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Prediction of vapor-liquid equilibrium in highly asymmetric paraffinic systems with new modified EOS-GE model
Journal of Petroleum Science and Engineering 159 (2017) 810–817
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Prediction of vapor-liquid equilibrium in highly asymmetric paraffinic systems with new modified EOS-GE model Juheng Yang a, b, Jing Gong a, c, *, Guoyun Shi a, Huirong Huang a, Dan Wang a, Wei Wang a, b, **, Qingping Li d, Bohui Shi a, Haiyuan Yao d a
National Engineering Laboratory for Pipeline Safety, China University of Petroleum, Beijing 102249, PR China MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, PR China Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, PR China d China National Offshore Oil Cooperation Research Center, Beijing 100027, PR China b c
A R T I C L E I N F O Keywords: Thermodynamics Vapor-liquid equilibrium Asymmetric paraffinic system EOS-GE UNIFAC Activity coefficient
A B S T R A C T
A new modified EOS-GE model is developed for the highly asymmetric paraffinic systems, where the volume translated Peng-Robinson EOS is adopted coupled with the LCVM mixing rule. In the new modified EOS-GE model, the original UNIFAC is replaced by a newly established UNIFAC where the nonlinear calculation of the segment fractions of molecules in γC (the combinatorial activity coefficient) is introduced to modify the traditional assumption that “all groups are isotropic in solution”. A total of 956 vapor-liquid experimental bubble points in highly asymmetric paraffinic systems including binary systems, ternary systems, quaternary systems and multiple systems are used to test the new developed EOS-GE model. Results show that the original UNIFAC and the improved UNIFAC both perform well if the molefractions of light components (CH4 or C2H6) are low; however, with the increase of the light components, the improved UNIFAC is remarkably superior to the original UNIFAC.
1. Introduction In petroleum production, the vapor-liquid equilibrium plays an extremely important role in the process of exploitation, gathering and transportation and rectification. Therefore, the accurate prediction of vapor-liquid equilibrium is critical for the petroleum industry. In the modeling of vapor-liquid equilibrium, the vapor phase is usually described by cubic Equation of State (Redlich and Kwong, 1949; Soave, 1972; Stryjek and Vera, 2010; Patel and Teja, 1982); while the liquid phase can be modeled in two different ways: Equation of State (EOS) (Redlich and Kwong, 1949; Soave, 1972; Stryjek and Vera, 2010; Patel and Teja, 1982) or activity coefficient method (GE) (Hildebrand, 1929; Flory, 1942; Huggins, 1942; Wilson, 1964; Renon and Prausnitz, 1968; Abrams and Prausnitz, 1975; Fredenslund et al., 1975). The objective with the use of EoS/GE models is to combine the “advantages” of cubic EOS and of the local composition activity coefficient models incorporated (Kontogeorgis and Coutsikos, 2012). The EOS can deal with the influence of large pressure variation, but it is not suitable for the systems with high polarity and high asymmetry. On the contrary, the activity coefficient method (GE) can describe the non-ideality of polar and asymmetric
systems well without the capability to reflect the effects of pressure variation. Therefore, it is necessary to build a bridge between EOS and GE. To introduce GE into EOS, the traditional linear mixing rule is modified. Huron and Vidal (1979) assumed that fluid systems were in a state of liquid or nearly liquid at the infinite pressure and established the HV mixing rule which predicts well for some complicated fluid systems. Mollerup (1981), Heidemann and Kolal (Heidemann and Kokal, 1990) proved the possibility that GE and EOS could be integrated at zero or low pressure. Later, Michelsen et al (Michelsen, 1990a, 1990b; Dahl and Michelsen, 1990). developed the mixing rule at zero pressure (MHV1) and amplified the application range of EOS-GE. Boukouvalas et al. (1994). found that as the size difference of tested systems increases, the HV mixing rule led to underprediction and the MHV1 mixing rule led to overprediction. Therefore, Boukouvalas et al. (1994). proposed the LCVM mixing rule (Linear Combination of the Vidal and Michelsen mixing rules) to make up for the weakness of the HV mixing rule and the MHV1 mixing rule. These EOS/GE mixing rules realized two progresses (Kontogeorgis and Coutsikos, 2012). First of all, the cubic EOS is successfully applied to the mixtures of compounds of wide complexity and
* Corresponding author. National Engineering Laboratory for Pipeline Safety, China University of Petroleum, Beijing 102249, PR China. ** Corresponding author. National Engineering Laboratory for Pipeline Safety, China University of Petroleum, Beijing 102249, PR China. E-mail addresses: [email protected] (J. Gong), [email protected] (W. Wang). https://doi.org/10.1016/j.petrol.2017.10.015 Received 10 April 2017; Received in revised form 1 October 2017; Accepted 4 October 2017 Available online 7 October 2017 0920-4105/© 2017 Published by Elsevier B.V.
J. Yang et al.
Journal of Petroleum Science and Engineering 159 (2017) 810–817 2=3
xi r ϕi ¼ P i 2=3 xj rj
Table 1 Parameters of Eq. (11). Constants
Values
A
In 1993, Gmehling et al (Weidlich and Gmehling, 1987). developed another way to improve the calculation method for the combinatorial part:
4.1034 31.723 0.0531 188.68 0.0057 20,196 0.0003
k0 k1 k2 k3 k4 k5 k6
ϕ ϕ ln γ Ci ¼ 1 ϕ0i þ ln ϕ0i 5qi 1 i þ ln i θi θi
(3)
3=4
r ϕ0i ¼ P i 3=4 rj xj
B 0.3489 28.547 0.0687 817.73 0.0007 65.067 0.0076
k0 k1 k2 k3 k4 k5 k6
(4)
j
Similarly, to improve the phase equilibria prediction of the highly asymmetric paraffinic systems, this work develops a new modified EOSGE model where the volume translated PR-EOS with the LCVM mixing rule is adopted. And GE is calculated by the newly improved UNIFAC with the nonlinear calculation of the molecule's segment fractions to consider the anisotropy of groups in solution.
