PETROLEUM SCIENCE & ENGINEERING

Journal of Petroleum Science and Engineering 13 ( 1995) 233-245

Prediction of the critical points of natural gas mixtures by rigorous and semi-empirical methods

Qiang Huang, Tian-Min Guo * University of Petroleum, P.O. Box 902, Beijing IOOO83, P.R. China

Received 25 May 1994; accepted 8 March 1995

Abstract

In the first part of this work, the modified Pate]-Teja equation-of-state was applied to the rigorous critical point calculation method developed by Heidemann and Khalil. Two binary interaction coefficients (k,, and kbij) were used to fit the binary critical point data and the composition dependency of &-values is evident. Considering the composition characteristics of natural gas mixtures, an optimal kij-matrix is recommended. The test on 46 natural gases (including a rich gas condensate), and other multicomponent hydrocarbon mixtures, shows a satisfactory prediction of critical properties.

In the second part, the simpler semi-empirical methods based on excess critical property functions were systematically studied and a new interaction parameter (A,) matrix designed for natural gas mixtures was developed. Test results on 93 binary and multicomponent mixtures are reported. Critical point data of mixtures are important parameters for phase behavior measurements of reservoir fluids.

1. Introduction

Critical point data of mixtures are important in stud-

ying the phase behavior of near-critical reservoir fluids,

gas-injected enhanced oil recovery processes and many

high-pressure processing technologies. The critical properties of a mixture depend on its composition,

therefore, it is tedious work to perform systematic experimental measurements. The development of a reliable prediction method is highly desirable.

The critical point of a given mixture can be deter- mined by the following three methods: ( 1) an indirect method, through a pressure-temperature (P-T) phase

envelope construction, and (2) a direct calculation: (a) a rigorous method, and (b) a semi-empirical method.

The P-T phase envelope construction is time-con- suming and convergence problems often occurred in

* Corresponding author.

0920-4105/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO920-4105(95)00013-5

the near-critical region. In this work, the rigorous and semi-empirical methods were extensively studied, with emphasis on the prediction of the critical properties of

multicomponent natural gas mixtures (including gas

condensates). Optimal binary interaction coefficient

matrices for both methods were proposed.

2. Rigorous method

Based on the Gibbs critical state criteria for mixtures

and equation of state (EOS), algorithms were devel- oped for direct calculation of mixture critical proper- ties.

2.1. Previous work

Peng and Robinson ( 1976, 1977) and Heidemann and Khalil ( 1980) developed rigorous methods based

234 (2. Huang, T.-M. Guo / Journal of Peiroleum Science and Engineering 13 (1995) 233-245

Table I Summary of the comparison of the prediction results of the critical

properties for natural gas related binary mixtures based on six equa-

tions-of-state (k, = 0)

System Critical

property

AAD (%) based on various EOS

PTV PT SRK SW MKS DG

C,,, + Nz T, 2.61 2.63 2.93 2.96 2.01 2.56 m=l6 P, 9.01 8.88 11.42 8.34 12.44 10.01 C,,, + CO2 T, 3.02 3.23 4.82 2.99 3.14 2.91 m=I-6 P, 5.56 5.59 6.71 4.39 5.09 4.12

I Ic 9.02 9.86 18.10 9.00 8.01 8.15 C,,, + HIS T, 1.04 1.45 1.21 1.01 2.00 1.08 m=lL6 P, 4.38 3.90 5.39 5.34 5.70 5.08

I’, 11.95 14.69 21.00 13.90 4.81 8.66 Nz+C02 T, 0.31 0.31 0.27 0.28 0.92 0.30

p, 10.40 9.68 8.55 8.17 6.05 5.14

;

10.40 9.37 11.50 7.41 10.60 8.79 CO,+H,S 2.29 2.18 2.19 2.30 3.29 1.14

P, 2.25 2.25 2.29 2.35 8.25 2.31 llc 5.64 6.25 12.30 7.34 2.17 4.64

C, +C,,, T, 1.05 0.99 3.30 1.97 2.67 2.14 m=2-7 P, 5.34 4.67 4.41 5.09 6.25 5.39

I‘, 8.04 6.79 12.40 14.80 8.62 8.08 Cz+C,,, T, 0.78 1.08 0.80 0.92 1.37 1.34 m=3-7 P, 3.59 3.98 3.27 3.21 4.70 2.91

Oc 6.78 10.30 14.30 8.08 4.09 5.90 G-t-C,,, T, 0.45 0.43 0.53 0.38 1.50 1.13 m=4-8, 10 P, 1.94 2.56 1.36 1.90 6.17 1.56

