derivation and fugacity coefficients - Ufrgs

A Self-Consistent Gibbs Excess Mixing Rule for Cubic
Equations of State: derivation and fugacity coefficients
Paula B. Staudt, Rafael de P. Soares ∗
Departamento de Engenharia Química, Escola de Engenharia, Universidade Federal do Rio Grande do
Sul, Rua Engenheiro Luis Englert, s/n, Bairro Farroupilha, CEP 90040-040, Porto Alegre, RS, Brazil
Abstract
The extension of the applicability of cubic equations of state (EoS) with Gibbs excess
models to the prediction of high-pressure/high-temperature vapor–liquid equilibria of
polar and/or asymmetric is well known. In a recent work (http://dx.doi.org/10.
1016/j.fluid.2012.06.029) we have proposed the so called Self–Consistent Mix-
ing Rule (SCMR). The method was derived solely based on the assumption of a zero
excess volume liquid-like phase. Tests with substances dissimilar in size, shape and
chemical nature have shown that any cubic equation of state coupled with the pro-
posed mixing rule can reproduce the underlying liquid activity model at low pressures,
showing that the method is self-consistent. Further, the method was extended for high
pressures/temperatures by assuming a constant thermal expansion coefficient liquid–
like phase as the reference state. Very good results were obtained when the proposed
method was coupled with Wilson, UNIQUAC, UNIFAC and a COSMO-based model
in liquid–liquid and vapor–liquid equilibrium examples. The present document con-
tains a minimum description of the SCMR method, its derivation and the equations for
the fugacity coefficient.
Key words: cubic equations of state, EoS/G E mixing rules, Gibbs excess models,
vapor-liquid equilibrium, liquid-liquid equilibrium.
∗ Corresponding author. Tel.:+55 51 33083528; fax: +55 51 33083277
Email address: rafael@enq.ufrgs.br (Paula B. Staudt, Rafael de P. Soares)
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1. Introduction
In a recent work (http://dx.doi.org/10.1016/j.fluid.2012.06.029) a new
G E –based mixing rule for cubic equations of state was developed. The main advantage
of the proposed method is that the combined model reproduces very well the G E γ model
it is based on without any additional empirical correction.
For a detailed description of the method as well as comparison with similar methods
and experimental data, please refer to the full–length manuscript available at http://
dx.doi.org/10.1016/j.fluid.2012.06.029. In this document only the mixing
rule derivation and the fugacity coefficient equations are shown.
2. Mixing rule derivation
Most of the cubic equations of state (EoS) available today are special cases of a
general cubic equation [1], which can be written as:
P = RT
V − b −
a (T)
(V + ɛb) (V + σb)
where P is the pressure, T is the temperature, V is the molar volume, ɛ and σ are con-
stants for all substances and depend on the particular EoS (see Table 1) and a (T) and b
are, respectively, the attractive and co-volume parameters specific for each substance.
The attractive a (T) and co-volume b parameters are usually determined using gen-
eralized correlations based on critical properties and acentric factor, according to:
a (T) = Ψ α (Tr, ω) R 2 T 2 c
Pc
b = Ω RTc
Pc
where Tc is the critical temperature, Pc is the critical pressure, ω is the acentric factor,
Tr = T/Tc the reduced temperature and the other symbols are shown in Table 1.
2.1. Mixing rule
When dealing with mixtures, the expressions for the attractive a and co-volume b
parameters should be computed as a function of the pure substances values ai and bi
through mixing rules.
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Table 1: Specific cubic equation parameters.
