High-Pressure Fluid Phase Equilibria

Abstract

One of the crucial aspects in the design of processes of this millennium is the use of environmentally benign technologies. The introduction of supercritical fluids (SCF) and, in addition, their use with other solvents, such as ionic liquids, further diversify the applications of these fluids. SCF are powerful solvents with many unique properties. They have the mobility of gases and the dissolving power of liquid solvents, resulting in efficient high mass transfer rates and penetration into porous matrices. However, reliable and versatile mathematical models of phase equilibrium thermodynamics are needed for use in process design and viability studies. This chapter reviews experimental procedures for obtaining high-pressure phase equilibria data. In addition, phase diagrams describing binary mixtures and thermodynamic models capable of determining the conditions at phase equilibria at high pressures are considered.

1. Introduction

Phase behavior of mixtures is essential in supercritical fluid (SCF) process design. In these processes, feasibility and optimal conditions can only be established if phase equilibrium and solubility data are available. For this reason, there are many studies that have been carried out to elucidate the phase behavior in systems involving SCF and, particularly, carbon dioxide + solute systems. Numerous experimental methods for investigating high-pressure phase equilibria have been described and reviews since the 80s to the present day have been complicated. These reviews include the experimental method used and data from vapor-liquid equilibria (VLE), liquid-liquid equilibria (LLE), vapor-liquid-liquid equilibria (VLLE), solubility of high boiling point substances in SCF, and gas solubility in liquids [1, 2, 3, 4, 5, 6].

Although the effectiveness of SCF such as supercritical water and supercritical carbon dioxide in variety of process is very promising, any industrial supercritical application to be designed depends on the possibility of modeling and predicting the phase equilibria in the systems involved. Only through this knowledge, engineers will be able to estimate feasibility and choose conditions to optimize the process. In the same way, processes with little chance of success can be identified and rejected. Therefore, the design of supercritical processes requires a sufficiently detailed understanding of the actual molecular process in SCF mixtures, and using this knowledge to search for correlations and develop reliable, versatile predictive models. However, this task is not easy. There are some aspects of SCF behavior that make them especially difficult to handle in the lab and to model. One of them is derived from the operating conditions to achieve the supercritical state, the other is the proximity of the conditions to the critical point, and another, the asymmetry of most of the SCF systems of interest is in terms of size and the attractive forces of the molecules involved.

The purpose of this chapter is to achieve a brief review of the experimental and analytical procedures in phase equilibrium thermodynamics of SCF systems.

2. Experimental acquisition of high-pressure fluid phase equilibrium data

This section consists of a brief review of the techniques for measuring the high-pressure phase equilibria. A large number of experimental methods have been developed for the measurement of the phase equilibrium of fluids at high pressure [5]. This is due to high-pressure phase behavior is often complex and difficult to predict, and hence, no single method is appropriate to study all different systems. As a general classification, the used methods can be divided into analytical or direct methods, and synthetic or indirect methods, depending on how the composition of the phases is measured. Some extensive reviews on the experimental methods used in previous works can be found in the literature [3, 4, 5].

2.1 Analytical methods

Analytical methods involve the analytical determination of the compositions of the co-existing phases. They can be subdivided into static, recirculating, and flow methods, depending on the technique used to achieve phase equilibrium.

The static method is represented in Figure 1 [7]. A liquid-vapor mixture is inside the cell, whose pressure and temperature are controlled. A stirring system is normally used to facilitate phase equilibrium. Once this equilibrium is reached, small samples are taken from both the vapor phase and the liquid phase and their composition is analyzed. The main drawback of this method is that the sampling can alter the equilibrium of the phases. Static analytical methods have been widely used in the literature for measuring the phase equilibria [8, 9, 10].

Figure 1.

Static method.

The second analytical method is the recirculation method, which is shown in Figure 2. In this method, the liquid phase and the vapor phase are recirculated in parallel flow to achieve a better mixing between phases and to guarantee that the phase equilibrium is reached. It uses a closed cell similar to that of the static method. The composition of the phases is determined by collecting the corresponding samples through the sampling valves that are on-line connected to an analysis equipment. Then, main drawbacks of this method are that undesirable pressure gradients across the equilibrium cell can be provoked by the circulation pump and the need for a uniform temperature field to avoid partial condensation or vaporization in the recirculation lines [5]. This method has been widely described in previous works [11, 12, 13].

