Vapor–liquid equilibria of LNG and gas condensate mixtures by the Nasrifar–Moshfeghian equation of state

Fluid Phase Equilibria 200 (2002) 203–216

Vapor–liquid equilibria of LNG and gas condensate mixtures by the Nasrifar–Moshfeghian equation of state Kh. Nasrifar a , M. Moshfeghian b,∗ a b

Institute of Petroleum Engineering, University of Tehran, Tehran, Iran Department of Chemical Engineering, Shiraz University, Shiraz, Iran Received 4 September 2001; accepted 22 January 2002

Abstract The Nasrifar–Moshfeghian (NM) equation of state (EOS) is used to predict vapor–liquid equilibria (VLE) of multi-component mixtures. The systems under study consist of liquefied natural gases (LNG), gas condensates, an asymmetric system, slightly polar systems and gas/water systems. van der Waals mixing rules are used and no pure component parameter is adjusted; however, the predictions compare well with experimental data, in particular, the solubility of CO2 , H2 S, CO and some hydrocarbon gases in water are calculated quite accurately. The average absolute error was found to be 7% for calculating gas solubility in water. For some systems, the volumetric properties were also predicted. The saturated liquid density of LNG systems is predicted with an average absolute error less than 0.6%. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Mixture; Liquid density; Gas solubility

1. Introduction Petroleum industry often relies on cubic equations of state (EOS) for predicting vapor–liquid equilibria (VLE) of multi-component systems. These systems consist of a diversity of components, hence showing complex phase behavior. The popular Peng–Robinson (PR) EOS [1] or Soave–Redlich–Kwong (SRK) EOS [2] is usually used to predict the phase behavior of petroleum mixtures. Using these EOS, the K-values and vapor densities are predicted quite accurately; however, the saturated liquid density is predicted incorrectly with an average error about 10% [3]. For calculating saturated liquid density, correlations are usually used. Unfortunately, this is a source of inconsistency in calculations, e.g. in simulating the production history of reservoirs using material balance equations. Consequently, the use ∗

Corresponding author. Present address: Department of Chemical Engineering, University of Qatar, P.O. Box 2713, Doha, Qatar. Tel.: +974-485-2120; fax: +98-21-871-4432. E-mail address: [email protected] (M. Moshfeghian). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 2 ) 0 0 0 2 8 - 6

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of an EOS capable of predicting phase densities in addition to VLE is desirable. The most desirable would be a cubic EOS able to predict accurate liquid densities and with similar accuracy in predicting VLE as the PR or SRK EOS; however, requiring the least number of input properties for a compound (TC , pC , ω). Recently, Nasrifar–Moshfeghian (NM) [4] developed a new cubic EOS. This EOS quite accurately predicts pure component properties. Considering this ability, one can expect to get similar quality for multi-component mixtures. In this paper, we shall demonstrate the ability of the NM EOS to predict the phase behavior of multi-component mixtures occurring in petroleum industry. Systems including liquefied natural gases (LNG), gas condensates, an asymmetric system of methane + n-decane, slightly polar systems and gas/water systems are examined. Only van der Waals type mixing rules are applied to the NM EOS; the results are remarkable, however. The solubility of gases in water is correlated accurately. Considering this ability only, the accuracy of the NM EOS becomes warrant. Comparisons with the PR EOS for predicting the VLE of LNG and gas condensates reveal further outstanding characteristics of the NM EOS.

2. The NM equation of state The NM equation of state (EOS) [4] is a two-constant EOS. It is expressed by p=

RT a − 2 v − b v + 2bv − 2b2

(1)

where a and b are determined at the critical point using the critical point constraints for the EOS. For a pure compound aC = 0.497926

R 2 TC2 pC

(2)

bC = 0.094451

RTC pC

(3)

and

At other conditions, a and b are calculated by √ a = aC [1 + ma (1 − θ )]2

(4)

and b = bC [1 + mb (1 − θ )] where θ is the reduced temperature which is defined by   T − Tpt , T > T pt θ = TC − Tpt  0 T < Tpt

(5)

(6)

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The parameter Tpt is the pseudo triple point temperature, a characteristic of a compound. It is calculated from Tpt = 0.2498 + 0.3359ω − 0.1037ω2 TC where ω is the Pitzer’s acentric factor (ω ≤ 0.5). The parameters ma and mb are determined from  apt ma = −1 aC

(7)

(8)

and mb =

bpt −1 bC

(9)

where apt and bpt are the molecular attraction and molecular co-volume parameters at the pseudo triple point temperature. They are calculated from apt = 29.7056 bpt RTpt