I,C 6.10 5.73 13.40 6.65 4.09 4.29 C,+C,,, T, 0.49 0.42 0.59 0.44 1.74 1.08

m=5-8, 10 PC 1.50 1.75 1.25 1.01 6.20 1.02 G+C,,, T, 0.42 0.43 0.43 0.46 1.64 1.35

m=6-10 PC 1.19 1.15 1.02 1.11 8.52 1.12 T, 1.25 1.32 1.71 1.37 2.09 1.50

Overall P, 4.51 4.47 4.57 4.09 6.94 3.93 AAD, % ~1, 8.28 9.00 14.71 9.60 6.06 6.93

on the critical state criteria expressed through Gibbs free energy G (P and T as independent variables) and Helmholtz free energy A (T, V as independent varia- bles), respectively. Because EOS is usually expressed as an explicit function of pressure P, the latter method

is simpler for practical applications, and it is briefly described in the following section.

Using a Tayler series expansion of Helmholtz free energy A, Heidemann and Khalil (1980) derived the

following criteria for the critical point of a mixture with a composition specified by the mole number vector:

QAn = 0, AnTAn = 1 (1)

and the cubic form:

C=CCCAniAnjAn i j k

Eq. 1 demands that matrix {BOLD (Q}) have a zero

determinant, i.e.:

Q=Det(Q) =0 (3)

The elements in the quadratic form of the critical cri-

teria are found from:

From basic thermodynamic relationships we have:

(5)

Table 2

The effect of composition on k,, and k,,, for some typical binary

systems (PTV EOS)

Binary system ( 1) + (2) Mole fraction of ( 1) k,,, k,,

C,+N, 0.0198 0.0361 0.0200

0.6831 0.0801 0.0501

0.7507 0.0601 0.0401

0.8499 0.0750 0.0647

c, + co2 0.0600 0.005 1 0.0105

0.2100 0.005 1 0.0105

0.4000 0.1315 0.1021

0.4500 0.1444 0.1024

0.5930 0.1586 0.1526

0.6400 0.1600 0.1600

C, +H,S 0.0670 0.0921 0.0955

0.1000 0.0620 0.0662

0.2090 0.0893 0.0840

0.2290 0.1932 0.1710

0.3000 0.1255 0.1063

0.4000 0.1986 0.1547

0.9650 0.1186 0.0645

C,+C, 0.1700 0.0661 0.0361

0.3228 0.0500 0.0500

0.5882 0.0512 0.0544

0.7450 0.0641 0.0472

0.9460 0.0360 0.0264

c,+c, 0.2950 0.1360 0.1456 0.6720 0.2031 0.0988

0.7665 0.1194 0.0584

0.8236 0.1698 0.0923

c,+c, 0.4600 0.0999 0.0959

0.5820 0.0999 0.0959

0.7320 0.2169 0.1614

0.8150 0.2097 0.0977 0.8500 0.2100 0.0750

Q. Huang T.-M. Guo / Journal of 'Petroleum Science and Erqjneerin~ 13 ( 1995) 233-245 235

O k alj

30

20

0 D

x

,’ 10 Y o k

alj

1 \ ,

0.0 0.2 0.4 t).h L.8 1.0

~olc fractlor of TH4

Fig. 1. Variation of k,,, and k,,, with composition of CH,-CZH, Fig. 3. Variation of k,,, and k,,, with composition of CH,-n&H,,

system system.

, ,

4 7 I \

Y 10 o k

a1.I

5 /

n.o 0.2 0.4 11.6 0.8 1.0

Mole fraction of CH4

Fig. 2. Variation of k,,, and k,,, with composition of CH,-n-C,H,,,

system.

0.5 1 . 0

15,

13- 0 0

0

z II- /f---O x

.:: Y

9- o k t3lJ

0.n 0.2 0.4 0.6 Ll.8 1.0

Mole fraction of 1‘U2

Fig, 4. Variation ofk,,, and k,,, with composition of COZ-CH, system.

236

1

.c

Fig. 5. Variation of k,,,, and k,,, with composirion of CCI-n-C,H,,, system.

4 ,

0

0 0 0

r- 2 x 1

7

rl

Y

Ii . .

-I 0.0 0.2 0.4 0.6 0.R 1 . II

Molt fraction of Np

Fig. 6. Variation of k,, and kh,, with composition of N2-CH, system.

(6)

The fugacities cf;) can be calculated from a suitable equation of state.

The Heidemann and Khalil method is modified by Huang et al. ( 1990) and they extended the application

to polar mixtures by using the cubic chain-of-rotators

(CCOR) equation of state (Guo et al., 1985) and CCOR-II EOS (Leet et al., 1986) ; significant improve-

ments of critical volume predictions were observed.