Figure 2.

Recirculation method.

In the flow method, which is represented in Figure 3, high-pressure pumps are in charge of feed the preheated components into a static mixer where the equilibrium is attained. The feed stream from the mixer is separated into a vapor and liquid phases in an equilibrium cell. A liquid or heavy phase is continuously withdrawn from the bottom of the equilibrium cell, while the vapor or light phase is withdrawn from the top. Both phases are then depressurized, properly collected, and analyzed. Flow methods have the advantage that sampling does not alter the equilibrium. In addition, large quantities of sample for analysis can be generated when components are in very low concentrations because the run time of the experiment can be extended to accumulate more material. This method is preferred when working with compounds that are temperature-sensitive due to the short residence time of the components in the equipment. As a drawback, this method can be only used for systems that need short times to achieve phase equilibrium [5]. Several authors have used flow methods to obtain equilibrium data [14, 15, 16].

Figure 3.

Flow method.

2.2 Synthetic methods

Synthetic methods are based on the preparation of a mixture of precisely known composition, the observation of the phase behavior in an equilibrium cell, and the measurement of the properties in the equilibrium state, for example, pressure and temperature. Therefore, no sampling is necessary and hence, no alteration of the equilibrium can occur [5]. Figure 4 shows an example of the experimental device.

Figure 4.

Scheme of a possible experimental device for a synthetic method.

Synthetic methods can be used for systems with or without phase transition:

• Synthetic methods with phase transition: Temperature and pressure values are adjusted so that the mixture is homogeneous, that is, there is only a phase. After a variation of pressure or temperature, the formation of a new phase can be observed. The new phase can be detected by visual observation [17, 18, 19] or by changes in certain physical properties [20, 21, 22].

• Synthetic methods without phase transition: These methods are less frequent than synthetic methods with phase transition. Equilibrium properties such as pressure, temperature, phase volumes, and densities are measured and phase compositions are calculated using the material balance. They can be isothermal or isobaric [23, 24, 25].

Synthetic methods can be used when the analytical methods are not suitable. For instance, when the coexisting phases have a similar density, as occurs in the critical region [3]. However, visual observation of a new phase is difficult for those systems in which both phases have approximately the same refractive index [26]. This method is not practical for multicomponent systems because the tie lines cannot be determined without carrying out additional experiments [5]. The great advantages of synthetic methods are that the experimental procedures are usually easy and quick and that analytical equipment and complex sampling are not required.

3. Classification of the phase diagrams for binary mixtures

When a system consists of more than one substance, it is important to determine how the composition of the phases in equilibrium varies with temperature, pressure, and/or the initial composition of the system. These changes are key to carry out the well-known unit operations used industrially, such as distillation and extraction.

In isolated systems without applied external fields, composite systems tend to homogenize (although if the limit of stability is reached, separate phases of different composition are segregated, but always of homogeneous chemical potential in equilibrium). Whether a binary mixture remains a stable homogeneous fluid or is divided into two or more phases in equilibrium is determined by its thermodynamic stability. A mixture can be considered stable when its Gibbs- or Helmholtz-free energy is at a minimum.

In the case of two component systems, three variables are required to graphically represent the stability field of a homogeneous region (single-phase), which makes necessary a three-dimensional diagram. For convenience, a constant variable (pressure, temperature, or composition) is usually maintained and two-dimensional phase diagrams are represented, which are the cross sections of the three-dimensional representation.

The phase diagram to describe binary mixtures depends on the behavior of the species. At high pressures, a wide variety of phase behaviors can occur. Van Konynenburg and Scott [27] classified the behavior of the phases of binary mixtures into six types of pressure-temperature diagrams, considering van der Waals equation of state and quadratic mixing rules. Figure 5 presents the different types of phase diagrams [28].

Figure 5.

Different types of phase behavior in binary fluid systems. C: Critical point, V: Vapor, L: Liquid, C1: Critical point of the most volatile component, C2: Critical point of least volatile component, LCEP: Lower critical end point. UCEP: Upper critical end point. Dashed curves are critical [28].