(10)

bpt = 1 − 0.1519ω − 3.9462ω2 + 7.0538ω3 bC

(11)

and

The linear temperature dependence for the parameter b does not lead to unphysical behaviors for calculating some properties at extreme conditions. For example, heat capacities are predicted with good accuracy for a wide range of pressure and temperature [4]. To predict mixture properties, the following van der Waals mixing rules are used  b= xj bj (12) j

and a=

 xi xj aij i

(13)

j

√ where aij = aii ajj (1 − kij ) and kij = 0 for i = j . If Eqs. (12) and (13) are used as mixing rules, the fugacity coefficient for component i in a mixture is calculated from      √ 2 j xj aij bi bi A Z + (1 + 3)B ln φˆ i = (Z − 1) − ln(Z − B) − √ (14) − ln √ a b b 2 3B Z + (1 − 3)B where Z is the compressibility factor and A = ap/R 2 T 2 and B = bp/RT.

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3. Results and discussion The NM EOS is a cubic EOS, requiring three input properties for a compound (TC , PC , ω). To illustrate the ability of this EOS for predicting the properties of multi-component mixtures, its results are compared with experimental data and the PR EOS as well. The PR EOS has been chosen because it is cubic and needs the same number of input properties. Further, the PR EOS is known as a strong tool for predicting the properties of hydrocarbon mixtures. Table 1 presents the analyses of LNG and gas condensates used in this work. Some mixtures need C7+ characterization. In this paper, the Shariati–Peters–Moshfeghian (SPM) characterization method [5] is used. In order to have an unbiased comparison between the NM and PR EOS, both are applied to the mixtures using pure component properties only (kij = 0). Otherwise, it was possible that some EOS drawbacks were absorbed by the binary interaction parameters. The effect of kij on the calculations of binary and ternary mixtures is illustrated later on. Table 2 presents experimental and predicted liquid density for mixtures A–E. As can be seen, liquid density is predicted quite accurately using the NM EOS. Note that no adjustable parameter was used and no volume translation method was incorporated; however, the average absolute error for predicting saturated liquid density was found to be 0.58% for the NM EOS and 10.57% for the PR EOS. In this respect, the NM EOS is as accurate as liquid density correlations. This ability is attributed to the temperature dependence Table 1 Analyses (mol%) of LNG and gas condensate mixtures used in this work Component

CO2 N2 CH4 C 2 H6 C 3 H8 i-C4 H10 n-C4 H10 i-C5 H12 n-C5 H12 n-C6 H14 n-C7 H16 n-C10 H22 C7+ a

Mixture Aa

Ba

Ca

Da

Ea

Fb

Gc

Hd

Ie

– – 85.34 7.90 4.73 0.85 0.99 0.1 0.09 – – – –

– – 75.44 15.40 6.95 0.98 1.06 0.09 0.08 – – – –

– 0.86 75.70 13.59 6.74 1.34 1.33 0.22 0.22 – – – –

– 0.80 74.27 16.51 6.55 0.84 0.89 0.07 0.07 – – – –

– 0.60 90.07 6.54 2.20 0.29 0.28 0.01 0.01 – – – –

– – 91.35 4.03 1.53 – 0.82 – 0.34 0.39 – – 1.54f

0.61 0.46 68.64 13.90 6.89 0.66 2.66 0.62 0.94 1.14 – – 3.48g

2.44 0.08 82.10 5.78 2.87 0.56 1.23 0.52 0.60 0.72 – – 3.10h

0.15 0.48 80.64 5.93 2.98 – – – 4.30 – 3.08 2.44 –

[6]. [7]. c [8]. d [9]. e [10]. f C7+ specification: SG (60/60) = 0.7961, MW = 138.78, T b = 444.7 K. g C7+ specification: SG (60/60) = 0.7763, MW = 152.30, T b = 461.25 K. h C7+ specification: SG (60/60) = 0.7740, MW = 132, T b = 430.96 K. b

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Table 2 Experimental and predicted bubble point pressure and liquid density for mixtures A–E using the NM and PR EOS T (K)

Experimental pbubble (MPa)

