ELSEVIER Fluid Phase Equilibria 140 (1997) 87-95

On volume translations in equations of state

M.S. Zabaloy, E.A. Brignole *

PL.APiQUI/ UNS/ CONICET, CC 717, Bahia Blanca 8000, Argentinu

Received 18 November 1996; accepted 20 June I997

Abstract

Translations are often applied on equations of state with the aim of improving the description of molar volumes. In the present note an alternative presentation of the ideas developed in the pioneering work of Ptneloux and Rauzy [PCneloux, A., E. Rauzy, A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilib., 8 (1982) 7-23.1 is provided and some issues are discussed in a deeper level of detail. 0 1997 Elsevier Science B.V.

Key~~wds: Model; Equation of state; Volume; Vapor pressure; Mixture: Translations

1. Introduction

In the present work we consider the pioneering contribution of PCneloux and Rauzy (1982), on translated equations of state. These equations make possible to separate the problem of reproducing the experimental relation between equilibrium temperature, pressure and phase compositions from the problem of reproducing the experimental phase densities. The article of PCneloux and Rauzy (1982), by showing the invariance of the phase equilibria conditions under volume translations, has had much influence on the further development of equations of state. However, in our opinion, some steps in the original presentation were not sufficiently specified, e.g., the application of the general theory to the case of Van der Waals type equations of state. Besides, since the appearance of the PCneloux and Rauzy ( 1982) work, some researchers have applied the volume translation approach without using the original formalism. These different ways of presenting translated equations of state may lead to confusion. Therefore, the goals of this note are to propose a somewhat alternative more detailed derivation of the results, to discuss some points at a greater level of detail and to show the consistency between the original and further formalisms. The present communication may help other researchers in the field to avoid errors in the development of applications of the volume translation approach.

* Corresponding author

0378.3812/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. /‘II s037x-~x12(97)00185-4

88 M.S. Zubaloy, E.A. Brignole/ Fluid Phase Equilibria 140 (1997) 87-95

2. Effect of translations performed along the volume axis

In the following derivation we avoid the introduction of the concepts of pseudo volume and pseudo equation of state used in the original derivation by Peneloux and Rauzy (1982). We instead reach their conclusions by comparing two different interrelated models.

Let us compare the phase equilibria predictions of two different models, A and B. Both of them satisfy the following equation for mixtures:

P = E(T,U,n) (1)

where P is the absolute pressure, T is the absolute temperature, n is a vector for which the ith element is ni (the number of moles of the it,, component), and E is a well defined functional form. The argument U has units of total volume. The relation between the argument U and the total volume of the mixture is what distinguishes the model A from the model B. For the model A the variable U is equal to the total volume V of the mixture. Thus, we write

u, = v, (2) For the model B the variable U has the following expression:

u, = v, -I- c (3) where

c = f: cini (4) i= 1

where p is the number of components in the mixture and ci is a parameter independent of IZ (i.e., specific of component i) and independent of the pressure P, but it may depend on temperature. The parameter ci has units of molar volume. The pressure of the model B represents a translation of the pressure of the model A along the volume axis.

From Eq. (1) it is evident that at fixed T and P and n the value of U, is equal to the value of UB. Thus, we write:

VA = V, + 2 cqnq (at fixed T, P and n) (5) 4= I

The partial molar volume of component i is defined as follows:

where Ui is the partial molar volume of component i. From Eqs. (5) and (6), we write

(6)

;. 1.A = Ui,B + ci (at fixed T, P and n) (7)

M.S. Zabaloy, E.A. Brignole/ Fluid Phase Equilibria 1-N f 1997) 87-95 89

Eq. (7) states that the partial molar volumes of models A and B are related by the parameters c,. From exact thermodynamics, the fugacity coefficient of component i, in the mixture, is related to

the partial molar volume by the following equation:

where 6; is the fugacity coefficient of component i in the mixture. Following Peneloux and Rauzy (1982). Eqs. (7) and (8) are combined to obtain the relationship between the fugacity coefficients of models A and B:

(at fixed T, P and n) (9)