asymmetry in size and energies. Secondly, the incorporated activity coefficient models established at low pressure can be extrapolated to the higher pressure conditions. For the high prediction accuracy of the vapor-liquid equilibrium, the calculation method for GE should be improved. UNIFAC is an approximation method where the properties of the practical systems are predicted by the properties of the composed chemical groups. The original UNIFAC is not suitable for systems with large size difference due to the assumption that “all groups are isotropic in solution” (Deiters, 1989). Deiters et al. (Deiters, 1989) proposed a nonlinear calculation for the volume fractions of non-spherical molecules. Li et al. (1998). employed the concept of effective R*k and Qk* to describe the local characteristic of groups. Sayegh and Vera (1980) pointed out that the Staverman-Guggenheim correction may give unrealistic large corrections to the combinatorial excess entropy. Consequently, Larsen et al. (1987). and Kikic et al. (1980), dropped the Staverman-Guggenheim correction in the UNIFAC model:
ln γ Ci ¼ ln
(2)
j
ϕi ϕ þ1 i xi xi
2. The equation of state The PR-EOS with the consideration of the volume translation effects (Mathias et al., 1989) is used:
P¼
RT a V þ t b ðV þ tÞðV þ t þ bÞ þ bðV þ t bÞ
(5)
where P is the pressure of the system; T is the temperature of the system; R is the gas universal constant; V is the volume of gas or liquid; t is the translation volume. For pure component, parameter a is calculated by (Stryjek and Vera, 2010):
a ¼ 0:45724
(1)
R2 Tc2 ⋅αðTÞ pc
(6)
2 αðTÞ ¼ 1 þ k 1 Tr0:5
(7)
k ¼ 0:37464 þ 1:54226ω 0:26992ω2
(8)
Table 2 The tested pressure range and the composition of the systems. Tested system Binary system C1þC10 C1þC12 C1þC16 C1þC17 C1þC20 C1þC24 C1þC32 C2þC10 C2þC16 C2þC20 C2þC22 C2þC28 C2þC36 C2þC44 Ternary system C1þC10 þ C32 Quaternary system C1þC16 þ C17 þ C18 Multiple system C1þC10þ Multiple-paraffin Total number of points
light component content range
Pressure range (MPa)
Temperature range(K)
Number of points
Reference
0.1–0.55 0.1–0.6 0.1–0.6 0.2–0.6 0.025–0.65 0.1–0.7 0.1–0.325 0.017–0.995 0.2–0.875 0.07–0.47 0.05–0.9 0.1–0.5 0.087–0.531 0.1–0.52
1–32 1–23 2–26 4–27 0–32 1–30 1.5–7 0.1–11 0.5–16 0.5–4 0.1–9.5 0.5–5.5 0.3–5 0.3–3.5
243.15–313.15 255–320 285–360 293–373 303–370 315–450 343.15 277.6–510.9 260–450 373.75–572.85 290–370 348–423 373–423 373–423
61 38 84 29 83 127 10 107 148 11 110 24 13 15
(Rijkers et al., 1992a) (Rijkers et al., 1992b) (Glaser et al., 1985) (Pauly et al., 2007) (Van der Kooi et al., 1995) (Fl€ oter et al., 1997) (Cordeiro et al., 1973) (Reamer and Sage, 1962) (De Goede et al., 1989) (Huang et al., 1988) (Peters et al., 1988) (Gasem et al., 1989) (Gasem et al., 1989) (Gasem et al., 1989)
0.025–0.4
0.1–10
330–340
28
(Cordeiro et al., 1973)
0.2–0.6
3–30
293–373
14
(Pauly et al., 2010)
C1:0.436–0.440 C10:0.458–0.462
12–16
293–423
54
(Daridon et al., 1996)
956
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4. Modification of UNIFAC model
where Tc is the critical temperature; pc is the critical pressure; Tr is the reduced temperature; ω is the acentric factor. Although Soave α function (Eq. (7)) is doubted because of its inconsistency in supercritical region (Le Guennec et al., 2016a; Segura et al., 2003; Le Guennec et al., 2016b), it is certainly the most employed α function in the oil and gas industries. Thus, for the hydrocarbon mixture in this work, Eq. (7) is still used. For pure component, parameter b is obtained from (Stryjek and Vera, 2010):
RTc b ¼ 0:07780 pc
4.1. The original UNIFAC The UNIFAC model is developed by applying the group contribution method to the UNIQUAC model (Universal Quasichemical) (Fredenslund et al., 1975). And the calculation of activity coefficient is as follows:
ln γ i ¼ ln γ Ci þ ln γ Ri
(9)
whereγ Ci represents the entropy contribution, the differences in size and shape between the molecules and γ Ri is the enthalpy contribution, the energetic interactions between the components. The calculation of γ Ci is the same as that in UNIQUAC:
The translation volume, t, is suggested by Baled et al. (2012):
t ¼ A þ B:Tr
(10)
A; B ¼ f ðM; ωÞ
1 1 1 þ k3 exp þ k5 exp ¼ k0 þ k1 exp k2 Mω k4 Mω k6 Mω
(16)
ln γ Ci ¼ ln
(11)
m ϕi z θi ϕ X þ qi ln þ li i xj lj xi 2 ϕi xi j¼1
z li ¼ ðri qi Þ ðri 1Þ; 2
The valuesof k0 , k1 , k2 , k3 , k4 , k5 , k6 are shown in Table 1: While it must be acknowledge that several studies recommended temperature-independent volume-translation parameters and proved that the temperature-dependent volume-translation parameters may results in unrealistic predictions at extreme conditions (Le Guennec et al., 2016b; Kalikhman et al., 2010; Privat et al., 2016). However, Baled's study (Baled et al., 2012) covers short- and long-chain alkanes ranging from CH4 to C40H82 at pressures between 7 and 276 MPa and temperatures between 278 and 533 K. So Baled's translation volume can be used for the investigated mixtures which are included in Baled's research (shown as Table 2). The critical properties of pure components, such as Pc and Tc, are proposed by Twu (1984). And the acentric factor is calculated by Edmister correlation (Edmister, 1958).
qi xi θi ¼ P ; qj xj j
(17)
z ¼ 10
(18)
ri xi ϕi ¼ P rj xj
(19)
j
In these equations, xi is the mole fraction of component i; θi is the area fraction; ϕi is the segment fraction which is similar to the volume fraction; ri and qi are respectively the molecular van der Waals volume and the molecular surface area for pure component which are calculated as the sum of the group volume and area parameters,Rk and Qk , obtained from the van der Waals group volume and surface areas given by Bondi (1968):
ri ¼
X ðiÞ vk Rk ;
qi ¼
X ðiÞ vk Qk
k
3. The LCVM mixing rule
(20)
k
ðiÞ
The LCVM mixing rule (Boukouvalas et al., 1994) is the linear combination of HV mixing rule and MHV1 mixing rule and adopted:
where vk is the number of groups of type k in molecule i. However, the solution-of-groups concept is introduced into the calculation of γ Ri in UNIFAC:
α ¼ λ:αV þ ð1 λÞ:αM
(12)
ln γ Ri ¼
1 GE X ⋅ þ xi ai AV RT i
(13)
αV ¼
" # X 1 GE X b þ þ αM ¼ ⋅ xi In xi ai AM RT b i i i
α¼
a ¼ bRT
X λ 1 λ GE 1 λ X b ⋅ þ þ þ ⋅ xi In xi ai AV AM AM i bi RT i
h i ðiÞ ðiÞ vk ln Γ k ln Γ k
(21)
k all groups
ðiÞ
where Γ k is the group residual activity coefficient; Γ k is the group residual activity coefficient of group k in the solution which only contains molecule i. Γ k is obtained from:
"
(14)
X ln Γ k ¼ Qk 1 ln Θm ψ mk
Combine Eqs. (12)–(14), one obtains the expression of the LCVM mixing rule:
X
!
m
X
, !# X Θn ψ nm Θm ψ km
m
n
(22)
ðiÞ
Eq. (22) is also suitable for Γ k . In Eq. (22), Θm is the area fraction of group m and calculated by:
(15)
Qm Xm Θm ¼ P Qn Xn
For PR-EOS, AV ¼ 0:623and AM ¼ 0:52 (Boukouvalas et al., 1994). The mixing rule yields the HV and the MHV1 mixing rule, respectively for λ ¼ 1and λ ¼ 0. In this work, λ ¼ 0:36 suggested by Boukouvalas (Boukouvalas et al., 1994) is used. However, Polishuk et al. (2002). proved that UNIFAC model (temperatures outside 250–425 K) was able to generate non-realistic phase diagrams. In Table 2, it is evident that the investigated temperature range covers the most of the asymmetric paraffinic systems except C2-C10 and C2-C20. Unfortunately, there is little UNIFAC group interaction parameters fitted from the experimental data at temperatures above 500 K. Therefore, the UNIFAC group interaction parameters covering temperature range from 250 to 425 K is extrapolated to the cases of C2-C10 and C2-C20.
(23)
n
Xm is the mole fraction of group m in the mixture. ψ mn Is the group interaction parameter and given by:
a mn ψ mn ¼ exp T
(24)
The parameters amn are fitted from the experimental data. Note that amn ≠anm and that if n ¼ m, then amn ¼ 0 and ψ mn ¼ 1.
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Table 3 The AAREs by the original UNIFAC and the improved UNIFAC in binary systems. Binary system
Improved UNIFAC (AARE%)
Original UNIFAC (AARE%)
C1þC10 C1þC12 C1þC16 C1þC17 C1þC20 C1þC24 C1þC32 C2þC10 C2þC16 C2þC20 C2þC22 C2þC28 C2þC36 C2þC44
2.92 5.36 3.40 3.72 7.23 5.44 7.83 4.18 7.69 3.62 10.57 5.49 5.78 7.74
7.93 7.30 9.80 11.47 8.92 11.41 9.23 6.27 11.65 6.15 16.37 9.54 10.93 10.20
Global average AARE%
5.78
9.80
Fig. 2. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C16.