EoS α(Tr) σ ɛ Ω Ψ
van der Waals (vdW) 1 0 0 1/8 27/64
Redlich–Kwong (RK) T −1/2
r 1 0 0.08664 0.42748
Soave–Redlich–Kwong (SRK) αS RK(Tr; ω) a 1 0 0.08664 0.42748
Peng-Robinson (PR) αPR(Tr; ω) b 1 + √ 2 1 − √ 2 0.07780 0.45724
aαS RK(Tr; ω) = � 1 + (0.48 + 1.574ω − 0.176ω2 ) � 1 − √ ��2 Tr
bαPR(Tr; ω) = � 1 + (0.37464 + 1.54226ω − 0.26992ω2 ) � 1 − √ ��2 Tr
The van der Waals (vdW) or classic mixing rule, present in most professional pro-
cess simulation systems, is given by:
a =
b =
N�
i=1
N�
i=1
√
xixj aia j(1 − ki j) (4)
xibi
where xi is the mole fraction of the substance i and ki j is the binary interaction param-
eter, introduced to improve the correlation of phase equilibrium of mixtures.
G E –based mixing rules, in contrast to the classic mixing rule, obtain the interac-
tion information from excess Gibbs energy G E γ models, originally developed for the
prediction of liquid activity coefficients γi.
is [2]:
One possible expression for computing the Gibbs excess energy from a cubic EoS
G E φ
RT
�
= ln φ −
i
xi ln φi
where φ is the mixture fugacity coefficient and φi is the fugacity coefficient of the pure
substance i, all in the same conditions of temperature and pressure.
The fugacity coefficient considered in Equation 6 for any cubic EoS in the generic
form (Equation 1) is given by [3]:
ln φ = (Z − 1) − ln(Z − β) + qI (7)
3
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where Z ≡ PV/RT is the compressibility factor and the other auxiliary variables are:
β ≡ Pb/RT, q ≡ a/bRT, I ≡ I0 ln V+ɛb
V+σb , and I0 is a constant given by 1/(σ − ɛ).
Using Equation 7 one can compute ln φ as well as ln φi by exchanging the mixture
properties (Z, β, q, and I) by the pure substance properties (Zi, βi, qi, and Ii). Thus, by
combining Equation 6 and Equation 7, the Gibbs excess energy for a cubic EoS can be
computed by:
G E φ
RT
�
= (Z − 1) − ln(Z − β) + qI − xi ((Zi − 1) − ln(Zi − βi) + qiIi) (8)
i
Using the definition of excess volume V E ≡ V− � i xiVi and recalling that ln(Z−β) =
ln P
RT + ln(V − b) one can obtain:
G E φ
RT
E PV
=
RT +
� � �
Vi − bi
�
xi ln + qI −
V − b
i
i
xiqiIi
The expression given by Equation 9 contains no simplification assumptions and can
be used to get a fully consistent mixing rule if we make G E φ = GE γ . Although exact, the
practical use of Equation 9 is limited because it is an implicit mixing rule (the mixture
volume V depends on q and vice–versa). Now, by assuming that the excess volume is
negligible (V E = 0, V = V Id = � i xiVi) the following expression is obtained:
G E φ
RT =
�
i
�
Vi − bi
xi ln
V Id �
+ qI
− b
Id −
�
i
xiqiIi
where b should be computed with the mixing rule Equation 5, the volume of the pure
substances Vi, as well as I Id and Ii, should be computed using the liquid-like root of
the pure fluids at the system temperature and pressure.
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However, for cubic equations, the attractive parameter a should depend on tem-
perature and composition only. Since most G E γ are developed for near–atmospheric
pressure, in this work the liquid–like root required for the determination of Vi is ob-
tained at 1 bar. The results would be essentially the same if a zero pressure is taken
as reference. Finally, by making G E φ = GE γ a new explicit mixing rule is obtained by
isolating q in Equation 10:
q = 1
IId ⎛
G
⎜⎝
E γ
RT −
� �
Vi − bi
xi ln
V Id � �
+
− b
i
4
i
xiqiIi
⎞
⎟⎠
(11)

The mixing rule given by Equation 11 will reproduce the G E γ model as long as the
the system pressure is not too far from 1 bar and the zero excess volume assumption
holds. For all tests considered (see the full-length manuscript) the GE φ reproduced the
G E γ very well. Thus, the mixing rule given by Equation 11 was referred as the Self–
Consistent Mixing Rule (SCMR). The derivation of fugacity coefficients of substances
in mixture according to the SCMR is given in Section 3.