2.2. Ne~v developments

Selection of’ equation of stute Based on the Heidemann-Khalil ( 1980) method, six

cubic equations of state: ( I ) PT (Pate1 and Teja, 1982); (2) PTV (Valderrama, 1990); (3) SRK

(Soave, 1972) ; (4) SW (Schmidt and Wenzel, 1980) ; (5) MKS (Chu et al., 1992); and (6) DG (Guo and

Du, 1989) were tested for their ability to predict (with

k,, =0) the critical properties of binary mixtures. A summary of the prediction results for 45 natural gas

IL,

10

0 0

11 I I , I I 0.0 0.2 0.4 0.6 0.8 1.0

+101e fraction of Np

Fig. 7. Variation of k,,, and k,,, with composition of N?-C2HI, system.

Q. Huang, T.-M. Guo /Journal of Petroleum Science and Engineering 13 (1995) 233-245 237

Table 3 Recommended k,, and k,,, values for dry natural gas and gas condensate critical point calculations (Rigorous method, PTV EOS)

Component C, C, C, C, CS C, C, C, C, C,,

C,

CO*

N,

H,S

0.0 0.0396 0.04101 0.2031 0.1698 0.1811 0.2097 0.2 183 0.2201 0.2274 0.0 0.0319 0.03014 0.0998 0.0923 0.075 1 0.0977 0.205 1 0.2111 0.2111 0.0809 0.0361 0.1083 0.1508 0.0558 0.0912 0.1000 0.1000 0.1000 0.1000 0.0749 0.1210 0.1024 0.1251 0.0310 0.0522 0.1000 0.1000 0.1000 0.1000 0.0208 0.0307 0.1986 0.1547 0.1018 0.0478 0.1000 0.1000 0.1000 0.1000 0.0083 0.0211 0.1547 0.0825 0.0507 0.025 1 0.1000 O.lcOO 0.1000 0.1000 0.1986 0.0361 0.0923 0.0501 0.0583 0.0511 0.1000 0.1000 0. IO00 0.1000 0.1547 0.0201 0.07 13 0.0900 0.0959 0.1000 0.1000 0.1000 0.1000 0.1000

For non-hydrocarbon pairs:

COZ-N, COZ-H,S N,-H,S

k a, khy

0.0544 0.0325 0.0540 0.0255 0.1210 0.0852

Table 4 The AAD’ of the predicted critical properties of multicomponent mixtures (Rigorous method, PTV EOS)

Mix. No.” AAD (%) Mix. No. AAD (%)

T‘ P, T, p,

11 3.98 12 4.24 13 5.08 14 5.30 15 I .05 16 2.01 17 I .08 18 1.23 19 1.3 I 20 1.3 1 21 0.4 I 22 0.3 I 23 0.56 24 2.36 25 0.9 I 26 0.07 28 0.24 29 2.20 30 I .69 31 0.0 1 32 0.46 33 2.49 34 1.26 Overall AAD, %

0.77 35 1.55 36 0.87 37 2.29 38 4.79 39 1.73 40 1.03 41 0.87 42 0.45 43 1.09 44 0.31 45 0.56 46 0.25 47 4.55 48 0.51 49 0.24 50 0.53 51 1.56 53 1.40 54 0.42 55” 2.25 57 2.38 58 0.72

1.33 0.72 I .39 2.48 I .53 6.83 1 .Oo 4.32 0.71 7.61 0.38 3.91 0.7 1 0.18 0.40 1.61 0.93 0.48 0.68 0.12 0.10 1 .I6 1.34 3.40 0.36 0.19 2.35 1.97 0.34 1.59 1.07 0.24 2.07 6.27 1.49 1.01 2.56 3.55 1.67 6.13 0.51 0.18 0.68 8.33

1.40 2.09

“Mixture numbers correspond to the numbers in Table 11. ‘A lean gas condensate. ‘AAD is defined as

Table 5

Effect of the number of pseudocomponents on the predicted critical

properties of a rich gas condensate (Mix. No. 56) [EOS: PTV,

T, - P, - w correlation: Pedersen et al. ( 1989) 1

Number of pseudocomponents AAD (%)

T, P<

1 12.25 52.74

2 8.25 36.77 4 1.42 3.07

related binary mixtures (a total of 352 data points) is listed in Table 1. Comparison of the test results indi-

cates that the three-parameterPI’VEOS generally yield

better predictions. The PVT EOS are, therefore, selected for further study of the prediction of the critical

points of multicomponent mixtures. The superiority of

PTV EOS for the phase behavior calculations for res- ervoir fluid also has been mentioned by Danesh et al. ( 1991). A brief description of the P7V EOS is given

in the Appendix.