5. Results and discussion In this work, 17 highly asymmetric paraffinic systems are investigated including C1-C10, C1-C12, C1-C16, C1-C17, C1-C20, C1-C24, C1-C32, C2-C10, C2-C16, C2-C20, C2-C22, C2-C28, C2-C36, C2-C44 as binary systems (Rijkers et al., 1992a, 1992b; Glaser et al., 1985; Pauly et al., 2007; Van der Kooi et al., 1995; Fl€ oter et al., 1997; Cordeiro et al., 1973; Reamer and Sage, 1962; De Goede et al., 1989; Huang et al., 1988; Peters et al., 1988; Gasem et al., 1989), C1-C10-C32 as ternary system (Cordeiro et al., 1973), C1-C16-C17-C18 as quaternary system (Pauly et al., 2010) and C1-C10-multiple paraffin as multiple systems (Daridon et al., 1996). A total of 956 experimental bubble points are used to evaluate the EOS-GE models respectively with the original UNIFAC and the improved UNIFAC. The specific information of the tested pressure range and the composition of the systems are demonstrated in Table 2. 5.1. Binary systems
Fig. 1. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C10.
The binary systems are divided into two groups: methane þ heavy nparaffin and ethane þ heavy n-paraffin. The AAREs (Absolute average relative error) of the predicted bubble points by the original UNIFAC and the improved UNIFAC are shown in Table 3. It is evident that, both for methane þ heavy n-paraffin and ethane þ heavy n-paraffin, the AAREs by the original UNIFAC are much higher than those by the improved UNIFAC, due to the inaccuracy assumption that “all groups are isotropic in solution”. The global average AARE by the improved UNIFAC is 5.84%, much lower than 9.62% predicted by the original UNIFAC. According to the research of Jiding Li et al. (1998), the concept of group contributions that the distribution of free groups is homogeneous in the solution is not suitable for the systems with large size difference and the idea of effectiveR*k and Qk* is suggested, which depends on the size of the molecules. In the investigated binary systems, the size difference between the two components is significantly large, and the original UNIFAC can not hold the highly asymmetric binary systems. For methane þ heavy n-paraffin, a detailed discussion on C1-C10, C1C16 and C1-C24 is conducted. In Fig. 1, the predictions by the original UNIFAC and the improved UNIFAC are compared to the experimental bubble points in C1-C10. It is evident that both the original UNIFAC and the improved UNIFAC give good predictions for C1-C10 at the low mole fraction of CH4. With the increase of CH4, the deviation between the original UNIFAC and the experimental data enlarges. However, the improved UNIFAC keeps much better consistent with the experimental data at the higher mole fraction of CH4. The variation trend that the growth rate of the bubble point with temperature becomes slower and
4.2. The improved UNIFAC In this work, to consider the anisotropy of the groups in solution, a new nonlinear calculation of the segment fractions of molecules is suggested. In UNIFAC, during the calculation of γ Ci , the general equation for the segment fractions of molecules ϕi is expressed as:
xi r m ϕi ¼ P i m xj rj
(25)
j
The original UNIFAC assumes that “all groups are isotropic in solution” and sets m ¼ 1, which results in the much lower predictions for the bubble points in the highly asymmetrical systems. To modify the weakness of the original UNIFAC, Eq. (1) is used for γ Ci and the exponent m is not regarded as one and fitted from the experimental data of vapor-liquid equilibrium in C1-C10, C1-C12 and C1-C16 binary systems with the mole fraction ratio as 1:1. Consequently, m ¼ 0:91and Eq. (25) is written as:
xi r 0:91 ϕi ¼ P i 0:91 xj rj
(26)
j
For the calculation of γ Ri , the method adopted in the original UNIFAC remains unchanged.
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Fig. 3. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C24.
Fig. 5. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C2-C36.
Fig. 4. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C2-C28.
Fig. 6. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C2-C44.
slower is also captured by the improved UNIFAC. The same advantage of the improved UNIFAC is also significant in C1-C16 and C1-C24, demonstrated respectively in Figs. 2 and 3. In C1-C16 (Table 3), the AAREs are 3.4% and 9.8% respectively for the improved UNIFAC and the original UNIFAC. And in C1-C24 (Table 3), the AAREs are 5.44% and 11.41% respectively for the improved UNIFAC and the original UNIFAC. For ethane þ heavy n-paraffin, the calculated results of C2-C28, C2C36 and C2-C44 are analysed. In Fig. 4, the variation of the bubble points in C2-C28 with the mole fraction of C2H6 at constant temperature is predicted by the original UNIFAC and the improved UNIFAC. It is evident that at low mole fraction of C2H6, the prediction accuracy of the original UNIFAC is comparable to that of the improved UNIFAC. However, at high mole fraction of C2H6, the calculated bubble points by the original UNIFAC are much lower than the experimental data. It is noteworthy that the improved UNIFAC still predicts the bubble points well at high mole fraction of C2H6. The similar results are obtained in C2-C36 and C2-C44 and shown respectively in Figs. 5 and 6. In all the binary systems, it is evident that the original UNIFAC can not deal with the highly asymmetric systems with high mole fraction of light components and gives much higher prediction error compared to the improved UNIFAC. The explanations are as follows: the original
UNIFAC (Fredenslund et al., 1975) assumed that “all groups are isotropic in solution”, which is theoretically proved to be improper for the mixtures containing non-spherical molecules (large size paraffins) (Deiters, 1989). Li et al. (1998). pointed out the same defect of the original UNIFAC and employed the concept of effective R*k and Qk* to describe the difference between the same groups which are located at different paraffinic molecules. Unfortunately, R*k and Qk* were only fitted for PSRK model and not suitable for PR-EOS. To realise the high prediction accuracy through EOS-GE model, it is desicive to give a precise prediction for GE. Shen et al.'s (Weiguo et al., 1990) studied the influence of the mole fraction of light component on GE in binary systems and found that the value of GE firstly increases and then decreases with the increase of light component. And the experimental data are show in Fig. 7. It is evident that the absolute value of GE approaches the maximum at the mole fraction of light component of about 0.5. According to Eq. (15), the calculation of parameter a in PR-EOS is related to GE. The larger absolute value of GE, the stronger influence of the activity coefficient method on parameter a. There are two contributions for the activity coefficient (Coutinho et al., 1995): enthalpy contribution (γ Ri ), the energetic interactions between the components, and entropy contribution (γ Ci ), the 814
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Journal of Petroleum Science and Engineering 159 (2017) 810–817 Table 4 The AAREs by the original UNIFAC and the improved UNIFAC in multiple systems. Multiple systems
Improved UNIFAC (AARE%)
Original UNIFAC (AARE%)
Mixture A Mixture B Mixture C Mixture D
1.62 1.22 1.32 2.06
7.93 9.98 9.49 10.61
Global AARE%
1.54
9.46
Fig. 7. The variation of GE with the mole fraction of light component in C6-C16 and C8C16 (Fl€ oter et al., 1997).
Fig. 10. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in Mixture A.
differences in size and shape between the molecules. However, Kikic et al. (1980). pointed out that the residual contribution may be expected to be zero for hydrocarbon mixtures. It means that accurate description of γ Ci is decisive for the calculation of γ i in the highly asymmetric paraffinic systems and consequently for the calculation of GE. The original UNIFAC neglect the local characteristics of groups in different size paraffins. Therefore, the original UNIFAC can not precisely describe the entropy contribution γ Ci (Deiters, 1989; Li et al., 1998) and consequently GE whose effects on parameter an increases with the increase of the light component. As a result, the original UNIFAC gives larger prediction errors with the increase of light component, as is shown in Figs. 1–6 and Table 3. On the contrary, the improved UNIFAC uses the nonlinear calculation of the segment fraction ϕi (Eq. (26)) to take the local characteristics of groups into account. Hence, the improved UNIFAC keeps good consistent with the experimental bubble points at both low and high mole fraction of the light component, as is demonstrated in Figs. 1–6 and Table 3.
Fig. 8. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C10-C32.
5.2. Ternary and quaternary systems In ternary and quaternary systems, C1-C10-C32 and C1-C16-C17-C18 are tested. The results are shown respectively in Figs. 8 and 9. As is shown in Fig. 8, it is evident that the improved UNIFAC behaves much better than the original UNIFAC. The AARE of the improved UNIFAC is 6.63% and only about 1/3 of the AARE (18.88%) of the original UNIFAC. In Fig. 9, the variation of bubble points with temperature at certain composition is studied. The same results in binary systems are also found in C1-C16-C17-C18 that the improved UNIFAC and the original UNIFAC have the comparable prediction accuracy at low mole fraction of the light component and that at high mole fraction of the light component, the improved UNIFAC always shows good performance with the original UNIFAC giving much lower predictions. And the AAREs are 3.83% and 15.30% respectively for the improved UNIFAC and the original UNIFAC.