If, for the model A, the necessary condition of equilibrium between a phase (Y and a phase p, is satisfied for a given set of values of T, P, n” and 12 p, the following set of equations is valid:

n ” &;T4 --L

ri” = &f,$ [at T,P,n* and np(i= l,p)]

where

and

From Eqs. (9) and ( 1 O), it is written:

We recognize in Eq. ( 13) the necessary condition of equilibrium written for the model B. Thus, if a given set of values of T, P, na and at p is the solution of the necessary condition of equilibrium for the model A, then the same set of values is the solution of the necessary condition of equilibrium for the model B. Therefore, the equilibrium values of temperature, pressure and composition of the phases do not depend on the values of the parameters ci. This is the well-known conclusion of PCneloux and Rauzy (1982). Models A and B are said to be consistent. Hence, we understand by consistent models those giving identical composition, temperature and pressure of the phases at equilibrium, but different phase densities.

The equation of state for model A is

P = E( T,V,,n) (14)

90 MS. Zabaloy, E.A. Brignole/ Fluid Phase Equilibria 140 Cl 997) 87-95

while the equation of state for model B is

P = E(T,V, + C,n) (15) Following Peneloux and Rauzy ( 1982) the equation of state for model B can be taken to the same

format of the equation of state for model A, i.e.,

P = Ea( T,V, ,n) (16)

From Eqs. (15) and (16) it is concluded that the form of the function E, is defined by the following equation

EB( T,V, ,n) = E( T,V, + 0) (17)

3. Van der Waals type equations of state

In the present section we study the application of the previous section results to the special case of Van der Waals type equations of state, and concentrate on the problem of the true covolume. Van der Waals type equations of state are the most widely used in practical applications. An example of them is the Redlich-Kwong-Soave (RKS) equation (Soave, 1972). In general, they can be expressed, for mixtures, as follows:

P = F(T,u,u,,a) (18) In Eq. (18) F is a well defined functional form, the argument u is equal to U of Eq. (1) divided by the total number of moles and it has units of molar volume, U, is the value of u at infinite pressure and a is an energetic parameter having units of bar cm6 mall*. Variables U, and a depend on composition and temperature and, respectively, on the values of U, and a for the pure compounds, which may depend on temperature. Eq. (18) is equivalent to Eq. (1) and evidently the function F is such that:

F(T,u,u,,a) = E(7Y-Ln) (19)

If models A and B of the previous section are expressed in the format of Eq. (18), then

u* = VA (20)

% =u,+c (21)

Additionally,

u* = llrj = u (at fixed T and P) (22) or equivalently,

VA =v,+c=u (at fixed T and P) (23) where v is the molar volume of the mixture, and c is equal to C of Eq. (4) divided by the total number of moles in the mixture. Eq. (23) is valid at any pressure, e.g., at the particular condition of pressure tending to infinity. Hence,

lim P_cavA = lim,,,v, + c = u, (at fixed T) (24)

M.S. Zabaloy, E.A. Brignole/ Fluid Phase Equilibriu 140 (IYY7) 87-9.5 91

since the translation mixture parameter c does not depend on pressure. The limit of G for pressure tending to infinity is conventionally identified as b and called covolume. Thus, from Eq. (24), we write

b,=b,+c=u, (atfixedT) (25)

Hence, when passing form model A to model B (Eq. (23)), the (true) covolume is shifted (Eq. (2.5)) by the same amount than the molar volume, i.e., c. The reason is that the covolume is just the molar volume under an special condition of pressure.

From Eqs. (18), (20) and (25), we write for model A:

P = F(T,u,,b,,a) (26:)

while from Eqs. ( 181, (2 1) and (25) we write for model B,

P = F(T,u, + c,b, + c,a) (27)

From Eqs. (25) and (27) we may express model b as follows

P = F(T,rl, + c,b/,,a) (28)

Eqs. (27) and (28) are equivalent formulations for model B. While variable bA represents the volume at infinite pressure within Eq. (26) ( covolume) it loses the character of covolume within Eq. (28). Thus, Eq. (27) seems to be more elegant than the equivalent Eq. (28) since it contains a symbol which represents the value of the volume at infinite pressure.