Evaluation of the binary interaction coefficients for

PTV EOS

The binary interaction coefficients (IQ and khjj) of

natural gas related binary systems (a total of 49 sys- tems) were systematically determined for PZ’VEOS by fitting the experimental binary critical point data reported in a review paper by Hicks and Young ( 1975). The modified Levenberg-Marquardt nonlinear least-

238 Q. Huang, T.-M. Guo / Journal of Petroleum Science and Engineering 13 (1995) 233-245

Table 6

Recommended binary parameters (A,? and AZ,) for the prediction of critical temperatures (K) (Semi-Empirical Method)

Component C, C? C3 C4 CS C6 C7 COZ NZ H2S

1 .ooo 1.305 1.325 1.327 1 .I34 0.645 1.300 1.238 1.275 0.715 1.000 1.127 1.164 1.201 0.990 1.469 1.524 1.012

0.665 0.869 1.000 1.160 1.257 1.289 1.490 1.699 0.441

0.635 0.829 0.847 1.000 1.196 1.256 1.311 1.519 0.376 0.366 0.779 0.758 0.817 1.000 1.267 1.242 0.583 0.202 1.202 0.935 0.717 0.761 0.763 1.000 1.227 0.890 0.101 0.558 0.549 0.562 0.711 0.777 0.794 1.000 0.623 0.113 0.778 0.603 0.490 0.575 1.436 1.032 1.330 1 .ooo 1.352 0.756 0.942 1.638 1.713 1.997 2.294 2.110 0.670 1.000

1.612 0.796 0.442 0.556 0.530 0.866 1.251 1.573 0.568

0.524

1.255

1.803

1.571

1.564

1.102

0.723

0.55 1

1.425

1.000

Table 7

Recommended binary parameters (A, 2 and AZ,) for the prediction of critical pressures (bar) (Semi-Empirical Method)

Component CL C? CT C, CS C6 C7 COZ N, H,S

C, I .ooo 1.487 1.903 2.066 2.291 2.698 2.684 1.130 1.189 1.623

C? 0.583 1.000 1.231 1.492 1.587 1.798 1.915 1.507 0.469 1.144

C, 0.332 0.784 1.000 1.228 1.352 1.499 1.631 I.466 0.097 1.593

C, 0.236 0.590 0.792 1.000 1.184 1.220 1.382 0.621 - 0.035 1.646

CY 0.117 0.523 0.692 0.830 1.000 1.047 1.166 0.553 0.039 0.677

Cii 0.035 0.399 0.588 0.797 0.953 1.000 1.222 0.289 0.035 0.571

C7 0.018 0.338 0.562 0.672 0.835 0.799 I.000 0.220 0.172 0.487

co2 0.840 0.602 0.535 1.432 1.508 2.021 2.182 1.000 0.560 0.581

N 0.815 1.596 2.344 2.675 2.490 2.314 1.946 1.413 1.000 3.629

H2S 0.444 0.874 0.552 0.516 I .361 1.462 1.532 1.528 - 0.337 1 .OOo

squares regression method was used for the data regres-

sion. The regression results indicate that both kaij and khjj

are composition-dependent, especially for those asym- metric systems whose coefficients are quite sensitive to mixture composition. Typical effects of composition

on k,j values are shown in Table 2 and Figs. l-7. In regard to the irregularity of the variation of k,

values with composition, it is not feasible to develop

generalized composition-dependent kti-correlations. Based on the typical natural gas composition range and the k, values determined for PTV EOS, the recom- mended kaij and kbji values are given in Table 3.

Prediction of the critical properties of multicomponent mixtures

Based on the recommended k, values listed in Table 3, the critical temperature (T,) and critical pressure (P,) of 45 multicomponent, nonhydrocarbon-hydro-

carbon and hydrocarbon-hydrocarbon mixtures were

predicted. The mole composition, critical point data and data source of the mixtures tested in this work are listed in the Appendix (Table 11) . The average abso-

lute deviations (AAD) of the predicted T, and P, are

given in Table 4.

Prediction of the critical properties of a rich gas con-

densate

Sample No. 56 in Table 11 is a typical rich gas condensate containing 9.8 1 mol% CT -fraction; its crit- ical temperature and critical pressure reported by Alonso and Nectoux (1984) are 398.2 K and 346.0 bar, respectively.

For predicting the critical properties of a gas con- densate, characterization of the Cq -fraction is involved and in this work the widely used characterization pro- cedure proposed by Pedersen et al. ( 1984) was chosen.

