Gibbs Tangent Plane Criterion Research Articles

Effect of vapor liquid equilibrium on product quality and yield in oil shale pyrolysis

Separation of the products from the reacting organic matter is necessary to ensure product quality and reduce oil degradation to solid reside (coke) when considering conversion of oil shale to oil by in-situ processes. Many effects on yield and quality due to operational conditions may be explained by considering their role in volatilization. A five-reaction network with vapor–liquid equilibrium has been created in MATLAB to examine the effect of heating rate and pressure on volatilization. The model uses the Peng-Robinson equation of state, phase stability using the Gibbs tangent plane criterion, and Rachford-Rice flash calculations. Component compositions may be changed with automatically adjusting stoichiometry and species properties. A new feature is that dew and bubble point pressure calculations are performed at each calculation-step and plotted to show the phase behavior of the changing product composition with conversion. Product expulsion is represented as a mass fraction of generated vapor. The results indicate that for 10 °C/min and atmospheric pressure, 50% removal is enough to obtain a liquid yield that is 97% (977 mg/g TOC) of the maximum yield with 95% removal. The operational conditions with the lowest oil yield (127 mg/g TOC) are 0.001 °C/min and 500 psi with 5% vapor removed. Additionally, changing from high to low fractions of removed vapor move the maximum light oil yield from low pressure and low heating rate to high pressure and high heating rate.

Computation of Critical Points of Mixtures Using Particle Swarm Optimization with Low-Discrepancy Sequences

The prediction of critical points of thermodynamic systems is an important tool for modeling many high-pressure processes of theoretical and practical interest. In this article, the calculation of critical points of multicomponent mixtures is treated as a global minimization problem of a modified merit function associated with the criticality conditions obtained from the Gibbs tangent plane criterion, designed to discriminate the scale of the problem. The methodology used to solve the optimization problem is based on two versions of the particle swarm optimization (PSO), equipped with low-discrepancy sequences to prevent the sensitivity of the swarm with respect to the location of the initial population. To avoid a rapid decrease in the weight inertia, and to prevent stagnations near undesirable local minimizers, we present a modification of the PSO method, which uses different search cycles with the same inertia weight. This new version developed here is a fast and robust algorithm for solving the critical-point problem, via global optimization.

Prediction of Tricritical Points in Ternary Mixtures by Global Optimization

Tricritical points of mixtures are characterized by thermodynamic coordinates where three coexisting phases become identical. Here, we formulate and solve for the first time the tricritical-point calculation as a global optimization problem. Using finite-difference formulas and the ordinary criticality criteria, we develop novel numerical approximations for tricriticality conditions associated with the derivatives of the fourth and fifth orders with respect to the compositions, obtained from the Gibbs tangent plane criterion for phase stability. Such approximations simplified the numerical implementations of these two conditions, avoiding propagation errors related to floating-point arithmetic without requiring a code with excessive numerical precision, which usually leads to high computational costs. This paper deals only with ternary mixtures. Thus, the present formulation is described as a minimization problem with strict inequality constraints, having four variables (two independent mole fractions, te...

A deduction of the multicriticality conditions of mixtures from the Gibbs tangent plane criterion

Here, we follow a classification proposed by Griffiths and Widom [1], where the order of a multicritical point in a mixture is equal to the number of phases which simultaneously become identical, considering m phases and assuming that m−1 of these phases become identical to a given test phase. Thus, employing Rolle's theorem and basic properties of the so-called tangent-plane distance function, we develop a deduction of the multicriticality conditions of mixture from Gibbs tangent plane criterion, which relies on the principle of mathematical induction, being appropriate for any m≥2.

Bilevel optimization formulation for parameter estimation in liquid–liquid phase equilibrium problems

Excess Gibbs free energy models contain parameters which for a given mixture are estimated from measurements of phase-splits. Local composition models are very flexible and can accurately predict complex phase behavior. However, in many cases it has been reported that use of local composition models leads to prediction of more phase splits or more phases than measured, modeling homogeneous azeotropes as heterogeneous, etc. Here, a formulation is proposed that addresses these limitations of current parameter estimation methods. The formulation is based on a bilevel program, i.e., an optimization problem embedded in another one. Minimizing the error between model predictions and measurements gives the upper-level program. The lower-level programs are given by the minimization of the Gibbs free energy, or equivalently the satisfaction of the Gibbs tangent plane criterion. Each of the experiments is cast as a separate lower-level program. Additional requirements on the phase behavior of the system, such as enforcing the correct number of phase splits and the correct number of phases in each phase split, are similarly formulated as additional lower-level programs. Global optimization techniques are necessary even to obtain a feasible point since the lower-level programs are nonconvex. The proposed formulation is applied to problems from the literature, in which inappropriate fitting of the parameters of the non-random two-liquid (NRTL) model to experimental data has been reported to result in significant model errors, such as the prediction of an additional spurious phase split. The discussion is restricted to binary mixtures, however, the formulation can in principle be applied also to multicomponent mixtures.

Heterogeneous Nonequilibrium Mole Fraction Curve Maps

A strategy for drawing mole fraction curve maps for heterogeneous mixtures, using the nonequilibrium model that involves rigorous mass-transfer calculations, is presented. The calculations involve the use of rigorous stability analysis in the form of the Gibbs tangent plane criterion and nonequilibrium liquid-liquid calculations when the mixture is determined to be unstable. This is followed by a nonequilibrium bubble-point calculation. A comparison of the equilibrium and nonequilibrium mole fraction curve maps for the mixture of ethanol, benzene, and water is presented.