Fig. 9. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C16-C17-C18.
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5.3. Multiple systems The improved UNIFAC and the original UNIFAC are also applied to multiple systems. The AAREs of the improved UNIFAC and the original UNIFAC are listed in Table 4. It is evident that for mixture A to D, the improved UNIFAC gives much more accurate predictions than the original UNIFAC. And the global AAREs are 1.54% and 9.46% respectively for the improved UNIFAC and the original UNIFAC. The detailed comparison between the experimental data and the predictions are shown in Figs. 10–13 respectively for Mixture A to D. Again, the original UNIFAC predicts much lower bubble points than the improved UNIFAC. The reasons are analysed in the binary systems. 6. Conclusions In this work, a modified EOS-GE model is developed for the vaporliquid equilibrium in the highly asymmetric paraffinic systems. The volume translated PR-EOS with the LCVM mixing rule is adopted and the original UNIFAC is improved by the nonlinear calculation of the segment fraction ϕi in the combinatorial activity coefficient γ Ci . A total of 956 experimental bubble points obtained from 17 highly asymmetric paraffinic systems are used to evaluate the original UNIFAC and the improved UNIFAC. Results show that the original UNIFAC and the improved UNIFAC give accurate predictions for bubble points under the condition of low mole fraction of light component. However, the improved UNIFAC still keeps great consistent with the experimental data at high mole fraction of the light component. It is the consideration of the local characteristics of groups in different molecules that ensures the high prediction accuracy of the improved UNIFAC. The original UNIFAC shows remarkably larger prediction errors with the increase of the light component, due to the assumption that “all groups are isotropic in solution”. It is evident that the improved UNIFAC is much superior to the original UNIFAC. Nevertheless, it must be acknowledge that the EOS-GE established in this work may generate non-realistic predictions at extreme conditions, due to the defects of α function (Le Guennec et al., 2016a; Segura et al., 2003; Le Guennec et al., 2016b), translation volume correlation (Le Guennec et al., 2016b; Kalikhman et al., 2010; Privat et al., 2016), and the application range of group interaction parameters (Polishuk et al., 2002). To develop a more general EOS-GE model, such problems should be resolved in the future work. In addition, the inapplicability of present model for dew points also needs improvement for the practical applications.
Fig. 11. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in Mixture B.
Fig. 12. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in Mixture C.
Acknowledgement The authors wish to thank the National Natural Science Foundation of China (51422406, 51534007), the National Science & Technology Specific Project (2016ZX05028-004-001), and the Science Foundation of China University of Petroleum Beijing (C201602) for providing support for this work. References Abrams, D.S., Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J. 21, 116–128. Baled, H., Enick, R.M., Wu, Y., McHugh, M.A., Burgess, W., Tapriyal, D., Morreale, B.D., 2012. Prediction of hydrocarbon densities at extreme conditions using volumetranslated SRK and PR equations of state fit to high temperature, high pressure PVT data. Fluid Phase Equilibria 317, 65–76. Bondi, A.A., 1968. Physical Properties of Molecular Crystals Liquids, and Glasses. Boukouvalas, C., Spiliotis, N., Coutsikos, P., Tzouvaras, N., Tassios, D., 1994. Prediction of vapor-liquid equilibrium with the LCVM model: a linear combination of the Vidal and Michelsen mixing rules coupled with the original UNIF. Fluid Phase Equilibria 92, 75–106. Cordeiro, D.J., Luks, K.D., Kohn, J.P., 1973. Process for extracting high-molecular-weight hydrocarbons from solid phase in equilibrium with liquid hydrocarbon phase. Ind. Eng. Chem. Process Des. Dev. 12, 47–51 (United States).
Fig. 13. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in Mixture D.
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Journal of Petroleum Science and Engineering 159 (2017) 810–817 Mathias, P.M., Naheiri, T., Oh, E.M., 1989. A density correction for the Peng—Robinson equation of state. Fluid Phase Equilibria 47, 77–87. Michelsen, M.L., 1990. A method for incorporating excess Gibbs energy models in equations of state. Fluid Phase Equilibria 60, 47–58. Michelsen, M.L., 1990. A modified Huron-Vidal mixing rule for cubic equations of state. Fluid Phase Equilibria 60, 213–219. Mollerup, J., 1981. A note on excess Gibbs energy models, equations of state and the local composition concept. Fluid Phase Equilibria 7, 121–138. Patel, N.C., Teja, A.S., 1982. A new cubic equation of state for fluids and fluid mixtures. Chem. Eng. Sci. 37, 463–473. Pauly, J., Coutinho, J., Daridon, J.L., 2007. High pressure phase equilibria in methaneþ waxy systems: 1. Methaneþ heptadecane. Fluid phase Equilibria 255, 193–199. Pauly, J., Coutinho, J.A.P., Daridon, J.L., 2010. High pressure phase equilibria in methaneþ waxy systems. 2. Methaneþ waxy ternary mixture. Fluid Phase Equilibria 297, 149–153. Peters, C.J., Spiegelaar, J., Arons, J.D.S., 1988. Phase equilibria in binary mixtures of ethaneþ docosane and molar volumes of liquid docosane. Fluid Phase Equilibria 41, 245–256. Polishuk, I., Stateva, R.P., Wisniak, J., Segura, H., 2002. Prediction of high-pressure phase equilibria using cubic EOS: what can Be learned? Can. J. Chem. Eng. 80, 927–942. Privat, R., Jaubert, J.N., Le Guennec, Y., 2016. Incorporation of a volume translation in an equation of state for fluid mixtures: which combining rule? which effect on properties of mixing? Fluid Phase Equilibria 427, 414–420. Reamer, H.H., Sage, B.H., 1962. Phase equilibria in hydrocarbon systems. Volumetric and phase behavior of the ethane-n-decane system. J. Chem. Eng. Data 7, 161–168. Redlich, O., Kwong, J.N.S., 1949. On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chem. Rev. 44, 233–244. Renon, H., Prausnitz, J.M., 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J. 14, 135–144. Rijkers, M., Malais, M., Peters, C.J., de Swaan Arons, J., 1992. Measurements on the phase behavior of binary hydrocarbon mixtures for modelling the condensation behavior of natural gas: Part I. The system methaneþ decane. Fluid phase Equilibria 71, 143–168. Rijkers, M., Maduro, V.B., Peters, C.J., de Swaan Arons, J., 1992. Measurements on the phase behavior of binary mixtures for modeling the condensation behavior of natural gas: Part II. The system methaneþ dodecane. Fluid phase Equilibria 72, 309–324. Sayegh, S.G., Vera, J.H., 1980. Lattice-model expressions for the combinatotial entropy of liquid mixtures: a critical discussion. Chem. Eng. J. 19, 1–10. Segura, H., Kraska, T., Mejía, A., Wisniak, J., Polishuk, I., 2003. Unnoticed pitfalls of soave-type alpha functions in cubic equations of state. Ind. Eng. Chem. Res. 42, 5662–5673. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 27, 1197–1203. Stryjek, R., Vera, J.H., 2010. PRSV: an improved Peng—Robinson equation of state for pure compounds and mixtures. Can. J. Chem. Eng. 64 (2), 323–333. Twu, C.H., 1984. An internally consistent correlation for predicting the critical properties and molecular weights of petroleum and coal-tar liquids. Fluid phase Equilibria 16, 137–150. Weidlich, U., Gmehling, J., 1987. A modified UNIFAC model. I: prediction of VLE, hE, and γ∞. Ind. Eng. Chem. Res. 26, 1372–1381. Weiguo, S., Qin, A.X., McElroy, P.J., Williamson, A.G., 1990. (Vapourþ liquid) equilibria of (n-hexaneþ n-hexadecane),(n-hexaneþ n-octane), and (n-octaneþ n-hexadecane). J. Chem. Thermodyn. 22, 905–914. Wilson, G.M., 1964. Vapor-liquid equilibrium. XI. A new expression for the excess free energy of mixing. J. Am. Chem. Soc. 86, 127–130.
