Addition of the Nitrogen Group to the PPR78 Model (Predictive 1978

Feb 27, 2008 - In our previous papers, 12 groups were defined: CH3, CH2, CH, C, CH4 ... (kij(T)) for the widely used Peng-Robinson equation of state. ...
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Ind. Eng. Chem. Res. 2008, 47, 2033-2048

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Addition of the Nitrogen Group to the PPR78 Model (Predictive 1978, Peng Robinson EOS with Temperature-Dependent kij Calculated through a Group Contribution Method) Romain Privat, Jean-Noe1 l Jaubert,* and Fabrice Mutelet Laboratoire de Thermodynamique des Milieux Polyphase´ s, Nancy-UniVersite´ , 1 rue GrandVille, B.P. 20451, F-54001 Nancy Cedex, France

In 2004, we started to develop a group contribution method aimed at estimating the temperature-dependent binary interaction parameters (kij(T)) for the widely used Peng-Robinson equation of state (EOS). This model was called PPR78 (predictive 1978, Peng Robinson EOS), because it relies on the Peng-Robinson EOS as published by Peng and Robinson in 1978 and because the addition of a group contribution method to estimate the kij value makes it predictive. In our previous papers, 12 groups were defined: CH3, CH2, CH, C, CH4 (methane), C2H6 (ethane), CHaro, Caro, Cfused aromatic rings, CH2,cyclic, CHcyclic ) Ccyclic, and CO2. Thus, it was possible to estimate the kij value for any mixture that contains alkanes, aromatics, naphthenes, and CO2, regardless of the temperature. In this study, the PPR78 model is extended to systems that contain nitrogen. To do so, the group N2 was added.

Introduction Nitrogen is a major component of petroleum fluids. Moreover, the injection of N2 into oil reservoirs is a currently used oil recovery technique. To simulate and design the processes that involve N2, it is necessary to predict the phase equilibrium of mixtures that contain N2 in both the subcritical and critical regions. To meet these requirements, Jaubert and co-workers1-7 developed a group contribution method that allowed estimation of the temperature-dependent binary interaction parameters (kij(T)) for the widely used Peng-Robinson equation of state (EOS). Because their model relies on the Peng-Robinson EOS as published by Peng and Robinson8 in 1978, and because the addition of a group contribution method to estimate the kij value makes it predictive, Jaubert et al. decided to call this new model PPR78 (predictive 1978, Peng-Robinson EOS). In our previous papers,1-4 12 groups were defined: CH3, CH2, CH, C, CH4 (methane), C2H6 (ethane), CHaro, Caro, Cfused aromatic rings, CH2,cyclic, CHcyclic ) Ccyclic, and CO2. Thus, it was possible to estimate the kij value for any mixture that contained saturated hydrocarbons (n-alkanes and branched alkanes), aromatic hydrocarbons, cyclic hydrocarbons (naphthenes), and carbon dioxide, regardless of the temperature. In this study, the PPR78 model is extended to systems that contain nitrogen. To do so, the N2 group was added. The interactions between this new group and the 12 groups previously defined (a total of 24 parameters) are determined. Today, it is thus possible to estimate, at any temperature, the kij value between two components in any mixture that contains paraffins, naphthenes, aromatics, CO2, and N2. Difficulties in Predicting the Phase Behavior of the N2 + Hydrocarbon Binary Mixtures All binary N2 + hydrocarbon systems develop, except for methane, Type III phase diagrams in the classification scheme * Author to whom correspondence should be addressed. Fax: +33 3 83 17 51 52. E-mail addresss: [email protected].

of van Konynenburg and Scott.9 Such systems are known to be extremely difficult to predict with a cubic EOS. Another major difficulty in the modeling of binary systems that contain N2 with a cubic EOS comes from the unusual shape of the isothermal (P,x,y) diagrams for such systems. Indeed, as shown in Figure 1a for the N2 + n-hexane system at T ) 377.9 K, in the vicinity of the critical point, the experimentally determined diagram becomes very flat (in a large composition range around the critical composition, the slope of the bubble and dew curves is maintained low). Using a cubic EOS, such behavior is impossible to reproduce and the slope of the bubble and dew curves in the vicinity of the critical composition will always be much steeper than that which is observed experimentally. This last point remains true regardless of the alpha function used (Soave, Twu Bluck Cunningham Coon, ...). Therefore, the choice of the best kij value for such systems becomes an issue. If we select, as in Figure 1b, a kij value that allows one to perfectly reproduce the critical pressure (in the present case, kij ) -0.03), the dew curve also will be accurately reproduced, but huge deviations will appear on the bubble curve, because, in a large composition range around the critical point, the calculated shape of the phase diagram is not correct. The only way to better predict the bubble curve is to increase the kij value. By doing so (see Figure 1c, 1d, 1e), we increase the critical pressure and we increase the deviations on the dew compositions. Figure 1 also clearly shows that, regardless of the kij value, the calculated critical composition is more or less always the same and always much higher than the experimental value. This means that it is impossible to find a kij value that would lead to a correct critical composition. As a conclusion, cubic EOS are unable to accurately reproduce the experimental shape of the isothermal (P,x,y) diagrams for binary systems that contain N2. In this study, we decided to define a compromise between an accurate critical pressure and an accurate bubble composition. As an example, Figure 1d shows the prediction obtained with the PPR78 model (kij ) 0.02). However, we are conscious that this kij value does not lead to very accurate results, but no kij value is able to do a better job!

