are calculated by application of continuous thermodynamics. Blends are composed of two copolymers, each consisting of two different monomer units. Both copolymers are characterized by divariate distribution functions with respect to molecular weight and to chemical composition. Based on a model for the excess Gibbs free energy of mixing, the necessary relations are derived for the calculation of spinoidal and critical points. The influences of the polydispersities of molecular weight and/or chemical composition on the instability limit are discussed. Miscibility maps are calculated on the basis of the critical point. Cloud-point curves are shown to be most sensitive to polydispersities if the excess Gibbs free energy of mixing exceeds only slightly the lowest value necessary for demixing.

The chiral transition in an SU(2) gauge-Higgs-fermion theory with Yukawa couplings

We study the (zero-temperature) chiral transition in an SU(2) lattice gauge theory with I = 1 2 Higgs fields, and fermion fields with I = 1 2 and I = 0 , including Yukawa couplings. Using analytic methods in the strong gauge coupling limit, we demonstrate that (i) if at least one of the Yukawa couplings is sufficiently weak, this theory undergoes a spontaneous chiral symmetry breaking phase transition; (ii) this transition is removed when the Yukawa couplings are sufficiently strong. The transition is determined to be of second order, and approximate critical point calculations are given. Some implications for the continuum limit of the SU(2) model, and for electroweak interactions, are discussed.

Monte Carlo calculation of free energy, critical point, and surface critical behavior of three-dimensional Heisenberg ferromagnets.

A transfer-matrix Monte Carlo technique is developed to compute the free energy of three-dimensional, classical Heisenberg ferromagnets. From the free energy of systems with periodic and antiperiodic boundary conditions, helicity moduli are calculated. From these the critical couplings for simple-cubic (sc) and face-centered-cubic lattices are estimated by use of finite-size scaling. For the simple-cubic lattice, the critical dimension of the surface magnetization is estimated with standard Monte Carlo methods, yielding a result in excellent agreement with the \ensuremath{\epsilon}-expansion work of Diehl and N\usser.

Calculation of tri-critical points

Procedures are presented for calculating higher order critical points in thermodynamic models. The procedures arise from an analysis of the Gibbs tangent plane criterion for phase stability. New methods are proposed for evaluating the functions that must be zero at critical points and the ease with which these functional evaluations can be carried out is the major element in permitting calculation of higher order critical points in mixtures of any number of components. Comparison is made between experimental data and tri-critical points for ternary mixtures as calculated with the Peng-Robinson and SRK equations.

Some aspects of multiphase calculations

Computation procedures for locating and constructing phase boundaries in multiphase systems, where a single equation of state is used for calculation of all thermodynamic properties, are described. Special emphasis is given to the calculation of the pressure-temperature bounderies between two-phase and three-phase regions in mixtures of fixed overall composition, and to the calculation of critical points, critical lines and tricritical points.

Numerical calculation of critical points for a slab with partial insulation

Numerical solutions of a nonlinear partial differential equation arising in thermal explosion theory are obtained for a finite rectangular strip domain with mixed boundary conditions. Critical points in the solution (marking loss of stability and loss of criticality) are computed directly for the case of a self-heating slab with its upper and lower surfaces partially insulated, retaining the full Arrhenius rate term. For the special case of the Frank-Kamenetskii approximation it is shown numerically that the critical value of the rate parameter satisfies λ c (ϵ)=λ c (0)(1+O(ϵ 2)), where λ c(0) ≅ 0.878 and ϵ is the ratio of the insulation width to slab thickness.

Calculation of critical points and phase boundaries in the critical region

A computationally efficient, general method is presented for calculating the critical temperature and pressure of a multicomponent mixture of specified composition. The procedure suggested also provides coefficients α T , α P and u relating changes in temperature, pressure, phase fraction β and composition in the immediate vicinity of the critical point. These variations are shown to be of the general form T − T c = α T (1 − 2β)δ P − P c = α P (1 − 2β)δ ln K i = u iz i − 1 2 δ where δ is the distance parameter of the phase-boundary expansion. Finally, an explicit first-order approximation is developed, based on derivatives of the Gibbs-energy surface, givin the phase fractions and equilibrium phase composition for a mixture under near-critical conditions.