Coutinho, J.A.P., Andersen, S.I., Stenby, E.H., 1995. Evaluation of activity coefficient models in prediction of alkane solid-liquid equilibria. Fluid Phase Equilibria 103, 23–39. Dahl, S., Michelsen, M.L., 1990. High-pressure vapor-liquid equilibrium with a UNIFACbased equation of state. AIChE J. 36, 1829–1836. Daridon, J.L., Xans, P., Montel, F., 1996. Phase boundary measurement on a methaneþ decaneþ multi-paraffins system. Fluid phase Equilibria 117, 241–248. Deiters, U.K., 1989. Extended 1-fluid theory for mixtures containing non-spherical molecules. Fluid phase Equilibria 48, 185–195. Edmister, W.C., 1958. Applied hydrocarbon thermodynamics, Part 4, compressibility factor and equations of state. Pet. Re Fin. 173–179. Flory, P.J., 1942. Thermodynamics of high polymer solutions. J. Chem. Phys. 10, 51–61. Fl€ oter, E., De Loos, T.W., de Swaan Arons, J., 1997. High pressure solid-fluid and vapourliquid equilibria in the system (methaneþ tetracosane). Fluid phase Equilibria 127, 129–146. Fredenslund, A., Jones, R.L., Prausnitz, J.M., 1975. Group-contribution estimation of activity coefficients in nonideal liquid mixtures. AIChE J. 21, 1086–1099. Gasem, K.A.M., Bufkin, B.A., Raff, A.M., Robinson Jr., R.L., 1989. Solubilities of ethane in heavy normal paraffins at pressures to 7. 8 MPa and temperatures from 348 to 423. Kappa. J. Chem. Eng. Data; (USA) 34, 187–191. Glaser, M., Peters, C.J., Van Der Kool, H.J., Lichtenthaler, R.N., 1985. Phase equilibria of (methaneþ n-hexadecane) and (p, V m, T) of n-hexadecane. J. Chem. Thermodyn. 17, 803–815. De Goede, R., Peters, C.J., Van Der Kooi, H.J., Lichtenthaler, R.N., 1989. Phase equilibria in binary mixtures of ethane and hexadecane. Fluid Phase Equilibria 50, 305–314. Le Guennec, Y., Lasala, S., Privat, R., Jaubert, J.N., 2016. A consistency test for α-functions of cubic equations of state. Fluid Phase Equilibria 427, 513–538. Le Guennec, Y., Privat, R., Jaubert, J.N., 2016. Development of the translated-consistent tc-PR and tc-RK cubic equations of state for a safe and accurate prediction of volumetric, energetic and saturation properties of pure compounds in the sub-and super-critical domains. Fluid Phase Equilibria 429, 301–312. Heidemann, R.A., Kokal, S.L., 1990. Combined excess free energy models and equations of state. Fluid Phase Equilibria 56, 17–37. Hildebrand, J.H., 1929. Solubility. XII. Regular Solutions1. J. Am. Chem. Soc. 51, 66–80. Huang, S.H., Lin, H.M., Chao, K.C., 1988. Solubility of carbon dioxide, methane, and ethane in n-eicosane. J. Chem. Eng. Data 33, 145–147. Huggins, M.L., 1942. Thermodynamic properties of solutions of long-chain compounds. Ann. N. Y. Acad. Sci. 43, 1–32. Huron, M.J., Vidal, J., 1979. New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria 3, 255–271. Kalikhman, V., Kost, D., Polishuk, I., 2010. About the physical validity of attaching the repulsive terms of analytical EOS models by temperature dependencies. Fluid Phase Equilibria 293, 164–167. Kikic, I., Alessi, P., Rasmussen, P., Fredenslund, A., 1980. On the combinatorial part of the UNIFAC and UNIQUAC models. Can. J. Chem. Eng. 58, 253–258. Kontogeorgis, G.M., Coutsikos, P., 2012. Thirty years with EoS/GE models-what have we learned? Ind. Eng. Chem. Res. 51, 4119–4142. Van der Kooi, H.J., Fl€ oter, E., De Loos, T.W., 1995. High-pressure phase equilibria of {(1x) CH4þ xCH3 (CH2) 18CH3}. J. Chem. Thermodyn. 27, 847–861. Larsen, B.L., Rasmussen, P., Fredenslund, A., 1987. A modified UNIFAC groupcontribution model for prediction of phase equilibria and heats of mixing. Ind. Eng. Chem. Res. 26, 2274–2286. Li, J., Fischer, K., Gmehling, J., 1998. Prediction of vapor–liquid equilibria for asymmetric systems at low and high pressures with the PSRK model. Fluid Phase Equilibria 143, 71–82.
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Prediction of vapor-liquid equilibrium in highly asymmetric paraffinic systems with new modified EOS-GE model Juheng Yang a, b, Jing Gong a, c, *, Guoyun Shi a, Huirong Huang a, Dan Wang a, Wei Wang a, b, **, Qingping Li d, Bohui Shi a, Haiyuan Yao d a
National Engineering Laboratory for Pipeline Safety, China University of Petroleum, Beijing 102249, PR China MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, PR China Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, PR China d China National Offshore Oil Cooperation Research Center, Beijing 100027, PR China b c
A R T I C L E I N F O Keywords: Thermodynamics Vapor-liquid equilibrium Asymmetric paraffinic system EOS-GE UNIFAC Activity coefficient
A B S T R A C T
A new modified EOS-GE model is developed for the highly asymmetric paraffinic systems, where the volume translated Peng-Robinson EOS is adopted coupled with the LCVM mixing rule. In the new modified EOS-GE model, the original UNIFAC is replaced by a newly established UNIFAC where the nonlinear calculation of the segment fractions of molecules in γC (the combinatorial activity coefficient) is introduced to modify the traditional assumption that “all groups are isotropic in solution”. A total of 956 vapor-liquid experimental bubble points in highly asymmetric paraffinic systems including binary systems, ternary systems, quaternary systems and multiple systems are used to test the new developed EOS-GE model. Results show that the original UNIFAC and the improved UNIFAC both perform well if the molefractions of light components (CH4 or C2H6) are low; however, with the increase of the light components, the improved UNIFAC is remarkably superior to the original UNIFAC.
1. Introduction In petroleum production, the vapor-liquid equilibrium plays an extremely important role in the process of exploitation, gathering and transportation and rectification. Therefore, the accurate prediction of vapor-liquid equilibrium is critical for the petroleum industry. In the modeling of vapor-liquid equilibrium, the vapor phase is usually described by cubic Equation of State (Redlich and Kwong, 1949; Soave, 1972; Stryjek and Vera, 2010; Patel and Teja, 1982); while the liquid phase can be modeled in two different ways: Equation of State (EOS) (Redlich and Kwong, 1949; Soave, 1972; Stryjek and Vera, 2010; Patel and Teja, 1982) or activity coefficient method (GE) (Hildebrand, 1929; Flory, 1942; Huggins, 1942; Wilson, 1964; Renon and Prausnitz, 1968; Abrams and Prausnitz, 1975; Fredenslund et al., 1975). The objective with the use of EoS/GE models is to combine the “advantages” of cubic EOS and of the local composition activity coefficient models incorporated (Kontogeorgis and Coutsikos, 2012). The EOS can deal with the influence of large pressure variation, but it is not suitable for the systems with high polarity and high asymmetry. On the contrary, the activity coefficient method (GE) can describe the non-ideality of polar and asymmetric
systems well without the capability to reflect the effects of pressure variation. Therefore, it is necessary to build a bridge between EOS and GE. To introduce GE into EOS, the traditional linear mixing rule is modified. Huron and Vidal (1979) assumed that fluid systems were in a state of liquid or nearly liquid at the infinite pressure and established the HV mixing rule which predicts well for some complicated fluid systems. Mollerup (1981), Heidemann and Kolal (Heidemann and Kokal, 1990) proved the possibility that GE and EOS could be integrated at zero or low pressure. Later, Michelsen et al (Michelsen, 1990a, 1990b; Dahl and Michelsen, 1990). developed the mixing rule at zero pressure (MHV1) and amplified the application range of EOS-GE. Boukouvalas et al. (1994). found that as the size difference of tested systems increases, the HV mixing rule led to underprediction and the MHV1 mixing rule led to overprediction. Therefore, Boukouvalas et al. (1994). proposed the LCVM mixing rule (Linear Combination of the Vidal and Michelsen mixing rules) to make up for the weakness of the HV mixing rule and the MHV1 mixing rule. These EOS/GE mixing rules realized two progresses (Kontogeorgis and Coutsikos, 2012). First of all, the cubic EOS is successfully applied to the mixtures of compounds of wide complexity and
* Corresponding author. National Engineering Laboratory for Pipeline Safety, China University of Petroleum, Beijing 102249, PR China. ** Corresponding author. National Engineering Laboratory for Pipeline Safety, China University of Petroleum, Beijing 102249, PR China. E-mail addresses: [email protected] (J. Gong), [email protected] (W. Wang). https://doi.org/10.1016/j.petrol.2017.10.015 Received 10 April 2017; Received in revised form 1 October 2017; Accepted 4 October 2017 Available online 7 October 2017 0920-4105/© 2017 Published by Elsevier B.V.