Prediction of critical points: A new methodology using global optimization

AbstractIn the present work we develop, implement, and analyze the results of a new methodology destined to calculate critical points of phase transitions. Our criticality conditions are developed from a slight modification of the Gibbs tangent plane criterion. We formulate the critical points calculation as an optimization problem. Such formulation offers many advantages, among them we can mention the capability of determining more than one critical point, the use of robust numerical methods, and the possibility of visualization of the critical phenomenon from a two‐variable (T and P) objective function. To solve the optimization problems we use the simulated annealing algorithm, a stochastic method whose convergence does not depend on initial guesses and gives preference to global optima. Several samples of mixtures were used to test the present methodology. The results of our simulations were compared, in a satisfactory way, with experimental data and with some results calculated in other approaches. © 2004 American Institute of Chemical Engineers AIChE J, 50: 1300–1314, 2004

Calculation of chemical and phase equilibrium based on stability analysis by QBB algorithm: application to NRTL equation

The Gibbs tangent plane criterion is of crucial importance to confirm the reliability of the solution obtained for the chemical and phase equilibrium (CPE) problem, and it consequently facilitates the search for the true equilibrium state if a postulated solution is thermodynamically unstable. However, the nonconvex and nonlinear natures of the thermodynamic models, which are the necessary conditions in order to describe the chemical and phase equilibrium problems, make the application of the deterministic global optimization techniques to minimize the tangent plane distance function (TPDF) become very difficult. In this paper, a general quadratic underestimation function based branch and bound (QBB) algorithm is developed by the construction of a rigorous underestimator for TPDF, which is the sum of the original convex part in the TPDF and a quadratic underestimation function of the generic nonconvex part upon the basis of the maximal eigenvalue estimation of its interval Hessian matrix. The linear constraints, especially the simplex feasible region, provide the novel compact partition in the above framework and mathematically guarantee the ε-global convergence of this global optimization algorithm together with the rigorous underestimator obtained by interval analysis. The phase equilibrium compositions are then calculated by the Newton–Raphson method. The preliminary results for three ternary systems with up to two or three phases described by NRTL activity coefficient equation showed that the novel QBB algorithm could solve the global stability problem effectively.

Extending the Gibbs tangent plane semicontinuous mixtures

The computation of phase equilibrium for semicontinuous mixtures results in a set of algebraic-functional equations which are commonly solved using the pseudocomponent approach or reducing the order of the functional equations to a set of algebraic equations by means of a Petrov–Galerkin approach. Both approaches can be taken as particular cases of the method of weighed residuals. In order to perform stability analysis of phase equilibrium, the classical tangent plane criterion algorithms can be used as the first approach. There is a lack of functional formulation for the stability criterion in the literature. In some special cases, such as mixtures containing continuous families of heavy-components, a new formulation of the tangent plane criterion is required. In the present work, functional extensions of the Gibbs tangent plane criterion are described. The use of the proposed criteria was exemplified by calculating solid–liquid phase equilibrium of a crude oil containing asphaltenes on titration with a precipitating agent, n-pentane.

Modeling of fluid multiphase equilibria in ternary systems of carbon dioxide as the near-critical solvent and two low-volatile solutes

A direct calculation method for critical endpoints has been developed on the basis of the algorithm for critical points from Heidemann and Khalil. Hereby, the Gibbs tangent plane criterion in the formulation of Michelsen was applied for the determination of the occurrence of an additional phase. Some results obtained with the Peng–Robinson equation of state for the ternary systems carbon dioxide+1-hexanol+tridecane and carbon dioxide+1-pentanol+tridecane are compared to experimental critical endpoint data. The difficulties encountered with variation of the binary interaction parameters are discussed. Qualitative agreement between the calculated and the experimental data was obtained.

Global optimization for the phase stability problem

AbstractThe Gibbs tangent plane criterion has become important in determining the quality of obtained solutions to the phase and chemical equilibrium problem. The ability to determine if a postulated solution is thermodynamically stable with respect to perturbations in any or all of the phases is very useful in the search for the true equilibrium solution. Previous approaches have focused on finding stationary points of the tangent plane distance function. Obtaining all stationary points, however, cannot be guaranteed due to the complex and nonlinear nature of the models used to predict equilibrium. Simpler formulations for the stability problem are presented for special problems where nonideal liquid phases can be adequately modeled using the NRTL and UNIQUAC activity coefficient equations. It shows how the global minimum of the tangent plane distance function can be obtained for these problems. A global optimization approach is advantageous because a nonnegative solution can be asserted to be the globally stable equilibrium one, unlike available local algorithms. For the NRTL equation, the GOP algorithm of Floudas and Visweswaran (1990, 1993) is used to guarantee obtaining ϵ ‐global convergence to the global minimum. For the UNIQUAC equation, a branch and bound algonthm based on that of Falk and Soland (1969) is used to guarantee convergence to the global solution. Computational results demonstrate the efficiency of both global optimization algorithms in solving various challenging problems.

A diverse approach for the solution of the isothermal multiphase flash problem. Application to vapor‐liquid‐liquid systems

AbstractThe paper presents a novel approach for the solution of the isothermal multiphase flash problem with particular application to systems exhibiting liquid‐liquid‐vapor equilibria. The approach includes a rigorous method for thermodynamic stability analysis as a first step and an efficient phase identification procedure. The stability analysis, exercised only once, uses a modification of the Gibbs tangent plane criterion. The identification procedure implements the results of the stability test in a sequence of liquid‐liquid and liquid‐vapor calculations only till the phase configuration with a minimum Gibbs energy is determined.The efficiency and reliability of the proposed new method is illustrated by solving three typical problems encountered in enhanced oil recovery, natural gas processing and petrochemical industry.

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.