%Deva in pbubble NM

PR

Experimental l ρbubble (kg m−3 )

l %Dev in ρbubble

NM

PR

Mixture A 110 115 120 125 130 %AADb

0.0787 0.1172 0.1686 0.2351 0.3210 –

−1.33 −1.22 −1.04 −0.80 −1.18 1.12

−2.50 −1.95 −1.47 −0.97 −1.06 1.59

484.09 477.32 470.57 463.69 456.64 –

−0.69 −0.30 0.09 0.46 0.80 0.47

10.87 10.92 10.90 10.84 10.73 10.85

Mixture B 110 115 120 125 %AAD

0.0723 0.1081 0.1549 0.2153 –

−1.54 −2.22 −1.89 −1.54 1.80

−4.05 −3.94 −3.12 −2.49 3.42

511.88 505.74 499.23 492.51 –

−1.24 −0.94 −0.55 −0.15 0.72

10.10 10.14 10.20 10.26 10.18

Mixture C 110 115 120 125 130 %AAD

0.1155 0.1595 0.2155 0.2873 0.3744 –

−20.95 −17.99 −15.11 −13.10 −11.16 15.66

−23.11 −19.47 −16.25 −13.98 −11.68 16.90

515.28 508.72 502.28 495.75 489.08 –

−1.17 −0.79 −0.42 −0.05 0.27 0.54

9.98 10.09 10.12 10.12 10.09 10.08

Mixture D 110 115 120 125 %AAD

0.1158 0.1584 0.2093 0.2853 –

−23.55 −19.87 −15.15 −15.04 18.40

−25.38 −21.04 −15.97 −15.56 19.49

512.97 506.68 499.88 493.27 –

−1.20 −0.89 −0.46 −0.092 0.66

10.22 10.27 10.38 10.40 10.32

Mixture E 115 120 125 130 %AAD

0.1456 0.2024 0.2762 0.3698 –

−7.99 −5.97 −4.78 −4.12 5.72

−7.66 −5.45 −4.08 −3.22 5.10

454.01 446.95 439.42 431.97 –

−0.01 0.36 0.77 1.08 0.55

11.65 11.54 11.45 11.24 11.47

a b

%Dev = 100[(calcd− expl)/expl]. %AAD = (100/n) ni=1 (|calcd − expl|/expl).

of molecular co-volume parameter in the NM EOS. Table 2 also indicates that the NM and PR EOS predict the bubble point pressures with comparable accuracy. The average absolute error is 8.53% for the NM EOS and 9.30% for the PR EOS. For predicting the bubble point pressure of mixtures A, B, C and D, the NM EOS is superior, while the PR EOS is superior for mixture E. For predicting the bubble point pressure of mixtures containing nitrogen, the error is large for both EOS. However, for systems that do

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Table 3 Experimental and predicted dew point results for mixture F at 367 K Component

CH4 C 2 H6 C 3 H8 n-C4 H10 n-C5 H12 n-C6 H14 C7+ a Dew point pressure (MPa) a

Vapor (mol%)

91.35 4.03 1.53 0.82 0.34 0.39 1.54 –

Liquid (mol%) Experimental

NM

PR

52.00 3.81 2.37 1.72 1.20 2.06 36.84 26.46

53.20 4.24 2.42 1.94 1.17 1.94 35.00 20.23

58.50 4.47 2.50 1.96 1.16 1.87 29.50 19.95

C7+ fraction was characterized by the SPM method [5] using the specification given in the footnote of Table 1.

not contain nitrogen (A and B) both EOS are accurate. This large error occurs because kij is relatively large for nitrogen + hydrocarbon binaries. In Table 3, the experimental and predicted dew point compositions for the Hoffmann–Crump–Hocott [7] gas condensate system (mixture F) are illustrated. Using the SPM method [5], the C7+ fraction is characterized (in mol%) by 0.51% n-decane, 0.52% n-propylbenzene and 0.51% n-hexylcyclopentane (altogether 1.54% as in the feed for C7+ fraction). The dew point pressure at 367 K was predicted to be 20.23 MPa for the NM EOS and 19.95 for the PR EOS. Considering the experimental value of 26.46 MPa, both EOS predict a 25% too low dew point pressure. The predicted compositions are in close agreement with the measured values for the NM EOS, but large errors are observed for the PR EOS, in particular for methane and C7+ fractions. Clearly, with respect to the PR EOS, the NM EOS is more consistent with the SPM characterization method. Because the predictions of dew point pressures were poor for both EOS, mixture G is examined for further evaluations. Mixture G is called the Donohoe–Buchanan [8] gas condensate system. The C7+ fraction is characterized (in mol%) by 1.94% n-undecane, 0.54% n-butylbenzene and 1% n-hexylcyclopentane (altogether 3.48%). The predicted dew point pressure at 363 K is 22.03 MPa for the NM EOS and 20.89 MPa for the PR EOS. The experimental value was reported 23.53 MPa. Evidently, the NM EOS is more accurate. Table 4 compares the experimental and predicted dew point pressure of the Firoozabadi–Hekim–Katz [9] gas condensate system (mixture H). For comparison, the calculated value from Firoozabadi et al. [9] is also included. Again the C7+ fraction has been characterized according to the SPM method. As can be Table 4 Dew point pressure of mixture H at 355.6 K Method

pdew (MPa)