10.1021/ie071524b CCC: $40.75 © 2008 American Chemical Society Published on Web 02/27/2008

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Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008

The PPR78 Model The Equation of State. In 1978, Peng and Robinson8 published an improved version of their well-known EOS, noted as PR78 in this paper. For a pure component, the PR78 EOS is

P)

ai(T) RT V - bi V(V + bi) + bi(V - bi)

(1)

with

R ) 8.314472 J‚mol-1‚K-1

(2a)

RTc,i bi ) 0.0777960739 Pc,i

(2b)

ai ) 0.457235529

{

( ) ( )[ ( x )] R2Tc,i2 1 + mi 1 Pc,i

T Tc,i

2

(2c)

mi ) 0.37464 + 1.54226ωi - 0.26992ωi2 (if ωi e 0.491) mi ) 0.379642 + 1.48503ωi - 0.164423ωi2 + 0.016666ωi3 (if ωi > 0.491)

}

(2d)

where P is the pressure, R the ideal gas constant, T the temperature, V the molar volume, Tc the critical temperature, Pc the critical pressure, and ω the acentric factor. In this paper, the PR78 EOS is used. To apply such an EOS to mixtures, mixing rules are used to calculate the a and b values of the mixtures. Classical mixing rules are used in this study: N

a)

added in this paper (group 13 ) N2), we must estimate the interactions between this new group and the 12 groups previously defined. Therefore, we must estimate 24 parameters (12 Akl values and 12 Bkl values). These parameters have been determined to minimize the deviations between calculated and experimental VLE data from an extended database. The corresponding Akl and Bkl values (expressed in MPa) are summarized in Table 1. An example of kij calculation may be found in our first article.1 Database and Reduction Procedure. Table 2 presents the list of pure components used in this study. The pure fluid physical properties (Tc, Pc, and ω) used in this study originate from Poling et al.10 Table 3 details the sources of the binary experimental data used in our evaluations,11-135 along with the temperature, pressure, and composition range for each binary system. Most of the data available in the open literature (3915 bubble points + 4148 dew points + 87 mixture critical points) have been collected. Our database includes VLE data on 36 binary systems. The 24 parameters (12 Akl and 12 Bkl) determined in this study (see Table 1) are those that minimize the following objective function:

Fobj )

where nbubble

∑ ∑ zizjxaiaj(1 - kij(T)) i)1 j)1 zibi ∑ i)1

1

2

k)1 l)1

(

(

-

bj

)

298.15 K

)}[

xai(T) xaj(T) bi

(

ndew

(Rik - Rjk)(Ril - Rjl)Akl

T

2

|∆x|

+

x1,exp

|∆x| x2,exp

)

(5a)

i

|∆x| ) |x1,exp - x1,cal| ) |x2,exp - x2,cal|

(3b)

Ng

(

with

where zk represents the mole fraction of component k in a mixture, and N is the number of components in the mixture. The term kij(T) is the so-called “binary interaction parameter” which characterizes molecular interactions between molecules i and j. In this paper, to obtain a predictive model and to define the PPR78 model (predictive 1978, PR EOS), kij, which is dependent on temperature, is calculated by a group contribution method through the following expression:

{ [∑ ∑

0.5

(3a)

N

kij(T) ) -

∑ i)1

Fobj,bubble ) 100

N

b)

Ng

Fobj,bubble + Fobj,dew + Fobj,crit. comp + Fobj,crit. pressure nbubble + ndew + ncrit + ncrit (5)

(Bkl/Akl)-1

]

]

xai(T)aj(T)

/ 2

b ib j

Fobj,dew ) 100

|∆y|

0.5 ∑ y i)1

+

1,exp

|∆y| y2,exp

)

(5b)

i

with

|∆y| ) |y1,exp - y1,cal| ) |y2,exp - y2,cal| ncrit

Fobj,crit. comp ) 100

∑ i)1

0.5

(

|∆xc|

xc1,exp

+

|∆xc| xc2,exp

)

(5c) i

-

with (4)