Calculation of critical branching points in two-parameter bifurcation problems

Calculation of critical branching points in two-parameter bifurcation problems

The calculation of critical points of fluid mixtures-effect of improved pure component critical point representation

The poor representation of mixture critical volumes by cubic equations of state is well known and has been widely attributed to the fact that the pure component critical compressibility calculated from the equation of state (designated as ζ ci) is, in general, not equal to the experimental critical compressibility (designated as Z ci expt) of most fluids. This work examines the behavior of two and three-constant cubic equations of state in the critical region and demonstrates that simply setting ζ ci equal to Z ci expt does not necessarily lead to more accurate prediction of mixture critical points. The improvement in the pure component critical point representation creates the necessity for an additional mixing rule which is not inherent in the usual mixing rules for the equation of state constants. The critical region behavior of two cubic equations of state was studied in this work. These equations, namely the Peng-Robinson and the Teja-Patel, were chosen as representative of two and three-constant equations of state.

Calculation of critical points from cubic two‐constant equations of state

A computational modification of the Heidemann-Khalil method for calculating the critical temperatures and pressures for general phase-equilibrium problems greatly reduced the computing time for simple two-constant cubic equations of state. For systems where the unlike binary interaction parameters can be derived from the pure-component parameters using the geometric mean values, a further simplification lowers the computing times to a few milliseconds, regardless of the number of components.

The calculation of critical points

AbstractGibb's classical theory of critical points leads to two simultaneous, nonlinear equations in the intensive variables of the critical phase. In this paper is presented a new procedure for evaluating the functions which appear in these nonlinear equations. The new procedure simplifies and permits the speeding up of the computation of critical points in multicomponent mixtures.Computations have been performed for critical points in binary and multicomponent mixtures described by the SRK equation. The methods developed will be equally applicable to other two‐constant equations of state. The equations to be solved are organized as two equations in the unknown critical temperature and specific volume for a mixture of known composition. One of the two equations, the determinant which establishes the stability limit for the mixture, is shown to be satisfied by more than one volume at a given temperature and by several temperatures at a given volume. A technique is proposed to assure that the correct temperature, volume solution can be found for this equation. For critical points in ordinary gas‐liquid systems, an overall computational procedure is suggested in which it proves to be unnecessary to provide initial guesses for either the temperature or the volume.Application has also been made to several systems with high density (liquid‐liquid) critical points.

Critical Point and Saturation Pressure Calculations for Multipoint Systems

Abstract Calculation of fluid properties and phase equilibria is important as a general reservoir engineering tool and for simulation of the carbon dioxide or rich gas multiple-contact-miscibility (MCM) mechanisms. Of particular interest in such simulations is the near-critical region, through which the compositional path must go in an MCM process. This paper describes two mathematical techniques that enhance the utility of an equation of state for phase equilibrium calculations. The first is an improved method of estimating starting parameters (pressure and phase compositions) for the iterative saturation pressure (bubble-point or dew-point) solution of the equation of state. Techniques previously have been presented for carrying out this iterative solution; however, the previously described procedure for obtaining initial parameter values was not satisfactory in all cases. The improved method utilizes the equation of state to estimate the parameter values. Since the same equation then is used to calculate the saturation pressure, the method is self-consistent and results in improved reliability. The second development is the use of the equation of state to calculate directly the critical point of a fluid mixture, based on the rigorous thermodynamic criteria set forth by Gibbs. 1 The paper presents an iterative method for solving the highly nonlinear equations. Methods of obtaining initial estimates of the critical temperature and pressure also are presented. The techniques described are illustrated with reference to a modified version of the Redlich-Kwong equation of state (R-K EOS); however, they are applicable to other equations of state. They have been used successfully for a wide variety of reservoir fluid systems, from a simple binary to complex reservoir oils.

Calculation of phase envelopes and critical points for multicomponent mixtures

Abstract An algorithm for the construction of complete vapour—liquid phase envelopes, capable of accurately determining the critical points and maxima in temperature and pressure is described. Recently developed methods for direct calculation of critical points are critically examined, and a computationally more convenient formulation is suggested.

New steam catapult for aircraft carrier

An algorithm for the construction of complete vapour—liquid phase envelopes, capable of accurately determining the critical points and maxima in temperature and pressure is described.Recently developed methods for direct calculation of critical points are critically examined, and a computationally more convenient formulation is suggested.