The Type I diagram (Figure 5) represents a phase behavior that frequently occurs when the two components of the mixture have critical properties of the same magnitude or a similar interaction energy and molecular size [29]. It is characterized by a continuous critical site between the two critical points of the two pure components and does not show any region of liquid-liquid immiscibility between the components [7]. An example of binary mixture showing type I behavior was found by Wei et al. [30] for the mixture of methane + ethane, in which both components are similar nonpolar molecules.

The Type II phase behavior is similar to Type I, but presents a zone of liquid-liquid immiscibility at low temperatures. As can be seen in Figure 5, there is a vapor-liquid-liquid (VLLE) line, along which the three phases (two liquids and one vapor) are in equilibrium, whose end is called the upper critical end point (UCEP). At that point, the two liquid phases come together and merge into a single liquid phase. The end of the liquid-liquid line is called the upper critical solution temperature (UCST), where the two liquids come together to form a single liquid phase when the system temperature increases. An example of this type of phase behavior can be found in the carbon dioxide-1-butanol system [31, 32]. The authors found that CO₂ solubility in 1-butanol decreases with increasing temperature, while increasing when pressure raises, although the solubility of 1-butanol in CO₂ is relatively low and shows no major changes as temperature or pressure varies, except in the vicinity of the critical point. It was found that vapor-liquid equilibrium exists in a large range of experimental conditions and that the liquid-liquid-vapor zone ends at an UCEP at 22.99 bar and 259.25 K.

Type III phase behavior is frequently found in mixtures of components with high immiscibility. In these systems, the liquid-liquid line rises to higher temperatures and eventually intersects with the vapor-liquid curve, resulting in a discontinuous critical curve. This critical curve begins at the critical point of the least volatile component C2 and spread to higher pressures, while the natural change from vapor-liquid to liquid-liquid occurs. The other critical curve runs from the critical point of the most volatile component C1 to the UCEP, in which the liquid and vapor phases are critically linked in a single fluid phase in the presence of another liquid phase. An example of the type III phase behavior is the carbon dioxide-1-hexanol mixture [33]. This binary system exhibits a phase behavior with three critical curves: a critical vapor-liquid curve that starts from the least volatile component, that is, 1-octanol, and continues with another liquid-liquid curve toward higher pressures, and the third, again a critical vapor-liquid curve, starting from the most volatile component, that is, CO₂, and ending at UCEP, where intersects with the three-phase equilibrium curve (VLLE).

Type IV and type V phase behaviors are quite similar to types II and I, respectively [34]. The main difference lies in the liquid-vapor critical line that no longer continuously joins the critical points of pure components. The critical line starting from pure component two ends in a three-phase line from which a second critical line arises, which is connected to the critical point of the lightest pure component 1. At the intersection between the three-phase line and the critical lines are two critical end points: The one with the lowest temperature is called the lower critical end point (LCEP) and another with the highest temperature UCEP. These phase behaviors usually occur when the critical properties of the two components are very different because there are large differences in structure, molecular size, or intermolecular forces. An example of Type IV phase behavior has been shown by Kodama et al. [35] for the mixture ethane +1-butanol. They studied the phase equilibria and saturated densities of this system at high pressures using a static circulation apparatus at 313.15 K. The CO₂ + nitrobenzene binary system can be mentioned as example of type V phase behavior. Hou et al. [36] studied this system in a high-pressure variable volume view cell using an analytical method. Phase boundaries were measured at temperatures of 298.15, 310.45, and 322.75 K under pressures between 2.76 and 12.83 MPa and they found that three-phase equilibria exist in a temperature range of 303.60 to 313.65 K. Experimental data could be correlated with the Peng-Robinson equation of state (PR EoS) and two binary parameters.

Type VI phase behavior is presented by some rare fluid mixtures. They are characterized because they show a three-phase line that begins in an LCEP and ends in a UCEP. The two critical end points are connected by a critical liquid-liquid curve showing an elliptical minimum-pressure critical point. This type of phase behavior is also characterized by a continuous vapor-liquid critical locus extending between the two critical points of pure components. The behavior of type VI is not derived from the Van der Waals equation so van Konynenburg and Scott [27] did not include it in their initial classification.