More information on eq 4 may be found in our first article.1 The PPR78 model is thus defined by eqs 1-4. In eq 4, T is the temperature and the parameters ai and bi are simply calculated by eq 2. Ng is the number of different groups defined by the method (for the time being, 13 groups are defined and Ng ) 13). The parameter Rik is the fraction of molecule i occupied by group k (occurrence of group k in molecule i divided by the total number of groups present in molecule i). Akl () Alk) and Bkl () Blk) (where k and l are two different groups) are constant parameters determined either in this study or in our previous papers1-4 (Akk ) Bkk ) 0). For the new group

|∆xc| ) |xc1,exp - xc1,cal| ) |xc2,exp - xc2,cal| ncrit

Fobj,crit. pressure ) 100

∑ i)1

(

)

|Pcm,exp - Pcm,cal| Pcm,exp

(5d) i

The parameters nbubble, ndew, and ncrit are the number of bubble points, dew points, and mixture critical points, respectively; x1 is the mole fraction in the liquid phase of the most volatile component, and x2 the mole fraction of the heaviest component (it is obvious that x2 ) 1- x1). Similarly, y1 is the mole fraction in the gas phase of the most volatile component, and y2 is the mole fraction of the heaviest component (it is obvious that y2 ) 1 - y1); xc1 is the critical mole fraction of the most volatile

Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008 2035

Figure 1. Isothermal dew and bubble curves for the N2 (1) + n-hexane (2) binary system at T ) 377.9 K, calculated with the Peng-Robinson equation of state (EOS), giving different kij values: (+) experimental bubble points, (*) experimental dew points, and (O) experimental critical points. Solid lines represent calculated curves. Panel a shows the experimental data points, panel b shows data for kij ) -0.03, panel c shows data for kij ) 0.0, panel d shows data for kij ) 0.02, and panel e shows data for kij ) 0.06.

component, and xc2 the critical mole fraction of the heaviest component. Pcm is the binary critical pressure.

and

∆x% )

Results and Discussion For all of the 8150 data points included in our database, the objective function defined by eq 5 is Fobj ) 8.70%. The value of the objective function for each binary system is given in Table 3. The average overall deviation on the liquid-phase composition is

The average overall deviation on the gas-phase composition is ndew

nbubble

∆x1 ) ∆x2 )

∑ i)1

∆y1 ) ∆y2 )

(|x1,exp - x1,cal|)i nbubble

∆x1% + ∆x2% Fobj,bubble ) 10.08% ) 2 nbubble

) 0.025 and

(|y1,exp - y1,cal|)i ∑ i)1 ndew

) 0.012

a

0 A67 ) 41.18 B67 ) 50.79 A68 ) -3.088 B68 ) 13.04 A69 ) -3.088 B69 ) 13.04 A6-10 ) 8.579 B6-10 ) 76.86 A6-11 ) 10.29 B6-11 ) -52.84 A6-12 ) 135.5 B6-12 ) 239.5 A6-13 ) 61.59 B6-13 ) 84.92

0 A56 ) 13.04 B56 ) 6.863 A57 ) 67.26 B57 ) 167.5 A58 ) 139.3 B58 ) 464.3 A59 ) 139.3 B59 ) 464.3 A5-10 ) 36.37 B5-10 ) 26.42 A5-11 ) 40.15 B5-11 ) 255.3 A5-12 ) 137.3 B5-12 ) 194.2 A5-13 ) 37.90 B5-13 ) 37.20

0 A45 ) 263.9 B45 ) 531.5 A46 ) 333.2 B46 ) 203.8 A47 ) 158.9 B47 ) 613.2 A48 ) 79.61 B48 ) -326.0 A49 ) 79.61 B49 ) -326.0 A4-10 ) 177.1 B4-10 ) 601.9 A4-11 ) 17.84 B4-11 ) -109.5 A4-12 ) 287.9 B4-12 ) 346.2 A4-13 ) 263.9 B4-13 ) 772.6

0 A34 ) -305.7 B34 ) - 250.8 A35 ) 145.2 B35 ) 301.6 A36 ) 174.3 B36 ) 352.1 A37 ) 103.3 B37 ) 191.8 A38 ) 6.177 B38 ) -33.97 A39 ) 6.177 B39 ) -33.97 A3-10 ) 101.9 B3-10 ) - 90.93 A3-11 ) -226.5 B3-11 ) -51.47 A3-12 ) 184.3 B3-12 ) 762.1 A3-13 ) 365.4 B3-13 ) 521.9

0 A23 ) 51.47 B23 ) 79.61 A24 ) 88.53 B24 ) 315.0 A25 ) 36.72 B25 ) 108.4 A26 ) 31.23 B26 ) 84.76 A27 ) 29.78 B27 ) 58.17 A28 ) 3.775 B28 ) 144.8 A29 ) 3.775 B29 ) 144.8 A2-10 ) 12.78 B2-10 ) 28.37 A2-11 ) -54.90 B2-11 ) -319.5 A2-12 ) 136.9 B2-12 ) 254.6 A2-13 ) 82.28 B2-13 ) 202.8