Q. Huang, T.-M. Guo /Journal of Petroleum Science and Engineering I3 (1995) 233-245 239

Table 8

Calculation results of critical properties for binary systems based on

the recommended A,z and A,, values

System Number of data points” AAD (%)

T, p,

c,-c2

G-c, c,-G c,-c, GG c,-c, G-C3 c,-c, c,-CT c2-c,

c,-c, G-G c,-c, c,-c, c3-c, G-G C-G GG G-C7 G-G GG” cs-cc, c5-c7 c49 G-G c,-cl! c,-cs C,_COI

c>-co2

cq-co> c,-co2

c5-co2 C,“_COZ

C,-N> CT-N2

C,-Nz Cd-N2 C,N> G-+b C,-H2S

C>-H$

C,-H?S

CpH2S

C5-HzS

N2-CO2

CO,-HZS Overall

14

17

9

3

II

6

12

7

3

10

6

5

11

4

9

9

14

8

5

3

6

9

9

9

6

2

14

II

5

8

5

4

3

4

7

6

3

3

II

3

15

334

0.28 0.73

0.23 0.51

0.96 0.16

0.62 3.32

2.36 4.61

1.99 1.13

0.13 1.25

0.21 0.33

0.44 0.92

0.52 0.52

0.42 I .45

0.02 0.11

0.10 0.43

0.07 0.37

0.13 0.13

0.03 0.70

0.05 0.78

0.03 0.28

0.07 0.13

0.05 0.18

0.04 0.05

0.03 0.13

0.03 1.07

0.30 0.41

0.04 0.38

0.04 0.46

0.05 0.50

0.34 0.87

0.01 0.03

0.37 2.23

0.58 9.61

3.20 3.33

1.69 4.92

0.06 0.41

0.18 4.45

0.41 2.75

I .41 2.70

0.00 0.004

2.09 2.09

0.63 0.57

0.05 1.11

0.02 0.86

0.03 1.89

0.65 2.13

0.18 0.70 0.13 0.33 0.46 1.35

“All experimental data were taken from Hicks and Young (1975).

The CT -fraction was subdivided into four pseudocom- ponents with equal weight fractions.

The critical properties ( Tcj, P,;) and acentric factor (Oi) of pseudocomponents are required for determin-

ing the PTV EOS parameters. The selection of the empirical T,-PC-u correlation has a significant effect

on the predicted critical properties of gas condensates. We have examined the following six sets of T,-PC-u correlations: ( 1) T,, PC - Riazi and Daubert ( 1980) ; w-Edmister(1958) (2) T,,P,--Cavett(1964);w

- Edmister (3) T,, P,, w - Kesler and Lee (1976)

(4) T,, P, - Kesler and Lee ( 1976) ; w - Edmister

(5) T,,P,-Cavett(1964);w-KeslerandLee(6)

Table 9

Comparison of critical properties prediction results based on semi-

empirical method for methane-hydrocarbon and nonhydrocarbon-

hydrocarbon multicomponent mixtures

Mix. No. AAD (%)

This work

T, P,

Teja et at.

T, P,

Peng -~

T‘ p,

1 0.56 5.13 0.13 12.7 2.47 7.37 2 2.48 4.11 1.17 14.3 4.44 6.88 3 3.28 3.57 1.40 16.5 5.29 6.36 4 0.14 4.95 0.84 9.81 1.14 10.3

5 2.51 8.82 0.73 15.9 3.89 15.2 6 6.23 12.5 3.07 21.4 8.13 16.3 I 1.01 9.35 0.01 8.41 1.45 1.50

8 0.54 6.00 1.36 3.66 I .48 1.13 9 0.68 3.75 0.89 2.87 2.33 3.21

10 2.48 1.79 0.95 1.00 4.39 4.69 11 0.73 1.40 0.54 8.25 1.67 0.44 12 0.26 3.23 0.13 8.94 1.74 0.75 13 1.22 3.03 1.25 9.87 2.30 1.83 14 0.42 5.98 0.91 11.9 2.34 3.58 29 0.42 4.15 1.33 19.0 0.98 6.34 33 2.65 10.8 8.37 4.41 0.38 0.15 41 0.16 1.33 0.77 13.9 0.14 4.08 42 0.38 0.85 0.72 7.32 0.36 2.14 43 0.44 2.77 0.91 5.85 0.45 4.11 47 0.22 3.75 0.26 7.15 0.21 1.30 48 0.61 22.6 0.40 8.56 2.01 3.97 49 0.82 1.12 0.30 I .65 0.93 1.10 50 0.37 5.63 0.69 2.69 1.43 8.64 51 0.95 21.3 0.94 11.8 I .79 20.5 52 2.19 14.7 10.3 10.7 5.28 0.15 53 0.91 4.53 3.06 8.78 1.53 2.12 54 0.03 1.42 1.49 26.5 0.63 0.55 55 3.42 20.8 10.7 17.7 3.06 24.4 Av. AAD 1.29 6.76 1.95 10.4 2.23, 5.68

240 Q. Huang. T.-M. Guo / Journal of Petroleum Science and Engineering 13 (1995) 233-245

Table 10

Comparison of the prediction results of the critical properties based

on semi-empirical methods for hydrocarbon-hydrocarbon multi-

component mixtures

Mix. No. AAD (%)

This work

T, P,

Teja et al.