Experimental Taken from [9] NM PR

28.1 26.6 26.83 24.88

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Table 5 Experimental and predicted flash yields of retrograde condensation for mixture H T (K)

278.15 278.15 318.15 318.15 a

p(MPa)

14.6 20.8 14.6 20.8

L/Fa Experimental

Firoozabadi

NM

PR

0.1106 0.0993 0.0659 0.0333

0.1296 0.0921 0.0820 0.0682

0.1108 0.0932 0.0778 0.0685

0.1585 0.1099 0.0963 0.0818

L/F is the liquid to feed molar ratio.

seen, the NM EOS and Firoozabadi’s method predict the dew point pressure more or less accurately. The result from the PR EOS is poor, however. Table 5 compares the experimental and predicted flash yields for retrograde condensation of mixture H. The three methods predict correctly the retrograde behavior of the system. The NM EOS and Firoozabadi’s method predict equally well, while the PR EOS is relatively poor.

Fig. 1. Experimental and predicted K-values for mixture I at 366.48 K using pure component properties only (kij = 0); (experimental data from Yarborough and Vogel [10]).

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Fig. 2. Experimental and correlated p–x–y diagram using the NM EOS for CH4 + n-C10 H22 mixture (experimental data from Lin et al. [11]).

Fig. 1 represents the experimental and predicted K-values of mixture I using the PR and NM EOS. As in the previous cases, no binary interaction parameter was used. Both EOS predict the K-values quite accurately; however, for small molecules (N2 and CO2 ) the NM EOS is more accurate and for the large molecules (n-C7 H16 and n-C10 H22 ), especially at high-pressures, the PR EOS. Consequently, considering the given comparisons so far, the NM EOS can be used as a powerful tool for predicting the properties of LNG and gas condensate systems. Fig. 2 illustrates the impact of using a binary interaction parameter (kij ) on the VLE of an asymmetric system (CH4 + n-C10 H22 ). The experimental data of the system is correlated using the NM EOS with kij = 0.08. The temperature ranges from 423 to 583 K. Except near the mixture critical points where the NM EOS underestimates the measured values, the agreement with measured values is good. To evaluate the impact of binary interaction parameters on the VLE of a ternary system (N2 + CO2 + n-C4 H10 ) at 311 K, each pair (N2 + CO2 , N2 + n-C4 H10 , CO2 + n-C4 H10 ) is correlated separately.

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Fig. 3. Experimental and correlated p–x–y diagram using the NM EOS for CO2 + n-C4 H10 and N2 + n-C4 H10 mixtures at 311 K (experimental data from Shibata and Sandler [12]).

Having determined the kij of each pair, the VLE of the ternary system are predicted. Fig. 3 indicates the experimental and correlated p–x–y of N2 + n-C4 H10 and CO2 + n-C4 H10 binary mixtures at 311 K. The agreement is clearly good, especially far from the critical point. The kij s were found to be 0.023 and 0.102 for N2 + n-C4 H10 and CO2 + n-C4 H10 , respectively. A value of 0 was found for N2 + CO2 . Adjusting the kij did not have any profound effect on the fit of the experimental data given by Yorizane et al. [13]. Shibata and Sandler [12] also used 0 for the kij of this system using the PR EOS. Fig. 4 shows the phase density of N2 + n-C4 H10 and CO2 + n-C4 H10 binary mixtures. Clearly, the agreement is excellent far from the mixtures critical points. Note that no adjustable parameter was used to fit the experimental phase density data; nevertheless, the predictions compare well with the experimental values. Also Fig. 4 clears that using kij s for correlating bubble point pressures does not lead to appreciable amount of errors in predicting saturated liquid densities. The experimental and predicted VLE of the ternary mixture N2 + CO2 + n-C4 H10 at 311 K and at two different pressures (3.45 and 13.8 MPa) are depicted in Figs. 5 and 6. The agreement between the

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Fig. 4. Experimental and predicted phase density using the NM EOS for CO2 + n-C4 H10 and N2 + n-C4 H10 mixtures at 311 K (experimental data from Shibata and Sandler [12]).