0 A12 ) 74.81 B12 ) 165.7 A13 ) 261.5 B13 ) 388.8 A14 ) 396.7 B14 ) 804.3 A15 ) 32.94 B15 ) -35.00 A16 ) 8.579 B16 ) -29.51 A17 ) 90.25 B17 ) 146.1 A18 ) 62.80 B18 ) 41.86 A19 ) 62.80 B19 ) 41.86 A1-10 ) 40.38 B1-10 ) 95.90 A1-11 ) 98.48 B1-11 ) 231.6 A1-12 ) 164.0 B1-12 ) 269.0 A1-13 ) 52.74 B1-13 ) 87.19 0 A78 ) -13.38 B78 ) 20.25 A79 ) -13.38 B79 ) 20.25 A7-10 ) 29.17 B7-10 ) 69.32 A7-11 ) -26.42 B7-11 ) -789.2 A7-12 ) 102.6 B7-12 ) 161.3 A7-13 ) 185.2 B7-13 ) 490.6

CHaro (group 7)

Cfused aromatic rings (group 9)

0 A9-10 ) 34.31 B9-10 ) 95.39 A9-11 ) -105.7 B9-11 ) -286.5 A9-12 ) 267.3 B9-12 ) 444.4 A9-13 ) 718.1 B9-13 ) 1892

Caro (group 8)

0 A89 ) 0.0 B89 ) 0.0 A8-10 ) 34.31 B8-10 ) 95.39 A8-11 ) -105.7 B8-11 ) -286.5 A8-12 ) 110.1 B8-12 ) 637.6 A8-13 ) 284.0 B8-13 ) 1892

Only the last line of this table, relative to N2, was determined in this study; the first 12 lines of this table were determined from our previous work.1-4

CH3 (group 1) CH2 (group 2) CH (group 3) C (group 4) CH4 (group 5) C2H6 (group 6) CHaro (group 7) Caro (group 8) Cfused aromatic rings (group 9) CH2,cyclic (group 10) CHcyclic or Ccyclic (group 11) CO2 (group 12) N2 (group 13)

C2H6 (group 6)

CH4 (group 5)

C (group 4)

CH (group 3)

CH2 (group 2)

CH3 (group 1)

Table 1. Group Interaction Parameters Akl () Alk) and Bkl () Blk) (Both Given in Units of MPa)a

0 A10-11 ) - 50.10 B10-11 ) - 891.1 A10-12 ) 130.1 B10-12 ) 225.8 A10-13 ) 179.5 B10-13 ) 546.6

CH2,cyclic (group 10)

CO2 (group 12)

0 A11-12 ) 91.28 0 B11-12 ) 82.01 A11-13 ) 100.9 A12-13 ) 98.42 B11-13 ) 249.8 B12-13 ) 221.4

CHcyclic or Ccyclic (group 11)

0

N2 (group 13)

2036 Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008

∆y% )

∆xc% )

∆Pc )

component

∆y1% + ∆y2% Fobj,dew ) 7.42% ) 2 ndew

The average overall deviation on the critical composition is ncrit

∆xc1 ) ∆xc2 )

∆Pc% )

nitrogen carbon dioxide methane ethane propane N2 CO2 1 2 3

2-methyl propane n-butane 2,2-dimethyl propane 2-methyl butane n-pentane n-hexane n-heptane 2,2,4-trimethyl pentane n-octane 2,2,5-trimethyl hexane n-nonane n-decane 2m3

n-dodecane

12

n-tetradecane

14

(|xc1,exp - xc1,cal|)i ∑ i)1 ncrit

ncrit

(|Pcm,exp - Pcm,cal|)i ∑ i)1 ncrit

abbreviation

) 0.033

and

∆xc1% + ∆xc2% Fobj,crit. comp ) ) 9.01% 2 ncrit

is The average overall deviation on the binary critical pressure

) 20.5 bar

and

Fobj,crit. pressure ) 6.68% ncrit

These results indicate that the PPR78 model remains an accurate predictive model, even if the deviations observed in this study are higher than those observed with hydrocarbons1-3 or with CO24 (for all the systems studied in our previous papers1-4 (i.e., for >50 000 data points), the objective function is Fobj ) 5.86%). As discussed at the beginning of this paper, in this study, it was necessary to find a compromise between a small deviation on the liquid-phase composition and a small deviation on the critical pressure. Indeed, a kij value that is able to simultaneously

Table 2. List of the 37 Pure Components Used in This Study component abbreviation

16 20 B mB 14mB

4 22m3 n-hexadecane n-eicosane benzene methyl benzene (toluene) 1,4-dimethyl benzene (para-xylene) 1,3-dimethyl benzene (meta-xylene) n-propyl benzene 1,3,5-trimethyl benzene

2m4

naphthalene

BB

5 6 7 224m5

tertiobutyl benzene 1-methyl naphthalene cyclopentane cyclohexane

tbuB 1mBB C5 C6

8 225m6

methyl cyclohexane ethyl cyclohexane

mC6 eC6

9 10

n-propyl cyclohexane tetralin (1,2,3,4-tetrahydro naphthalene) trans decalin (trans-decahydro naphthalene)

prC6 tet

13mB

prB 135mB

tCC6

0.0010-0.4990 0.0020-0.9947 0.0036-0.9957

x1 range (liquid mole fraction of first compound)