T, P,

Peng

T, P,

15 1.33 3.49 0.28 3.31 0.36 5.20

16 1.41 4.39 0.26 3.42 1.01 3.63

17 0.93 0.88 0.11 0.43 0.91 1.68

18 I .59 1.95 0.35 6.32 1.20 0.27

19 1.44 1.27 0.12 12.3 0.99 1.39

20 1.38 1.60 0.05 10.5 1.07 1.55 30 0.23 4.15 1.14 2.41 1.71 0.79

31 0.00 0.44 0.74 0.47 0.39 0.06

32 0.12 1.63 0.08 4.34 0.07 1.85

34 0.58 3.66 0.46 2.63 2.12 1.21

35 0.67 3.24 0.33 2.61 1.96 1.04

36 0.70 0.95 0.28 0.73 1.85 0.90

37 0.61 0.79 0.33 0.57 I .76 1.91

38 1.15 1.51 0.10 1.79 0.97 1.74

39 1.08 5.32 0.09 3.82 0.54 7.70

40 0.83 2.14 0.08 2.83 0.24 4.15

44 0.03 1.56 0.52 2.12 1.10 0.8 1 45 0.18 2.28 0.06 4.28 0.43 2.25

46 1.93 0.79 0.66 8.24 0.48 0.98

Av. AAD 0.85 2.21 0.32 3.85 1.02 2.06

“Mixture numbers correspond to the numbers in Table 11.

T,, P,, - Pedersen et al. ( 1989) Set (6) was found

best suited for PTV EOS. The prediction results of the critical properties of the given gas condensate are

shown in Table 5, in which the effect of the number of

pseudocomponents also is shown. Non-zero binary interaction coefficients (k,, and

khij) were used for methane-pseudocomponent (Ci-

C, + i) pairs with values given below (n = 7) :

c a+ I G+2 cn+3 cl+4 kaii 0.40 0.60 0.60 0.60 (7) khij 0.30 0.35 0.40 0.40

Since the experimental critical point data of real gas condensates are extremely scarce, further experimental studies are being conducted.

3. Semi-empirical method

The simpler semi-empirical method is based on the concept of excess critical property (@) proposed by Etter and Kay ( 1961), which is defined as:

(8)

(9)

where Qc and phi denote the critical property (T,, P,,

u,) of mixture and pure component i, respectively, and the superscript “id” stands for ideal mixture.

3.1. Previous work

Tejaet al. ( 1983) and Peng ( 1986) selected the well known excess Gibbs energy model proposed by Wilson (1964) to express the e function:

@ = - CCzi In Cz& (10) I j

For binary systems Eq. 10 becomes:

@= -Clz, ln(zl +A12z2) +z2 ln(z,+&z,)l

(11)

where: C is a constant (C = 2500) ; zi is the mole frac- tion of component i in a mixture; and A, is the binary

interaction parameter (for i = j, Aij= Aji= 1.0)

Teja et al. (1983) determined the A, values from experimental critical property data of binary mixtures, while Peng (1986) used the critical property values

calculated from the rigorous method (Peng and Rob- inson, 1977) as experimental data.

3.2. New Aij-matrix

In this work, a new A, matrix designed for predicting

the critical temperature and critical pressure of natural gas mixtures was developed and is presented in Tables 6 and 7, respectively. For those binary pairs with

reported experimental critical point data, parameter values were determined by using the modified Leven-

berg-Marquardt nonlinear least-squares regression

method. For those systems where experimental data are not available, the A, values were estimated through the calculated critical point data based on the rigorous method using PTV EOS.

3.3. Calculation results

Table 8 lists the absolute average deviations of the calculated critical temperature (T,) and critical pres-

Q. Huun~, T.-M. Guo /Journal of Petroleum Science and Engineering 13 (1995) 233-245 741

Table I1 Critical point and composition data ofthe multicomponentmixturestested in this work

Mix. Experimental data Composition,mole fraction Dat.1 source

T,(K) P, (bar) C, C? C, C, C, Cc, C, CO? NZ -__-

I 345.37 70. 12 0.0960 0.6220 0.2820 Formann and

Thodos ( 1962)

2 336.48 73.43 0.1410 0.5820 0.2770

3 327.59 77.08 0.1900 0.5600 0.2500

4 380.37 68.40 0.1120 0.3720 0.5161

5 362.04 83.98 0.2160 0.3210 0.4630

6 318.71 117.21 0.4520 0.2290 0.3190

7

8

9

IO

377.59 82.S3 0.3880 0.1224 0.4986 Wie:;e et al.