J. Yang et al.
Journal of Petroleum Science and Engineering 159 (2017) 810–817 2=3
xi r ϕi ¼ P i 2=3 xj rj
Table 1 Parameters of Eq. (11). Constants
Values
A
In 1993, Gmehling et al (Weidlich and Gmehling, 1987). developed another way to improve the calculation method for the combinatorial part:
4.1034 31.723 0.0531 188.68 0.0057 20,196 0.0003
k0 k1 k2 k3 k4 k5 k6
ϕ ϕ ln γ Ci ¼ 1 ϕ0i þ ln ϕ0i 5qi 1 i þ ln i θi θi
(3)
3=4
r ϕ0i ¼ P i 3=4 rj xj
B 0.3489 28.547 0.0687 817.73 0.0007 65.067 0.0076
k0 k1 k2 k3 k4 k5 k6
(4)
j
Similarly, to improve the phase equilibria prediction of the highly asymmetric paraffinic systems, this work develops a new modified EOSGE model where the volume translated PR-EOS with the LCVM mixing rule is adopted. And GE is calculated by the newly improved UNIFAC with the nonlinear calculation of the molecule's segment fractions to consider the anisotropy of groups in solution.
asymmetry in size and energies. Secondly, the incorporated activity coefficient models established at low pressure can be extrapolated to the higher pressure conditions. For the high prediction accuracy of the vapor-liquid equilibrium, the calculation method for GE should be improved. UNIFAC is an approximation method where the properties of the practical systems are predicted by the properties of the composed chemical groups. The original UNIFAC is not suitable for systems with large size difference due to the assumption that “all groups are isotropic in solution” (Deiters, 1989). Deiters et al. (Deiters, 1989) proposed a nonlinear calculation for the volume fractions of non-spherical molecules. Li et al. (1998). employed the concept of effective R*k and Qk* to describe the local characteristic of groups. Sayegh and Vera (1980) pointed out that the Staverman-Guggenheim correction may give unrealistic large corrections to the combinatorial excess entropy. Consequently, Larsen et al. (1987). and Kikic et al. (1980), dropped the Staverman-Guggenheim correction in the UNIFAC model:
ln γ Ci ¼ ln
(2)
j
ϕi ϕ þ1 i xi xi
2. The equation of state The PR-EOS with the consideration of the volume translation effects (Mathias et al., 1989) is used:
P¼
RT a V þ t b ðV þ tÞðV þ t þ bÞ þ bðV þ t bÞ
(5)
where P is the pressure of the system; T is the temperature of the system; R is the gas universal constant; V is the volume of gas or liquid; t is the translation volume. For pure component, parameter a is calculated by (Stryjek and Vera, 2010):
a ¼ 0:45724
(1)
R2 Tc2 ⋅αðTÞ pc
(6)
2 αðTÞ ¼ 1 þ k 1 Tr0:5
(7)
k ¼ 0:37464 þ 1:54226ω 0:26992ω2
(8)
Table 2 The tested pressure range and the composition of the systems. Tested system Binary system C1þC10 C1þC12 C1þC16 C1þC17 C1þC20 C1þC24 C1þC32 C2þC10 C2þC16 C2þC20 C2þC22 C2þC28 C2þC36 C2þC44 Ternary system C1þC10 þ C32 Quaternary system C1þC16 þ C17 þ C18 Multiple system C1þC10þ Multiple-paraffin Total number of points
light component content range
Pressure range (MPa)
Temperature range(K)
Number of points
Reference
0.1–0.55 0.1–0.6 0.1–0.6 0.2–0.6 0.025–0.65 0.1–0.7 0.1–0.325 0.017–0.995 0.2–0.875 0.07–0.47 0.05–0.9 0.1–0.5 0.087–0.531 0.1–0.52
1–32 1–23 2–26 4–27 0–32 1–30 1.5–7 0.1–11 0.5–16 0.5–4 0.1–9.5 0.5–5.5 0.3–5 0.3–3.5
243.15–313.15 255–320 285–360 293–373 303–370 315–450 343.15 277.6–510.9 260–450 373.75–572.85 290–370 348–423 373–423 373–423
61 38 84 29 83 127 10 107 148 11 110 24 13 15
(Rijkers et al., 1992a) (Rijkers et al., 1992b) (Glaser et al., 1985) (Pauly et al., 2007) (Van der Kooi et al., 1995) (Fl€ oter et al., 1997) (Cordeiro et al., 1973) (Reamer and Sage, 1962) (De Goede et al., 1989) (Huang et al., 1988) (Peters et al., 1988) (Gasem et al., 1989) (Gasem et al., 1989) (Gasem et al., 1989)
0.025–0.4
0.1–10
330–340
28
(Cordeiro et al., 1973)
0.2–0.6
3–30
293–373
14
(Pauly et al., 2010)
C1:0.436–0.440 C10:0.458–0.462
12–16
293–423
54
(Daridon et al., 1996)
956
811
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Journal of Petroleum Science and Engineering 159 (2017) 810–817
4. Modification of UNIFAC model
where Tc is the critical temperature; pc is the critical pressure; Tr is the reduced temperature; ω is the acentric factor. Although Soave α function (Eq. (7)) is doubted because of its inconsistency in supercritical region (Le Guennec et al., 2016a; Segura et al., 2003; Le Guennec et al., 2016b), it is certainly the most employed α function in the oil and gas industries. Thus, for the hydrocarbon mixture in this work, Eq. (7) is still used. For pure component, parameter b is obtained from (Stryjek and Vera, 2010):
RTc b ¼ 0:07780 pc
4.1. The original UNIFAC The UNIFAC model is developed by applying the group contribution method to the UNIQUAC model (Universal Quasichemical) (Fredenslund et al., 1975). And the calculation of activity coefficient is as follows:
ln γ i ¼ ln γ Ci þ ln γ Ri
(9)
whereγ Ci represents the entropy contribution, the differences in size and shape between the molecules and γ Ri is the enthalpy contribution, the energetic interactions between the components. The calculation of γ Ci is the same as that in UNIQUAC:
The translation volume, t, is suggested by Baled et al. (2012):
t ¼ A þ B:Tr
(10)
A; B ¼ f ðM; ωÞ
1 1 1 þ k3 exp þ k5 exp ¼ k0 þ k1 exp k2 Mω k4 Mω k6 Mω
(16)
ln γ Ci ¼ ln
(11)
m ϕi z θi ϕ X þ qi ln þ li i xj lj xi 2 ϕi xi j¼1
z li ¼ ðri qi Þ ðri 1Þ; 2
The valuesof k0 , k1 , k2 , k3 , k4 , k5 , k6 are shown in Table 1: While it must be acknowledge that several studies recommended temperature-independent volume-translation parameters and proved that the temperature-dependent volume-translation parameters may results in unrealistic predictions at extreme conditions (Le Guennec et al., 2016b; Kalikhman et al., 2010; Privat et al., 2016). However, Baled's study (Baled et al., 2012) covers short- and long-chain alkanes ranging from CH4 to C40H82 at pressures between 7 and 276 MPa and temperatures between 278 and 533 K. So Baled's translation volume can be used for the investigated mixtures which are included in Baled's research (shown as Table 2). The critical properties of pure components, such as Pc and Tc, are proposed by Twu (1984). And the acentric factor is calculated by Edmister correlation (Edmister, 1958).
