Thermodynamic consistency test of high pressure gas–liquid equilibrium data including both phases

Abstract

A method to test the thermodynamic consistency of high pressure gas–liquid equilibrium data in binary mixtures which considers data for a single phase has been extended to propose an overall test using the whole set of PTxy data. The method previously proposed by the authors was applied before to situations in which the concentration in the gas phase only was known (a solid dissolved in a high pressure gas) and to situations in which the concentration in the liquid phase only was known (ionic liquid and a high pressure gas). The extension is done here by proposing a combined analysis in which the test is applied to both phases while the modeling is done using bubble pressure calculations as recommended in the literature. Data for water + carbon dioxide mixtures at nine temperatures and for pressures ranging from 100 to 1500 bar and temperatures 383 to 598 K were used. Results indicate that the proposed method is reliable and can be used to check the thermodynamic consistency using all experimental phase equilibrium data available.

Introduction

The inaccuracies that arise in measuring experimental phase equilibrium properties has made it necessary to come up with methods to test inherent inaccuracies of such data. Although it is difficult to be absolutely certain about the correctness of a given set of experimental data, it is possible to check whether such data satisfy certain thermodynamic relationships, thereby establishing that the data are thermodynamically consistent or inconsistent. The thermodynamic relationship that is frequently used to analyze thermodynamic consistency of experimental phase equilibrium data is the fundamental Gibbs–Duhem equation. The Gibbs–Duhem equation relates the activity coefficients, the partial Gibbs free energy, or the fugacity coefficients of all components in a given mixture. Depending on the way in which the Gibbs–Duhem equation is handled, different consistency tests have been derived [1], [2], [3]. If the Gibbs–Duhem equation is not obeyed then the data are inconsistent and can be considered as incorrect. If the equation is obeyed, the data are thermodynamically consistent but not necessarily correct. More details and discussion on all these methods are given by Raal and Mühlbauer [4] and Poling et al. [5].

In previous communications the authors presented a thermodynamic consistency test and applied it to several situations: (i) Valderrama and Alvarez [6] applied the test to high-pressure phase equilibrium using the gas phase experimental data to determine the consistency; (ii) Valderrama and Zavaleta [7] applied the test to high pressure solid–gas equilibrium using the concentration of the solid in the gas phase to test consistency; (iii) Valderrama and Robles [8] applied the test to ternary solid + solid + gas phase equilibrium and used the concentration of the solids in the gas phase to determine consistency; Valderrama et al. [9] extended the method to binary mixtures containing an ionic liquid and high pressure CO₂ and used the concentration of CO₂ in the liquid phase to apply the test. These and other methods presented in the literature are described in Table 1.

As seen in Table 1 the different approaches presented in the literature not only use different equilibrium data but also different thermodynamic functions that are calculated either directly using experimental data or including thermodynamic models. For instance Mühlbauer [13], Jackson and Wilsak [14], Bertucco et al. [15] and Valderrama and Alvarez [6] use (P, y) data and fugacity coefficients. The more complex methods of Chueh et al. [10] and of Won and Prausnitz [11] both use (P, x, y) data and other derived thermodynamic properties (equilibrium ratios, saturation pressure, liquid volumes). In another approach, consistency tests that use the liquid phase concentration and that have been applied to high-pressure phase equilibrium were presented by Christiansen and Fredenslund [12] and more recently by Valderrama et al. [9]. The first method includes calculated variables such as excess enthalpy, liquid-phase volume and activity coefficients, while the latter uses the fugacity coefficient and compressibility factor of the liquid phase which are simultaneously calculated using an equation of state.

As explained in previous papers, the consistency method proposed by the authors can be considered as a modeling procedure. This is because a thermodynamic model that can accurately fit the experimental data must be used to apply the consistency test, based on the Gibbs–Duhem equation. The Gibbs–Duhem equation in terms of residual properties applied to any of the phases of a given mixture is [3]:

$$\sum {\xi }_{i}d\left[\frac{G^R}{RT}\right]=-\frac{H^R}{R{T}^2}dT+\frac{V^R}{RT}dP$$

$${\sum {\xi }_{i}d\text{Ln}\phi }_i=-\frac{H^R}{R{T}^2}dT+\frac{V^R}{RT}dP$$

In these equations, ξ_i is the concentration of component “i” in the liquid or gas phase, φ_i is the fugacity coefficient of component “i” in the corresponding phase, H^R is the residual enthalpy, V^R is the residual volume, T and P are the temperature and pressure of the system, respectively.

For high-pressure phase equilibrium, either gas–solid or gas–liquid, there are some general problems for testing experimental data: (i) the gas-phase non-idealities are important and a good model to evaluate the fugacity coefficients φ_i in eqn. (2) is needed; (ii) for isothermal data the term involving the residual enthalpy (H^R) vanishes, but the term involving the residual volume (V^R) cannot be ignored as done at low pressure; (iii) the data available do not cover the whole concentration range for both, the liquid and the gas phase in gas–liquid mixtures or the solid and the gas phases in gas–solid mixtures; and (iv) the concentration of one of the components in one of the phases is usually low and commonly unknown.

In some cases simplifications can be introduced to derive a consistency test that makes use of the incomplete data available. For instance in supercritical fluid applications such as extraction of substances from a liquid solution, the solute concentration in the gas phase is low while the gas concentration in the liquid phase could reach high values (PTx data available). In supercritical fluid applications such as extraction of substances from a solid matrix, the solute concentration in the gas phase is low while the gas concentration in the solid phase is negligible (PTy data available). In some situations, however, the concentration of all components in the different phases cover wider ranges of concentration and the whole set of data (PTxy) should be used to define a reasonable consistency test.

The authors have established certain requirements to define a good consistency test method to analyze high-pressure phase equilibrium data [7]. The test should fulfill the following ten basic requirements: (i) use the Gibbs–Duhem equation; (ii) use the fundamental equation of phase equilibrium, that is the equality of fugacities of a component in all phases; (iii) use for testing, all the experimental PTxy data available; (iv) does not necessarily require experimental data for the whole concentration range and be applicable for data in any range of concentration; (v) be able to correlate the data within acceptable limits of deviations, deviations that must be evenly distributed; (vi) requires few calculated properties; (vii) be able to detect erroneous experimental points; (viii) makes appropriate use of necessary statistical parameters; (ix) be simple to be applied, considering the complexity of the problem to be solved; and (x) be able to conclude about consistency if the defined criteria are not fulfilled.

The method proposed by the authors fulfills these basic requirements and can conclusively determine the consistency or inconsistency of data in most cases, as demonstrated in previous works. However, the method has not been used to test data using the whole set of PTxy data, as done in this paper.