Konynenburg and Scott’s classification was presented as a succession of unrelated phase diagram types; that is, a given binary system belongs to a single type (from I to VI) of fluid phase behavior. However, after the publication of Van der Waals’ thesis, it is known that there is a continuous path between liquid and vapor; therefore, there must be a continuous path between the six types of phase behavior. Transitions from one type to another are often observed experimentally in homologous series. For example, systems (CO₂ + n-C12), (CO₂ + n-C13), and (CO₂ + n-C14) exhibit phase diagrams of type II, type IV and type III, respectively. Then, there is a relationship between the size difference and the interactions and the type of resulting phase behavior. These phase transitions can be reproduced by an equation of state and modeling homologous series; however, this is not the best option because the transition from one type to another would be observed for a number of carbon atoms, which would not be an integer. To obviate this problem, it is better to choose a well-defined binary system and vary the binary interaction parameter between the components. In this way, the interactions between the two molecules change and the transitions between types of phase behavior can be continuously observed [28].

Figure 6 illustrates the evolution of phase diagrams. The transition between each type of phase diagram can be explained by considering the size effects of molecules and the repulsive interactions between them.

Figure 6.

Possible transitions between the different types of phase diagrams. Adapted from [28].

4. Thermodynamic modeling of high-pressure fluid phase equilibria

If a liquid mixture is in equilibrium with a vapor mixture being both at the same temperature (T) and pressure (P), for every component i in a mixture the condition of thermodynamic equilibrium is given by the following expression [37]:

$${\widehat{f}}_i^V={\widehat{f}}_i^L$$E1

where ${\widehat{f}}_i^V$ and ${\widehat{f}}_i^L$ are the fugacities of the component i in the vapor and liquid phases, respectively.

Fugacity can be expressed in terms of the fugacity coefficient so Eq. (1) can be expressed also as follows:

$$\begin{array}{c}{y}_i{\widehat{\varphi }}_i^{V}P=x_i{\widehat{\varphi }}_i^{L}P\end{array}$$E2

where $y_i$ and $x_i$ are the mole fractions of the component i in the vapor and liquid phase, respectively. The fugacity coefficients of the liquid and vapor phases can be calculated from the following thermodynamic expression [37]:

$$\begin{array}{c}\mathit{ln}{\widehat{\varphi }}_i=\frac{1}{\mathit{RT}}{\int }_V^{\infty }\left[{\left(\frac{\partial P}{\partial n_i}\right)}_{T,V,n_{i\ne j}}-\frac{\mathit{RT}}{V}\right]\mathit{dV}-\mathit{ln}\left(\frac{\mathit{Pv}}{\mathit{RT}}\right)\end{array}$$E3

where R is the gas constant, V is the total volume of the phase, v is the molar volume of the phase, and n_i and n_j are the moles of components i and j, respectively. The fugacities can be calculated from a simple equation of state.

4.1 Equations of state

An equation of state (EoS) is an algebraic relation between P, V, and T, which serves for describing some behavior of Nature. In this section, we will discuss the most common EoS models. The well-known van der Waals equation of state (vdW EoS) was proposed in 1873 and was the first equation able to represent liquid-vapor coexistence to binary mixtures [38]:

$$\begin{array}{c}P=\frac{\mathit{RT}}{v-b}-\frac{a}{v^2}\end{array}$$E4

The wdW EoS is a cubic EoS based on molecular interactions of attraction and repulsion. It can quantitatively predict most of the phase equilibrium behaviors exhibited by binary mixtures (39). However, the calculations are quantitatively inaccurate. Because of this, a large number of modifications to the vdW EoS have been developed, some of which (the most important) will be reviewed here.

For high-pressure phase equilibrium, the most widely used equations are the Peng-Robinson (PR) [40] and the Soave-Redlich-Kwong (SRK) [41] EoS.

The PR EoS has the following form for pure components:

$$\begin{array}{c}P=\frac{\mathit{RT}}{v-b}-\frac{a\left[T\right]}{v\left(v+b\right)+b\left(v-b\right)}\end{array}$$E5

where P, T, v, and R are the pressure, temperature, molar volume and the ideal gas constant, respectively, while a[T] and b are the PR constants that are obtained from the critical temperature (T_c), critical pressure (P_c) and acentric factor (ω) as describe below:

$$\begin{array}{c}a\left[T\right]=0.45724\left(\frac{R^2{T}_c^2}{P_c}\right)\alpha \left[T_r\right]\end{array}$$E6

$$\begin{array}{c}{T}_r=\frac{T}{T_c}\end{array}$$E7

$$\begin{array}{c}\alpha \left[T_r\right]={\left(1+m\left[1-T_r^{0.5}\right]\right)}^2\end{array}$$E8

$$\begin{array}{c}m=0.37464+1.54226\omega -0.26992{\omega }^2\end{array}$$E9

$$\begin{array}{c}b=0.0778\left(\frac{R{T}_c}{P_c}\right)\end{array}$$E10

The above equations closely reproduce vapor-liquid equilibria measurements for a number of different systems.

