Fluid Phase Equilibria at High Pressures - ACS Publications

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Fluid Phase Equilibria at High Pressures: Correlations and Predictions ULRICH K. DEITERS University of Bochum, Department of Chemistry, Bochum, Federal Republic of Germany

In fluid mixtures the limits between liquid-gas, liquid-liquid, and gas-gas equilibria are not clearly defined, and transitions occur at high pressures. This is demonstrated for binary mixtures of hydrocarbons with carbon tetrafluoride and for some inert gas mixtures. The experimental results are compared with calculations. These calculations make use of three mathematical relations: 1. an equation of state, from which the thermodynamic stability criteria are derived 2. mixing rules, which refer the characteristic parameters of a mixture to those of the pure substances 3. combining rules, which estimate binary interaction parameters from pure substance parameters These three relations are discussed; the reliability of the equation of state approach is demonstrated for several equations of state. The correlation of high pressure phase equilibria is shown to be a severe test for the quality of the mixing rules as well as for the usefulness of an equation of state.

A

S T E M P E R A T U R E AFFECTS T H E MOTION O F M O L E C U L E S , pressure affects

^ the average distances of molecules and therefore their average potential energy. Varying the pressure, in addition to varying the temperature, is therefore a second way to control the balance of kinetic and potential energies in a fluid system. This balance is of central importance for static as well as dynamic and transport properties. By varying the 0065-2393/83/0204-0353$06.00/0 © 1983 American Chemical Society In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.


354

MOLECULAR-BASED STUDY OF FLUIDS

pressure in supercritical fluid chromatography (SFC) (I) it is possible to affect activity and diffusion coefficients to obtain any intermediate state between gas chromatography and high pressure liquid chromatography. High pressure fluid extraction techniques permit the extraction of delicate organic substances without the need for high temperatures or toxic solvents (2, 3). Modern production of oil or natural gas is closely tied up with the understanding of high pressure phase equilibria. It has long been known that a rigid discrimination between vaporliquid equilibria and liquid-liquid equilibria cannot be maintained; investigations using high pressure techniques show continuous transitions between these two types of equilibria and eventually to a third type of fluid phase equilibrium, the so-called gas-gas equilibrium (4). A typical example of this class is shown in Figure 1. In the phase diagram of the system neon-krypton the critical line originating from the critical point of krypton shows a temperature minimum; for temperatures above this minimum phase separations can be achieved by raising the pressure of the system. In order to demonstrate the transitions between the three types of fluid phase equilibria, and in order to find correlations between equilibrium type and molecular parameters, several series of fluid systems have been investigated. Examples of this systematic research are studies of noble gas mixtures (5-7), methane-alkane mixtures (4), and carbon tetrafluoride-alkane mixtures (8). The critical lines of the latter systems are shown in Figure 2. With increasing chain length of the alkane component the liquid-liquid equilibrium critical line shifts more and more to higher temperatures, until it "overlaps" with the vapor-liquid equilibrium domain, thus giving rise to gas-gas equilibrium-like phase diagrams.

Thermodynamic Conditions Most methods for calculating phase equilibria are characterized by the use of activity coefficients by which the properties of a mixture are related to those of a perfect mixture or a perfect gas. These methods, while working very well and efficiently for low pressure vapor—liquid equilibria, are difficult to apply to high pressure phase equilibria because the Poynting corrections become very large and because—with supercritical components—no reference states of the pure component are available. In addition, critical coalescence of phases has to be accounted for. It is therefore advantageous to use one equation of state for the description of all phases of a fluid mixture, thereby assuming that the concept called the continuity of phases (9) holds. Because most equations of state are written as functions of molar volume and temperature, it is useful to regard the Helmholtz energy A as the central property of a mixture, from which all other properties may be derived. The Helmholtz energy

In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

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DEITERS

355

Fluid Phase Equilibria at High Pressures

MPa


150

100

50-

0U

*. 50

100

150

200 T/K

Figure 1. P-T diagram of the neon-krypton system. Key: —, critical line; —, vapor pressure curves; #, critical points of the pure substances; and +, experimental binary critical points. (Reproduced with permission from Ref. 18, Copyright 1982, Pergamon Press Ltd.)

