Journal of Thermodynamics

Volume 2016, Article ID 3806364, 21 pages

http://dx.doi.org/10.1155/2016/3806364

## Kelvin’s Dissymmetric Models and Consistency Conditions of Multicomponent Gas-Liquid Equilibrium and Capillary Condensation

^{1}Laboratoire d’Energétique et de Mécanique Théorique et Appliquée, CNRS-UMR 7563, Université de Lorraine, 54500 Nancy, France^{2}Laboratory of Fluid Dynamics and Seismics, Moscow Institute of Physics and Technology (State University), Dolgoprudny 141700, Russia

Received 3 October 2015; Revised 4 January 2016; Accepted 5 January 2016

Academic Editor: Mohammad Al-Nimr

Copyright © 2016 Mikhail Panfilov and Alexandre Koldoba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To describe phase equilibrium of mixtures, we develop a nonclassical approach based on using different equations of state for gas and liquid. We show that not all the types of EOS are admissible but only those which verify some specific conditions of consistency. We developed the mathematical theory of this new approach for pure cores and for mixtures, in presence and absence of capillary forces, which leads to explicit analytical relationships for phase concentrations of chemical components. Several examples of comparison with experimental data for binary and ternary mixtures illustrate the feasibility of the suggested approach.

#### 1. Introduction

The objective of the present paper is to develop simplified mathematical models of phase equilibria which would significantly reduce computation time, while retaining an acceptable level of precision. This problem is highly pertinent for multicomponent two-phase or three-phase fluids. The results of such simulations provide information about the chemical composition of each phase at any pressure and temperature and the fractional volume of each phase.

Modern thermodynamic models are essentially based on cubic equations of state (EOS) such as those developed by Van-der-Waals, Peng-Robinson, or Redlich-Kwong [1–4]. They ensure high quality in descriptions of gas-liquid coexistence, which determines the success of the modern thermodynamic theory. At the same time, when such models are used to simulate fluid flow in geological oil, gas, or water reservoirs, more than the half of the computation time is spent on the thermodynamic block. The iterative numerical procedure suggested in [5, 6] in a form called “*the flash calculation*” must be done at each time step and at each space point which is highly time-consuming.

Consequently a lot of papers have been published with the objective of reducing the time of flash calculation by reducing the number of primary variables. Michelsen in [7] reduced the number of parameters to three and accepted zero values for all binary interaction parameters (BIPs). By only taking into account the BIPs between a single component and the system of all other components, the number of parameters was reduced to five in [8]. More recently, in [9] the authors suggested the split of the BIPs into two parameters using a simple quadratic expression.

Unfortunately the simplest approaches based on empirical Henry’s or Raoulth’s laws only permit the calculation of the composition of one phase and are not adapted for use with multiphase multicomponent systems.

Another way to reduce the computational time for the general coupled hydrodynamic-thermodynamic problem consists of splitting the thermodynamics and hydrodynamics. An example of such splitting is published in [10, 11], where the authors developed the so-called HT-splitting, an asymptotic procedure providing nonclassic thermodynamic equations completely independent of hydrodynamic equations. This approach has lead to a new kind of thermodynamic model made up of traditional algebraic relationships and new differential thermodynamic equations.

Completely different technique of splitting the thermodynamics and hydrodynamics was developed in [12] for mono-dimensional two-phase -component flow. By a specific replacement of variables, the conservation laws are transformed into a reduced closed auxiliary system containing just thermodynamic variables and one lifting hydrodynamic equation containing the phase saturation.

A cubic EOS has to be excessively complicated in order to be capable of capturing not only the single-phase behaviour of gas and liquid but also the transition two-phase zone. However, just within this zone the cubic EOS is not used: its complicated nonmonotonic behaviour is simply replaced by an isobaric straight line (Maxwell’s line) which corresponds to the true gas-liquid equilibrium. Then our logic was the following: it is not necessary to require that an EOS would describe the transition zone; then it is possible to describe liquid and gas by two different EOS, which may represent very simple equations. This becomes the significant source of simplifying equilibrium equations.

