Page 285 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 265
ˆ
activity coefficient at infinite dilution for component 1 (γ )
get Using the relation between ˆμ i andf i given by Eq. (6.119) we x i → 0, will result in the following relation for calculation of
∞
1
in a binary system of components 1 and 2 at T and P [17]:
ˆ γ
ˆ β
ˆ α
(6.172) f =f =f = ··· for every i at constant T and P
i i i

b 1 Z 1 − B 1
∞
Equations (6.171) or (6.172) are the basis for formulation ln γ 1 = (Z 2 − 1) − (Z 1 − 1) + ln
b 2 Z 2 − B 2
of mixture phase equilibrium calculations. Application of
A 1 Z 1 + 2.414B 1
Eq. (6.172) to VLE gives + √ ln
2 2B 1 Z 1 − 0.414B 1
ˆ V
ˆ L
(6.173) f (T, P, y i ) = f (T, P, x i ) (6.178) − a 12 P √ 1 ln Z 2 + 2.414B 2
i
i
2
for all i components at constant T and P R T 2 2B 2 Z 2 − 0.414B 2

b 1 A 2 Z 2 + 2.414B 2
For SLE, Eq. (6.172) becomes + √ ln
2 2B 2 b 2 Z 2 − 0.414B 2
(6.174) f ˆ S T, P, x i S = f ˆ L T, P, x i L where a 12 = a 1/2 1/2 (1 − k 12 ) in which k 12 is the binary inter-
i
i
a
1
2
for all i components at constant T and P action parameter. Parameters a, b, A, and B for PR EOS are
given in Table 5.1. Z 1 and Z 2 are the compressibility factor
S
where x is the mole fraction of i in the solid phase. Simi-
i for components 1 and 2 calculated from the PR EOS.
larly Eq. (6.172) can be used in liquid–liquid equilibria (LLE),
solid–liquid–vapor equilibria (SLVE), or vapor–liquid–liquid The main difference between Eqs. (6.175) and (6.176) for
equilibria (VLLE). VLE calculations is in their applications. Equation (6.176)
V
L
is particularly useful when both ˆ φ and ˆ φ are calculated
i i
from equations of state. Cubic EOSs generally work well in
6.8.2 Vapor–Liquid Equilibria—Gas Solubility the VLE calculation of petroleum systems at high pressures
V
L
in Liquids through this equation. ˆ φ and ˆ φ may be calculated through
i
i
Eq. (6.126) with use of appropriate composition and Z; that
In this section general relations for VLE and specific relations is, x and Z must be used in calculation of ˆ φ , while y V i
L
L
L
i
i
developed for certain systems such as Raoult’s and Henry’s and Z are used in calculation of ˆ φ . Binary interaction coeffi-
V
V
i
laws are presented. For high pressure VLE calculations equi- cients (BIPs) given in Table 5.3 must be used when dissimilar
librium ratio (K i ) is defined and its methods of estimation for (very light and very heavy or nonhydrocarbon and hydrocar-
hydrocarbon systems are presented. bon) molecules exist in a mixture. However, as mentioned
earlier there is no need for use of volume translation or shift
6.8.2.1 Formulation of Vapor–Liquid parameter in calculation of ˆ φ and ˆ φ for use in Eq. (6.176).
V
L
i
i
Equilibria Relations At low and moderate pressures use of Eq. (6.175) with
Formulation of VLE calculations requires substitution of rela- activity coefficient models is more accurate than use of
L
tions forf ˆ V andf ˆ L from Eqs. (6.113) and (6.114). Combining Eq. (6.176) with an EOS. Assuming constant V and substi-
i
i
i
Eqs. (6.114) and (6.173) gives the following relation: tuting Eq. (6.136) into Eq. (6.175) we have
L
ˆ V
(6.175) y i φ P = x i γ i f L V (P − P sat )
i i (6.179) y i φ P = x i γ i φ sat P sat exp i i
ˆ V
i i i RT
L
where f is the fugacity of pure liquid i at T and P of the mix-
i
ture and it may be calculated through Eq. (6.136). The activity where the effect of pressure on the liquid molar volume is
coefficient γ i is also a temperature-dependent parameter in neglected and saturated liquid molar volume V sat may be used
i
addition to x i . Another general VLE relation may be obtained for V . As discussed in Section 6.5, the vapor pressure P sat
L
i
i
when both f ˆ V and f ˆ L are expressed in terms of fugacity coef- is a function of temperature and the highest temperature at
i
i
V
L
ficients ˆ φ and ˆ φ through Eq. (6.113) and are substituted in which P sat can be calculated is T c , where P sat = P c . Therefore,
i
i
i
i
Eq. (6.173): Eq. (6.179) cannot be applied to a component in a mixture at
which T > T c . For ideal liquid solutions or those systems that
ˆ V
ˆ L
(6.176) y i φ (T, P, y i ) = x i φ (T, P, x i )
i i follow Lewis rule (Section 6.6.3), the activity coefficient for
all components is unity (γ i = 1). If pressure P and saturation
where pressure P from both sides of the equation is dropped. sat
Equation (6.176) is essentially the same as Eq. (6.175) and pressure P i are low and the gas phase can be considered
V
sat
the activity coefficient can be related to fugacity coefficient as an ideal gas, then ˆ φ and φ i are unity and the Poynting
i
as [17] factor is also unity; therefore, the above relation reduces to
the following simple form:
(6.177) ln γ i = ln ˆ φ i (T, P, x i ) − ln φ i (T, P)
(6.180) y i P = x i P sat
i
where ˆ φ i (T, P, x i ) is the fugacity coefficient of i in the liquid
mixture and φ i (T, P) is fugacity coefficient of pure liquid i at This is the simplest VLE relation and is known as the Raoult’s
T and P of mixture. In fact one may use an EOS to calcu- law. This rule only applies to ideal solutions such as benzene–
late γ i by calculating ˆ φ i and φ i for the liquid phase through toluene mixture at pressures near or below 1 atm. If the gas
Eq. (6.126). Application of PR EOS in the above equation, at phase is ideal gas, but the liquid is not ideal solution then
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