of a binary mixture is given by the following equation (a detailed derivation is given elsewhere (10-12)) A = A?(V+,T) ni

+ n A (V-,T) 2

2

+

f' P dV JV

+

+ R T ^ ! ln x + n ln x ) x

2

2

(1)

The A terms denote the molar Helmholtz energies of the pure substances in the perfect gas state at temperature T and the very large volume V . The pressure P is given by the equation of state. The conditions of phase equilibrium are then represented by the f

+

+



Figure 2. P-T diagram of carbon tetrafluoride-alkane mixtures. Key: —, experimental critical lines; vapor pressure lines; O , critical points of the pure substances; and — , • , O , experiments and calculations of Mendonga (35). (Reproduced with permission from Ref. 8. Copyright 1982, Academic Press Inc. (London) Ltd.)


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357


following system of equations (different phases are denoted by ' and ")

(2)

?' = r 1 =


ix; = u.;'

1,2

The chemical potentials, u^, are obtained from the thermodynamic relation

* - (r)

»

3(

A binary critical point is defined by « „ \ dxf /

_

fe) \ dxf

Q

T P

.

0

( 4 )

This can be expressed in terms of A as =

~

^3x^-2v

~~ 3A A A

0

^2x^2v

v2x

vx

2c

(5) + 3A A* 2vx

x

—AAA 3v

2x

vx

= 0

Here each subscript xort; indicates a partial differentiation of the molar Helmholtz energy A with respect to V or x Once the dependence of the pressure on temperature, density, and composition is known, the determination of fluid phase equilibrium states or critical properties is accomplished by solving systems of nonlinear equations. Computer algorithms and the conditions under which Equations 5 hold and criteria for the elimination of physically unreasonable solutions are discussed elsewhere (10, 13). m

m

v

The Equation of State Any relation that permits the calculation of the pressure from density and temperature may serve as an equation of state. There are, however, several requirements that different PVT relations will meet to a different degree. Fluid mixtures are sometimes stable under conditions that cannot be approached by pure fluid substances. Calculation procedures based on the corresponding states principle or on a strictly empirical equation of state may be beyond their working ranges in such cases. Examples are mixtures of noble gases under high pressure. These mixtures can be



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in a fluid state, while a pure noble gas with the same reduced temperature and density would be a solid (14). Purely empirical equations of state are seldom valid beyond the range of density and temperature to which they have originally been fitted, and therefore cannot always be considered safe for high pressure calculations. On the other hand, there is as yet no purely theoretical equation of state with sufficient precision to cover wide ranges of temperature and density. It is possible, however, to combine the advantages of these two kinds of equations of state in so-called semiempirical equations. Even the simplest representative of this class, the van der Waals equation, is able to reproduce all of the kinds of fluid phase equilibria (15), although agreement with experimental data is qualitative only. Now, a large number of equations of state for the quantitative treatment of phase equilibria has become available, ranging from the rather simple Redlich-Kwong equation to the very sophisticated perturbed chain equation by Beret and Prausnitz (16). It is impossible to give here a complete list and evaluation of all equations of state; recently a comparison of cubic equations has been compiled by Peneloux (17), and of several noncubic equations by Vera and Prausnitz (18). The following equations of state have been used by us: 1. The equation of Redlich and Kwong (19) RT v

m

-

aT~ 0

b

V (V m

m

5

(6)

+ b)

2. The equation of Peng and Robinson (20) P =

_

v

« m

«n

i

- b

vjy

m

)

(7

+ b) + b(v

-b)

m

u

3. A new three-parameter equation of state derived from a square-well model of intermolecular interaction (12, 21)

° T=W

1 + cc

t (exp(t- ) V,2

Rab

eS

l

e

- 1)1,

(8)