qi xi θi ¼ P ; qj xj j
(17)
z ¼ 10
(18)
ri xi ϕi ¼ P rj xj
(19)
j
In these equations, xi is the mole fraction of component i; θi is the area fraction; ϕi is the segment fraction which is similar to the volume fraction; ri and qi are respectively the molecular van der Waals volume and the molecular surface area for pure component which are calculated as the sum of the group volume and area parameters,Rk and Qk , obtained from the van der Waals group volume and surface areas given by Bondi (1968):
ri ¼
X ðiÞ vk Rk ;
qi ¼
X ðiÞ vk Qk
k
3. The LCVM mixing rule
(20)
k
ðiÞ
The LCVM mixing rule (Boukouvalas et al., 1994) is the linear combination of HV mixing rule and MHV1 mixing rule and adopted:
where vk is the number of groups of type k in molecule i. However, the solution-of-groups concept is introduced into the calculation of γ Ri in UNIFAC:
α ¼ λ:αV þ ð1 λÞ:αM
(12)
ln γ Ri ¼
1 GE X ⋅ þ xi ai AV RT i
(13)
αV ¼
" # X 1 GE X b þ þ αM ¼ ⋅ xi In xi ai AM RT b i i i
α¼
a ¼ bRT
X λ 1 λ GE 1 λ X b ⋅ þ þ þ ⋅ xi In xi ai AV AM AM i bi RT i
h i ðiÞ ðiÞ vk ln Γ k ln Γ k
(21)
k all groups
ðiÞ
where Γ k is the group residual activity coefficient; Γ k is the group residual activity coefficient of group k in the solution which only contains molecule i. Γ k is obtained from:
"
(14)
X ln Γ k ¼ Qk 1 ln Θm ψ mk
Combine Eqs. (12)–(14), one obtains the expression of the LCVM mixing rule:
X
!
m
X
, !# X Θn ψ nm Θm ψ km
m
n
(22)
ðiÞ
Eq. (22) is also suitable for Γ k . In Eq. (22), Θm is the area fraction of group m and calculated by:
(15)
Qm Xm Θm ¼ P Qn Xn
For PR-EOS, AV ¼ 0:623and AM ¼ 0:52 (Boukouvalas et al., 1994). The mixing rule yields the HV and the MHV1 mixing rule, respectively for λ ¼ 1and λ ¼ 0. In this work, λ ¼ 0:36 suggested by Boukouvalas (Boukouvalas et al., 1994) is used. However, Polishuk et al. (2002). proved that UNIFAC model (temperatures outside 250–425 K) was able to generate non-realistic phase diagrams. In Table 2, it is evident that the investigated temperature range covers the most of the asymmetric paraffinic systems except C2-C10 and C2-C20. Unfortunately, there is little UNIFAC group interaction parameters fitted from the experimental data at temperatures above 500 K. Therefore, the UNIFAC group interaction parameters covering temperature range from 250 to 425 K is extrapolated to the cases of C2-C10 and C2-C20.
(23)
n
Xm is the mole fraction of group m in the mixture. ψ mn Is the group interaction parameter and given by:
a mn ψ mn ¼ exp T
(24)
The parameters amn are fitted from the experimental data. Note that amn ≠anm and that if n ¼ m, then amn ¼ 0 and ψ mn ¼ 1.
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Journal of Petroleum Science and Engineering 159 (2017) 810–817
Table 3 The AAREs by the original UNIFAC and the improved UNIFAC in binary systems. Binary system
Improved UNIFAC (AARE%)
Original UNIFAC (AARE%)
C1þC10 C1þC12 C1þC16 C1þC17 C1þC20 C1þC24 C1þC32 C2þC10 C2þC16 C2þC20 C2þC22 C2þC28 C2þC36 C2þC44
2.92 5.36 3.40 3.72 7.23 5.44 7.83 4.18 7.69 3.62 10.57 5.49 5.78 7.74
7.93 7.30 9.80 11.47 8.92 11.41 9.23 6.27 11.65 6.15 16.37 9.54 10.93 10.20
Global average AARE%
5.78
9.80
Fig. 2. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C16.
5. Results and discussion In this work, 17 highly asymmetric paraffinic systems are investigated including C1-C10, C1-C12, C1-C16, C1-C17, C1-C20, C1-C24, C1-C32, C2-C10, C2-C16, C2-C20, C2-C22, C2-C28, C2-C36, C2-C44 as binary systems (Rijkers et al., 1992a, 1992b; Glaser et al., 1985; Pauly et al., 2007; Van der Kooi et al., 1995; Fl€ oter et al., 1997; Cordeiro et al., 1973; Reamer and Sage, 1962; De Goede et al., 1989; Huang et al., 1988; Peters et al., 1988; Gasem et al., 1989), C1-C10-C32 as ternary system (Cordeiro et al., 1973), C1-C16-C17-C18 as quaternary system (Pauly et al., 2010) and C1-C10-multiple paraffin as multiple systems (Daridon et al., 1996). A total of 956 experimental bubble points are used to evaluate the EOS-GE models respectively with the original UNIFAC and the improved UNIFAC. The specific information of the tested pressure range and the composition of the systems are demonstrated in Table 2. 5.1. Binary systems
Fig. 1. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C10.
The binary systems are divided into two groups: methane þ heavy nparaffin and ethane þ heavy n-paraffin. The AAREs (Absolute average relative error) of the predicted bubble points by the original UNIFAC and the improved UNIFAC are shown in Table 3. It is evident that, both for methane þ heavy n-paraffin and ethane þ heavy n-paraffin, the AAREs by the original UNIFAC are much higher than those by the improved UNIFAC, due to the inaccuracy assumption that “all groups are isotropic in solution”. The global average AARE by the improved UNIFAC is 5.84%, much lower than 9.62% predicted by the original UNIFAC. According to the research of Jiding Li et al. (1998), the concept of group contributions that the distribution of free groups is homogeneous in the solution is not suitable for the systems with large size difference and the idea of effectiveR*k and Qk* is suggested, which depends on the size of the molecules. In the investigated binary systems, the size difference between the two components is significantly large, and the original UNIFAC can not hold the highly asymmetric binary systems. For methane þ heavy n-paraffin, a detailed discussion on C1-C10, C1C16 and C1-C24 is conducted. In Fig. 1, the predictions by the original UNIFAC and the improved UNIFAC are compared to the experimental bubble points in C1-C10. It is evident that both the original UNIFAC and the improved UNIFAC give good predictions for C1-C10 at the low mole fraction of CH4. With the increase of CH4, the deviation between the original UNIFAC and the experimental data enlarges. However, the improved UNIFAC keeps much better consistent with the experimental data at the higher mole fraction of CH4. The variation trend that the growth rate of the bubble point with temperature becomes slower and
4.2. The improved UNIFAC In this work, to consider the anisotropy of the groups in solution, a new nonlinear calculation of the segment fractions of molecules is suggested. In UNIFAC, during the calculation of γ Ci , the general equation for the segment fractions of molecules ϕi is expressed as:
xi r m ϕi ¼ P i m xj rj
(25)
j
The original UNIFAC assumes that “all groups are isotropic in solution” and sets m ¼ 1, which results in the much lower predictions for the bubble points in the highly asymmetrical systems. To modify the weakness of the original UNIFAC, Eq. (1) is used for γ Ci and the exponent m is not regarded as one and fitted from the experimental data of vapor-liquid equilibrium in C1-C10, C1-C12 and C1-C16 binary systems with the mole fraction ratio as 1:1. Consequently, m ¼ 0:91and Eq. (25) is written as:
xi r 0:91 ϕi ¼ P i 0:91 xj rj
(26)
j
For the calculation of γ Ri , the method adopted in the original UNIFAC remains unchanged.
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Fig. 3. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C24.
Fig. 5. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C2-C36.
Fig. 4. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C2-C28.