For its part, the SRK EoS is defined by the following equations:

$$\begin{array}{c}P=\frac{\mathit{RT}}{v-b}-\frac{a\left[T\right]}{v\left(v+b\right)}\end{array}$$E11

$$\begin{array}{c}a\left[T\right]=0.42747\left(\frac{R^2{T}_c^2}{P_c}\right)\alpha \left[T_r\right]\end{array}$$E12

$$\begin{array}{c}m=0.48+1.574\omega -0.176{\omega }^2\end{array}$$E13

$$\begin{array}{c}b=0.08664\left(\frac{R{T}_c}{P_c}\right)\end{array}$$E14

where T_r and α[T_r] are calculated with Eqs. (7) and (8), respectively.

SRK and PR EoS are able to precisely model phase equilibrium based solely on the critical properties and acentric factors of the pure compounds. As a drawback, both EoS have limitations for predicting the densities of saturated liquids and low accuracy when working with highly polar systems.

Some years later, Stryjek and Vera (SV) [42] developed a variation of PR EoS by modifying the temperature-dependent function, α[T_r], to broaden the range of application to polar compounds. The variation is shown in Eq. (15):

$$\begin{array}{c}m=\left(0.379+1.490\omega -0.171{\omega }^2+0.020{\omega }^3\right)+{\kappa }_1\left(1+{\left(\frac{T}{T_c}\right)}^{0.5}\right)\left(0.7-\left(\frac{T}{T_c}\right)\right)\end{array}$$E15

where ${\kappa }_1$ is an adjustable parameter, which is characteristic of each pure compound.

Patel and Teja (PT) also [43] proposed a new cubic EoS that reproduce the good features of the SRK and PR EoS for nonpolar fluids but, at the same time, overcomes some of the limitations of these equations for polar systems such as heavy hydrocarbons. The PT EoS is shown in Eq. (16):

$$\begin{array}{c}P=\frac{\mathit{RT}}{v-b}-\frac{a\left[T\right]}{v\left(v+b\right)+c\left(v-b\right)}\end{array}$$E16

where

$$\begin{array}{c}a\left[T\right]={\Omega }_a\left(\frac{R^2{T}_c^2}{P_c}\right)\alpha \left[T_r\right]\end{array}$$E17

$$\begin{array}{c}b={\Omega }_b\frac{R{T}_c}{P_c}\end{array}$$E18

$$\begin{array}{c}c={\Omega }_c\frac{R{T}_c}{P_c}\end{array}$$E19

$$\begin{array}{c}{\Omega }_c=1-{\zeta }_c\end{array}$$E20

$$\begin{array}{c}{\Omega }_a=3{\zeta }_c^2+3\left(1-2{\zeta }_c\right){\Omega }_b+{{\Omega }_b}^2+1-3{\zeta }_c\end{array}$$E21

and ${\Omega }_b$ is the smallest positive root of the following equation:

$$\begin{array}{c}{{\Omega }_b}^3+\left(2-3{\zeta }_c\right){{\Omega }_b}^2+3{{\zeta }_c}^2{\Omega }_b-{{\zeta }_c}^3=0\end{array}$$E22

For $\alpha \left[T_r\right]$, the same function of reduced temperature that was described for the SRK and PR was used (see Eq. (8)).