predictions using the NM EOS with fitted binary interaction parameters for each pair and experimental values are excellent. Another interesting feature of the NM EOS is that it can be used to correlate the solubility of gases in water. Few cubic EOS provide this facility. Even, for the outstanding EOS, the pure component parameters are tuned and volume-dependent mixing rules [14] or GE -mixing rules [15] are used. However, in this work, the pure component parameters are not tuned and the conventional van der Waals mixing rules, Eqs. (12) and (13), are used. Only kij is proposed to be temperature-dependent according to kgw = m1 +

m2 T

(15)

where m1 and m2 are two adjustable parameters determined from a fit of solubility data. For the solubility of eight gases in water m1 and m2 are provided in Table 6. Table 6 also presents the average absolute error for the fit of the eight solubility data using the NM EOS. The average absolute error was found to be 7.04%. Except for the solubility of CO2 and H2 S in water where chemical reactions occur, the results are promising.

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Fig. 5. Experimental (symbol) and predicted (solid line) phase diagram using the NM EOS for the ternary mixture of N2 + CO2 + n-C4 H10 at 311 K and 3.45 MPa (experimental data from Shibata and Sandler [12]).

In Fig. 7, the NM EOS is used to extrapolate the solubility of CH4 and N2 in water at very high-pressures. The EOS was applied to pressures up to 40 MPa for N2 /water system and up to 50 MPa for CH4 /water system. Predictions are getting worse with increasing pressure but qualitatively the trends are correct. For higher pressures convergence problems occur.

Fig. 6. Experimental (symbol) and predicted (solid line) phase diagram using the NM EOS for the ternary mixture of N2 + CO2 + n-C4 H10 at 310.9 K and 13.8 MPa (experimental data from Shibata and Sandler [12]).

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Table 6 Calculated solubility of gases in water using the NM EOS Gas

CO2 b H2 Sc,d N2 d Cod CH4 c C 2 H6 e C 3 H8 e n-C4 H10 f

n

T-range (K)

12 24 16 18 16 38 31 16

273–298 311–588 311–588 311–588 323–588 311–444 288–427 344–444 n

%AAD = (100/n) [16]. c [17]. d [18]. e [19]. f [20]. a

b

i=1 [|xCalcd

p-range (MPa)

1.01–4.56 0.34–20.68 0.34–13.78 0.34–13.78 1.38–16.89 2.76–55.16 0.69–3.44 0.26–3.44

%AADa

13.79 15.13 5.60 5.15 6.56 4.90 3.53 5.79

kgw = m1 +

m2 T

m1

m2

0.192 0.335 0.808 0.628 0.890 0.597 0.442 0.407

−126.092 −139.932 −491.128 −442.700 −415.088 −289.391 −245.874 −242.711

− xExpl |/xExpl ]i .

Fig. 7. Experimental and predicted solubility of N2 and CH4 in water at 375.15 K using the NM EOS (experimental data from O’Sullivan and Smith [21]).

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4. Conclusions Considering the accurate results obtained for predicting the properties of LNG and gas condensate systems, in particular, the saturated liquid density, the NM EOS is introduced as an accurate EOS for use in petroleum industry. Another remarkable conclusion is that selecting temperature dependence for the molecular co-volume in the NM EOS not only has improved the prediction of saturated liquid densities for pure compounds [4] but for mixtures as well. The NM EOS can also be used to correlate the solubility of gases in water and to some extent can be used to predict high-pressure gas solubility data. When the mixture temperature is less than the pseudo triple point temperature of one of the components that constitutes the mixture, θ becomes negative for the component and Eq. (4) is no longer applicable. For this case, we select a 0 value for the θ . This note is given explicitly by Eq. (6). List of symbols a attraction parameter (MPa m6 kmol−2 ) A dimensionless attraction parameter b molecular co-volume parameter (m3 kmol−1 ) B dimensionless molecular co-volume parameter F feed flow rate (mol s−1 ) kij binary interaction parameter K K-value L condensate rate (mol s−1 ) m1 , m2 constants ma , mb parameters given by Eqs. (8) and (9), respectively n number of points p pressure (MPa) R universal gas constant (8.314 × 10−3 MPa m3 kmol−1 K−1 ) T temperature (K) V volume (m3 kmol−1 ) x liquid phase composition (also phase composition) Z compressibility factor Greek letters φˆ partial fugacity coefficient θ reduced temperature ρ density (kg m−3 ) ω acentric factor Subscripts i, j C g pt w

components i and j critical gas pseudo triple point water

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