0.83-219.18 0.0090-0.5500 1.01-207.74 0.0050-0.4620 1.58-290.00 0.0036-0.7000 31.10-833.00 0.0600-0.7750 1.83-207.77 0.0010-0.4370 2.50-1932.00 0.0022-0.8100 9.60-533.80 0.0111-0.7797 12.00-998.50 0.0080-0.8400 20.00-630.00 0.0267-0.7480 20.50-509.80 0.0204-0.7908 66.17-85.62 19.70-497.50 0.0255-0.7659 1.01-346.40 0.0012-0.3980 10.30-346.90 0.0143-0.3490 85.09-2020.04 0.1150-0.8010 19.96-588.40 0.0380-0.8370 38.30-172.30 0.0610-0.2121 23.20-1357.00 0.0116-0.7044 13.81-1000.00 0.0050-0.5000 53.93-1685.48 0.0506-0.7387 11.50-1001.00 0.0190-0.3500 6.00-396.70 0.0150-0.3600 11.00-998.00 0.0180-0.3400 62.20-1559.46 0.0250-0.3007 67.18-101.73 20.27-254.00 0.0120-0.2884 13.63-312.83 0.0210-0.3740 17.53-275.93 0.0094-0.2906 4.36-900.00 0.0050-0.8090 4.34-203.90 0.0049-0.3279 20.20-997.00 0.0204-0.4590 20.74-255.60 0.0152-0.5734 37.40-145.70 0.0292-0.1105 total number of points:

12.77-167.26 0.21-50.68 0.21-134.17

pressure range (bar)

Given in the following format: first compound-second compound.

173.15-365.20 120.00-394.26 153.15-410.93 211.40-293.60 277.59-377.37 172.00-447.90 310.93-488.40 251.78-523.70 323.15-452.95 293.25-543.50 298.15-373.15 261.70-543.40 263.15-411.10 297.15-423.15 320.00-430.00 323.15-703.40 323.20-423.20 288.15-473.15 241.15-548.15 360.00-440.00 313.20-584.60 313.20-473.20 313.20-472.60 373.15-423.15 323.15-398.15 344.30-703.30 366.40-410.20 366.50-433.15 310.93-497.15 310.95-477.55 313.60-472.90 463.60-662.80 344.30-410.90

N2-3 N2-2m3 N2-4 N2-22m3 N2-2m4 N2-5 N2-6 N2-7 N2-224m5 N2-8 N2-225m6 N2-9 N2-10 N2-12 N2-14 N2-16 N2-20 N2-B N2-mB N2-14mB N2-13mB N2-prB N2-135mB N2-BB N2-tbuB N2-1mBB N2-C5 N2-C6 N2-mC6 N2-eC6 N2-prC6 N2-tet N2-tCC6

a

218.15-298.80 78.37-187.20 90.69-301.46

temperature range (K)

N2-CO2 N2-1 N2-2

binary systema

Table 3. Binary Systems Database

0.9925-0.9995 0.4684-0.9953 0.6010-0.9030 0.7133-0.9721 0.1110-0.9951 0.6141-0.9989 0.8440-0.9991 0.4937-0.9886

0.7044-0.9990 0.0950-0.9999 0.8015-0.9434 0.3771-0.9990 0.9200-0.9994 0.7800-0.9992

0.2624-0.9990

0.0600-0.9965 0.0900-0.9995 0.0638-0.9993 0.7450-0.9540 0.1630-0.9940 0.1035-0.9979 0.1316-0.9890 0.1186-0.9998 0.8490-0.9962 0.1849-0.9975 0.9861-0.9992 0.3639-0.9999 0.9268-0.9999 0.9958-1.0000

0.0020-0.8470 0.0120-0.9970 0.0437-0.9998

y1 range (gas mole fraction of first compound)

143 102 208 81 47 131 121 338 32 99 0 82 118 68 40 124 20 120 112 62 52 24 30 24 0 45 31 53 89 41 33 25 18 3915

299 759 344

number of bubble points (T, P, x)

376 212 245 30 47 131 121 347 33 78 9 76 136 31 0 106 0 150 128 25 44 34 30 0 9 27 31 18 101 41 33 25 0 4148

280 681 513

number of dew points (T, P, y)

3 0 1 14 0 15 7 10 0 4 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 87

14 12 4

number of binary critical points (Tcm, Pcm, xc)

7.27 6.43 10.3 12.6 6.34 8.92 10.6 12.1 14.2 10.5 12.2 11.3 10.6 14.2 18.4 10.0 10.6 12.6 17.5 17.5 15.8 14.4 21.0 12.5 18.9 11.8 10.4 7.27 12.9 5.17 10.2 12.7 10.0 F h obj ) 8.70%