$$\begin{array}{c}{\zeta }_c=\frac{P_c{V}_c}{R{T}_c}\end{array}$$E23

The critical compressibility factor ${\zeta }_c$ is treated as an empirical parameter. The optimal values of ${\zeta }_c$ and m correspond to the minimum deviation in saturated liquid densities and the equilibrium condition of equality of fugacities. Therefore, the proposed equation using optimum values of ${\zeta }_c$ and m can provide good prediction of liquid phase densities and vapor-liquid equilibria [43]. For nonpolar substances, ${\zeta }_c$ and m are calculated as follows:

$$\begin{array}{c}{\zeta }_c=0.329-0.077\omega +0.021{\omega }^2\end{array}$$E24

$$\begin{array}{c}m=0.452+1.310\omega -0.296{\omega }^2\end{array}$$E25

4.2 Mixing rules

In order to adapt the above EoS to a binary mixture, the characteristics parameters a, b, and c must be obtained using mixing rules. Commonly, the “one fluid” wdW mixing rules are used when the mixture is assumed to behave as a pure component with appropriate parameters. It can be expressed as follows:

$$\begin{array}{c}{a}_{\mathit{mix}}={\sum }_i{\sum }_j{x}_i{x}_j{a}_{\mathit{ij}}\end{array}$$E26

$$\begin{array}{c}{a}_{\mathit{ij}}=\sqrt{a_i{a}_j}\left(1-k_{\mathit{ij}}\right)\end{array}$$E27

where a_ij is a cross-parameter and k_ij is the binary interaction parameter obtained from the correlation of the experimental data.

$$\begin{array}{c}b={\sum }_i{x}_i{b}_i\end{array}$$E28

$$\begin{array}{c}c={\sum }_i{x}_i{c}_i\end{array}$$E29

For normal fluids, the above equations can give reasonable correlation with the experimental data. However, modifications of the wdW mixing rules for polar or highly non-ideal compounds are required. For instance, Panagiotopoulos and Reid [44] defined a two-parameter mixing rule, for the a_ij term and another binary interaction parameter k_ji (Eq. (30)):

$$\begin{array}{c}{a}_{\mathit{ij}}=\sqrt{a_i{a}_j}\left(1-k_{\mathit{ij}}+\left(k_{\mathit{ij}}-k_{\mathit{ji}}\right)x_i\right)\end{array}$$E30

The Panagiotopoulos and Reid mixing rule and others [42, 45] allow that the EoS would be adapted at different compositions. More recently, Mathias et al. [46] proposed the well-known Mathias-Klotz-Prausnitz (MKP) mixing rule that is shown below:

$$\begin{array}{c}a={\sum }_i{\sum }_j{x}_i{x}_j\sqrt{a_i{a}_j}\left(1-k_{\mathit{ij}}\right)+{\sum }_i{x}_i{\left.{\left({\sum }_j{x}_j\right.\left(\sqrt{a_i{a}_j}\right)}^{\frac{1}{3}}{l}_{\mathit{ji}}^{\frac{1}{3}}\right)}^3\end{array}$$E31

where k_ij is a symmetric binary parameter and l_ji is antisymmetric.

With this approach, the MKP mixing rules has contributed to a better representation of the phase behavior of highly non-ideal mixtures and systems formed by two very similar components and a very different third one [46].

5. Conclusion

From the review of experimental methods and thermodynamic models to describe the behavior of high-pressure systems, it can be concluded that the applicability of SCF depends largely on the availability of models that allow us to predict phase equilibria in the systems involved. There is an extensive amount of experimental data on high-pressure phase equilibria in the literature, although SCF systems are particularly difficult due to the high compressibility and asymmetry of most systems of interest.

A large number of experimental methods have been developed for the measurement of the phase equilibria of fluids at high pressure, both analytical, involving the analytical determination of the compositions of the co-existing phases, and synthetic, which are based on the preparation of a mixture of precisely known composition, the observation of phase behavior in an equilibrium cell, and the measurement of properties in the equilibrium state.

For high-pressure binary systems, a wide variety of phase behaviors can occur. Van Konynenburg and Scott’s classification (six types of pressure-temperature diagrams, considering the van der Waals equation of state and the rules of quadratic mixture) have been used to analyze the phase behavior of binary mixtures.

An equation of state (EoS) is an algebraic relation between P, V, and T, which serves for describing some behavior of Nature. The most common EoS models have been described in this chapter. The cubic wdW EoS, based on molecular interactions of attraction and repulsion, can quantitatively predict most of the phase equilibrium behaviors exhibited by binary mixtures. Mixing rules are necessary to adapt EoS to binary mixtures.