( 19?0) 344.26 100.53 0.5295 0.1882 0.2832

310.93 121.62 0.6961 0.0607 0.2432

27759 123.14 0.7835 0.0433 0.1732

II 322.03 86.75 0.4150 0.5420

I2 322.03 92.05 0.3600 0.5450

13 313.70 92.32 0.4.530 0.5005

I4 313.70 97.98 0.4115 0.5030

0.0430 Yarborough

and :?mith

(1970) 0.09.50

0.0465

0.0855

IS 438.15 66.13 0.4290 0.3730 0.1980

16 385.92 76.06 0.7260 0.1710 0.1030

17 400.37 62.79 0.5140 0.4120 0.0740

Ekinerand

Thoclos (1966)

18 391.49 81.01 0.8010 0.0640 0.1350

19 421.48 71.57 0.6120 0.2710 0.1170

20 415.93 70.61 0.6150 0.2960 0.0890

Ekincrand

Thodos (1968)

21 303.60 72.56 0.0123 0.0306

22 303.90 72.51 0.0063 0.0186

23 303.20 72.63 0.0193 0.0063

24 302.00 69.37 0.0482 0.0445

25 306.10 72.12 0.0445 0.0440

26 309.10 72.31 0.0443 0.0441

27 304.50 72.79 0.0127

28 305.20 72.78 0.0132

0.0127

0.0129

0.9751

0.9751

0.9744

0.9073

0.9115

0.9116

0.9746

0.9739

Morrison and

Kincaid

( 198.1)

29 224.21 68.50 0.8511 0.1007 0.0482 Parlkh et al.

(1984)

30 397.15 56.02

31 428.81 41.88

32 450.20 38.80

0.3414 0.3421 0.3165 Etter :and Kay

(1961)

0.3276 0.3398 0.3326

0.6449 0.2359 0.1192

33 3 10.92 82.74 0.0700 0.6160 0.314" Kobinson and

Bailey (1957)

242 Q. Huang T.-M. Guo / Journal c$Perroleurn Science and Engitieering 13 (1995) 233-245

Table I I (continued)

Mix. Experimental data Composition, mole fraction Data source

No.

T, (K) P, (bar) C, C? C, C, C5 Gl C7 CO2 N?

34 366.44 67.20 0.7500

35 366.44 66.20 0.7390

36 366.44 63.80 0.7230

37 366.44 61.20 0.6920

38 394.22 61.00 0.5450

39 42 I .99 60.70 0.4300

40 421.99 51.90 0.2980

0.0380 0.2120 Mehra and

Thodos ( 1963)

0.0700 0.1910

0.1120 0.1650

0.1880 0.1200

0.2350 0.2200

0.1100 0.4600

0.4100 0.2920

41 199.44 s3.40

42 194.06 49.35

43 196.72 51.81

0.9100 0.0560 0.0012 0.0328 Gonzales and

Lee ( 1968)

0.9590 0.0260 0.0001 0.0150

0.9500 0.0260 0.0078 0.0160

44 405.87 51.13

45 4 17.92 45.06

46 423.15 74.12

47 304.35 72.62

0.2542 0.2547 0.2554 0.2357 Etter and Kay

(1961)

0.4858 0.3316 0.1213 0.0613

0.6168 0.1376 0.0726 0.1730

0.0081 0.0084 0.0085 0.9750 Morrison and

Kin&d

( 1984)

48 387.03 72.20

49 385.42 56.24

0.2019 0.2029 0.2033 0.2038 0.1881 Etter and Kay

(1961)

0.3977 0.2926 0.1997 0.0713 0.0369

SO 199.72 54.58 0.9450

51 20 I .09 55.78 0.9430

0.0260 0.0081 0.0052 0.0 160 Gonzales and

Lee (1968)

0.0270 0.0074 0.0049 0.0010 0.0027 0.0140

52 310.53 137.48

53 376.42 65.36

0.6626 0.1093 0.1057 0.0616 0.0605 Peng and Rob-

inson ( 1977)

0.1015 0.3573 0.2629 0.1749 0.0657 0.0332

54 313.70 78.46 0.3160 0.3880 0.2230 0.0430 0.0080 0.0220 Yarborough

and Smith

( 1970)

s5 188.87 67.08 0.6870 0.0330 0.0144 0.0070 0.0026 0.0011 0.0014” 0.0091 0.2441 Davis et al.