4.41 3.29 6.51

Fobj (from eq 5) for each binary system (%)

reference(s) 11-29 30-49 13, 30, 31, 34, 44, 45, 47, 50-59 11, 30, 50, 60-64 60, 65-67 12, 60, 68-75 76 77 78-81 82,83 84-94 84, 95, 96 88, 95, 97, 98 99 88, 91, 100 88, 96, 99, 101-106 88, 99, 107-109 110 111-113 103 88, 114-120 88, 96, 121-124 120 121, 125 126 121 119, 120 99 119, 127 128 119, 129, 130 84, 89, 131, 132 133 134 135 119

Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008 2037

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Ind. Eng. Chem. Res., Vol. 47, No. 6, 2008

Figure 2. Prediction of the corresponding critical locus of isothermal dew and bubble curves for two binary systems that contain N2 and an n-alkane using the PPR78 model: (+) experimental bubble points, (*) experimental dew points, and (O) experimental critical points. Solid line represents predicted curves with the PPR78 model, and the dashed lines represent the vaporization curves of the pure compounds. Panel a shows data for the N2 (1)/methane (2) system at six different temperatures (T1 ) 90.68 K (kij ) 0.0314), T2 ) 110.93 K (kij ) 0.0326), T3 ) 130.00 K (kij ) 0.0338), T4 ) 150.00 K (kij ) 0.0349), T5 ) 170.00 K (kij ) 0.0360), and T6 ) 183.15 K (kij ) 0.0367). Panel b shows data for the critical locus of the N2/methane system. Panel c shows data for the N2 (1)/ethane (2) system at two different temperatures (T1 ) 129.72 K (kij ) 0.0535), and T2 ) 230.00 K (kij ) 0.0362)). Panel d shows data for the N2 (1)/ethane (2) system at two different temperatures (T1 ) 172.04 K (kij ) 0.0453), and T2 ) 290.00 K (kij ) 0.0278). Panel e shows the critical locus of the N2/ethane system.

predict the liquid-phase composition and the critical pressure does not exist. To illustrate the accuracy and the limitations of the proposed model, it was decided that several families of binary systems would be defined. It is indeed impossible to show the results for all the studied systems.

Results for Mixtures of Nitrogen + n-Alkanes. The binary system N2 + methane exhibits Type I phase behavior in the classification scheme of van Konynenburg and Scott.9 As a consequence,1-4 the PPR78 model is able to reproduce the phase behavior of this system perfectly, whatever the temperature (see Figure 2a and 2b). For this system, the binary interaction

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Figure 3. Prediction of the corresponding critical locus of isothermal dew and bubble curves for two binary systems that contain N2 and a n-alkane, using the PPR78 model: (+) experimental bubble points, (*) experimental dew points, (O) experimental critical points. Solid lines represent predicted curves with the PPR78 model, and the dashed lines represent the vaporization curves of the pure compounds. Panel a shows data for the N2 (1)/n-pentane (2) system at two different temperatures (T1 ) 255.40 K (kij ) 0.0557) and T2 ) 398.30 K (kij ) -0.00511)). Panel b shows data for the N2 (1)/n-pentane (2) system at two different temperatures: T1 ) 344.30 K (kij ) 0.0154) and T2 ) 447.90 K (kij ) -0.0235). Panel c shows data for the critical locus of the N2/n-pentane system. Panel d shows data for the N2 (1)/n-hexane (2) system at two different temperatures (T1 ) 344.60 K (kij ) 0.0330), T2 ) 444.90 K (kij ) -0.00474)). Panel e shows data for the N2 (1)/n-hexane (2) system at two different temperatures (T1 ) 377.90 K (kij ) 0.0200), T2 ) 488.40 K (kij ) -0.0207)). Panel f shows data for the critical locus of the N2/n-hexane system.

parameter (kij) increases slightly with temperature. All the other binary mixtures of the N2 + n-alkane system exhibit Type III phase behavior. From our experience,1-4 it is thus very difficult to predict accurately (see Figures 2-4) the phase behavior of such systems with a cubic equation of state even with temperature-dependent kij values. For all these other systems, the kij

value decreases with temperature. Moreover, regardless of the length of the n-alkane, the accuracy of the PPR78 is constant. As can be seen in Figures 2e, 3c, 3f, and 4c, the PPR78 predicts (with an acceptable accuracy) the critical loci of these binary systems. In return, the liquid-phase composition generally is systematically overestimated. We believe that the inevitable