The subscript associated to the molar volume in Eqs. (26)-(28) can be ignored since its presence or absence will not affect the relation between pressure, temperature and volume within the universe of the chosen model. Thus, we can safely write for model A

P = F(T,c,b,,a) (29)

and for model B

P = F(T,u + c,b, + c,a) (30)

or equivalently

P = F(T,u + c,b,,a) (31)

On the other hand the subscript corresponding to variable b cannot be ignored if consistent translations are wanted. If that is anyway done the consistency between Eqs. (29) and (3 1) is not affected while Eq. (30) does become inconsistent with respect to Eqs. (29) and (3 11, as illustrated in Appendix A. Soave (1984) chose to work with expressions such as Eqs. (29) and (3 1) (hence keeping consistency), for the case of an improved version of the Van der Waals equation of state, without using any subscript. The same approach was chosen by Watson et al. (1986) and Magoulas and Tassios (1990).

92 M.S. Zabaloy, E.A. Brignole/ Fluid Phase Equilibria 140 (1997187-95

4. Pure compound critical point

Now we address the problem of the critical point for a pure compound. The question to answer is whether model B gives a critical state at temperature and pressure values respectively equal to the critical temperature and critical pressure of model A, for pure compounds. From Eq. (23) it is evident that

PJ)T = WT (32) both, for model A and model B, since for a pure compound the parameter c is a function of T only. Therefore, the Van der Waals critical conditions can be expressed as follows for any of the models:

0 = [P -F(T,~,w)]~ (33)

where the subscript c denotes the critical point and F, and F,, are, respectively, the first and second partial derivatives of function F with respect to variable u at constant temperature. The system of three Eqs. (33)-(35) has five unknowns. If as usual the experimental values of critical temperature and pressure are introduced into Eqs. (33)--(35), then solution values for a, u and U, are obtained at T, and P,.. Since at any given pressure-temperature combination the variables a, u and U, have identical values for models A and B, both of these models depict a critical behavior at the same T, P coordinates. Hence, volume translations do not change the pure compound critical temperature and pressure. Application of Eqs. (23) and (25) at the critical temperature and pressure makes possible to relate the critical volumes and critical (true) covolumes of both models.

‘c = 'A,. = 'B, + c,

%c < = bA = b,< + c,

In conclusion, the defining equation of model B, i.e., Eq. including (T, P + a) (Eq. (25)), CT,., PC) where the critical both models (Eq. (36)), and (T,, P + a)(Eq. (37)).

(36)

5. Conclusions

(37) (23), is valid at any (T, P) combination, temperature an pressure are the same for

In the present note we have provided a more detailed derivation of the results of Peneloux and Rauzy (1982) on translated equations of state and have discussed some related issues at a greater level of detail, including the effect of translations on the covolume and on the critical point. We have also shown the consistency between the different formalisms which have been used in the practical application of the volume translation approach.

6. List of Symbols

cl

h c 17

I’ P R T

1’

Greek letters @ w

A

cy, p

Subscript.\ A.B c i

M.S. Zuhaloy. E.A. Brigrzo/e/ Fluid Phtrse Equilihritr 140 (19971 87-9.5

Energetic parameter Covolume parameter Translation parameter Number of moles Number of components Absolute pressure Universal gas constant Absolute temperature Molar volume Total volume

Fugacity coefficient Acentric factor

Partial molar property In the mixture Identification of phases

Identification of models Critical ith component

Appendix A

Consider the Redlich-Kwong-Soave (RKS) (Soave, 1972) equation of state, for a pure compound, which we write in the format of Eq. (29) of the text:

RT a p=_-

I’ - bA /I( 1’ + b*) (A-1 1

where R is the universal gas constant. The energetic parameter a and the covolume parameter bA are specific of the compound considered. For the original RKS equation of state (I is a function of T while h, is constant.