2.2. Extension for high pressure/temperature
In the mixing rule proposed in the present work, the pure fluid liquid–like volume
of each substance in mixture is necessary. At high temperature conditions, usually
above Tr ≡ T/Tc = 0.7, one can have problems with finding a liquid–like root from the
cubic EoS.
To circumvent this problem, an alternative procedure is adopted in this work to
compute the pure fluid liquid-like molar volume to be used in Equation 11. From the
definition of the volumetric thermal expansion coefficient of a pure fluid βi:
βi ≡ 1
� �
∂Vi
∂T
Vi
P
and assuming a constant βi, evaluated at a reference temperature T ◦ i , the molar volume
of a pure substance can be obtained by the following expression:
ln Vi
V ◦ i
(12)
= βi(T − T ◦ i ) (13)
The pure fluid thermal expansion βi, according to an EoS, is easily determined by
its definition (Equation 12). In this work, the reference temperature T ◦ i
chosen was
that to correspond to a Ti,r = 0.5. This temperature corresponds, approximately, to
the normal boiling temperature. This reference temperature assures a valid liquid–like
root, consequently no problems will occur to evaluate Vi.
Then, in the SCMR mixing rule, the pure fluid liquid–like root Vi is always eval-
uated by Equation 13, allowing its application to high pressure/temperature and/or su-
percritical systems.
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2.3. Polymer solutions
Specially for polymer components, the required pure component liquid-like vol-
ume Vi = bi/ui was computed by considering a constant and universal value for the
inverse packing fraction ui = 1.288. This value, taken from Sanchez and Cho [4],
corresponds to the average value of the ratio of the van der Waals’ density and the
characteristic density ρ ∗ , which is very chose to the Bondi constant 1.3[5]. With this
assumption the liquid-like volume value, to be used by the mixing rule, is constant for
pure polymers. The solvent liquid-like volume is still calculated using the volumetric
thermal expansion coefficient, as explained in subsection 2.2.
3. Fugacity coefficients from SCMR
The fugacity coefficient of a substance i in a mixture, for any cubic EoS given by
Equation 1, can be obtained by [3]:
ln ˆφi = ¯bi
b (Z − 1) − ln(Z − β) + ¯qiI (14)
where ¯bi and ¯qi are partial molar properties defined by:
� �
∂nT k
¯ki ≡
∂ni
with nT = � l nl.
T,P,n j�i
(15)
For the SCMR mixing rule, the co–volume parameter b is given by the linear mix-
ing rule, Equation 5, then ¯bi = bi.
In order to simplify the notation in the derivation of ¯qi, let us introduce the quantity
α for the SCMR mixing rule (Equation 11):
α ≡ qI Id = GE γ
RT −
� �
Vi − bi
xi ln
V Id � �
+
− b
which leads to:
¯qi = q + 1
IId ⎛
⎜⎝ ¯αi − α
i
Īi Id
I Id
⎞
⎟⎠
i
xiqiIi
and the remaining partial molar properties are:
�
Vi − bi
¯αi = ln γi − ln
V Id �
+
− b
Vi − bi
V Id − 1 + qiIi
(18)
− b
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Īi Id = I Id + I0
�
Vi + σbi
V Id + σb − Vi + ɛbi
V Id �
+ ɛb
where the activity coefficient γi should be computed by the chosen Gibbs excess model
G E γ .
References
[1] J. O. Valderrama, Ind. Eng. Chem. Res. 42 (8) (2003) 1603–1618.
[2] K. Fischer, J. Gmehling, Fluid Phase Equilib. 121 (1-2) (1996) 185–206.
[3] J. M. Smith, H. C. V. Ness, M. M. Abbott, Introduction to Chemical Engineering
Thermodynamics, McGraw-Hill, New York, 2005.
[4] I. C. Sanchez, J. Cho, Polymer 36 (15) (1995) 2929–2939.
[5] A. Bondi, van der Waals Volumes and Radii, J. Phys. Chem. 68 (3) (1964) 441–
451.
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