Fig. 6. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C2-C44.
slower is also captured by the improved UNIFAC. The same advantage of the improved UNIFAC is also significant in C1-C16 and C1-C24, demonstrated respectively in Figs. 2 and 3. In C1-C16 (Table 3), the AAREs are 3.4% and 9.8% respectively for the improved UNIFAC and the original UNIFAC. And in C1-C24 (Table 3), the AAREs are 5.44% and 11.41% respectively for the improved UNIFAC and the original UNIFAC. For ethane þ heavy n-paraffin, the calculated results of C2-C28, C2C36 and C2-C44 are analysed. In Fig. 4, the variation of the bubble points in C2-C28 with the mole fraction of C2H6 at constant temperature is predicted by the original UNIFAC and the improved UNIFAC. It is evident that at low mole fraction of C2H6, the prediction accuracy of the original UNIFAC is comparable to that of the improved UNIFAC. However, at high mole fraction of C2H6, the calculated bubble points by the original UNIFAC are much lower than the experimental data. It is noteworthy that the improved UNIFAC still predicts the bubble points well at high mole fraction of C2H6. The similar results are obtained in C2-C36 and C2-C44 and shown respectively in Figs. 5 and 6. In all the binary systems, it is evident that the original UNIFAC can not deal with the highly asymmetric systems with high mole fraction of light components and gives much higher prediction error compared to the improved UNIFAC. The explanations are as follows: the original
UNIFAC (Fredenslund et al., 1975) assumed that “all groups are isotropic in solution”, which is theoretically proved to be improper for the mixtures containing non-spherical molecules (large size paraffins) (Deiters, 1989). Li et al. (1998). pointed out the same defect of the original UNIFAC and employed the concept of effective R*k and Qk* to describe the difference between the same groups which are located at different paraffinic molecules. Unfortunately, R*k and Qk* were only fitted for PSRK model and not suitable for PR-EOS. To realise the high prediction accuracy through EOS-GE model, it is desicive to give a precise prediction for GE. Shen et al.'s (Weiguo et al., 1990) studied the influence of the mole fraction of light component on GE in binary systems and found that the value of GE firstly increases and then decreases with the increase of light component. And the experimental data are show in Fig. 7. It is evident that the absolute value of GE approaches the maximum at the mole fraction of light component of about 0.5. According to Eq. (15), the calculation of parameter a in PR-EOS is related to GE. The larger absolute value of GE, the stronger influence of the activity coefficient method on parameter a. There are two contributions for the activity coefficient (Coutinho et al., 1995): enthalpy contribution (γ Ri ), the energetic interactions between the components, and entropy contribution (γ Ci ), the 814
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Journal of Petroleum Science and Engineering 159 (2017) 810–817 Table 4 The AAREs by the original UNIFAC and the improved UNIFAC in multiple systems. Multiple systems
Improved UNIFAC (AARE%)
Original UNIFAC (AARE%)
Mixture A Mixture B Mixture C Mixture D
1.62 1.22 1.32 2.06
7.93 9.98 9.49 10.61
Global AARE%
1.54
9.46
Fig. 7. The variation of GE with the mole fraction of light component in C6-C16 and C8C16 (Fl€ oter et al., 1997).
Fig. 10. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in Mixture A.
differences in size and shape between the molecules. However, Kikic et al. (1980). pointed out that the residual contribution may be expected to be zero for hydrocarbon mixtures. It means that accurate description of γ Ci is decisive for the calculation of γ i in the highly asymmetric paraffinic systems and consequently for the calculation of GE. The original UNIFAC neglect the local characteristics of groups in different size paraffins. Therefore, the original UNIFAC can not precisely describe the entropy contribution γ Ci (Deiters, 1989; Li et al., 1998) and consequently GE whose effects on parameter an increases with the increase of the light component. As a result, the original UNIFAC gives larger prediction errors with the increase of light component, as is shown in Figs. 1–6 and Table 3. On the contrary, the improved UNIFAC uses the nonlinear calculation of the segment fraction ϕi (Eq. (26)) to take the local characteristics of groups into account. Hence, the improved UNIFAC keeps good consistent with the experimental bubble points at both low and high mole fraction of the light component, as is demonstrated in Figs. 1–6 and Table 3.
Fig. 8. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C10-C32.
5.2. Ternary and quaternary systems In ternary and quaternary systems, C1-C10-C32 and C1-C16-C17-C18 are tested. The results are shown respectively in Figs. 8 and 9. As is shown in Fig. 8, it is evident that the improved UNIFAC behaves much better than the original UNIFAC. The AARE of the improved UNIFAC is 6.63% and only about 1/3 of the AARE (18.88%) of the original UNIFAC. In Fig. 9, the variation of bubble points with temperature at certain composition is studied. The same results in binary systems are also found in C1-C16-C17-C18 that the improved UNIFAC and the original UNIFAC have the comparable prediction accuracy at low mole fraction of the light component and that at high mole fraction of the light component, the improved UNIFAC always shows good performance with the original UNIFAC giving much lower predictions. And the AAREs are 3.83% and 15.30% respectively for the improved UNIFAC and the original UNIFAC.
Fig. 9. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in C1-C16-C17-C18.
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5.3. Multiple systems The improved UNIFAC and the original UNIFAC are also applied to multiple systems. The AAREs of the improved UNIFAC and the original UNIFAC are listed in Table 4. It is evident that for mixture A to D, the improved UNIFAC gives much more accurate predictions than the original UNIFAC. And the global AAREs are 1.54% and 9.46% respectively for the improved UNIFAC and the original UNIFAC. The detailed comparison between the experimental data and the predictions are shown in Figs. 10–13 respectively for Mixture A to D. Again, the original UNIFAC predicts much lower bubble points than the improved UNIFAC. The reasons are analysed in the binary systems. 6. Conclusions In this work, a modified EOS-GE model is developed for the vaporliquid equilibrium in the highly asymmetric paraffinic systems. The volume translated PR-EOS with the LCVM mixing rule is adopted and the original UNIFAC is improved by the nonlinear calculation of the segment fraction ϕi in the combinatorial activity coefficient γ Ci . A total of 956 experimental bubble points obtained from 17 highly asymmetric paraffinic systems are used to evaluate the original UNIFAC and the improved UNIFAC. Results show that the original UNIFAC and the improved UNIFAC give accurate predictions for bubble points under the condition of low mole fraction of light component. However, the improved UNIFAC still keeps great consistent with the experimental data at high mole fraction of the light component. It is the consideration of the local characteristics of groups in different molecules that ensures the high prediction accuracy of the improved UNIFAC. The original UNIFAC shows remarkably larger prediction errors with the increase of the light component, due to the assumption that “all groups are isotropic in solution”. It is evident that the improved UNIFAC is much superior to the original UNIFAC. Nevertheless, it must be acknowledge that the EOS-GE established in this work may generate non-realistic predictions at extreme conditions, due to the defects of α function (Le Guennec et al., 2016a; Segura et al., 2003; Le Guennec et al., 2016b), translation volume correlation (Le Guennec et al., 2016b; Kalikhman et al., 2010; Privat et al., 2016), and the application range of group interaction parameters (Polishuk et al., 2002). To develop a more general EOS-GE model, such problems should be resolved in the future work. In addition, the inapplicability of present model for dew points also needs improvement for the practical applications.
Fig. 11. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in Mixture B.
Fig. 12. The comparison between the experimental data and the predictions by the original UNIFAC and the improved UNIFAC in Mixture C.
Acknowledgement The authors wish to thank the National Natural Science Foundation of China (51422406, 51534007), the National Science & Technology Specific Project (2016ZX05028-004-001), and the Science Foundation of China University of Petroleum Beijing (C201602) for providing support for this work. References Abrams, D.S., Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J. 21, 116–128. Baled, H., Enick, R.M., Wu, Y., McHugh, M.A., Burgess, W., Tapriyal, D., Morreale, B.D., 2012. Prediction of hydrocarbon densities at extreme conditions using volumetranslated SRK and PR equations of state fit to high temperature, high pressure PVT data. Fluid Phase Equilibria 317, 65–76. Bondi, A.A., 1968. Physical Properties of Molecular Crystals Liquids, and Glasses. Boukouvalas, C., Spiliotis, N., Coutsikos, P., Tzouvaras, N., Tassios, D., 1994. Prediction of vapor-liquid equilibrium with the LCVM model: a linear combination of the Vidal and Michelsen mixing rules coupled with the original UNIF. Fluid Phase Equilibria 92, 75–106. Cordeiro, D.J., Luks, K.D., Kohn, J.P., 1973. Process for extracting high-molecular-weight hydrocarbons from solid phase in equilibrium with liquid hydrocarbon phase. Ind. Eng. Chem. Process Des. Dev. 12, 47–51 (United States).
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