(1954)

56 398.15 346.00 0.6807’ 0.0853 0.0407 0.0320 0.0234 0.0147 0.0981” 0.025 1 Alonso and

Nectoux

(1984)

Q. Huang, T.-M. Guo / Journal of Petroleum Science nnd Engineering 13 (1995) 233-245 243

Table 1 1 (continued)

Mix. Exp. data Composition, mole fraction Data source

No.

T,(K) P, (bar) iCs nC, C, C7 CS C, C 1”

57 541.26 30.93 0.2465 0.2176 0.1925 0.1779 0.1656 Hissong, 1968 58 543.37 353.90 0.1989 0.1963 0.1483 0.1344 0.1213 0.1137 0.1142

59 509.90

60 512.80

61 s17.40

_

_

-

0.9782 0.0055 0.0055 0.0054 0.0054 Anselme and

Teja ( 1990)

0.9500 0.0125 0.0126 0.0124 0.0125

0.9047 0.0239 0.0240 0.0237 0.0237

“C++ -fraction.

“Mole fraction of H,S.

‘CL+ NZ.

“C7 -fraction, mol. wt. = 200, sp. gr. = 0.806.

sure (P,) of 46 binary nonhydrocarbon-hydrocarbon critical properties of conventional natural gas mixtures.

and hydrocarbon-hydrocarbon mixtures (a total of 334 However, the applicability to gas condensates contain-

data points). The overall average deviations of the cal- ing significant amount of Cq-fraction needs further

culated T, and P, are 0.46% and 1.35%, respectively. investigation.

The new Ati-matrix was applied to predict the critical

properties of 47 multicomponent mixtures. The com-

parisons of the prediction results with those based on the A, values reported by Teja et al. ( 1983), and Peng ( 1986) are given in Tables 9 and 10. Table 9 indicates

that the A,, values recommended in this work yield

significant improvements of the critical temperature prediction for methane-rich/nonhydrocarbon-contain-

ing natural gas mixtures; however, comparable results

were observed for hydrocarbon-hydrocarbon mixtures,

as indicated in Table 10.

5. List of symbols

AAD

a

A

A,

b

&OR

EOS

f” G

k,,j

average absolute deviation

equation-of-state parameter

Helmholtz free energy binary (i-j) interaction parameter in the

semi-empirical method

equation-of-state parameter

equation-of-state parameter cubic chain-of-rotators

equation-of-state constant in Eq. 10

fugacity Gibbs free energy binary interaction coefficient for parameter a in PTV EOS

binary interaction coefficient for parameter b in PTV EOS

number of moles

vector of mole numbers

4. Conclusions

The rigorous method of Heidemann and Khalil ( 1980) as coupled with the PTV equation of state and the recommended binary interaction coefficient matrix

was tested on 46 multicomponent mixtures and satis- factory results were observed for the prediction of the

critical temperature and critical pressure of conven-

tional natural gas mixtures and a rich gas condensate. A new A;,-matrix designed for natural gas mixtures

was established for the semi-empiricalmethod. The test results on 47 multicomponent mixtures indicate that this method is simpler and adequate for predicting the

n

n P

4ij

R

pressure elements in the quadratic form of the critical criteria gas constant

244 Q. Huang, T.-M. Guo / Journal of Petroleum Science and Engineering 13 (1995) 233-245

T temperature u mole volume V total volume

mole fraction

critical compressibility factor

Greek letters

a! temperature coefficient of parameter a in PTV

EOS

Qn excess property 6J acentric factor

Subscripts

c critical property

i component i

j componentj

Superscripts

E excess property

id ideal state

Acknowledgements

Financial supports received from the China National

Petroleum Corporation and the PhD Student Research

Fund Foundation are gratefully acknowledged.

J&=0.66121 -0.761052, (A5)

.n, = 0.02207 + 0.208682, (A6)

fit = 0.57765 - 1.870802, (A7)

(Y[T~,wXZ,]=[~+F(~-T;~)]* (A8)

F= 0.46283 + 3.58230( wZ,)

+S.l9417(wZ,)* (A9)

For nonpolar substances, the critical compressibility

factor Z, can be correlated with the acentric factor as follows:

Z,=0.29740-0.12136w+0.03349w2 (AlO)

Conventional mixing rules are used for determining mixture parameters. In this work, two binary interac-

tion coefficients (k, and kbv) are introduced for para- meters a and b, i.e.:

i j (All)

av= (a,~~)~~~( 1 - kajj)

b = C.&q.xjbij i j

b,=0.5(bi+bj)(1-k,,)

c = cxic;

(‘412)

(A13)

(A14)

(Al5)

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where a, b and c are the three EOS parameters:

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