In this equation £ denotes a reduced density, T a reduced effective temperature, and I a polynomial representing a first-order perturbation contribution. The repulsive part of Equation 8 is a Carnahan-Starling function with two modifications, which have been discussed in detail eff

x


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359


(21); c is a constant accounting for deviations from rigid core repulsion, and c is a shape parameter for nonspherical molecules. Equation 8 is shown to be valid even for pressures beyond 100 MPa for several simple molecules and to yield good vapor pressure data. In addition, it can be fitted to real critical compressibility factors (22). For calculations of properties of mixtures we assume that a mixture may be considered as a hypothetical pure substance, with the same equation of state as a pure substance, but with the parameters a and b (and c in Equation 8) depending on composition. The following mixing rules are used:


0

For the Redlich-Kwong and Peng-Robinson equations a = a xf + 2a x x n

l2

b = b xf

l

+ a xi

2

+ 2b x x

n

l2

l

^

22

+ b x

2

22

2

For Equation 8 a = xa x

u

+ xa 2

22

2x s q Aa l

1

2

b = b xf

+ 2b x x

n

C

l2

^l^i

l

- \)

"4* (-

4

p

+ b x

2

22

(1

o,

2

CX 2

2

In this equation the s terms denote contact numbers per molecule, and the q contact fractions. The harmonic mean of the s is s . The formula for a is an extension of the mixing functions for a strictly regular solution according to Guggenheim (23); it can be applied to mixtures of spherical molecules of different size. The contact numbers per molecule are not proportional to the "surface area" of a molecule, but roughly to the power 2.4 of the diameter. This is shown by studies of the maximum number of molecules that can be grouped around a central molecule of given size (24). The mixing rules, Equations 9 and 10, enable us to calculate the parameters of the equations of state for any composition of the mixture under consideration, provided that parameters for the pure substances and for unlike interaction (a , b ) are available. Pure substance parameters are calculated from the critical data or from vapor pressure data. {

i9

{

l2

l2

l2


360

M O L E C U L A R - B A S E D

STUDY O F

FLUIDS

The unlike interaction parameters are linked to the pure substance parameters by combining rules:

= (1 - i)\(b

b

12


2iS

u

=

S

l2

( 1

_

d

)

(11)

+ b) 22

/f!li ^ V

$1

(12) $2

(For the Redlich-Kwong and Peng-Robinson equations, s = s = 5 x

2

1 2

= i.) Parameters -& and £ are adjustable. Their values are calculated from one equilibrium state of the mixture under consideration. If £ is set to zero, b becomes the arithmetic mean of b and b , and thefo-mixingrules in Equation 9 and 10 degenerate to linear mixing rules. Linear mixing rules for b are widely adopted in literature, and are usually sufficient for vapor-liquid equilibrium calculations. The influence of deviations from linearity increases with density, however, and therefore a quadratic mixing rule is useful for calculations of high pressure fluid phase equilibria. It has been shown that the introduction of £ greatly improves the representation of critical curves using the Redlich-Kwong equation (10, 11). The mixing rules (Equation 10) have been specifically designed for spherical molecules. Mixing theories for more complicated systems have been discussed elsewhere (25, 26). l2

n

22

Application to Mixtures When several equations of state are to be compared, one must realize that most of the modern equations of state are of nearly equal precision when it comes to the calculation of vapor-liquid equilibria for mixtures of simple molecules. As examples we quote an experimental and computational investigation of the systems carbon dioxide-dimethyl ether (27) and methane-krypton (28). In both cases the Redlich-Kwong and Peng-Robinson equations and Equation 8 have been used to correlate the experimental vapor-liquid equilibria data (up to 5.2 MPa). The interaction parameters had been fitted to one isotherm. All three equations of state are able to represent this isotherm with 0.3% deviation in pressure, which is comparable to the scatter of the experimental data. The other isotherms could be predicted within 1% deviation in pressure by all equations of state; however, Equation 8 is shown to be superior for the prediction of supercritical phase equilibria. The real test for equations of state is the correlation of high pressure phase equilibria. Figure 3 shows three experimental isotherms of the



14.