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Figure 4. Prediction of the corresponding critical locus of isothermal dew and bubble curves for two binary systems that contain N2 and a n-alkane, using the PPR78 model: (+) experimental bubble points, (*) experimental dew points, and (O) experimental critical points. Solid lines represent predicted curves with the PPR78 model, and the dashed lines represent the vaporization curves of the pure compounds. Panel a shows data for the N2 (1)/n-heptane (2) system at two different temperatures (T1 ) 305.45 K (kij ) 0.0658), T2 ) 452.90 K (kij ) 0.00853)). Panel b shows data for the N2 (1)/n-heptane (2) system at two different temperatures (T1 ) 376.35 K (kij ) 0.0362), T2 ) 523.70 K (kij ) -0.0168)). Panel c shows data for the critical locus of the N2/n-heptane system. Panel d shows data for the N2 (1)/n-hexadecane (2) system at two different temperatures (T1 ) 473.15 K (kij ) 0.101), T2 ) 623.15 K (kij ) 0.0665)). Panel e shows data for the N2 (1)/n-hexadecane (2) system at two different temperatures (T1 ) 523.15 K (kij ) 0.0894), T2 ) 703.40 K (kij ) 0.0466)). Panel f shows data for the critical locus of the N2/n-hexadecane system.

compromise between a correct critical pressure and an accurate liquid-phase composition has been properly determined in the PPR78 model. Results for Mixtures of Nitrogen + Branched Alkanes. Our databank (see Table 3) contains VLE data for only five binary mixtures that contain nitrogen and a branched alkane. All these systems exhibit Type III phase behavior. Consequently (see Figure 5), the critical pressure is always overestimated and the liquid-phase composition is not very accurately predicted.

As shown in Figure 5f, the overestimation of the critical pressure is generally not too high (15 bar on average). For all these systems, the kij value is a monotonous decreasing function of temperature. Results for Mixtures of Nitrogen + Aromatic Compounds. VLE data are only known for nine binary systems that contain N2 and an aromatic compound. The 916 experimental data points available for these systems unfortunately do not contain any critical points. Moreover, VLE data are generally only available

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Figure 5. Prediction of isothermal dew and bubble curves for three binary systems that contain N2 and a branched alkane, using the PPR78 model (prediction of the critical locus for one system): (+) experimental bubble points, (*) experimental dew points, and (O) experimental critical points. Solid lines represent predicted curves with the PPR78 model, and the dashed lines represent the vaporization curves of the pure compounds. Panel a shows data for the N2 (1)/2-methyl propane (2) system at three different temperatures (T1 ) 255.37 K (kij ) 0.113), T2 ) 310.87 K (kij ) 0.105), T3 ) 366.32 K (kij ) 0.0993)). Panel b shows data for the N2 (1)/2-methyl propane (2) system at three different temperatures (T1 ) 283.21 K (kij ) 0.109), T2 ) 338.71 K (kij ) 0.102), T3 ) 394.26 K (kij ) 0.0964)). Panel c shows data for the N2 (1)/2-methyl butane (2) system at two different temperatures (T1 ) 310.82 K (kij ) 0.0956), T2 ) 377.37 K (kij ) 0.0837)). Panel d shows data for the N2 (1)/2,2,4-trimethyl pentane (2) system at T ) 376.45 K (kij ) 0.0887). Panel e shows data for the N2 (1)/2,2,4-trimethyl pentane (2) system at T ) 413.15 K (kij ) 0.0806). Panel f shows data for the critical locus of the N2/2,2-dimethyl propane (22m3) system.

at low pressure, and finding the six parameters (three Akl and three Bkl) that better correlate the entire data is not an easy task. As can be seen in Figure 6, where four aromatic molecules were selected, the PPR78 model predicts the systems that contain an

aromatic compound less accurately than those which contain an alkane. The predicted phase diagrams have the same drawbacks as those previously shown: the liquid-phase composition is not accurately predicted and the shape of the

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Figure 6. Prediction of isothermal dew and bubble curves for four binary systems that contain N2 and an aromatic compound, using the PPR78 model: (+) experimental bubble points, (*) experimental dew points. Solid lines represent predicted curves with the PPR78 model. Panel a shows data for the N2 (1)/benzene (2) system at two different temperatures (T1 ) 348.15 K (kij ) 0.0612), T2 ) 398.15 K (kij ) 0.00358)). Panel b shows data for the N2 (1)/benzene (2) system at two different temperatures (T1 ) 373.15 K (kij ) 0.0314), T2 ) 423.15 K (kij ) -0.0230)). Panel c shows data for the N2 (1)/toluene (2) system at two different temperatures (T1 ) 478.15 K (kij ) -0.0696), T2 ) 545.20 K (kij ) -0.123)). Panel d shows data for the N2 (1)/toluene (2) system at two different temperatures (T1 ) 498.15 K (kij ) -0.0860), T2 ) 548.15 K (kij ) -0.125)). Panel e shows data for the N2 (1)/meta-xylene (2) system at two different temperatures (T1 ) 472.60 K (kij ) -0.0643), T2 ) 584.60 K (kij ) -0.140)). Panel f shows data for the N2 (1)/1-methyl naphthalene (2) system at two different temperatures (T1 ) 624.00 K (kij ) -0.0802), T2 ) 703.30 K (kij ) -0.144)).

calculated diagram in the critical region is not similar to the experimental diagram. Once again, all these systems exhibit Type III phase behavior, and, once again, the kij is a monotonous

decreasing function of temperature. Generally, the kij value is positive at low temperature and becomes negative at higher temperature.