The system of equilibrium equations and two different EOS for gas and liquid will be called the dissymmetric equilibrium model.

The approach based on dissymmetric EOS is not new and is frequently applied in chemistry for various purposes. A lot of other papers exist when the dissymmetric models are applied to calculate some thermodynamic parameters of mixtures. Their application to calculate phase equilibria is less known. This was done first probably by Kelvin, who used the model of the ideal gas and an incompressible liquid to describe the equilibrium controlled by capillary forces for a monocomponent fluid. The dissymmetric EOS for two-component fluids was applied in [13] to calculate the two-phase envelop. In [14] the generalization of the Kelvin equation was obtained for multicompositional mixtures by applying the real-gas EOS and an incompressible liquid and by introducing the new concept of a “*mixed volume*.” The resulting equation is valid only near the dew point.

Another kind of dissymmetric models, known as the phi-gamma approach, has been used for decades in chemical engineering thermodynamics [15]. In this approach vapor is described by an EOS, while liquid is described by the activity coefficient. The model used for the activity coefficient represents some semiempirical approximations but never an EOS.

The main difference of the present paper from all other papers in which a kind of dissymmetric approaches was applied consists of the following:(i)Within the present paper we use two different EOS for gas and liquid to calculate phase equilibria.(ii)We have shown that not any couple of EOS is consistent with the two-phase equilibrium.(iii)We have determined the general theoretical constraints to the choice of two EOS consistent with the gas-liquid equilibrium.

In all the papers which used the dissymmetric approach the selection of the individual EOS is arbitrary and not constrained. However, once we tried to develop dissymmetric EOS models and to extend them to multicomponent mixtures, we found that several EOS lead to nonexistence of two-phase states. In particular, it is possible to show that the incompressible liquid is unable to capture two-phase states whatever the EOS for gas is.

The capacity of an EOS to capture two-phase states is called consistency conditions. Thus, the selection of the individual dissymmetric EOS for gas and liquid is not arbitrary but must satisfy the consistency conditions which represent mathematically the conditions of solution existence for the equilibrium equations.

The development of the consistency conditions and their use in order to determine the adjustable parameters of the dissymmetric EOS is the second key element of the paper.

The developed approach gives a form of equilibrium equations which can be solved analytically with respect to phase concentrations even for multicomponent nonideal systems, which significantly reduces the computational time.

#### 2. Principle Idea: One-Component Fluid

The best way of presenting the principles of our approach is to analyze a simple fluid consisting of one chemical component.

##### 2.1. Main Equilibrium Relationship

The gas-liquid equilibrium may be described using the system of one equilibrium equation, (1a), and two equations of state, (1b) and (1c), which have the following form:where is the pressure, is the temperature, is the molar volume, is the chemical potential, and are two arbitrary reference values of pressure, is the equilibrium pressure at given , and indexes and correspond to gas and liquid, respectively.

The definition of the chemical potential, the fluid volume, and the molar volume through the Gibbs energy iswhere is the number of moles of fluid.

*Proof of (1a). *The most general conditions of equilibrium for simple fluid (without capillarity) areUsing definition (2) and the equivalence, , we obtain which yields the well known explicit relationship for the chemical potential:Substituting (5) in the equilibrium conditions (3) we obtain (1a).

In system ((1a)–(1c)) the chemical potentials and are known at reference pressures and .

Then, for any , (1a) represents the nonlinear system of one equation with respect to which is the pressure of gas-liquid coexistence.

The classic approach consists of using the same EOS for all fluids, whereas our approach consists of using different EOS for gas and for liquid.

##### 2.2. Pseudoliquid and Pseudogas

Within the framework of the suggested approach, it is necessary to introduce the following definitions.

*The pseudoliquid* and* the pseudogas* are the hypothetical fluids which are authorized to exist at any and , with each of them being defined only by a formal EOS, (1b) or (1c).

This means that both pseudophases can coexist at any and , but this kind of coexistence is not proved by any equilibrium conditions.

Figure 1 presents two curves, ADE and ABM, which are the graphs of the EOS for pseudogas and pseudoliquid, respectively.