DE ITERS


361

hydrogen-methane system together with calculated curves (29). Again, all equations of state have been fitted to the middle isotherm (for the Peng-Robinson equations and Equation 8 only ft has been adjusted), and the same set of parameters has been used to predict the other isotherms. Hydrogen causes special problems in calculations; because of quantum effects its pure substance parameters had to be extracted from PVT data rather than calculated from critical or vapor pressure data. Again it is evident from Figure 3 that with all equations of state a similar agreement between experimental and computed data is achieved

100 K

Figure 3. P-x diagram of the hydrogen-methane system. Key: O , +, experimental data; —, calculated with Equation 8; — , calculated with the Redlich-Kwong equation; and calculated with the Peng-Robinson equation.



362


at moderate pressures, although the Peng-Robinson equation proves to be superior to the Redlich-Kwong equation. At high pressures, only Equation 8 leads to an agreement with the experimental data. The Redlich-Kwong equation produces very large errors in pressure, and the Peng-Robinson equation has no solutions beyond 60 MPa. Similar results have been obtained for the systems hydrogen-carbon monoxide and hydrogen-carbon dioxide (30, 31). The gas-gas equilibrium in the system neon-krypton (5) is represented quite well by Equation 8. Calculated and experimental critical points agree very well (Figure 1). Although the interaction parameters for the calculation had been fitted to an equilibrium state at 178.15 K and 20 MPa, the critical double point is predicted correctly within 3 K. Figure 4 shows three isotherms of this system. The agreement with the experimental data is very good (18). The dashed curve in this diagram had been calculated without size and nonrandomness corrections to the mixing rule; the importance of these refinements to the mixing rule is evident. An especially interesting way of checking the validity of the equation of state approach is the study of series of mixtures, e.g., carbon tetrafluoride with a series of homologous alkanes (Figure 2). In this case, one expects a correlation for the interaction parameters with the chain length of the alkane. The first four critical curves in Figure 2 have therefore also been calculated with the Redlich-Kwong equation (Equation 6). The agreement of calculated and experimental data is very good; the curves virtually coincide for large pressure ranges (8, II). It must be noted, however, that the ft parameter for the RK calculation varies from 0.05 to the rather large value of 0.21 in the carbon tetrafluoride-alkane series, whereas it varies for Equation 8 only from 0.02 to 0.05. In the mixtures of carbon tetrafluoride with alkanes, a continuous transition from liquid-liquid equilibrium to a gas-gas equilibrium-like phase diagram takes place. A similar transition is found for carbon dioxidealkane mixtures (4). Transitions from gas-gas equilibrium of the second kind to gas-gas equilibrium of the first kind have been reported for wateralkane mixtures (32) or for helium-noble gas mixtures (33, 34). From a theoretical point of view, the use of adjustable binary interaction parameters might be considered as a weak point of the equation of state approach. There is indeed the danger that the adjustable parameters will not only take care of deviations from the Lorentz-Berthelot combining rules (Equations 11 and 12), but also absorb inadequacies of the mixing rules, the equation of state, and the thermodynamic assumptions inherent in Equation 1. This compensation effect, however, makes itself felt in physically unreasonable values of the adjustable parameters. In addition, the investigation of series of mixtures (as men-



14.

DEITERS


363

Figure 4. P-x diagram of the neon-krypton system. Key: —, calculated with nonrandom mixing rules; —, calculated with random mixing rules; + , experimental data at 133.16 K; O , at 163.15 K; and •, at 178.15 K. (Reproduced with permission from Ref. 18. Copyright 1982, Pergamon Press Ltd.)

tioned earlier) will not lead to useful correlations of these parameters. It is one of the advantages of Equation 8 over the Redlich-Kwong equation that its parameters are less prone to unrealistic variations. Furthermore, the adjustable parameters of Equation 8 are less temperaturedependent than those of the Redlich-Kwong equation. Equations of state must be regarded as useful tools for the calculation of high pressure phase equilibria. In spite of many improvements of the experimental techniques, calculations with equations of state have kept up with the precision of the experimental data and with the recent efforts to extend our knowledge of intermolecular interactions and statistical mechanics. It is to be hoped that they will keep up in the future.


364


Acknowledgment The author expresses his gratitude to G. M . Schneider for friendly support of this work and many helpful discussions.


Literature

Cited

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for review January 27, 1982. ACCEPTED for publication September

28, 1982.


Fluid Phase Equilibria at High Pressures - ACS Publications

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