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Figure 7. Prediction of isothermal dew and bubble curves for five binary systems that contain N2 and a naphthenic compound, using the PPR78 model: (+) experimental bubble points, (*) experimental dew points. Solid lines represent predicted curves with the PPR78 model. Panel a shows data for the N2 (1)/cyclopentane (2) system at two different temperatures (T1 ) 366.40 K (kij ) 0.0733), T2 ) 410.20 K (kij ) 0.0183)). Panel b shows data for the N2 (1)/methyl cyclohexane (2) system at two different temperatures (T1 ) 352.59 K (kij ) 0.118), T2 ) 453.15 K (kij ) 0.0185)). Panel c shows data for the N2 (1)/methyl cyclohexane (2) system at two different temperatures (T1 ) 376.45 K (kij ) 0.0914), T2 ) 497.15 K (kij ) -0.0190)). Panel d shows data for the N2 (1)/ethyl cyclohexane (2) system at two different temperatures (T1 ) 394.25 K (kij ) 0.101), T2 ) 477.55 K (kij ) 0.0286)). Panel f shows data for the N2 (1)/n-propyl cyclohexane (2) system at two different temperatures (T1 ) 393.20 K (kij ) 0.106), T2 ) 472.90 K (kij ) 0.0384)). Panel f shows data for the N2 (1)/tetralin (2) system at two different temperatures (T1 ) 544.00 K (kij ) -0.144), T2 ) 662.80 K (kij ) -0.253)).

Results for Mixtures of Nitrogen + Naphthenic Compounds (Also Called Naphthenes or Cycloparaffins). Our databank (see Table 3) contains VLE data for seven binary mixtures that contain nitrogen and a naphthene. The 539 experimental data points available for these systems unfortu-

nately do not contain any critical points. Figure 7 shows predicted phase diagrams with the PPR78 model for five binary systems. The accuracy of our model is similar to that observed with paraffins and aromatics. All these systems exhibit Type III phase behavior. As a consequence, the calculated critical

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Figure 8. Prediction of isothermal dew and bubble curves, prediction of the critical locus, and temperature dependence of kij for the binary system N2 + CO2, using the PPR78 model: (+) experimental bubble points, (*) experimental dew points, and (O) experimental critical points. Solid lines represent predicted curves with the PPR78 model, and the dashed lines represent the vaporization curves of the pure compounds. Panel a shows data for T1 ) 218.15 K (kij ) -0.00856) and T2 ) 253.15 K (kij ) -0.0325). Panel b shows data for T1 ) 232.85 K (kij ) -0.0194) and T2 ) 273.20 K (kij ) -0.0437). Panel c shows data for T1 ) 240.00 K (kij ) -0.0242) and T2 ) 288.20 K (kij ) -0.0514). Panel d shows data for T1 ) 248.15 K (kij ) -0.0294) and T2 ) 298.80 K (kij ) -0.0564). Panel e shows data for the critical locus. Panel f shows a graph of kij vs T (the square indicates the temperature at which kij ) 0).

pressure is often >2000 bar (see Figures 7b and 7e). For all these systems, kij is a monotonous decreasing function of temperature. Results for Mixtures of Nitrogen + Carbon Dioxide. Mixtures of nitrogen and carbon dioxide have been measured extensively (see Table 3), and there is a vast amount of reliable experimental phase equilibria and critical data (593 VLE data points were determined for this system). Although it exhibits Type III behavior (see Figure 8e), the PPR78 model is able to predict, with high accuracy, the phase behavior of this system.

A few examples may be seen in Figure 8. For this system, the objective function defined by eq 5 is Fobj ) 4.41%, which is smaller than the objective function obtained for the entire database (by a factor of 2). In the vicinity of the Tc value for carbon dioxide, the critical locus is perfectly predicted with the PPR78 model. By decreasing the temperature, our model has a tendency to overestimate the critical pressure. Once again, the binary interaction parameter is a decreasing function of the temperature (see Figure 8f); kij is positive at temperatures of 208 K.

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Conclusion In two of our previous papers,3,4 we concluded that, even with a temperature-dependent kij expression, the Peng-Robinson equation of state (EOS) was not able to predict the critical loci of Type III systems accurately, according to the classification scheme of van Konynenburg and Scott.9 This was particularly true at low temperature (in the vicinity of the upper critical end point), where the slope of the critical curve is often very steep (a small change of the temperature induces a large change of pressure). This paper has shown that the phase behavior of all binary N2 + hydrocarbon fluid mixtures develop, except for methane, Type III phase diagrams. Therefore, it is not amazing that the results obtained in this paper are less accurate than those previously published.1-4 This paper also gives proof that the N2 + hydrocarbon binary systems do not behave classically. Because of the fact that, in a large composition range around the critical point, the slopes of the dew curve and the bubble curve remain low, the PPR78 model overestimates the liquidphase composition and the critical pressure. However, an objective function of