A translation of Eq. (A-l) can be written in the format of Eq. (30) of the text or equivalently in the format of Eq. (3 1) of the text. Hence, we write:

RT a p=.-.-.--

r>-h, (r+c)(u+b,+2c) (A-2)

RT a P= -

(I’ + c) - bA (u+c)(u+c+b,) (A-3)

94 M.S. Zubaloy. E.A. Brignole/ Fluid Phase Equilibria 140 (1997) 87-95

For pure compounds c is either constant or it depends only on temperature. Now, we artificially ignore the subscripts associated to the variable b. Thus, we rewrite Eqs. (A-l),

(A-2) and (A-3)

RT a p=--

u-b v(v + b) (A-4)

RT a p=--

v-b (v+c)(v+b+2c)

RT a P=

(v+c)-b - (v+c)(v+c+b)

(A-5)

(A-6)

Now we calculate the vapor pressure of n-propane at three values of temperature using Eqs. (A-5) and (A-6) for varying values of the parameter c. In both cases we used expressions for the parameters a and b corresponding to the original RKS model (Soave, 1972). The values of the critical temperature, critical pressure and acentric factor used for the computations were T, = 369.8 K, PC = 41.9 atm and o = 0.152, respectively.

---x-- 230 K Eq. (A-6) -230 K Eq. (A-5) ,.

B 260 K Eq. (A-6) - 260 K Eq. (A-5) mm+-- 290 K Eq. (A-6) - 290 K Eq. (A-5)

e! : 8 Ii b iit >

--g---~-_ --*----_o---

_.-J(_._._.*._-__. I Y

-8 -6 -4 -2 0 2 4 6 8

c I (cm3/mol)

Fig. 1. Vapor pressure for pure n-propane as a function of the parameter c for different values of temperature. Comparison of Eqs. (A-5) and (A-6).

M.S. Zubnloy, E.A. Brignolr/ FluidPhase Equilibria 140 (lYY7) X7-95 ‘)j

Fig. 1 shows the results for the calculation of the vapor pressure of propane as a function of the parameter c, using Eqs. (A-5) and (A-6) for different values of temperature. The value of the parameter h is independent from temperature and equal to 62.74843 cm3 mol- ‘. From Fig. 1 it is clear that the vapor pressure does depend on the value of c when Eq. (A-5) is used. On the other hand, the vapor pressure is independent from the value of c if Eq. (A-6) is applied. Note that for c = 0 both models are the same, i.e., the original RKS model, Eq. (A-4), and hence they give the same vapor pressure value. Fig. 1 shows that Eq. (A-5) is inconsistent with respect to Eq. (A-4), because the difference between the variable b of Eq. (A-4) and the variable h of Eq. (A-5) was ignored. In conclusion a practical recipe to safely obtain a consistent translated equation of state from an original untranslated equation of state consists of just changing I‘ by (1. + c) within the untranslated expression. However, the resulting expression will not contain a symbol representing the true covolume.

Consistent translations affect the equilibrium phase densities without changing the equilibrium temperature, pressure and phase compositions. There are also other properties that cannot be changed by using these volume translations, e.g., the pure compound vaporization enthalpy. For a saturated pure compound any equation of state complies with the Clapeyron equation. This equation comes from setting the equality between the molar Gibbs energies for the saturated phases. i.e., from a condition equivalent to the isofugacity criterion used for calculating vapor pressures using equations of state. The Clapeyron equation of state is the following:

(A-7 1

where P’“’ is the saturation pressure at temperature T, AH vnp is the heat of vaporization at temperature T and superscripts V and L correspond to saturated vapor and liquid respectively.

Both the translated and untranslated model give the same saturation pressure and hence the same derivative of saturation pressure with respect to temperature. They give different saturated molar volumes but the difference between the vapor and liquid volumes is independent from c. as it is evident from Eq. (23) applied consecutively to the vapor and liquid phases. Therefore, from Eq. (A-7) we conclude that the parameter c does not change the vaporization enthalpy value with respect to that of the untranslated model.

References

Magoulas, K., Tassios, D., 1990. Thermophysical properties of n-alkanes from Cl to C20 and their prediction for higher ones. Fluid Phase Equilib. 56, 119-140.

PCneloux. A., Rauzy, E., 1982. A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilib. 8, 7-23. Soave. G.. 1984. Improvement of the Van der Waals equation of state. Chem. Eng. Sci. 39, 357-369. Soave. G.. 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 27. I l97- 1203. Watson, P., Cascella, M., May, D., Salerno, S., Tassios, D., 1986. Prediction of vapor pressures and saturated molar volumes

with a simple cubic equation of state: Part II. The Van der Waals-711 EOS. Fluid Phase Equilib. 27. 35-53.